Paper Title: Rotational influence on fermions within negative curvature wormholes.




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Rotational influence on fermions within negative curvature wormholes
     Abdullah Guvendi^ 1[ E-mail: abdullah.guvendi@erzurum.edu.tr  ] ,  Semra Gurtas Dogan^ 2[ E-mail: semragurtasdogan@hakkari.edu.tr (Corr. Auth.) ] ,  R. L. L. Vitória^ 3[ E-mail: ricardo-luis91@hotmail.com ]   

	^ 1 Department of Basic Sciences, Erzurum Technical University, 25050, Erzurum, Türkiye

	^ 2 Department of Medical Imaging Techniques, Hakkari University, 30000, Hakkari, Türkiye

	^ 3 Faculdade de Física, Universidade Federal do Pará, Av. Augusto Corrêa, Guamá, Belém, PA 66075-110, Brazil
    
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§ ABSTRACT 

In this research, we examine relativistic fermions within the rotating frame of negative curvature wormholes. Initially, as is typical in our context, we introduce the wormholes by embedding a curved surface into a higher dimensional flat Minkowski spacetime. Subsequently, we derive the spacetime metric that characterizes the rotating frame of these wormholes. We then investigate analytical solutions of the generalized Dirac equation within this framework. Through exploring a second-order non-perturbative wave equation, we seek exact solutions for fermions within the rotating frame of hyperbolic and elliptic wormholes, also known as negative curvature wormholes. Our analysis provides closed-form energy expressions, and we generalize our findings to Weyl fermions. By considering the impact of the rotating frame and curvature radius of wormholes, we discuss how these factors affect the evolution of fermionic fields, offering valuable insights into their behavior.








Keywords: Fermions; Non-inertial effects; Wormhole; Rotating reference frame; Graphene; Weyl fermions	
















§ INTRODUCTION



Landau and Lifshitz introduced rotation to Minkowski spacetime, analyzing two specific effects: the impact on clocks within a rotating body and the singular behavior at significant distances within a uniformly rotating frame <cit.>. Notably, the latter effect imposes a limitation on the radial coordinate. Within the domain of relativistic quantum theory, investigating the impact of a rotating frame involves understanding the principles of special relativity, where the quantized nature of fields and particle behaviors are crucial considerations. This exploration unveils intriguing and unconventional effects absent in classical Sagnac-type descriptions, representing the coupling between angular momentum and rotation angular frequency of a uniformly rotating frame, emphasized across various references <cit.>. In the non-relativistic context, this effect is known as the Page-Werner et al. term <cit.>. The incorporation of relativistic corrections within this framework underscores the potential for phase shifts induced by relative motion between an observer and the rotating frame. Analyzing relativistic quantum systems within such a rotating reference frame unveils a realm of interesting results. Additionally, for quantum fields within curved spaces, the modified effective gravitational field, influenced by the centrifugal force, alters the energy levels of quantum entities, especially in confined or bound states <cit.>. In this context, several studies explore the impact of nontrivial rotating spacetimes on quantum mechanical systems. These investigations encompass various scenarios such as scalar bosons within cosmic string spacetime <cit.>, a scalar field within a Kaluza-Klein theory <cit.>, and a scalar field within time-dislocation spacetime <cit.>. Predicting the effects of a rotating frame on relativistic quantum particles is theoretically plausible; however, practically observing these minute alterations demands precision, control, and advanced techniques in quantum optics, atomic physics, or condensed matter physics, alongside cutting-edge technology <cit.>. Directly observing these effects necessitates trapping and controlling particles within the rotating frame, potentially detectable through variations in energy profiles, trajectories, or alterations in total angular momentum or spin. While extensive theoretical studies have scrutinized the effects of rotating reference frames on physical systems within non-relativistic and relativistic frameworks <cit.>, there is a noticeable gap in announced results regarding the evolution of relativistic fermions within the rotating frame of graphene or a graphene wormhole characterized by a two-dimensional curved surface of constant negative curvature. This unexplored domain requires further investigation to comprehend the intricate interplay between relativistic fermionic fields and specific curved geometries such as graphene structures.

In astrophysics, wormholes emerge from solutions in general relativity, serving as connections between two distinct regions <cit.>. These phenomena may manifest in both astrophysical contexts and condensed matter systems (by embedding curved surfaces into higher-dimensional spacetime (see also <cit.>)). However, the latter depiction differs because it involves two-dimensional curved surfaces, with time remaining unaffected by the background curvature <cit.>. Wormholes are actively researched to understand the fundamental properties of spacetime and whether they can exist within our current understanding of physics. Current research on wormholes spans from astrophysics to condensed matter contexts <cit.>. Research examining quantum systems evolving within curved spaces has garnered considerable interest <cit.>, exemplified by investigations into Weyl particles in optical backgrounds <cit.>, fermion-antifermion pairs in magnetized elliptic wormholes <cit.>, and more <cit.>. No results were found on fermionic fields in the rotating frame of negative curvature wormholes. This research addresses a gap in the literature by offering non-perturbative insights into fermions within rotating graphene and wormhole-like graphene structures characterized by a rotating surface with negative Gaussian curvature. Pursuing exact solutions holds promise in enabling the control or observation of compelling physical properties, potentially fostering novel phenomena, validating fundamental theories, and facilitating technological advancements. Our focus is on investigating relativistic fermion evolution within the rotating framework of negative curvature wormholes by seeking exact solutions to the corresponding Dirac equation.

Our examination of Dirac particles within the rotating frame of negative curvature wormholes is structured as follows: In section <ref>, we unveil a general and non-perturbative wave equation, exploring the intricate realm of relativistic fermions in the rotating frame of negative curvature wormholes. Through a thorough analysis, we present the main wave equation governing their behavior in these environments. In section <ref>, we uncover precise and exact outcomes concerning the evolution of relativistic fermions within two distinct rotating frames: the hyperbolic wormhole and the elliptic wormhole. By investigating these specific scenarios, we reveal compelling insights into the behavior and dynamics of particles under these conditions. Finally, in section <ref>, we summarize our findings, encapsulating the essence of our results. Additionally, we provide a comprehensive discussion that explores the implications of our results across various physically plausible scenarios and limits. This reflective section aims to highlight the significance and broader ramifications of our work within the realm of theoretical physics.



§ WAVE EQUATION
 



In this section, we formulate a radial equation system for relativistic fermions in the rotating frame of negative curvature wormholes. This type of wormhole is characterized geometrically by points on its surface, which are expressed parametrically as shown below (also referenced in <cit.>):

    r⃗(u,v)=x(u,v)î+y(u,v)ĵ+zk̂,

where x(u,v)=χ( u )cos(v), y(u,v)=χ( u )sin(v), and z(u)=∫√(1-χ_,u^2) du, with _,u indicating the derivative with respect to x^u. The shape of the wormhole is generally described by the function χ(u). The constraints on z arise from the relationship:

    ds^2=dx^2+dy^2+dz^2=du^2+χ^2( u )dv^2,

which results in the Hilbert horizon at χ_,u=1 <cit.>. Embedding a three-dimensional wormhole into a (3+1)-dimensional spacetime generates effective gravity or curvature within the altered spacetime, leading to the experience of spacetime curvature by any particle or quasi-particle present in this modified background spacetime. Now, let us define the (2+1)-dimensional wormhole spacetime coordinates as dx^μ= { cdT, du, dv } and the (3+1)-dimensional Minkowski spacetime coordinates as dx^μ^'= { cdT, dx, dy, dz }, where Greek indices denote the coordinates within the curved space. The transformation between these coordinates can be accomplished via the following matrix:

    ∂ x^μ^'/∂ x^ν=([           1           0           0           0;           0  χ_,ucos(v)  χ_,usin(v) √(1-χ_,u^2);           0 -χ_,usin(v)  χ_,ucos(v)           0 ]).

Consequently, the metric of the considered wormhole spacetime takes the following form (with a negative signature) <cit.>:

ds^2=c^2dT^2-du^2-χ^2( u )dv^2.  

A wormhole of this nature is geometrically depicted as shown in Fig. <ref>. In Eq. (<ref>), c symbolizes the speed of light, and the function χ(u) takes the form χ( u )=a cosh( u/r_0) for a hyperbolic wormhole and χ( u )=b sinh( u/r_0) for an elliptic wormhole. Here, a (b) denotes the radius of the associated wormhole at the midpoint (u=0) between two ends, while r_0 represents the radius of curvature of the wormhole surface û. It is noteworthy that the Gaussian curvature for the spacetime background as expressed in Eq. (<ref>) is K =-χ_,uu/χ. Furthermore, Eq. (<ref>) describes a curved surface exhibiting negative Gaussian curvature, making such curved surfaces suitable for describing a monolayer graphene sheet and graphene wormholes. These surfaces can also be derived using the optical metric of the Banados-Teitelboim-Zanelli (BTZ) black hole <cit.>, referred to as negative curvature wormholes (generally) or hyperbolic/elliptic wormholes depending on the shape of the radius function χ( u ) (for further details, see <cit.>). In this scenario, we examine the rotating frame of spacetime given by Eq. (<ref>). This can be achieved through transformations provided by T=t, u=ρ, and v=ϕ+ω_rf T, where ω_rf denotes the angular frequency of the rotating frame. These transformations allow us to derive the spacetime interval describing a uniformly rotating frame of negative curvature wormholes (also see <cit.>).

ds^2=(c^2-ω_rf^2 χ^2)dt^2-2ω_rf χ^2  dt dϕ-dρ^2-χ^2 dϕ^2,

for which the Gaussian curvature remains unchanged but for u →ρ. In this context, it is crucial to observe that the coordinate ρ needs to adhere to a condition derived from the relationship χ(ρ)< c/ω_rf for the fermions to reside within the light-cone. This introduces an additional requirement, indicating that the corresponding wave function must vanish at the point determined by the condition χ(ρ)= c/ω_rf. Referring to the line element Eq. (<ref>), one can derive the covariant (g_μν) and contravariant (g^μν) metric tensors

   g_μν=([ c^2-ω_rf^2 χ^2              0      -ω_rf χ^2;              0             -1              0;      -ω_rf χ^2              0          - χ^2 ]), 

   g^μν=([            1/c^2                0        -ω_rf/c^2;                0               -1                0;        -ω_rf/c^2                0 ω_rf^2/c^2- χ^-2 ]).

Now, let us express the Dirac equation in its generalized form within this specific background <cit.>

   [γ^μ𝒟_μ+im̃ℐ_2]Ψ(x^μ)=0,   μ=t,ρ,ϕ,

   𝒟_μ=∂_μ-Γ_μ,  m̃=mc/ħ.

In this context, m denotes the fermion's rest mass, ħ represents the conventional Planck constant, c stands for the speed of light, ℐ_2 signifies the 2-dimensional identity matrix, Ψ(x^μ) symbolizes the Dirac field dependent on the spacetime position vector x^μ, and γ^μ denotes the space-dependent Dirac matrices, derived from the relation: γ^μ=e^μ_kγ^k, where e^μ_k are the inverse tetrad fields (k=0,1,2.) and γ^k denote the free Dirac matrices. The space-independent (free) Dirac matrices are represented in terms of the Pauli spin matrices (σ^x,σ^y,σ^z) as follows: γ^0=σ^z, γ^1=iσ^x, γ^2=iσ^y, in accordance with the signature of the line element <cit.>. In Equation (<ref>), Γ_μ refers to the spinorial affine connections for the Dirac field, attainable through the relation: Γ_λ=1/4[e^k_ν_,λe^τ_k-Γ_νλ^τ]𝒮^μν, where _,λ indicates the derivative with respect to x^λ. Here, Γ_νλ^τ represents the Christoffel symbols Γ_νλ^τ=1/2g^τϵ[∂_ν g_λτ+∂_λ g_ϵν-∂_ϵ g_νλ], e^τ_k denotes the inverse tetrad fields, and 𝒮^μν symbolizes the spin operators, defined as 𝒮^μν=1/2[γ^μ,γ^ν] <cit.>. Now, the tetrads (and their inverses) can be obtained as follows (also refer to <cit.>):

e^k_μ=([      c      0      0;      0      1      0; ω_rf χ      0      χ ]), e^μ_k=([     1/c       0       0;       0       1       0; -ω_rf/c       0    χ^-1 ])

since g_μν=e^k_μe^l_νη_kl and e^μ_k=g^μνe^l_νη_kl where η_kl is the Minkowski tensor, η_kl=diag(1,-1,-1). Accordingly, we have γ^t= σ^z/c, γ^ρ= i σ^x, γ^ϕ=-ω_rf/c σ^z+ i χ^-1σ^y <cit.>. Also, non-vanishing components of the the spinorial affine connections are Γ_t=i/2ω_rfχ_,ρσ^z and Γ_ϕ=i/2χ_,ρσ^z. Hence, one can verify that <cit.>

γ^μΓ_μ=-i/2χ_,ρ/χ σ^x.

The metric Eq. (<ref>) allows us to factorize the Dirac field as the following Ψ(x^μ)=e^-iω te^is ϕ(ψ_1(ρ) ψ_2(ρ))^T where ω is the relativistic frequency, s is the spin and T means transpose of the ρ-dependent spinor. Finally, a system of coupled equations can be derived

   [ϖ-m̃]ψ_1-[∂_ρ+1/2χ_,ρ/χ+s/χ]ψ_2=0,

   [ϖ+m̃]ψ_2+[∂_ρ+1/2χ_,ρ/χ-s/χ]ψ_1=0.

in which, ϖ=ω/c+s  ω_rf/c, describing fermions within the rotating frame of negatively curved wormholes (see also <cit.>). In the subsequent section, we aim to uncover analytical solutions to this wave equation within two specific contexts: (i) hyperbolic wormholes and (ii) elliptic wormholes.



§ EXACT RESULTS
 

By decoupling the equations presented in Eq. (<ref>), a non-perturbative second-order wave equation for the component ψ_2 is obtained in the subsequent format

[∂_ρ^2+χ_,ρ/χ∂_ρ
 +Q̂ +(ϖ^2-m̃^2)]ψ_2(ρ)=0,

where

Q̂=1/2χ_,ρρ /χ-s^2/χ^2-sχ_,ρ/χ^2 -1/4χ_,ρ^2/χ^2.

Now, we have the ability to explore the precise solution of this wave equation under two distinct scenarios.



 §.§ Relativistic fermions in the rotating frame of hyperbolic wormhole


If we examine a fermionic field within the relativistic framework of a rotating hyperbolic wormhole, the equation (<ref>) transforms into the following form

    ψ_2_,ρρ+tanh( ρ/r_0)/r_0 ψ_2_,ρ+[λ̃+Q̂_1] ψ_2=0,
       Q̂_1=1/2r_0^2-s^2/a^2/cosh^2( ρ/r_0)-s/a/r_0 tanh( ρ/r_0)/cosh( ρ/r_0)-tanh^2( ρ/r_0)/4 r^2_0,

where λ̃=ϖ^2-m̃^2. To transform this unfamiliar equation into a more recognizable form, we can employ an assumed function, ψ_2(ρ)=(sinh(ρ/r_0)+i)^p(sinh(ρ/r_0)-i)^p^*ψ(ρ), where p=3/2-isr_0/2a and ^* denotes the complex conjugate. Then, by introducing a new variable, ξ=1/2[1-i sinh(ρ/r_0)], we arrive at the following wave equation

   ξ(ξ-1)ψ_,ξξ-[γ-(α+β+1)ξ]ψ_,ξ+αβψ=0,

   α=1-ir_0√(λ̃),   β=α^*,   γ=3/2+isr_0/a, 

in which α, β and γ are constants. This equation is the renowned Gauss' hypergeometric equation, and its solution in the vicinity of the origin can be expressed as ψ(ξ)=𝒩  _2F_1(α, β ; γ; ξ) <cit.>. Here, 𝒩 denotes an arbitrary constant. The polynomial condition associated with this function is α=-n <cit.>, where n signifies the principal quantum number (n=0,1,2..). This criterion leads to the quantization principle governing the formation of the system under consideration, consequently yielding the subsequent expression

ℰ_ns=-s  ħω_rf ±ħc√(m^2c^2/ħ^2-(n+1)^2/r_0^2), 

for the relativistic energy of the system under scrutiny. In this context, it becomes evident that the energy of such a system relies on the properties of particles, including spin (assuming ω_rf≠0), particle's rest mass, and the curvature radius of the hyperbolic wormhole, in addition to fundamental constants like ħ and c. The background under consideration is a spacetime with constant negative Gaussian curvature, characteristic of structures like monolayer graphene sheets or graphene wormholes since K=-1/r_0^2 (refer to <cit.>). Consequently, there is potential utility in adjusting the outcomes for Weyl fermions by setting m=0 and substituting c with ν_f, where ν_f denotes the Fermi velocity, approximately equal to c/300. Notably, equation (<ref>) transforms accordingly

ℰ_ns=-s ħω_rf ±iħν_f (n+1)/r_0,

if m=0. The Eq. (<ref>) indicates that the energy of this system possesses a complex nature. Consequently, these states cannot remain stable, and the associated modes either decay or grow over time since Ψ∝e^-iℰ_ns/ħt. Additionally, the real oscillation of a Weyl particle is solely influenced by the coupling between the particle's spin and the rotational frequency of the frame. Our findings suggest that the stability of the background spacetime is compromised under perturbations caused by massless particles, while it remains stable under perturbations from Weyl antiparticles (also see <cit.>) when ω_rf=0. Notably, the coupling s ω_rf might induce symmetry breaking around the Dirac point when m≠ 0. Furthermore, it appears conceivable to measure, in principle, only the energy stemming from the sω_rf coupling if the Dirac particle possesses a critical (or effective) mass value, m_c=ħ (n+1)/c r_0. Such a scenario could arise particularly in condensed matter systems, especially when r_0 is on the order of a few nanometers (also see <cit.>).



 §.§ Relativistic fermions in the rotating frame of elliptic wormhole


In this scenario, we derive precise outcomes for relativistic fermions within the rotating frame of an elliptical wormhole. Consequently, the wave equation presented in Eq. (<ref>) manifests as follows:

    ψ_2_,ρρ+( ρ/r_0)/r_0 ψ_2_,ρ+[λ+Q̂_2] ψ_2=0,
       Q̂_2=1/2r_0^2-s^2/b^2/sinh^2( ρ/r_0)-s/b/r_0 ( ρ/r_0)/sinh( ρ/r_0)-^2( ρ/r_0)/4 r^2_0.

Let us attempt to reformulate this equation using a new variable substitution, z=1/2[1+cosh(ρ/r_0)], aiming to eliminate the hyperbolic functions. By expressing Eq. (<ref>) in terms of the variable z, we can render it free of hyperbolic functions, facilitating an exploration of its asymptotic behaviors. Consequently, employing an ansatz function, ψ_2(z)=z^2sr_0-b/4b(z-1)^-2sr_0+b/4bψ(z), the resulting equation can be transformed into the following wave equation

   z(z-1)ψ_,zz-[γ̃-(α̃+β̃+1)z]ψ_,z+α̃β̃ψ=0,  , 

   α̃=-ir_0√(λ̃),   β̃=α̃^*,   γ̃=1/2+sr_0/b. 
In this context, α̃, β̃, and γ̃ represent fundamental constants within Gauss hypergeometric wave equation, as extensively documented in references <cit.>. The regular solution of this equation in the vicinity of the origin can be elegantly expressed as ψ(z )=𝒞 _2F_1(α̃, β̃ ; γ̃; z ), where 𝒞 stands for an arbitrary constant. Remarkably, when the parameter α̃ assumes the value -n, with n representing the radial quantum number, the solution function ψ(z) transforms into a polynomial of degree n concerning the variable z. This establishes a link between the polynomial nature of the solution function and the quantum characteristics of the system, offering valuable insights into the system's energy behavior at different quantum states. This condition establishes an intriguing connection to the system's energy spectrum, unveiling the following set of energy levels

    ℰ_ns=-s  ħω_rf±ħ c√(m^2c^2/ħ^2-n^2/r_0^2).

While bearing resemblance to the expression presented in Eq. (<ref>), the relativistic energy pertaining to a fermion precisely equals the rest mass energy ± mc^2 under specific conditions where ω_rf=0 and n=0. This distinction stands as the fundamental dissimilarity between the outcomes delineated in Eqs. (<ref>) and (<ref>). Our investigations substantiate that within the rotating frame of an elliptic wormhole, the energy of a relativistic fermion manifests an intriguing independence from the curvature radius of the wormhole as the system converges toward its ground state. Notably, our observations assert that analogous deliberations, explicated in sec. <ref>, remain viable and generalizable when transitioning solely through n+1⟶ n. The inference arises from the energy profiles derived in both situations, suggesting that the baseline energy level for relativistic fermions in the rotating frame of a hyperbolic wormhole appears akin to the first excited state for those in the rotating frame of an elliptic wormhole. Moreover, this correlation remains intact even when ω_rf=0.



§ SUMMARY AND DISCUSSIONS


In this study, we analyse the influence of non-inertial effects arising from the uniformly rotating reference frame of negative curvature wormholes, such as the hyperbolic wormhole and elliptic wormhole, on relativistic fermions. Our approach involves the investigation of analytically permissible solutions of the associated Dirac equation. Initially, we derive a comprehensive non-perturbative second-order wave equation. Subsequently, we proceed to ascertain the exact solutions of this wave equation within two distinct scenarios. In the first scenario, we focus on a background that characterizes the hyperbolic wormhole. Within this context, we determine the solution function expressed in terms of Gauss hypergeometric function. Consequently, we derive a non-perturbative energy spectrum in a closed mathematical form. The expression for this energy spectrum is provided by equation (<ref>).

The findings reveal an intriguing interdependence between the energy states of relativistic fermions, the curvature radius of a wormhole, and the angular frequency of the uniformly rotating frame. Alongside the intrinsic quantum attributes of the fermionic field, our investigation underscores the direct influence of the particle's spin on the energy manifestation. Notably, this interaction may induce symmetry breaking centered around the Dirac point (zero energy), with energy levels diminishing until they reach a critical curvature radius of the wormhole, precisely at r_0=ħ(n+1)/mc, where n signifies the principle quantum number. Importantly, the system's stability is contingent upon the relationship between r_0 and ħ(n+1)/mc; instability ensues if r_0 surpasses ħ(n+1)/mc, irrespective of whether the angular frequency of the rotating frame is null or not. Our investigation further reveals intriguing insights into Weyl fermions (m=0), suggesting that states associated with these fermions may not exhibit stability due to potential damped modes characterized by a decay time τ_n=ħ/|ℰ_ns_Im|, where ℰ_ns_Im denotes the imaginary component of the energy. In an alternative scenario involving an elliptic wormhole background in spacetime, our analysis of the primary equation yields precise solutions for the relativistic energy within the system under consideration. Surprisingly, the ground state energy identified in the earlier scenario appears to correspond to the first excited state of relativistic fermions within the rotating frame of an elliptic wormhole. Intriguingly, when the angular velocity of the rotating frame reaches zero, our results demonstrate that the relativistic energy of the system aligns with the rest mass energy of the fermion upon reaching the ground state (n=0). This alignment reflects an inevitable outcome given our understanding that the system will ultimately settle into its ground state. Furthermore, our outcomes suggest the possibility of measuring energy solely derived from the interplay between the particle's spin and the angular frequency of the uniformly rotating frame. This measurement becomes feasible if the particle possesses a critical (or effective) mass value represented by m_c=ħ (n+1)/c r_0 and m_c=ħ n/c r_0 for the first and second scenarios, respectively. Particularly intriguing is the potential manifestation of this specialized case in condensed matter systems, where c can be equated to ν_f (Fermi velocity). Notably, this scenario appears plausible when the curvature radius of the wormhole approximates a few nanometers, assuming the particle's mass is the typical electron mass.

Recently, there has been a notable focus on performing thermodynamic evaluations using energy levels from quantum systems in complex environments. Several studies in the field have investigated this, setting the stage for future exploration of thermodynamic properties <cit.>. Our precise findings could be valuable for analyzing the thermal properties of graphene structures, presenting an intriguing avenue for future research.







§ DATA AVAILABILITY

This manuscript does not contain any associated data.



§ CONFLICTS OF INTEREST STATEMENT


The authors have disclosed no conflicts of interest.



§ FUNDING


This research has not received any funding.

99

landau L. D. Landau, E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics, vol. 2, 4th edn. (Elsevier, Oxford, 1980).

ref1 E.J. Post, "Sagnac effect".  https://doi.org/10.1103/RevModPhys.39.475Reviews of Modern Physics 39, 475 (1967).

ref2 J. Anandan, "Gravitational and rotational effects in quantum interference". https://doi.org/10.1103/PhysRevD.15.1448Physical Review D 15, 1448 (1977).

ref3 J.J. Sakurai, "Comments on quantum-mechanical interference due to the Earth's rotation". https://doi.org/10.1103/PhysRevD.21.2993Physical Review D 21, 2993 (1980).

ref4 B.R. Iyer, "Dirac field theory in rotating coordinates". https://doi.org/10.1103/PhysRevD.26.1900Physical Review D 26, 1900 (1982).

rot L.A. Page, "Effect of Earth's rotation in neutron interferometry". https://doi.org/10.1103/PhysRevLett.35.543Physical Review Letters 35, 543 (1975).

ref12 S.A. Werner, J.L. Staudenmann, R. Colella, "Effect of Earth's rotation on the quantum mechanical phase of the neutron". https://doi.org/10.1103/PhysRevLett.42.1103Physical Review Letters 42, 1103 (1979).

rot2 F.W. Hehl, W.-T. Ni, "Inertial effects of a Dirac particle". https://doi.org/10.1103/PhysRevD.42.2045Physical Review D 42, 2045 (1990).

ref5 B. Mashhoon, "Neutron interferometry in a rotating frame of reference". https://doi.org/10.1103/PhysRevLett.61.2639Physical Review Letters 61, 2639 (1988).

rot3 L.B. Castro, "Noninertial effects on the quantum dynamics of scalar bosons". https://doi.org/10.1140/epjc/s10052-016-3904-4European Physical Journal C 76, 61 (2016).

rot4 E.V.B. Leite, H. Belich, R.L.L. Vitória, "Effects of rotation on a scalar field in a Kaluza–Klein theory". https://doi.org/10.1142/S0217732320502831Modern Physics Letters A 35, 2050283 (2020).

rot5 K. Bakke, V.B. Bezerra, R.L.L. Vitória, "Scalar field in a uniformly rotating frame in the time-dislocation space–time". https://doi.org/10.1142/S0217751X20501298International Journal of Modern Physics A 35, 2050129 (2020).

exp1 M. Toroš, M. Cromb, M. Paternostro, D. Faccio, "Generation of entanglement from mechanical rotation".  https://doi.org/10.1103/PhysRevLett.129.260401Physical Review Letters 129, 260401 (2022).

exp2 M. Cromb, S. Restuccia, G.M. Gibson, M. Toroš, M.J. Padgett, D. Faccio, "Mechanical rotation modifies the manifestation of photon entanglement".  https://doi.org/10.1103/PhysRevResearch.5.L022005Physical Review Research 5, L022005 (2023).

exp3 S. Restuccia, M. Toroš, G.M. Gibson, H. Ulbricht, D. Faccio, M.J. Padgett, "Photon bunching in a rotating reference frame". https://doi.org/10.1103/PhysRevLett.123.110401Physical Review Letters 123, 110401 (2019).

ricardo-1 R.L.L. Vitória, "Noninertial effects on a scalar field in a spacetime with a magnetic screw dislocation". https://doi.org/10.1140/epjc/s10052-019-7359-2European Physical Journal C 79, 844 (2019).

ricardo-2 R.L.L. Vitória, K. Bakke, "Rotating effects on the scalar field in the cosmic string spacetime, in the spacetime with space-like dislocation and in the spacetime with a spiral dislocation". https://doi.org/10.1140/epjc/s10052-018-5658-7European Physical Journal C 78, 1–6 (2018).

ref6 F.W. Hehl, W.T. Ni, "Inertial effects of a Dirac particle". https://doi.org/10.1103/PhysRevD.42.2045Physical Review D 42, 2045 (1990).

ref7 S.M. Cui, H.H. Xu, "Berry’s phase in rotating systems". https://doi.org/10.1103/PhysRevA.44.3343Physical Review A 44, 3343 (1991).

ref8 K. Bakke, C. Furtado, "Bound states for neutral particles in a rotating frame in the cosmic string spacetime". https://doi.org/10.1103/PhysRevD.82.084025Physical Review D 82, 084025 (2010).

ref9 L.C.N. Santos, C.C. Jr Barros, "Relativistic quantum motion of spin-0 particles under the influence of noninertial effects in the cosmic string spacetime"., https://doi.org/10.1140/epjc/s10052-017-5476-3European Physical Journal C 78, 1–8 (2018).

ref10 S. Zare, H. Hassanabadi, M. de Montigny, "Non-inertial effects on a generalized DKP oscillator in a cosmic string space-time". https://doi.org/10.1007/s10714-020-02676-0General Relativity and Gravitation 52, 1–20 (2020).

ref13 U.R. Fischer, N. Schopohl, "Hall state quantization in a rotating frame". https://doi.org/10.1209/epl/i2001-00273-1Europhysics Letters 34, 502 (2001).

ref14 L-H. Lu, You-Q. Li, "Effects of an optically induced non-Abelian gauge field in cold atoms". https://doi.org/10.1103/PhysRevA.76.023410Physical Review A 76, 023410 (2007).

ref15 J-Q. Shen, S-L. He, "Geometric phases of electrons due to spin-rotation coupling in rotating C 60 molecules".  https://doi.org/10.1103/PhysRevB.68.195421Physical Review B 68, 195421 (2003).


ref19 R.R. Cuzinatto, M. de Montigny, P.J. Pompeia, "Non-commutativity and non-inertial effects on a scalar field in a cosmic string space-time: I. Klein–Gordon oscillator". https://doi.org/10.1088/1361-6382/ac51bbClassical and Quantum Gravity 39, 075006 (2022).

cjp F. Ahmed, " Aharonov-Bohm and non-inertial effects on a Klein-Gordon oscillator with potential in the cosmic string space-time with a spacelike dislocation". https://doi.org/10.1016/j.cjph.2020.06.012Chinese Journal of Physics 66, 587 (2020).

einstein-rosen A. Einstein, N. Rosen, "The Particle Problem in the General Theory of Relativity". https://doi.org/10.1103/PhysRev.48.73Physical Review  48, 73 (1935).

wheeler C.W. Misner, J.A. Wheeler, "Classical physics as geometry". https://doi.org/10.1016/0003-4916(57)90049-0Annals of physics  2, 525–603 (1957).

ellis H.G. Ellis, "Ether flow through a drainhole: A particle model in general relativity". https://doi.org/10.1063/1.1666161Journal of Mathematical Physics  14, 104–118 (1973).

ES1 F.S. Azevedo, J.D. de Lima, A. de Pádua Santos, F. Moraes, "Optical wormhole from hollow disclinations" . https://doi.org/10.1103/PhysRevA.103.023516Physical Review A  103, 023516  (2021).

ES2 R. Dandoloff, A. Saxena, B. Jensen, "Geometry-induced potential on a two-dimensional section of a wormhole: Catenoid". https://doi.org/10.1103/PhysRevA.81.014102Physical Review A  81, 014102  (2010).

wormhole T. Rojjanason, P. Burikham, K. Pimsamarn, "Charged fermion in (1+2)-dimensional wormhole with axial magnetic field".  https://doi.org/10.1140/epjc/s10052-019-7156-yEuropean Physical Journal C  79, 1–17  (2019).

ap1 A. Malik, T. Naz, A. Qadeer, M.F. Shamir, Z. Yousaf, "Investigation of traversable wormhole solutions in modified f(R) gravity with scalar potential".  https://doi.org/10.1140/epjc/s10052-023-11704-7European Physical Journal C  83, 1–12  (2023).

ap2 S. Antonini, P. Simidzija, B. Swingle, M. Van Raamsdonk, "Accelerating cosmology from a holographic wormhole".  https://doi.org/10.1103/PhysRevLett.130.221601Physical Review Letters  130, 221601  (2023).

ap3 F. Ahmed, "Three-dimensional wormhole with cosmic string effects on eigenvalue solution of non-relativistic quantum particles".  https://doi.org/10.1038/s41598-023-40066-zScientific Reports  13, 12953  (2023).

ap4 F. Ahmed, "Harmonic oscillator problem in the background of a topologically charged Ellis-Bronnikov–type wormhole (a)".  https://doi.org/10.1209/0295-5075/acbb21Europhysics Letters 141, 54001  (2023).

ap5 F. Ahmed, A. Guvendi, "Rainbow gravity’s effects on scalar field in wormhole background with cosmic strings".  https://doi.org/10.1016/j.cjph.2023.11.028Chinese Journal of Physics 87, 174–184  (2024).

ap6 A. Guvendi, S.G. Dogan, "Magnetic flux-driven modulation of Weyl pair dynamics on catenoid bridge: A theoretical analysis".  https://doi.org/10.1016/j.cjph.2024.04.034Chinese Journal of Physics 89, 1538-1544  (2024).

ap7 S. Arshad, U. Sheikh, "String Fluid as a Source of Traversable Rainbow Wormholes".  https://doi.org/10.1007/s10773-024-05624-9International Journal of Theoretical Physics 63, 1–10  (2024).

ap8 O. Mustafa, A. Guvendi, "Klein-Gordon oscillators in (2+ 1)-dimensional traversable wormhole spacetime in rainbow gravity".  https://doi.org/10.48550/arXiv.2401.09342 arXiv:2401.09342 [gr-qc] (2024).

ap9 F. Naderi, K. Hasanirokh, "Quantum transport of massless Dirac fermions through wormhole-shaped curved graphene in presence of constant axial magnetic flux".  https://doi.org/10.1038/s41598-024-57718-3Scientific Reports 14, 7763  (2024).

ap10 A. Guvendi, S.G. Dogan, "Vector bosons in the rotating frame of negative curvature wormholes". https://doi.org/10.1007/s10714-024-03213-zGeneral Relativity and Gravitation 56, 32  (2024).

ap11 J. Chandra, "Euclidean wormholes for individual 2d CFTs".  https://doi.org/10.1007/JHEP04(2024)051Journal of High Energy Physics 2024, 1–42  (2024).

ap12 S. Alfaro, P.A. González, D. Olmos, E. Papantonopoulos, Y. Vásquez, Yerko, "Quasinormal modes and bound states of massive scalar fields in wormhole spacetimes".  https://doi.org/10.1103/PhysRevD.109.104009Physical Review D 109, 104009  (2024).

ap13 G. Clément, D. Gal'Tsov, "Rotating traversable wormholes in Einstein-Maxwell theory". https://doi.org/10.1016/j.physletb.2023.137677Physics Letters B 838, 137677  (2023).

ap14 M.M. Roberts, T. Wiseman, "Analog gravity and continuum effective theory of the graphene tight-binding lattice model". https://doi.org/10.1103/PhysRevB.109.045425Physical Review B 109, 045425  (2024).


Gibbons M. Cvetič, G.W. Gibbons, "Graphene and the Zermelo optical metric of the BTZ black hole". https://doi.org/10.1016/j.aop.2012.05.013Annals of Physics  327, 2617–2626 (2012).

wormhole2 S. Zare, M. de Montigny, H. Chen, H. Hassanabadi, "Lorentz violation in a family of (1+ 2)-dimensional wormhole". https://doi.org/10.48550/arXiv.2209.05630arXiv:2209.05630v2 [math-ph] (2022).


2 S.G. Dogan, A. Guvendi, "Weyl fermions in a 2+1 dimensional optical background of constant negative curvature".  https://doi.org/10.1140/epjp/s13360-023-04101-2European Physical Journal Plus 138, 452 (2023). 

2a A. Guvendi, H. Hassanabadi, "Fermion-antifermion pair in magnetized optical wormhole background". https://doi.org/10.1016/j.physletb.2023.138045Physics Letters B 843, 138045 (2023). 

2b S.G. Dogan, "Dirac pair in magnetized elliptic wormhole". https://doi.org/10.1016/j.aop.2023.169344Annals of Physics 454, 169344 (2023). 

new1 M. Burgess, B. Jensen, "Fermions near two-dimensional surfaces". https://doi.org/10.1103/PhysRevA.48.1861Physical Review A  48, 1861  (1993).

new2 M.V. Entin, L.I. Magarill, "Spin-orbit interaction of electrons on a curved surface". https://doi.org/10.1103/PhysRevB.64.085330Physical Review B  64, 085330  (2001).

new3 M. Cariglia, G.W. Gibbons, "Lévy-Leblond fermions on the wormhole". https://doi.org/10.48550/arXiv.1806.05047arXiv preprint arXiv:1806.05047  (2018).

new4 A. Lherbier, H. Terrones, J.C. Charlier, "Three-dimensional massless Dirac fermions in carbon schwarzites". https://doi.org/10.1103/PhysRevB.90.125434Physical Review B  90, 125434  (2014).

new5 J. Gonzalez, J. Herrero, "Graphene wormholes: A condensed matter illustration of Dirac fermions in curved space". https://doi.org/10.1016/j.nuclphysb.2009.09.028Nuclear physics B  825, 426–443  (2010).

22c A. Guvendi, O. Mustafa, "Fermion-antifermion pairs in a magnetized space-time with non-zero cosmological constant". https://doi.org/10.1016/j.nuclphysb.2024.11657Nuclear physics B 1004, 116571 (2024). 

22d A. Guvendi, "Evolution of an interacting fermion–antifermion pair in the near-horizon of the BTZ black hole". https://doi.org/10.1140/epjc/s10052-024-12542-xEuropean Physical Journal C 84, 1–7 (2024). 

RF A. Guvendi, H. Hassanabadi, "Noninertial effects on a composite system". https://doi.org/10.1142/S0217751X21502535International Journal of modern Physics A 36, 2150253 (2021).

new7 S. Zare, H. Hassanabadi, A. Guvendi, "Relativistic Landau quantization for a composite system in the spiral dislocation spacetime".  https://doi.org/10.1140/epjp/s13360-022-02802-8European Physical Journal Plus 137, 589 (2022).

new8 A. Guvendi, S. Zare, H. Hassanabadi, "Exact solution for a fermion–antifermion system with Cornell type nonminimal coupling in the topological defect-generated spacetime". https://doi.org/10.1016/j.dark.2022.101133Physics of the Dark Universe 38, 101133 (2022).


new9 A. Guvendi, Y. Sucu, "An interacting fermion-antifermion pair in the spacetime background generated by static cosmic string". https://doi.org/10.1016/j.physletb.2020.135960Physics Letters B 811, 135960 (2020).

Abramowitz M. Abramowitz  and I.A. Stegun, Handbook of Mathematical Functions (without numerical tables). NBS, 10 ed. (1972).

ter H. Hassanabadi, M. Hosseinpour, "Thermodynamic properties of neutral particle in the presence of topological defects in magnetic cosmic string background". https://doi.org/10.1140/epjc/s10052-016-4392-2European Physical Journal C 76, 1–7 (2016).

ter1 A.N. Ikot, B.C. Lutfuoglu, M.I. Ngwueke, M.E. Udoh, S. Zare, H. Hassanabadi, "Klein-Gordon equation particles in exponential-type molecule potentials and their thermodynamic properties in D dimensions". https://doi.org/10.1140/epjp/i2016-16419-5European Physical Journal Plus 131, 1–17 (2016).

ter2 M. Eshghi, H. Mehraban, "Study of a 2D charged particle confined by a magnetic and AB flux fields under the radial scalar power potential". https://doi.org/10.1140/epjp/i2017-11379-xEuropean Physical Journal Plus 132, 1–13 (2017).

ter4 R.L.L. Vitória, T. Moy, H. Belich, "Thermodynamics Properties of a Quantum Particle Confined into Two Elastic Concentric Spheres". https://doi.org/10.1007/s00601-022-01752-2Few-Body Systems 63, 51 (2022).

ag1 A. Guvendi, A. Boumali, "Superstatistical properties of a fermion–antifermion pair interacting via Dirac oscillator coupling in one-dimension". https://doi.org/10.1140/epjp/s13360-021-02083-7European Physical Journal Plus 136, 1098 (2021).

ag2 A. Guvendi, A. Boumali, "Landau levels for a Weyl pair in a monolayer medium and thermal quantities". https://doi.org/10.31349/RevMexFis.69.061701Revista Mexicana de Física 69, 061701–1 (2023).


ag3 A. Guvendi, A. Boumali, "Thermodynamic properties of an interacting fermion-antifermion pair in a magnetized spacetime with a non-zero cosmological constant". https://doi.org/10.1209/0295-5075/ad49d2Europhysics Letters 146, 51003 (2024).







 Paper Title: Hot and Dense QCD Shear Viscosity at (almost) NLO.


=1
















Λ_QCD
#1Eq. (<ref>)
 Re 
 Im 

Li_2
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ff̅h
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d^3/2||   (2π)^3 
Π_00
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22

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m_ eff,f
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T_F
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n_B
n_q
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    𝒞#1Appendix <ref>#1Section <ref>#1Ref. <cit.>#1Fig. <ref>Institut für Kernphysik, Technische Universität Darmstadt
    
    Schlossgartenstraße 2, D-64289 Darmstadt, Germanyidanhoni@theorie.ikp.physik.tu-darmstadt.de,guy.moore@physik.tu-darmstadt.deThe next-to-leading order weak-coupling shear viscosity of QCD was computed 6 years ago.
     However, these results have never been applied at finite baryon chemical potential μ, even though intermediate-energy heavy ion collisions and merging neutron stars may explore the Quark-Gluon Plasma in a regime where baryon chemical potentials are large.
     Here, we extend the next-to-leading order shear viscosity calculations to finite μ, and we show that, while the convergence of the weak-coupling expansion is questionable for achievable plasmas, it is somewhat better at μ > T than at μ=0.
    
    Hot and Dense QCD Shear Viscosity at (almost) NLO
        Isabella Danhoni, Guy D. Moore
        August 5, 2024
    =================================================
    
    
    
    § INTRODUCTION
    
    
    The deconfined phase of hot and dense QCD is known as the quark-gluon plasma (QGP). Evidence shows that this phase was present in the early universe and that it has been recreated at microscopic scales in heavy ion collision experiments<cit.>.
    These results have sparked large interest in this subject over the past few decades, and experiments indicate that the QGP in such collisions is a strongly coupled fluid<cit.> with behavior close to that of a perfect fluid<cit.>.
    On the other hand, sufficiently early in the history of the early Universe, the running coupling should have been small enough that a perturbative treatment may be valid.
    For thermodynamical quantities, suitably resummed perturbative calculations work starting at temperatures a few times that of the chiral symmetry restoration transition<cit.>.
    
    
    
    
    
    In lower-energy collisions, quark-gluon plasma is generated with a significant net baryon number, such that the baryon chemical potential μ_B is several times higher than the temperature.
    This regime cannot be directly studied on the lattice due to the sign problem (or complex action problem).
    For a chemical potential μ_B≠ 0 the fermion determinant (or Monte Carlo action) becomes complex, which ruins the probabilistic interpretation of this quantity used in lattice QCD calculations<cit.>.
    For this reason, the high-density region of the QCD phase diagram is much less well understood than the μ_B=0 axis.
    As experimental efforts focus on bringing clarity to this region, such as RHIC-BES <cit.>, HADES<cit.> and CBM at
    FAIR<cit.>, the need for parallel theoretical development becomes clear.
    
    At high temperatures, we claim that the perturbative series should work better at a high chemical potential, such that μ_B > T > T_c, than in the case of μ_B=0<cit.>. This is a consequence of the fact that scattering from quarks is enhanced by a factor of μ_B^2 / T^2.
    The failure of perturbation theory in hot QCD occurs before the coupling becomes large, due to the strongly-coupled nature of soft gluons.
    These play a central role in the breakdown of the perturbative expansion for the shear viscosity <cit.>.
    But at high chemical potential, where quarks play a larger role, one expects the perturbative series to be better behaved.
    This means that perturbation theory has a better chance of working at high μ/T than at μ=0.
    Although the (μ,T) values which can be explored in experiment are unlikely to extend into the region where perturbation theory is directly predictive, the perturbative results can still be used in modeling the (μ,T) behavior of the shear viscosity phenomenologically, as was recently done using leading-log results in Ref. <cit.>.
    Essentially, perturbation theory serves as a limiting behavior so that η/s can be interpolated based on high-density perturbative, low-density hadron-resonance gas, and intermediate density data-driven estimates.
    
    Previously Arnold, Moore, and Yaffe have computed the shear viscosity at leading perturbative order in the absence of a chemical potential
    <cit.>, and Ghiglieri et al have extended this calculation to “almost” next-to-leading order (NLO) <cit.>.
    Separately, we have presented the shear viscosity at finite chemical potential at leading-log order <cit.>.
    The goal of this paper is to work to almost next-to-leading order at finite chemical potential, both to obtain the most reliable estimates for the viscosity in this regime that we can, and to test how perturbation theory converges in the high-μ regime.
    We start in Section <ref> by reviewing the kinetic theory and computational approach that Arnold, Moore, and Yaffe used in <cit.>.
    In Section <ref>, we review how to extend the collision operator to next-to-leading order and describe the computational tools required. 
    In Section <ref>, we show what changes must be made to the calculation to include chemical potentials.
    The analysis of our calculations and final results are shown in
    Section <ref>.
    We end in Section <ref> with a short discussion and summary.
    But we can present the main conclusions very briefly now.
    The shear viscosity, rescaled as η T/(ε + P), is a weak function of μ_q/T (with μ_q=μ_B/3 the chemical potential of a quark).
    The convergence of the perturbative series improves when
    μ_q^2 > N_c π^2 T^2, but the improvement is somewhat modest.
    For more complete details, see Section <ref>.
    
    
    
    
    
    § KINETIC THEORY
    
    In this section, we briefly summarize how shear viscosity is defined and how it can be computed to leading order (LO) in the Effective Kinetic Theory (EKT).
    Shear viscosity is a property relevant for systems with space-nonuniform flow velocity, meaning that the energy and momentum densities vary through space, but in a manner which is sufficiently smooth that the system is locally near equilibrium.
    In this case we can exchange the conserved densities T^μ 0 for the local 4-velocity and energy density, defined with the Landau-Lifshitz convention <cit.>:
    u^μ(x,t) is the timelike unit eigen(four)vector of T^μν and ε is its eigenvalue,
    T^μν u_ν = ε u^μ
    with[We use a mostly-positive metric tensor g_μν = Diag[-1,+1,+1,+1],
    and natural units, c=1, ħ = 1. Capital letters P,K are 4-vectors, lower case letters p⃗,k⃗ or p_i,k_i are the space components, p^0,k^0 are the time components, and p,k are the magnitudes of the space components.]u^μ u_μ = -1.
    For uniform u^μ,ε the system will be in equilibrium with
    T^μν = ε u^μ u^ν + P g^μν,
    where P the pressure is determined from ε by the equilibrium equation of state.
    
    However, for nonuniform u^μ the system will depart from equilibrium and T^μν will not follow its ideal form.
    Defining the projection operator into the local rest frame,
    
        Δ^μν≡ g^μν + u^μ u^μ ,
    
    the shear tensor is defined as
    
        σ^μν≡∇^μ u^ν + ∇^ν u^μ
            - 2/3Δ^μν∇_α u^α .
    
    For small and slowly varying σ_ij, linear response theory predicts that the stress tensor should change by an amount linear in the shear tensor:  in the local rest frame,
    
        T_ij = P g_ij - η σ_ij
    
    with P the pressure and η, the shear viscosity, defined as the coefficient of the linear response of T_ij to nonvanishing σ_ij.
    If the fluid flow also has a divergence θ = ∂_i u_i, then an additional term T_ij = … - ζθΔ_ij would be present, with ζ the bulk viscosity.
    Bulk viscosity requires a different analysis <cit.> which we will not attempt here.
    
    Many (not all) theories have a kinetic description, and this should include nonabelian gauge theories in the weak coupling regime.
    In this case one can give a coarse-grained description of the state of the system in terms of the occupancies of each field type at each point in time and phase space, f^a(p⃗,x⃗,t).
    In equilibrium and at lowest order in the coupling, we have
    
        f_0^a(p⃗,x⃗,t) = 1/exp(γβ ( p^0 - u_i p^i - μ_a ) ∓ 1 
            = 1/exp (β (-u_μ P^μ - μ_a)) ∓ 1.
    
    Here u^μ = (γ , γv⃗) is the local velocity 4-vector as before, γ is the associated relativistic gamma-factor, and β=β(x⃗,t) = 1/T is the local temperature.
    The sign ∓ 1 is - for bosons and + for fermions (the upper sign will always apply to bosons), and μ_a is the chemical potential for species a, which will be nonzero for quark species and opposite for a particle and its antiparticle.
    In this work, we will consider SU(3) gauge theory with  vectorlike quark species each with the same chemical potential.
    
    
    Shear viscosity, as well as other transport coefficients, is theoretically defined in a Kubo-type formulae relating them to the zero-frequency transport limit of the spectral functions of two-point correlators of the appropriate operators.
    Performing a perturbative evaluation at leading order using the diagrammatic approach based directly on these Kubo formulas would require complex resummations of numerous sub-diagrams to all orders.
    This has been done successfully for scalar field theory
    <cit.> and for QED for electrical conductivity
    <cit.> and shear viscosity <cit.>,
    and in each case the result proved to be the same as a kinetic description.
    In the case of full QCD such a diagrammatic calculation has never been successfully attempted, but physical arguments indicate that the result should again justify a kinetic description <cit.>.
    Therefore, we adopt the linearized version of the EKT developed in <cit.>.
    
    One starts as usual with a Boltzmann equation
    
        [ v⃗_p ·∂/∂x⃗] f^a(p⃗,x⃗,t) = - ^a[f],
    
    where f^a refers to the nonequilibrium form of the statistical function,
    which we linearize about the equilibrium form
    
        f^a(p⃗,x⃗) = f_0^a(p⃗, x⃗) + f_0(1 ± f_0)  f_1^a(p⃗,x⃗)  ,
    
    according to the Landau-Lifshitz conventions under which f_0 contributes all of the energy, momentum density, and net particle numbers in the system.
    The choice to normalize f_1 with a factor of f_0(1± f_0) will be convenient in what follows.
    For a more detailed description one can refer to <cit.> which developed the leading-log application of Boltzmann equations to the evaluation of the QCD shear viscosity,
    <cit.>, which extended the treatment to leading order and established most of our notation and techniques,
    and <cit.>, which extended the treatment to finite chemical potentials, albeit at leading-logarithmic order.
    We will save the discussion of the collision operator ^a[f] for the next section.
    
    The departure from equilibrium must first arise at linear order in σ_ij.
    Using rotational invariance, one can then write the departure from equilibrium as
    
        f_1^a(p⃗,x⃗) = β^2 X_ij(x⃗) χ^a_ij(p⃗) = β^2 X_ij(x⃗) I_ij(p⃗) χ^a(p)  .
    
    Here χ^a(p) is a species-dependent, pure scalar function of p which characterizes the departure from equilibrium in the presence of shear stress, and
    
        X_ij   ≡1/√(6)σ_ij
        
            I_ij(p⃗)    ≡√(3/2)( p̂_i p̂_j - 1/3δ_ij)  .
    
    The normalization is explained in Ref. <cit.> and is chosen so that I_ij is a properly normalized projection operator.
    Evaluating the space derivatives on the lefthand side of boltz
    one finds to lowest order,
    
        v⃗_p ·∂/∂x⃗
            f^a_0(p⃗,x⃗,t) = β^2 X_ij S^a_ij
    
    with
    
        S^a_ij(p⃗)  ≡ p^0   T   f_0^a(1± f_0^a) I_ij(p⃗)  .
    
