Paper Title: Brane-vector dark matter and its connection to inflation and primordial gravitational waves.








nam.caohoang@phenikaa-uni.edu.vn (corresponding author) 
Phenikaa Institute for Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam

hung.tranngoc@phenikaa-uni.edu.vn  
Phenikaa Institute for Advanced Study and Faculty of Fundamental Sciences, Phenikaa University, Yen Nghia, Ha Dong, Hanoi 12116, Vietnam


§ ABSTRACT 
The scalar mode describing the fluctuation of the 3-brane (the observable universe) in a five-dimensional bulk spacetime compactified on a circle is absorbed by the Kaluza-Klein U(1) gauge field, leading to a massive brane-vector living on the 3-brane. The brane-vector can be responsible for dark matter because it is odd under a Z_2 symmetry, neutral under the Standard Model (SM) symmetries, and couples extremely weak to the SM particles due to its gravitational origin. Interestingly, the brane-vector dark matter could leave particular imprints on the cosmic microwave background (CMB) and the primordial gravitational waves. Hence, the precise measurements of the CMB and the observations of the primordial gravitational waves generated during the inflation can provide a potential way to probe the extra-dimensions and branes which are the main ingredients of string/M theory.



Brane-vector dark matter and its connection to inflation and primordial gravitational waves
    Tran N. Hung
    August 5, 2024
===========================================================================================



§ INTRODUCTION


Extra-dimensions and branes (hypersurfaces embedded in a higher-dimensional spacetime or the bulk) play a fundamental role in string/M theory which has been considered as a promising candidate for quantum gravity. String/M theory predicts that the spacetime has spatial extra-dimensions wrapped upon a compact internal space and open strings describing the non-gravitational sector with their endpoints are attached to branes. Whereas, closed strings describing the gravitational sector can propagate freely in the bulk. The low energy regime of string/M theory hence can imply a scenario that our observable universe is actually a 3-brane on which the Standard Model (SM) fields are confined, the directions transverse to the 3-brane are compact, and only the gravitational degrees of freedom and other exotic ones can move freely in the bulk <cit.>. In such a scenario, the constraints on the size of the extra-dimensions could be obtained from deviations from the usual Newton's law <cit.> and the experiments of gravitational waves (GW) <cit.>. Therefore, the experimental bounds of the size of the extra-dimensions can be relaxed, where the size of the extra-dimensions is as large as the micron scale. Interestingly, the compactified extra-dimensions can provide a possible solution to the hierarchy problem where the fundamental Planck scale can be much lower than the observed Planck scale <cit.>.

The brane tension is an important parameter that characterizes the typical size of the brane fluctuations along the extra-dimensions. If the brane tension is large enough, corresponding to the branes being too heavy, the branes can be considered as rigid objects. On the contrary, when the brane tension is much lower than the fundamental scale, the branes can fluctuate along the extra-dimensions. The fluctuations corresponding to the oscillating brane are realized as the Nambu-Goldstone bosons, also well-known as the branons, which arise as a result of the spontaneously broken isometry of the internal space due to the presence of the brane <cit.>. The branons are the physical scalar excitations in the low energy spectrum of the effective field theory generically only in the global case where the physical degrees of freedom describing the higher-dimensional gravitation are negligible. 

From the Kaluza-Klein (KK) theory, we know that the isometry of the internal space leads to the gauge transformations where the gauge fields are the off-diagonal components of the bulk metric <cit.> or the connection on the spacetime fiber bundle with the typical fiber being the internal manifold <cit.>. This implies when the isometry of the internal space is broken spontaneously by the presence of the brane, the KK gauge fields would absorb the branons as their longitudinal modes, corresponding to a gravitational Brout-Englert-Higgs mechanism. This leads to massive vector fields appearing on the brane, as well-known as brane-vector <cit.>. The mass of the brane-vector is determined by the brane tension, and hence it is a free parameter. Because the brane tension is much lower than the fundamental Planck scale for the case of the flexible branes, we expect that the mass of the brane-vector can range from the eV regime to the TeV regime. The brane-vector interacts with the SM fields via their energy-momentum tensor with the coupling determined by the observed Planck scale as a result of the fact that the brane-vector has a gravitational origin as new dynamic degrees of freedom describing the behavior of the spacetime at the short-distances <cit.>.