    Using this and inserting chidef into the RHS, boltz reduces to
    
        β^2 X_ij  S^a_ij(p⃗) = ^a[β^2 X_ij χ^a_ij(p⃗)]
                
            S^a_ij(p⃗) = χ^a_ij(p⃗) ,
    
    where χ^a_ij(p⃗) means the result of factoring β^2 X_ij out of the collision operator.
    
    This equation is solved using a variational method.
    This strategy was used in <cit.> in leading log and leading order calculations. 
    First one defines an inner product on departures from equilibrium:
     
        (f,g) = β^3 ∑^_a∫_p⃗ f^a(p⃗)g^a(p⃗)
    
    It can be shown that the collision operator is Hermitian with respect to this inner product.
    We define
     
        𝒬[χ] = (χ_ij, S_ij)- 1/2(χ_ij,χ_ij)  ,
    
    which can be viewed as a functional over the space of χ^a(p) values.
    That is, the functional 𝒬 takes its maximum value when the Boltzmann equation is satisfied.
    Furthermore, its value at this maximum,
     
        =1/2(χ_ij, S_ij)=1/2 (χ_ij,χ_ij) = 1/2 ( S_ij,^-1 S_ij )  ,
     
    determines the viscosity:
     
        η = 2/15.
    
    As known from other variational problems, one can obtain highly accurate approximate results by performing a restricted extremization within a well-chosen finite dimensional subspace.
    For this purpose, we expand the χ(p) functions into a finite basis set:
    
        χ(p) =∑_m=1^N a_m ϕ^(m)(p) .
    
    We will use the variational Ansatz proposed
    [To our knowledge this variational Ansatz was first used in Ref. <cit.>, but it is not actually presented there; we thank Larry Yaffe for private communications on this point.]
    in Ref. <cit.>:
    
        ϕ^(m)=p(p/T)^m/(1+p/T)^N-1,  m=1,...,N  .
    
    The subspace spanned by this basis strictly increases as N is increased.
     
    
    
    
    § COLLISION OPERATORS FOR AN NLO EVALUATION
    
    
    At vanishing chemical potential, the perturbative framework suggests that the QGP predominantly comprises quasiparticle excitations with momenta of the order of the temperature.  
    Their dispersion and occupancy are corrected only at (g^2), which for an (g)-subleading calculation means that we can ignore corrections both to the lowest-order entropy density s and to the lowest-order particle dispersion relations (except for splitting, see below).
    Therefore NLO corrections will only be needed in the collision integral, which we turn to next.
    The LO calculation <cit.> requires two types of scattering processes, the  scatterings with all hard ((T)) external particles and  effective splitting processes between hard participants. 
    
    At NLO, all the scatterings and corrections to be added are the ones suppressed by a single power of g.  As shown in detail in <cit.>, there are only a
    few such (g) subleading effects, which can be described as follows. The rate of soft
     scattering is modified to be considered as an
    additional momentum-diffusion coefficient δq̂.  This along with an (g) correction to the medium corrections to dispersion provides an (g) shift in the  splitting rate.  Next, the  splitting rate must be corrected wherever one participant of the process is "soft", p ∼ gT, or the opening angle is less collinear.  Finally, the numerical implementation of the LO scattering kernel <cit.> already resums a small amount of these NLO effects, requiring a subtraction to δq̂ and the longitudinal momentum diffusion, δq̂_L. 
    This section will briefly review the collision operator at LO and NLO at vanishing chemical potential.
    
    
    
     §.§ Collision Integrals at LO
    
    
    The leading-order collision operator encodes the contribution of tree-level 
    scattering processes, with Hard-Loop resummed
    propagators in the soft-sensitive channels,
    as well as collinear, effective  processes resumming the effect
    of splitting caused by any number of soft scatterings.
    Both processes contribute to order g^4T to the collision operator; a subset of C_a^[f]
    is logarithmically enhanced, g^4Tln(1/g), due to the sensitivity
    to the soft scale gT.
    C_a^[f] and C_a^[f] are described in detail
    in <cit.>. Therefore, the collision term has the form:
    
        𝒞_a[f] = 𝒞_a^2↔ 2[f] + 𝒞_a^1↔ 2[f]
    
    
    
    The matrix elements that contribute to 2 ↔ 2 processes at leading order come from the diagrams shown in Fig. <ref>. In terms of the collision operator, these contributions can be written as:
    
        ^a[f](p⃗) =1/2∑_bcd∫_k⃗,p⃗,k⃗'⃗   |ℳ_abcd(P,K,P',K')|^2/2p^0  2k^0  2p'^0  2k'^0
            (2π)^4 δ^4(P+K-P'-K') 
           ×{ f^a(p⃗)f^b(k⃗)[1± f^c(p⃗'⃗)][1± f^d(k⃗'⃗)] 
           ×{ -  f^c(p⃗'⃗)f^d(k⃗'⃗)[1± f^a(p⃗)][1± f^b(k⃗)]}
    
    where the incoming/outgoing momenta p⃗,k⃗ and p⃗',k⃗' are all on shell, p^0=p. The squared matrix element 
    |ℳ_cd^ab| is hard thermal loop (HTL) resummed
    <cit.>
    and is summed over all spin polarizations and colors.  We write ∫_k⃗ = ∫d^3k⃗/(2π)^3 to simplify the notation. 
    
    
    Diagrams (D) and (E) are processes involving t (or u) channel fermion exchange.
    These processes have a logarithmic infrared divergence when computed using free propagators.
    To cut off this divergence, one can include the retarded HTL fermionic thermal self-energy, as was originally done in Ref. <cit.>.
    It was later shown in Ref. <cit.> that there is a simpler treatment which is also correct at the leading order.
    The most important feature of the HTL self-energy insertion is that it correctly reproduces the total scattering rate for this process; since the process is nearly isotropic, the detailed angle dependence is not relevant.
    A mass-like insertion, with the correctly chosen coefficient, reproduces this rate and is much simpler to use in calculation.
    Specifically for a soft fermion exchange in the t or u channel, one can simply use:
    
        u-s/t→u-s/tq^2/q^2 + ξ_q^2 m^2_∞,q
    
    where q is the exchanged momentum, ξ_q = e/2 ≃ 1.359, and m^2_∞,q is the medium-induced effective quark mass-squared for quarks, given by
    
        m^2_∞,q    = 2g^2∫d^3 p/2p^0(2π^3)[2 f_g(p) +  (f_q(p)+ f_q̅(p) )]
         
            = g^2 /4( T^2 + μ^2/π^2) = g^2/3( T^2 + μ^2/π^2).
    
    We have allowed ourselves to make this simplification – also because the cumulative effect of these diagrams is rather small and is exponentially suppressed ∝ e^-μ/T in the large μ/T region.
    
    On the other hand, diagrams (A),(B), and (C) represent processes involving t or u-channel gauge boson exchange that require the inclusion of the thermal
    gauge boson self-energy on the internal propagator to cut off the infrared sensitivity of these processes. The matrix element of interest is (s^2+u^2)/t^2 which can be written as:
    
        1/8D_μα^retD_νβ^adv(pγ^μkγ^ν)(p'γ^αk'γ^β)
    
    where D_μα^ret and D_μα^adv, are the retarded and advanced boson propagators, respectively.
    Following the same notation as <cit.>, the plasma frame frequency and momentum
    carried by the gauge boson propagator are denoted as ω and q. The retarded gauge boson propagator can be conveniently expressed in Coulomb gauge as:
    
        D_00^ret(ω,q)=-1/q^2 - ^ret(ω,q)
           D_ij^ret(ω,q)=δ_ij- q̂_iq̂_j/q^2 - ω^2 +^ret(ω,p)
           D_0i^ret(ω,q)=D_i0^ret(ω,q)=0 .
    
    The equilibrium transverse and longitudinal gauge boson self-energies in the HTL approximation are <cit.>
        ^ret(ω,)     =  ^2
        	{ω^2/2q^2 
        	+
        	    ω  ( q^2 -ω^2 )/4 q^3[
        		ln( q+ω/q-ω) - i  π]
        	}  , 
        
        Π^ret(ω,)     =  ^2
        	{
        	    1
        	    -
        	    ω/2q[
        		ln(q+ω/q-ω) - i  π]
        	}  ,
    
    where we have assumed |ω| < q,
    which is the only case of relevance. 
    Here ^2 is the squared Debye mass, which we will give for the case of nonzero μ,T in mDebye.
    The advanced propagators are the complex conjugates of the retarded ones.
    
    The remaining diagrams and the interference terms are all finite and can be computed directly. One should refer to <cit.> for more details on these calculations.
    
    The second type of contribution to leading order comes from 1 ↔ 2 nearly collinear processes. These can be approximated as strictly collinear, which only produces relative changes suppressed by at least one power of g.  Corrections at this order will be treated in the next subsection.
    The leading-order computation proceeds as follows
    <cit.>:
    
        𝒞_a^1↔ 2[f](p⃗) =    (2π)^3/2|p⃗ |^2ν_a∑_bc∫_0^∞ dp' dk δ(|p|-p'-k')γ^a_bc(p;p',k') 
           {f^a(p)[1± f^b(p')][1 ± f^c(k)]-f^b(p') f^c(k)[1 ± f^a(p)]} 
        
            +   (2π)^3/2|p⃗ |^2ν_a∑_bc∫_0^∞ dk dp' δ(|p|+k-p')γ^c_ab(p';p,k) 
           {f^a(p)f^b(k)[1 ± f^c(p')]-f^c(p') [1 ± f^a()][1 ± f^b(k)]},
    
    where γ_bc^a represents the splitting rate, and is given by <cit.>:
    
        γ_bc^a(p;p-k,k)= g^2d_R_bC_R_b/64π^4{[ p^4+k^4+(p-k)^4/p^3k^3(p-k)^3                        g ↔ gg;     p^2+(p-k)^2/p^2k^3(p-k)^2                        q ↔ qg;     p^2+(p-k)^2/p^3k^2(p-k)^2                        g ↔ qq;                               ]}×∫dh^2/(2π)^22 h·F_b,
    
    where h= p×k is a transverse, two-dimensional vector related to the transverse momentum picked up during the splitting process. And F_b can be obtained through the following integral equation:
    
        2=     iδ E(h)F_b(h) + ∫dq_⊥^2/(2π)^2C(q_⊥) {(C_R_b-C_A/2)[F_b(h)-F_b(F-kq_⊥)]
            C_A/2[F_b(h)-F_b(h+pq_⊥)] + C_A/2[F_b(h)-F_b(h-(p-k)q_⊥)]}.
    
    
    For the case of g ↔ q q̅, C_R_b - C_A/2 =  - C_A/2 multiplies the term with F_b(h-pq_⊥)
    rather than F_b(h-kq_⊥). Here C(q_⊥) represents the  leading-order transverse scattering kernel:
    
        C(q_⊥)=g^2T^2/q_⊥^2(q_⊥^2+^2),
    
    and δ E(h) is the energy difference between initial and final collinear particles, expanded to first nontrivial order in thermal masses and transverse momenta:
    
        δ E(h) = h^2/2pk(p-k)+m^2_∞,c/2k+m^2_∞,b/2(p-k) - m^2_∞,a/2p.
    
    We encountered m^2_∞,q already in thermalquarkmass.
    For gluons, m^2_∞,g = ^2/2.
    
    
    
    
     §.§ Next-to-Leading Order corrections
    
    
    We now focus on the next-to-leading order corrections.
    Treating kinetic theory to NLO was pioneered in
    Ref. <cit.> and applied to QCD shear viscosity in Ref. <cit.>.
    In this work, the authors observed that the analytically complex parts of the collision operator do not receive (g) corrections, which instead only modify collinear exchange, incorporated by changing γ^a_bc,
    and soft exchange, which can be modeled as momentum-diffusion and identity-changing processes.
    We briefly review the extended treatment that captures these O(g) corrections, and then we summarize it and present an explicit form of the collision operator at NLO.  
    
    
    
    
      §.§.§ Momentum diffusion
    
    
    The small momentum exchange part of diagrams (A), (B), (C) can be well approximated by momentum diffusion for the particles on each line.
    Beyond leading order, there are additional small-momentum-exchange effects which are only suppressed by (g) relative to LO, which must be incorporated.
    One such effect is the possibility for the gluon in diagrams (A), (B) to carry a soft momentum.
    Other effects involve the interference between multiple soft scatterings, soft gluon emission during scattering, and nonabelian interactions between scattering processes, such as those depicted in Figure <ref>.
    
    
    
    Because the momentum exchanges involved in these processes are small (that is the reason they can occur at (g) and not (g^2)), these terms can be adequately described in terms of Langevin dynamics, that is, in terms of momentum drag and longitudinal and transverse momentum diffusion.
    This is analyzed in the context of heavy quark dyamics in Ref. <cit.> and jet energy loss in Ref. <cit.>, and applied to shear viscosity in Ref. <cit.>.
    As shown there, the effective collision term due to soft scatterings, cut off at a scale μ_⊥, is
     
        (C_diff^ f_1)^a() =
            -1/2∂/∂ p^i q̂^ij_a    f_0^a(p) (1 ± f_0^a(p))
            ∂ f_1^a()/∂ p^j  +   ,
    
    where
    
        q̂^ij_a = q̂_a,Lp̂^i p̂^j
          +  12q̂_a (δ^ij - p̂^i p̂^j )
    
    represents the momentum diffusion parallel and perpendicular to the particle's momentum through q̂_L and q̂ respectively.
    
    Contracting with the departure from equilibrium and integrating over , the contribution to the inner product of Qdef takes the form
    
     (f_1, C^2 → 2_diff f_1)  ≡. (f_1, C^2 → 2_diff f_1)|_ loss
     +
     . (f_1, C^2 → 2_diff f_1)|_ gain  ,
     
     where
     . (f_1, C^2 → 2_ diff f_1) |_ loss
    = 1/2β^3 ∑_aν_a  ∫_  f_0^a(p)(1 ± f_0^a(p))  q̂^ij_a ∂ f_1^a() /∂ p^i∂ f_1^a() /∂ p^j   ,
    
    
    The gain collision term takes the form
    
        =    1/2∂/∂ p^i (
           f_0^a(p) (1 ± f_0^a(p))   ∑_bν_b ∫_^ij_ab (·̨̂)   f_0^b(k) (1 ± f_0^b(k))
           ∂ f_1^b()̨/∂ k^j )
        
        . (f_1, C^2 → 2_diff f_1)|_ gain 
        =      -β^3/2∑_abν_a ν_b ∫_
        f_0^a(p)(1 ± f_0^a(p)) f^b_0(k) (1 ± f_0^b(k))
        
           ×_ab^ij (·̨̂) ∂ f_1^a() /∂ p^i∂ f_1^b()̨/∂ k^j,
    
    where the angular function is
    _ab^ij(·̨̂)
    =  g^4 C_R_a C_R_b/d_A∫d^4Q/(2π)^4   |G_μν^R(Q) v_^μ v_^̨ν|^2     2πδ(v_· Q) 2πδ(v_·̨Q) q^i q^j   ,
    
    which is related to q̂^ij_a through
    q̂^ij_a =  ∑_b ν_b ∫_f̨_0^b(k) (1 ± f_0^b(k))  _ab^ij(·̨̂)   .
    
    Beyond leading order it turns out to be much easier to determine q̂^ij_a than ^ab_ij(·̨̂),
    since light-cone techniques work for the former but not the latter.
    Therefore, in practice we will not be able to compute this gain term.
    Since our best estimates based on leading-order analyses indicate that angular cancellations make this term small, we will instead drop it.
    This is why we generally refer to our results as “almost next-to-leading order.”
    
    Examining the expression for
    . (f_1, C_diff^f_1) |_ loss,
    it becomes evident that it involves a single light-like vector denoted as v_. 
    Using the causality and KMS properties of such light-like correlators <cit.>, these leading-order
    soft contributions to q̂ and q̂_L  can be evaluated
    in closed form <cit.>,
    
        q̂^a|_soft=g^2C_R_a T
        	^2/2πlnμ_⊥/,	    q̂_ L^a|_soft=g^2C_R_a T
        	^2/4πln√(2)μ_⊥/.
    
    In these expressions, gT≪μ_⊥≪ T is a cutoff on the q_⊥≡√(q^2 - ω^2)
    integration, which separates the soft from the hard scale. In the region where ω,q≳ T, and the bare matrix elements can be used to compute the hard contribution to the collision operator C^, there is a cancellation on this cutoff dependence. Since light-like separated points are effectively causally disconnected
    as far as the soft gauge fields are concerned,  q̂ and q̂_L have a simple form.
    Using the explicit form of the angular dependence of
    f_1^a() = β^2 χ_ij() X_ij,
    
    straightforward analysis shows that the loss term  reduces to
    
        . ( χ_ij, C^_lossshort χ_ij) |_ loss =
         	 β^3/2∑_a ν_a∫_ f^a_0(p)
         	 (1± f_0^a(p)) ×
        [(χ^a(p)')^2  q̂^a_ L|_soft
         	+
         	ℓ(ℓ+1)χ^a(p)^2/2p^2q̂^a|_soft]   ,
    
    which is the most useful form for evaluating the transport coefficients
    numerically.
    
    The next-to-leading order value of q̂^a has been computed <cit.>.
    This result can be simply included by adding it to the above results <cit.>:
      
        δq̂^a=g^4C_R_a C_A  T^2 /32π^2(3π^2+10-4ln 2)  .
    
    the shift included in the soft gluon exchange diffusion rate gives
     
        ( χ_ij, C^δq χ_ij) =
         	 β^3/2∑_aν_a δq^a
         	 ∫_   f^a_0(p)
         	[1± f_0^a(p)]
         	 ℓ(ℓ+1)χ^a(p)^2/2p^2 .
    
    
    As discussed in Ref. <cit.>, the precise procedure described in the last subsection for including HTL effects in the leading-order calculation also introduces some NLO effects in that calculation.
    In particular, the phase space and statistical functions for scattering from soft gluons incorporate some of the (g) effects which we have just described.
    Therefore, the leading-order procedure we use will not in fact reproduce qhatql, but will reproduce that expression plus some (g) additional terms.
    These contributions must then be subtracted from the (g) contribution we include here, in order to avoid overcounting such effects.
    Fortunately, since the effects arise from scattering off of gluons, they are unmodified by the inclusion of quark chemical potentials, and we can take over the analysis of
    Ref. <cit.>.
    The resulting counterterm can be written as
     
        ( χ_ij, C^_(g) finite g χ_ij)    =   d_AC_Ag^4 /32π^5 T∑_a T_R_a∫_0^∞ dp  p^2 f^a_0(p)[1± f_0^a(p)]
              ×{4.2695[(χ^a(p)')^2+(χ^g(0)')^2]
        	  +7.1769ℓ(ℓ+1)/2p^2[(χ^a(p))^2+(pχ^g(0)')^2]
              +
        	  18.0669 δ_ℓ2[χ^g(0)']^2
        	  },
    
    and the soft fermion exchange counterterm is given by
    
        ( χ_ij, C^_(g) finite q χ_ij) =     d_F ^2 g^4
        m_∞/32π^5T^2 9.95268 ∫_0^∞ dk k 
         f^q_0(k)	[1+ f_0^g(k)]
           ×[(χ^q(k)-χ^g(k))^2+(χ^q(0))^2] .
    
    One should refer to <cit.> for a detailed derivation of these counterterms.
    
    
    
    
      §.§.§ Collinear and semi-collinear contributions 
    
    
    The collinear splitting process, collop21 to deltaEeq, involves the same momentum diffusion effects we just described.
    And as we just described, the strength of momentum diffusion receives (g) corrections.
    Furthermore, deltaEeq shows that collinear splitting is sensitive to thermal effective masses.
    These also turn out to receive (g) corrections.
    Therefore we need to analyze how these two corrections impact the collinear splitting rate.
    To conduct the analysis, we will follow Ref. <cit.> and Fourier transform collop21 from transverse-momentum _⊥ to transverse impact parameter _⊥ space.
    This not only diagonalizes the collision kernel C(k_⊥), but also turns an integral equation into a differential equation.
    Furthermore,  the source on the left-hand side becomes a boundary condition at b=0
    and the desired final integral becomes a boundary
    value of the ODE solution.
    Hence, defining
    
        F() = ∫d^2h/(2π)^2 e^i· hF(h) ,
    
    we have
    
        ∫d^2h/(2π)^2 2  ·F(q)
        = ( 2 ∇_ b·F(0)) ,
    
    and split_int becomes
    
        -2iδ^2()   =   i/2pω(p-ω)(p(p-ω)m_∞ ω^2+
        p ω m_∞ p-ω^2-ω(p-ω)m_∞ p^2-∇_^2)F()
              +
        ('_R(|ω| b)-'_A(|ω| b)/2
        +'_A(| p| b)/2+'_A(| p-ω| b)/2)F(),
    
    with
    
        _R'(|ω| b) ≡∫d^2k_⊥/(2π)^2( 1 - e^i ω·k_⊥) _R(k_⊥) .
    
    As noted above, we need to incorporate the change in the collision operator,
    
        '_R  , LO+NLO(b) = _R'(b) + δ_R'(b),
    
    where the _⊥-space correction of _R is the unintegrated version of δq̂^a detailed in qhatnlo.
    The details appear in Appendix C of Ref. <cit.>.
    We must also incorporate a shift in the thermal masses:
    
        m^2_∞ a, LO+NLO    = m^2_∞ a +δ m^2_∞ a,     δ m^2_∞ q    = -gT/2π,    δ m^2_∞ g    = -gC_AT/2π.
    
    These shifts are incorporated into bspace by solving for their effects perturbatively, by treating F()
    formally as an expansion in powers of δ m_∞, δ;
    F() = F_0()+F_1()+…  to first order. For details on the boundary conditions and the numerical evaluation of these
    equations, one should refer to 
    <cit.>
    
    The above analysis considers (g) corrections to the regime in which the external states are collinear up to (g^2) corrections.
    This allowed for simplifications in the analysis, in particular in expressing deltaEeq in terms of the leading mass and transverse-momentum corrections.
    But this is not always the case.
    Integrating over opening angles, deltaEeq breaks down if the external momenta P,K obey P· K ∼ gT^2,
    and this regime represents an (g) part of the splitting processes considered.
    Similarly, for hard scattering, if the transfer momentum Q becomes nearly lightlike, our treatment also breaks down, and this again represents an (g) effect.
    Both regimes feature small angles, but not as small as in the collinear splitting processes; so we refer to them as semi-collinear.
    The analysis of this kinematic region was carried out in Ref. <cit.>, and we will follow the development there and in Ref. <cit.>.
    
    The good news is that, in treating this regime, the kinematic variable h becomes large, namely h≫ gT^2.
    The collision operator then becomes subdominant, ≪δ E, and one can perturb in it to first order.
    The lowest-order and first nontrivial-order solutions are
    
        F_b1(h)     = 2/iδ E(h)  , 
        F_b2(h)     = i/δ E(h)∫d^2 q_⊥/(2π)^2C̅(q_⊥) {( C_R_b - C_A/2)
               [F_b1(h) - F_b1(h-kq_⊥)] .
               
             . + C_A/2 [F_b1(h) - F_b1(h+pq_⊥)]
               + C_A/2 [ F_b1(h) - F_b1 (q-(p-k)q_⊥)]
               } .
    
    This is the same as treating the emission in the Bethe-Heitler limit.
    However, at large h, q^- = δ E ∼ gT ∼ q_⊥, which changes the kinematics of the scattering. Therefore,
    C̅(q_⊥) must be recomputed.  A more accurate form for
    C̅(q_⊥) in this regime is
    <cit.>
        C̅_NLO(q_⊥,δ E) =
          g^2 T ^2/(q_⊥^2 + δ E^2)(q_⊥^2 + δ E^2 + ^2)
               + 2 g^2 T δ E^2/q_⊥^2(q_⊥^2 + δ E^2) .
    
    Physically, this is a consequence of the
    splitting being induced by an elastic scattering or by the
    absorption of a soft on-shell particle, as observed in <cit.>. Therefore,
    the already-computed LO  contribution, corresponding to the small δ E
    limit, and the already-included LO  contribution <cit.> need to be subtracted as:
    
        δC̅(q_⊥,δ E) =
          g^2 T ^2/(q_⊥^2 + δ E^2)(q_⊥^2 + δ E^2 + ^2)
               - g^2 T ^2/q_⊥^2(q_⊥^2+^2) .
    
    The second term is the LO collinear form for C̅ (the small
    δ E limit of CLONLO).
    
    The semi-collinear contribution is found by substituting CNLO
    into Fseries and using it to evaluate split_int and finally
    collop21, which results in
    
        γ^a_bc|_semi(p;p-k,k)
              = g^2/32π^4{[ d_AC_Ap^4+k^4+(p-k)^4/p^3k^3(p-k)^3                               g↔ gg;       d_F p^2+(p-k)^2/p^2(p-k)^2k^3                              q↔ q g;       d_F (p-k)^2+k^2/(p-k)^2k^2p^3                              g↔ qq̅ ].
        ∫d^2h/(2π)^2∫d^2/(2π)^2δC̅(,δ E)
           ×[(C_R-C_A/2)(h/δ E(h)-
        h-kq_⊥/δ E(h-kq_⊥))^2+C_A/2(h/δ E(h)-
        h+pq_⊥/δ E(h+pq_⊥))^2.
           .+C_A/2(h/δ E(h)-
        h-(p-k)q_⊥/δ E(h-(p-k)q_⊥))^2],
    
    with (C_R-C_A/2) appearing on the h + pq_p term for
    g ↔ qq̅ processes.  This is then inserted into
    collop21, resulting in
    
        (f_1,𝒞^semif_1)
        	   ≡   2π/T^3∑_abc∫_0^∞ dp
                ∫_0^p dk γ^a_bc|_semi (p;p-k,k)
        	f^a_0(p)[1± f^b_0(k)][1± f^c_0(p-k)]
              ×[f_1^a()-f_1^b(k)-
        	f_1^c((p-k))]^2.
    
    
      §.§.§ Summary
    
    
    In summary, the corrections to be added to the strict leading-order contributions
    of the previous section produce a collision operator that is fully
    NLO.  At NLO, the collision term will be given as
    
        ( f_1, C_NLO  f_1 )
            =
            ( f_1, C_LO  f_1 ) +
            ( f_1, δ C   f_1 )  ,
    
    where
    
        ( f_1, δ C   f_1 )
          ≡   ( f_1, C^δq  f_1 )-
          ( f_1, C^_(g) finite  f_1 )		  +
          ( f_1, C^  f_1 )  +
          ( f_1, δ C^  f_1 ).
    
    Here, the first two terms are the contribution from the soft gluon exchange and its counterterm. The last two terms are the contribution from semi-collinear processes and the correction to the splitting rate, respectively. All of these corrections were carefully studied in <cit.>, and what was found is that the main difference between LO and NLO results arises from the soft gluon exchange. 
    
    
    
    § FROM ZERO TO NONZERO CHEMICAL POTENTIAL
    
    
    In this section, we delve into the differences between the description of the QGP at vanishing chemical potential, μ=0, and at high densities, μ>0.
    Temperature, density, and screening enter the calculation, and we must make sure that we include them correctly each place they both arise.
    The structure fits within the general effective kinetic framework of Ref. <cit.>, for which finite μ and T case is a special case.
    
    When introducing a chemical potential we must shift the equilibrium occupancies from their simple thermal values; we have already done this in f0.
    This change propagates into the Boltzmann equation and the definition of the extremization problem, Qdef, in a straightforward way.
    There is no need to change the basis of variational functions, but more terms may be needed because the most important momentum range is likely to be higher, p ∼μ rather than p ∼ T.
    
    The matrix elements depend on the chemical potential indirectly through the strength of screening effects.
    As shown in Ref. <cit.>, the form of the HTLs are identical for high density as for high temperature.
    The two effects simply combine in the Debye screening mass squared,
    
        ^2 = N_c/3 g^2T^2 + /6 g^2(T^2 + 3/π^2μ^2).
    
    The last term represents the increase in the screening due to the quark abundance induced by the chemical potential.
    The strength of quark screening is similarly enhanced; one uses the occupancies with chemical potentials in evaluating thermalquarkmass.
    The resulting expression is already written in that equation.
    
    In evaluating splitting rates, a pivotal role was played by the transverse scattering rate, described by the kernel C̅(_⊥).
    Remarkably, scattkernel still holds at finite chemical potential, provided one uses the adjusted value of the Debye mass squared.
    This is discussed in Ref. <cit.>.
    Essentially, the shifted ^2 value accounts for the change in the number of scattering “targets” in the medium and therefore correctly adjusts the strength of medium scattering which induces the splitting.
    Similarly, the calculation of δ E for splitting, deltaEeq, must use the form of the asymptotic masses which accounts for chemical potentials.
    
    Similarly, at NLO we must correctly account for the strength of scattering in the medium, but generally the calculations presented so far remain true after one does so.
    The light-cone methods which were used to derive qhatql and qhatnlo are based on integrating out the quarks and incorporating their effects in the HTLs or equivalently within the effective theory of EQCD.
    This proceeds the same with μ as without, except that the value of ^2 must be adjusted using mDebye.
    By similar reasoning, mshift and CLONLO apply as written, but using the corrected form of ^2.
    
    Next, let us briefly analyze the formal convergence of perturbation theory.
    At weak coupling ≪ 1 and μ = 0 hot QCD has three scales: the hard scale π T, the soft scale gT, and the ultrasoft scale g^2 T.
    At leading order the scale π T participates as the dominant propagating degrees of freedom, while the gT scale is relevant as the momentum-exchange scale for soft scattering relevant in both  and  processes.
    NLO corrections occur primarily because these gT scale degrees of freedom have occupancies f(q) ∼ 1/g and loop corrections are only suppressed by a single power of g.
    Therefore the structure of NLO (g) corrections relative to LO is often as a ratio g^2 T/.
    Including color and other numerical factors, these corrections often prove to be quite large.
    
    The inclusion of a chemical potential with μ > π T raises the hard scale from π T to μ, and it increases the Debye scale as one sees in mDebye:
    ∼ g μ rather than gT.
    Therefore the separation between the soft and ultrasoft scales is increased, and (g) corrections are expected to scale as g^2 T / gμ instead of g^2 T/gT.
    This underlies the expectation that NLO effects should be milder in the large-μ/T case.
    
    In practice, the largest NLO effect is the shift in the transverse momentum diffusion coefficient, qhatnlo.
    This effect is linear in , while the leading-order part is proportional to ^2.
    The ratio δq̂ / q̂∼ g^4  T^2/g^2 ^2 T is therefore of order g^2 T/, as anticipated above.
    For parametrically large μ/T, one indeed finds an improvement in the NLO convergence by a single factor of T/μ.
    In the next section we will see to what extent this really improves the convergence in practice.
    
    
    
    
    § RESULTS
    
    In the last sections, we have presented the ingredients necessary for the computation of shear viscosity at leading order, discussed the relevant next-to-leading-order corrections, and presented a good basis set for these calculations.
    Using this computational apparatus, we compute the shear viscosity as a function of μ/T at both leading order and (almost) next-to-leading order.
    
    
    
    In figure <ref> we plot the shear-viscosity to entropy density ratio η/s for LO and NLO, and in figure <ref> we
    show the kinematic viscosity, that is, the ratio of the shear viscosity to the enthalpy density, η T /(e + P ).
    [Raw data for all plots are included in the ancillary materials.]
    Note that the chemical potential μ we refer to is that for a quark, and we treat it to be the same for all quark species.
    The baryon number chemical potential is then 3 times larger.
    We have used the EQCD-inspired scale setting by choosing the MS renormalization point to be
    =2.7T, following the treatment from <cit.>, and we run the coupling using the 2-loop beta function and the value = 0.118 at the Z-pole.
    We assume that when  reaches a quark mass, one crosses the quark-mass threshold and the quark becomes active both in the medium and in the running of the coupling constant.
    So for temperatures below 2.7T=m_b we use =4 and for temperatures above 2.7T=m_b we use =5 (and similarly for the charm and top quarks).
    The plots clearly show that the difference between LO and NLO is smaller at large μ than at μ=0, particularly at high temperatures where the coupling is smaller and  is larger.
    
    
    
    
    Since one motivation to study high-density regimes comes from the expectation that perturbation theory should show better convergence than at μ=0, we present a more detailed comparison between LO and NLO results in Fig. <ref>.
    We plot the ratio between LO and NLO calculations as a function of the temperature for each fixed chemical potential.
    We reproduce the previous result <cit.> that the NLO corrections at μ=0 are large, more than a factor of 2, up to 10^3 GeV temperatures.
    Once one goes to higher densities, the NLO effects become less dominant, falling below a factor of 2 by T=500 MeV for μ/T = 6.
    This suggests that the expansion in g is better behaved at large μ/T.
    Nevertheless, NLO effects are rather important at all temperatures and chemical potentials we consider.
    
    
    
    
    
    
    
    
    
    
    § SUMMARY AND DISCUSSION
    
    
    We have extended the investigation of shear viscosity for hot QCD with chemical potentials to both leading and (almost) next-to-leading order.
    We find that the viscosity-to-entropy ratio η/s is larger at finite μ than at zero μ for all temperatures we consider.
    However, the kinematic viscosity η T/(e+P) is more weakly dependent on μ, falling below the μ=0 result except at NLO below T=1 GeV, where the μ/T dependence almost vanishes.
    We argued previously that, at a given temperature or coupling strength, the convergence of perturbation theory should be better in the large μ/T regime than for μ=0.
    Our results show that this is indeed the case, but the improvement is not dramatic.
    NLO effects are a factor of 2 at μ/T=0 for
    temperatures of hundreds of GeV,
    whereas for μ=6T they are a factor of 2 at about 500 MeV temperature.
    Perturbation theory does not converge at experimentally achievable combinations of (μ,T), but it comes closer at high μ than at small μ.
    
    It would be straightforward to apply the techniques developed here to the more physical case in which different quark species experience different chemical potentials.
    This could allow the development of an interpolating estimate for the viscosity based on experimental data, hadron-resonance modeling, and NLO perturbative input, similar to what was recently developed by one of us <cit.>.
    
    
    
    § ACKNOWLEDGEMENTS
    
    The authors acknowledge the support by the State of Hesse within the Research Cluster ELEMENTS (Project ID 500/10.006), and support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through the CRC-TR 211 'Strong-interaction matter under extreme conditions'– project number 315477589 – TRR 211. The authors also thank Nicolas Wink, Jacopo Ghiglieri, and Kevin Ingles for instructive conversations.
    
    
    
    
    
    
    § SELF-ENERGIES BEYOND THE HARD THERMAL LOOP APPROXIMATION
    
    As discussed in <cit.>, the t-channel exchange process is subtle, since the use of vacuum matrix elements leads to logarithmic divergences in the “transport” cross-section and in the inverse viscosity.
    Baym et al already identified the correct procedure to fix this problem in the context of viscosity <cit.>; one should amend the matrix element through the inclusion of the thermal self-energy.
    At the leading-log level, relevant in Refs. <cit.>, it is sufficient to identify the energy scale where this occurs.
    At leading order it is necessary to really include the self-energy, at least within an approximation which is adequate in the formal kinematic regime (p,k) ∼ T but (q,ω) ∼ gT ≪ T.
    This is precisely the regime where the HTL approximation should be valid.
    
    At large chemical potentials we can try to do better.
    In this regime, the self-energy is dominated by fermionic loops.
    For these, the self-energy is gauge invariant and the full expression, not just the hard-loop limit, is known <cit.>.
    Specifically, the medium-induced part of the fermion-loop contribution to the gluonic self-energy is
    <cit.>
        = g^2  T^2/3H(ω,q),  = g^2  T^2/3(-1/2H(ω,q) + q^2/2(q^2-ω^2)G(ω,q))
    
    where H(ω,q) and G(ω,q) are, writing 2ω_+=ω+q and 2ω_-=ω-q:
    
        G(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)(4k+q^2-ω^2/2qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)]) 
         H(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)(2k-4k^2+ω^2-q^2/4qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)] 
           -2kω/qlnω_+/ω_--kω/qln[(k+ω_-)(k-ω_+)/(k-ω_-)(k+ω_+)]) 
         G(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)q^2-ω^2/2q[Θ(k+ω_+)Θ(-k-ω_-)-Θ(-k+ω_+)Θ(k-ω_-)] 
         H(ω,q)   = 3/π^2T^2∫_0^∞ dk f(k)[-(2k+ω)^2-q^2/4qΘ(k+ω_+)Θ(-k-ω_-)
           +(2k-ω)^2-q^2/4qΘ(k-ω_-)Θ(-k+ω_+)+2kω/qΘ(ω_+)Θ(-ω_-)]
    
    The references only present the real parts, but the imaginary parts are easily obtained by using the retarded prescription ω→ω+i0 and carefully interpreting the logarithms.
    The hard thermal loop approximation corresponds to expanding the k integrals in the approximation k ≫ (q,ω), in which case the functions simplify:
    
        H_HTL(ω/q) = 1 - ω/2qlnω_+/ω_-+iπω/2qΘ(ω_+)Θ(-ω_-)
           G_HTL(ω/q) = 1 .
    
    
    One concern regarding the AMY leading-order calculation <cit.> has always been the use of hard thermal loop self-energies, rather than full thermal self-energies.
    At high chemical potential, where the hard loop is dominated by fermions, the above self-energies accurately represent the correct loop screening effect at any energy and momentum, without the need for an HTL approximation.
    Therefore we can find out how well the HTL approximation works by comparing how the leading-order shear viscosity changes when we “improve” the screening effects by replacing the HTL approximation, GH-HTL, with the full self-energy, GH.
    
    We make this comparison in figure <ref>.
    The figure considers a single collision process, diagram (C), and computes the collision integral (χ_ij,χ_ij) for the single-parameter Ansatz.
    We then take the ratio of this collision rate for the case of the full self-energy over the collision rate using the HTL self-energy.
    The Debye screening strength is chosen to vary with μ as it would if the coupling strength g^2 is held fixed.
    
    The figure shows, first and foremost, that the effect of replacing the HTL self-energy with the full self-energy is not that large.
    Also, the HTL approximation actually becomes more reliable as the chemical potential is increased.
    This indicates that the use of the HTL approximation for the self-energies is justified and represents a smaller effect than the NLO (and, presumably, unknown NNLO) corrections.
    
    
    JHEP2
    
    § TESTS AND CHECKS 
    
    Since previous results were computed using different numerical integration techniques, it is convenient to compare our results with AMY leading order results. This is done in figure <ref>. The plot shows that there are small differences due to different integration methods and base choice, but they are compatible, especially at small values of , where one expects that the consequences of these effects should have a small role on the transport coefficient calculation.
    
    We also verified that a basis of size N=4 is enough to give accurate results, especially at high chemical potential, where the most important contribution comes from the diagram (C). This is shown in figure <ref>, where we compare the results given by a basis of size N=1 and N=4 to a basis of size N=6, which was considered to give accurate results for LL calculations.


 Paper Title: Cosmologies in $f(R,\mathcal{L}_m)$ theory with non-minimal coupling between geometry and matter.























theoremTheorem[section]
lemma[theorem]Lemma
proposition[theorem]Proposition
corollary[theorem]Corollary
remark[theorem]Remark
proof[1][Proof.]
      
  #1
     definition[1][Definition]
     
  #1
     example[1][Example]
     
  #1
abbrvnat

































sergiobravom@javeriana.edu.co

Departamento de Física,
 Pontificia Universidad Javeriana, Cra.7
No.40-62, Bogotá, Colombia


marek.nowakowski@ictp-saifr.org

 ICTP-South American Institute for Fundamental Research,
Rua Dr. Bento Teobaldo Ferraz 271, 01140-070 São Paulo, SP Brazil


contact@rvlobato.com

  ICRANet, Piazza della Repubblica 10, Pescara, 65122, Italy



davide.batic@ku.ac.ae


Department of Mathematics, 

Khalifa University of Science and Technology,
Sas Al Nakhl Campus,
P.O. Box 2533 Abu Dhabi,
United Arab Emirates



             

§ ABSTRACT 
Among the recent extensions to standard General Relativity, f(R,ℒ_m) gravity has risen an interest given the possibility of coupling between geometry and matter.
We examine the simplest model with non-minimal coupling in the context
of cosmology. We pay special attention to the question of how far this
model could reproduce the observational fact of our universe.



 
                             

                              
Cosmologies in f(R,ℒ_m) theory with non-minimal coupling between geometry and matter
    Davide Batic
    August 5, 2024
====================================================================================





§ INTRODUCTION

Since the discovery of the accelerated expansion of the universe, cosmologists have developed the Λ CDM model, a standard cosmological model rooted in Einstein's general relativity <cit.>. This model incorporates a positive cosmological constant Λ <cit.>, the Friedmann-Robertson-Walker metric <cit.>, and the concept of Dark Matter (DM) <cit.>. The otherwise robust model is not completely without problems.
To start with, despite numerous efforts, a candidate for DM has not yet been found. The
second problem is coined as Hubble tension, careful phrasing of the fact that different measurements of the Hubble constant yield different results <cit.>. In the future, as we seek to explain these issues, we may require a new theoretical model. This could involve modifying Einstein's gravity <cit.>, replacing the cosmological constant with different models of Dark Energy (DE) <cit.>, and exploring alternative candidates for DM <cit.>. The number of extensions of general relativity as well as the number of DE models is, of course, quite large <cit.>.

Despite the remarkable success of the Λ Cold Dark Matter (ΛCDM) model in explaining a wide array of cosmological observations, from the cosmic microwave background (CMB) anisotropies to the large-scale structure of the universe, several significant challenges remain unresolved. The nature of dark energy, epitomized by the cosmological constant Λ, poses a profound theoretical conundrum, known as the "cosmological constant problem" <cit.>. The astonishingly small value of Λ, as required to explain the observed acceleration of the universe, stands in stark contrast to theoretical predictions from quantum field theory, typically larger by many orders of magnitude <cit.>.

Moreover, the ΛCDM model does not escape from the Hubble tension, a growing discrepancy between the values of the Hubble constant, H_0, measured directly from local astronomical observations and those inferred from the CMB under the ΛCDM framework <cit.>. These inconsistencies point to potential physics beyond the standard model, either in the form of new particles or fields, or through modifications to the theory of general relativity itself.

In response to these challenges, a plethora of modified gravity theories have been proposed as alternatives to general relativity, aiming to provide a more comprehensive theoretical framework that can naturally incorporate the phenomena attributed to dark energy and dark matter. Theories such as f(R) gravity, where the Ricci scalar R in the Einstein-Hilbert action is replaced by a function f(R), offer promising ways to explain cosmic acceleration without the need for a cosmological constant <cit.>.

Expanding further, f(R,T) theories introduce a coupling between matter, represented by the trace of the energy-momentum tensor T, and geometry, providing a framework to explore the effects of such couplings on the dynamics of the universe <cit.>. The f(R,ℒ_m) models we focus on in this paper extend this idea by incorporating the matter Lagrangian ℒ_m directly into the gravitational action, potentially offering new insights into the interaction between dark matter and dark energy, and their impact on the evolution of cosmic structures <cit.>.

Our motivation for selecting f(R,ℒ_m) gravity stems from its ability to unite the geometrical and material sectors of the universe in a single, coherent theoretical framework. This approach not only allows for the exploration of the cosmic acceleration and dark matter problems from a new angle but also provides a platform for testing the limits of Einstein's general relativity on cosmological scales. By investigating the cosmological implications of f(R,ℒ_m) gravity, we aim to contribute to the ongoing discussion on viable alternatives to ΛCDM, exploring whether these theories can offer a more satisfactory explanation of observational phenomena without some of the fine-tuning issues that plague the standard model.

It is therefore a priori not clear
which class of models is theoretically preferred over the others.
We think that simplicity of the modification or, in other words,
Occam's razor, could serve us here as guiding principle.
Within a range of models, this possibility seems quite likely. Among the wide class of models based on a Lagrangian function of the type f(R), where R is the Ricci scalar <cit.>, one could choose to focus on R^2 gravity <cit.> and the resulting cosmological models <cit.>. Recently, another class has been examined, based on
f(R, ℒ_m) where ℒ_m <cit.>
represents the matter
Lagrangian, often taken to be the energy density ρ, pressure p, or the trace of the
energy-momentum tensor T-with all these quantities being diffeomorphically invariant. In this class, an additional term in the Einstein-Hilbert Lagrangian of the form σ Rℒ_m would be considered a mild extension of the previous model. Moreover, it provides a straightforward coupling of the geometry encoded in R with matter represented
astrophysically by ℒ_m.  The
astrophysical and other implications of such a theory have been examined in
<cit.> with consequences for Neutron star and
White Dwarf physics. Some cosmological aspects with specific choices
of ℒ_m have been considered in <cit.>.
Except for <cit.>, the choice of the function
f(R, L_m) is different from ours. The
f(R,ℒ_m) theories have their own theoretical
peculiarities that require careful examination. One notable aspect is
that choosing
ℒ_m=±ρ, ± p, T does not
lead, through standard metric variation to the energy-momentum tensor of a perfect fluid T_μν <cit.>. On the other hand, a constrained variation
based on the Lagrangian with constraints can result in
the prefect fluid energy-momentum tensor. However, this comes at the cost of  T_μν no longer being conserved.  We show that explicitly using
the new Friedmann equation in f(R, ℒ_m).


In solving the new Friedmann equations, we try to make the model resemble our present universe. While it seems possible to achieve this by choosing adequate initial values, the evolution into the past and future can bring some surprises, as will be shown below.  One of the simplest and model independent feature of our
universe is the lower limit on its age as given for example by the
redshift of the oldest galaxies <cit.>.
A second constraint independent of the model comes from uranium decays  <cit.>.
If a cosmological model, as
appealing as it may be, cannot
reproduce these facts, it is then certainly not a good candidate to
describe our universe. We will pay a special attention to these
questions while examining the details of the model under discussion.

The paper is organized as follows. In Section II, we outline the basics
of f(R,ℒ_m) theory with minimal coupling, denoted as σ
Rℒ_m. In this section, we also present the new Friedmann equations with a cosmological constant in a dimensionless form suitable for numerical integration.
In Section III, we present the generalization of one of the
Friedmann equations. In its standard form, it reads
Ω_Λ,0 +Ω_m,0=1. However, in the context of f(R,ℒ_m) theory, it generalizes to
Ω_Λ,0 +Ω_m,0+ Ω_σ, 0=1. In the same section, we discuss the possible initial values, a necessary undertaking, since the new Friedmann equation contains the second derivative of the density. We believe it is necessary to provide a brief overview of the standard cosmological model in its analytical form (see Section IV). This is important because in Section V, we will present the numerical results and compare the new model with the standard one. In Section VI, we will draw our conclusions.





§ THE F(R,ℒ_M) THEORY

As a generalization of the standard Einstein-Hilbert Lagrangian in General Relativity, i.e.

    ℒ_GR=1/2κR+ℒ_m,

where κ=8π G_N, a new Lagrangian has been proposed by
making use of the invariants R and ℒ_m.  It makes use
of a general function, f(R,ℒ_m), and reads simply
with an action <cit.>

    S=∫ d^4x √(-g) f(R,ℒ_m).

From the formal variation of the action (δ_gS=0) with respect
to the metric, the following field equations are obtained <cit.>

    f_RR_μν + (g_μν-∇_μ∇_ν)f_R-1/2(f-f_L_mℒ_m)g_μν=1/2f_L_mT_μν,

where f_R=∂ f/∂ R, f_L_m=∂
  f/∂ L_m and the energy-momentum tensor T_μν is
given by

    T_μν=-2/√(-g)δ(√(-g)ℒ_m)/δ
      g^μν.

An alternative form of these field equations may be obtained by taking the trace of (<ref>) and solving for f_R, namely

    f_R=1/6f_L_mT+2/3(f-f_L_mL_m)-1/3f_RR.

When replacing it back into (<ref>), we obtain the equation

    f_RG_μν+1/6(f+f_RR - f_L_mL_m)g_μν-1/2f_L_m( T_μν-1/3T g_μν)-∇_μ∇_νf_R=0.

Here, G_μν=R_μν-1/2g_μνR represents the Einstein
tensor. As mentioned in the Introduction, the particular form of
f(R,ℒ_m) we wish to work with is the same as the one presented in <cit.>

    f(R,ℒ_m)=R/2κ+ℒ_m+σ Rℒ_m,

where σ is the parameter which determines the coupling between
matter and geometry.

With respect to the choice of the matter Lagrangian ℒ_m
that reproduces the perfect fluid Energy-Momentum tensor there is a
current debate <cit.>. While some authors make the
choice ℒ_m=±ρ, others suggest ℒ_m=± P <cit.> with particular values for the polytropic index. With the choice in <cit.>, i.e. ℒ_m=-P, the field equations read

    (1/2κ-σ P)R_μν-σ[g_μν-∇_μ∇_ν]P-1/4κR g_μν=1/2( 1+σ R)T_μν,

or in the alternative form












    (1-2κσ P) G_μν+1/3Rg_μν-κσ P/3Rg_μν-κ (1+σ R) (T_μν-1/3Tg_μν)+2κσ∇_μ∇_νP=0,

which is consistent with the equations displayed above.  The conservation
of the energy-momentum tensor is given in general by <cit.>

    ∇^μT_μν=2∇^μln[f_L_m(R,ℒ_m) ]∂ℒ_m/∂ g^μν.

This suggests that choosing the matter Lagrangian to be proportional to
ρ or P leads to the conservation of the energy-momentum
tensor. However, this is only the case when relying directly on
equation (<ref>). Unfortunately, given the choices of the
matter Lagrangian as mentioned above, it does not reproduce the
perfect fluid energy-momentum tensor. To illustrate this point, if we
start with ℒ_m=±ρ, the energy-momentum tensor
defined in (<ref>) will not contain the pressure term.
On the other hand, by using a constrained variation in the metric <cit.>  or alternatively, a
constrained Lagrangian formalism, we can explicitly show that, for example, choosing the energy density as the matter Lagrangian yields a perfect fluid energy-momentum tensor. However, its conservation no longer holds, as we introduce the metric through the constraints.

There is a certain confusion on this issue in the literature
<cit.>. This is a point of divergence between our approach and others. Therefore, we explicitly demonstrate the non-conservation of T_μν. The
equations of motion, as presented in
(<ref>), (<ref>), and (<ref>), remain valid when using the
constrained formalism. In this formalism, T_μν is the
energy-momentum tensor of perfect fluid, which, in general, is not conserved and is defined as

    T_μν=(ρ+P)u_μu_ν-Pg_μν.

As mentioned earlier, the model under discussion displays a
non-conservation of the energy-momentum tensor, specifically ∇_μT^μν≠ 0 (at least in general). This
non-conservation is also a characteristic of other gravity models such as Rastall Gravity <cit.>, and has been advocated in modular gravity models <cit.>.
From the field equations (<ref>) we can
obtain the modified Friedmann equations in the flat (k=0)
Friedmann–Lemaitre–Robertson–Walker metric

    ds^2=g_μνdx^μdx^ν=dt^2-a^2(t)(dr^2+r^2dθ^2+r^2sin^2θ dϕ^2).

By taking the 0-0 components of the new field equations, one obtains <cit.>

    3H^2f_R+1/2(f-f_RR-f_L_mℒ_m)+3Hḟ_R=1/2f_L_mρ,

where H is the Hubble parameter given by H=ȧ/a.
The space components (i.e. μ=i, ν=j) give

    Ḣf_R+3H^2f_R-f̈_R-3Hḟ_R+1/2(f_L_mℒ_m-f)=1/2f_L_mP,

where the f̈_R term comes from the double derivatives in the field equations ( and ∇_μ∇_ν). For the model we have chosen, we have

    f_R=1/2κ+σℒ_m,      f_L_m=1+σ R,     ḟ_R=σρ̇,     f̈_R=σρ̈.

Together with the choice ℒ_m=ρ, the new Friedmann equations read







    H^2   =   κρ/3+σκ[ 4Ḣρ+6H^2ρ-2Hρ̇],
    ä/a   =   -κ/6(ρ + 3P )-σκ [ Ḣ(3P+ρ)+6H^2P +ρ̈+2Hρ̇].

In general relativity, the two Friedmann equations lead to the
conservation law ρ̇+3H(ρ+P)= 0. It is known that the
same conservation law follows from ∇^μT_μν=0
<cit.>.  It is straightforward to derive from (<ref>)
and (<ref>) the following relation

    ρ̇+3H(ρ+P)=-6σ[HḢ(3P+11ρ)+ρ̇(3H^2+Ḣ)+6H^3(P+ρ) +2Ḧρ],

which clearly gives back the conservation law (and therefore, the
conservation of the energy-momentum tensor) in the case
σ=0. The nature of the non-conservation of the energy-momentum
tensor is explicitly proved here. It should have been addressed also in
<cit.> where the authors
consider cosmological models of f(R, L_m) theories.

Commonly, equations (<ref>) and (<ref>) are known as (modified)
Friedmann equations. In some papers <cit.> equation (<ref>) is also
called Raychaudhuri equation. Both equations are related by (<ref>).

In rewriting the Friedmann equations in a dimensionless form, specific choices were made to align the theoretical framework with observational benchmarks and simplify the numerical analysis. By introducing χ = ρ/ρ_ crit, we scale the density to the critical density, which is pivotal in determining the curvature of the universe. Similarly, scaling the Hubble parameter by its current value H_0 through h^2 = H^2/H_0^2 allows us to examine the evolution of the expansion rate relative to its present value. The introduction of σ̃ = H_0^2 σ helps in assessing the impact of the coupling constant σ within the typical energy scales of the current universe. Lastly, by using η = H_0 t, time derivatives are normalized to the current expansion rate, making them dimensionless and easier to handle numerically. These transformations are not only mathematically convenient but also provide a direct link between theoretical predictions and observable quantities. To this end, we introduce

    χ≡ρ/ρ_ crit =κρ/3H_0^2,      h^2≡H^2/H_0^2,
        σ̃≡ H_0^2σ,    η=H_0 t,
        ρ_crit=3H_0^2/κ,

where ρ_0 is a reference value of ρ at the present epoch.
Similar comment applies to the Hubble parameter H_0. In addition, we
write the time derivatives in terms of derivatives with respect to the
dimensionless parameter η by means of

    ∘A≡dA/dη=dA/dtdt/dη=Ȧ1/H_0.


In the context of our model, the barotropic index γ
characterizes the equation of state (EOS) of the cosmological fluid,
namely P = (γ - 1) ρ. This parameter is crucial for
defining the thermodynamic properties of the universe's contents,
where γ = 1 corresponds to a universe dominated by
non-relativistic matter (the choice we make here for the calculations), γ = 4/3 to one dominated by radiation, and γ = 0 to the cosmological constant scenario with vacuum energy. Each value of γ distinctly affects the evolution dynamics of the universe, as reflected in the modified Friedmann equations. By means of the aforementioned (EOS), equations
(<ref>), (<ref>) and (<ref>) take the form

    h^2   =   χ + σ̃[ 18h^2χ + 12∘hχ-6h∘χ],
    
    h^2+∘h   =   -1/2(3γ-2)χ-σ̃[3(3γ-2)∘hχ+18(γ-1)h^2χ+3∘∘χ+6h∘χ],
    
    3∘χ+9γ hχ   =    -6σ̃[(9γ+24)h∘hχ+3∘χ(3h^2+∘h)+18γ h^3χ+6∘∘hχ].

Since we want to compare the modified cosmological model with standard
cosmology  we should also include the cosmological constant
Λ. This can be done without much effort by considering the action

    S_f(R,ℒ_m)+Λ=∫ d^4x √(-g)(f_Λ(R,ℒ_m)-Λ/κ)

with

    f_Λ(R,ℒ_m)=R/2κ+ℒ_m+σ Rℒ_m.

This gives the new Einstein equations

    f_RR_μν+(g_μν -∇_μ∇_ν)f_R-1/2(f-f_L_mL_m)g_μν+Λ/2κg_μν=1/2f_L_mT_μν

as well as the generalized Friedmann equations

    3H^2f_R+1/2(f-f_L_mℒ_m-f_RR)+3Hḟ_R-Λ/2κ   =   1/2f_L_mρ,
    Ḣf_R+3H^2f_R-f̈_R-3Hḟ_R+1/2(f_L_mℒ_m-f)+Λ/2κ   =   1/2f_L_mP.

With the explicit choice of f_Λ the modified Friedmann equations
simplify to

    H^2   =   κρ/3+σκ[ 4Ḣρ+6H^2ρ-2Hρ̇] + Λ/3,
    ä/a   =   -κ/6(ρ + 3P )-σκ [ Ḣ(3P+ρ)+6H^2P +ρ̈+2Hρ̇]+Λ/3.

Their dimensionless form can be written by using the dimensionless parameter Ω_Λ=ρ_ vac/ρ_ crit as follows

    h^2   =   χ + σ̃[ 18h^2χ + 12∘hχ-6h∘χ]+Ω_Λ,
    
    h^2+∘h   =   -1/2(3γ-2)χ-σ̃[3(3γ-2)∘hχ+18(γ-1)h^2χ+3∘∘χ+6h∘χ]+Ω_Λ.

These equations are now suitable for numerical integration.

Normally, the inclusion of Λ in a modified gravity theory
might not be well justified as the modification itself can
account
for Dark Energy. But it appears that this is a too global statement
and the role of Λ can vary from model to model. In the
section on numerical results we will demonstrate that the cosmological
constant does not affect the characteristic feature of the model,
i.e., its shortcoming to produce a reasonable lifetime of the universe.



§ THE INITIAL VALUES AND THE ∑_I Ω_I=1 RELATION

The Friedman equations are now first order in h and second order in
χ. Therefore, we need three initial values, which we provide at the time t_0 corresponding to the present epoch. We have

    h_0≡ h(t_0)=H_0/H_0=1,   χ_0≡χ (t_0)=κρ_0/3H_0^2,   ∘h_0≡∘h(t_0)=Ḣ_0/H^2_0,   ∘χ_0≡∘χ(t_0)=κρ̇_0/3 H_0^3.

We look for model independent measurements for the density of the Universe (ρ_0), the Hubble parameter (H_0) and its first and second derivatives (Ḣ_0, Ḧ_0). For Ḣ we use the deceleration parameter q, given by <cit.>

    q≡-äa/ȧ^2,   →  Ḣ/H^2=-(1+q)

and thus,

    Ḣ_0=-H_0^2(1+q_0),

where q_0=q(t=t_0) has been obtained in a model independent way
<cit.>. The jerk parameter will be useful for
Ḧ_0 appearing in the second modified Friedmann equation. This is given by <cit.>

    j=⃛a/aH^3→ j=Ḧ/H^3+3Ḣ/H^2+1

and therefore,

    Ḧ_0=(j_0-1)H_0^3-3Ḣ_0H_0=(j_0+q_0+2)H_0^3.

To ensure that our exploration of the modified f(R, ℒ_m) gravity model remains grounded in observational reality, we adopt standard cosmological parameters that are widely accepted within the community. This choice allows us to rigorously test whether the modifications introduced by the theoretical framework can provide a viable alternative to the ΛCDM model, particularly in light of recent tensions and discrepancies such as those in measurements of the Hubble constant. By using these parameters, the model's predictions can be directly compared with those derived from both local and cosmological scales, providing a comprehensive evaluation of its empirical adequacy. For the numerical values, it is important to note that due to recent
discrepancies in measurements of the Hubble constant, known as the
Hubble Tension <cit.>, there are several values for H_0. For example we have
H^+_0= 73,52±1,62 km/s/Mpc and H^-_0= 67,4 ±0,5
km/s/Mpc, respectively. As for the current energy density
of the Universe and the deceleration parameter, we use values found in
<cit.>:
χ_0=ρ_0/ρ_ crit=0.285±0.012, q_0 =
-0.545 ± 0.107, j_0=1.30± 0.37 with the value for
the Hubble parameter taken as H_0=71.34±1.74 km s^-1
Mpc^-1. It is worth noting that these are specific parameter
choices, but other sets of parameters are available in
<cit.>. Possible values for these parameters are
displayed in Table <ref>. This particular choice of parameters that as of today we are as close as possible to the standard cosmology.

For a standard reference value of ∘χ_0, we consider the standard cosmological model and use the continuity equation (<ref>) with σ̃=0, even in the presence of Λ≠0. This leads to

    3∘χ+9γ h χ=0.

For the initial value we then obtain

    ∘χ_0=-3γ h_0χ_0=-3γχ_0.

Since h_0=1, in the case of dust γ=1, we have

    ∘χ_0=-3χ_0.

In mathematics, when solving differential equations, one typically
assumes that all parameters and functions entering the equation, as
well as the initial values are
known and given. However, in physics, the situation can become more
complex because these parameters and initial values may not be
precisely known. When considering a differential equation at the point
where the initial values are applied, it leads to a relationship
between
these initial values and
the parameters of the equation. The Friedmann equations provide a notable example of this phenomenon.

The first modified Friedmann equation can be expressed in terms of the density
parameters Ω_i. To this purpose, let us start by defining the Ω_m parameter as

    Ω_m=ρ/ρ_ crit=κρ/3H^2.

This, in turn, implies that χ=Ω_m and ρ=Ω_mρ_ crit=Ω3H^2/κ. Thus, our first Friedmann equation can be reformulates as follows

    H^2=Ω_m H^2+Ω_m H^2(18σ H^2)+Ω_m
    H^2 (12σḢ)-σκ H ρ̇ +Λ/3.

Recognizing this equation as

    Ω_m + Ω_Λ+ σ̃[18 Ω_m H^2+12 Ω_mḢ-κρ̇/H]=1,

we can express it in a more compact form. Finally, defining the term proportional to σ as Ω_σ, we arrive at

    Ω_m,0 +Ω_Λ, 0+Ω_σ,0=1,

where we have taken the values at t=t_0. Making use of (<ref>) and
 (<ref>) the above equation reduces to

    Ω_m,0 + Ω_Λ,0+σ̃Ω_m,0 (15-12 q_0)=1.

Equation (<ref>) establishes a relationship between initial
values and parameters within the modified Friedmann equation. Moreover, it
introduces the deceleration parameter q_0, which is a measured
quantity,
though not strictly
necessary for solving the first order modified Friedmann equation in H.
Consequently, there are generally two approaches to solve the new
Friedmann equations.  The first one involves specifying h_0=1, χ_0,
∘χ_0, and Ω_Λ while varying σ̃. Equation (<ref>) can then be used to determine q_0. The second approach relies on the measured value of q_0 and fixes the other parameters
as previously described. In this case, equation (<ref>) provides
the parameter σ̃. In particular, we note that in the case without coupling between R and ℒ_m, the relative densities today read

    Ω_m,0+Ω_Λ,0=1.


Numerically, however, there is an uncertainty in the obtained values,
reflected in the relation Ω_m,0+Ω_Λ,0=1±Δ, where Δ represents this uncertainty. We can exploit it in our model by setting, as shown in equation (<ref>),

    σ̃=±Δ/Ω_m,0(15-12q_0).














§ THE STANDARD COSMOLOGY

Before embarking on a discussion of the numerical solutions it is
convenient to briefly outline the analytical picture of the standard
cosmology. This makes sense since we will be comparing the two scenarios.


The dimensionless solution for h_Λ+ for the standard
cosmological model can be obtained by taking σ̃=0
in equations (<ref>) and (<ref>) with γ=1, namely

    h^2   =   χ +Ω_Λ,
    
    h^2+∘h   =   -1/2χ +Ω_Λ.

Combining the above equations, we arrive at a Riccati equation of the form

    ∘h=-3/2 h^2 + 3/2Ω_Λ.

We can solve the differential equation above by setting

    ∫_h'=1^h'=hdh'/Ω_Λ-(h')^2=3/2∫_η'= η_0^η'=η dη',

where η_0=H_0t_0 is the time at which the
Hubble parameter is H_0, which, in turn, is related to the present values of the parameters used. To solve the integral above, we redefine x=h'/√(Ω_Λ) and write

    1/√(Ω_Λ)∫_x=1/√(Ω_Λ)^x=h/√(Ω_Λ)dx/1-x^2=3/2(η-η_0).

One possible solution for the integral on the left-hand side is

    ∫dx/1-x^2=1/2ln(1+x/1-x).

However, such a solution is valid for -1<x<1. This would imply that h<√(Ω_Λ) and thus, χ=h^2-Ω_Λ<0, which would suggest an unphysical negative density. Another solution is

    ∫dx/1-x^2=1/2ln(x+1/x-1).

It is valid whenever |x|>1. Moreover, it implies h^2>Ω_Λ and, consequently χ=h^2-Ω_Λ>0, ensuring a positive and physically meaningful density. If we also use the fact that

    ^-1x=1/2ln(x+1/x-1)

and solve for the integral (<ref>), we end up with the result

    1/√(Ω_Λ)[^-1(h/√(Ω_Λ))-^-1(1/√(Ω_Λ)) ]=3/2(η-η_0).

Solving for h in the γ=1 case leads to

    h_Λ+(η)=√(Ω_Λ)[3/2√(Ω_Λ)(η-η_0)+^-1(1/√(Ω_Λ)) ].

For the general γ case, the equation above reads

    h_Λ+(η)=√(Ω_Λ)[3γ/2√(Ω_Λ)(η-η_0)+^-1(1/√(Ω_Λ)) ].

In the standard notation of the scale factor a_Λ(t) and the Hubble parameter H_Λ(t), we get

    a_Λ+(t)   =   (1-Ω_Λ/Ω_Λ)^1/3γsinh^2/3γ(μ t+ν),
    
    H_Λ+(t)   =   √(Λ/3)(μ t+ν),

where

    μ=γ√(3Λ)/2,  ν=1/2ln(1+√(Ω_Λ)/1-√(Ω_Λ))-μ t_0

for a positive cosmological constant Λ. For completeness, we
also give the solution for Λ<0 which reads

    a(t)_Λ-   =   a_0[cos(α-β t)/cos(α-β t_0)]^2/3γ,
    
    H_Λ-(t)   =   √(-Λ/3)tan(α-β t)

with

    α=β t_0+tan^-1(√(-3/Λ)H_0),  β=γ√(-3Λ)/2.


Without loss of generality we can set t_0=0. Then from (<ref>)
we can infer that 0 <α < π/2 and hence cos(α)>0. To
ensure that the scale factor is positive it suffices to choose t ∈
[α -π/2/β, α +π/2/β]
  corresponding to Big Bang and collapse.

We will explore later how this simple solutions evolves when we introduce a non-zero σ.

Two important model-independent restrictions on a realistic
cosmological model relate to the universe lifetime. In the
standard cosmological model, we can calculate the lifetime (neglecting
the short radiation period) by setting the argument of the Hubble
function to zero, which results in a singularity. Choosing η_0=0,
we obtain

    |η_univ|=
      1/3√(Ω_Λ)ln(1+√(Ω_Λ)/1- √(Ω_Λ)),

we get |η_univ| ≃ 0.96 for Ω≃ 0.7. This
lifetime is not in conflict with model-independent estimates,
such as those from the oldest galaxies <cit.>
(which indicate the existence of galaxies some 400 million years after
the Big Bang) and <cit.>,
which points to a lower limit of 12.5Gy.







§ NUMERICAL RESULTS

Assuming σ≠ 0, we have converted the system under discussion
into a first order system which we solved by standard numerical
methods. As an independent check, we used MAPLE18 routines and found that the
results coincide.

When solving the modified Friedmann equations numerically, the question arises:
What values should we take for σ̃? The standard
cosmology has two constants, the Newtonian one G_N and the
cosmological constant. Interestingly, the dimension of σ is the same as that of the Newtonian constant.  If we take σ as
nG_N with n some number of order 1, σ̃ would turn out extremely small. Therefore, it makes sense to initially try more moderate values for σ̃. In the subsequent figures, the blue curve represents the standard cosmological model prediction.


In Figure <ref>, we have chosen the initial values as discussed
in Section III, with Ω_Λ=0.7. The chosen values of
σ̃ are relatively small to illustrate that the
dimensionless Hubble function closely follows the one of the standard
cosmological model around η close to zero.
Several notable features can be observed in the plot, which are likely to persist for other values of σ̃. First of all, the Hubble function reaches the singularity much earlier than in the standard model. This poses a challenge if we aim to adapt the model under discussion as a viable cosmological model of our universe. As mentioned earlier, one of the constraints on any cosmological model is its ability to be roughly compatible with a lower limit on the lifetime of the universe. This constraint arises from evidence such as the existence of galaxies just 400 million years after the Big Bang <cit.>. Therefore, any cosmological model must strive to be in agreement with the estimated lifetime of the universe as predicted by the standard cosmological model.
Secondly, we notice another interesting feature. The Hubble function becomes negative which implies a contraction of the universe. Therefore, the dimensionless density χ starts increasing
again. This is known as the Big Crunch scenario which happens also
in f(R, L_m) models with the simple choice L_m=ρ. The Big Crunch here is a result of the model and cannot be avoided.




In Figure <ref>, we have plotted the same functions for two
values of σ̃ appearing also in figure
<ref>, but over a larger scale of time. One can appreciate the
steep rise of h at a relatively modest negative values of η.




In Figure <ref>, we have increased the values of
σ̃ but there is  no visible effect on the time of the
Big-Bang. However, for larger values of the parameter, the density becomes unphysically negative.




This is confirmed in Figure <ref> which shows the same functions,
but over a larger timescale. The differences between the models
become then visible at large η.





Increasing the values of σ̃ does not bring any new
insight, but for the first time there appears a cusp for
σ̃=5. MAPLE18 interpretes this as singularities. This is
demonstrated in Figure <ref>.



In Figure <ref>, we return to relative small values of
σ̃ to show that the cusp behavior is quite common in the
model. This is interesting from a purely theoretical point of view, but
leaves doubts about the viability of the model over large time scales.



Finally, we turn our attention the the behaviour of h and χ for
negative values of σ̃ which is depicted in Figures 
<ref>, <ref>, and <ref>.



In Figure <ref>, we encounter a universe whose contraction
slows down to turn into a expanding universe at η around
zero. Again many cusps accompany this behavior which could well mean
that universe undergoes a singularity there.



For small and very small values of negative σ̃ we have
again negative densities.

We have also explored a second approach for selecting the parameter
σ̃. More precisely, we utilized (<ref>)
and considered the error bars associated with measured values in standard cosmology.
Figure <ref> provides an example where we calculated
σ̃ with high precision, staying within the range of
uncertainty. This allowed us to examine how sensitive the model is to the choice of parameters. The lifetime of the universe comes out
approximately
|η_univ| ≃ 0.4, which is too small to account for the
existence of the first galaxies in our universe. The Big
Crunch appears to be unavoidable again.




In the third approach, we used equation (<ref>) while keeping
the standard value for Ω_m,0, but varying Ω_Λ
and q_0. We then solved the equation to find σ̃. An
example is illustrated in Figure <ref>.

This variation led to a slight increase in the universe lifetime. Motivated by this observation, we
explored scenarios with a negative cosmological constant.  Two such
examples are noteworthy. Setting q_0=-0.5 and Ω_Λ=-50 resulted in σ̃≃ 10 and a lifetime of |η_univ| ≃ 0.51. Alternatively, when choosing a  deceleration parameter of q_0=-1, we obtained  σ̃≃ 7.8 and a lifetime of  0.56. However, it appears challenging to significantly extend the model lifetime beyond these values.

In summary, this model represents a mild extension of
    Einstein gravity, particularly when viewed from the perspective of
    its Lagrangian. Surprisingly, the cosmology given by this model for a
spatially flat, homogenous universe is quite different from the
standard case based on Einstein equations.





§ CONCLUSIONS

Proposing a new gravity theory which goes beyond Einstein leads inevitably
to the examination of the new model in a cosmological context. In this context, theoretical advancements revealing new phenomena are equally important as empirical efforts to evaluate the model potential to supersede the standard cosmological framework.
We have emphasized both aspects studying the extension f(R, 
  L_m) theory with an explicit coupling between geometry and matter
in the form σ R ρ with ρ being the energy density. This
appears as a natural choice used also in astrophysical context. We
explored the model over a wide range of the coupling constant σ
with initial data mimicking our present universe.  Given that the model predicts a universe lifespan that is drastically shorter than the observed age of our universe (approximately 13.8 billion years), it fails to meet a fundamental criterion for a viable cosmological model. This discrepancy leads us to conclude that the model, in its current form, cannot adequately describe the observed universe. As such, the potential occurrence of a Big Crunch, while theoretically interesting, becomes a secondary concern because the model is already ruled out based on its inability to account for the observed age of the universe.

All in all, the findings highlight
challenges in adapting the model to realistically represent the
universe, particularly regarding the universe lifetime and the issue
of negative densities in some scenarios.

Before comparing our work with existent literature, let us point out
that once we derive the field equations from the Lagrangian principle,
the Lagrangian should contain a matter part L_m from which the
perfect energy-momentum tensor can be derived. The latter seems only
possible via a constrained variation with respect to the metric. The
conservation of the energy-momentum tensor in theories of the type
f(R, L_m) is not guaranteed and requires special attention. In
<cit.> the choice of the Lagrangian is f(R, L_m)=R/2+L_m
+σ RL_m with L_m=-p.  It is then difficult to study the dust
scenario with p=0 as it implies L_m=0. Furthermore, the
energy-momentum tensor is not conserved in this choice in contrast to
the authors' claim.  In <cit.> the choice is f(R,L_m)=R/2
+L_m^α with α being some number. It is not clear how we
can derive from that a hydrodynamical energy-momentum tensor and, as
explained above, whether we obtain a conservation law.  The choice of
the Lagrangian in <cit.> is also quite different from ours
as it lacks the direct coupling between geometry (represented by R)
and matter (represented by L_m). This shows that such coupling has
special consequences. In <cit.> the Lagrangian reads
f(R,L_m)=R/2 +α L_m^n - β with α and β constants. Remarks similar
to the case of <cit.> would apply also here.










Einstein
S. Weinberg, Gravitation and Cosmology: Principles and Applications
of the General Theory of Relativity, Wiley & Sons, New York (1972).

Lambda
S. Weinberg, The cosmological constant problem, Rev. Mod. Phys. 61, 1 (1989); T. Padmanabhan, Cosmological Constant-the Weight of the Vacuum, Phys. Rept. 380, 235 (2003).

Verde
L. Verde, T. Treu, and A.G. Riess, Tensions between the Early and the Late Universe, Nat. Astron. 3, 891 (2019).

Sotiriou T. P. Sotiriou and V. Faraoni, f(R) theories of
  gravity, Rev. Mod. Phys. 82, 451 (2010).

DeFelice A. De Felice and S. Tsujikawa, f(R) theories,
  Living Rev. Rel. 13: 3, (2010).

HLNO T. Harko, F. S.N. Lobo, S. Nojiri and
  S. D. Odintsov. f(R,  T) gravity, Phys. Rev.  D84:024020
  (2011).

FRLm
T. Harko, and F.S.N. Lobo, f(R,L_m) gravity Eur. Phys. J. C. 70, 373 (2010).

FRW
H. P. Robertson, Kinematics and World-Structure, ApJ 82, 284 (1935); A.G. Walker, On Milne's Theory of World-Structure, Proc. Lond. Math. Soc., 42, 90 (1936).

DM
G. Bertone, Particle Dark Matter: Observations, Models and
Searches, Cambridge University Press, Cambridge (2013).

HubbleTension
E. Di Valentino, O. Mena, S. Pan, L. Visinelli,
We. Yang, A. Melchiorri, D. F. Mota, A. G. Riess, and J. Silk, In the Realm of Hubble Tension-a Review of Solutions, Class. Quantum Grav. 38, 153001 (2021); W.D. Kenworthy, D. Scolnic, and A. Riess, The Local Perspective on the Hubble Tension: Local Structure Does Not Impact Measurement of the Hubble Constant, ApJ 875, 145 (2019).

Modifications
S. Capozziello, and V. Faraoni, Beyond Einstein Gravity: a Survey of
Gravitational Theories for Cosmology and Astrophysics,  Springer (2013).

DE
V. Motta, M. A. García-Aspeitia, A. Hernández-Almada, J. Magaña,
T. Verdugo, Taxonomy of Dark Energy Models, Universe 7, 163 (2021).

Alternatives
N. C. M. Martens and M. King, Doing More with Less: Dark Matter and
Modified Gravity in N. Mills Boyd, S. de Baerdemaeker, K. Heng and V. Matarese (eds.), Philosophy of Astrophysics: Stars,
Simulations and the Struggle to Determine What is Out There, Springer (2023).


DeFelice A. De Felice and S. Tsujikawa, f(R) theories,
  Living Rev. Rel. 13: 3, (2010).
  
Olmo G. J. Olmo Palatini Approach to Modified Gravity:
f(R) Theories and Beyond, Int. J. Mod. Phys.D20:413-462, (2011).

  Myrzakulov R. Myrzakulov, FRW cosmology in F(R,T)
    gravity, Eur. Phys. J. bf C 72, 2203 (2012).
JWu J. Wu, G. Li, T. Harko and S. Liang, Palatini
  formulation of f(R, T) gravity theory, and its cosmological
  implications, Eur. Phys. J. C 78, 430 (2018).

Katirci N. Katirci and M. Kavuk, f(R,T_μνT^μν  gravity and Cardassian-like expansion as one of its consequences, Eur. Phys. J. Plus 129: 163 (2014).
Roshan M. Roshan and F. Shojai, Energy-Momentum Squared
  Gravity, Phys. Rev. D 94, 044002 (2016).
Board C. V. R. Board and J.D. Barrow, Cosmological Models
  in Energy-Momentum-Squared Gravity, Phys. Rev. D 96, 123517 (2017).


Cai Y. Cai, S. Capozziello, M. De Laurentis and
  E. N. Saridakis, f(T) teleparallel gravity and cosmology,
  Rept.Prog.Phys. 79 no.4, 106901 (2016).
RYang R. Yang, New types of f(T) gravity,
Eur. Phys. J. C 71, 1797 (2011)
Capozziello S. Capozziello, V. F. Cardone, H. Farajollahi
and A. Ravanpak, Cosmography in f(T)-gravity, Phys.Rev.D84 :043527 (2011).


Heisenberg L. Heisenberg, Review on f(Q) Gravity,
Phys.Rept. 1066 (2024) 1-78.
Khyllep W. Khyllep, J. Dutta, E. N. Saridakis,
  K. Yesmakhanova, Cosmology in f(Q) gravity: A unified dynamical
  system analysis at background and perturbation levels,
  Phys. Rev. D 107, 044022 (2023).
Koussour K. Koussour, K. El Bourakadi, S.H. Shekh,
  S.K.J. Pacif and M. Bennai, Late-time acceleration in f(Q)
  gravity: Analysis and constraints in an anisotropic background,
  Ann. Physics, 445 (2022), 169092.
Jimenez  J. B. Jimenez, L. Heisenberg, T. Koivisto, and
  S. Pekar, Cosmology in f(Q) geometry, Phys. Rev. D 101, 103507 (2020).
Guangjie  Y. X., Guangjie Li, T. Harko and Shi-Dong Liang
  f(Q,T) gravity, Eur. Phys. J.C 79, 708 (2019).
Shiravand  M. Shiravand, S. Fakhry, and M, Farhoud
  Cosmological Inflation in f(Q, T) Gravity, Physics of the Dark
  Universe 37 101106 (2022).
XHarko Y. Xu, T. Harko, S. Shahidi and S. Liang  Weyl type
  f(Q, T) gravity, and its cosmological implications
  Eur. Phys. J. C 80, 449 (2020).


Bahamonde  S. Bahamonde et al. Teleparallel Gravity: From
  Theory to Cosmology, Rep. Prog. Phys. 86 026901 (2023)
Obukhov Yu. N. Obukhov and J. G. Pereira, Metric-affine
  approach to teleparallel gravity, Phys. Rev. D 67, 044016 (2003)
HeisenbergKuhn L. Heisenberg, M. Hohmann and S. Kuhn
  Homogeneous and isotropic cosmology in general teleparallel
  gravity, Eur. Phys. J. C 83, 315 (2023).

Odintsov
S. Nojiri, S.D. Odintsov, and V.K. Oikonomou, Modified Gravity
  Theories on a Nutshell: Inflation, Bounce and Late-time Evolution,
  Phys. Rept. 692, 1 (2017).

fR2
F. Payadeh, and M. Fathi, R^2 Theory of Gravity,  J. Phys. Conf. Ser.  442, 012053 (2013).

fR2cosm
E. V. Arbuzova, A. D. Dolgov, and L. Reverberi, Cosmological evolution
in R^2 gravity,  JCAP 02, 049 (2012).


FRLm2
T. Harko, and F.S.N. Lobo, Extensions of f(R) Gravity: Curvature-Matter Couplings and Hybrid Metric-Palatini Theory, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge (2018).

FRLm3
T. Harko, and F.S.N. Lobo, Generalized Curvature-Matter Couplings in Modified Gravity, Galaxies 2(3), 410 (2014).

Montelongo N. Montelongo Garcia and F. S. N. Lobo,
“Wormhole geometries supported by a nonminimal curvature-matter
coupling”,
Phys. Rev. D 82, 104018 (2010)
Lobato
R.V. Lobato, A. Carvalho, N.G. Kelkar, and M. Nowakowski, Massive white dwarfs in f(R,L_m) gravity, Eur. Phys. J. C. 82, 540 (2022).

Lobato2
R. V. Lobato, G. A. Carvalho, N. G. Kelkar, and M. Nowakowski, In-Stabilities of massive white dwarfs in modified gravity, Acta Phys. Pol. 142, 315 (2022).

Lobato3
R. V. Lobato, G. A. Carvalhob, and C. A. Bertulani, Neutron stars in f(R, L_m) gravity with realistic equations of state: joint-constrains with GW170817, massive pulsars, and the PSR J0030+0451 mass-radius from NICER data, Eur. Phys. J. C 81, 1013 (2021).

CosmologyFRLm
L.V. Jaybhe, R. Solanki, S. Mandal, and P.K. Sahoo, Cosmology in f(R,L_m) gravity, Phys. Lett. B 831, 137148 (2022).

TransitFRLm
A. Pradhan, D.C. Maurya, G. K. Goswani, and A. Beesham, Modeling transit Dark Energy in f(R,ℒ_m)-gravity,  Int. J. Geom. Methods Mod. 20, 2350105 (2023).

AcceleratingFRLm
D.C. Mauyura, Accelerating scenarios of massive universe in f(R,ℒ_m)-gravity, New Astron. 100, 101974 (2023).

ViscousDM
L.V. Jaybhaye, R. Solanki, S. Mandal, and P.K. Sahoo, Constraining Viscous Dark Energy Equation of State in f(R,L_m) Gravity, Universe 9, 163 (2023).


ConstrainFRLm
N.S. Kavya, V. Venkatesha, S. Mandal, and P.K. Sahoo, Constraining Anisotropic Comological Model in f(R,L_m) Gravity, Phys. Dark Universe  38, 101126 (2022).

Gonclaves
B.S. Gonclaves, P.H.R.S. Moraes, and B. Mishra, Cosmology from Non-Minimal Geometry-Matter Coupling,. Fortschr. Phys. 71, 2200153 (2023).

EnergyMomentumHarko
T. Harko, The matter Lagrangian and the energy-momentum tensor in modified gravity with non-minimal coupling between matter and geometry, Phys. Rev. D81, 044021, (2010).

MatterLagrangian
S. Mendoza, and S. Silva, The matter Lagrangian of an ideal fluid, Int. J. Geom. Methods Mod. 18, 2150059 (2020).

MinazzolliHarko
O. Minazolli, and T. Harko, New derivation of the Lagrangian of a perfect fluid with a barotropic equation of state, Phys. Rev. D 86, 087502 (2012).

Brown
D. Brown, Action Functionals for Relativistic Perfect Fluids, Class. Quant. Grav. 10, 1579 (1993).

galaxies
B. E. Robertson et al., Identification and properties of intense
star-forming galaxies at redshifts z > 10, Nat. Astron. 7, 6
(2023); E. Curtis-Lake et al., Spectroscopic confirmation of four
metal-poor galaxies at z = 10.3–13.2, Nat. Astron. 7, 622 (2023).

uranium
C. Sneden, The age of the universe, Nature 409, 643 (2001); R Cayrel et al., Measurement of stellar age from uranium decay, Nature 409, 691 (2001).

Harko2014
T. Harko, Thermodynamic interpretation of the generalized gravity models with geometry-matter coupling, Phys. Rev. D 90, 044067 (2014).

Rastall
H. Moradpour, Y. Heydarzade, F. Darabi, and I. G. Salako, A Generalization to the Rastall Theory and Cosmic Eras, Eur. Phys. J. C 77, 259 (2017).

Rastall1
N. Katirci, S. Kumar, R. C. Nunes, B. Ozturk, and S. Sharma, Rastall gravity extension of the standard ΛCDM model: theoretical features and observational constraints, Eur. Phys. J. C 80, 1050 (2020).

Sudarsky
T. Maudlin, E. Okon, and D. Sudarsky, On the Status of Conservation Laws in Physics: Implications for Semiclassical Gravity, Stud. Hist. Phil. Sci. B  69, 67 (2020).

Sudarsky1
S. J. Landau, M. Benetti, A. Perez, and D. Sudarsky, Cosmological constraints on unimodular gravity models with diffusion,  https://arxiv.org/abs/2211.07424arXiv:2211.07424 [astro-ph.CO] (preprint).

Singh Parampreet Singh and S. K. Soni,
  “On the relationship between the modifications to the Raychaudhuri
  equation and the canonical Hamiltonian structures”,
  Class. Quant. Grav 33 125001 (2016)

Cosmographic
A. Mehrabi, and M. Rezaei, Cosmographic parameters in model-independent approaches, ApJ 923, 274, (2021).

JerkSnap
M. Visser, Jerk, snap and the cosmological equation of state, Class. Quant. Grav. 21, 2603 (2004).

VisserM
M. Visser, Cosmography: Cosmology without the Einstein equations, Gen. Rel. Grav. 37, 1541, (2005).





 Paper Title: Comparing a gauge-invariant formulation and a "conventional complete gauge-fixing approach" for $l=0,1$ mode perturbations on the Schwarzschild background spacetime.









equationsection



















  Gravitational-Wave Science Project,
  National Astronomical Observatory of Japan,

  2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan











§ ABSTRACT 
  Comparison of the gauge-invariant formulation for l=0,1-mode
  perturbations on the Schwarzschild background spacetime proposed in
  [K. Nakamura, Class. Quantum Grav. 38 (2021), 145010.] and a
  “conventional complete gauge-fixing approach” in which we use the
  spherical harmonic functions Y_lm as the scalar harmonics from
  the starting point is discussed.
  Although it is often said that “gauge-invariant formulations in
  general-relativistic perturbations are equivalent to complete
  gauge-fixing approaches,” as the result of this comparison, we conclude
  that the derived solutions through the proposed gauge-invariant
  formulation and those through a “conventional complete gauge-fixing
  approach” are different.
  It is pointed out that there is a case where the boundary conditions
  and initial conditions are restricted in a conventional complete
  gauge-fixing approach.



  Comparing a gauge-invariant formulation and a “conventional
  complete gauge-fixing approach” for l=0,1 mode perturbations on the
  Schwarzschild background spacetime

    
  Kouji Nakamura
  [E-mail address: dr.kouji.nakamura@gmail.com]

    August 5, 2024
==========================================================================================================================================================================



















§ INTRODUCTION











Through the ground-based gravitational-wave detectors <cit.>,
many events of gravitational waves, mainly from black hole-black hole
coalescences, have now been detected.
We are now at the stage where there is no doubt about the existence of
gravitational waves due to their direct observations.
One of the future directions of gravitational-wave astronomy will be a
precise science through the statistics of many events.
Toward further development of gravitational-wave science, the projects
of future ground-based gravitational-wave
detectors <cit.> are also
progressing to achieve more sensitive detectors and some projects of
space gravitational-wave antenna are also
progressing <cit.>.
Although there are many targets of these detectors, the
Extreme-Mass-Ratio-Inspiral (EMRI), which is a source of gravitational
waves from the motion of a stellar mass object around a supermassive
black hole, is a promising target of the Laser Interferometer Space
Antenna <cit.>.
Since the mass ratio of this EMRI is very small, we can describe the
gravitational waves from EMRIs through black hole
perturbations <cit.>.
Furthermore, the sophistication of higher-order black hole
perturbation theories is required to support gravitational-wave
physics as a precise science.
The motivation of our series of papers,
Refs. <cit.>
and this paper, is in this theoretical sophistication of black hole
perturbation theories toward higher-order perturbations.





Although realistic black holes have their angular momentum and we have
to consider the perturbation theory of a Kerr black hole for direct
applications to the EMRI, we may say that further sophistications are
possible even in perturbation theories on the Schwarzschild background
spacetime.
From the pioneering works by Regge and
Wheeler <cit.> and
Zerilli <cit.>,
there have been many studies on the perturbations in the Schwarzschild
background
spacetime <cit.>.
In these works, perturbations are decomposed through the spherical
harmonics Y_lm because of the spherical symmetry of the background
spacetime, and l=0,1 modes should be separately treated.
These modes correspond to the monopole and dipole perturbations.
Due to these separate treatments, “gauge-invariant” treatments
for l=0 and l=1 modes were unclear.





Owing to this situation, in the previous
papers <cit.>,
we proposed the strategy of the gauge-invariant treatments of these
l=0,1 mode perturbations, which is declared as
Proposal <ref>
in Sec. <ref>
of this paper below.
One of the important premises of our gauge-invariant perturbations is
the distinction between the first-kind gauge and the second-kind
gauge.
The first-kind gauge is essentially the choice of the coordinate
system on the single manifold, and we often use this first-kind gauge
when we predict or interpret the measurement results of experiments
and observations.
On the other hand, the second-kind gauge is the choice of the
point identifications between the points on the physical spacetime
_ϵ and the background spacetime .
This second-kind gauge has nothing to do with our physical spacetime
_ϵ.
Although this difference is extensively explained in the Part I
paper <cit.>, we also emphasize this difference
in Sec. <ref> of this paper.
The proposal in the Part I paper <cit.> is a
part of our developments of the general formulation of a higher-order
gauge-invariant perturbation theory on a generic background spacetime
toward unambiguous sophisticated nonlinear general-relativistic
perturbation theories <cit.>.
Although we have been applied this general framework to cosmological
perturbations <cit.>,
we applied it to black hole perturbations in the series of papers,
i.e.,
Refs. <cit.>
and this paper.
Even in cosmological perturbation theories, the same problem as the
above l=0,1-mode problem exists as gauge-invariant treatments of
homogeneous modes of perturbations.
In this sense, we can expect that the proposal in the previous
paper <cit.> will be a clue to the same problem
in gauge-invariant perturbation theory on the generic background
spacetime.





In the Part I paper <cit.>, we also derived the
linearized Einstein equations in a gauge-invariant manner following
Proposal <ref>.
Perturbations on the spherically symmetric background spacetime are
classified into even- and odd-mode perturbations.
In the same paper <cit.>, we also gave the strategy
to solve the odd-mode perturbations including l=0,1 modes.
Furthermore, we also derived the formal solutions for the l=0,1
odd-mode perturbations to the linearized Einstein equations following
Proposal <ref>.
In the Part II paper <cit.>, we gave the
strategy to solve the even-mode perturbations including l=0,1 modes,
and we also derive the formal solutions for the l=0,1 even-mode
perturbations following
Proposal <ref>.
In the Part III paper <cit.>, we find the fact
that the derived solutions in the Part II
paper <cit.> realize the linearized version of
two exact solutions: one is the Lemaître-Tolman-Bondi (LTB)
solution <cit.> and the other is the
non-rotating
C-metric <cit.>.
Due to this fact, we conclude that the solutions for even-mode
perturbations derived in the Part II
paper <cit.> are physically reasonable.
This series of papers is the full paper version of our short
paper <cit.>.
Furthermore, brief discussions on the extension to the higher-order
perturbations are given in the short
paper <cit.>.





On the other hand, it is well-known the fact that we cannot construct
gauge-invariant variables for l=0,1 modes in a similar manner
to l≥ 2 modes if we decompose the metric perturbations through
the spherical harmonics as the scalar harmonics from the starting
point.
For this reason, we usually use gauge-fixing approaches.
Furthermore, it is often said that “gauge-invariant formulations in
general-relativistic perturbations are equivalent to complete
gauge-fixing approaches.”
In this paper, we check this statement through the comparison of our
proposed gauge-invariant formulation, in which we introduce singular
harmonics at once and regularize them after the derivation of the
mode-by-mode Einstein equation, and a “conventional complete
gauge-fixing approach”, in which we use the spherical harmonic
functions Y_lm as the scalar harmonics S_δ from the
starting point.
As the result of this comparison, we conclude that our gauge-invariant
formulation and the above “conventional complete gauge-fixing
approach” are different, though these two formulations lead similar
solutions for l=0,1-mode perturbations.
More specifically, there is a case where the boundary conditions and
initial conditions are restricted in a “conventional complete
gauge-fixing approach” where we use the decomposition of the metric
perturbation by the spherical harmonics Y_lm from the starting point.





The organization of this paper is as follows.
In Sec. <ref>, we briefly review the
premise of our series of
papers <cit.>,
which are necessary for the ingredients of this paper.
We also emphasize the difference between the concepts of the first-
and the second-kind gauges and summarize the linearized Einstein
equations for l=0,1 modes and their formal solutions in
Sec. <ref>.
In Sec. <ref>, we specify the rule of our
comparison between our gauge-invariant formulation and a conventional
gauge-fixing approach and summarize the gauge transformation rules for
the metric perturbations of l=0,1 modes, because the above statement
 “gauge-invariant formulations in
general-relativistic perturbations are equivalent to complete
gauge-fixing approaches” includes some ambiguous.
In Sec. <ref>, we discuss the
linearized Einstein equations for l=1 odd-modes and their solutions
in the conventional gauge-fixing approach.
In Sec. <ref>, we discuss the
linearized Einstein equations for l=1 even-modes and their solutions
in the conventional gauge-fixing approach.
In Sec. <ref>, we derive the solution
to the linearized Einstein equations for l=0 modes through the
complete gauge-fixing and discuss the comparison with linearized LTB
solution.
Sec. <ref> is devoted to the summary of
this paper and discussions based on our results.





We use the notation used in the previous
papers <cit.>
and the unit G=c=1, where G is Newton's constant of gravitation
and c is the velocity of light.











§ BRIEF REVIEW OF A GAUGE-INVARIANT TREATMENT OF L=0,1 MODES











In this section, we briefly review the premises of our series of
papers <cit.>
which are necessary for the ingredients of this paper.
In Sec. <ref>, we review
our framework of the gauge-invariant perturbation
theory <cit.>.
This is an important premise of the series of our
papers <cit.>
and this paper.
In Sec. <ref>, we review the
gauge-invariant perturbation theory on spherically symmetric
spacetimes which includes our proposal in
Refs. <cit.>.
In Sec. <ref>, we summarize
the l=1 odd-mode linearized Einstein equations and their “formal”
solutions.
In Sec. <ref>, we summarize
the l=0,1 even-mode linearized Einstein equations and their
“formal” solutions.
The equations and their solutions in
Sec. <ref> and
Sec. <ref> are derived
based on our proposal in Refs. <cit.>.
These are necessary for the arguments within this paper.










 §.§ General framework of gauge-invariant perturbation theory










In any perturbation theory, we always treat two spacetime manifolds.
One is the physical spacetime (_ ph,g̅_ab),
which is identified with our nature itself, and we want to describe
this spacetime (_ ph,g̅_ab) by perturbations.
The other is the background spacetime (,g_ab),
which is prepared as a reference by hand.
Note that these two spacetimes are distinct.
Furthermore, in any perturbation theory, we always write equations
for the perturbation of the variable Q as follows:

    Q(“p) = Q_0(p) + δ Q(p).

Equation (<ref>) gives a
relation between variables on different manifolds.
Actually, Q(“p) in
Eq. (<ref>) is a variable on
the physical spacetime _ϵ=_ ph, whereas
Q_0(p) and δ Q(p) are variables on the background spacetime
.
Because we regard Eq. (<ref>) as
a field equation, Eq. (<ref>)
includes an implicit assumption of the existence of a point
identification map _ϵ :
→_ϵ :
p∈↦“p∈_ϵ.
This identification map is a gauge choice in
general-relativistic perturbation theories.
This is the notion of the second-kind gauge pointed out by
Sachs <cit.>.
Note that this second-kind gauge is a different notion from the
degree of freedom of the coordinate transformation on a single
manifold, which is called the first-kind
  gauge <cit.>.
This distinction between the first- and the second-kind of gauges
extensively explained in the Part I paper <cit.> and
is also important to understand the ingredients of this paper.





To compare the variable Q on _ϵ with its
background value Q_0 on , we use the pull-back
_ϵ^* of the identification map
_ϵ : →_ϵ and
we evaluate the pulled-back variable _ϵ^*Q on the
background spacetime .
Furthermore, in perturbation theories, we expand the pull-back
operation _ϵ^* to the variable Q with respect
to the infinitesimal parameter ϵ for the perturbation as

    _ϵ^*Q
      =
      Q_0
      + ϵ^(1)_ Q
      + O(ϵ^2)
      .

Equation (<ref>) are evaluated on
the background spacetime .
When we have two different gauge choices _ϵ and
_ϵ, we can consider the gauge transformation,
which is the change of the point-identification
_ϵ→_ϵ.
This gauge transformation is given by the diffeomorphism
Φ_ϵ :=
(_ϵ)^-1∘_ϵ
:  → .
Actually, the diffeomorphism Φ_ϵ induces a pull-back from
the representation _ϵ^*Q_ϵ to the
representation _ϵ^*Q_ϵ as
_ϵ^*Q_ϵ(q)=Φ_ϵ^*_ϵ^*Q_ϵ(q)
at any point q∈.
From general arguments of the Taylor
expansion <cit.>,
the pull-back Φ_ϵ^* is expanded as

    _ϵ^*Q_ϵ(q)
         =   _ϵ^*Q_ϵ(q)
      + ϵ_ξ_(1)_ϵ^*Q_ϵ(q)
      + O(ϵ^2),

where ξ_(1)^a is the generator of Φ_ϵ.
From Eqs. (<ref>) and
(<ref>), the gauge transformation for the first-order
perturbation ^(1)Q is given by

    ^(1)_ Q(q) - ^(1)_ Q(q)    =   _ξ_(1)Q_0(q)

at any point q∈.
We also employ the order by order gauge invariance as a
concept of gauge invariance <cit.>.
We call the kth-order perturbation ^(k)_Q as
gauge invariant if and only if

    ^(k)_ Q(q) = ^(k)_ Q(q)

for any gauge choice _ϵ and _ϵ at any
point of q∈.





Based on the above setup, we proposed a formulation to construct
gauge-invariant variables of higher-order
perturbations <cit.>.
First, we expand the metric on the physical spacetime
_ϵ, which was pulled back to the background
spacetime  through a gauge choice _ϵ as

    ^*_ϵg̅_ab   =   
      g_ab + ϵ_h_ab
      + O(ϵ^2).

Although the expression (<ref>) depends
entirely on the gauge choice _ϵ, henceforth,
we do not explicitly express the index of the gauge choice
_ϵ in the expression if there is no
possibility of confusion.
The important premise of our formulation of higher-order
gauge-invariant perturbation theory was the following
conjecture <cit.> for the linear
metric perturbation h_ab:

  
  If the gauge transformation rule for a perturbative pulled-back
  tensor field h_ab from the physical spacetime _ϵ
  to the background spacetime  is given by _h_ab
  - _h_ab = _ξ_(1)g_ab with the
  background metric g_ab, there then exist a tensor field
  _ab and a vector field Y^a such that h_ab is
  decomposed as h_ab =: _ab + _Yg_ab,
  where _ab and Y^a are transformed as
  __ab - __ab = 0 and
  _Y^a - _Y^a = ξ^a_(1)
  under the gauge transformation, respectively.

We call _ab and Y^a as the
gauge-invariant and gauge-dependent parts
of h_ab, respectively.





The proof of Conjecture <ref> is
highly
nontrivial <cit.>,
and it was found that the gauge-invariant variables are essentially
non-local.
Despite this non-triviality, once we accept
Conjecture <ref>,
we can decompose the linear perturbation of an arbitrary tensor field
_^(1)Q, whose gauge transformation is given by
Eq. (<ref>), through the gauge-dependent part Y_a of
the metric perturbation in
Conjecture <ref> as

    _^(1)Q = ^(1) + __YQ_0,

where ^(1) is the gauge-invariant part of the perturbation
_^(1)Q.
As examples, the linearized Einstein tensor
_^(1)G_a^  b and the linear perturbation of the
energy-momentum tensor _^(1)T_a^  b are also
decomposed as

    _^(1)G_a^  b
      =
      ^(1)_a^  b[] + __YG_a^  b
      ,
        _^(1)T_a^  b
      =
      ^(1)_a^  b + __YT_a^  b
      ,

where G_ab and T_ab are the background values of the Einstein
tensor and the energy-momentum tensor,
respectively <cit.>.
Using the background Einstein equation
G_a^  b=8π T_a^  b, the linearized Einstein equation
_^(1)G_ab=8π_^(1)T_ab is
automatically given in the gauge-invariant form

    ^(1)_a^  b[]
      =
      8 π^(1)_a^  b[,ϕ]

even if the background Einstein equation is nontrivial.
Here, “ϕ” in Eq. (<ref>)
symbolically represents the matter degree of freedom.





Finally, we comment on the coordinate
transformation induced by the gauge transformation
Φ_ϵ of the
second-kind <cit.>.
To see this, we introduce the coordinate system
{O_α,ψ_α} on the background spacetime ,
where O_α are open sets on the background spacetime and
ψ_α are diffeomorphisms from O_α to ^4
(4=).
The coordinate system {O_α,ψ_α} is the set of
collections of the pair of open sets O_α and diffeomorphism
ψ_α:O_α↦^4.
If we employ a gauge choice _ϵ of the second kind, we
have the correspondence of the physical spacetime
_ϵ and the background spacetime .
Together with the coordinate system ψ_α on ,
this correspondence _ϵ between _ϵ and
 induces the coordinate system on _ϵ.
Actually, _ϵ(O_α) for each α is an
open set of _ϵ.
Then, ψ_α∘_ϵ^-1 becomes a
diffeomorphism from an open set _ϵ(O_α)⊂_ϵ to ^4.
This diffeomorphism ψ_α∘_ϵ^-1
induces a coordinate system of an open set on _ϵ.
When we have two different gauge choices _ϵ and
_ϵ of the second kind,
ψ_α∘_ϵ^-1 :
_ϵ ↦ ^4 ({x^μ}) and
ψ_α∘_ϵ^-1 : _ϵ
↦ ^4 ({y^μ}) become different coordinate systems
on _ϵ.
We can also consider the coordinate transformation from the coordinate
system ψ_α∘_ϵ^-1 to another
coordinate system ψ_α∘_ϵ^-1.
Because the gauge transformation
_ϵ→_ϵ is induced by the
diffeomorphism Φ_ϵ :=
(_ϵ)^-1∘_ϵ, this
diffeomorphism Φ_ϵ induces the coordinate transformation
as

    y^μ(q) := x^μ(p) = ((Φ_ϵ^-1)^*x^μ)(q)

in the passive point of
view <cit.>,
where p, q ∈  are identified to the same point
“p ∈ _ϵ by the gauge choices
_ϵ and _ϵ, respectively.
If we represent this coordinate transformation in terms of the Taylor
expansion (<ref>), we have the coordinate transformation

    y^μ(q) = x^μ(q) - ϵξ^μ_(1)(q) + O(ϵ^2).