In this paper, we explore the brane-vector as a candidate for dark matter (DM) whose stability is guaranteed by
a Z_2 parity as a result of the 4D general coordinate invariance on the brane. The coupling of the brane-vector to the SM particles is strongly suppressed by the observed Planck scale, hence the relic abundance of the brane-vector DM could not generated by the thermal processes that appeared after the inflation. On the other hand, the brane-vector DM which couples extremely weak to the SM particles could be produced by the gravitational mechanism due to the quantum fluctuations in the inflationary stage <cit.>. The transverse polarizations of a massive vector field that couples minimally to 4D gravity are conformally invariant in analogy to the massless vector field, since its production is suppressed and could be negligible during the inflation <cit.>. In this sense, the observed relic abundance of the brane-vector DM should result from the production of its longitudinal polarization.

The Friedmann equation describing the evolution of the brane universe consists of linear and quadratic terms in terms of the energy density. The quadratic term is inversely proportional to the mass of the brane-vector or the brane tension since it plays an important role in the early universe. (Whereas, the linear term is dominant when the energy density is much smaller than the brane tension corresponding to the late time of the universe evolution.) This novel feature of the modified Friedmann equation combined with the production of the brane-vector DM during the inflation leads to a prediction of the tensor-to-scalar ratio and the reheating temperature in terms of the brane-vector mass. Note that, the reheating temperature can be estimated by the primordial GW spectrum <cit.>. In this way, the brane-vector DM could leave specific imprints on the cosmic microwave background (CMB) and the primordial GWs. Therefore, the precise measurements of the CMB as well as the future detections of the primordial GWs and the B-mode polarization in the CMB can allow us to probe the extra-dimensions and branes.

This paper is organized as follows. In Sec. <ref>, we study the four-dimensional effective field theory derived from a flexible 3-brane which propagates in a bulk spacetime compactified on a circle S^1 and on which the SM and inflaton fields are confined. We show that the low-energy spectrum of the effective theory includes a brane-vector that possesses a discrete Z_2 parity and hence could be responsible for the DM. Then, we compute the relic abundance of the brane-vector DM based on the production of its longitudinal mode during the inflation. In Sec. <ref>, we study the imprints of the brane-vector on the spectrum of the CMB and the primordial GWs carrying the information of the inflation era for two typical potentials of the inflation: i) α-attractor potential and ii) natural potential. Finally, we present a brief conclusion in Sec. <ref>.





§  BRANE-VECTOR DM


For simplicity without loss of generality, we consider the dynamics of a 3-brane propagating in a five-dimensional bulk spacetime compactified on a circle S^1. The action of the system is given by a sum of the bulk action and the brane action as follows

    S   =   S_bulk+S_brane,
    
    S_bulk   =   M^3_5/2∫ d^5X√(-G)(ℛ^(5)-2Λ_5),
    
    S_brane   =   ∫ d^4x√(-g)[-f^4+ℒ_SM+1/2g^μν∂_μϕ∂_νϕ-V(ϕ)].

In the bulk action (<ref>), M_5 is the five-dimensional Planck scale, ℛ^(5) is the Ricci scalar of the bulk spacetime, G is the determinant of the bulk metric, X^M with M=0,1,2,3,4 are the coordinates of the points on the bulk spacetime, and Λ_5 refers to the bulk cosmological constant which is necessarily negative to restore the conventionally Hubble expansion law in the low-energy regime <cit.>. In the brane action (<ref>), f is the tension of the 3-brane, the coordinates of the points on the 3-brane are parameterized by x^μ with μ=0,1,2,3, and g is the determinant of the induced metric g_μν on the 3-brane. The SM fields that are described by the Lagrangian ℒ_SM and the inflaton field ϕ are confined to the 3-brane. 