We should emphasize that the coordinate transformation
(<ref>) is not the starting
point of the gauge transformation but a result of the above framework.





On the other hand, we may consider the point replacement by the
one-parameter diffeomorphism s=Ψ_λ(r), where
s,r∈_ϵ and λ is an infinitesimal parameter
satisfying Ψ_λ=0(r)=r.
The pull-back Ψ_λ^* of any tensor field Q on
_ϵ is given by

    Q(s)
         =   
          Q(Ψ_λ(r)) = (Ψ_λ^*)Q(r)
          
       =   
          Q(r) + λ._ζQ|_λ=0(r)
          + O(λ^2)
          ,

where ζ^a is the generator of the pull-back
Ψ_λ^*.
Equation (<ref>) is just the formula of
the Taylor expansion on the manifold _ϵ without using
any coordinate system.
At the same time, Equation (<ref>) is the
definitions of the Lie derivative _ζ and its generator
ζ^a.
In many literatures, this formula is derived from the coordinate
transformation

    y^μ(s) := x^μ(r) + λζ^μ(r) + O(λ^2)

as explained in the Part I paper <cit.>.
The formula (<ref>) of the Taylor
expansion defined without any coordinate system and the second
term in the right-hand side of Eq. (<ref>)
represents the actual difference of the tensor fields Q(s) and
Q(r) in the different point s≠ r and its physical meaning.





If we consider a formal metric decomposition
g̅_ab=^(0)g_ab+λ h_ab+O(λ^2) within
_ϵ as an example of the tensor field Q in
Eq. (<ref>), the formula
(<ref>) is given by

    ^(0)g_ab(s) + λ h_ab(s)
      =
      ^(0)g_ab(r) + λ(
      h_ab(r) + ._ζ^(0)g_ab|_r)
      + O(λ^2).

Since the formula of the Taylor expansion
(<ref>) is derived from the infinitesimal
coordinate transformation rule
(<ref>) in many
literatures, the term ._ζ^(0)g_ab|_r
in the right-hand side of
Eq. (<ref>) is often
called “the degree of freedom of the coordinate transformation” and
“unphysical degree of freedom”.
However, since the formula
(<ref>) is just the
re-expression of the Taylor expansion
(<ref>) defined without any coordinate
system and the term ._ζ^(0)g_ab|_r
in Eq. (<ref>) should have
its physical meaning.
If we insist that the term
._ζ^(0)g_ab|_r which appears due
to the point replacement on the single manifold is “unphysical”,
this directly leads the statement that “the famous arguments of the
Killing vector and the symmetry of the spacetime are physically
meaningless.”
For this reason, we have to emphasize that we cannot regard the
second term of the right-hand side of
Eq. (<ref>) as an “unphysical degree of
freedom.”





In our series of
papers <cit.>
and in this paper, we classify the term
._ζ^(0)g_ab|_r in
Eq. (<ref>) as one of
gauge-degree of freedom of the first kind, since this term can
be eliminate the infinitesimal coordinate transformation
(<ref>) which is the
coordinate transformation within the single manifold _ϵ.
Furthermore, Equation
(<ref>) does not mean
h_ab(s) = h_ab(r) +
._ζ^(0)g_ab|_r but it just means
^(0)g_ab(s) = ^(0)g_ab(r) +
λ._ζ^(0)g_ab|_r
+ O(λ^2).
Moreover, we also have to emphasize that the infinitesimal
coordinate transformation
(<ref>) is
essentially different from the infinitesimal coordinate transformation
(<ref>).
Actually, the infinitesimal coordinate transformation
(<ref>) is the
replacement of the point within the single manifold
_ϵ.
On the other hand, the coordinate transformation
(<ref>) is just the change
of the coordinate label at the same point in the background spacetime
.










 §.§ Linear perturbations on spherically symmetric background spacetime










Here, we consider the 2+2 formulation of the perturbation of a
spherically symmetric background spacetime, which originally proposed
by Gerlach and Sengupta <cit.>.
Spherically symmetric spacetimes are characterized by the direct
product =_1× S^2 and their metric is

    g_ab   =   
      y_ab + r^2γ_ab
      , 
    
      y_ab   =    y_AB (dx^A)_a(dx^B)_b,   γ_ab = γ_pq (dx^p)_a (dx^q)_b,

where x^A = (t,r), x^p=(θ,ϕ), and γ_pq is the
metric on the unit sphere.
In the Schwarzschild spacetime, the metric
(<ref>) is given by

    y_ab   =   
          - f (dt)_a(dt)_b
          +
          f^-1 (dr)_a(dr)_b
          ,
            
          f = 1 - 2M/r
      ,
    γ_ab   =   
          (dθ)_a(dθ)_b
          +
          sin^2θ(dϕ)_a(dϕ)_b
          =
          θ_aθ_b + ϕ_aϕ_b
          ,
      
    θ_a   =   
          (dθ)_a,   ϕ_a
          =
          sinθ (dϕ)_a
          .






On this background spacetime (,g_ab), the components of
the metric perturbation is given by

    h_ab
      =
      h_AB (dx^A)_a(dx^B)_b
      +
      2 h_Ap (dx^A)_(a(dx^p)_b)
      +
      h_pq (dx^p)_a(dx^q)_b
      .

Here, we note that the components h_AB, h_Ap, and
h_pq are regarded as components of scalar, vector, and
tensor on S^2, respectively.
In the Part I paper <cit.>, we showed the
linear independence of the set of harmonic functions

    {
      S_δ,
      D̂_pS_δ,
      ϵ_pqD̂^qS_δ,
      1/2γ_pqS_δ,
      (D̂_pD̂_q-1/2γ_pq)S_δ,
      2ϵ_r(pD̂_q)D̂^rS_δ}
      ,

where D̂_p is the covariant derivative associated with
the metric γ_pq on S^2,
D̂^p=γ^pqD̂_q,
ϵ_pq=ϵ_[pq]=2θ_[pϕ_q] is the totally
antisymmetric tensor on S^2.
In the set of harmonic function (<ref>), the
scalar harmonic function S_δ is given by

    S_δ
      =
      {[      Y_lm                                   l≥ 2;; k_(Δ̂+2)m                                    l=1;;    k_(Δ̂)                                    l=0. ].

Here, functions k_(Δ̂) and k_(Δ̂+2)m are
the kernel modes of the derivative operator Δ̂ and
[Δ̂+2], respectively, and we employ the explicit form of
these functions as

    k_(Δ̂)   =   
          1 + δln(1-cosθ/1+cosθ)^1/2,
            δ∈
          ,
          
    
      k_(Δ̂+2,m=0)   =   cosθ
          +
          δ(1/2cosθln1+cosθ/1-cosθ -1)
          ,
           δ∈
         ,
         
    
      k_(Δ̂+2,m=±1)   =   [
          sinθ
          +
          δ(
          + 1/2sinθln1+cosθ/1-cosθ
          + θ)
          ]
          e^± iϕ
          .






Then, we consider the mode decomposition of the components
{h_AB,h_Ap,h_pq} as follows:

    h_AB   =   ∑_l,mh̃_AB S_δ
          ,
      
    
      h_Ap   =   
          r ∑_l,m[
          h̃_(e1)AD̂_pS_δ
          +
          h̃_(o1)Aϵ_pqD̂^qS_δ]
          ,
      
    
      h_pq   =   
          r^2∑_l,m[
          1/2γ_pqh̃_(e0) S_δ
          +
          h̃_(e2)(
          D̂_pD̂_q - 1/2γ_pqD̂^rD̂_r) S_δ
          +
          2 h̃_(o2)ϵ_r(pD̂_q)D̂^r S_δ]
          .

Since the linear independence of each element of the set of harmonic
function (<ref>) is guaranteed, the
one-to-one correspondence between the components {h_AB, h_Ap,
h_pq} and the mode coefficients {h̃_AB,
h̃_(e1)A, h̃_(o1)A, h̃_(e0),
h̃_(e2), h̃_(o2)} with the decomposition
formulae (<ref>)-(<ref>) is guaranteed
including l=0,1 mode if δ≠ 0.
Then, the mode-by-mode analysis including l=0,1 is possible when
δ≠ 0.
However, the mode functions
(<ref>)–(<ref>)
are singular if δ≠ 0.
When δ=0, we have k_(Δ̂)∝ Y_00 and
k_(Δ̂+2)m∝ Y_1m, but the linear dependence of
the set of harmonics (<ref>) is lost in this
case.
Because of this situation, we proposed the following strategy:

  
  We decompose the metric perturbation h_ab on the background
  spacetime with the metric
  (<ref>)–(<ref>)
  through Eqs. (<ref>)–(<ref>) with the
  harmonic function S_δ given by
  Eq. (<ref>).
  Then, Eqs. (<ref>)–(<ref>) become
  invertible including l=0,1 modes.
  After deriving the mode-by-mode field equations such as linearized
  Einstein equations by using the harmonic functions S_δ, we
  choose δ=0 as the regular boundary condition for solutions
  when we solve these field equations.






As shown in the Part I paper <cit.>, once we accept
Proposal <ref>,
the Conjecture <ref> becomes the
following statement:

  
  If the gauge-transformation rule for a perturbative pulled-back
  tensor field h_ab to the background spacetime  is
  given by _h_ab - _h_ab =
  _ξ_(1)g_ab with the background metric g_ab
  with spherically symmetry, there then exist a tensor field
  _ab and a vector  field Y^a such that h_ab is
  decomposed as h_ab =: _ab +
  _Yg_ab, where _ab and Y^a are
  transformed into __ab -
  __ab = 0 and _Y^a
  - _Y^a = ξ^a_(1) under the gauge
  transformation, respectively.

Actually, the gauge-dependent variable Y_a is given by

    Y_a   :=   ∑_l,mỸ_A S_δ (dx^A)_a
           +
           ∑_l,m(
           Ỹ_(e1)D̂_pS_δ
           +
           Ỹ_(o1)ϵ_pqD̂^qS_δ)
           (dx^p)_a
           ,

where

    Ỹ_A   :=   
           r h̃_(e1)A
           - r^2/2D̅_Ah̃_(e2)
           ,
      
    Ỹ_(e1)   :=   r^2/2h̃_(e2)
           ,
      
    Ỹ_(o1)   :=   
           - r^2h̃_(o2)
           .

Furthermore, including l=0,1 modes, the components of the
gauge-invariant part _ab of the metric perturbation
h_ab is given by

    _AB   =   ∑_l,mF̃_AB S_δ
          ,
      
    _Ap   =   
          r ∑_l,mF̃_Aϵ_pqD̂^qS_δ,   D̂^p_Ap = 0
          ,
      
    _pq   =   1/2γ_pq r^2∑_l,mF̃ S_δ
          ,

where F̃_AB, F̃_A, and F̃ are given by

    F̃_AB   :=   h̃_AB
           - 2 D̅_(AỸ_B)
           ,
      
    F̃_A   :=   h̃_(o1)A
           + r D̅_Ah̃_(o2)
           ,
      
    F̃   :=   h̃_(e0)
           - 4/rỸ_AD̅^Ar
           + h̃_(e2) l(l+1)
      .

Thus, we have constructed gauge-invariant metric perturbations on
the Schwarzschild background spacetime including l=0,1 modes.





Furthermore, from
Eqs. (<ref>)–(<ref>)
for the gauge-invariant variables _AB, _Ap, and
_pq and gauge-dependent variables Y_a defined by
Eqs. (<ref>)–(<ref>), the original
components {h_AB,h_Ap,h_pq} of the metric perturbation
(<ref>)–(<ref>) are given by

    h_ab   =   _ab + _Yg_ab .

This is the assertion of
Theorem <ref>.





The aim of our proposal is to add a greater degree of freedom of the
metric perturbation so that the decomposition
(<ref>) is guaranteed even in the case of
l=0,1 modes.
Therefore, it is impossible to reach the final expression
(<ref>) if we treat the metric perturbation
by using S_δ=0=Y_lm in the decomposition formulae
(<ref>)–(<ref>) from the starting point.





To see this more specifically, we show an explicit example of the
l=1 odd mode perturbation.
If we apply our proposal to the decomposition formulae
(<ref>)–(<ref>), l=1 odd-mode
perturbation is given by

    h_ab
      =
      2 r h̃_(o1)A^(l=1)ϵ_pqD̂^qS_δ
      (dx^A)_(a (dx^p)_b)
      +
      2 r^2h̃_(o2)^(l=1)ϵ_r(pD̂_q)D̂^r S_δ
      (dx^p)_(a (dx^q)_b)
      .

If we use  S_δ=0=Y_lm from the starting point,
2ϵ_r(pD̂_q)D̂^r S_δ=0.
So, the mode coefficient h̃_(o2)^(l=1) does not appear.
When δ≠ 0, the gauge-transformation rule of the variables
h̃_(o1)A^(l=1) and h̃_(o2)^(l=1) are given
by

    _h̃_(o1)A^(l=1)
      -
      _h̃_(o1)A^(l=1)
      =
      r D̅_A(1/rζ_(o))
      ,
        _h̃_(o2)^(l=1)
      -
      _h̃_(o2)^(l=1)
      =
      - 1/rζ_(o)
      ,

where the generator of the gauge transformation is given by
ξ_A=0 and ξ_p=r ζ_(o)ϵ_pqD̂^qS_δ.
If we use  S_δ=0=Y_lm from the starting point, the
gauge-transformation rule for the variable h̃_(o2)^(l=1)
does not appear, either.
However, in our case, this gauge transformation appears due to the fact
ϵ_r(pD̂_q)D̂^r S_δ≠ 0.
Inspecting these gauge-transformation rules, we rewrite
Eq. (<ref>) as

    h_ab   =   
          2 r [
          (
          h̃_(o1)A^(l=1)
          + r D̅_Ah̃_(o2))
          - r D̅_Ah̃_(o2)] ϵ_pqD̂^qS_δ
          (dx^A)_(a (dx^p)_b)
          
          +
          2 r^2h̃_(o2)^(l=1)ϵ_r(pD̂_q)D̂^r S_δ
          (dx^p)_(a (dx^q)_b)
       =   
          2 r [
          F̃_A
          - r D̅_A(1/r^2Ỹ_(o1))
          ] ϵ_pqD̂^qS_δ
          (dx^A)_(a (dx^p)_b)
          
          +
          2 r^2(1/r^2Ỹ_(o1)) ϵ_r(pD̂_q)D̂^r S_δ
          (dx^p)_(a (dx^q)_b)
          .

In Eq. (<ref>), F̃_A
is the gauge-invariant variable for the metric perturbation defined by
Eq. (<ref>) and Ỹ_(o1) is
the gauge-dependent part of the metric perturbation defined by
Eq. (<ref>).
The terms of Ỹ_(o1) in
Eq. (<ref>) can be written in the
form of the Lie derivative of the background metric as in
Eq. (<ref>).
When δ=0, Eq. (<ref>) is
given by

    h_ab   =   
          2 r [
          F̃_A
          - r D̅_A(1/r^2Ỹ_(o1))
          ] ϵ_pqD̂^qS_δ
          (dx^A)_(a (dx^p)_b)
          .

This is also given in the form (<ref>).





When we construct the gauge-invariant variable F̃_A in
Eqs. (<ref>) and
(<ref>), the existence of the
coefficient h̃_(o2)^(l=1)=Ỹ_(o1)/r^2 is
essential which does not appear in the treatment using the spherical
harmonics Y_lm from the starting point.
Actually, we cannot define the gauge-invariant variable for l=1
odd-mode perturbation in the same manner as l≥ 2-mode case.
Our
proposal <ref>,
make the degree of freedom of the metric perturbations increase.
We have to emphasize that we can reach the
decomposition (<ref>) owing to this
additional degree of freedom.





To discuss the linearized Einstein equation
(<ref>) and the linear perturbation of
the continuity equation

    ∇_a^(1)_b^  a = 0

of the gauge-invariant energy-momentum tensor
^(1)_b^  a:=g^ac^(1)_bc on a
vacuum background spacetime, we consider the mode-decomposition of the
gauge-invariant part ^(1)_bc of the linear perturbation
of the energy-momentum tensor through the set
(<ref>) of the harmonics as follows:

    ^(1)_ab   =   ∑_l,mT̃_AB
          S_δ
          (dx^A)_a(dx^B)_b
          +
          r
          ∑_l,m{T̃_(e1)AD̂_pS_δ
          +
          T̃_(o1)Aϵ_prD̂^rS_δ}
          2 (dx^A)_(a(dx^p)_b)
          
         +
         ∑_l,m{T̃_(e0)1/2γ_pq S_δ
         +
         T̃_(e2)(
         D̂_pD̂_qS_δ
         -
         1/2γ_pqD̂_rD̂^rS_δ)
         .
         
                .
         +
         T̃_(o2)
         2 ϵ_r(pD̂_q)D̂^rS_δ}
         (dx^p)_a(dx^q)_b
         .






In the Part I paper <cit.>, we derived the
linearized Einstein equations, discussed the odd-mode perturbation
F̃_Ap in
Eq. (<ref>), and derived the
l=1 odd-mode solutions to these equations.
The Einstein equation for even mode F̃_AB and F̃ in
Eqs. (<ref>) and
(<ref>) also derived in the
Part I paper <cit.>, and l=0,1
even-mode solutions are derived in the Part II
paper <cit.>.
Since these solutions include the Kerr parameter perturbation and the
Schwarzschild mass parameter perturbation of the linear order in the
vacuum case, these are physically reasonable.
Then, we conclude that our proposal is also physically reasonable.
Furthermore, we also checked that our derived solutions include the
linearized LTB solution and non-rotating C-metric with the
Schwarzschild background in the Part III
paper <cit.>.





The purpose of this paper is to compare our proposed gauge-invariant
treatments for l=0,1-mode perturbations on the Schwarzschild
background spacetime with conventional “complete gauge-fixing
treatments” in which we use the spherical harmonics Y_lm from the
starting point.
For this purpose, the linearized Einstein equations for the l=0,1-modes
and their solutions based on our proposal are necessary.
Therefore, we review them below.










 §.§ l=1 odd-mode linearized Einstein equations and solutions










As derived in the Part I paper <cit.>, the
l=1 odd-mode part in the linearized Einstein equations are
simplified as the constraint equation

    D̅_D(rF̃^D)
      =
      0
      ,

and the evolution equation

    - [ D̅^DD̅_D - 2/r^2] (rF̃_A)
         - 2/r^2 (D̅^Dr) (D̅_Ar) (rF̃_D)
         + 2/r (D̅^Dr) D̅_A(rF̃_D)
         
       =   
      16 π r T̃_(o1)A
      ,

where we choose T̃_(o2)=0 by hand.
Furthermore, we have the continuity equation

    D̅^CT̃_(o1)C
      + 3/r (D̅^Dr) T̃_(o1)D
      =
      0

for the l=1 odd-mode matter perturbation which is derived from the
divergence of the first-order perturbation of the energy-momentum
tensor.





We also note that the constraint
(<ref>) comes from the
traceless part of the (q,p)-component of the Einstein equation
(<ref>), which is the coefficient of
the mode function 2ϵ_r(pD̂_q)D̂^pS_δ.
If δ=0, i.e., the scalar harmonics S_δ is the
spherical harmonics Y_lm,
2ϵ_r(pD̂_q)D̂^pS_δ=0 for l=1.
Therefore, the constraint
(<ref>) does not appear when
we use the spherical harmonics Y_lm from the starting point.





The explicit strategy to solve these odd-mode perturbations and
l=0,1 mode solutions was discussed in the Part I
paper <cit.>.
As the result, we obtained the l=1 odd-mode “formal” solution as
follows:

    2 _Ap (dx^A)_(a(dx^p)_b)
         =
         6 M r^2[∫ dr a_1(t,r)/r^4]
         sin^2θ
         (dt)_(a (dϕ)_b)
         +
         _Vg_ab
         ,
      
          
         V_a
         =
         (β_1t + β_0 + W_(o)(t,r)) r^2sin^2θ (dϕ)_a
         ,

where

    a_1(t,r)
         =   
          - 16 π/3M r^3 f ∫ dt T̃_(o1)r + a_10
       =   
          - 16 π/3M∫ dr r^31/fT̃_(o1)t + a_10
          .

The constant a_10 in Eq. (<ref>)
corresponds to the Kerr parameter.
Furthermore, the function W_(o)(t,r) in
Eq. (<ref>) is related
to the solution to the l=1-mode Regge-Wheeler equation

    ∂_t^2Z_(o)
      -  f ∂_r( f ∂_rZ_(o))
      + 1/r^2 f [ l(l+1) - 3 (1-f) ] Z_(o)
      =
      16 π f^2T̃_(o1)r
      ,

where Z_(o) = r f ∂_rW_(o).
We have to solve Eq. (<ref>) to
obtain the function W_(o)(t,r).
In this sense, the solution
(<ref>)  should be
regarded as the “formal” one.










 §.§ l=0,1 even-mode linearized Einstein equations and solutions










The l=0,1 even-mode part of the linearized Einstein equation
(<ref>) is summarized as follows:

    F̃_D^   D   =   
          0
          ,
      
    D̅^D_AD
      - 1/2D̅_AF̃   =   
          16 π r T̃_(e1)A
          ,

where the variable _AB is the traceless part of the
variable F̃_AB defined by

    _AB := F̃_AB - 1/2 y_ABF̃_C^  C

and we choose the component of the energy-momentum tensor so that
T̃_(e2)=0 by hand.
Owing to the same choice T̃_(e2)=0, we also have the
evolution equations

    (
         D̅_DD̅^D
         + 2/r (D̅^Dr) D̅_D
         -  (l-1)(l+2)/r^2) F̃
         - 4/r^2 (D̅_Cr) (D̅_Dr) ^CD=
         16 π S_(F)
         ,
         
          
      S_(F)
      :=
           T̃_C^   C
           + 4 (D̅_Dr) T̃_(e1)^D
         ,
         
          [
         -  D̅_DD̅^D
         -  2/r (D̅_Dr) D̅^D
         + 4/r (D̅^DD̅_Dr)
         + l(l+1)/r^2]
         _AB
          
         + 4/r (D̅^Dr) D̅_(A_B)D
         -  2/r (D̅_(Ar) D̅_B)F̃
           =
          16 π S_()AB
          ,


    S_()AB   :=   T̃_AB - 1/2 y_ABT̃_C^   C
          - 2 ( D̅_(A(r T̃_(e1)B)) - 1/2 y_ABD̅^D(r T̃_(e1)D) )
          
          
          + 2 y_AB (D̅^Cr) T̃_(e1)C
         ,

for the variable F̃ and the traceless variable
_AB with l=0,1.
We also have to take into account the even-mode part of the
continuity equation as follows:

    D̅^CT̃_C^  B
         + 2/r (D̅^Dr)T̃_D^   B
         -  1/r l(l+1) T̃_(e1)^B
         -  1/r (D̅^Br) T̃_(e0)
         =
         0
         ,
         
          D̅^CT̃_(e1)C
         + 3/r (D̅^Cr) T̃_(e1)C
         + 1/2rT̃_(e0)
         =
         0
         ,

for l=0,1.





We also note that the constraint
(<ref>) comes from the
traceless part of the (q,p)-component of the Einstein tensor
(<ref>), which is the coefficient of
the mode function
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S_δ.
If δ =0, i.e., the scalar harmonics S_δ is the
spherical harmonics Y_lm,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S_δ=0
for l=1.
Therefore, the constraint
(<ref>) does not appear when
we use the spherical harmonics Y_lm from the starting point.









  §.§.§ l=1 even-mode solution









In the Part II paper <cit.>, we derived the
l=1 solution to
Eqs. (<ref>),
(<ref>),
(<ref>),
(<ref>),
(<ref>), and
(<ref>) with l=1.
For m=0 mode, in the Part II paper <cit.>, we
derived the following “formal” solution to the linearized Einstein equation

    _ab   =   _Vg_ab
          - 16 π r^2/3(1-f)[
          f^2{1+f/2T̃_rr
          + r f ∂_rT̃_rr
          -  T̃_(e0)
          -  4 T̃_(e1)r}  (dt)_a(dt)_b.
          
                            .
          + 2r/f{∂_tT̃_tt
          - 3f(1-f)/2rT̃_tr} (dt)_(a(dr)_b).
          
                            .
          + r/f{∂_rT̃_tt
          - 3(1-3f)/2rfT̃_tt} (dr)_a(dr)_b.
          
                            .
          + r^2T̃_ttγ_ab] cosθ
          ,

where the vector field V_a is given by

    V_a   :=   
           -  r ∂_tΦ_(e)cosθ (dt)_a
           + ( Φ_(e) - r ∂_rΦ_(e)) cosθ (dr)_a
           -  r Φ_(e)sinθ (dθ)_a
           .

Here, the variable Φ_(e) is a solution to the l=1
Zerilli equation

    -  ∂_t^2Φ_(e)
         + f ∂_r[ f ∂_rΦ_(e)]
         -
         f(1-f)/r^2Φ_(e)
       =   4π r/3(1-f)[
         3(1-3f) T̃_tt
         + (1+f)f^2T̃_rr
         -  2 r f ∂_rT̃_tt.
          
                    .
         + 2 r f^3∂_rT̃_rr
         -  2 f^2T̃_(e0)
         -  8 f^2T̃_(e1)r]
         .

If we obtain the solution to the l=1 Zerilli equation
(<ref>), we can write the explicit
form of the solution (<ref>)
through the generator V_a defined by
Eq. (<ref>).
In this sense, we have to regard the solution
(<ref>) is a “formal” one.









  §.§.§ l=0 even-mode solution









On the other hand, for the l=0-mode, we may choose
T̃_(e1)A=0 in
Eqs. (<ref>),
(<ref>),
(<ref>)–(<ref>)
with l=0.
Then, we derived the l=0 mode solution

    _ab   =   2/r(M_1+4π∫ dr r^2/f T_tt)
          ((dt)_a(dt)_a+ 1/f^2 (dr)_a(dr)_a)
          
          
          + 2 [4 π r ∫ dt (1/fT̃_tt + f T̃_rr)] (dt)_(a(dr)_b)
          + _Vg_ab
          ,

where

    V_a
      =
      (
      f/4Υ
      + rf/4∂_rΥ
      -  r Ξ(r)/(1-3f)
      + f ∫ dr 2 Ξ(r)/f(1-3f)^2) (dt)_a
      +
      1/4f r ∂_tΥ (dr)_a
      .

Here, the variable F̃=:∂_tΥ must satisfy the
equation

    - 1/f∂_t^2Υ
         + ∂_r(f ∂_rΥ)
         + 3(1-f)/r^2Υ
         - 8/r^3∫ dt m_1(t,r)
         - 4/1-3f∂_rΞ(r)
         
       =   
          16 π∫ dt ( - 1/fT̃_tt + f T̃_rr)
          ,

where m_1(t,r) is given by

    m_1(t,r) = 4 π∫ dr r^2/fT̃_tt + M_1
      = 4 π∫ dt r^2 f T̃_tr + M_1,

M_1 is the constant corresponding to the Schwarzschild mass
perturbation, and Ξ(r) is an arbitrary function of r.










§ RULE OF COMPARISON AND GAUGE-TRANSFORMATION RULES IN A
  CONVENTIONAL GAUGE-FIXING










The main purpose of this paper is to check the following statement.
“Gauge invariant formulations of perturbations are equivalent to
  complete gauge fixing approaches.”
However, this statement is too ambiguous to check if we have the
background knowledge explained in
Sec. <ref>.
For example, there is no explanation of the terminology “gauge” in this statement.
Therefore, we have to clarify this statement as follows:







  Our rule for comparison 
  First of all, we assume the terminology “gauge” in the
  statement “Gauge invariant formulations of perturbations are
    equivalent to complete gauge fixing approaches” is the second-kind
  gauge which is explained in
  Sec. <ref>.
  Therefore, the gauge-transformation rule for this statement is given
  by (<ref>).
  Furthermore, the degree of freedom that changes under this
  gauge-transformation rule is regarded as “unphysical.”
  The “gauge fixing” in the above statement is a specification
  of some perturbative variables through the degree of freedom of the
  generator ξ_(1)^a.
  Furthermore, “complete gauge fixing” in the above statement is
  a specification of some perturbative variables through the “entire” degree of freedom of the generator ξ_(1)^a.






These are all rules of our game to compare our gauge-invariant
formulation and a “conventional complete gauge-fixing approach” in
which we use the spherical harmonics Y_lm from the starting point.









 §.§ Metric perturbations









Based on the above conceptual premise, we consider the metric
perturbation h_ab on the background Schwarzschild spacetime
(,g_ab) whose metric g_ab is given by Eqs. (<ref>)–(<ref>).
We consider the components of the metric perturbation h_ab as
Eq. (<ref>) and the decomposition of
these components {h_AB, h_Ap, h_pq} as
Eqs. (<ref>)–(<ref>) but we concentrate
on the case S_δ=Y_lm=:S, where Y_lm is the conventional
spherical harmonics.





Since we choose S=Y_lm, we have D̂_pS =
ϵ_pqD̂^qS = 0,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0 for l=0
modes.
For l=1 modes,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0.
Due to these facts, we cannot construct gauge-invariant variables for
l=0,1 modes in a similar manner to the derivation of
Eqs. (<ref>)–(<ref>)
for l≥ 2 modes.
Then, we cannot use the gauge-invariant formulation reviewed in
Sec. <ref>.
Instead, we have to fix the second-kind gauge degree of freedom to
exclude the “unphysical” degree of freedom.









 §.§ Conventional gauge-transformation rules for l=0,1 metric perturbations









Here, we consider the gauge-transformation rule

    _h_ab
      -
      _h_ab
      =
      _ξg_ab
      =
      2 ∇_(aξ_b)
      .

From this gauge-transformation rule, we can derive the
gauge-transformation rule for the components
{h_AB,h_Ap,h_qp} defined by
Eq. (<ref>) is given by

    _h_AB
      -
      _h_AB   =   _ξg_AB
          =
          ∇_Aξ_B
          +
          ∇_Bξ_A
          =
          D̅_Aξ_B
          +
          D̅_Bξ_A
          ,
      
    _h_Ap
      -
      _h_Ap   =   _ξg_Ap
          =
          ∇_Aξ_p
          +
          ∇_pξ_A
          =
          D̅_Aξ_p
          +
          D̂_pξ_A
          -
          2/r (D̅_Ar) ξ_p
          ,
      
    _h_pq
      -
      _h_pq   =   _ξg_pq
          =
          ∇_pξ_q
          +
          ∇_qξ_p
          =
          D̂_pξ_q
          +
          D̂_qξ_p
          +
          2 r (D̅^Ar) γ_pqξ_A

from the formulae summarized in Appendix B of the Part I
paper <cit.>.
Here, we consider the Fourier transformation of
ξ_a=:ξ_A(dx^A)_a+ξ_p(dx^p)_a as

    ξ_A = ∑_l,mζ_A S,   ξ_p = r ∑_l,m(
      ζ_(e)D̂_pS
      +
      ζ_(o)ϵ_pqD̂^pS
      )
      .

From these Fourier transformation rules,
Eqs. (<ref>)–(<ref>)
are given by

    _h_AB
      -
      _h_AB   =   D̅_Aξ_B
          +
          D̅_Bξ_A
          =
          ∑_l,m 2 D̅_(Aζ_B) S
          ,
      
    _h_Ap
      -
      _h_Ap   =   D̅_Aξ_p
          +
          D̂_pξ_A
          -
          2/r (D̅_Ar) ξ_p
       =   
          r
          ∑_l,m[
          (
          1/rζ_A
          + D̅_Aζ_(e)
          - 1/r (D̅_Ar) ζ_(e)) D̂_pS
          .
          
                  .
         +
         (
         D̅_Aζ_(o)
         - 1/r (D̅_Ar) ζ_(o)) ϵ_pqD̂^pS
         ]
         ,
      
    _h_pq
      -
      _h_pq   =   D̂_pξ_q
          +
          D̂_qξ_p
          +
          2 r (D̅^Ar) γ_pqξ_A
       =   
          r^2∑_l,m[
          4/r(
          -  l (l+1)/2ζ_(e)
          + (D̅^Ar) ζ_A)
          1/2γ_pq S
          .
          
                  .
         +
         2/rζ_(e)(
         D̂_pD̂_qS
         -
         1/2γ_pqD̂^rD̂_rS
         )
          .
          
                  .
         -
         1/rζ_(o) 2ϵ_r(qD̂_p)D̂^rS
         ]
         .

Here, we used the property of the mode function S is an eigen
function of the Laplacian
D̂_rD̂^r:=γ^rsD̂_rD̂_s as 

    D̂_rD̂^rS = - l(l+1) S,    S=Y_lm.









  §.§.§ l≥ 2 perturbations








For l≥ 2 modes, the set of mode functions

    {
      S,
      D̂_pS,
      ϵ_pqD̂^qS,
      1/2γ_pq S,
      (
      D̂_pD̂_q - 1/2γ_pqD̂_rD̂^r) S,
      2 ϵ_r(pD̂^rD̂_q) S
      }

is a linear-independent set of the tensor field of the rank 0, 1, and
2, even if S=Y_lm.
Then, we may compare
Eqs. (<ref>)–(<ref>) and
Eqs. (<ref>)–(<ref>).
As the result of this comparison, we obtain the gauge-transformation
rules for the mode functions
{h̃_AB,h̃_(e1)A,h̃_(o1)A,h̃_(e0),h̃_(e2),h̃_(o2)}
as

    _h̃_AB
      -
      _h̃_AB   =   
          2 D̅_(Aζ_B)
          ,
      
    _h̃_(e1)A
      -
      _h̃_(e1)A   =   1/rζ_A
          + D̅_Aζ_(e)
          - 1/r (D̅_Ar) ζ_(e)
          ,
      
    _h̃_(o1)A
      -
      _h̃_(o1)A   =   D̅_Aζ_(o)
          - 1/r (D̅_Ar) ζ_(o)
          ,
      
    _h̃_(e0)
      -
      _h̃_(e0)   =   4/r(
          -  l (l+1)/2ζ_(e)
          + (D̅^Ar) ζ_A)
          ,
      
    _h̃_(e2)
      -
      _h̃_(e2)   =   2/rζ_(e)
          ,
      
    _h̃_(o2)
      -
      _h̃_(o2)   =   
         -
         1/rζ_(o)
          .

From these gauge-transformation rules, it is well-known that we can
construct gauge-invariant variables and the gauge-dependent variables
and Conjecture <ref> is valid for
l≥ 2 modes.








  §.§.§ l=1 perturbations







As shown in the Appendix A in Ref. <cit.>, for
l=1 modes,

    D̂_pY_1m≠ 0≠ϵ_pqD̂^qY_1m

but

    (
      D̂_pD̂_q - 1/2D̂_rD̂^r) Y_1m
      =
      0
      =
      2 ϵ_r(pD̂_q)D̂^rY_1m
      .

In this case, the Fourier transformation
(<ref>)–(<ref>)
of the metric perturbation with S_δ=S=Y_lm is given by

    h_AB   =   ∑_mh̃_AB S
          ,
      
    
      h_Ap   =   ∑_m r h̃_(e1)AD̂_pS
          +
          ∑_m r h̃_(o1)Aϵ_pqD̂^qS
          ,
      
    
      h_pq   =   ∑_mr^2/2γ_pqh̃_(e0) S
          .

The gauge-transformation rules
(<ref>)–(<ref>)
for each m mode are given by

    _h̃_AB
      -
      _h̃_AB   =   
          2 D̅_(Aζ_B)
          ,
      
    _h̃_(e1)A
      -
      _h̃_(e1)A   =   1/rζ_A
          + D̅_Aζ_(e)
          - 1/r (D̅_Ar) ζ_(e)
          ,
      
    _h̃_(o1)A
      -
      _h̃_(o1)A   =   D̅_Aζ_(o)
          - 1/r (D̅_Ar) ζ_(o)
          ,
      
    _h̃_(e0)
      -
      _h̃_(e0)   =   
          -  4/rζ_(e)
          + 4/r (D̅^Ar) ζ_A
          .

Comparing the gauge-transformation rules
(<ref>)–(<ref>)
for l≥ 2 with the gauge-transformation rules
(<ref>)–(<ref>),
it is easy to find the gauge-transformation rule
(<ref>) and
(<ref>) for l≥ 2 do not appear in
the l=1 mode gauge transformation.
This is due to the fact
Eqs. (<ref>), and the reason
why we cannot construct gauge-invariant variables for l=1 mode
perturbations in a similar method to the l≥ 2 case.








  §.§.§ l=0 perturbations








The spherical harmonic function S=Y_lm is constant when l=0.
In this case, the Fourier transformation
(<ref>)–(<ref>)
of the metric perturbation with S_δ=S=Y_lm is given by

    h_AB   =   h̃_AB S
          ,
      
    
      h_Ap   =   
          0
          ,
      
    
      h_pq   =   r^2/2γ_pqh̃_(e0) S
          .

The gauge-transformation rules
(<ref>)–(<ref>)
are given by

    _h̃_AB
      -
      _h̃_AB   =   
          2 D̅_(Aζ_B)
          =
          D̅_Aζ_B
          +
          D̅_Bζ_A
          ,
      
    _h̃_(e0)
      -
      _h̃_(e0)   =   4/r (D̅^Ar) ζ_A
          .

It is easy to see that many gauge-transformation rules in
(<ref>)–(<ref>)
are missing.
This is due to the fact

    D̂_pS = ϵ_pqD̂^qS = 0,   (D̂_qD̂_p- 1/2γ_pqD̂_rD̂^r)S
      =
      2 ϵ_r(pD̂_q)D̂^r S = 0

This is the reason why we cannot construct gauge-invariant variables
for l=0 mode perturbations in a similar method to the l≥ 2 case.










§ L=1 ODD-MODE PERTURBATION IN THE CONVENTIONAL APPROACH










As discussed in
Sec. <ref>, l=1 mode
perturbations are given by
Eqs. (<ref>)–(<ref>).
In particular, among these expressions of the l=1 mode perturbation, 
odd-mode perturbation is given by

    h_AB   =   
          0
          ,   
      h_Ap
      =
          r h̃_(o1)Aϵ_pqD̂^qS
          ,   
      h_pq
      =
          0
          .

As the property of the l=1 spherical harmonics, we obtain the
condition (<ref>).
For odd-mode perturbation we have

    2 ϵ_r(pD̂_q)D̂^rY_1m
      =
      0
      .






Here, we apply the notation which is introduced by
Eqs. (<ref>)–(<ref>) and we obtain

    h̅_AB = 0,   h̅_A^  D = 0,   h̅^pD := γ^pq y^DE r h̃_(o1)Eϵ_qrD̂^rS
    ,   h̅_p^  q = 0,   h̅^pq = 0,

for l=1 odd mode perturbations.
Through this notation of the metric perturbation, the linear
perturbations
(<ref>)–(<ref>)
of the Einstein tensor are given by

    ^(1)G_A^  B   =   
          0
          ,


    ^(1)G_A^  q   =   1/2r[
          D̅_AD̅^Ch̃_(o1)C
          -  D̅_CD̅^Ch̃_(o1)A
          -  2/r (D̅^Cr) D̅_Ch̃_(o1)A.
          
            .
          + 3/r (D̅^Cr) D̅_Ah̃_(o1)C
          -  1/r (D̅_Ar) D̅^Ch̃_(o1)C
          + 1/2r^2 (D̅_Cr) (D̅^Cr) h̃_(o1)A.
          
            .
          -  2/r^2 (D̅_Ar)  (D̅^Cr) h̃_(o1)C
          + 3/2r^2h̃_(o1)A]
          ϵ^qrD̂_rS
          
         ,


    ^(1)G_p^  q   =   
          - 1/2r^2D̅^C( r h̃_(o1)C)
          γ^qr
          2
          ϵ_s(pD̂_r)D̂^sS
      =
          0
          .

The final equality in
Eq. (<ref>) is due to
the property of the spherical harmonics Y_lm
(<ref>).
Furthermore, the expression
(<ref>) is gauge
invariant under the gauge transformation rule
(<ref>).
On the other hand, the D̅^C( r h̃_(o1)C)
in Eq. (<ref>) is not
gauge invariant as shown in below.
However, the gauge-invariance of the linear-order Einstein tensor
^(1)G_a^  b is guaranteed by the identity
2ϵ_s(pD̂_r)D̂^sS=0 for the l=1 odd-mode
perturbations.





For the l=1 odd-mode perturbations, the components of the linearized
energy-momentum tensor are summarized as

    ^(1) T_A^   B
      =
          0
          ,
    
        ^(1) T_A^  q
      =
          1/rT̃_(o1)Aϵ^qrD̂_rS
          ,
    
        ^(1) T_p^  q
      =
          0
         .

Then, the linearized Einstein equations for the odd-mode perturbations
are given by

    D̅_AD̅^Ch̃_(o1)C
         -  D̅_CD̅^Ch̃_(o1)A
           
          -  2/r (D̅^Cr) D̅_Ch̃_(o1)A
          + 3/r (D̅^Cr) D̅_Ah̃_(o1)C
          -  1/r (D̅_Ar) D̅^Ch̃_(o1)C
           
         + 1/2r^2 (D̅_Cr) (D̅^Cr) h̃_(o1)A
         -  2/r^2 (D̅_Ar)  (D̅^Cr) h̃_(o1)C
         + 3/2r^2h̃_(o1)A
       =   
          16πT̃_(o1)A
          ,






Although the equation

    D̅^C( r h̃_(o1)C) = 0

does not appear from
Eq. (<ref>) due to the
fact that 2ϵ_s(pD̂_r)D̂^sY_1m=0,
we may use
Eq. (<ref>) as a
gauge condition.
As noted in
Sec. <ref>, the
gauge-transformation rule for the variable h̃_(o1)C is
given by Eq. (<ref>).
From the gauge transformation rule
(<ref>), we consider the
gauge-transformation rule for the left-hand side of
Eq. (<ref>) as

    D̅^A(r _h̃_(o1)A)
      -
      D̅^A(r _h̃_(o1)A)
         =   
          r D̅^AD̅_Aζ_(o)
          - (D̅^AD̅_Ar) ζ_(o)
          .

Using the background Einstein equation (Eq. (B67) of Appendix B in
Ref. <cit.>), we consider the equation

    r D̅^CD̅_Cζ_(o)
      - 1/r(
      1 - (D̅^Cr)(D̅_Cr)
      ) ζ_(o)
      =
      - D̅^C(r _h̃_(o1)C)

More explicitly, Eq. (<ref>)

    -    ∂_t^2ζ_(o)
      + f ∂_r(f∂_rζ_(o))
      - f/r^2(
      1 - f
      ) ζ_(o)
      =
      - f/rD̅^C(r _h̃_(o1)C)

If we choose ζ_(o) as a special solution to
Eq. (<ref>) or equivalently
Eq. (<ref>),
Eq. (<ref>) is
regarded as a gauge condition in the -gauge, i.e.,

    D̅^C(r _h̃_(o1)C)
         =   
          0
          .






We have to emphasize that this is not a complete gauge fixing, since
there is a room of the choice of ζ_(o) which satisfy the
homogeneous equation of
Eq. (<ref>):

    r D̅^CD̅_Cζ_(o)
      - 1/r(
      1 - (D̅^Cr)(D̅_Cr)
      ) ζ_(o)
      =
      0
      ,

i.e.,

    -    ∂_t^2ζ_(o)
      + f ∂_r(f∂_rζ_(o))
      - f/r^2(
      1 - f
      ) ζ_(o)
      =
      0
      .






Under the gauge choice (<ref>),
Eq. (<ref>) is given by

    [
      D̅_CD̅^C
      - 2/r^2] (r h̃_(o1)A)
      -  2/r^2 (D̅_Ar) (D̅^Cr) (r h̃_(o1)C)
      + 2/r (D̅^Cr) D̅_A(r h̃_(o1)C)
       
       =   
      16π r T̃_(o1)A
      .

Although the variable rh̃_(o1)C is a gauge-dependent
variable which is different from the gauge-invariant variable
rF̃_A, we can obtain the equation
(<ref>) if we
replace the variable rF̃_A in
Eq. (<ref>) with the
variable rh̃_(o1)C.





We also note that the gauge condition
(<ref>) coincides with
Eq. (<ref>), though we have
to replace the gauge-invariant variable r F̃^D with the
gauge-dependent variable r h̃_(o1)C in
Eq. (<ref>) to confirm this
coincidence.
Furthermore, we also take into account the continuity equation
(<ref>) of the l=1 odd-mode
perturbation of the energy-momentum tensor.
Thus, the equations to be solved for the l=1 odd-mode perturbation
are the gauge condition
(<ref>), the
evolution equation
(<ref>), and the
continuity equation (<ref>).
These equations coincide with
Eqs. (<ref>)–(<ref>)
except for the fact that the variable to be obtained is not the
gauge-invariant rF̃^D variable but the gauge-dependent
variable r h̃_(o1)C.
Then, through the same logic in Ref. <cit.>, we
obtain the solution to
the gauge condition (<ref>),
the evolution equation
(<ref>), and the
continuity equation (<ref>) as follows:

    2 r h̃_(o1)Csin^2θ (dx^C)_(a (dϕ)_b)
         =
         6 M r^2[∫ dra_1(t,r)/r^4]
         sin^2θ (dt)_(a(dϕ)_b)
                                      
         + _Vg_ab,
         
         , 
          
         V_a = (β_1t + β_0 + W_(o)(t,r)) r^2sin^2θ (dϕ)_a,

Here, we concentrate only on the m=0 solution, β_1 and
β_0 are constant, and Z_(o)=rf∂_rW_(o)(t,r) is
an arbitrary function which satisfies the equation

    ∂_t^2Z_(o)
      -
      f∂_r(f∂_rZ_(o))
      +
      1/r^2 f [
      2 - 3 (1-f)
      ]
      Z_(o)
      =
      16π f^2T̃_(o1)r
      .

a_1(t,r) in Eq. (<ref>) is
given by Eq. (6.44) in Ref. <cit.>, i.e.,

    a_1(t,r)
         =   
          - 16π/3M r^3 f ∫ dt T̃_(o1)r + a_10
       =   
          - 16π/3M∫ dr r^31/fT̃_(o1)t + a_10
          ,

where a_10 is the perturbative Kerr parameter.





Note that the formal solution described by
Eqs. (<ref>)–(<ref>)
has the same form as the formal solution described by
Eqs. (<ref>)–(<ref>).
However, we have to emphasize that the variable _Ap in
Eq. (<ref>) is gauge invariant
as noted by Eq. (<ref>) and the
vector field V_a in
Eq. (<ref>) is also gauge
invariant.
On the other hand, the variable h̃_(o1)C is still
gauge-dependent.
Actually, there are remaining gauge degrees of freedom of the
generator ζ_(o) which satisfy
Eq. (<ref>) as noted
above.
This is clear from the fact that the gauge transformation rule
(<ref>) with
Eq. (<ref>) gives a still
non-trivial transformation rule.
This remaining gauge degree of freedom is so-called “residual gauge”.
For this reason, there is a possibility that V_a in
Eq. (<ref>) includes this
residual gauge.





To clarify whether V_a in
Eq. (<ref>) includes the
“residual gauge”, we have to confirm
Eq. (<ref>).
Within our rules to compare our gauge-invariant formulation with a
conventional gauge-fixed approach, we regard the terms in V_a in
Eq. (<ref>) which
satisfies Eq. (<ref>) is
the second-kind gauge degree of freedom and we regard these degrees of
freedom as “unphysical degree of freedom.”
To clarify this “unphysical degree of freedom,” we introduce the
indicator function _(o)[*] as

    _(o)[ζ_(o)]
      :=
      -    ∂_t^2ζ_(o)
      + f ∂_r(f∂_rζ_(o))
      - f/r^2(
      1 - f
      ) ζ_(o)
      .

If _(o)[ζ_(o)]=0, we should regard ζ_(o) is
the second-kind gauge and we regard ζ_(o) as “unphysical
degree of freedom.”





We can easily confirm that

    _(o)[(β_1t+β_0)r] = 0
      .

Then, according to our above rule of comparison, we have to conclude
that the rigid rotating term (β_1t+β_0)r in V_a in
Eq. (<ref>) should be regarded
as the second-kind gauge degree of freedom which is “unphysical degree
of freedom.”





Next, we consider the term W_(o)(t,r)r in
Eq. (<ref>).
In this case, the direct calculation of the indicator
_(o)[W_(o)(t,r)r] is useless.
However, it is useful to consider the r-derivative of
_(o)[W_(o)(t,r)r] and we can show that

    r f ∂_r(
      1/r_(o)[W_(o)(t,r)r]
      )
         =   
          -  ∂_t^2Z_(o)
          + f ∂_r(f ∂_rZ_(o))
          -  f/r^2( 2 - 3 (1-f) ) Z_(o)
          ,

where we used Z_(o)=rf∂_rW_(o)(t,r).
Since the left-hand side of
Eq. (<ref>) coincides with
the right-hand side of Eq. (<ref>).
Through Eq. (<ref>), we obtain

    r f ∂_r(
      1/r_(o)[W_(o)(t,r)r]
      )
         =   
          - 16 π f^2T̃_(o1)r
          .

or, equivalently,

    _(o)[W_(o)(t,r)r]
         =   
          - 16 π r ∫ dr f/rT̃_(o1)r
         .

Then, if we consider the case where T̃_(o1)r≠ 0, we
have nonvanishing _(o)[W_(o)(t,r)r].
This means that the W_(o)(t,r) term does not belong to the
second-kind gauge, and we have to regard the degree of freedom
W_(o)(t,r) in Eq. (<ref>)
is a “physical one”.





On the other hand, even in the case where T̃_(o1)r=0,
Eq. (<ref>) is valid.
If W_(o)(t,r) belongs to the second-kind gauge, i.e.,
_(o)[W_(o)(t,r)r]=0,
Eq. (<ref>) yields

    -  ∂_t^2Z_(o)
      + f ∂_r(f ∂_rZ_(o))
      -  f/r^2( 2 - 3 (1-f) ) Z_(o)
      =
      0
      .

However, even if Z_(o) is a solution to
Eq. (<ref>), we cannot directly yield
_(o)[W_(o)(t,r)r]=0.
Therefore, considering the following two sets of function
W_(o)(t,r)r as

    _(o)   :=   {.
           r W_(o)(t,r)
           |
           _(o)[W_(o)(t,r)r] = 0
           }
           ,
      
    _(o)   :=   {.
           r W_(o)(t,r)
           |
           Z_(o) = rf ∂_rW_(o)(t,r),
           .
           
                      .
           - ∂_t^2Z_(o)
           + f ∂_r(f ∂_rZ_(o))
           -
           f/r^2( 2 - 3(1-f) ) Z_(o)
           = 0
           .
           }
           ,

we obtain the relation

    _(o)⊂_(o).

This indicates that a part of solutions to
Eq. (<ref>) should be regarded
as a gauge degree of freedom of the second kind which is the
“unphysical degree of freedom.”





Furthermore, to obtain the explicit solution W_(o)(t,r), we have
to solve Eq. (<ref>) with
appropriate boundary conditions.
Equation (<ref>) is an inhomogeneous
second-order linear differential equation for Z_(o) and its boundary conditions
are adjusted by the homogeneous solutions to
Eq. (<ref>), i.e., the element of the
set of function _(o).
However, a part of this homogeneous solution to
Eq. (<ref>) should be regarded as
an unphysical degree of freedom in the “complete gauge fixing approach”
as mentioned above.
Therefore, in the conventional “complete gauge fixing approach”, the
boundary conditions for Eq. (<ref>) is
restricted.
In this sense, a conventional “complete gauge-fixing approach”
includes a stronger restriction than our proposed gauge-invariant
formulation.










§ L=1 EVEN-MODE PERTURBATION IN THE CONVENTIONAL APPROACH










As discussed in
Sec. <ref>, l=1 mode
perturbations are given by
Eqs. (<ref>)–(<ref>).
In particular, among these expressions of the l=1 mode
perturbations, even-mode perturbation is given by

    h_AB
      =
          h̃_AB S
          ,
        
      h_Ap
      =
          r h̃_(e1)AD̂_pS
          ,
        
      h_pq
      =
          r^2/2γ_pqh̃_(e0) S
          .

As the property of the l=1 spherical harmonics, we obtain the
condition (<ref>).
For even-mode perturbation, we have

    (
      D̂_pD̂_q - 1/2γ_pqD̂_rD̂^r) Y_1m
      =
      0
      .






The gauge-transformation rules for the l=1 even-mode perturbations
are given by Eqs. (<ref>),
(<ref>), and
(<ref>).
Inspecting the gauge-transformation rules
(<ref>) and
(<ref>), we define the
variable H̃ by

    H̃   :=   h̃_(e0) - 4 (D̅^Ar) h̃_(e1)A
          .

The gauge-transformation rule of H̃ is given by

    _H̃
      -
      _H̃   =   
          - 4 (D̅^Ar) D̅_Aζ_(e)
          -  4/r( 1 - (D̅^Ar) (D̅_Ar) ) ζ_(e)
          .

Furthermore, inspecting gauge-transformation rules
(<ref>) and
(<ref>) we also define
the variable H̃_AB by

    H̃_AB   :=   h̃_AB
           -
           D̅_A( r h̃_(e1)B)
           -
           D̅_B( r h̃_(e1)A)
           .

The gauge-transformation of H̃_AB is given by

    _H̃_AB
      -
      _H̃_AB   =   
          - 2 r D̅_AD̅_Bζ_(e)
          + 2 (D̅_AD̅_Br) ζ_(e)
          .






The definitions (<ref>)
and (<ref>) of the
variables H̃ and H̃_AB, respectively, are
analogous to the gauge-invariant variable F̃ and
F̃_AB define by Eqs. (<ref>)
and (<ref>) for l≥ 2 modes,
respectively.
However, the variables H̃ and H̃_AB are not gauge
invariant.
The employment of the variables H̃ and H̃_AB
corresponds to the gauge fixing of the gauge degree of freedom of the
generator ζ_A and the specification of the component
h̃_(e1)A of the metric perturbation.
On the other hand, the gauge-transformation rules
(<ref>),
(<ref>), and
(<ref>) includes the gauge
degree of freedom of the generator ζ_(e).
This gauge degree of freedom appears in the gauge-transformation rules
(<ref>) and
(<ref>).





The linearized Einstein tensor for l=1 even modes in terms of
H̃_AB and H̃ are given as

    y_CB^(1)G_A^  C/S
         =   [
          -  1/2D̅_CD̅^C
          + 3/r^2
          -  1/r (D̅^Cr) D̅_C] H̃_AB
          + D̅_(AD̅^CH̃_B)C
          -  1/2D̅_AD̅_BH̃_C^  C
          
         +  2/r (D̅^Cr) [
         D̅_(AH̃_B)C
         -  1/r (D̅_Cr) H̃_AB]
         -  1/2D̅_AD̅_BH̃
         -  1/r (D̅_(Ar) D̅_B)H̃
          
         + 1/2 y_AB[
         [
         D̅_CD̅^C
         + 3/r (D̅^Cr) D̅_C] H̃
         +
         {D̅_DD̅^D
         -  5/r^2}H̃_C^  C.
         
                  .
         + 1/r (D̅^Cr) {
         2 D̅_CH̃_D^  D
         + 3/r (D̅_Cr) H̃_D^  D}
         -  D̅^CD̅^DH̃_CD.
         
                  .
         - 2/r (D̅^Dr) {
         2 D̅^CH̃_CD
         + 1/r (D̅^Cr) H̃_CD}]
         ,


    ^(1)G_A^  q   =   [
          1/2r^2D̅^CH̃_AC
          -  1/2r^2D̅_AH̃_C^  C
          + 1/2r^3 (D̅_Ar) H̃_C^  C
         -  1/4r^2D̅_AH̃] D̂^qS
         ,


    γ_qr^(1)G_p^  r   =   [
          + 1/4D̅_CD̅^CH̃
          + 1/2r (D̅^Cr) D̅_CH̃
         + 1/2{
         + D̅_CD̅^C
         + 1/r (D̅^Cr) D̅_C
         -  1/r^2}H̃_D^  D.
         
            .
         - 1/2{D̅_C
         + 2/r (D̅_Cr)
         }D̅_DH̃^CD] γ_pq S
          
          
         -  1/2r^2H̃_C^  C(
         D̂_pD̂_qS
         - 1/2γ_pqD̂^rD̂_r S
         )
         .

Since there is the identity
(<ref>) for l=1 mode
perturbations, the last term of
Eq. (<ref>)
does not appear for l=1 even modes.
Furthermore, the expressions
(<ref>) and
(<ref>) are
gauge invariant under the gauge-transformation rules
(<ref>) and
(<ref>).
Even in
Eq. (<ref>)
the component γ_qr^(1)G_p^  r is gauge invariant
except for the last term in
Eq. (<ref>).
Actually, the variable H̃_C^  C is gauge dependent as
shown below.
However, the gauge-invariance γ_qr^(1)G_p^  r is
guaranteed by the identity
(<ref>). 





On the other hand, the l=1 even components of them are given by

    ^(1) T_A^   B   =   
          S
          T̃_A^   B
          ,


    γ_qr^(1) T_A^  r   =   1/rT̃_(e1)AD̂_qS
          ,


    γ_qr^(1) T_p^  r   =   T̃_(e0)1/2γ_pq S
         +
         T̃_(e2)(
         D̂_pD̂_qS
         -
         1/2γ_pqD̂_rD̂^rS
         )

Here again, we note that
Eq. (<ref>) satisfies as
a mathematical identity for l=1 mode perturbations.
Therefore, the last term of
Eq. (<ref>) does not appear
for l=1 even modes.





If equation (<ref>) is
not satisfied, we have the equation

    [
         -  1/2r^2H̃_C^  C]
         (
         D̂_pD̂_qS
         - 1/2γ_pqD̂^rD̂_r S
         )
      =
        8πT̃_(e2)(
         D̂_pD̂_qS
         -
         1/2γ_pqD̂_rD̂^rS
         )

as one of the components of the linearized Einstein equation.
However, Eq. (<ref>)
implies that this equation is identically satisfied.
Therefore, this equation does not restrict H̃_C^  C nor
T̃_(e2) and there is no other restriction of them.
Here, we note that the tensor field H̃_AB is not
gauge-invariant as shown in
Eq. (<ref>).
Since we may freely choose H̃_C^  C and
T̃_(e2), we choose T̃_(e2)=0 by hand and choose

    H_C^  C = 0.

as a gauge condition.





Actually, from the gauge-transformation rule
(<ref>), we obtain

    y^AB_H̃_AB
      -
      y^AB_H̃_AB   =   
          - 2 r [
          -  1/f∂_t^2ζ_(e)
          + ∂_r( f ∂_rζ_(e))
          - 1-f/r^2ζ_(e)]
          .

Then, if we choose ζ_(e) as a special solution to the equation

    1/2r y^AB_H̃_AB   =   
          -  1/f∂_t^2ζ_(e)
          + ∂_r( f ∂_rζ_(e))
          - 1-f/r^2ζ_(e)
          ,

we may regard that _H̃_C^C=0 in the
-gauge.
We have to note that this gauge fixing
(<ref>) is not complete
gauge fixing.
There is a remaining degree of freedom in the choice of ζ_(e)
as the homogeneous solution ζ_(e)h to the wave equation

    -  1/f∂_t^2ζ_(e)h
      + ∂_r( f ∂_rζ_(e)h)
      - 1-f/r^2ζ_(e)h
      =
      0
      .






Due to the gauge condition (<ref>),
the components
(<ref>)–(<ref>)
of the linearized Einstein tensor and components
(<ref>)–(<ref>)
of the linear-order energy-momentum tensor yield the linearized
Einstein equations.
The (A,p)-components (equivalently, (p,B)-components) of the
linearized Einstein equations are given by

    D̅^CH̃_AC
      -  1/2D̅_AH̃
      =
      16 π r
      T̃_(e1)A
      .

Since the tensor H̃_AC is traceless due to the gauge
condition (<ref>), equation
(<ref>)
coincides with Eq. (<ref>).





From the trace part of the (p,q)-component of the linearized
Einstein equation and
Eq. (<ref>)
yields the component (<ref>) of the
continuity equation of the linearized energy-momentum tensor.





The trace part of (A,B)-component of the linearized Einstein
equation with
Eq. (<ref>)
gives

    D̅_CD̅^CH̃
         + 2/r (D̅_Cr) D̅^CH̃
         -  4/r^2 (D̅^Cr) (D̅^Dr) H̃_CD
      =
          16 π(
          T̃_C^  C
          +  4 (D̅^Dr) T̃_(e1)D)
          .

Since H̃_CD is a traceless tensor,
Eq. (<ref>)
coincides with the l=1 version of
Eq. (<ref>) with the source term
(<ref>).
However, we have to emphasize that H̃ nor H̃_CD
are not gauge invariant as shown above.





Finally, the traceless part of (A,B)-component of the linearized
Einstein equation with
Eqs. (<ref>) and
(<ref>)
yields

    [
          -  D̅_CD̅^C
          -  2/r (D̅^Cr) D̅_C
          + 4/r (D̅^CD̅_Cr)
          + 2/r^2] H̃_AB
         + 4/r (D̅^Cr) D̅_(AH̃_B)C
          
         -  2/r (D̅_(Ar) D̅_B)H̃
       =   
          16 π[
          T̃_AB
          -
          1/2 y_ABT̃_C^  C
         -  2 (
          D̅_(A(r T̃_(e1)B))
          - 1/2 y_ABD̅^C(r T̃_(e1)C)
          )
          .
          
                .
         + 2 y_AB (D̅^Dr) T̃_(e1)D]
          ,

Since the variable H_AB is traceless due to the gauge condition
(<ref>),
Eq. (<ref>)
coincides with the l=1 version of
Eq. (<ref>) with the
source term (<ref>).





In addition to these linearized Einstein equations, the following
linearized perturbations of the continuity equation should be
satisfied

    D̅^CT̃_C^  B
         + 2/r (D̅^Dr)T̃_D^   B
         -  2/rT̃_(e1)^B
         -  1/r (D̅^Br) T̃_(e0)
         =
         0
         ,
         
          D̅^CT̃_(e1)C
         + 3/r (D̅^Cr) T̃_(e1)C
         + 1/2rT̃_(e0)
         =
         0
         .






Together with the gauge condition
(<ref>),
Eqs. (<ref>)–(<ref>)
coincide with
Eqs. (<ref>)–(<ref>)
with l=1 in our gauge-invariant formulation developed in Refs. <cit.>.
In Ref. <cit.>, we derived the formal solution to
Eqs. (<ref>)–(<ref>)
with l=1 as Eq. (<ref>) with
Eqs. (<ref>),
(<ref>), and m=0.
Due to the coincidence of the set of equations, this formal solution
should be the m=0 formal solution to
Eqs. (<ref>), and Eqs. (<ref>)–(<ref>).
Then, as the m=0 solution, we obtain

    _ab   :=   
           H_ABcosθ (dx^A)_(a(dx^B)_b) + r^2/2 H cosθγ_ab
       =   _Vg_ab
          - 16π r^2/3(1-f)[
          f^2{1+f/2T̃_rr + r f ∂_rT̃_rr -
          T̃_(e0) - 4 T̃_(e1)r} (dt)_a (dt)_b.
          
                            .
          + 2r/f{∂_tT̃_tt - 3f(1-f)/2rT̃_tr} (dt)_(a(dr)_b).
          
                            .
          + r/f{∂_rT̃_tt - 3(1-3f)/2rfT̃_tt} (dr)_a (dr)_b)
          .
          
                            .
          + r^2T̃_ttγ_ab] cosθ
         ,

where γ_ab = γ_pq(dx^p)_a(dx^q)_b,
V_a=g_abV^b is the vector field defined by

    V_a := - r ∂_tΦ_(e)cosθ (dt)_a +
      (Φ_(e) - r ∂_rΦ_(e)) cos (dr)_a - r Φ_(e)sinθ (dθ)_a

and Φ_(e) is a solution to the equation

    - 1/f∂_t^2Φ_(e) + ∂_r[f ∂_rΦ_(e)]
      - 1-f/r^2Φ_(e)
      =
      16π r/3(1-f) S_(Φ_(e)),
      
          
         S_(Φ_(e)) = 1/2 r ∂_t T_tr - 1/2
         r ∂_rT̃_tt + 1-4f/2fT̃_tt
         - f/2T̃_rr - f T̃_(e1)r
         .






As in the case of the l=1 odd-mode perturbation, the tensor field
_ab in Eq. (<ref>) is
gauge invariant.
Then, V_a in Eq. (<ref>),
which defined by
Eq. (<ref>) is also
gauge invariant.
Therefore, we regarded V_a in
Eq. (<ref>) as the first kind gauge.
On the other hand, the tensor field _ab in
Eq. (<ref>) is gauge-dependent in
the sense of the second kind due to the gauge-transformation rules
(<ref>) and
(<ref>) for the components
H̃ and H̃_AB, respectively.
Actually, there is a remaining gauge degree of freedom of the generator
ζ_(e) which satisfies
Eq. (<ref>) as noted
above.
This remaining gauge degree of freedom is so-called “residual
gauge”.
For this reason, there is a possibility that V_a in
Eq. (<ref>) includes this “residual
gauge”, while V_a in
Eq. (<ref>) is gauge invariant.





Since we already fixed the gauge-degree of freedom ζ_(e) so that
Eq. (<ref>) is satisfied, the
remaining gauge degree of freedom ζ_(e)h must satisfy the
equation (<ref>).
Therefore, to clarify whether V_a in
Eq. (<ref>) includes the “residual
gauge,” we have to confirm
Eq. (<ref>).
Within our rules to compare our gauge-invariant formulation with a
conventional gauge-fixed approach, we regard the term in V_a in
Eq. (<ref>) satisfies
Eq. (<ref>) is the
second-kind gauge degree of freedom and we regard this degree of
freedom as an “unphysical degree of freedom.”
As in the case of the l=1 odd-mode perturbations, we introduce the
indicator function _(e)[*] as

    _(e)[ζ_(e)h]
      :=
      -  1/f∂_t^2ζ_(e)h
      + ∂_r( f ∂_rζ_(e)h)
      - 1-f/r^2ζ_(e)h
      .

If _(e)[ζ_(e)]=0, we should regard ζ_(e) is
the second-kind gauge degree of freedom and we regard ζ_(e) as
an “unphysical degree of freedom.”





To clarify the second-kind gauge degree of freedom, we consider the
gauge-transformation rule of the tensor field _ab through the
gauge-transformation rules
(<ref>) and
(<ref>) as

    __ab
      -
      __ab
       :=   (
           _H_AB
           -
           _H_AB) cosθ (dx^A)_(a(dx^B)_b)
           + r^2/2(
           _H
           -
           _H
           ) cosθγ_ab


    =   
          - 2 r [
          ∂_t^2ζ_(e)h
          -  f(1-f)/2r∂_rζ_(e)h
          +  f(1-f)/2r^2ζ_(e)h] cosθ
          (dt)_a(dt)_b
          
          - 4 r [
          ∂_t∂_rζ_(e)h
          -  1-f/2rf∂_tζ_(e)h] cosθ
         (dt)_(a(dr)_b)
          
          - 2 r [
          ∂_r^2ζ_(e)h
          + 1-f/2rf∂_rζ_(e)h
          -  1-f/2r^2fζ_(e)h] cosθ
         (dr)_a(dr)_b
          
         - 2 r^2(
         f ∂_rζ_(e)h
         + 1/r (1-f) ζ_(e)h) cosθγ_ab
       =   _Wg_ab
          ,

where

    W_a   =   
          - r ∂_tζ_(e)hcosθ (dt)_a
          + ( ζ_(e)h - r ∂_rζ_(e)h) cosθ(dr)_a
          - r ζ_(e)hsinθ (dθ)_a
          .

Comparing Eq. (<ref>) with the
generator (<ref>) and
Eq. (<ref>) with the generator
(<ref>), there is the
possibility that the Lie derivative term _Vg_ab in the
solution (<ref>) is “residual gauge
degree of freedom” with the identification

    Φ_(e) = ζ_(e)h.

For this reason, we check the indicator
(<ref>).
From Eq. (<ref>), we obtain

    _(e)[Φ_(e)]
      :=
      16π r/3(1-f) S_(Φ_(e))
      .

Therefore, the variable Φ_(e) should be regarded as the
second-kind gauge degree of freedom and is the “unphysical degree of
freedom” if S_(Φ_(e))=0, while in the non-vacuum case
S_Φ_(e)≠ 0, the indicator
(<ref>) yields
_(e)[Φ_(e)]≠ 0.
This means that Φ_(e) is not the second-kind gauge degree of
freedom but is the “physical degree of freedom” in the non-vacuum
case S_Φ_(e)≠ 0.
Due to this existence of the source term, the identification
(<ref>) is impossible in the case of the
non-vacuum situation.
Rather, this term is gauge-invariant in the sense of the second kind,
and we regard this term as a gauge-degree of freedom of the first
kind in the non-vacuum case.





Although we have _(e)[Φ_(e)]= 0 in the case where
S_Φ_(e)=0 and we should regard Φ_(e) is a “gauge
degree of freedom of the second kind”,
Eq. (<ref>) indicates that

    _(e)[Φ_(e)]
      =
      -  1/f∂_t^2Φ_(e)
      + ∂_r( f ∂_rΦ_(e))
      - 1-f/r^2Φ_(e)
      =
      0
      .

This coincides with the left-hand side of
Eq. (<ref>).
Therefore, as in the case of the l=1 odd-mode perturbations, we
consider the following two sets of function Φ_(e) as

    _(e)   :=   {.
           Φ_(e)|
           _(o)[Φ_(e)] = 0
           }
           ,
      
    _(e)   :=   {.
           Φ_(e)|
           -  1/f∂_t^2Φ_(e)
           + ∂_r( f ∂_rΦ_(e))
           - 1-f/r^2Φ_(e)
           =
           0
           }
           .

From Eq. (<ref>), we
obtain the relation

    _(e) = _(e).

This indicates that any homogeneous solution (without source term) to
Eq. (<ref>) should be
regarded as a gauge degree of freedom of the second kind which is
the “unphysical degree of freedom.”





On the other hand, to obtain the explicit solution Φ_(e) in the
case S_Φ_(e)≠ 0, which is regarded as a “physical degree
of freedom,” we have to solve
Eq. (<ref>) with appropriate
boundary conditions.
Equation (<ref>) is an inhomogeneous
second-order linear differential equation for Φ_(e) and its boundary conditions
are adjusted by the homogeneous solutions to
Eq. (<ref>), i.e., the
element of the set of function _(e).
However, any homogeneous solution to
Eq. (<ref>) should be
regarded as an “unphysical degree of freedom” in the “complete
gauge fixing approach” as mentioned above.
Therefore, in the conventional “complete gauge fixing approach”, we
have to impose the boundary conditions for
Eq. (<ref>) using a homogeneous
solution, which is regarded as an “unphysical degree of freedom,” to
obtain a “physical solution” to
Eq. (<ref>) with nonvanishing source
term S_Φ_(e)≠ 0.
This situation is a dilemma.
In this sense, as in the case of l=1 odd-mode perturbations, a
conventional “complete gauge-fixing approach” includes a stronger
restriction than our proposed gauge-invariant formulation.










§ L=0 MODE PERTURBATION IN THE CONVENTIONAL APPROACH














 §.§ Einstein equations for l=0 mode perturbations









Here, we consider the l=0 mode perturbations which are described by
Eqs. (<ref>)–(<ref>).
Substitution of
Eqs. (<ref>)–(<ref>)
into
Eqs. (<ref>)–(<ref>),
we obtain the l=0 mode perturbations of the linearized Einstein
tensor.
We use the background Einstein equations (Eq. (B67) and (B68) of
Appendix B in the Part I paper <cit.>).
Then, we obtain the non-trivial components of the l=0 mode Einstein
equation ^(1)G_a^  b=8π^(1)T_a^  b are  given by

    -  1/2D̅_CD̅^Ch̃_A^  B
          + 1/2D̅^BD̅^Ch̃_AC
          + 1/2D̅_AD̅_Ch̃^BC
          -  1/2D̅_AD̅^Bh̃_C^  C
          
         -  1/r (D̅^Cr) D̅_Ch̃_A^  B
         + 1/r (D̅^Cr) D̅^Bh̃_AC
         +  1/r (D̅_Cr) D̅_Ah̃^BC
         -  2/r^2 (D̅^Cr) (D̅_Cr) h̃_A^  B
          
         -  1/2r (D̅^Br) D̅_Ah̃_(e0)
         -  1/2r (D̅_Ar) D̅^Bh̃_(e0)
         -  1/2D̅_AD̅^Bh̃_(e0)
         + 2/r^2h̃_A^  B
          
         + y_A^  B(
          -  1/2D̅_CD̅_Dh̃^CD
         + 1/2D̅_CD̅^Ch̃_D^  D
         -  2/r (D̅_Dr) D̅_Ch̃^CD
         + 1/r (D̅^Cr) D̅_Ch̃_D^  D.
         
                  .
         -  1/r^2 (D̅_Cr) (D̅_Dr) h̃^CD
         + 3/2r^2 (D̅^Dr) (D̅_Dr) h̃_C^  C
         -  3/2r^2h̃_C^  C.
         
                  .
         + 1/2r^2h̃_(e0)
         + 3/2r (D̅^Cr) D̅_Ch̃_(e0)
         + 1/2D̅_CD̅^Ch̃_(e0))
         
       =   
             8 πT̃_A^   B
             ,
         
          D̅_CD̅^Ch̃_D^  D
         -  D̅_CD̅_Dh̃^CD
         -  2/r (D̅_Cr) D̅_Dh̃^CD
         + 1/r (D̅^Cr) D̅_Ch̃_D^  D
          
         + 1/2D̅_CD̅^Ch̃_(e0)
         + 1/r (D̅^Cr) D̅_Ch̃_(e0)
       =   
         8 πT̃_(e0)
         .






Here, we decomposition of the component h̃_AB as

    h̃_AB =: _AB + 1/2 y_ABh̃_C^  C.

In terms of the variables defined by
Eq. (<ref>), the linearization of
the Einstein equations for l=0 mode are summarized as follows.
The trace of ^(1)G_A^  B=8π^(1)T_A^  B
for l=0 mode is given by

    -  1/r^2h̃_C^  C
         - 2/r (D̅_Cr) (
         + D̅_D^DC
         + 1/r (D̅_Dr)  ^CD)
         
          
         + 1/2D̅_CD̅^Ch̃_(e0)
         + 2/r (D̅_Cr) D̅^Ch̃_(e0)
         + 1/r^2h̃_(e0)
         =
          8 πT̃_C^   C
          .

The traceless part of ^(1)G_A^  B=8π^(1)T_A^  B for l=0 mode is given by

    - 1/2[
         D̅_CD̅^C
         + 2/r (D̅^Cr) D̅_C
         + 4/r^2 (D̅^Cr) (D̅_Cr)
         - 4/r^2] _AB
          
         + D̅_(AD̅^C_B)C
         -  1/2 y_ABD̅_CD̅_D^CD
         + 2/r (D̅^Cr) (
         D̅_(A_B)C
         -  1/2 y_ABD̅^D_DC)
         
          
         + 1/r(
         (D̅_(Ar) D̅_B)h̃_E^  E
         -  1/2 y_AB (D̅^Cr) D̅_Ch̃_D^  D)
         - 1/2[
         D̅_AD̅_Bh̃_(e0)
         - 1/2 y_ABD̅_CD̅^Ch̃_(e0)]
         
          
         - 1/r[
         (D̅_(Ar) D̅_B)h̃_(e0)
         -  1/2 y_AB (D̅^Cr) D̅_Ch̃_(e0)]
         =
          8 π[
          T̃_AB - 1/2 y_ABT̃_C^   C]
          .

Furthermore, ^(1)G_p^  q=8π^(1)T_p^  q
for l=0 mode is given by

    1/2D̅_CD̅^Ch̃_(e0)
          + 1/r (D̅^Cr) D̅_Ch̃_(e0)
         - D̅_CD̅_D^CD
         -  2/r (D̅_Cr) D̅_D^CD
         + 1/2D̅_CD̅^Ch̃_D^  D
       =   
          8 πT̃_(e0)
          .










 §.§ Gauge-fixing for l=0 mode perturbations









We consider the gauge-transformation rules
(<ref>) and
(<ref>) of the mode
coefficient h̃_AB and h̃_(e0).
If these gauge-transformation rules are those of the second kind, we
should exclude these gauge degrees of freedom through some
gauge-fixing procedure, because the degree of freedom of the
second-kind gauge is the unphysical degree of freedom.





In the static chart in which the metric y_AB is given by
Eq. (<ref>).
Through the static chart
(<ref>), the
gauge-transformation rule
(<ref>) is given by

    _h̃_(e0)
      -
      _h̃_(e0)
      =
      4/r f  ζ_r
      .

Then we may choose ζ_r so that

    _h̃_(e0)   =   _h̃_(e0)
          + 4/r f  ζ_r
          =
          0
          ,

i.e.,

    ζ_r
      =
      - r/4f_h̃_(e0)
      .

Through this gauge-fixing, we may regard h̃_(e0)=0 in the
-gauge.





As the gauge-fixing for h̃_AB, we fix the gauge so that
h̃_AB is traceless.
This gauge-fixing so that h̃_AB is traceless makes easy to
compare with our gauge-invariant expression in Refs. <cit.>.
The trace of the gauge transformation rue
Eq. (<ref>), we obtain

    y^AB_h̃_AB
      -
      y^AB_h̃_AB   =   _h̃_C^  C
          -
          _h̃_C^  C
          =
          2
          D̅^Cζ_C
          .

In the static chart
(<ref>) of the
Schwarzschild spacetime, the gauge transformation rule
(<ref>) is given by

    _h̃_C^  C
      -
      _h̃_C^  C   =   
          - 2 f^-1∂_tζ_t
          + 2 ∂_r(fζ_r)
          .

From this gauge-transformation rule and the gauge fixing
(<ref>), we may fix the gauge
degree of freedom ζ_t so that

    0
         =   _h̃_C^  C
       =   _h̃_C^  C
          - 2 f^-1∂_tζ_t
          + 2 ∂_r(
          - r/4_h̃_(e0))
          .

Through this gauge-fixing, we may regard h̃_C^  C=0.
However, we have to emphasize that the degree of freedom of the choice
of the generator ζ_t remains the degree of freedom of an
arbitrary function of r.





Thus, through the gauge-fixing
(<ref>) and
(<ref>), we
may regard

    h̃_(e0) = h̃_C^  C = 0

Under the gauge-fixing condition
(<ref>) linearization of the Einstein equations
for l=0 mode are summarized as

    - (D̅_Cr) D̅_D^DC
      -  1/r (D̅_Cr) (D̅_Dr)  ^CD
      =
      4 π r T̃_C^   C
      ,


    - 1/2[
         D̅_CD̅^C
         + 2/r (D̅^Cr) D̅_C
         + 4/r^2 (D̅^Cr) (D̅_Cr)
         - 4/r^2] _AB
         + D̅_(AD̅^C_B)C
          
         -  1/2 y_ABD̅_CD̅_D^CD
         + 2/r (D̅^Cr) (
         + D̅_(A_B)C
         -  1/2 y_ABD̅^D_DC)
         
       =   
          8 π[
          T̃_AB - 1/2 y_ABT̃_C^   C]
          ,

and

    - D̅_CD̅_D^CD
      - 2/r (D̅_Cr) D̅_D^CD
      =
      8 πT̃_(e0)
      .










 §.§ Component expression of the linearized l=0 gauge-fixed
  field equations









Here, we consider the component representations of
Eqs. (<ref>)–(<ref>).
To do this, we consider the components of the traceless tensor
_AB as

    _AB
      =:
      X_(e)[(dt)_A (dt)_B + f^-2 (dr)_A(dr)_B]
      +
      2 Y_(e) (dt)_(A (dr)_B)
      .

Here, we use the background Einstein equation (B65) of Appendix B in
the Part I paper <cit.>, i.e.,

    ∂_rf = 1-f/r.

Then, we obtain

    ∂_r^2f
      =
      ∂_r(1-f/r)
      =
      - 2(1-f)/r^2






Through Eqs. (<ref>) and
(<ref>),
Eq. (<ref>) is given by

    r f ∂_tY_(e)
      - r f ∂_rX_(e)
      -  f X_(e)
      =
      - 4 π r^2( T̃_tt - f^2T̃_rr)
      .

The (A,B)=(t,t) and (A,B)=(r,r) components of
Eq. (<ref>) yield the
same equation as

    r f ∂_tY_(e)
      =
      4 π r^2( T̃_tt + f^2T̃_rr)
      .

The (A,B)=(t,r) component of
Eq. (<ref>) is given by

    ∂_tX_(e)
      =
      8 π r f T̃_tr
      .

In terms of the components (<ref>),
Eq. (<ref>) is
given by

    -  ∂_t^2X_(e)
      -  f^2∂_r^2X_(e)
      -  2/r f^2∂_rX_(e)
      + 2 f^2∂_t∂_rY_(e)
      + 1/r f(1+f) ∂_tY_(e)
      =
      8 π f^2T̃_(e0)
      .






On the other hand, the even-mode perturbation of the divergence of the
energy-momentum tensor is given by Eqs. (<ref>)
and (<ref>).
However, Eq. (<ref>) does not appear due to
D̂_pS=ϵ_pqD̂^qS=0.
Then, the non-trivial l=0 components of the divergence of the
energy-momentum tensor are given by

    D̅^CT̃_C^  B
      + 2/r (D̅^Dr)T̃_D^   B
      -  1/r (D̅^Br) T̃_(e0)
      =
      0
      .

The B=t component of
Eq. (<ref>) is given by

    ∂_tT̃_tt
      -  f^2∂_rT̃_rt
      - 1/r f (1+f) T̃_rt
      =
      0
      .

and B=r component of
Eq. (<ref>) is given by

    -  f ∂_tT̃_tr
      + f^3∂_rT̃_rr
      + 1/2r (1-f) T̃_tt
      + 1/2r f^2 (3+f) T̃_rr
      -  1/r f^2T̃_(e0)
      =
      0
      .






Substituting
Eq. (<ref>)
into
Eq. (<ref>),
we obtain

    ∂_r(r X_(e))
      =
      r ∂_rX_(e)
      + X_(e)
      =
      8 πr^2/fT̃_tt
      .

Substituting
Eqs. (<ref>),
(<ref>), and (<ref>)
into
Eq. (<ref>),
we obtain

    0
         =   
          -  f ∂_tT̃_tr
          + f^3∂_rT̃_rr
          + 1/2r (1-f) T̃_tt
          + 1/2r f^2 (3+f) T̃_rr
          -  1/r f^2T̃_(e0)
          .

This coincides with
Eq. (<ref>)
which indicates that
Eq. (<ref>)
does not give any new information other than
Eq. (<ref>).





Here, we consider the integrability of
Eq. (<ref>)
and
(<ref>) as follows:

    ∂_r(∂_t(rX_(e)))
      -
      ∂_t(∂_r(rX_(e)))
         =   ∂_r(
          8 π r^2 f T̃_tr)
          -
          ∂_t(
          8 πr^2/fT̃_tt)
          
       =   
          - 8 π r^2[
          + ∂_tT̃_tt
          - f^2∂_rT̃_tr
          - 1/r f (1+f) T̃_tr]
          
       =   
          0.

Here, we used
Eq. (<ref>) in
the last equality.
This means that the t-component
(<ref>) of the
continuity equation guarantees the integrability of
Eqs. (<ref>)
and
(<ref>).
Then, from
Eq. (<ref>),
we may write the solution to
Eqs. (<ref>)
and
(<ref>)
under the integrability condition
(<ref>) as

    X_(e)
      =
      1/r[2M_1 + 8π∫ dr r^2/fT̃_tt]
      ,

where M_1 is the constant of integration.
This M_1 corresponds to the perturbation of the Schwarzschild mass
parameter.





On the other hand, in the linearized Einstein equation, there is no
equation for ∂_rY_(e) which guarantees the integrability
condition for
Eq. (<ref>).
However, we may write the solution to the equation
(<ref>) as

    Y_(e)
      =
      4π r/f∫ dt ( T̃_tt + f^2T̃_rr)
      +
      Y_(e)0(r)
      .

where Y_(e)0(r) is an arbitrary function of r.
There is no equation that determines the arbitrary function
Y_(e)0(r) of r within the linearized Einstein equations.





From the definition (<ref>) of the metric
perturbation _AB and the solutions
(<ref>) and
(<ref>), we
obtain

    _AB   =   2/r(M_1 + 4π∫ dr r^2/fT̃_tt)
          ((dt)_A (dt)_B + f^-2 (dr)_A(dr)_B)
          
          
          2 [
          4π r ∫ dt ( 1/fT̃_tt + f T̃_rr)
          +
          Y_(e)0(r)
          ] (dt)_(A (dr)_B)
          .

Here, the components of the energy-momentum tensor satisfy the
continuity equations
(<ref>) and
(<ref>).





To interpret the term of Y_(e)0(r) in
Eq. (<ref>), we consider the term
_Vg_ab with the generator V_a whose components are
given by

    V_a = V_t(r) (dt)_a.

Then, the nonvanishing components of _Vg_ab are given
by

    _Vg_tr   =   
          f ∂_r( 1/f V_t)
          .

Choosing V_t so that

    f ∂_r( 1/f V_t)
      =
      Y_(e)0(r)
      ,   
      V_t
      =
      f ∫ dr 1/f Y_(e)0(r)
      ,

we obtain

    _Vg_tr   =   
          Y_(e)0(r)
          .

Then, the solution (<ref>) is given by

    _ab   =   2/r(M_1 + 4π∫ dr r^2/fT̃_tt)
          ((dt)_a (dt)_b + f^-2 (dr)_a(dr)_b)
          
          
         +
         8π r ∫ dt ( 1/fT̃_tt + f T̃_rr)
         (dt)_(a (dr)_b)
         +
         _Vg_ab
         ,

where

    V_a
      =
      (
      f ∫ dr 1/f Y_(e)0(r)
      ) (dt)_a
      .

As noted just after
Eq. (<ref>),
there is still the remaining degree of freedom of gauge whose
generator is given by

    ζ_a = ζ_t(r) dt_a,

where ζ_t(r) is an arbitrary function of r.
Therefore, we may regard the degree of freedom of V_a given in
Eq. (<ref>) can be
eliminated as the second-kind gauge degree of freedom
(<ref>).
According to our rule of the comparison between our gauge-invariant
formulation and a “conventional complete gauge-fixing approach,” we
have to regard that the degree of freedom of V_a given in
Eq. (<ref>) is
“unphysical.”





Thus, as the “physical solution” for l=0-mode linearized Einstein
equation based of a “conventional complete gauge-fixing approach,” we
obtain

    h_ab   =   2/r(M_1 + 4π∫ dr r^2/fT̃_tt)
          ((dt)_a (dt)_b + f^-2 (dr)_a(dr)_b)
          
          
         +
         8π r ∫ dt ( 1/fT̃_tt + f T̃_rr)
         (dt)_(a (dr)_b)
         .

This coincides with the solution obtained in
Refs. <cit.> except for the
terms of the Lie derivative of the background metric g_ab.
Therefore, we conclude that the l=0 solution except for the
terms of the Lie derivative of the background metric g_ab
obtained in Refs. <cit.> can
be also obtained as
(<ref>)
through a complete gauge-fixing approach.





The difference between our solution in
Refs. <cit.> is in the term
of the Lie derivative of the background metric.
As shown above, according to our rule of comparison between our
gauge-invariant formulation and a “conventional complete gauge-fixing
approach,” all terms of the Lie derivative of the background metric
should be regarded as the second-kind gauge degree of freedom,
and these are “unphysical degree of freedom” in the above solution
(<ref>).
On the other hand, in our gauge-invariant formulation developed in
Refs. <cit.>,
l=0,1-mode metric perturbations are given in a gauge-invariant form
and the terms of the Lie derivative of the background metric is
included in these gauge-invariant variables.
Since the second-kind gauge degree of freedom is completely excluded
in our gauge-invariant formulation, we cannot regard these terms of
the Lie derivative of the background metric as an “unphysical degree
of freedom.”
Therefore, we regard the terms of the Lie derivative of the
background metric in these gauge-invariant variables as
first-kind gauges that have some physical meaning.
This point is the essential difference of the solutions in
Refs. <cit.>
and the above solutions based on a “conventional complete
gauge-fixing approach.”
This difference leads to the confusion of the interpretation of the
perturbative Tolman Bondi solution, as shown in the next subsection.









 §.§ Comparing with Lemaître-olman-Bondi solution












  §.§.§ Perturbative expression of the LTB solution on
  Schwarzschild background spacetime








Here, we consider the Lemaître-Tolman-Bondi (LTB)
solution <cit.> which is an exact solution to
the Einstein equation with the matter field

    T_ab = ρ u_au_b,    u_a = - (dτ)_a,

and the metric

    g_ab   =   
          - (dτ)_a (dτ)_b
          + (∂_Rr)^2/1+f(R) (dR)_a(dR)_b
          + r^2γ_ab
          ,
            
          r = r(τ,R)
         .

This solution is a spherically symmetric solution to the Einstein equation.
The function r=r(τ,R) satisfies the differential equation

    (∂_τr)^2
      =
      F(R)/r
      +
      f(R)
      .

Here, we note that F(R) is an arbitrary function of R representing
the dust matter's initial distribution.
f(R) is also an arbitrary function of R that represents the
initial distribution of the energy of the dust field in the Newtonian
sense.
The solution to Eq. (<ref>) is given in the
three cases


  (i) f(R)>0 : 
  
    r = F(R)/2f(R)(coshη - 1),   τ_0(R) - τ  = F(R)/2f(R)^3/2 (sinhη -η)
        ,


  (ii) f(R)<0 : 
  
    r = F(R)/-2f(R)(1-cosη),   τ_0(R) - τ = F(R)/2(-f(R))^3/2 (η - sinη)
        ,


  (iii) f(R)=0 : 
  
    r = (9F(R)/4)^1/3[τ_0(R)-τ]^2/3
        .


The energy density ρ is given by

    8πρ = ∂_RF/(∂_Rr)r^2.

The LTB solution includes the three arbitrary functions f(R),
F(R), and τ_0(R).





Here, we consider the vacuum case ρ=0.
In this case, from Eq. (<ref>), we have

    ∂_RF = 0,    F = 2M
      ,

where M is the Schwarzschild metric with the mass parameter M.
Furthermore, we consider the case f(R)=0.
Here, we chose τ_0=R, i.e., ∂_Rτ_0=1.
In this case, Eq. (<ref>) yields

    (dR)_a   =   
          (dτ)_a
          +
          (2M/r)^-1/2
          (dr)_a
          .

Then, the metric (<ref>) is given by

    g_ab   =   
          - (dτ)_a (dτ)_b
          + (∂_Rr)^2 (dR)_a(dR)_b
          + r^2γ_ab
       =   
          -
          f
          (dt)_a(dt)_b
         +
         f^-1
         (dr)_a
         (dr)_b
         + r^2γ_ab
         ,
            
          f = 1 - 2M/r
          ,

where (dt)_a is defined by

    (dt)_a
      :=
      (dτ)_a
      -
      f^-1(1-f)^1/2
      (dr)_a
      .

We also note that the degree of freedom of the choice of the
coordinate R is completely fixed through the choice τ_0=R,
though there remains a degree of freedom of the choice of
R=R(R̃) in the exact solution (<ref>).





Now, we consider the perturbation of the Schwarzschild spacetime which
is derived by the exact LTB solution (<ref>) so that

    F(R)
         =   
          2 [
          M
          +
          ϵ
          m_1(R)
          ]
          + O(ϵ^2)
          ,
          
    
      f(R)
         =   
          0
          +
          ϵ
          f_1(R)
          + O(ϵ^2)
          ,
          
    τ_0(R)
         =   
          R
          +
          ϵτ_1(R)
          + O(ϵ^2)
          .

Through these perturbations
(<ref>)–(<ref>), we
consider the perturbative expansion of the function r which is
determined by Eq. (<ref>):

    r(τ,R) = r_s(τ,R) + ϵ r_1(τ,R)
      + O(ϵ^2)
      .

Here, the function r_s(τ,R) is given by
Eq. (<ref>), i.e.,

    r_s(τ,R)
      =
      r(τ,R)
      =
      (9M/2)^1/3[R-τ]^2/3
      .

In Eqs. (<ref>) and
(<ref>), we chose the background
value of the function τ_0(R) to be R.





Through this perturbative expansion, we evaluate
O(ϵ^1) perturbation of Eq. (<ref>)
through Eq. (<ref>) as

    (1-f)^1/2
         (∂_τr_1)
         +
         m_1(R)/r
         -
         M/r^2
         r_1
         +
         1/2
         f_1(R)
         =
         0
         ,

where we used Eq. (<ref>)
and the replacement r_s→ r.
The solution to Eq. (<ref>) is
given by

    r_1   =   (M/6)^1/3m_1(R)/M[R-τ]^2/3
          -
          3/20(6/M)^1/3
          f_1(R)
          [R-τ]^+4/3
          
          +
          B(R)
          [R-τ]^-1/3
          .

From the comparison with Eq. (<ref>),
B(R) is the perturbation of the τ_1(R) as
τ_0(R)=R+τ_1(R) in the exact solution
(<ref>)–(<ref>).
Furthermore, the solution
(<ref>) can be also derived
from the exact solution
(<ref>)–(<ref>).
From Eq. (<ref>), the perturbative dust energy
density given by

    8πρ = 2∂_Rm_1(R)/(∂_Rr)r^2.






Through the perturbative solution
(<ref>), the metric
(<ref>) is given by

    g_ab   =   
          - (dτ)_a (dτ)_b
          +
          (∂_Rr)^2
          (dR)_a(dR)_b
          +
          r^2γ_ab
          
         +
         ϵ[
         (
         2(∂_Rr_1)
         -
         f_1
         (∂_Rr)
         )
         (∂_Rr)
         (dR)_a(dR)_b
         +
         2
         r
         r_1γ_ab]
         + O(ϵ^2)
         
       =:   
           g_ab^(0)
           +
           ϵ_h_ab
           + O(ϵ^2)
           .

As shown in Eq. (<ref>),
the background metric g_ab^(0) is given by the Schwarzschild
metric in the static chart.
On the other hand, the linear order perturbation _h_ab
(in the gauge _ϵ) is given by

    _h_ab
      :=
      (
      2
      (∂_Rr_1)
      -
      f_1(R)
      (∂_Rr)
      )
      (∂_Rr)
      (dR)_a(dR)_b
      +
      2
      r
      r_1γ_ab
      .

Here, we fixed the second-kind gauge so that

    _ϵ :
      (τ,R,θ,ϕ)∈_ph↦
      (τ,R,θ,ϕ)∈.

Through this second-kind gauge choice and the choice of the background
radial coordinate R in
Eq. (<ref>), the degree of
freedom of the choice of the radial coordinate R=R(R̃) is
also completely fixed even in the linearized version of the exact
solution (<ref>), which
though there remains a degree of freedom of the choice of
R=R(R̃) in the exact solution (<ref>).





Of course, if we employ the different gauge choice
_ϵ from the above gauge-choice
_ϵ, we obtain the different expression of the
metric perturbation _h_ab.
Actually, we may choose _ϵ as the identification of

    _ϵ :
      (τ+ϵξ^τ(τ,R),R+ϵξ^R(τ,R),θ,ϕ)∈_ph↦
      (τ,R,θ,ϕ)∈.

In this identification, the metric on _ph pulled back to
 is given by

    __ϵg_ab   =   __ϵg_ab
          +
          ϵ_ξg_ab^(0)
          +
          O(ϵ^2)
          
       =   
          g_ab^(0)
          +
          ϵ(
          _h_ab
          +
          _ξg_ab^(0))
          +
          O(ϵ^2)
           ,

where
ξ^a=ξ^τ(∂_τ)^a+ξ^R(∂_R)^a is
the generator of second-kind gauge transformation
_ϵ→_ϵ.








  §.§.§ Expression of the perturbative LTB solution in static chart








Here, we consider the expression of the linear perturbation
_h_ab given by
Eq. (<ref>).
From Eqs. (<ref>) and
(<ref>) with F=2M, we obtain

    (dR)_a   =   
          (dt)_a
          +
          f^-1
          (1-f)^-1/2
          (dr)_a
          ,
            
          f = 1 - 2M/r
          ,
          
    
      (dτ)_a   =   
          (dt)_a
          +
          f^-1
          (1-f)^1/2
          (dr)_a
          .






First, we consider the perturbation of the energy-momentum tensor of
the matter field.
In the case of the LTB solution, the matter field is characterized by
the dust field whose energy-momentum tensor
(<ref>) is given by

    T_ab = ρ u_a u_b,    u_a = - (dτ)_a,
         u^a = (∂_τ)^a.

In our case, the linearized Einstein equation gives
Eq. (<ref>), i.e.,

    8πρ
      =
      2∂_Rm_1(R)/(∂_Rr)r^2
      =
      ∂_Rm_1(R)/4π r^2 (1-f)^-1/2
      .

On the other hand, substituting
Eq. (<ref>) into
Eq. (<ref>), we obtain

    T_ab   =   ρ (dτ)_a (dτ)_b
       =   ρ(
          (dt)_a
          +
          f^-1
          (1-f)^1/2
          (dr)_a)
          (
          (dt)_b
          +
          f^-1
          (1-f)^1/2
          (dr)_b)
          
       =   ρ
          (dt)_a
          (dt)_b
          +
          ρ(1-f)^1/2/f
          2
          (dt)_(a (dr)_b)
          +
          ρ1-f/f^2
          (dr)_a
          (dr)_b
          .

Then, we obtain the components of the energy-momentum tensor for the
static coordinate (t,r) as

    T̃_tt
      =
      ρ
      ,
        T̃_tr
      =
      (1-f)^1/2/fρ
      ,
        T̃_rr
      =
      1-f/f^2ρ
      ,
        T̃_(e0)
      =
      0
      .

From this component, we can confirm the continuity equations (<ref>)
and (<ref>).





As derived in Ref. <cit.>, the linearized
Tolman-Bondi solution (<ref>)
with the Schwarzschild background is given by

    _h_ab   =   2m_1(R)/r[
          (dt)_a (dt)_b + 1/f^2 (dr)_a (dr)_b]
          +
          2-f/f (1-f)^1/2m_1(R)/r
          2 (dt)_(a (dr)_b)
          
         + _V_(LTB)g_ab
         ,

where V_(LTB)a are given by

    V_(LTB)a
      :=
      [(1-f)^1/2 r_1 + 1/2 f ∫ dt f_1(R)] (dt)_a
      +
      r_1/f (dr)_a
      .

As shown in Ref. <cit.>, the l=0 solution
(<ref>) with
the components (<ref>) of the linearized
energy-momentum tensor realize the first line of
Eq. (<ref>).
In this sense, the solutions
(<ref>) of
the l=0 mode perturbations are justified by the LTB solutions.





However, we emphasize that the linearized LTB solution
(<ref>) does have the
term _V_LTBg_ab.
As noted by Eqs. (<ref>) and
(<ref>), we can always
eliminate the terms of the Lie derivative of the background metric as
the second-kind gauge-degree of freedom.
Therefore, according to our rule of the comparison, the term
_V_(LTB)g_ab should be regarded as the second-kind
gauge-degree of freedom in the “conventional gauge-fixing approach”
discussed in this paper.
In this case, we have to regard that the perturbation f_1(R) of
the initial distribution f(R) of the energy of dust field in
Eq. (<ref>) is an “unphysical degree of
freedom” in spite that the behavior of the LTB solution crucially
depends on the signature of the function f(R) as shown in
Eqs. (<ref>)–(<ref>).










§ SUMMARY AND DISCUSSIONS










In this paper, we have discussed comparison of our gauge-invariant
formulation for l=0,1 perturbations on the Schwarzschild background
spacetime proposed in the series of papers <cit.>
and a “conventional complete gauge-fixing approach.”
It is well-known that we cannot construct gauge-invariant variables
for l=0,1 mode perturbations through a similar manner to the
construction of the gauge-invariant variables for l≥ 2 mode
perturbations if we use the decomposition formula
(<ref>)–(<ref>) with the spherical
harmonic functions Y_lm as the scalar harmonics S_δ.
In our gauge-invariant formulation for l=0,1 perturbations on the
Schwarzschild background spacetime, we proposed the introduction of
the singular harmonic function at once.
Due to this, we can construct gauge-invariant variables for
l=0,1-mode perturbations in a similar manner to the l≥ 2
modes of perturbations.
After deriving the mode-by-mode perturbative Einstein equations in
terms of the gauge-invariant variables, we impose the regularity on
the introduced singular harmonics when we solve the derived Einstein
equations.
Based on this proposal, we derived formal solutions to the
l=0,1-mode linearized Einstein equations without the specification
of the components of the linear perturbation of the energy-momentum tensor <cit.>.
Our proposal enables us to develop higher-order perturbations of the
Schwarzschild spacetime <cit.>.
Furthermore, we check that our derived solutions realized the
linearized version of the LTB solution and non-rotating
C-metric <cit.>.
In this sense, we conclude that our proposal is physically
reasonable.
On the other hand, it is often said that “gauge-invariant
formulations in general-relativistic perturbations are equivalent to
complete gauge-fixing approaches.”
For this reason, we check this statement through the comparison of our
gauge-invariant formulation and a “conventional complete gauge-fixing
approach” in which we use the spherical harmonic functions Y_lm as
the scalar harmonics S_δ from the starting point.





After reviewing the concept of “gauges” in general relativistic
perturbation theories and our proposed gauge-invariant formulation for
the l=0,1-mode perturbations and our derived l=0,1-mode solutions,
we considered l=1 odd-mode perturbations, l=1 even-mode
perturbations, and l=0 even-mode perturbations, separately.
As a result, it is shown that we can actually derive similar solutions
even in the treatment of the “conventional complete gauge-fixing
approach.”
However, we have to emphasize that the derived solutions are slightly
different from those derived based on our gauge-invariant formulation
from a conceptual point of view.





In the case of l=1 odd-mode perturbations, we obtained the
formal solution to the linearized Einstein equation through our proposed
gauge-invariant formulation.
This formal solution includes the term of the Lie derivative of the
background metric.
In our gauge-invariant formulation, we describe the solutions
only through gauge-invariant variables.
Therefore, we should regard the term of the Lie derivative of the
background metric is gauge invariant.
On the other hand, in the conventional gauge-fixing approach where we
use the spherical harmonics Y_lm from the starting point, we
cannot construct gauge-invariant variable for l=1 odd-mode
perturbations in a similar manner to those for l≥ 2 mode
perturbations.
For this reason, we have to treat gauge-dependent variables for
perturbations.
However, the linearized Einstein equations and the continuity
equations of the linearized energy-momentum tensor for l=1 odd-mode
perturbations in terms of these gauge-dependent variables have the
completely same form that is derived through our gauge-invariant
formulation.
Therefore, the same formal solutions derived through our
gauge-invariant formulation should be the formal solutions to these
linearized Einstein equations and the continuity equation of the
linearized energy-momentum tensor in terms of gauge-dependent
variables.
As noted above, we treat gauge-dependent variables in the
conventional gauge-fixing approach.
Therefore, the above Lie derivative terms of the background metric may
include the gauge degree of freedom of the second kind, which should be
regarded as an “unphysical degree of freedom.”
Checking the residual gauge degree of freedom, we conclude that the above
Lie derivative terms of the background metric include the second-kind
gauge degree of freedom.
However, in our formal solution, there is a variable that should be
obtained by solving the l=1 Regge-Wheeler equation.
We conclude that the solution to this l=1 Regge-Wheeler equation is
not the gauge degree of freedom of the second kind but a physical
degree of freedom.
Furthermore, we have to impose appropriate boundary conditions to
solve this l=1 Regge-Wheeler equation.
Since the l=1 Regge-Wheeler equation is an inhomogeneous linear
second-order partial differential equation, the boundary conditions
for an inhomogeneous linear second-order partial differential equation
are adjusted by the homogeneous solutions to this linear second-order
partial differential equation.
According to the check of the residual gauge degree of freedom, we
have to conclude that a part of homogeneous solutions to this equation
is the gauge degree of freedom of the second kind.
We have to eliminate this part from our consideration because this
gauge degree of freedom is “unphysical.”
This is the restriction of the boundary conditions of the linearized
Einstein equations.