The presence of the 3-brane could be created spontaneously by a proper physical process, such as the tachyon condensation when turning on the magnetic fluxes <cit.>. This configuration should be realized as the ground state. As a result, the formation of the 3-brane would break spontaneously the local isometry of the bulk spacetime in the compactified extra dimension. As we see below, the Nambu-Goldstone (NG) mode describing the excitation of the 3-brane would be eaten by the KK gauge field, corresponding to the local U(1) isometry of the circle S^1, hence it becomes massive. 



 §.§ Dimensional reduction of the bulk action


Let us study the compactification of the five-dimensional bulk spacetime on the circle S^1 that is parametrized by an angle θ. The bulk metric can be decomposed into the four-dimensional fields as follows

    G_MN=(
    [ g_μν+ρ^2 A_μ A_ν          ρ^2 A_μ;          ρ^2 A_ν              ρ^2;                  ]
    ),

where g_μν is the four-dimensional metric tensor, A_μ is the KK gauge field corresponding to the local isometry in the fifth compactified dimension, and ρ is a scalar field whose vacuum expectation value (vev) determines physically the radius of the circle S^1. With this decomposition, we can write the bulk action (<ref>) in terms of the four-dimensional fields as follows.

    S_bulk   =   M^2_P/2⟨ρ⟩∫ d^4x√(-g)ρ(ℛ^(4)-2Λ_5-ρ^2/4F_μνF^μν+2/3∂^μρ∂_μρ/ρ^2),

where g is the determinant of the four-dimensional metric g_μν, ℛ^(4) is the four-dimensional Ricci scalar, F_μν≡∂_μ A_ν-∂_ν A_μ is the field strength tensor of the KK gauge field, and M_P is the observed Planck scale determined as M^2_P=2π⟨ρ⟩ M^3_5 with ⟨ρ⟩ denoting the vev of the scalar field ρ or the radius of the circle S^1. We can check that the gravitational action (<ref>) is invariant under the following transformation 

    θ   ⟶   θ+α(x),
    
    A_μ   ⟶   A_μ-∂_μα(x).

A set of such transformations forms the local isometry in the compactified dimension associated with a particular class of general coordinate transformations. In particular, the off-diagonal element of the bulk metric transforms as a gauge field of the U(1) local symmetry.

Reducing the bulk action (<ref>) on the internal manifold S^1, we obtain an effective action that consists of the four-dimensional Einstein gravity plus a cosmological constant coupled to the massless U(1) gauge field and the scalar field as follows

    S_4D   ⊃   ∫ d^4x√(g)[M^2_P/2(ℛ^(4)-2Λ_5)-1/4κ^2F_μνF^μν+1/2∂^μ r∂_μ r+⋯],

where we have expanded the scalar field ρ around its vev as ρ=⟨ρ⟩(1+√(3)r/2) with r to be the radion field, κ is the KK gauge coupling given as 

    κ≡√(2)⟨ρ⟩^-1/M_P,

and the ellipse refers to the terms related to the self-couplings of the radion field and its couplings to the four-dimensional metric and the KK gauge field.



 §.§ Effective action on the 3-brane


The induced metric on the 3-brane can be determined by the embedding geometry as ds^2_brane=g_μνdx^μ dx^ν where g_μν=G_MN∂_μ Y^M∂_ν Y^N with Y^M(x) being the embedding function which describes the position of the 3-brane in the bulk spacetime. In the static gauge, the embedding function is given by Y^M(x)=(x^μ,Y^θ(x)). With the ansatz of the bulk metric as given by Eq. (<ref>), the induced metric g_μν is found as follows

    g_μν   =   g_μν-ρ^2(A_μ+∂_μ Y^θ)(A_ν+∂_ν Y^θ),
       =   g_μν-⟨ρ⟩^2(A_μ+∂_μ Y^θ)(A_ν+∂_ν Y^θ)+𝒪(r).