In the case of l=1 even-mode perturbations, the situation is
worse than the l=1 odd-mode case.
As in the l=1 odd-mode case, we obtained the
formal solution to the linearized Einstein equation through our proposed
gauge-invariant formulation.
This formal solution includes the term of the Lie derivative of the
background metric, which is gauge-invariant within our proposed
gauge-invariant formulation.
On the other hand, in the conventional gauge-fixing approach where we
use the spherical harmonics Y_lm from the starting point, we
cannot construct gauge-invariant variable for l=1 even-mode
perturbations in a similar manner to those for l≥ 2 mode
perturbations.
For this reason, we have to treat gauge-dependent variables for
perturbations.
As in the l=1 odd-mode case, the linearized Einstein equations and
the continuity equations of the linearized energy-momentum tensor for
l=1 even-mode perturbations in terms of these gauge-dependent
variables have the completely same form that is derived through our
gauge-invariant formulation.
Therefore, the same formal solutions derived through our
gauge-invariant formulation should be the formal solutions to these
linearized Einstein equations and the continuity equation of the
linearized energy-momentum tensor in terms of gauge-dependent
variables.
As noted above, we treat gauge-dependent variables in the
conventional gauge-fixing approach as in the l=1 odd-mode case.
Therefore, the above Lie derivative terms of the background metric may
include the second-kind gauge degree of freedom, which should be
regarded as an “unphysical degree of freedom.”
However, in our formal solution, there is a variable that should be
obtained by solving the l=1 Zerilli equation.
We conclude that the solution to this l=1 Zerilli equation is
not the gauge degree of freedom of the second kind but the physical
degree of freedom in the non-vacuum case.
Furthermore, we have to impose appropriate boundary conditions to
solve this l=1 Zerilli equation in the non-vacuum case.
Since the l=1 Zerilli equation is an inhomogeneous linear
second-order partial differential equation.
The boundary conditions for an inhomogeneous linear second-order
partial differential equation are adjusted by the homogeneous
solutions to this linear second-order partial differential equation.
According to the check of the residual gauge degree of freedom, we
have to conclude that all homogeneous solutions to this equation
are the gauge degree of freedom of the second kind, i.e., all
homogeneous solution to the l=1 Zerilli equation is “unphysical.”
We have to eliminate these homogeneous solutions from our
consideration because these are regarded as “unphysical.”
Due to this situation, we have to say that we have to impose the
boundary conditions using these “unphysical degrees of freedom” to
obtain the “physical solution” through l=1 Zerilli equation.
This is a dilemma.





In the case of l=0 even-mode perturbations, we obtain the complete
gauge fixed solution.
This solution does not include any term of the Lie derivative of the
background metric.
All terms of the Lie derivative of the background metric are regarded
as the gauge degree of freedom of the second kind.
On the other hand, the solution that is derived through our proposed
gauge-invariant formulation includes the terms of the Lie derivatives
of the background metric which is regarded as the first-kind gauge in
our gauge-invariant formulation, i.e., these are “physical.”
This difference leads a problem when we compare the derived solution
with the linearized LTB solution.
In the linearized LTB solution, the initial energy distribution
f_1(R) of the dust field in the Newtonian sense is included in the
terms of the Lie derivative of the background metric.
When we realize this linearized exact solution by the solution
obtained by our gauge-invariant formulation, this initial energy
distribution is regarded as a physical degree of freedom.
On the other hand, when we realize this linearized exact solution
through the solution obtained by a conventional complete gauge-fixing
approach, we have to regard this initial energy distribution of the dust
field as the gauge degree of freedom of the second kind and have to
regard it “unphysical.”
Since the behavior of the exact LTB solution crucially depends on this
initial energy distribution of the dust field, it is not natural to
regard that this initial degree of freedom is unphysical.





In summary, we have to conclude that there is a case where the
boundary conditions and initial conditions are restricted in the
conventional complete gauge-fixing approach where we use the
decomposition of the metric perturbation by the spherical harmonics
Y_lm from the starting point.
On the other hand, such a situation does not occur in our proposed
gauge-invariant formulation.
As a theory of physics, this point should be regarded as the
incompleteness of the conventional complete gauge-fixing approach where
we use the decomposition of the metric perturbation by the spherical
harmonics Y_lm from the starting point.
This is the main result of this paper.





Let us discuss why such a difference occurs between our proposed
gauge-invariant formulation and a conventional complete gauge-fixed
approach where we use the decomposition of the metric perturbation by
the spherical harmonics Y_lm from the starting point.
The aim of the introduction of the singular harmonic functions in our
proposed gauge-invariant formulation is to increase the degree of
freedom to clarify the distinction between the gauge degree of freedom
of the second kind and the physical degree of freedom.
As emphasized in Sec. <ref>, if we choose
S=Y_lm, we have D̂_pS = ϵ_pqD̂^qS =
0,
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0 for l=0
modes and
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
= 2 ϵ_r(pD̂_q)D̂^rS = 0 for l=1 modes.
This is the essential reason for the fact that we cannot construct
gauge-invariant variables for l=0,1 mode metric perturbation in a
similar manner to the derivation of gauge-invariant variables for
l≥ 2 modes.
Due to these vanishing vector- or tensor-harmonics, the associated
mode coefficients do not appear, and we cannot construct
gauge-invariant variables for l=0,1-modes in a similar manner to the
case of l≥ 2 modes.
In our series of
papers <cit.>,
we regarded that this is due to the lack of the degree of freedom.
For this reason, in our proposed gauge-invariant formulation, we
introduced singular harmonic functions k_Δ̂ and
k_Δ̂+̂2̂m for l=0 and l=1 modes, respectively.
Owing to the introduction of these singular harmonic functions, we have
D̂_pS ≠ 0 ≠ ϵ_pqD̂^qS and
(D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
≠ 0 ≠ 2 ϵ_r(pD̂_q)D̂^rS for l=0
modes and (D̂_pD̂_q-1/2γ_pqD̂^rD̂_r)S
≠ 0 ≠ 2 ϵ_r(pD̂_q)D̂^rS for l=1
modes.
Thanks to these non-vanishing vector- or tensor-harmonics, the
associated mode coefficients do appear, and we can construct
gauge-invariant variables for l=0,1-modes in a similar manner
to the case of l≥ 2 modes.
This leads to the development of the gauge-invariant perturbation
theory for all
modes <cit.>
and the development of the higher-order gauge-invariant
perturbations <cit.>.





As noted in the Part I paper <cit.>, the
decomposition using the spherical harmonics Y_lm from the starting
point corresponds to the imposition of the regular boundary condition
on S^2 on the metric perturbations from the starting point.
In this sense, our introduction of the singular harmonic functions
corresponds to the change of the boundary conditions on S^2.
As shown in the Part I paper <cit.>, if we
introduce this change of boundary conditions on S^2, we can
clearly distinguish the gauge degree of freedom of the second kind and
the physical degree of freedom, i.e., we can easily construct
gauge-invariant variables for l=0,1-mode perturbations.
This indicates that the imposition of the boundary conditions on
S^2 and the construction of the gauge-invariant variables does not
commute as the procedure of calculations.
This is the appearance of the non-locality of l=0,1-mode
perturbations as pointed out in Ref. <cit.>.
Due to these reasons, we reached the conceptual difference between
the l=0,1-mode solutions discussed in this paper.
Namely, in the conventional complete gauge-fixing approach where we use
the decomposition by the harmonic function Y_lm from the starting
point, the degree of freedom of the metric perturbations is not
sufficient so that we can distinguish the gauge degree of freedom of
the second kind, i.e., “unphysical modes” and “physical modes.”
In spite of this situation of the lack of degree of freedom, if we
dare to carry out the “complete gauge-fixing” as the “elimination
of unphysical modes,” this “complete gauge-fixing” leads to the
restriction of the boundary conditions and initial conditions as
shown in this paper.





On the other hand, in our proposed gauge-invariant formulation, there
are no conceptual difficulties, such as the restriction of the boundary
conditions and initial conditions pointed out in this paper, due to the
sufficient degree of freedom of the metric perturbations.
Incidentally, due to this sufficient degree of freedom of the metric
perturbations through the introduction of the singular harmonic
functions, our proposed gauge-invariant variables are equivalent to
variables of the complete gauge-fixing within our proposed formulation
in which the degree of freedom of the metric perturbations is
sufficiently extended.
Furthermore, we can develop higher-order perturbation theory without
gauge-ambiguities if we apply our proposal and there are wide
applications of the higher-order gauge-invariant perturbations on
the Schwarzschild background spacetime as briefly discussed in
Ref. <cit.>.
We leave these further developments of the application of our
formulation to concrete problems of perturbations on the Schwarzschild
background spacetime as future works.




















§ LINEARIZED EINSTEIN TENSOR










In this Appendix, we derive components of the linearized Einstein
tensor.
Although similar formulae are derived in Appendix C in the Paper
I <cit.> in the gauge-invariant form.
However, in this paper, we have to derive the components of the
Einstein tensor with the spherically background spacetime without any
gauge fixing.
In this case, the formulation of the “gauge-ready formulation”
proposed in Ref. <cit.> in the context of cosmological
perturbation theories is an appropriate formulation.
Therefore, in this Appendix, we derive the components of the
linear-order Einstein tensor without any gauge fixing following the
philosophy of the “gauge-ready formulation” 'in
Ref. <cit.>.
The derived formulae in the philosophy of “gauge ready formulation”
is useful in the ingredient of this manuscript.





Here, we consider the metric perturbation as

    _ Xg̅_ab
      =
      g_ab + ϵ_ Xh_ab + O(ϵ^2).

The connection C^c_  ab between the covariant derivative
∇̅_a associated with the metric g̅_ab and the
covariant derivative ∇_a associated with the metric g_ab
is given by

    C^c_  ab = 1/2g̅^cd(
      ∇_ag̅_db + ∇_bg̅_da - ∇_dg̅_ab)
      ,

where g̅^ab is the inverse of g̅_ab.
Here, we expand the connection C^c_  ab with respect to
ϵ as

    C^c_  ab =
      ϵ^(1)C^c_  ab
      +
      O(ϵ^2)
      .

Then, we have

    ^(1)C^c_  ab
      =
      1/2 g^cd(
      ∇_ah_db + ∇_bh_da - ∇_dh_ab)
      .

The relation between the Riemann curvature
R̅_abc^      d associated with the metric
g̅_ab and the curvature R_abc^      d associated
with the background metric g_ab is given by

    R̅_abc^      d
      =
      R_abc^      d
      - 2 ∇_[a^C^d_  b]c
      + 2 C^e_  c[aC^d_  b]e,

Then, we have

    R̅_abc^      d
      =
      R_abc^      d
      - 2 ϵ∇_[a^^(1)C^d_  b]c
      +
      O(ϵ^2)
      .

The perturbative expansion of the Ricci tensor R̅_ac is given
by

    R̅_ac
      =
      R_ac
      - 2 ϵ∇_[a^^(1)C^b_  b]c
      +
      O(ϵ^2)
      .

The perturbative expansion of the curvature R̅_a^  c is
given by

    R̅_a^  d   =   g̅^cdR̅_ac
       =   g̅^cdR̅_ac
          + ϵ(
          - 2 g^cd∇_[a^^(1)C^b_  b]c
          -  h^cd R_ac)
          +
          O(ϵ^2)
          .

The perturbation of the scalar curvature is given by

    R̅   =   g̅^acR̅_ac
       =   
          R
          + ϵ(
          - h^ac R_ac
          - 2 g^ac∇_[a^^(1)C^b_  b]c)
          +
          O(ϵ^2)
          .

Then, the perturbative expansion of the Einstein tensor
G̅_a^  d is given by

    G̅_a^  d   =   R̅_a^  d - 1/2δ_a^  dR̅
       =   
          G_a^  d
          + ϵ(
          - 2 g^cd∇_[a^^(1)C^b_  b]c
          +  δ_a^  d  g^ec∇_[e^^(1)C^b_  b]c
          -  h^cd R_ac
          +  1/2δ_a^  d h^ec R_ec)
          +
          O(ϵ^2)
          
       =:   
          G_a^  d
          + ϵ^(1)G_a^  d
          +
          O(ϵ^2)
          .

Namely, the 1st-order perturbation of the Einstein tensor is given by

    ^(1)G_a^  d   =   
          -  2 g^cd∇_[a^^(1)C^b_  b]c
          + δ_a^  d  g^ec∇_[e^^(1)C^b_  b]c
          -  h^cd R_ac
          + 1/2δ_a^  d h^ec R_ec
          .

Substituting Eq. (<ref>) into
Eq. (<ref>), we obtain

    ^(1)G_a^  d   =   
          -  1/2∇_b∇^bh_a^  d
          + R_abc^      d h^bc
          
          + 1/2(
          g_ac∇^d_
          - 2 δ_[a^  d∇_c]^) ∇_bh^cb
          - 1/2(
          ∇_a∇^d - δ_a^  d∇_c∇^c) h_b^  b
          
          -  1/2 R_ac h^dc
          + 1/2 R^df h_af
          + 1/2δ_a^  d h^ec R_ec
          .

Since we consider the vacuum background spacetime R_ab=0, we
obtain

    ^(1)G_a^  d   =   
          -  1/2∇_b∇^bh_a^  d
          + R_abc^      d h^bc
          
          + 1/2(
          g_ac∇^d_
          - 2 δ_[a^  d∇_c]^) ∇_bh^cb
          - 1/2(
          ∇_a∇^d - δ_a^  d∇_c∇^c) h_b^  b
          .






To derive the component of ^(1)G_a^  d, we denote the
components of the metric perturbation and the derivative operator as

    h̅_AB := h_AB,   h̅_A^  D := y^DE h_AE,   h̅^pD := γ^pq y^DE h_qE,   
          h̅_p^  q := γ^qr h_pr,   h̅^pq := γ^prγ^qs h_rs,

and

    D̅^C = y^CED̅_E,   D̂^p = γ^pqD̂_q.






Now, the components of ^(1)G_a^  d in terms of the
variable defined by Eq. (<ref>) as follows:

    ^(1)G_A^  B   =   
          -  1/2D̅_CD̅^Ch̅_A^  B
         -  1/2r^2D̂_pD̂^ph̅_A^  B
         -  2/r^2 (D̅^Cr) (D̅_Cr) h̅_A^  B
         + 2/r^2h̅_A^  B
          
          + 1/2D̅^BD̅^Ch̅_AC
          + 1/2D̅_AD̅_Ch̅^BC
         -  1/r (D̅^Cr) D̅_Ch̅_A^  B
          -  1/2D̅_AD̅^Bh̅_C^  C
          
         + 1/r (D̅^Cr) D̅^Bh̅_AC
         +  1/r (D̅_Cr) D̅_Ah̅^BC
          
         + 1/2r^2D̅^BD̂^ph̅_Ap
         + 1/2r^2D̅_AD̂_ph̅^Bp
          
         -  1/2r^2D̅_AD̅^Bh̅_r^  r
         + 1/2r^3 (D̅_Ar) D̅^Bh̅_r^  r
         + 1/2r^3 (D̅^Br) D̅_Ah̅_r^  r
         -  1/r^4 (D̅_Ar) (D̅^Br) h̅_r^  r
          
         + y_A^  B(
          -  1/2D̅_CD̅_Dh̅^CD
         + 1/2D̅_CD̅^Ch̅_D^  D
         + 1/2r^2D̂_pD̂^ph̅_C^  C
         -  2/r (D̅_Dr) D̅_Ch̅^CD.
         
                .
         + 1/r (D̅^Cr) D̅_Ch̅_D^  D
         -  1/r^2 (D̅_Cr) (D̅_Dr) h̅^CD
         + 3/2r^2 (D̅^Dr) (D̅_Dr) h̅_C^  C.
         
                .
         -  1/r^2D̅_CD̂_ph̅^pC
         -  1/r^3 (D̅_Cr) D̂_ph̅^Cp.
         
                .
         + 1/2r^2D̅_CD̅^Ch̅_r^  r
         + 1/2r^4D̂_pD̂^ph̅_r^  r
         -  1/2r^3 (D̅^Cr) D̅_Ch̅_r^  r
         -  1/2r^4D̂_pD̂_sh̅^ps.
         
                .
         + 1/2r^4 (D̅_Cr) (D̅^Cr) h̅_r^  r
         -  3/2r^2h̅_C^  C)
         .


    ^(1)G_A^  q   =   1/2r^2D̂^qD̅^Ch̅_AC
          -  1/2r^2D̅_AD̂^qh̅_C^  C
          + 1/2r^3 (D̅_Ar) D̂^qh̅_C^  C
          
          -  1/2r^2D̅_CD̅^Ch̅_A^  q
          -  1/2r^4D̂_rD̂^rh̅_A^  q
          + 1/2r^4h̅_A^  q
          + 1/2r^4D̂^qD̂^ph̅_Ap
          
          + 1/2r^2D̅_AD̅_Dh̅^qD
          -  1/r^3 (D̅_Ar) D̅_Dh̅^qD
          -  1/r^4 (D̅_Ar) (D̅_Dr) h̅^qD
          + 1/r^3 (D̅_Dr) D̅_Ah̅^qD
          
          + 1/2r^4D̅_AD̂_sh̅^qs
          -  1/r^5 (D̅_Ar) D̂_sh̅^qs
          -  1/2r^4D̅_AD̂^qh̅_r^  r
          + 1/r^5 (D̅_Ar) D̂^qh̅_r^  r
          ,


    ^(1)G_p^  q   =   
          -  1/2r^2D̅_CD̅^Ch̅_p^  q
          -  1/2r^4D̂_sD̂^sh̅_p^  q
          + 1/r^3 (D̅_Cr) D̅^Ch̅_p^  q
          -  2/r^4 (D̅_Cr) (D̅^Cr) h̅_p^  q
          + 2/r^4h̅_p^  q
          
         -  1/2r^2D̂_pD̂^qh̅_C^  C
          + 1/2r^2D̂_pD̅_Ch̅^qC
          + 1/2r^2D̂^qD̅^Ch̅_pC
          
         + 1/2r^4D̂_pD̂_rh̅^qr
         + 1/2r^4D̂^qD̂^rh̅_pr
         -  1/2r^4D̂_pD̂^qh̅_r^  r
          
          + γ_p^  q(
          -  1/2D̅_CD̅_Dh̅^CD
          -  1/r (D̅_Cr) D̅_Dh̅^CD
          + 1/2D̅_CD̅^Ch̅_D^  D
          + 1/2r (D̅^Cr) D̅_Ch̅_D^  D.
          
                .
          + 1/2r^2D̂_sD̂^sh̅_C^  C
          -  1/r^2D̅_CD̂_rh̅^Cr.
          
                .
          -  1/2r^4D̂_rD̂_sh̅^sr
          -  3/2r^4h̅_r^  r
          + 1/2r^2D̅_CD̅^Ch̅_r^  r
          -  1/r^3 (D̅^Cr) D̅_Ch̅_r^  r.
          
                .
          + 2/r^4 (D̅_Cr) (D̅^Cr) h̅_r^  r
         + 1/2r^4D̂_sD̂^sh̅_r^  r)
          .













99
LIGO-home-page
  LIGO Scientific Collaboration 2024 home page: https://www.ligo.org/
Virgo-home-page
  Virgo 2024 home page : https://www.virgo-gw.eu/
KAGRA-home-page
  KAGRA 2024 home page: https://gwcenter.icrr.u-tokyo.ac.jp/en/
LIGO-INDIA-home-page
  LIGO INDIA 2024 home page : https://www.ligo-india.in/
ET-home-page
  Einstein Telescope 2024 home page : https://www.et-gw.eu/
CosmicExplorer-home-page
  Cosmic Explorer 2024 home page : https://cosmicexplorer.org/
LISA-home-page
  LISA 2024 home page : https://lisa.nasa.gov/
DECIGO-PTEP-2021
  S. Kawamura, et al., Prog. Theor. Exp. Phys. 2021 (2021), 05A105.
TianQin-PTEP-2021
  J. Mei, et al., Prog. Theor. Exp. Phys. 2020 (2020), 05A107.
Taiji-PTEP-2021
  Z. Luo, et al., Prog. Theor. Exp. Phys. 2020 (2020),05A108.
L.Barack-A.Pound-2019
  L. Barack and A. Pound, Rep. Prog. Phys. 82 (2019) 016904.
K.Nakamura-2021-CQG
  K. Nakamura, “Proposal of a gauge-invariant treatment of
  l=0,1-mode perturbations on Schwarzschild background spacetime”,
  Class. Quantum Grav. 38 (2021), 145010.
K.Nakamura-2021-LHEP
  K. Nakamura, “Formal Solutions of Any-Order Mass, Angular-Momentum,
  anda Dipole Perturbations on the Schwarzschild Background
  Spacetime”, Letters in High Energy Physics 2021 (2021), 215.
K.Nakamura-2021-PartI
  K. Nakamura, “Gauge-invariant perturbation theory on the
  Schwarzschild background spacetime Part I: — Formulation and
  odd-mode perturbations —”, arXiv:2110.13408v8 [gr-qc].
K.Nakamura-2021-PartII
  K. Nakamura, “Gauge-invariant perturbation theory on the
  Schwarzschild background spacetime Part II: — Even-mode
  perturbations —”, arXiv:2110.13512v5 [gr-qc].
K.Nakamura-2021-PartIII
  K. Nakamura, “Gauge-invariant perturbation theory on the
  Schwarzschild background spacetime Part III: — Realization of
  exact solutions —”, arXiv:2110.13519v5 [gr-qc].
T.Regge-J.A.Wheeler-1957
  T. Regge and J. A. Wheeler, Phys. Rev. 108 (1957), 1063.
F.Zerilli-1970-PRL
  F. Zerilli, Phys. Rev. Lett. 24 (1970), 737.
F.Zerilli-1970-PRD
  F. Zerilli, Phys. Rev. D 2 (1970), 2141.
H.Nakano-2019
  H. Nakano, Private note on “Regge-Wheeler-Zerilli formalism” (2019).
V.Moncrief-1974a
  V. Moncrief, Ann. Phys. (N.Y.) 88 (1974), 323.
V.Moncrief-1974b
  V. Moncrief, Ann. Phys. (N.Y.) 88 (1974), 343.
C.T.Cunningham-R.H.Price-V.Moncrief-1978
  C. T. Cunningham, R. H. Price, and V. Moncrief, Astrophys. J. 224 (1978), 643.
Chandrasekhar-1983
  S. Chandrasekhar, The mathematical theory of black holes (Oxford:
  Clarendon Press, 1983).
Gerlach_Sengupta-1979a
  U.H. Gerlach and U.K. Sengupta, Phys. Rev. D 19 (1979), 2268.
Gerlach_Sengupta-1979b
  U.H. Gerlach and U.K. Sengupta, Phys. Rev. D 20 (1979), 3009.
Gerlach_Sengupta-1979c
  U.H. Gerlach and U.K. Sengupta, J. Math. Phys. 20 (1979), 2540.
Gerlach_Sengupta-1980
  U.H. Gerlach and U.K. Sengupta, Phys. Rev. D 22 (1980), 1300.
T.Nakamura-K.Oohara-Y.Kojima-1987
  T. Nakamura, K. Oohara, Y. Kojima,
  Prog. Theor. Phys. Suppl. No. 90 (1987), 1.
Gundlach-Martine-Garcia-2000
  C. Gundlach and J.M. Martín-García, Phys. Rev. D61
  (2000), 084024.
Gundlach-Martine-Garcia-2001
  J.M. Martín-García and C. Gundlach, Phys. Rev. D64
  (2001), 024012.
A.Nagar-L.Rezzolla-2005-2006
  A. Nagar and L. Rezzolla, Class. Quantum Grav. 22 (2005),
  R167; Erratum ibid. 23 (2006), 4297.
K.Martel-E.Poisson-2005
  K. Martel and E. Poisson, Phys. Rev. D 71 (2005), 104003.
K.Nakamura-2003
  K. Nakamura, Prog. Theor. Phys. 110, (2003), 723.
K.Nakamura-2005
  K. Nakamura, Prog. Theor. Phys. 113 (2005), 481.
K.Nakamura-2011
  K. Nakamura, Class. Quantum Grav. 28 (2011), 122001.
K.Nakamura-IJMPD-2012
  K. Nakamura, Int. J. Mod. Phys. D 21 (2012), 124004.
K.Nakamura-2013
  K. Nakamura, Prog. Theor. Exp. Phys. 2013 (2013), 043E02.
K.Nakamura-2014
  K. Nakamura, Class. Quantum Grav. 31 (2014), 135013.
K.Nakamura-2010
  K. Nakamura, Advances in Astronomy, 2010 (2010), 576273.
K.Nakamura-2020
  K. Nakamura, “Second-order Gauge-invariant Cosmological
  Perturbation Theory: Current Status Updated in 2019”, Chapter 1 in
  Theory and Applications of Physical Science, Vol.3
  Ed. M. Rafatullah, (Book Publisher International, 2020), ISBN
  978-93-89816-24-2 (Print); ISBN 978-93-89816-25-9 (eBook),
  Preprint arXiv:1912.12805 [gr-qc].
A.J.Christopherson-K.A.Malik-D.R.Matravers-K.Nakamura-2011
  A. J. Christopherson, K. A. Malik, D. R. -Matravers,
  K. Nakamura, Class. Quantum Grav. 28 (2011), 225024.

K.Nakamura-2006
  K. Nakamura, Phys. Rev. D 74 (2006), 101301(R).
K.Nakamura-2007
  K. Nakamura, Prog. Theor. Phys. 117 (2007), 17.
K.Nakamura-2008
  K. Nakamura, ““Gauge” in General Relativity: – Second-order
  general relativistic gauge-invariant perturbation theory –”,
  in Lie Theory and its Applications in Physics VII
  ed. V. K. Dobrev et al, (Heron Press, Sofia, 2008)
K.Nakamura-2009a
  K. Nakamura, Phys. Rev. D 80 (2009), 124021.
K.Nakamura-2009b
  K. Nakamura, Prog. Theor. Phys. 121 (2009), 1321.
L.Landau-E.Lifshitz-1962
  L. Landau and E. Lifshitz, The Classical Theory of Fields
  (Addison-Wesley, Reading, Mass., 1962).
W.Kinnersley-M.Walker-1970
  W. Kinnersley and M. Walker, Phys. Rev. D 2 (1970), 1359.
J.B.Griffiths-P.Krtous-J.Podolsky-2006
  J. B. Griffiths, P. Krtous, and J. Podolsky, Class. and Quantum
  Grav. 23 (2006), 6745.
R.K.Sachs-1964
  R. K. Sachs, “Gravitational Radiation”, in Relativity,
    Groups and Topology ed. C. DeWitt and B. DeWitt, (New York:
  Gordon and Breach, 1964).
J.M.Stewart-M.Walker-1974
  J. M. Stewart and M. Walker, Proc. R. Soc. London A 341
  (1974), 49.
J.M.Stewart-M.Walker-1990
  J. M. Stewart, Class. Quantum Grav. 7 (1990), 1169.
J.M.Stewart-1990
  J. M. Stewart, Advanced General Relativity (Cambridge
  University Press, Cambridge, 1991).
M.Bruni-S.Matarrese-S.Mollerach-S.Sonego-1997
  M. Bruni, S. Matarrese, S. Mollerach and S. Sonego,
  Class. Quantum Grav. 14 (1997), 2585.
M.Bruni-S.Sonego-CQG1999
  M. Bruni and S. Sonego, Class. Quantum Grav. 16 (1999), L29.
S.Sonego-M.Bruni-1998
  S. Sonego and M. Bruni, Commun. Math. Phys. 193 (1998), 209.
Noh-Hwang-2004
  H. Noh and J. Hwang, Phys. Rev. D 69 (2004), 104011.





 Paper Title: Axion-like Particle Effects on Photon Polarization in High-Energy Astrophysics.





§ INTRODUCTION


The Standard Model (SM) of particle physics provides a satisfactory theoretical baseline able to explain all known processes acting on elementary
particles. However, the SM cannot be considered as the ultimate theory of fundamental interactions since it does not really unify the electroweak and the strong interaction and does not contain gravity. Moreover, the SM describes neither dark matter nor dark energy, which together make up the majority of our Universe. Therefore, the SM represents a low-energy manifestation of a more fundamental theory describing and unifying all four fundamental interactions at the quantum level.

Several proposals have been presented with the aim of extending and completing the SM, such as
multidimensional Kaluza–Klein theories <cit.>, four-dimensional unsymmetrical models <cit.>, and superstring and superbrane theories <cit.>. A remarkable fact is that all these theories invariably predict the existence of axion-like particles (ALPs, for a review, see, e.g., <cit.>).

ALPs represent a generalization of the axion, the pseudo-Goldstone boson originating from the breakdown of the global Peccei–Quinn symmetry U(1)_ PQ, introduced to solve the strong CP problem (see, e.g., <cit.>). The axion is characterized by a strict relationship between its mass and its coupling to photons, and it interacts also with gluons and fermions so that the Peccei–Quinn mechanism can work. Instead, two aspects differentiate ALPs from axions: (i) the ALP mass m_a is unrelated to the two-photon-to-ALP coupling g_aγγ, and (ii) ALPs primarily interact with photons only, while other possible interactions can safely be neglected in the present context. ALPs are currently considered among the best candidates for being the long-sought dark matter <cit.>. Furthermore, ALPs produce two significant effects in the presence of external magnetic fields: (i) photon–ALP oscillations <cit.>, which are similar to the oscillations of massive neutrinos with different flavors with the difference that, in the ALP case, the external magnetic field is necessary for spin compensation between spin-0 ALPs and spin-1 photons, and (ii) the change of the polarization state of . Due to their faint interaction with photons, ALPs can hardly be detected over the limited distances of laboratory experiments. Instead, the astrophysical context allows us to overcome this restriction with photons that can originate at cosmological distances and can cross several magnetized media, where photon–ALP interaction can take place. Therefore, the astrophysical environment appears as the most promising way to study the ALP physics, especially in the high-energy (HE) and in the very-high-energy (VHE) band (see, e.g., <cit.> for a review). 

In particular, photon–ALP oscillations occur in different magnetized media, such as inside active galactic nuclei (AGN; see, e.g., <cit.> and Section <ref>). Specifically, photon–ALP oscillations explain the detection of photons with energies of up to ∼400   GeV originated from flat spectrum radio quasars (FSRQs, an AGN class), while conventional physics prevents any emission for energies above ∼20   GeV (see also Section <ref>): this fact represents the first hint at ALP existence <cit.>. Moreover, photon–ALP interaction solves the anomalous redshift dependence of the spectra of BL Lacs (another AGN class) <cit.>. In particular, within conventional physics, the values assumed by the BL Lac VHE-emitted spectral indexes decrease as the redshift grows, while no redshift dependence is expected, since cosmological effects are not present and selection biases are excluded, as discussed in <cit.>. The introduction of photon–ALP conversion inside the extragalactic space increases the Universe transparency <cit.>, partially avoiding the absorption of VHE photons due to their interaction with the extragalactic background light (EBL, see also Section <ref> ), and makes BL Lac VHE-emitted spectral indexes redshift independent. Since this is the only possibility according to the physical expectation, this result represents the second hint 
 at ALP existence <cit.>. Moreover, photon–ALP conversion inside the blazar jet, in the host galaxy, in the extragalactic space and in the Milky Way produces two features <cit.>: (i) spectral irregularities and (ii) photon excess in blazar spectra at VHE, which can be detected by current and planned observatories, such as ASTRI <cit.>, CTA <cit.>, GAMMA-400 <cit.>, HAWC <cit.>, HERD <cit.>, LHAASO <cit.>, and TAIGA-HiSCORE <cit.>. Furthermore, ALP-induced spectral irregularities produced inside the galaxy cluster turbulent magnetic field have been used to place constraints on the ALP parameter space <cit.>. In addition, photon–ALP interaction provokes modifications on stellar evolution <cit.>; it has been employed inside galaxy clusters as an explanation of the spectral distortions of the continuum thermal emission (T∼2  keV–8   keV) <cit.> and of the unexpected spectral line at 3.55   keV, viewed as dark matter decay into ALPs with their subsequent oscillations into photons <cit.>. ALPs have also been used in order to explain a blazar line-like feature <cit.>. The recent detection of the gamma-ray burst GRB 221009A by the LHAASO Collaboration <cit.> up to 18   TeV strongly challenges conventional physics, while the photon–ALP scenario naturally explains the VHE observation of this GRB, which represents the third hint and the firmest indication to date at ALP existence <cit.> (see also Section <ref> and <cit.>). Moreover, all three above-mentioned hints are derived with the same ALP parameters, which further strengthens the indication itself.

In this review, we concentrate on the other main effect of the photon–ALP interaction, i.e., the change of the polarization state of photons. This ALP effect has attracted less interest so far with respect to that producing spectral modifications. However, the existing and proposed polarimetric missions in both the X-ray band such as IXPE <cit.> (already operative), eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, XPP <cit.> and in the HE range like COSI <cit.> (approved to launch), e-ASTROGAM <cit.>, AMEGO <cit.> allow polarization studies to be carried out with unprecedented sensitivity. Therefore, these observatories represent a unique opportunity to perform additional studies on ALP physics. In particular, the ALP effects on the polarization of photons produced in GRBs have been studied in <cit.>. Attention to various ALP-induced polarization topics regarding different astrophysical sources, such as AGN, white dwarfs, and neutron stars, has been paid in <cit.>. Moreover, the photon–ALP interaction has been demonstrated as a unique way to measure the emitted photon polarization <cit.>. Quite recently, the impact of the photon–ALP interaction on the polarization of photons produced in the central region of typical galaxy clusters or generated at the jet base of generic blazars has been analyzed in <cit.>. Results obtained in <cit.> have been applied to concrete cases: (i) to Perseus and Coma concerning galaxy clusters, as shown in <cit.>, and (ii) to OJ 287, BL Lacertae, Markarian 501, and 1ES 0229+200 regarding blazars, as discussed in <cit.>, showing that, in all the cases, the initial photon polarization is strongly modified by the photon–ALP interaction with the production of features observable by present and planned missions <cit.>. The above-cited studies concerning blazars <cit.> and especially galaxy clusters <cit.> demonstrate that the measure of the polarization degree of photons produced by these astrophysical sources may provide us a new and an additional way to study ALPs and their effects. In fact, whenever the expectations of conventional physics about the photon polarization degree of astrophysical sources are in tension with possible future observations performed by missions in the X-ray band <cit.> and in the HE range <cit.>, this fact would represent a hint at ALP existence. This is especially true regarding galaxy clusters. As discussed below, photons diffusely produced in the central zone of galaxy clusters are expected to be unpolarized both in the X-ray and in the HE band within conventional physics, but the photon–ALP conversion makes them partially polarized, as shown in <cit.>. Therefore, a signal of partially polarized photons diffusely produced in a galaxy cluster would be a strong indication at ALP existence. Concerning blazars, we could achieve a similar conclusion if the detected polarization degree were extremely high, above the predictions of standard hadronic models (see below and <cit.>). In case no tension was found, the ALP parameter space (m_a, g_aγγ) could still be limited further. In this review, our aim is thus to describe the ALP-induced effects on photon polarization, which may be detected by missions in the X-ray band <cit.> and in the HE range <cit.> and which may provide us new ways to make progress in the ALP physics.

In particular, we introduce ALPs and the photon–ALP system in Section <ref>, and we illustrate polarization in Section <ref>. Section <ref> describes the astrophysical media crossed by the photon–ALP beam. In Section <ref>, the ALP-induced polarization effects are reported both in the X-ray and in the HE range for Perseus as an example for galaxy clusters and for OJ 287 as an example for blazars. In Section <ref>, the above-mentioned polarization effects are discussed along with their real detectability. Section <ref> is also devoted to describing possible future perspectives in the VHE range, where strong ALP signatures could be detected, along with a discussion about GRBs. Finally, we draw our conclusions in . Throughout this review, we employ the natural Lorentz–Heaviside (rationalized) units with ℏ = c = k_B = 1.



§ AXION-LIKE PARTICLES


ALPs are neutral, spin-zero pseudo-scalar bosons, whose primary interaction with photons (other interactions with fermions or gluons may exist but are subdominant and negligible in the present context) is described by the Lagrangian 

    ℒ_ ALP =  1/2 ∂^μ a  ∂_μ a - 1/2  m_a^2   a^2 -  1/4  g_aγγ  F_μνF̃^μν a  = 1/2 ∂^μ a  ∂_μ a - 1/2  m_a^2   a^2 + g_aγγ  E· B a,

where a denotes the ALP field, while E and B respectively represent the electric and magnetic components of the electromagnetic tensor F_μν, whose dual is F̃^μν.

Many bounds about the photon–ALP coupling g_a γγ and the ALP mass m_a have been derived in the literature <cit.>. The most reliable ALP limit is represented by g_a γγ < 0.66 × 10^- 10  GeV^- 1 for m_a < 0.02   eV at the 2 σ level from no detection of ALPs from the Sun <cit.>. Other bounds are more or less affected by the description of the astrophysical environment <cit.>.

Furthermore, the QED one-loop vacuum polarization effects, which become important in the presence of a strong external magnetic field, are accounted by the Heisenberg–Euler–Weisskopf (HEW) effective Lagrangian <cit.>

    ℒ_ HEW = 2 α^2/45 m_e^4 [ ( E^2 -  B^2 )^2 + 7 ( E· B)^2 ],

where α represents the fine-structure constant, while m_e is the electron mass.

We consider a photon–ALP beam of energy E propagating in the y-direction in a magnetized medium with an external magnetic field denoted by B and entering ), where E describes the photon electric field. Because of the off-diagonality of the mass matrix of the γ - a system, the propagation eigenstates differ from the interaction eigenstates, producing γ↔ a oscillations. This is similar to the oscillations of different flavor massive neutrinos, but in the case of the photon–ALP system, the external B field is essential to compensate for the spin mismatch between photons and ALPs. In particular, the functional expression of the photon–ALP coupling in Equation (<ref>) shows that a couples only with the component B_T of B transverse to the photon momentum k <cit.>. Furthermore, the short-wavelength approximation <cit.> holds in the present analysis since E ≫ m_a (we consider extremely light ALPs), so that the propagation of an unpolarized photon–ALP beam arising from ℒ_ ALP of Equation (<ref>) is described by the Von Neumann-like equation

    i d ρ (y)/d y = ρ (y)  ℳ^† ( E, y) - ℳ ( E, y)  ρ (y),

where ℳ (E,y) represents the photon–ALP mixing matrix, while ρ (y) is the polarization density matrix of the photon–ALP system. By defining ϕ, the angle that B_T forms with the z axis, ℳ reads as follows:

    ℳ (E,y) ≡(
    [        Δ_xx (E,y)        Δ_xz (E,y)  Δ_a γ(y)   sin ϕ;        Δ_zx (E,y)        Δ_zz (E,y)  Δ_a γ(y)   cos ϕ;  Δ_a γ(y)   sin ϕ Δ_ a γ(y)   cos ϕ         Δ_a a (E);                   ]),

with

    Δ_xx (E,y) ≡Δ_ (E,y)   cos^2  ϕ + Δ_∥ (E,y)   sin^2  ϕ,


    Δ_xz (E,y) = Δ_zx (E,y) ≡(Δ_∥ (E,y) - Δ_ (E,y) )  sin ϕ  cos ϕ,


    Δ_zz (E,y) ≡Δ_ (E,y)   sin^2  ϕ + Δ_∥ (E,y)   cos^2  ϕ,


    Δ_a γ(y) = 1/2g_aγγB_T(y),


    Δ_aa (E) = - m_a^2/2 E,

and

    Δ_ (E,y) = i/2  λ_γ (E,y) - ω^2_ pl(y)/2 E   + 2 α/45 π(B_T(y)/B_ cr)^2 E + ρ_ CMBE,


    Δ_∥ (E,y) = i/2  λ_γ (E,y) - ω^2_ pl(y)/2 E  + 7 α/90 π(B_T(y)/B_ cr)^2 E + ρ_ CMBE ,

[-15]where B_ cr≃ 4.41 × 10^13  G is the critical magnetic field and ρ_ CMB=0.522 × 10^-42.  describes the photon–ALP interaction, while Equation (<ref>) accounts for the ALP mass effect. The first term in Equations (<ref>) and (<ref>) describes photon absorption (e.g., due to the EBL <cit.>) with λ_γ being the γγ→ e^+e^- mean free path. In the second term of Equations (<ref>) and (<ref>), ω_ pl=(4 πα n_e / m_e)^1/2 represents the plasma frequency with n_e denoting the electron
number density. The third term in Equations (<ref>) and (<ref>) describes the QED one-loop vacuum polarization coming from ℒ_ HEW of Equation (<ref>), producing polarization variation and birefringence 
on the beam, and the fourth term expresses the photon dispersion on the CMB <cit.>. 

The solution of Equation (<ref>) can be expressed in terms of the transfer matrix 𝒰(E;y,y_0) of the photon–ALP system as

    ρ ( y ) = 𝒰(E; y, y_0 )  ρ_0  𝒰^†(E; y, y_0 ),

where ρ_0 is the density matrix at position y_0. The probability to find the photon–ALP beam in the final state ρ at position y starting in
the initial state ρ_0 at position y_0 reads as follows:

    P_ρ_0 →ρ (E,y) =  Tr[ρ 𝒰 (E; y, y_0)  ρ_0  𝒰^† (E; y, y_0) ],

with Tr ρ_0 =  Tr ρ =1 <cit.>.

We concentrate now on a photon–ALP system with the following characteristics: (i) fully polarized photons, (ii) no photon absorption (λ_γ→∞), (iii) homogeneous medium, and (iv) constant B field. Therefore, we can choose the z axis along the direction of B_T, resulting in ϕ=0 in Equation (<ref>). With these assumptions, the γ→ a conversion probability reads as follows:

    P_γ→ a (E, y) = (g_aγγB_T   l_ osc (E)/2π)^2  sin^2 (π (y-y_0)/l_ osc (E)),

with

    l_ osc (E) ≡2 π/[(Δ_zz (E) - Δ_aa (E) )^2 + 4  Δ_aγ^2 ]^1/2

being the photon–ALP beam oscillation length. The different energy dependence of the several Δ terms in Equation (<ref>) produces various regimes, which can be experienced by the photon–ALP beam. Hence, we can introduce the low-energy threshold E_L and the high-energy threshold E_H reading as follows:

    E_L ≡|m_a^2 - ω^2_ pl|/2 g_a γγ  B_T,

and 

    E_H ≡ g_a γγ  B_T [7 α/90 π(B_T/B_ cr)^2 + ρ_ CMB]^- 1,
 
respectively. In the energy interval E_L ≲ E ≲ E_H, the photon–ALP system lies in the strong-mixing regime, where the plasma, the ALP mass, the QED one-loop, and the photon dispersion on the CMB effects are all negligible with respect to the photon–ALP mixing term. In this case, P_γ→ a is maximal and energy independent and reduces to

    P_γ→ a (y) =  sin^2 ( g_aγγ B_T/2  (y-y_0) ).

Instead, both in the case E ≲ E_L, where the plasma contribution and/or the ALP mass term are prominent, and in the case E ≳ E_H, where the QED one-loop effect and/or the photon dispersion on the CMB dominate, the photon–ALP beam propagates in the weak-mixing regime, where P_γ→ a becomes energy dependent and progressively vanishes.

The usual situation of a generic photon initial polarization, structured magnetic fields, and non-homogeneous media represents a generalization of the above-considered case, which is reported here for the simplicity of the analytical expressions of the involved equations. However, in all our following calculations, we employ the appropriate photon polarization and the proper expressions of the magnetization, dispersion, and absorption properties of the crossed media.




§ POLARIZATION EFFECTS


The generalized polarization density matrix ρ associated with the photon–ALP beam can be expressed as 

    ρ (y) = ([ A_x (y); A_z (y);   a (y) ])
    ⊗([ A_x (y)   A_z (y)  a (y) ])^*,

where A_x (y) and A_z (y) represent the photon linear polarization amplitudes along the x and z axis, respectively, and a(y) is the ALP amplitude. Equation (<ref>) allows us to describe totally polarized (pure states), unpolarized, and partially polarized beams at once. In particular, pure photon states polarized in the x and z direction are expressed as

    ρ_x = (
    [ 1 0 0; 0 0 0; 0 0 0;   ]),         ρ_z = (
    [ 0 0 0; 0 1 0; 0 0 0;   ]),

respectively, and the ALP state as

    ρ_a = (
    [ 0 0 0; 0 0 0; 0 0 1;   ]),

whereas an unpolarized photon state is described by

    ρ_ unpol = 1/2(
    [ 1 0 0; 0 1 0; 0 0 0;   ]).

Partially 
 polarized photons are characterized by a polarization density matrix with an intermediate functional expression between Equations (<ref>) and (<ref>). 

The photonic part of the polarization density matrix of Equation (<ref>) can be expressed in terms of the Stokes parameters as <cit.>

    ρ_γ = 1/2(
    [  I+Q U-iV; U+iV  I-Q;      ]),

while the photon degree of linear polarization Π_L and the polarization angle χ are defined <cit.> as

    Π_L ≡(Q^2+U^2)^1/2/I = [ (ρ_11-ρ_22)^2+(ρ_12+ρ_21)^2]^1/2/ρ_11+ρ_22,

and

    χ≡1/2 arctan(U/Q)=1/2 arctan(ρ_12+ρ_21/ρ_11-ρ_22),

respectively, where ρ_ij with i,j=1,2 are the photon polarization density matrix elements. In the following Sections, we will show the behavior of the final Π_L, which turns out to be strongly modified by the photon–ALP interaction with respect to the initial photon degree of the linear polarization Π_L,0.

As shown by some theorems enunciated and demonstrated in <cit.>, in the absence of photon absorption, there exists a relationship between photon survival probability in the presence of photon–ALP interaction P_γ→γ and initial Π_L,0 (P_γ→ a=1-P_γ→γ due to the absence of photon absorption), and in particular, the following hold: 




  * In case of no photon absorption and a starting condition of only photons with an initial degree of linear polarization Π_L,0, the inequalities P_γ→ a≤ (1+Π_L,0)/2 and P_γ→γ≥ (1-Π_L,0)/2 hold. In case of initially unpolarized photons (Π_L,0=0), we observe P_γ→ a≤ 1/2 and P_γ→γ≥ 1/2.


  * In the previous conditions, Π_L,0 represents the measure of the overlap between the values assumed by P_γ→ a and P_γ→γ. In case of initially unpolarized photons (Π_L,0=0), P_γ→ a and P_γ→γ possess the common value of 1/2, at most.



We have employed the inequalities described in item 1 to verify the correctness of the behavior of P_γ→γ reported in several figures of this review (see the following sections).

As shown in <cit.>, the results presented in item 2 can be used to measure the initial Π_L,0 when only spectral data of an astrophysical source are known. The strategy works in the energy band in which the photon–ALP system lies in the weak mixing regime, where P_γ→ a and P_γ→γ are energy dependent and their assumed values satisfy the inequalities described in item 1. 

The procedure can be sketched as follows. We consider systems in an energy range where photon absorption is negligible. Starting from observational data of an astrophysical source with the observed spectrum Φ_ obs, we can infer P_γ→γ as 

    P_γ→γ=Φ_ obs/Φ_ em,

where Φ_ em represents the emitted spectrum, which can be either known or derivable from Φ_ obs (see <cit.> for more details). Furthermore, the absence of photon absorption by hypothesis ensures that P_γ→ a = 1 - P_γ→γ. Now, Π_L,0 can be obtained by measuring the interval of overlap between the values assumed by P_γ→γ and P_γ→ a.

In Figure <ref>, we show how the previous procedure works: in the left panels, an application in the optical-X-ray band is reported, while the right panels address a similar study but in the MeV range. We show how an observer would proceed to infer the initial Π_L,0 from spectral data. Note that cosmological effects are taken into account and E in Figure <ref> represents the photon energy, as observed from the Earth.



In the optical-X-ray band, we consider a high-frequency peaked blazar (HBL) placed inside a poor galaxy cluster at redshift z = 0.1 with an initial Π_L,0=0.3, while in the MeV range, we deal with an HBL but now placed inside a quite rich galaxy cluster at redshift z = 0.05 with Π_L,0=0.1. We assume the phenomenological emitted spectra derived in <cit.> and typical values concerning the magnetization properties of the astrophysical media crossed by the photon–ALP beam both in the optical-X-ray and in the MeV energy band.

In particular, in the optical-X-ray band, the photon–ALP interaction turns out to be negligible inside the poor galaxy cluster (see also Section <ref>), where the blazar is supposed to be located and inside the hosting elliptical galaxy, whose typical properties are described in Section <ref>. Instead, we have efficient photon–ALP conversion inside the jet of the blazar, whose characteristics are described in Section <ref>: For the magnetic field in the jet B^ jet, we assume a toroidal profile of Equation (<ref>) with magnetic field strength at the emission distance y_ em = 3 × 10^16  cm equal to B_0^ jet≡ B^ jet(y_ em)= 0.5   G, and for the electron number density n_e^ jet, we consider the profile expressed by Equation (<ref>), taking n_e,0^ jet≡ n_e^ jet(y_ em)=5 × 10^4   cm^-3. We hypothesize an efficient photon–ALP interaction inside the extragalactic space by assuming an extragalactic magnetic field
strength B_ ext = 1   nG and the B_ ext coherence properties described in Section <ref>. Lastly, the photon–ALP interaction turns out to be efficient also inside the Milky Way, where we adopt the Jansson and Farrar model <cit.> concerning the magnetic field B_ MW and that presented in <cit.> regarding the electron number density in the Galaxy n_ MW,   e (see also Section <ref>). For more details, we send the reader to <cit.>.



Instead, in the MeV range, we have an efficient photon–ALP conversion in the cluster since the blazar is supposed to be located inside a rich galaxy cluster, whose properties are reported in Section <ref>: we assume the cluster magnetic field B^ clu profile of Equation (<ref>), where n_e^ clu represents the cluster electron number density described by means of the β model expressed by Equation (<ref>). Concerning the parameters entering Equations (<ref>) and (<ref>), we assume k_L = 0.1   kpc^-1, k_H = 3   kpc^-1, q = -11/3, B^ clu_0 = 20   G, n_e,0^ clu = 0.1   cm^-3, η_ clu = 0.75, β_ clu = 2/3, a cluster core radius r_ core = 150   kpc, and a cluster radius of 1   Mpc (see Sections <ref> and <ref> for the definition of the parameters). In the present situation, we hypothesize a negligible photon–ALP conversion inside the extragalactic space by assuming B_ ext < 10^-15  G (see also Section <ref>). Concerning the photon–ALP beam propagation inside the blazar jet, host galaxy, and Milky Way, we consider the same models and parameter values assumed in the previous case (optical-X-ray band) apart from the jet magnetic field strength at the emission distance, which is now B_0^ jet≡ B^ jet(y_ em)= 0.1   G (see also Sections <ref>, <ref>, and <ref>). We send the reader to <cit.> for more details.



In both of the previous cases (optical-X-ray and MeV band), a typical realization of P_γ→γ is evaluated and reported in the top panels of Figure <ref>: we assume g_aγγ = 0.5 × 10^-11  GeV^-1 for both of the situations, but we take m_a = 5 × 10^-14  eV for the first blazar in the optical-X-ray band, while we deal with an alternative scenario by taking m_a = 2 × 10^-10  eV for the second blazar in the MeV range. The central panels of Figure <ref> show the observed binned spectra Φ_ obs (with a typical energy resolution <cit.>), simulated by multiplying the emitted spectra by P_γ→γ in the presence of photon–ALP interaction. In the bottom panels of Figure <ref>, we show P_γ→γ and P_γ→ a with P_γ→γ inferred from the simulated observed spectra Φ_ obs, reported in the central panels, by means of Equation (<ref>), where Φ_ em can be reconstructed by fitting the upper spectral bins in the central panels, since photon–ALP interaction produces an energy-dependent dimming in the observed spectra (see also <cit.>). We recall that P_γ→ a = 1 - P_γ→γ due to the absence of photon absorption. Finally, by employing the results presented in item 2, we can measure the initial Π_L,0, and we obtain Π_L,0 = 0.288 ± 0.016 in the X-ray band and Π_L,0 = 0.090 ± 0.018 in the MeV range. We observe that the inferred values are very close to the initially assumed ones, which demonstrates the power of the present method.

We want to stress that the above-described procedure is the only established possibility to measure the initial polarization of photons produced by astrophysical sources. Furthermore, this procedure can be employed in a wide energy range, where photon absorption is negligible, and only needs spectral observations of the source, so that flux-measuring observatories also become polarimeters.




§ PHOTON–ALP BEAM PROPAGATION IN DIFFERENT MEDIA


In the present section, we describe the main properties of the media crossed by the photon–ALP beam starting where photons are produced up to their arrival at the Earth: blazar, host galaxy, galaxy cluster, extragalactic space, and Milky Way. We concentrate mainly on the characteristics that are crucial for the photon–ALP system, such as the magnetization and absorption properties of the media, while we send the reader to the quoted papers for an exhaustive discussion.



 §.§ Blazars


Blazars are powerful astrophysical sources emitting in the whole electromagnetic spectrum, from radio waves up to the VHE band. They are a particular class of AGN that are basically extragalactic supermassive black holes (SMBHs) accreting matter from the surroundings. In these systems, two collimated relativistic jets can develop in opposite directions. In the accidental case where one of these jets turns out to be  in the Earth direction, the AGN is called blazar, while in other situations, the AGN shows a different phenomenology, as displayed in Figure <ref>, where the AGN structure is also shown. 



Blazars are divided into two sub-classes: flat spectrum radio quasars (FSRQs) and BL Lac objects (BL Lacs). FSRQs are powerful sources, where accretion from the disk surrounding the central SMBH is efficient. Furthermore, they show intense broad emission lines generated by clouds of gas in the so-called
broad-line region (BLR) encompassing the central SMBH and photoionized by the
ultraviolet photons produced by the disk. Moreover, FSRQs display absorption regions for VHE photons, which can interact with the optical/ultraviolet radiation emitted by the above-mentioned BLR and with the infrared radiation produced by the dusty torus (see Figure <ref>), preventing the observation of photons with energy above ∼20   GeV <cit.>. However, photons of up to ∼400   GeV have been observed from FSRQs <cit.>. Only ad hoc solutions have been proposed inside conventional physics, while photon–ALP oscillations provide a natural solution of the problem by partially preventing BLR absorption: this fact represents a hint at ALP existence <cit.>. Instead, BL Lacs are less powerful, and they show neither absorption regions nor significant emission lines due to the presence of a radiatively inefficient
accretion flow (RIAF) onto the central SMBH <cit.>.

In the following, we will concentrate on BL Lacs, where the emission region of photons is located at a distance of about y_ em= (10^16– 10^17)   cm from the central SMBH. Once produced at y_ em, photons propagate inside the BL Lac jet up to a distance y_ jet≃ 1   kpc, where they enter the host galaxy. During their path inside the jet, these photons can oscillate into ALPs inside the jet magnetic field B^ jet. The toroidal part of B^ jet, which is transverse to the jet axis <cit.>, turns out to be the dominant one at the distances under consideration, and its profile reads as follows:

    B^ jet ( y ) = B^ jet_0 (y_ em/y),

where B^ jet_0 denotes the magnetic field strength of the jet at y_ em. Another crucial information in order to evaluate photon–ALP conversion inside BL Lacs is the electron number density profile in the jet n_e^ jet, which, due to the conical shape of
the jet, is expected to be 

    n^ jet_e ( y ) = n^ jet_e,0(y_ em/y)^2,

where n^ jet_e,0 represents the electron number density at y_ em. Synchrotron Self Compton (SSC) diagnostics applied to blazar spectra suggests assuming n^ jet_e,0=5 × 10^4   cm^-3 <cit.>.

Two competing photon emission mechanisms have been proposed in order to explain blazar phenomenology: (i) the leptonic model (see, e.g., <cit.>) and (ii) the hadronic model (see, e.g., <cit.>). Both scenarios describe photon production in the optical up to the X-ray band as generated by electron synchrotron emission. At higher energies, leptonic models explain photon emission with inverse Compton scattering of synchrotron radiation or of external photons from the disk and/or clouds with the synchrotron producing relativistic electrons <cit.>. Instead, hadronic models attribute the high-energy photon generation to proton–synchrotron emission or photomeson production <cit.>. Hadronic models require higher values of B^ jet_0 and y_ em with respect to leptonic scenarios. Moreover, hadronic models predict a higher initial degree of linear polarization Π_L,0 with respect to leptonic scenarios, as shown in Figure <ref> for two particular BL Lacs.



The propagation of the photon–ALP beam is computed in the comoving frame of the jet so that a transformation E →γ E, with γ being the Lorentz factor, is required to pass to the fixed frames of the subsequently encountered regions. 

The values assumed by B^ jet_0, y_ em, γ and Π_L,0 affect the calculation of the photon–ALP beam propagation in the jet and will be discussed in Section <ref> for both leptonic and hadronic models. Then, the photon–ALP beam transfer matrix in the jet 𝒰_ jet can be evaluated by following the calculation reported in <cit.>.




 §.§ Host Galaxy


The effectiveness of the photon–ALP conversion inside the host galaxy depends on its nature and characteristics. Ideal galaxies in order to have an efficient photon–ALP conversion are spirals and even more starbursts. The reason lies in the high magnetic field strength inside the host B^ host, which can reach values of B^ host=𝒪(5-10)  μ G for spirals <cit.> and B^ host=𝒪(20-50)  μ G for starbursts <cit.>, and in the large coherence length L_ dom^ host of B^ host, which is around L_ dom^ host =𝒪(1-4)   kpc for both spiral and starburst galaxies outside the central region <cit.>. The magnetic field coherence can be modeled by employing a Kolmogorov-type turbulence power spectrum in a similar way as discussed in the following subsection concerning galaxy clusters (for more details about the parameter values, see also <cit.>). A possible example of photon–ALP conversion inside spirals or starbursts is represented by photons produced by GRBs, which are usually located in the central region of these galaxies <cit.>. 

In particular, the location of GRB 221009A—the brightest GRB to date—in the central region of a disk-like galaxy <cit.> allows for an efficient photon–ALP conversion inside the host, which is able to justify the detection by the LHAASO Collaboration of this GRB above 10   TeV <cit.>. The latter fact represents a strong hint at ALP existence, as demonstrated in <cit.> (see also Section <ref> and <cit.>).

Instead, blazars are often hosted inside elliptical galaxies, where B^ host is described by means of a domain-like model with the typical values B_ host≃ 5   G and L_ dom^ host≃ 150   pc <cit.>. In such a situation, the photon–ALP oscillation length turns out to be much larger than L_ dom^ host, resulting in an inefficient photon–ALP conversion, as shown in <cit.>. However, whatever the nature and the properties of the host galaxy are, we evaluate the associated photon–ALP beam transfer matrix 𝒰_ host.



 §.§ Galaxy Clusters


Galaxy clusters consist of 30 to more than 1000 galaxies with a total mass within the range (10^14–10^15)   M_⊙, making them the largest gravitationally bound structures in the Universe. Properties such as galactic content, density profile, shape, concentration, and symmetry allow galaxy clusters to be classified into three main classes: (i) regular clusters, (ii) intermediate clusters, and (iii) irregular clusters <cit.>. The approximate spherical symmetry and the satisfying knowledge of the density profile of regular clusters allow us to model their electron number density n_e^ clu and the linked magnetic field B^ clu carefully so that we can properly calculate the photon–ALP conversion. 

The strength of B^ clu is set to B^ clu = 𝒪 (1-10)   G by Faraday rotation measurements and synchrotron radio emission <cit.>. The morphology of B^ clu is turbulent and modeled with a Kolmogorov-type turbulence power spectrum M(k)∝ k^q, with index q=-11/3 and where k represents the wave number in the interval [k_L,k_H] with k_L = 2π/Λ_ max and k_H = 2π/Λ_ min being the minimal and maximal turbulence scales, respectively <cit.>. Consequently, the behavior of B^ clu can be modeled as <cit.>

    B^ clu(y)=ℬ( B_0^ clu,k,q,y ) ( n_e^ clu(y)/n_e,0^ clu)^η_ clu,

where ℬ represents the spectral function describing the Kolmogorov-type turbulence of the cluster magnetic field (for more details, see, e.g., <cit.>), η_ clu is a cluster parameter, and B_0^ clu and n_e,0^ clu are the magnetic field strength and the electron number density at the center of the cluster, respectively.

Several models are employed to describe the profile of n_e^ clu, such as the single β model typically used for non-cool-core (nCC) clusters, the double β model for cool-core (CC) clusters, or a modified version of them <cit.>. We will fix the behavior of n_e^ clu in Section <ref>, where we will concentrate on real situations.

Photons can be produced by some astrophysical source such as a blazar or a GRB and can pass through a galaxy cluster, or they can be diffusely generated in the central region of the cluster itself. In the latter situation, it is necessary to discuss their initial Π_L,0. In the X-ray band, photons are produced in the cluster central region via thermal Bremsstrahlung <cit.> with photons that turn out to be therefore unpolarized. In the HE range, several processes are proposed to explain possible photon emission, such as the cascade of VHE photons, inverse Compton scattering, and neutral pion decay (see, ). Even in the HE range, photons turn out to be effectively .

The transfer matrix 𝒰_ clu of the photon–ALP beam propagating from the central region of the cluster up to its virial radius can be evaluated by following the calculation reported in <cit.>. In the specific case in which photons are not diffusely produced in the central region of a rich cluster but are generated by another astrophysical source that does not lie inside a rich cluster—such a situation is frequent concerning blazars—the photon–ALP conversion in this region turns out to be inefficient.



 §.§ Extragalactic Space


The effectiveness of the photon–ALP conversion inside the extragalactic space is determined by the strength and morphology of the extragalactic magnetic field B_ ext. However, the properties of B_ ext are still poorly understood. While many configurations for B_ ext exist in the literature <cit.>, its limits are in the interval 10^- 7  nG≲ B_ ext≲ 1.7   nG on the scale of 𝒪 (1)   Mpc <cit.>.

The shape of B_ ext is modeled as a domain-like network, where B_ ext is homogeneous over a whole domain of size L_ dom^ ext, which is equal to its coherence length, and with B_ ext randomly and discontinuously changing its direction from one domain to the subsequent one, but maintaining approximately the same strength <cit.>. Since the exact orientation of B_ ext in each domain is unknown, the photon–ALP beam propagation becomes a stochastic process, and many realizations of it can be evaluated by varying the B_ ext orientations, in order to obtain the statistic properties of the system. However, only a single realization of the photon–ALP beam propagation process can be physically observed.

The domain-like model proposed for B_ ext is justified by several scenarios of outflows from primeval galaxies, further amplified by turbulence <cit.>. According to the latter scenarios, the usual values for B_ ext are B_ ext = 𝒪 (1)   nG and L_ dom^ ext = 𝒪 (1)   Mpc (for details, see <cit.>). In this fashion, we assume B_ ext = 1   nG and L_ dom^ ext in the interval (0.2–10)   Mpc with ⟨ L_ dom^ ext⟩ = 2   Mpc.

However, the commonly used discontinuous model for B_ ext produces unphysical results when the photon–ALP beam oscillation length l_ osc becomes smaller than L_ dom^ ext due to the photon dispersion on the CMB <cit.>, and consequently, the system turns out to be sensitive to the B_ ext substructure. In order to solve such a problem, we employ a more complex model developed in <cit.>, where the components of B_ ext change continuously, crossing from a domain to the subsequent one.

[-15]In the UV-X-ray and HE bands, photon absorption due to the interaction with EBL photons—we assume the conservative EBL model of Franceschini and Rodighiero <cit.>—is negligible, but it becomes substantial for E > 𝒪(100)   GeV <cit.>. Therefore, for , propagation in the extragalactic space produces additional photon–ALP oscillations when the photon–ALP conversion is efficient. Instead, for E > 𝒪(100)   GeV, where we have a large EBL absorption, the effect of the photon–ALP beam propagation in the extragalactic space is also to mitigate the EBL absorption of VHE photons. The reason lies in the fact that when photons become ALPs, they do not undergo EBL absorption so that the effective optical depth is reduced and more photons can reach the Earth.

In both of the previous situations, with large or negligible EBL absorption and in the case of both efficient or ineffective photon–ALP conversion, the transfer matrix 𝒰_ ext of the photon–ALP system in the extragalactic space can be calculated by following the analysis performed in <cit.>.



 §.§ Milky Way


Photon–ALP conversion in the Milky Way can be computed once the morphology and strength of the magnetic field B_ MW and the profile of the electron number density n_ MW,   e in the Galaxy are known.

Concerning B_ MW, we employ the model developed by Jansson and Farrar <cit.>, which includes a disk and a halo component, both parallel to the Galactic plane, and a poloidal `X-shaped' component at the galactic center. The most relevant component of B_ MW for the photon–ALP conversion turns out to be its regular part, to which we add for completeness the turbulent one developed in <cit.>. The alternative B_ MW model of Pshirkov et al. <cit.> does not substantially produce a different photon–ALP conversion in the Milky Way. Still, we prefer the model by Jansson and Farrar <cit.> since it determines the Galactic halo component with a higher level of accuracy. 

Regarding n_ MW,   e we employ the model presented in <cit.>, where the Galaxy is modeled with an extended thick disk standing for the warm interstellar medium, a thin disk accounting for the Galactic molecular ring, a Galactic Center disk, spiral arms and features for the Gum Nebula, Galactic Loop I, and the Local Bubble.

By following the procedure developed in <cit.>, we can evaluate the transfer matrix 𝒰_ MW of the photon–ALP system in the Milky Way.



 §.§ Overall Photon–ALP Beam Propagation


Once all the transfer matrices discussed in the previous subsections are known, we can combine them in order to calculate the total transfer matrix 𝒰 describing the photon–ALP beam propagation starting from the photon emission zone up to the Earth. When photons are diffusely generated in the central zone of a galaxy cluster, 𝒰 reads as follows:

    𝒰=𝒰_ MW 𝒰_ ext 𝒰_ clu,

while when photons are produced in the jet of a blazar, 𝒰 is expressed by

    𝒰=𝒰_ MW 𝒰_ ext 𝒰_ clu 𝒰_ host 𝒰_ jet.


In both of the previous cases, we can calculate the photon survival probability in the presence of photon–ALP interaction by specializing Equation (<ref>) as

    P_γ→γ  = ∑_i = x,z Tr[ρ_i  𝒰 ρ_ in 𝒰^†],

where ρ_x and ρ_z read from Equations (<ref>), and ρ_ in represents the beam initial polarization density matrix, which is specified in the following subsections for concrete cases, for which the initial photon degree of linear polarization Π_L,0 is known. Moreover, by recalling Equation (<ref>) with ρ_0 ≡ρ_ in, we obtain the final photon degree of the linear polarization Π_L and the polarization angle χ, thanks to Equations (<ref>) and (<ref>), respectively (see also <cit.>).




§ ALP EFFECTS ON PHOTON POLARIZATION


Hereafter, we show how photon–ALP interaction modifies the final degree of linear polarization Π_L and polarization angle χ for photons diffusely generated in the galaxy cluster central region or produced in the blazar jet. Here, we report two concrete examples: one for galaxy clusters, and specifically, we concentrate on Perseus, and one for BL Lacs and in particular, we focus on OJ 287. The latter two sources represent good observational targets to study ALP effects inside galaxy clusters <cit.> and blazars <cit.>, respectively. We send the interested reader to the above-quoted papers, where a deeper discussion is presented along with other concrete examples of both galaxy clusters and blazars <cit.>. We now describe the properties of Perseus and of OJ 287, which are crucial to evaluate the photon–ALP interaction effects on Π_L and χ.

Perseus is a deeply studied CC rich regular cluster located at redshift z = 0.01756 and represents the brightest galaxy cluster in the X-ray band. Its electron number density n_e^ clu can be expressed by

    n_e^ clu(y)=n_e,01^ clu( 1+y^2/r_ core,1^2)^-3/2β_ clu,1+ 
    
    n_e,02^ clu( 1+y^2/r_ core,2^2)^-3/2β_ clu,2,

with n_e,01^ clu=3.9 × 10^-2  cm^-3, r_ core,1=80   kpc, β_ clu,1=1.2, n_e,02^ clu=4.05 × 10^-3  cm^-3, r_ core,2=280   kpc, and β_ clu,2=0.58 <cit.>. The Perseus magnetic field B^ clu is described by Equation (<ref>) by taking B_0^ clu = 16   G, which represents an average estimate among those reported in the literature <cit.>. Due to the similarity of the stochastic properties of the Perseus B^ clu with those of the Coma cluster, we assume the following values regarding Perseus: q=-11/3, Λ_ min= 2   kpc and Λ_ max= 34   kpc <cit.>, and the average value η_ clu=0.5 entering Equation (<ref>). Finally, we take an initial degree of linear polarization Π_L,0=0 both in the UV-X-ray and in the HE band according to the discussion in Section <ref>. Then, the photon–ALP beam propagation can be evaluated as discussed in Section <ref>, and its transfer matrix reads from Equation (<ref>).

OJ 287 represents a typical low-frequency peaked blazar (LBL) located at redshift z=0.3056. Due to its high flux in both the X-ray and HE bands, OJ 287 turns out to be a favorable observational target for polarimetric studies in both energy ranges <cit.>. In the following, we consider the typical values regarding the LBL parameters: for the leptonic model, we take B^ jet_0 = 1   G, y_ em = 3 × 10^16  cm and γ = 10, while for the hadronic model, we assume B^ jet_0 = 20   G, y_ em = 10^17  cm and γ = 15 <cit.>. The profiles of B^ jet and n_e^ jet are represented by Equations (<ref>) and (<ref>), respectively. As discussed in Section <ref>, the initial Π_L,0 for OJ 287 is plotted in Figure <ref> for both the leptonic and the hadronic scenario. OJ 287 is hosted inside an elliptical galaxy with a resulting inefficient photon–ALP conversion in the host, as discussed in Section <ref>. Moreover, OJ 287 is not found within a rich galaxy cluster with a consequent negligible ALP effect also in this zone (see ). However, for completeness, we consider all the contributions for the evaluation of the total photon–ALP beam transfer matrix, which is expressed by Equation (<ref>). A remark regarding photons emitted by blazars should be kept in mind: due to the limited spatial resolution of real polarimeters, photons coming from different zones inside the transverse section of the blazar jet cannot be distinguished, and all photons are then collected together. Although the latter fact might in principle wash out the ALP-induced polarization features, the studies developed in <cit.> demonstrate that this is not the case in realistic situations. In order to be conservative, the magnetic field profile expressed by Equation (<ref>) and a propagation distance of 1   pc inside the jet are assumed. We send the reader to the papers quoted above <cit.> for a deeper discussion regarding the latter problem and the adopted strategy to take the polarimeter technological limitations into account. 

We can now show the final photon survival probability P_γ→γ, the corresponding photon degree of the linear polarization Π_L, and the polarization angle χ arising from photon–ALP oscillations inside the magnetized media analyzed in Section <ref> (blazar jet, host galaxy, galaxy cluster, extragalactic space, Milky Way) for the two above-discussed scenarios: (i) photons are diffusely generated in the central zone of a galaxy cluster—we take Perseus as an example, and (ii) photons are produced at the jet base of a BL Lac—we consider OJ 287 as an illustration. In order to assess the robustness of the results, several realizations of the photon–ALP beam propagation process are evaluated by considering different magnetic field configurations due to limited knowledge especially regarding B^ clu and B_ ext. The probability density function f_Π associated with the final Π_L can be computed by collecting all the values assumed by Π_L in the various realizations and gives us information about the most probable scenarios.

Regarding the ALP parameters, we assume g_aγγ=0.5 × 10^-11  GeV^-1 and the two ALP masses: (i) m_a ≲ 10^-14  eV and (ii) m_a = 10^-10  eV, which are within the CAST bound <cit.>. For m_a ≲ 10^-14  eV, the plasma term dominates over the ALP mass term for the chosen model parameters, while the opposite is true in the case m_a = 10^-10  eV. Results about P_γ→γ, Π_L and χ are obtained for photons produced at redshift z with emitted energy E and observed energy E_0=E/(1+z) in the two intervals: (i) UV-X-ray band and 

Note that, in the following figures, binned data about Π_L and χ are also shown in order to provide an idea of how a possible observation of ALP-induced features may look like. The binning procedure is based on the realistic energy resolution of current and/or planned polarimeters both in the X-ray band <cit.> and in the HE range <cit.> (see <cit.> for more details).



 §.§ UV-X-ray Band


In the UV-X-ray band, the photon–ALP beam propagates in the weak mixing regime due to the strength of the plasma and/or ALP mass term inside the mixing matrix of Equation (<ref>). As a result, P_γ→γ, Π_L and χ turn out to possess an energy-dependent behavior in the case of both diffuse photon production in the central region of the Perseus cluster and photon emission at the jet base of OJ 287.

In particular, in the case m_a = 10^-10  eV, the ALP mass term is so strong that it dominates over the photon–ALP mixing term. As a result, the photon–ALP conversion turns out to be inefficient in the UV-X-ray band for m_a = 10^-10  eV, apart from a moderate dimming of Π_L,0 for OJ 287 in the case of the hadronic emission model due to the high central value of B^ jet (see <cit.> for more details). This is the reason why we only consider the case m_a ≲ 10^-14  eV in this subsection. 

Regarding the Perseus cluster, we report P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> and the corresponding f_Π associated with the final Π_L in the bottom subfigure (see <cit.> for more details). Similarly, concerning OJ 287, we show P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> and the linked f_Π in the bottom subfigure, but we now consider the two possible emission mechanisms inside blazars (see Section <ref>): (i) the leptonic model in the left panels and (ii) the hadronic model in the right panels (see <cit.> for more details). In both cases, i.e., for both Perseus and OJ 287, we concentrate on two benchmark energies in the evaluation of f_Π: (i) E_0 = 1   keV and (ii) E_0 = 10   keV.




In the case considered here, i.e., g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV, the energy dependence of P_γ→γ, Π_L, and χ in the top subfigures of Figures <ref> and <ref> is caused by the strength of the plasma term, which is not negligible with respect to the photon–ALP mixing term, especially inside the cluster in the case of Perseus and in the BL Lac jet regarding OJ 287. As a result, the photon–ALP beam propagates in the weak mixing regime, which protracts for a few energy decades due to the large variation with respect to the distance of B^ clu and n_e^ clu regarding Perseus and of B^ jet and n_e^ jet concerning OJ 287 (see  and <cit.> for details). In addition, the previous figures demonstrate that the final Π_L turns out to be heavily modified with respect to the initial one Π_L,0 for both Perseus and OJ 287. The binned data in the top Subfigures of Figures <ref> and <ref> concerning Π_L and χ show that the ALP-induced features can be detected by IXPE <cit.>, eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, and XPP <cit.> for E_0 ≳ 2   keV regarding Perseus and for E_0 ≳ 0.5   keV concerning OJ 287. Regarding OJ 287, the top subfigure of Figure <ref> shows that the hadronic model produces a larger energy variability of Π_L and χ with respect to the leptonic scenario due to the higher value of B^ jet_0 and the wider extension of the jet region.



The behavior of f_Π reported in the bottom subfigures of Figures <ref> and <ref> shows a broadening of the final Π_L with respect to the initial Π_L,0 for both Perseus and OJ 287. Perseus appears as a strong target to study ALP effects on photon polarization, since the final value Π_L = 0 is never the most probable expected result, while conventional physics predicts Π_L = Π_L,0 = 0. Instead, for OJ 287, the most probable final value of Π_L is not very different from the initial Π_L,0 apart from the case of the leptonic model at E_0 = 10   keV, where the most probable result turns out to be Π_L ≳ 0.8, which is extremely difficult to explain within conventional physics. For more details, we send the reader to the discussion developed in <cit.> concerning Perseus and to that presented in <cit.> regarding OJ 287.
















 §.§ High-Energy Band


In the HE band, both the scenario with ALP parameters g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and that with g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV produce sizable ALP effects on P_γ→γ, Π_L and χ in the case both of photon diffuse production in the central zone of the Perseus cluster and of photon emission at the jet base of OJ 287.

Concerning the Perseus cluster, we show the behavior of P_γ→γ, Π_L and χ in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and in the top subfigure of Figure
 <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV. The associated f_Π are reported in the bottom subfigures of Figures <ref> and <ref>, respectively. Regarding OJ 287, we likewise report P_γ→γ, Π_L, and χ in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV and in the top subfigure of Figure <ref> for the case g_aγγ=0.5 × 10^-11  GeV^-1,  m_a = 10^-10  eV. The corresponding f_Π are shown in the bottom subfigures of Figures <ref> and <ref>, respectively. For OJ 287, we assume (i) the leptonic emission model in the left panels of Figures <ref> and <ref> and (ii) the hadronic one in the right panels of Figures <ref> and <ref> (see Section <ref> and <cit.> for more details). Moreover, f_Π is evaluated for both Perseus and OJ 287 at the two benchmark energies: (i) E_0 = 300   keV and (ii) E_0 = 3   MeV.




We first consider the scenario g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV. In the present situation, the photon–ALP beam propagates in the strong mixing regime, since the photon–ALP mixing term dominates over all other terms inside the mixing matrix of Equation (<ref>) (see Section <ref>). Consequently, P_γ→γ, Π_L, and χ are energy independent in almost all the considered energy band in the case both of photon production in the central region of Perseus, as shown by the top subfigure of Figure <ref>, and of photon generation in the jet of OJ 287, as reported in the top subfigure of Figure <ref>. We can also infer that, in all the considered cases, the final Π_L turns out to be strongly modified with respect to the initial Π_L,0. The binned data show a lower uncertainty than in the UV-X-ray band due to fact that the photon–ALP system lies in the strong mixing regime, which makes both Perseus and OJ 287 good observational targets for missions such as COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.>. Moreover, regarding OJ 287, we do not observe substantial differences between the leptonic and the hadronic emission model.




As far as the behavior of f_Π is concerned, for both Perseus and OJ 287, the bottom subfigures of Figures <ref> and <ref> respectively show that the most probable final value of Π_L is larger than the initial one Π_L,0 with a substantial broadening. For both Perseus and OJ 287, the most promising energy range to study ALP effects on photon polarization appears around E_0 = 3   MeV, where the most probable value for Π_L is Π_L ≳ 0.8, which is larger than that predicted by conventional physics in all considered cases.





We now move to the scenario g_aγγ=0.5 × 10^-11  GeV^-1, m_a = 10^-10  eV. In this situation, the photon–ALP system lies in the weak mixing regime, since the ALP mass term turns out to be non-negligible with respect to the photon–ALP mixing term inside the mixing matrix of Equation (<ref>) (see Section <ref>). As a result, P_γ→γ, Π_L, and χ become energy dependent, as shown in the top subfigure of Figure <ref> regarding Perseus and in the top subfigure of Figure <ref> concerning OJ 287. Furthermore, we observe that the final Π_L turns out to be altered compared with the initial Π_L,0 especially in the case of Perseus and in that of the leptonic model for OJ 287. As already observed in the X-ray band for the case g_aγγ=0.5 × 10^-11  GeV^-1, m_a ≲ 10^-14  eV, the wide energy range, where the photon–ALP system lies in the weak mixing regime, is due to the high variation of the magnetic field and electron number density profiles in the crossed media (see also Sections <ref> and <ref> and <cit.> for more details). The binned data in the top subfigures of Figures <ref> and <ref> show that observatories like COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.> may detect ALP-induced features on Π_L for E_0 ≳ 3   MeV concerning Perseus, and similar features for E_0 ≳ 0.2   MeV regarding OJ 287 in the case both of the leptonic and of the hadronic model.










The behavior of f_Π reported in the bottom subfigure of Figure <ref> for Perseus and in the bottom subfigure of Figure <ref> for OJ 287 shows a broadening of the final Π_L with respect to the initial Π_L,0 in all the cases. Moreover, the best energy interval for studies of ALP-induced features on Π_L appears to be E_0 ≳ 3   MeV concerning the Perseus cluster, where the most probable value for the final Π_L is Π_L ≳ 0.2, while conventional physics predicts Π_L = Π_L,0 = 0. Even OJ 287 represents a good observational target, since the most probable values of the final Π_L are larger than the initial ones, but in this case, conventional physics could justify moderately high values of Π_L albeit with some difficulty.



§ DISCUSSION AND FUTURE PERSPECTIVES


In Section <ref>, the ALP impact on the polarization of photons coming from Perseus has been reported as an example of ALP effects on galaxy clusters in the X-ray and in the HE band (for details, see <cit.>). Similarly, OJ 287 has been presented as an illustration of the ALP-induced features on photon polarization in BL Lacs (see <cit.> for details). Present and planned observatories in the X-ray band <cit.> and in the HE range <cit.> possess the capability to detect possible ALP signatures as those analyzed in Section <ref>, as shown by the realistic binned data reported therein. An analysis about ALP effects on photon polarization in the VHE band has been performed in <cit.>: we will briefly discuss this topic in Section <ref> along with its future crucial importance for the possible identification of an ALP signal.

The ALP-induced features on the final photon degree of linear polarization Π_L discussed in Section <ref> mainly originate inside the galaxy cluster central region and/or in the blazar jet, while the effect of other crossed media is subdominant, as shown . Specifically, a negligible photon–ALP conversion inside the extragalactic space (when B_ ext < 10^-15  G) does not substantially change the results apart from a smaller broadening of Π_L shown in the behavior of its associated probability density function f_Π <cit.>.

In particular, photons diffusely emitted in the central region of galaxy clusters are expected to be observed as unpolarized within conventional physics both in the X-ray and in the HE range (i.e., with Π_L = 0, see Section <ref>). Instead, the photon–ALP interaction model consistently predicts Π_L > 0 for them in both energy ranges. Therefore, galaxy clusters appear among the best observational targets for studies concerning ALP-induced effects on photon polarization. Specifically, a possible future detection of photons diffusely emitted from galaxy clusters with Π_L > 0 would represent a hint at ALP existence in addition to the already-existing ones <cit.>, since conventional physics along with other scenarios like Lorentz invariance violation <cit.> is unable to explain Π_L > 0 (see <cit.> for details). BL Lacs are also promising targets especially in the cases where the photon–ALP interaction model predicts Π_L > 0.8 (see Section <ref>), since, in this case, even the hadronic emission model can hardly justify such high polarization degrees within conventional physics (for details, see <cit.>). However, in case of lower Π_L values detected from BL Lacs, the exclusion of an explanation within conventional physics and the demonstration of the need for ALPs may be less robust. Note that the magnetic field in the jet B^ jet is expected to be quite turbulent, which implies a lower initial photon degree of linear polarization Π_L,0 especially within the leptonic emission model, as shown by a recent theoretical study <cit.> and confirmed by IXPE results <cit.>. Consequently, the final Π_L predicted by the photon–ALP interaction model is reduced in the cases where the initial Π_L,0 remains the most probable final value, as shown by the behavior of the corresponding f_Π in Section <ref>.

Other possible targets to evaluate ALP effects on photon polarization are represented by (i) white dwarfs, where ALP bounds can be derived from polarization measurements <cit.>; (ii) neutron stars, where ALPs produce polarization signatures detectable in the X-ray band <cit.>; and (iii) GRBs, where the photon–ALP conversion over cosmological distances strongly modifies the initial photon polarization degree, as shown in the pioneering study performed in <cit.>. We will focus our attention in Section <ref> on GRBs due to their importance also for the Universe transparency problem regarding the recent detection by the LHAASO Collaboration of GRB 221009A up to 18   TeV <cit.>, which is difficult to justify within conventional physics, but is instead explained by the photon–ALP interaction (see Section <ref> and <cit.> for details).

[-25]As far as the ALP parameter space (m_a, g_aγγ) is concerned, we have analyzed two cases: (i) m_a ≲ 10^-14  eV , g_aγγ = 0.5 × 10^-11  GeV^-1 and (ii) m_a = 10^-10  eV, . Both the previous possibilities are within the CAST bound <cit.>. The former ALP scenario, i.e., the case with m_a ≲ 10^-14  eV, is disfavored by some recent bounds <cit.>, but this possibility cannot be excluded, as discussed in <cit.>. Instead, the latter choice, i.e., the case with m_a = 10^-10  eV, meets all existing bounds and is compatible with the ALP parameters employed in previous studies, where ALP hints have been found <cit.>. This is the reason why the case m_a = 10^-10  eV, g_aγγ = 0.5 × 10^-11  GeV^-1 appears to be more promising. As discussed in <cit.>, other possibilities, such as an ALP with mass m_a = 10^-11  eV and coupling to photons g_aγγ= 0.5 × 10^-11  GeV^-1, will produce an intermediate behavior between the two cases considered in Section <ref> and can meet all current ALP bounds. Note that, while an increment (decrement) of the strength of g_aγγ implies a corresponding increase (decrease) of the ALP effects on the final Π_L and χ at a given energy, a variation in the value of m_a produces the modification of the energy threshold, starting from which the ALP-induced polarization features become relevant. In particular, as shown in Section <ref>, for m_a ≲ 10^-14  eV down to extremely low ALP masses, we do not expect modifications in the ALP-induced effects on Π_L and χ, since the ALP mass term turns out to be subdominant with respect to the plasma term in Equation (<ref>). For 10^-14  eV≲ m_a ≲ 10^-10  eV, the ALP mass term starts to become important and progressively overcomes the strength of the plasma term with a resulting translation at higher energies of the threshold above which the ALP-induced effects become sizable. For the above-mentioned ALP mass m_a = 10^-11  eV, substantial effects on Π_L and χ are expected starting from E_0 ≳𝒪(2-5)   keV, while, as shown in Section <ref>, for m_a = 10^-10  eV, the ALP impact on Π_L and χ is relevant for E_0 ≳𝒪(0.2-0.5)   MeV. Lastly, for m_a progressively increasing above 10^-10  eV, the ALP-induced effects on Π_L and χ will correspondingly appear at larger and larger energies up to disappear in the HE range and to become sizable only in the VHE band and beyond.

As far as polarization measurements are concerned, unfortunately, we do not presently have many observational data regarding the astrophysical sources considered in this review, i.e., galaxy clusters, blazars, and GRBs. In particular, polarization data in both the X-ray and HE bands regarding diffuse emission from galaxy clusters are substantially missing. Since, as discussed above, these latter astrophysical [-15]sources are extremely important for analyzing ALP-induced effects on photon polarization and for studying ALP physics in general, a dedicated observational campaign by IXPE <cit.> could provide vital information in this regard. Concerning blazars, we now have recent polarimetric data obtained in the X-ray band by IXPE (see, e.g., <cit.>), which promises to provide us with further important results in the near future. Regarding GRBs, some polarization measurements have been obtained in the X-ray band <cit.>. For all the astrophysical sources discussed in this review, further polarimetric data are expected from new missions proposed both in the X-ray band <cit.> and in the HE range <cit.>.






 §.§ Very High Energy Band


In the VHE band, we consider the case of a BL Lac emitting within the leptonic model and placed inside a rich galaxy cluster, as studied in <cit.>; therefore, this represents an alternative possibility with respect to those contemplated in Sections <ref> and <ref>. In the present situation, photon–ALP interaction inside the galaxy cluster is relevant, while the conversion inside the host is subdominant (see Section <ref> for more details). The total transfer matrix of the photon–ALP beam is represented by Equation (<ref>). In the following, two possibilities are evaluated: (i) a BL Lac with average properties (see below) is located at a redshift z=0.03; (ii) the same BL Lac is placed at z=0.4. The BL Lac redshift represents a crucial parameter to evaluate the photon–ALP system behavior in the VHE band: since, for E_0 ≳ 100   GeV, the absorption of VHE photons due to their interaction with the EBL is not negligible, as it occurs instead for lower energies, as in Sections <ref> and <ref> (see also Section <ref>), a larger redshift means a higher photon absorption.

In the present scenario, the profiles of the magnetic field in the jet B^ jet and of the electron number density n_e^ jet are provided by Equations (<ref>) and (<ref>), respectively, by taking the following parameter values: B^ jet_0 = 0.5   G, y_ em = 3 × 10^16  cm, n_e,0^ jet=5 × 10^4   cm^-3, and γ = 15. Photons are assumed to be produced within the leptonic model as unpolarized, i.e., with the initial degree of the linear polarization Π_L,0=0 (see <cit.> for more details). Moreover, we consider a CC galaxy cluster, hosting the BL Lac, with the electron number density profile

    n_e^ clu(y)=n_e,0^ clu( 1+y^2/r_ core^2)^-3/2β_ clu,

with the typical parameter values n_e,0^ clu=5 × 10^-2  cm^-3, r_ core= 100   kpc, and β_ clu=2/3 <cit.>. The cluster magnetic field B^ clu reads from Equation (<ref>) by taking the following typical parameter values: B_0^ clu=15   G, η_ clu=0.75, q=-11/3, k_L=0.2   kpc^-1, and k_H=3   kpc^-1 <cit.>. Furthermore, we assume an extragalactic magnetic field strength B_ ext < 10^-15  G, resulting in an inefficient photon–ALP conversion inside the extragalactic space. Concerning the ALP parameters, we take g_aγγ=0.5 × 10^-11  GeV^-1 and m_a = 10^-10  eV. 

For a BL Lac with the above-discussed properties, we show P_γ→γ and Π_L in the top subfigure of Figure <ref> and the corresponding f_Π associated with the final Π_L in the bottom subfigure. The left panels of Figure <ref> report the results for the BL Lac placed at z=0.03, while the right panels show the same results but for the BL Lac located at z=0.4 (see <cit.> for more details). 




The top subfigure of Figure <ref> shows that P_γ→γ and Π_L are energy dependent, since the photon–ALP system lies in the weak mixing regime due to the strength of the QED one-loop term, which is not negligible at high energies especially inside the BL Lac jet. Specifically, the final Π_L turns out to be strongly modified with respect to the initial one Π_L,0 in both cases of the BL Lac placed at z=0.03 and at z=0.4. Now, a new aspect must be considered in the VHE band with respect to the cases at lower energies discussed in Sections <ref> and <ref>: the top subfigure of Figure <ref> shows that P_γ→γ considerably decreases due to the EBL absorption starting from E_0 ≳ 3   TeV at z=0.03 and from E_0 ≳ 200   GeV at z=0.4. Correspondingly, Π_L starts to grow up to its limit value Π_L=1, which is reached at E_0 ≳ 30   TeV for z=0.03 and at E_0 ≳ 5   TeV for z=0.4, where photons are fully polarized and P_γ→γ≲ 10^-2. The different behavior with respect to the redshift is due to the fact that the EBL absorption grows as both the energy and the redshift increase. As deeply discussed in <cit.>, we have therefore found a peculiar feature: in the presence of photon–ALP interaction, photons, observed at energies where EBL absorption is very strong, are fully polarized. The full explanation of this important feature regarding Π_L, and shown in the top subfigure of Figure <ref>, is provided in <cit.>: in short, after photon–ALP interaction inside the source, we have both photons and ALPs entering the extragalactic space. Then, when EBL absorption is extremely high, all remaining photons are lost during their propagation in the extragalactic space, and we therefore collect only those photons that are reconverted back from ALPs in the Milky Way magnetic field B_ MW. Still, only the coherent component of B_ MW is efficient in producing photon–ALP conversion so that photons reconverted back from ALPs inside the Milky Way turn out to be fully polarized.

The behavior of f_Π in the bottom subfigure of Figure <ref> confirms the robustness of the previous discussion. In particular, upper panels deal with the case of small EBL absorption, while the lower ones describe the situation illustrated above, in which Π_L tends to its limit value Π_L=1 with a concomitant high EBL absorption. Note that the behavior of f_Π in the left panels is similar to that in the right ones, but different energies are considered (see the legend inside the panels) due to the above-described behavior of the EBL absorption as a function of energy and redshift. Since conventional physics cannot explain the scenario Π_L=1, a possible detection of fully polarized VHE photons not only would be an additional hint at ALPs in addition to those discussed in <cit.>, but also would represent the proof of ALP existence. Therefore, this ALP-induced feature identified in <cit.> producing fully polarized photons at VHE possesses a high discovery power. However, this possibility nowadays represents only a hope for the future, as currently known techniques cannot measure photon polarization in the VHE band but are limited to tens of GeV at most <cit.>.



 §.§ Gamma-Ray Bursts


ALP effects on the polarization of photons produced by GRBs have been analyzed in a pioneering study <cit.>, where photon–ALP conversion is evaluated inside the source, in the extragalactic space, inside possibly crossed galaxy clusters, and in the Milky Way within the tens of KeV-few MeV band. In particular, the benchmark energy E_0 = 100   keV and the ALP parameters g_aγγ= 10^-11  GeV^-1, m_a = 10^-13  eV are taken into account.

GRBs are extremely intense explosions originating at cosmological distances and are characterized by a prompt and an afterglow phase. By considering the typical parameters of the prompt phase—which is analyzed in <cit.>—regarding the magnetic field strength and the electron number density, the photon–ALP conversion inside the source turns out to be negligible. Regarding the magnetic field structure in extragalactic space, galaxy clusters, and Milky Way, domain-like models with different parameter values inside the several crossed media are assumed in <cit.>. In particular, by calling B_ ext, B_ clu, and B_ MW the magnetic field strengths in extragalactic space, galaxy clusters, and Milky Way, respectively, and L_ dom^ ext, L_ dom^ clu, and L_ dom^ MW the corresponding magnetic field coherence lengths, the following values are assumed: B_ ext=𝒪(1)   nG, B_ clu=𝒪(1)  μ G, B_ MW=𝒪(1-5)   G and , L_ dom^ clu=𝒪(10)   kpc, L_ dom^ MW=𝒪(10)   kpc (see <cit.> for more details).

For the chosen parameter values concerning both the photon–ALP system and the crossed media, the photon–ALP beam propagates in the strong mixing regime in all the previous zones within the considered energy band. Consequently, both P_γ→γ and Π_L turn out to be energy independent with Π_L which is modified with respect to the initial Π_L,0. Therefore, the only informative quantity in the present situation is represented by f_Π, which is shown in Figure <ref> for photon–ALP conversion inside the extragalactic space. In particular, the behavior of f_Π for different initial values of Π_L,0 is shown in the top subfigure of Figure <ref> for a GRB placed at redshift z=0.3 and in the bottom subfigure for a GRB located at z=2. Note that, in Figure <ref>, the initial degree of linear polarization is denoted by Π_0 and the photon–ALP coupling by g_aγ, but these quantities possess the same meaning of Π_L,0 and g_aγγ, respectively, which are used in the whole review.



Figure <ref> shows that the initial Π_L,0 (we recall the different notation here with ) is strongly modified and broadened in a similar way as it occurs in the systems considered in Sections <ref> and <ref>. In particular, the bottom subfigure shows that this effect increases with the redshift. Moreover, as shown in <cit.>, the photon–ALP beam propagation inside galaxy clusters and in the Milky Way further increases the broadening effect of the initial Π_L,0 to a point that, at high redshift, the most probable value of the final Π_L is no more the initial one, apart from the case of the initial Π_L,0≡Π_0 = 1.

These findings demonstrate that GRBs are very important astrophysical sources where ALP-induced effects on photon polarization can be studied, 
in addition to galaxy clusters and BL Lacs (see Sections <ref> and <ref>). The recent detection by the LHAASO Collaboration of photons with energy of up to 18   TeV produced by GRB 221009A <cit.>—which is difficult to justify in the context of conventional physics—shows the need to extend the analysis of the ALP impact on GRB polarization even at higher energy bands in a similar way as discussed in Sections <ref> and <ref>. Moreover, the explanation of the GRB 221009A detection up to 18   TeV within the photon–ALP interaction model developed in <cit.> demonstrates the importance of the photon–ALP conversion also inside the galaxy hosting the GRB. Therefore, an analysis similar to that performed in <cit.>, by considering the photon–ALP beam propagation also in the host galaxy and updated models for the photon–ALP conversion inside extragalactic space, galaxy clusters, and Milky Way, appears very promising in order to discover further ALP-induced features on photon polarization and even possible new hints at ALP existence or to improve constraints on the ALP parameter space.



§ CONCLUSIONS


In this review, we have described the ALP-induced effects on photon polarization, which can provide a different and complementary way to study the ALP physics with respect to the analysis of the ALP consequences on astrophysical spectra. As mentioned above, there exist three hints at ALP existence linked to the ALP effects on astrophysical spectra, with the most recent and strongest indication coming from the detection by the LHAASO Collaboration of GRB 221009A up to 18   TeV <cit.>, which can hardly be justified within conventional physics but is naturally explained by photon–ALP interaction <cit.> (see also <cit.>). In the near future, we expect that new spectral data from observatories such as ASTRI <cit.>, CTA <cit.>, GAMMA-400 <cit.>, HAWC <cit.>, HERD <cit.>, LHAASO <cit.>, and TAIGA-HiSCORE <cit.> will be able to increase our knowledge of the ALP properties.

The study of the ALP effects on the polarization of photons generated by astrophysical sources such as galaxy clusters, blazars, GRBs, neutron stars, and white dwarfs can add further information by shedding light on the ALP parameter space with current spectral hints suggesting m_a = 𝒪(10^- 10)   eV and g_a γγ = 𝒪(10^- 11)   GeV^- 1. As discussed in this review, galaxy clusters represent very promising observational targets in this respect. In fact, according to conventional physics, photons are emitted unpolarized in the central region of galaxy clusters both in the X-ray and in the HE band, resulting in an expected zero polarization degree for the observed photons, while the photon–ALP interaction makes them partially polarized with ALP parameters in the same ballpark of those producing spectral effects. Therefore, a possible future signal of partially polarized photons diffusely produced inside galaxy clusters would be a further strong hint at ALP existence. A negative signal would instead be useful to constrain the ALP parameter space further. In any case, as discussed in this review, future observations of the photon polarization degree from other astrophysical sources like blazars and GRBs are also very promising. Missions such as IXPE <cit.> (already operative), eXTP <cit.>, XL-Calibur <cit.>, NGXP <cit.>, XPP <cit.> in the X-ray band and like COSI <cit.>, e-ASTROGAM <cit.>, and AMEGO <cit.> in the HE range may test our findings about ALP effects on photon polarization. 

Finally, an ALP direct detection could be achieved by laboratory experiments such as ALPS II <cit.>, IAXO <cit.> and STAX <cit.>, thanks to the techniques developed by Avignone and collaborators <cit.> and possibly with the ABRACADABRA experiment <cit.>.



 




























The work of the author is supported by a contribution through Grant No. ASI-INAF 2023-17-HH.0 and by the INAF Mini Grant `High-energy astrophysics and axion-like particles', PI: Giorgio Galanti.

Data used in this paper can be asked from the authors of the quoted papers. 










The author thanks Giacomo Bonnoli, Silvano Molendi, and Lara Nava for the discussions and all collaborators in this field, especially Enrico Costa, Alessandro De Angelis, Marco Roncadelli, and Fabrizio Tavecchio.

The author declares no conflict of interest. 