When the 3-brane is created at a certain point Y^θ_0 in the fifth compactified dimension which minimizes the brane action (<ref>) and corresponds to the ground state. In this situation, the presence of the 3-brane breaks spontaneously the U(1) local isometry in the fifth compactified dimension. The fluctuations of the 3-brane around the ground state in the fifth compactified dimension can be parametrized as Y^θ(x)=Y^θ_0+π(x) where π(x) is the NG mode associated with the excitation of the 3-brane along the direction of the U(1) local isometry breaking. As a result, the KK gauge field would absorb this NG mode as its longitudinal mode, leading to the brane-vector. This brane-vector is expressed by the definition X_μ≡ A_μ+∂_μπ. The field strength tensor of the brane-vector is given by 

    X_μν≡∂_μ X_ν-∂_ν X_μ=F_μν.


It should be noted that the NG mode describing the fluctuation of the 3-brane in the fifth compactified dimension would be the physical excitation in the decoupling limit of the KK gauge field corresponding to take the limit κ→0. This physical excitation is well known as the branon which is massive in the situation that the U(1) global isometry in the fifth compactified dimension is realized as an approximate symmetry. Interestingly, it was indicated that the branon can be responsible for the DM candidate <cit.>. However, as indicated in Ref. <cit.>, this limit leads to the violation of the Sublattice Weak Gravity Conjecture <cit.>, the Swampland Distance Conjecture <cit.>, and the Festina Lente bound <cit.> which are recognized as the constraints of quantum gravity on the low-energy effective theories. On the other hand, this limit would belong to the Swampland which consists of the effective field theories that cannot be ultraviolet completed in quantum gravity. 

In this way, we can rewrite the induced metric on the 3-brane in terms of the four-dimensional metric and the brane-vector as follows

    g_μν=g_μν-⟨ρ⟩^2X_μ X_ν+𝒪(r).

Using this result, we can expand the square root of the determinant of the induced metric on the 3-brane as follows

    √(-g)=√(-g)(1-⟨ρ⟩^2/2X^μ X_μ+⋯),

where g refers to the determinant of the four-dimensional metric g_μν. With Eqs. (<ref>) and (<ref>), we can find an effective action on the 3-brane as follows

    S_4D   ⊃   ∫ d^4x√(-g)(-f^4+m^2_X/2κ^2X_μ X^μ+λ/κ^2X^μ X^ν T^mt_μν+ℒ_SM+1/2g^μν∂_μϕ∂_νϕ-V(ϕ)+⋯),

where m_X=√(2)f^2/M_P is the mass of the brane-vector which is acquired due to the spontaneous breaking of the U(1) local isometry in the fifth compactified dimension, T^mt_μν is the energy-momentum tensor of the matte fields (the SM and inflation fields) which is given by

    T^mt_μν=2δℒ_mt/δ g^μν-g_μνℒ_mt,

with ℒ_mt denote the Lagrangian of the matter fields living on the 3-brane, and λ=1/M^2_P is the coupling constant of the brane-vector to the matter fields.

From the actions (<ref>) and (<ref>), we observe that first, the brane-vector is odd under a discrete symmetry Z_2. This means that the effective action (<ref>) is invariant under the transformation X_μ→-X_μ, which means that the brane-vector always appears in the pairs. Hence, this discrete symmetry would guarantee the stability of the brane vector. Second, in analogy to the graviton, the brane-vector couples to the energy-momentum tensor of the matter fields with the coupling strength suppressed strongly by the observed Planck scale. This is due to the fact that the brane vector has a geometric or gravitational origin associated with the off-diagonal component of the bulk metric. Because the brane-vector is stable, neutral, and couples very weakly to the matter fields, it behaves as the DM. 



 §.§ Production of brane-vector particles during the inflation


Because the brane-vector is a massive vector field, it has three physical polarizations corresponding to two transverse components and one longitudinal component which is the NB mode describing the fluctuation of the 3-brane in the fifth compactified dimension. However, the production of the transverse polarizations of the brane-vector by the quantum fluctuations is suppressed during the inflation because the transverse polarizations are conformally invariant. This implies that the contribution of the transverse polarizations to the production of the energy density of the brane-vector can be negligible. Whereas, unlike transverse polarizations, the production of the longitudinal polarization is unsuppressed because it is not conformally invariant. Therefore, the relic abundance of the brane-vector DM which is observed today is essentially due to the production of its longitudinal polarization during the inflation.

The relic abundance of the brane-vector observed today is

    Ω_X=ρ_X/ρ_c,

where ρ_c is the critical density and ρ_X is the present energy density of the brane-vector. As discussed above, the contribution to the present energy density of the brane-vector is approximately that of its longitudinal mode and it is calculated as follows <cit.>

    ρ_X≃ρ_X,L=H^2_i/8π^2a_*k^2_*/a^3_0∫ dln kk^2/k^2_*[a_0/a_*|X_L(k,t_0)/X_0(k)|^2+a^3_0/a_*(k^2+a^2_0m^2_X)|∂_tX_L(k,t_0)/X_0(k)|^2],

where H_i refers to the Hubble parameter during the inflation, a_0 and a_* are the values of the scale factor at the present time and the moment of H=m_X, respectively, the wavenumber k_* is defined as k_*=a_*m_X. X_L(k,t_0) is the solution of the equation of motion for the longitudinal polarization of the brane-vector at the present time, given by

    (∂^2_t+3k^2+a^2m^2_X/k^2+a^2m^2_XH∂_t+k^2/a^2+m^2_X)X_L(k,t)=0,

with an initial condition at the horizon exit given as

    X_L(k,t)|_exit=X_0(k)(1-ik/aH_i)exp{ik/aH_i}.

By solving numerically Eq. (<ref>), we can find X_L(k,t) and then substitute it into the integral given in Eq. (<ref>) to obtain an approximate result which is equal to 1.2 <cit.>. As a result, we derive the energy density of the brane-vector observed at the present time as follows

    ρ_X≃3m^2_XH^2_i/20π^2(a_*/a_0)^3.

From the expansion law along the 3-brane as <cit.>

    H^2+k^2/a^2=(ρ_m+M^2_Pm^2_X/2/6M^3_5)^2+Λ_5/6,

and the relation a_*≃ a_mre√(H_mre/m_X) with a_mre and H_mre denoting the scale factor and the Hubble parameter at the time of the matter-radiation equality, we can determine the ratio a_*/a_0 in Eq. (<ref>) as

    a_*/a_0=(ρ_r,0/(√(7)-1)M^2_P)^1/41/√(m_X),

where ρ_r,0 is the energy density of the radiation at the present time. In addition, using the tensor power spectrum amplitude in the brane-world scenario as A_t≃12x_0[H_i/(2π M_P)]^2 with x_0≡2√(3H^2_i M^2_P/(2f^4)) <cit.> and the relation between the tensor power spectrum amplitude A_t and the scalar power spectrum amplitude A_s given by the tensor-to-scalar ratio r as r=A_t/A_s, we can determine the Hubble parameter H_i in terms of the inflationary quantities that can be observed in the CMB spectrum as

    H^3_i=π^2A_sr/6√(3)m_XM^2_P.

Finally, we find an expression for the relic abundance of the brane-vector DM in terms of its mass, the tensor-to-scalar ratio and the scalar power spectrum amplitude as

    Ω_Xh^2   ≃   0.16×(rA_sM^2_P/10^26GeV^2)^2/3(m_X/10^3 GeV)^7/6.


Using the constraints from both the Planck data on the observed DM relic abundance Ω_CDMh^2≃0.12 <cit.> and the 95% CL upper limit on the tensor-to-scalar ratio as r<0.056 <cit.>, we depict the allowed parameter space for the tensor-to-scalar ratio r and the brane-vector mass m_X in Fig. <ref>.