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References














999

kaluzaklein1 Chang, S.; Tazawa, S.; Yamaguchi, M. Axion model in extra dimensions with TeV scale gravity. Phys. Rev. D 2000, 61, 084005. [http://doi.org/10.1103/PhysRevD.61.084005CrossRef]

kaluzaklein2 Dienes, K.R.; Dudas, E.; Gherghetta, T. Invisible axions and large-radius compactifications. Phys. Rev. D 2000, 62, 105023. [http://dx.doi.org/10.1103/PhysRevD.62.105023CrossRef]

turok Turok, N. Almost-Goldstone Bosons from Extra-Dimensional Gauge Theories. Phys. Rev. Lett. 1996, 76, 1015. [http://dx.doi.org/10.1103/PhysRevLett.76.1015CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/10061612PubMed]

susy1 Corianò, C.; Irges, N. Windows over a new low energy axion. Phys. Lett. B 2007, 651, 298–305. [http://dx.doi.org/10.1016/j.physletb.2007.06.022CrossRef]

susy2 Corianò, C.; Irges, N.; Morelli, S. Stückelberg axions and the effective action of anomalous abelian models 1. A unitarity analysis of the Higgs-axion mixing. J. High Energy Phys. 2007, 2007, 008. [http://dx.doi.org/10.1088/1126-6708/2007/07/008CrossRef]

susy3 Baer, H.; Krami, S.; Sekmen, S.; Summy, H. Dark matter allowed scenarios for Yukawa-unified SO(10) SUSY GUTs. J. High Energy Phys. 2008, 2008, 056. [http://dx.doi.org/10.1088/1126-6708/2008/03/056CrossRef]

susy4 Baer, H.; Summy, H. SO(10) SUSY GUTs, the gravitino problem, non-thermal leptogenesis and axino dark matter. Phys. Lett. B 2008, 666, 5–9. [http://dx.doi.org/10.1016/j.physletb.2008.06.072CrossRef]

susy5 Baer, H.; Haider, M.; Kraml, S.; Sekmen, S.; Summy, H.  Cosmological consequences of Yukawa-unified SUSY with mixed axion/axino cold and warm dark matter. J. Cosmol. Astropart. Phys. 2009, 2009, 002. [http://dx.doi.org/10.1088/1475-7516/2009/02/002CrossRef]

string1 Witten, E. Some properties of O(32) superstrings. Phys. Lett. B 1984, 149, 351–356. [http://dx.doi.org/10.1016/0370-2693(84)90422-2CrossRef]

string2 Conlon, J.P. The QCD axion and moduli stabilisation. J. High Energy Phys. 2006, 2006, 078. [http://dx.doi.org/10.1088/1126-6708/2006/05/078CrossRef]

string3 Svrcek, P.; Witten, E. Axions in string theory. J. High Energy Phys. 2006, 2006, 051. [http://dx.doi.org/10.1088/1126-6708/2006/06/051CrossRef]

string4 Conlon, J.P. Seeing an Invisible Axion in the Supersymmetric Particle Spectrum. Phys. Rev. Lett. 2006, 97, 261802. [http://dx.doi.org/10.1103/PhysRevLett.97.261802CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/17280417PubMed]

string5 Choi, K.-S.; Kim, I.-W.; Kim, J.E. String compactification, QCD axion and axion–photon–photon coupling. J. High Energy Phys. 2007, 2007, 116. [http://dx.doi.org/10.1088/1126-6708/2007/03/116CrossRef]

axiverse Arvanitaki, A.; Dimopoulos, S.; Dubovsky, S.; Kaloper, N.; March-Russell, J. String axiverse. Phys. Rev. D 2010, 81, 123530. [http://dx.doi.org/10.1103/PhysRevD.81.123530CrossRef]

abk2010 Acharya, B.S.; Bobkov, K.; Kumar, P. An M theory solution to the strong CP-problem, and constraints on the axiverse. J. High Energy Phys. 2010, 11, 105. [http://dx.doi.org/10.1007/JHEP11(2010)105CrossRef]

cicoli2012 Cicoli, M.; Goodsell, M.; Ringwald, A. The type IIB string axiverse and its low-energy phenomenology. J. High Energy Phys. 2012, 10, 146. [http://dx.doi.org/10.1007/JHEP10(2012)146CrossRef]

alp1 Jaeckel, J.; Ringwald, A. The Low-Energy Frontier of Particle Physics. Ann. Rev. Nucl. Part. Sci. 2010, 60, 405–437. [http://dx.doi.org/10.1146/annurev.nucl.012809.104433CrossRef]

alp2 Ringwald, A. Exploring the role of axions and other WISPs in the dark universe. Phys. Dark Univ. 2012, 1, 116–135. [http://dx.doi.org/10.1016/j.dark.2012.10.008CrossRef]

axionrev1 Kim, J.H. Light pseudoscalars, particle physics and cosmology. Phys. Rep. 1987, 150, 1–177. [http://dx.doi.org/10.1016/0370-1573(87)90017-2CrossRef]

axionrev2 Cheng, H.Y. The strong CP problem revisited. Phys. Rep. 1988, 158, 1–89. [http://dx.doi.org/10.1016/0370-1573(88)90135-4CrossRef]

axionrev3 Kim, J.E.; Carosi, G. Axions and the strong CP problem. Rev. Mod. Phys. 2010, 82, 557. [http://dx.doi.org/10.1103/RevModPhys.82.557CrossRef]

axionrev4 Marsch, D.J.E. Axion cosmology. Phys. Rep. 2016, 643, 1. [http://dx.doi.org/10.1016/j.physrep.2016.06.005CrossRef]

preskill Preskill, J.; Wise, M.B.; Wilczek, F. Cosmology of the invisible axion. Phys. Lett. B 1983, 120, 127–132. [http://dx.doi.org/10.1016/0370-2693(83)90637-8CrossRef]

abbott Abbott, L.F.; Sikivie, P. A cosmological bound on the invisible axion. Phys. Lett. B 1983, 120, 133–136. [http://dx.doi.org/10.1016/0370-2693(83)90638-XCrossRef]

dine Dine, M.; Fischler, W. The not-so-harmless axion. Phys. Lett. B 1983, 120, 137–141. [http://dx.doi.org/10.1016/0370-2693(83)90639-1CrossRef]

arias2012 Arias, P.; Cadamuro, D.; Goodsell, M.; Jaeckel, J.; Redondo, J.; Ringwald, A. WISPy cold dark matter. J. Cosmol. Astropart. Phys. 2012, 2012, 013. [http://dx.doi.org/10.1088/1475-7516/2012/06/013CrossRef]

mpz Maiani, L.; Petronzio, R.; Zavattini, E. Effects of nearly massless, spin-zero particles on light propagation in a magnetic field. Phys. Let. B 1986, 175, 359–363. [http://dx.doi.org/10.1016/0370-2693(86)90869-5CrossRef]

raffeltstodolsky Raffelt, G.G.; Stodolsky, L. Mixing of the photon with low-mass particles. Phys. Rev. D 1988, 37, 1237. [http://dx.doi.org/10.1103/PhysRevD.37.1237CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/9958803PubMed]

irastorzaredondo Irastorza, I.G.; Redondo, J. New experimental approaches in the search for axion-like particles. Prog. Part. Nucl. Phys. 2018, 102, 89–159. [http://dx.doi.org/10.1016/j.ppnp.2018.05.003CrossRef]

grRew Galanti, G.; Roncadelli, M. Axion-like Particles Implications for High-Energy Astrophysics. Universe 2022, 8, 253. [http://dx.doi.org/10.3390/universe8050253CrossRef]

trg2015 Tavecchio, F.; Roncadelli, M.; Galanti, G. Photons to axion-like particles conversion in Active Galactic Nuclei. Phys. Lett. B 2015, 744, 375–379. [http://dx.doi.org/10.1016/j.physletb.2015.04.017CrossRef]

trgb2012 Tavecchio, F.; Roncadelli, M.; Galanti, G.; Bonnoli, G. Evidence for an axion-like particle from PKS 1222+216? Phys. Rev. D 2012, 86, 085036. [http://dx.doi.org/10.1103/PhysRevD.86.085036CrossRef]

grdb Galanti, G.; Roncadelli, M.; De Angelis, A.; Bignami, G.F. Hint at an axion-like particle from the redshift dependence of blazar spectra. Mon. Not. Astron. R. Soc. 2020, 493, 1553. [http://dx.doi.org/10.1093/mnras/stz3410CrossRef]

drm2007 De Angelis, A.; Roncadelli, M.; Mansutti, O. Evidence for a new light spin-zero boson from cosmological gamma-ray propagation? Phys. Rev. D 2007, 76, 121301. [http://dx.doi.org/10.1103/PhysRevD.76.121301CrossRef]

dgr2011 De Angelis, A.; Galanti, G.; Roncadelli, M. Relevance of axionlike particles for very-high-energy astrophysics. Phys. Rev. D 2011, 84, 105030; Erratum in Phys. Rev. D 2013, 87, 109903. [http://dx.doi.org/10.1103/PhysRevD.84.105030CrossRef]

grExt Galanti, G.; Roncadelli, M. 
Extragalactic photon–axion-like particle oscillations up to 1000 TeV.  J. High Energy Astrophys. 2018, 20, 1–17. [http://dx.doi.org/10.1016/j.jheap.2018.07.002CrossRef]

dgr2013 De Angelis, A.; Galanti, G.; Roncadelli, M. Transparency of the Universe to gamma rays. Mon. Not. Astron. R. Soc. 2013, 432, 3245. [http://dx.doi.org/10.1093/mnras/stt684CrossRef]

franceschinirodighiero Franceschini, A.; Rodighiero, G. The extragalactic background light revisited and the cosmic photon-photon opacity. Astron. Astrophys. 2017, 603, A34. [http://dx.doi.org/10.1051/0004-6361/201629684CrossRef]

gptr Galanti, G.; Piccinini, F.; Roncadelli, M.; Tavecchio, F. Estimating γγ absorption for ultrahigh-energy photons with lepton and hadron production. Phys. Rev. D 2020, 102, 123004. [http://dx.doi.org/10.1103/PhysRevD.102.123004CrossRef]

gtre2019 Galanti, G.; Tavecchio, F.; Roncadelli, M.; Evoli, C. Blazar VHE spectral alterations induced by photon–ALP oscillations. Mon. Not. Astron. R. Soc. 2019, 487, 123. [http://dx.doi.org/10.1093/mnras/stz1144CrossRef]

gtl2020 Galanti, G.; Tavecchio, F.; Landoni, M. Fundamental physics with blazar spectra: A critical appraisal. Mon. Not. Astron. R. Soc. 2020, 491, 5268. [http://dx.doi.org/10.1093/mnras/stz3411CrossRef]

astri Vercellone, S.; Bigongiari, C.; Burtovoi, A.; Cardillo, M.; Catalano, O.; Franceschini, A.; Lombardi, S.; Nava, L.; Pintore, F.; Stamerra, A.; et al. ASTRI Mini-Array core science at the Observatorio del Teide. J. High Energy Astrophys. 2022, 35, 1–42. [http://dx.doi.org/10.1016/j.jheap.2022.05.005CrossRef]

cta  Available online: <https://www.cta-observatory.org/> (accessed on 10 June 2024).

g400 Egorov, A.E.; Topchiev, N.P.; Galper, A.M.; Dalkarov, O.D.; Leonov, A.A.; Suchkov, S.I.; Yurkin, Y.T. Dark matter searches by the planned gamma-ray telescope GAMMA-400. J. Cosmol. Astropart. Phys. 2020, 11, 049. [http://dx.doi.org/10.1088/1475-7516/2020/11/049CrossRef]

hawc Available online: <https://www.hawc-observatory.org/> (accessed on 10 June 2024).

herd Huang, X.; Lamperstorfer, A.S.; Tsai, Y.L.S.; Xu, M.; Yuan, Q.; Chang, J.; Dong, Y.W.; Hu, B.L.; Lü, J.G.; Wang, L.; et al. Perspective of monochromatic gamma-ray line detection with the High Energy cosmic-Radiation Detection (HERD) facility onboard China's space station. Astropart. Phys. 2016, 78, 35–42. [http://dx.doi.org/10.1016/j.astropartphys.2016.02.003CrossRef]

LHAASOsens  Cao, Z.; della Volpe, D.; Liu, S.; Bi, X.; Chen, Y.; D'Ettorre Piazzoli, B.; Feng, L.; Jia, H.; Li, Z.; Ma, X.; et al. The Large High Altitude Air Shower Observatory (LHAASO) Science Book (2021 Edition). Chin. Phys. C 2022,  46, 035001–035007.

desy Available online: <https://taiga-experiment.info/taiga-hiscore/> (accessed on 10 June 2024).

fermi2016 Ajello, M. et al. [Fermi-LAT collaboration]. Search for spectral irregularities due to photon–axionlike-particle oscillations with the Fermi Large Area Telescope. Phys. Rev. Lett. 2016, 116, 161101. [http://dx.doi.org/10.1103/PhysRevLett.116.161101CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/27152783PubMed]

CTAfund Abdalla, H.; Abe, H.; Acero, F.; Acharyya, A.; Adam, R.; Agudo, I.; Aguirre-Santaella, A.; Alfaro, R.; Alfaro, J.; Alispach, C.; et al. Sensitivity of the Cherenkov Telescope Array for probing cosmology and fundamental physics with gamma-ray propagation. J. Cosmol. Astropart. Phys. 2021, 2021, 048. [http://dx.doi.org/10.1088/1475-7516/2021/02/048CrossRef]

straniero Ayala, A.; Domínguez, I.; Giannotti, M.; Mirizzi, A.; Straniero, O. Revisiting the Bound on Axion-Photon Coupling from Globular Clusters. Phys. Rev. Lett. 2014, 113, 191302. [http://dx.doi.org/10.1103/PhysRevLett.113.191302CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/25415896PubMed]

thermal Conlon, J.P.; Marsh, M.C.D.; Powell, A.J. Galaxy cluster thermal x-ray spectra constrain axionlike particles. Phys. Rev. D 2016, 93, 123526. [http://dx.doi.org/10.1103/PhysRevD.93.123526CrossRef]

DMdecay Jaeckel, J.; Redondo, J.; Ringwald, A. 3.55 keV hint for decaying axionlike particle dark matter. Phys. Rev. D 2014, 89, 103511. [http://dx.doi.org/10.1103/PhysRevD.89.103511CrossRef]

wang Zhou, J.; Wang, Z.; Huang, F.; Chen, L. A possible blazar spectral irregularity case caused by photon-axionlike-particle oscillations. J. Cosmol. Astropart. Phys. 2021, 2021, 007. [http://dx.doi.org/10.1088/1475-7516/2021/08/007CrossRef]

LHAASO LHAASO Collaboration. GCN Circular n. 32677. 2022. Available online: <https://gcn.gsfc.nasa.gov/gcn3/32677.gcn3> (accessed on 10 June 2024).

LHAASOspectrumHigh Cao, Z. et al. [The LHAASO Collaboration]. Very high-energy gamma-ray emission beyond 10 TeV from GRB 221009A. Sci. Adv. 2023, 9, eadj2778.

gnrtb2023 Galanti, G.; Nava, L.; Roncadelli, M.; Tavecchio, F.; Bonnoli, G. Observability of the Very-High-Energy Emission from GRB 221009A. Phys. Rev. Lett. 2023, 131, 251001. [http://dx.doi.org/10.1103/PhysRevLett.131.251001CrossRef] [http://www.ncbi.nlm.nih.gov/pubmed/38181368PubMed]

ALPinGRB2 Galanti, G.; Roncadelli, M.; Tavecchio, F. Assessment of ALP scenarios for GRB 221009A. arXiv 2022, arXiv:2211.06935.

ixpe Weisskopf, M.C.; Soffitta, P.; Baldini, L.; Ramsey, B.D.; O’Dell, S.L.; Romani, R.W.; Matt, G.; Deininger, W.D.; Baumgartner, W.H.; Bellazzini, R.; et al. Imaging X-ray Polarimetry Explorer: Prelaunch. J. Astron. Telesc. Instrum. Syst. 2022, 8, 026002. [http://dx.doi.org/10.1117/1.JATIS.8.2.026002CrossRef]

extp  Zhang, S.; Santangelo, A.; Feroci, M.; Xu, Y.; Lu, F.; Chen, Y.; Feng, H.; Zhang, S.; Brandt, S.; Hernanz, M.; et al. The enhanced X-ray Timing and Polarimetry mission–eXTP. Sci. China Phys. Mech. Astron. 2019, 62, 29502. [http://dx.doi.org/10.1007/s11433-018-9309-2CrossRef]

xcalibur Abarr, Q.; Awaki, H.; Baring, M.G.; Bose, R.; De Geronimo, G.; Dowkontt, P.; Errando, M.; Guarino, V.; Hattori, K.; Hayashida, K.;  et al. XL-Calibur—A second-generation balloon-borne hard X-ray polarimetry mission. Astropart. Phys. 2021, 126, 102529. [http://dx.doi.org/10.1016/j.astropartphys.2020.102529CrossRef]

ngxp Soffitta, P.; Bucciantini, N.; Churazov, E.; Costa, E.; Dovciak, M.; Feng, H.; Heyl, J.; Ingram, A.; Jahoda, K.; Kaaret, P.;  et al. A polarized view of the hot and violent universe. Exp. Astron. 2021, 51, 1109–1141. [http://dx.doi.org/10.1007/s10686-021-09722-yCrossRef]

xpp Jahoda, K.; Krawczynski, H.; Kislat, F.; Marshall, H.; Okajima, T.; Agudo, I.; Angelini, L.; Bachetti, M.; Baldini, L.; Baring, M.; et al. The X-ray Polarization Probe mission concept. arXiv 2019, arXiv:1907.10190.

cosi Yang, C.-Y.; Lowell, A.; Zoglauer, A.; Tomsick, J.; Chiu, J.-L.; Kierans, C.; Sleator, C.; Boggs, S.; Chang, H.-K.; Jean, P.; et al. The polarimetric performance of the Compton Spectrometer and Imager (COSI). Proc. SPIE 2018, 10699, 642.

eastrogam1 De Angelis, A.; Tatischeff, V.; Tavani, M.; Oberlack, U.; Grenier, I.; Hanlon, L.; Walter, R.; Argan, A.; von Ballmoos, P.; Bulgarelli, A.; et al. The e-ASTROGAM mission. Exp. Astron. 2017, 44, 25–82. [http://dx.doi.org/10.1007/s10686-017-9533-6CrossRef]

eastrogam2 Tatischeff, V. et al. [e-ASTROGAM Collaboration]. The e-ASTROGAM gamma-ray space observatory for the multimessenger astronomy of the 2030s. Proc. SPIE 2018, 10699, 106992J.

amego Kierans, C.A. et al. [AMEGO Collaboration]. AMEGO: Exploring the Extreme Multimessenger Universe. Proc. SPIE 2020, 11444, 1144431.

bassan Bassan, N.; Mirizzi, A.; Roncadelli, M. Axion-like particle effects on the polarization of cosmic high-energy gamma sources. J. Cosmol. Astropart. Phys. 2010, 2010, 010. [http://dx.doi.org/10.1088/1475-7516/2010/05/010CrossRef]

ALPpol1 Jain, P.; Panda, S.; Sarala, S. Electromagnetic polarization effects due to axion-photon mixing. Phys. Rev. D 2002, 66, 085007. [http://dx.doi.org/10.1103/PhysRevD.66.085007CrossRef]

ALPpol2 Agarwal, N.; Kamal, A.; Jain, P. Alignments in quasar polarizations: Pseudoscalar-photon mixing in the presence of correlated magnetic fields. Phys. Rev. D 2011, 83, 065014. [http://dx.doi.org/10.1103/PhysRevD.83.065014CrossRef]

ALPpolMWD Gill, R.; Heyl, J.S. Constraining the photon-axion coupling constant with magnetic white dwarfs. Phys. Rev. D 2011, 84, 085001. [http://dx.doi.org/10.1103/PhysRevD.84.085001CrossRef]

ALPpol3 Payez, A.; Cudell, J.R.; Hutsemekers, D. Can axionlike particles explain the alignments of the polarizations of light from quasars? Phys. Rev. D 2011, 84, 085029. [http://dx.doi.org/10.1103/PhysRevD.84.085029CrossRef]

ALPpol4 Agarwal, N.; Aluri, P.K.; Jain, P.; Khanna, U.; Tiwari, P. A complete 3D numerical study of the effects of pseudoscalar–photon mixing on quasar polarizations. Eur. Phys. J. C 2012, 72, 1928. [http://dx.doi.org/10.1140/epjc/s10052-012-1928-yCrossRef]

ALPpol5 Perna, R.; Ho, W.C.G.; Verde, L.; Adelsberg, M. Jimenez, R. Signatures of photon-axion conversion in the thermal spectra and polarization of neutron stars. Astrophys. J. 2012, 748, 116. [http://dx.doi.org/10.1088/0004-637X/748/2/116CrossRef]

day Day, F.; Krippendorf, S. Searching for axion-like particles with X-ray polarimeters. Galaxies 2018, 6, 45. [http://dx.doi.org/10.3390/galaxies6020045CrossRef]

galantiTheorems Galanti, G. Photon-ALP interaction as a measure of initial photon polarization. Phys. Rev. D 2022, 105, 083022. [http://dx.doi.org/10.1103/PhysRevD.105.083022CrossRef]

galantiPol Galanti, G. Photon-ALP oscillations inducing modifications to photon polarization. Phys. Rev. D 2023, 107, 043006. [http://dx.doi.org/10.1103/PhysRevD.107.043006CrossRef]

grtcClu Galanti, G.; Roncadelli, M.; Tavecchio, F.; Costa, E. ALP induced polarization effects on photons from galaxy clusters. Phys. Rev. D 2023, 107, 103007. [http://dx.doi.org/10.1103/PhysRevD.107.103007CrossRef]

grtPolBla Galanti, G.; Roncadelli, M.; Tavecchio, F. ALP-induced polarization effects on photons from blazars. Phys. Rev. D 2023, 108, 083017. [http://dx.doi.org/10.1103/PhysRevD.108.083017CrossRef]

cast CAST Collaboration. New CAST limit on the axion–photon interaction. Nat. Phys. 2017, 13, 584–590. [http://dx.doi.org/10.1038/nphys4109CrossRef]

payez2015 Payez, A.; Evoli, C.; Fischer, T.; Giannotti, M.; Mirizzi, A.; Ringwald, A. Revisiting the SN1987A gamma-ray limit on ultralight axion-like particles. J. Cosmol. Astropart. Phys. 2015, 2015, 006. [http://dx.doi.org/10.1088/1475-7516/2015/02/006CrossRef]

berg Berg, M.; Conlon, J.P.; Day, F.; Jennings, N.; Krippendorf, S.; Powell, A.J.; Rummel, M. Constraints on axion-like particles from X-ray observations of NGC1275. Astrophys. J. 2017, 847, 101. [http://dx.doi.org/10.3847/1538-4357/aa8b16CrossRef]

conlonLim Conlon,  J.P.; Day, F.; Jennings, N.; Krippendorf, S.; Rummel, M. Constraints on axion-like particles from non-observation of spectral modulations for X-ray point sources. J. Cosmol. Astropart. Phys. 2017, 2017, 005. [http://dx.doi.org/10.1088/1475-7516/2017/07/005CrossRef]

meyer2020 Meyer, M. et al. [Fermi-LAT Collaboration]. Search for Axionlike-Particle-Induced Prompt γ-Ray Emission from Extragalactic Core-Collapse Supernovae with the Fermi Large Area Telescope. Phys. Rev. Lett. 2020, 124, 231101; Erratum in Phys. Rev. Lett. 2020, 125, 119901. [http://dx.doi.org/10.1103/PhysRevLett.124.231101CrossRef]

limFabian Reynolds, C.S.; Marsh, M.C.D.; Russell, H.R.; Fabian, A.C.; Smith, R.; Tombesi, F.; Veilleux, S. Astrophysical limits on very light axion-like particles from Chandra grating spectroscopy of NGC 1275. Astrophys. J. 2020, 890, 59. [http://dx.doi.org/10.3847/1538-4357/ab6a0cCrossRef]

limJulia Sisk-Reynés, J.; Matthews, J.H.; Reynolds, C.S.; Russell, H.R.; Smith, R.N.; Marsh, M.C.D. New constraints on light axion-like particles using Chandra transmission grating spectroscopy of the powerful cluster-hosted quasar H1821+643. Mon. Not. Astron. R. Soc. 2022, 510, 1264. [http://dx.doi.org/10.1093/mnras/stab3464CrossRef]

limKripp Schallmoser, S.; Krippendorf, S.; Chadha-Day, F.; Weller, J. Updated Bounds on Axion-Like Particles from X-ray Observations. Mon. Not. Astron. R. Soc. 2022, 514, 329. [http://dx.doi.org/10.1093/mnras/stac1224CrossRef]

limRey2 Matthews, J.H.; Reynolds, C.S.; Marsh, M.C.D.; Sisk-Reynés, J.; Rodman, P.E. How do Magnetic Field Models Affect Astrophysical Limits on Light Axion-like Particles? An X-ray Case Study with NGC 1275. Astrophys. J. 2022, 930, 90. [http://dx.doi.org/10.3847/1538-4357/ac5625CrossRef]

mwd Dessert, C.; Dunsky, D.; Safdi, B.R. Upper limit on the axion-photon coupling from magnetic white dwarf polarization. Phys. Rev. D 2022, 105, 103034. [http://dx.doi.org/10.1103/PhysRevD.105.103034CrossRef]

hew1 Heisenberg, W.; Euler, H. Folgerungen aus der diracschen theorie des positrons. Z. Phys. 1936, 98, 714–732. [http://dx.doi.org/10.1007/BF01343663CrossRef]

hew2 Weisskopf, V.S. Über die elektrodynamik des vakuums auf grund des quanten-theorie des elektrons. K. Dan. Vidensk. Selsk. Mat. Fys. Medd. 1936, 14, 6.

hew3 Schwinger, J. On Gauge Invariance and Vacuum Polarization. Phys. Rev. 1951, 82, 664. [http://dx.doi.org/10.1103/PhysRev.82.664CrossRef]

raffelt2015 Dobrynina, A.; Kartavtsev, A.; Raffelt, G. Photon-photon dispersion of TeV gamma rays and its role for photon-ALP conversion. Phys. Rev. D 2015, 91, 083003; Erratum in Phys. Rev. D 2015, 91, 109902. [http://dx.doi.org/10.1103/PhysRevD.91.083003CrossRef]

poltheor1 Kosowsky, A. Cosmic Microwave Background Polarization. Ann. Phys. 1996, 246, 49. [http://dx.doi.org/10.1006/aphy.1996.0020CrossRef]

poltheor2 Rybicki, G.B.; Lightman, A.P. Radiative Processes in Astrophysics; Wiley: New York, NY, USA, 1979.

blazarSeq Ghisellini, G.; Righi, C.; Costamante, L.; Tavecchio, F. The Fermi blazar sequence. Mon. Not. Astron. R. Soc. 2017, 469, 255. [http://dx.doi.org/10.1093/mnras/stx806CrossRef]

jansonfarrar1 Jansson, R.; Farrar, G.R. A New Model of the Galactic Magnetic Field. Astrophys. J. 2012, 757, 14. [http://dx.doi.org/10.1088/0004-637X/757/1/14CrossRef]

jansonfarrar2 Jansson, R.; Farrar, G.R. The galactic magnetic field. Astrophys. J. 2012, 761, L11. [http://dx.doi.org/10.1088/2041-8205/761/1/L11CrossRef]

yaomanchesterwang2017 Yao, J.M.; Manchester, R.N.; Wang, N. A New Electron-density Model for Estimation of Pulsar and FRB Distances. Astrophys. J. 2017, 835, 29. [http://dx.doi.org/10.3847/1538-4357/835/1/29CrossRef]

optical 
Roming, P.W.A.; Kennedy, T.E.; Mason, K.O.; Nousek, J.A.; Ahr, L.; Bingham, R.E.; Broos, P.S.; Carter, M.J.; Hancock, B.K.; Huckle, H.E.; et al. The Swift Ultra-Violet/Optical Telescope. Space Sci. Rev. 2005, 120, 95. [http://dx.doi.org/10.1007/s11214-005-5095-4CrossRef]

swift Burrows, D.N.; Hill, J.E.; Nousek, J.; Kennea, J.A.; Wells, A.; Osborne, J.P.; Abbey, A.F.; Beardmore, A.; Mukerjee, K.; Short, A.D.T.;  et al. The Swift X-ray Telescope. Space Sci. Rev. 2005, 120, 165. [http://dx.doi.org/10.1007/s11214-005-5097-2CrossRef]

fermiSens Available online: <https://www.slac.stanford.edu/exp/glast/groups/canda/lat_Performance.htm> (accessed on 10 June 2024).

AGNfig1 Urry, C.M.; Padovani, P. Unified schemes for radio-loud active galactic nuclei. Pub. Astron. Soc. Pacific 1995, 107, 803. [http://dx.doi.org/10.1086/133630CrossRef]

AGNfig2 Collmar, W. News from CGRO-COMPTEL. Mem. Soc. Astron. Ital. 2002, 73, 99.

AGNfig3 Torres, D.F.; Anchordoqui, L.A. Astrophysical origins of ultrahigh energy cosmic rays. Rep. Prog. Phys. 2004, 67, 1663. [http://dx.doi.org/10.1088/0034-4885/67/9/R03CrossRef]

FSRQabs1 Liu, T.H.; Bai, J.M. Absorption of 10–200 GeV gamma rays by radiation from broad-line regions in blazars. Astrophys. J. 2006, 653, 1089. [http://dx.doi.org/10.1086/509097CrossRef]

FSRQabs2 Tavecchio, F.; Mazin, D. Intrinsic absorption in 3C 279 at GeV–TeV energies and consequences for estimates of the extragalactic background light. Mon. Not. Astron. R. Soc. 2009, 392, L40. [http://dx.doi.org/10.1111/j.1745-3933.2008.00584.xCrossRef]

FSRQabs3 Poutanen, J.; Stern, B. GeV Breaks in Blazars as a Result of Gamma-ray Absorption within the Broad-Line Region. Astrophys. J. 2010, 717, L118. [http://dx.doi.org/10.1088/2041-8205/717/2/L118CrossRef]

gltcTorus Galanti, G.; Landoni, M.; Tavecchio, F.; Covino, S. Probing the absorption of gamma-rays by IR radiation from the dusty torus in FSRQs with the Cherenkov telescope array. Mon. Not. Astron. R. Soc. 2020, 495, 3463. [http://dx.doi.org/10.1093/mnras/staa1391CrossRef]

FSRQvhe1 Albert, J.; Aliu, E.; Anderhub, H.; Antonelli, L.A.; Antoranz, P.; Backes, M.; Baixeras, C.; Barrio, J.A.; Bartko, H.; Bastieri, D.; et al. Very-High-Energy Gamma Rays from a Distant Quasar: How Transparent Is the Universe? Science 2008, 320, 1752. [http://dx.doi.org/10.1126/science.1157087CrossRef]

FSRQvhe2 Aleksić, J.; Antonelli, L.A.; Antoranz, P.; Backes, M.; Barrio, J.A.; Bastieri, D.; Becerra González, J.; Bednarek, W.; Berdyugin, A.; Berger, K.; et al. MAGIC Discovery of Very High Energy Emission from the FSRQ PKS 1222+21. Astrophys. J. 2011, 730, L8. [http://dx.doi.org/10.1088/2041-8205/730/1/L8CrossRef]

FSRQvhe3 Wagner, S.J.; Behera, B. Hess Observations of Quasars. In Proceedings of the 10th HEAD Meeting, BAAS, Honolulu, HI, USA, 5 July 2010; Volume 41.

riaf1 Ghisellini, G.; Maraschi, L.; Tavecchio, F. The Fermi blazars’ divide. Mon. Not. Astron. R. Soc. 2009, 396, L105–L109. [http://dx.doi.org/10.1111/j.1745-3933.2009.00673.xCrossRef]

riaf2 Righi, C.; Tavecchio, F.; Inoue, S. Neutrino emission from BL Lac objects: The role of radiatively inefficient accretion flows. Mon. Not. Astron. R. Soc. 2019, 483, L127–L131. [http://dx.doi.org/10.1093/mnrasl/sly231CrossRef]

bbr1984 Begelman, M.C.; Blandford, R.D.; Rees, M.J. Theory of extragalactic radio sources. Rev. Mod. Phys. 1984, 56, 255–351. [http://dx.doi.org/10.1103/RevModPhys.56.255CrossRef]

ghisellini2009 Ghisellini, G.; Tavecchio, F. Canonical high-power blazars. Mon. Not. Astron. R. Soc. 2009, 397, 985–1002. [http://dx.doi.org/10.1111/j.1365-2966.2009.15007.xCrossRef]

pudritz2011 Pudritz, R.E.; Hardcastle, M.J.; Gabuzda, D.C. Magnetic Fields in Astrophysical Jets: From Launch to Termination. Space Sci. Rev. 2012, 169, 27–72. [http://dx.doi.org/10.1007/s11214-012-9895-zCrossRef]

tavecchio2010 Tavecchio, F.; Ghisellini, G.; Ghirlanda, G.; Foschini, L.; Maraschi, L. TeV BL Lac objects at the dawn of the Fermi era. Mon. Not. Astron. R. Soc. 2010, 401, 1570–1586. [http://dx.doi.org/10.1111/j.1365-2966.2009.15784.xCrossRef]

Maraschi92 Maraschi, L.; Ghisellini, G.; Celotti, A. A jet model for the gamma-ray emitting blazar 3C 279. Astrophys. J. 1992, 397, L5–L9. [http://dx.doi.org/10.1086/186531CrossRef]

Sikora94 Sikora, M.; Begelman, M.G.; Rees, M.J. Comptonization of diffuse ambient radiation by a relativistic jet: The source of gamma rays from blazars? Astrophys. J. 1994, 421, 153–162. [http://dx.doi.org/10.1086/173633CrossRef]

ssc1 Bloom, S.D.; Marscher, A.P. An analysis of the synchrotron self-compton model for the multi–wave band spectra of blazars. Astrophys. J. 1996, 461, 657. [http://dx.doi.org/10.1086/177092CrossRef]

mannheim1 Mannheim, K.  The proton blazar. Astron. Astrophys. 1993, 269, 67–76.

mannheim2 Mannheim, K. TeV gamma-rays from proton blazars. Space Sci. Rev. 1996, 75, 331–340. [http://dx.doi.org/10.1007/BF00195043CrossRef]

Muecke2003 Mücke, A.; Protheroe, R.J.; Engel, R.; Rachen, J.P.; Stanev, T. BL Lac objects in the synchrotron proton blazar model. Astropart. Phys. 2003, 18, 593. [http://dx.doi.org/10.1016/S0927-6505(02)00185-8CrossRef]

zhangBott Zhang, H.; Böttcher, M. X-ray and Gamma-Ray Polarization in Leptonic and Hadronic Jet Models of Blazars. Astrophys. J. 2013, 774, 18. [http://dx.doi.org/10.1088/0004-637X/774/1/18CrossRef]

SpiralBrev Beck, R. Magnetic fields in spiral galaxies. Astron. Astrophys. Rev. 2016, 24, 4. [http://dx.doi.org/10.1007/s00159-015-0084-4CrossRef]

Fletcher2010 Fletcher, A. Magnetic Fields in Nearby Galaxies. The Dynamic Interstellar Medium: A Celebration of the Canadian Galactic Plane Survey. Astron. Soc. Pac. Conf. Ser. 2011, 438, 197.

Thompson2006 Thompson, T.A.; Quataert, E.; Waxman, E.; Murray, N.;  Martin, C.L. Magnetic Fields in Starburst Galaxies and the Origin of the FIR-Radio Correlation. Astrophys. J. 2006, 645, 186. [http://dx.doi.org/10.1086/504035CrossRef]

LopezRodriguez2021 Lopez-Rodriguez, E.; Guerra, J.A.; Asgari-Targhi, M.; Schmelz, J.T. The Strength and Structure of the Magnetic Field in the Galactic Outflow of Messier 82. Astrophys. J. 2021, 914, 24. [http://dx.doi.org/10.3847/1538-4357/abf934CrossRef]

GRBposition Blanchard, P.K.; Berger, E.; Fong, W. The Offset and Host Light Distributions of Long Gamma-Ray Bursts: A New View from HST Observations of Swift Bursts. Astrophys. J. 2016, 817, 144. [http://dx.doi.org/10.3847/0004-637X/817/2/144CrossRef]

GRBposition2 Lyman, J.D.; Levan, A.J.; Tanvir, N.R.; Fynbo, J.P.U.; McGuire, J.T.W.; Perley, D.A.; Angus, C.R.; Bloom, J.S.; Conselice, C.J.; Fruchter, A.S.;  et al. The host galaxies and explosion sites of long-duration gamma-ray bursts: Hubble Space Telescope near-infrared imaging. Mon. Not. Astron. R. Soc. 2017, 467, 1795–1817. [http://dx.doi.org/10.1093/mnras/stx220CrossRef]

GRB221009Ahost Levan, A.J.; Lamb, G.P.; Schneider, B.; Hjorth, J.; Zafar, T.; de Ugarte Postigo, A.; Sargent, B.; Mullally, S.E.; Izzo, L.; d’Avanzo, P.;  et al. The First JWST Spectrum of a GRB Afterglow: No Bright Supernova in Observations of the Brightest GRB of all Time, GRB 221009A. Astrophys. J. Lett. 2023, 946, L28. [http://dx.doi.org/10.3847/2041-8213/acc2c1CrossRef]

moss1996 Moss, D.; Shukurov, A. Turbulence and Magnetic Fields in Elliptical Galaxies. Mon. Not. Astron. R. Soc. 1996, 279, 229–239. [http://dx.doi.org/10.1093/mnras/279.1.229CrossRef]

GalCluClass Bahcall, N.A. Clusters and superclusters of galaxies. In Formation of Structure in the Universe; Dekel, A., Ostriker, J.P., Eds.; Cambridge University Press: Cambridge, UK, 1999.

cluB1 Govoni, F.; Feretti, L. Magnetic fields in clusters of galaxies. Int. J. Mod. Phys. D 2004, 13, 1549–1594. [http://dx.doi.org/10.1142/S0218271804005080CrossRef]

cluB2 Feretti, L.; Giovannini, G.; Govoni, F.; Murgia, M. Clusters of galaxies: Observational properties of the diffuse radio emission. Astron. Astrophys. Rev. 2012, 20, 54. [http://dx.doi.org/10.1007/s00159-012-0054-zCrossRef]

cluFeretti Bonafede, A.; Feretti, L.; Murgia, M.; Govoni, F.; Giovannini, G.; Dallacasa, D.; Dolag, K.; Taylor, G.B. The Coma cluster magnetic field from Faraday rotation measures. Astron. Astrophys. 2010, 513, A30. [http://dx.doi.org/10.1051/0004-6361/200913696CrossRef]

clu2 Kuchar, P.; Enßlin, T.A. Magnetic power spectra from Faraday rotation maps. Astron. Astrophys. 2011, 529, A13. [http://dx.doi.org/10.1051/0004-6361/200913918CrossRef]

meyerKolm Meyer, M.; Montanino, D.; Conrad, J. On detecting oscillations of gamma rays into axion-like particles in turbulent and coherent magnetic fields. J. Cosmol. Astropart. Phys. 2014, 2014, 003. [http://dx.doi.org/10.1088/1475-7516/2014/09/003CrossRef]

cluValues Hudson, D.S.; Mittal, R.; Reiprich, T.H.; Nulsen, P.E.J.; Andernach, H.; Sarazin, C.L. What is a cool-core cluster? A detailed analysis of the cores of the X-ray flux-limited HIFLUGCS cluster sample. Astron. Astrophys. 2010, 513, A37. [http://dx.doi.org/10.1051/0004-6361/200912377CrossRef]

mitchell1979 Mitchell, R.J.; Dickens, R.J.; Bell Burnell, J.S.; Culhane, J.L. The X-ray spectra of clusters of galaxies and their relationship to other cluster properties. Mon. Not. Astron. R. Soc. 1979, 189, 329. [http://dx.doi.org/10.1093/mnras/189.2.329CrossRef]

cluGammaEm1 Houston, B.P.; Wolfendale, A.W.; Young, E.C.M. Cosmic gamma rays from galaxy clusters. Phys, J. G Nucl. Phys. 1984, 10, L147–L150. [http://dx.doi.org/10.1088/0305-4616/10/6/007CrossRef]

cluGammaEm2 Dennison, B. Formation of radio halos in clusters of galaxies from cosmic-ray protons. Astrophys. J. 1980, 239, L93–L96. [http://dx.doi.org/10.1086/183300CrossRef]

cluGammaEm3 Colafrancesco, S.; Marchegiani, P. Warming rays in cluster cool cores. Astron. Astrophys. 2008, 484, 51–65. [http://dx.doi.org/10.1051/0004-6361:20078428CrossRef]

cluGammaEm4 Timokhin, A.N.; Aharonian, F.A.; Neronov, A.Y. On the non-thermal high energy radiation of galaxy clusters. Astron. Astrophys. 2004, 417, 391. [http://dx.doi.org/10.1051/0004-6361:20040004CrossRef]

polarRev Lei, F.; Dean, A.J.; Hills, G.L. Compton Polarimetry in Gamma-Ray Astronomy. Space Sci. Rev. 1997, 82, 309. [http://dx.doi.org/10.1023/A:1005027107614CrossRef]

FermiPol Giomi, M.; Bühler, R.; Sgrò, C.; Longo, F.; Atwood, W.B.;  on behalf of the Fermi LAT Collaboration. Estimate of the Fermi large area telescope sensitivity to gamma-ray polarization. AIP Conf. Proc. 2017, 1792, 0022.

kronberg1994 Kronberg, P.P. Extragalactic magnetic fields. Rept. Prog. Phys. 1994, 57, 325–382. [http://dx.doi.org/10.1088/0034-4885/57/4/001CrossRef]

grassorubinstein Grasso, D.; Rubinstein, H.R. Magnetic fields in the early Universe. Phys. Rep. 2001, 348, 163–266. [http://dx.doi.org/10.1016/S0370-1573(00)00110-1CrossRef]

wanglai Wang, C.; Lai, D. Axion-Photon Propagation in Magnetized Universe. J. Cosmol. Astropart. Phys. 2016, 6, 006. [http://dx.doi.org/10.1088/1475-7516/2016/06/006CrossRef]

neronovvovk Neronov, A.; Vovk, I. Evidence for Strong Extragalactic Magnetic Fields from Fermi Observations of TeV Blazars. Science 2010, 328, 73. [http://dx.doi.org/10.1126/science.1184192CrossRef]

durrerneronov Durrer, R.; Neronov, A. Cosmological magnetic fields: Their generation, evolution and observation. Astron. Astrophys. Rev. 2013, 21, 62. [http://dx.doi.org/10.1007/s00159-013-0062-7CrossRef]

upbbext Pshirkov, M.S.; Tinyakov, P.G.; Urban, F.R. New Limits on Extragalactic Magnetic Fields from Rotation Measures. Phys. Rev. Lett. 2016, 116, 191302. [http://dx.doi.org/10.1103/PhysRevLett.116.191302CrossRef]

reessetti1968 Rees, M.J.; Setti, G. Model for the Evolution of Extended Radio Sources. Nature 1968, 219, 127. [http://dx.doi.org/10.1038/219127a0CrossRef]

hoyle1969 Hoyle, F. Magnetic Fields and Highly Condensed Objects. Nature 1969, 223, 936. [http://dx.doi.org/10.1038/223936a0CrossRef]

kronbergleschhopp1999 Kronberg, P.P.; Lesch, H.; Hopp, U. Magnetization of the Intergalactic Medium by Primeval Galaxies. Astrophys. J. 1999, 511, 56. [http://dx.doi.org/10.1086/306662CrossRef]

furlanettoloeb2001 Furlanetto, S.; Loeb, A. Intergalactic Magnetic Fields from Quasar Outflows. Astrophys. J. 2001, 556, 619–634. [http://dx.doi.org/10.1086/321630CrossRef]

grSM Galanti, G.; Roncadelli, M. Behavior of axionlike particles in smoothed out domainlike magnetic fields. Phys. Rev. D 2018, 98, 043018. [http://dx.doi.org/10.1103/PhysRevD.98.043018CrossRef]



BMWturb Beck, M.C.; Beck, A.M.; Beck, R.; Dolag, K.; Strong, A.W.;  Nielaba, P. New constraints on modelling the random magnetic field of the MW. J. Cosmol. Astropart. Phys. 2016, 2016, 056. [http://dx.doi.org/10.1088/1475-7516/2016/05/056CrossRef]

pshirkov2011 Pshirkov, M.S.; Tinyakov, P.G.; Kronberg, P.P.; Newton-McGee, K.J. Deriving the Global Structure of the Galactic Magnetic Field from Faraday Rotation Measures of Extragalactic Sources. Astrophys. J. 2011, 738, 192. [http://dx.doi.org/10.1088/0004-637X/738/2/192CrossRef]

nePerseus Churazov, E.; Forman, W.; Jones, C.; Böhringer, H. XMM-Newton Observations of the Perseus Cluster. I. The Temperature and Surface Brightness Structure. Astrophys. J. 2003, 590, 225. [http://dx.doi.org/10.1086/374923CrossRef]

BpersHigh Taylor, G.B.; Gugliucci, N.E.; Fabian, A.C.; Sanders, J.S.; Gentile, G.;  Allen, S.W. Magnetic fields in the centre of the Perseus cluster. Mon. Not. Astron. R. Soc. 2006, 368, 1500–1506. [http://dx.doi.org/10.1111/j.1365-2966.2006.10244.xCrossRef]

BpersLow Aleksić, J. et al. [MAGIC Collaboration]. Constraining cosmic rays and magnetic fields in the Perseus galaxy cluster with TeV observations by the MAGIC telescopes. Astron. Astrophys. 2012, 541, A99. [http://dx.doi.org/10.1051/0004-6361/201118502CrossRef]

LeptHadrBott Böttcher, M.; Reimer, A.; Sweeney, K.; Prakash, A. Leptonic and Hadronic Modeling of Fermi-detected Blazars. Astrophys. J. 2013, 768, 54. [http://dx.doi.org/10.1088/0004-637X/768/1/54CrossRef]

LIVpol Kostelecký, V.A.; Mewes, M. Electrodynamics with Lorentz-violating operators of arbitrary dimension. Phys. Rev. D 2009, 80, 015020. [http://dx.doi.org/10.1103/PhysRevD.80.015020CrossRef]

blazarPolNew Zhang, H.; Böttcher, M.; Liodakis, I. Revisiting High-Energy Polarization from Leptonic and Hadronic Blazar Scenarios. arXiv 2024, arXiv:2404.12475.

IXPEmrk421 Di Gesu, L.; Donnarumma, I.; Tavecchio, F.; Agudo, I.; Barnounin, T.; Cibrario, N.; Di Lalla, N.; Di Marco, A.; Escudero, J.; Errando, M.; et al. The X-ray Polarization View of Mrk 421 in an Average Flux State as Observed by the Imaging X-ray Polarimetry Explorer. Astrophys. J. Lett. 2022, 938, L7. [http://dx.doi.org/10.3847/2041-8213/ac913aCrossRef]

IXPEmrk501 Liodakis, I.; Marscher, A.P.; Agudo, I.; Berdyugin, A.V.; Bernardos, M.I.; Bonnoli, G.; Borman, G.A.; Casadio, C.; Casanova, V.; Cavazzuti, E.; et al. Polarized blazar X-rays imply particle acceleration in shocks. Nature 2022, 611, 677. [http://dx.doi.org/10.1038/s41586-022-05338-0CrossRef]

IXPEblLacertae Middei, R.; Liodakis, I.; Perri, M.; Puccetti, S.; Cavazzuti, E.; Di Gesu, L.; Ehlert, S.R.; Madejski, G.; Marscher, A.P.; Marshall, H.L.;  et al. X-ray Polarization Observations of BL Lacertae. Astrophys. J. Lett. 2023, 942, L10. [http://dx.doi.org/10.3847/2041-8213/aca281CrossRef]

IXPEblLacertae2 Peirson, A.L.; Negro, M.; Liodakis, I.; Middei, R.; Kim, D.E.; Marscher, A.P.; Marshall, H.L.; Pacciani, L.; Romani, R.W.; Wu, K.; et al. X-ray Polarization of BL Lacertae in Outburst. Astrophys. J. Lett. 2023, 948, L25. [http://dx.doi.org/10.3847/2041-8213/acd242CrossRef]

IXPEpg1553 Middei, R.; Perri, M.; Puccetti, S.; Liodakis, I.; Di Gesu, L.; Marscher, A.P.; Cavero, N.R.; Tavecchio, F.; Donnarumma, I.; Laurenti, M.; et al. IXPE and Multiwavelength Observations of Blazar PG 1553+113 Reveal an Orphan Optical Polarization Swing. Astrophys. J. Lett. 2023, 953, L28. [http://dx.doi.org/10.3847/2041-8213/acec3eCrossRef]

IXPE1ES0229 Ehlert, S.R.; Liodakis, I.; Middei, R.; Marscher, A.P.; Tavecchio, F.; Agudo, I.; Kouch, P.M.; Lindfors, E.; Nilsson, K.; Myserlis, I.; et al. X-ray Polarization of the BL Lacertae Type Blazar 1ES 0229+200. Astrophys. J. 2023, 959, 61. [http://dx.doi.org/10.3847/1538-4357/ad05c4CrossRef]

IXPEmrk421-2 Di Gesu, L.; Marshall, H.L.; Ehlert, S.R.; Kim, D.E.; Donnarumma, I.; Tavecchio, F.; Liodakis, I.; Kiehlmann, S.; Agudo, I.; Jorstad, S.G.; et al. Discovery of X-ray polarization angle rotation in the jet from blazar Mrk 421. Nature Astronomy 2023, 7, 1245. [http://dx.doi.org/10.1038/s41550-023-02032-7CrossRef]

IXPE1ES1959 Errando, M.; Liodakis, I.; Marscher, A.P.; Marshall, H.L.; Middei, R.; Negro, M.; Peirson, A.L.; Perri, M.; Puccetti, S.; Rabinowitz, P.L.; et al. Detection of X-ray Polarization from the Blazar 1ES 1959+650 with the Imaging X-ray Polarimetry Explorer. Astrophys. J. 2024, 963, 5. [http://dx.doi.org/10.3847/1538-4357/ad1ce4CrossRef]

IXPEPKS2155 Kouch, P.M.; Liodakis, I.; Middei, R.; Kim, D.E.; Tavecchio, F.; Marscher, A.P.; Marshall, H.L.; Ehlert, S.R.; Di Gesu, L.; Jorstad, S.G.; et al. IXPE observation of PKS 2155-304 reveals the most highly
polarized blazar. arXiv 2024, arXiv:2406.01693.

GRBpol1 Kole, M.; De Angelis, N.; Berlato, F.; Burgess, J.M.; Gauvin, N.; Greiner, J.; Hajdas, W.; Li, H.C.; Li, Z.H.; Pollo, A.;  et al. The POLAR gamma-ray burst polarization catalog. Astron. Astrophys. 2020, 644, A124. [http://dx.doi.org/10.1051/0004-6361/202037915CrossRef]

GRBpol2 Gill, R.; Kole, M.; Granot, J. GRB Polarization: A Unique Probe of GRB Physics. Galaxies 2021, 9, 82. [http://dx.doi.org/10.3390/galaxies9040082CrossRef]

GRBpol3 Chattopadhyay, T.; Gupta, S.; Iyyani, S.; Saraogi, D.; Sharma, V.; Tsvetkova, A.; Ratheesh, A.; Gupta, R.; Mithun, N.P.S.; Vaishnava, C.S.; et al. Hard X-ray Polarization Catalog for a Five-year Sample of Gamma-Ray Bursts Using AstroSat CZT Imager. Astrophys. J. 2022, 936, 12. [http://dx.doi.org/10.3847/1538-4357/ac82efCrossRef]

IXPE-GRB221009A Negro, M.; Di Lalla, N.; Omodei, N.; Veres, P.; Silvestri, S.; Manfreda, A.; Burns, E.; Baldini, L.; Costa, E.; Ehlert, S.R.;  et al. The IXPE View of GRB 221009A. Astrophys. J. Lett. 2023, 946, L21. [http://dx.doi.org/10.3847/2041-8213/acba17CrossRef]

polLimit Depaola, G.O.; Kozameh, C.N.; Tiglio, M.H. A method to determine the polarization of high energy gamma rays. Astropart. Phys. 1999, 10, 175. [http://dx.doi.org/10.1016/S0927-6505(98)00054-1CrossRef]

alps2 Bähre, R.; Döbrich, B.; Dreyling-Eschweiler, J.; Ghazaryan, S.; Hodajerdi, R.; Horns, D.; Januschek, F.; Knabbe, E.-A.; Lindner, A.; Notz, D.; et al. Any light particle search II—Technical design report. J. Instrum. 2013, 8, T09001. [http://dx.doi.org/10.1088/1748-0221/8/09/T09001CrossRef]





iaxo Armengaud, E.; Attié, D.; Basso, S.; Brun, P.; Bykovskiy, N.; Carmona, J.M.; Castel, J.F.; Cebrián, S.; Cicoli, M.; Civitani, M.; et al. Physics potential of the international axion observatory (IAXO). J. Cosmol. Astropart. Phys. 2019, 2019, 047. [http://dx.doi.org/10.1088/1475-7516/2019/06/047CrossRef]

stax Capparelli, L.M.; Cavoto, G.; Ferretti, J.; Giazotto, F.; Polosa, A.D.; Spagnolo, P. Axion-like particle searches with sub-THz photons. Phys. Dark Univ. 2016, 12, 37. [http://dx.doi.org/10.1016/j.dark.2016.01.003CrossRef]

avignone1 Avignone, F.T., III. Potential for large germanium detector arrays for solar-axion searches utilizing the axio-electric effect for detection. Phys. Rev. D 2009, 79, 035015. [http://dx.doi.org/10.1103/PhysRevD.79.035015CrossRef]

avignone2 Avignone, F.T., III; Crewick, R.J.; Nussinov, S.  Can large scintillators be used for solar-axion searches to test the cosmological axion–photon oscillation proposal? Phys. Lett. B 2009, 681, 122–124. [http://dx.doi.org/10.1016/j.physletb.2009.09.011CrossRef]

avignone3 Avignone, F.T., III; Crewick, R.J.; Nussinov, S. The experimental challenge of detecting solar axion-like particles to test the cosmological ALP-photon oscillation hypotheses. Astropart. Phys. 2011, 34, 640–642. [http://dx.doi.org/10.1016/j.astropartphys.2010.12.012CrossRef]

abracadabra Kahn, Y.; Safdi, B.R.; Thaler, J. Broadband and resonant approaches to axion dark matter detection. Phys. Rev. Lett. 2016, 117, 141801. [http://dx.doi.org/10.1103/PhysRevLett.117.141801CrossRef]






