We observe that these constraints impose a lower bound on the brane-vector mass as m_X≳262.45 GeV. This bound can be transferred to an upper bound on the radius of the compactified extra-dimension. First, we need to determine the brane-vector mass in terms of the radius of the compactified extra-dimension. We observe that the expansion law on the 3-brane as given by Eq. (<ref>) would lead to the standard expansion law in the low energy limit of ρ_m≪ M^2_Pm^2_X/2 and the bulk cosmological constant satisfying 

    Λ_5=-6M^6_5/M^4_P.

Using this relation, we find an expression for the brane-vector mass in terms of ⟨ρ⟩ as follows

    m_X=√(3)/π⟨ρ⟩.

Finally, we determine the constraint on the radius of the compactified extra-dimension as ⟨ρ⟩≲4.14×10^-19 m in order for the brane-vector responsible for the DM candidate. Obviously, this upper bound satisfies the constraint coming from the fifth-force experiments <cit.> which set an upper bound ⟨ρ⟩≲𝒪(40) μm. In addition, we find that the brane-vector mass is about a few hundred GeV for the region r∈[𝒪(10^-3),𝒪(10^-2)] which is predicted in some potential inflation models and is experimentally accessible in the near future/present experiments.




§  CONNECTION OF BRANE-VECTOR DM TO INFLATION AND PRIMORDIAL GRAVITATIONAL WAVE

As discussed above, the brane-vector DM has the geometric origin by which its coupling to the SM particles would be strongly suppressed by the observed Planck scale. Hence, it is extremely hard to detect the signatures of the brane-vector DM at the colliders. However, the inflation and the primordial GWs generated during the inflation are very sensitive to the presence of the brane-vector DM. This can be seen from the non-standard expansion law given by Eq. (<ref>) with the allowed mass region of m_X≳262.45 GeV. The very different behavior of the expansion rate compared to the usual one due to the presence of the brane-vector DM becomes significant in the energy density region around/above 5.32×10^39 GeV^4. Therefore, the production of the brane-vector DM due to the quantum fluctuations during the inflation would leave imprints in the spectrum of the CMB and the primordial GWs whose accurate measurements allow us to probe the brane-vector DM and hence the compactified extra dimension and the brane universe. 

In the following, we assume that the brane-vector particles constitute almost the observed relic abundance by which the tensor-to-scalar ratio can be expressed in terms of the brane-vector mass as follows

    r≃(50.55 GeV/m_X)^7/4.

From this relation, we observe that if the tensor-to-scalar ratio is not too small, the energy density during the inflation is much larger than M^2_Pm^2_X/2. As a result, the expansion law during the inflation is given by

    H_i≃V(ϕ)/√(3)M^2_Pm_X.

The equation of motion which plays the behavior of the inflaton during the inflation in the slow-roll approximation reads

    3H_iϕ̇≃-V'(ϕ).

Also, we can determine the slow-roll parameters in terms of the inflation potential, its derivatives, and the brane-vector mass as follows

    ϵ≃M^4_Pm^2_X/V(ϕ)( V'(ϕ)/V(ϕ))^2,   η≃M^4_Pm_X^2/V(ϕ)V”(ϕ)/V(ϕ).

These slow-roll parameters are related to the scalar spectral index n_s and the tensor-to-scalar ratio r which can be measured in the CMB spectrum as

    n_s-1=-6ϵ+2η,   r=16ϵ.

The number of e-folds between the moment that the comoving mode with the wave number k exits the horizon and the end of the inflation is computed as 

    N_k   =   ∫^ϕ_e_ϕ_kH_i/ϕ̇dϕ
       =   -1/M^4_Pm^2_X∫^ϕ_e_ϕ_kV^2(ϕ)/V'(ϕ)dϕ,

where ϕ_k and ϕ_e refer to the values of the inflaton at the time of the mode k exiting the horizon and the end of the inflation, respectively. We can determine ϕ_e from ϵ≈1 which occurs at the end of the inflation. 

With the equations related to the inflation and the brane-vector DM production during the inflation, we can predict the scalar spectral index n_s and the tensor-to-scalar ratio r in terms of the number of e-folds N_k. The relation between r and N_k is given by

    r^15/7/1078.47=M^4_PGeV^2/V(ϕ_k)( V'(ϕ_k)/V(ϕ_k))^2,

where ϕ_k=ϕ_k(N_k,r) which is a function of the number of e-folds and the tensor-to-scalar ratio and can be determined from Eqs. (<ref>) and (<ref>).
Whereas, the relation between n_s and N_k is determined by 

    n_s=1-3/8r(N_k)+2η(N_k).




 §.§ α-attractor potential

The inflation with the α-attractor potential is given by <cit.>

    V(ϕ)=Λ^4[ 1-exp( -ϕ/M̅) ] ^2n,

where ϕ is the inflaton field coupled minimally to gravity on the 3-brane, Λ is related to the inflationary energy scale, M̅≡√(3α/2)M_P with α to be a parameter determining the inverse curvature of the Kähler manifold, and n is a parameter. Note that, in the case of α=1 and n=1, the α-attractor potential leads to the Starobinsky model <cit.>.

By substituting the potential (<ref>) into Eq. (<ref>), we obtain the slow-roll parameters as follows

    ϵ   =   [2nβ e^-ϕ/M̅/(1-e^-ϕ/M̅)^n+1]^2,
    η   =   (1-e^ϕ/M̅/2n)ϵ,

where

    β≡M_P^2m_X/M̅Λ^2.

At the moment that the mode k crosses the comoving Hubble horizon corresponding to X_k=exp(-ϕ_k/M̅)≪ 1, we obtain the following approximation

    X_k   ≃   √(ϵ)/2nβ,
    η   ≃   ϵ-β√(ϵ).

As a result, we can find approximate expressions for the scalar spectral index n_s and the tensor-to-scalar ratio r as follows

    ϵ   ≃   1/16[√(4(1-n_s)+β^2)-β]^2,
    
    r   ≃   [√(4(1-n_s)+β^2)-β]^2.

Note that, from Eqs. (<ref>) and (<ref>), we can express the parameter β in terms of the scalar spectral index n_s and the brane-vector mass m_X as follows

    β≃ 0.02[4(1-n_s)-958.48(GeV/m_X)^7/4](m_X/GeV)^7/8.


The value of the inflaton at the end of the inflation corresponding to ϵ≃ 1 reads

    X_e≃1+n+2nβ-√(1-n^2+4n(1+n)β+4n^2β^2)/n(1+n),

where 
X_e≡exp(-ϕ_e/M̅). Then, the number of e-folds is calculated by

    N_k=1/2nβ^2[B(1-X_k,2+2n,-1)-B(1-X_e,2+2n,-1)],

where B(z,p,q)=∫^z_0t^p-1(1-t)^q-1dt is the incomplete beta function.


In order to find the relation between N_k and n_s, we solve numerically Eq. (<ref>) for various values of the number of e-folds. The result is shown in Fig. <ref> where n is taken the values from 1 to 10 and the mass of the brane-vector DM is considered as 350 GeV and 1000 GeV.

We observe that the brane-vector DM would be consistent with the inflation which requires 50 - 60 e-folds for its high mass region. The low mass region of the brane-vector DM may be favorable to the inflation if the parameter n in the inflation potential is large enough.

By solving numerically Eq. (<ref>) and using Eq. (<ref>), we obtain the relation between the number of e-folds N_k and the mass of the brane-vector DM, as depicted in Fig. <ref>, with n=1 and n=10. Our prediction is shown by the purple bands with the scalar spectral index taking the value from 0.961 to 0.969. In adition, we show the 68% and 95% CL experimental constraints for n_s and r <cit.> corresponding to the (dashed) blue and (dashed) red contours.  

We see that the number of e-folds required by the inflation is satisfied in the high mass region of the brane-vector DM. Whereas, in order for the low mass region to meet this criterion, the parameter n in the inflation potential needs to be large enough. These are completely consistent with what is observed in Fig. <ref>.

We arrive to investigate the reheating constraint for the brane-vector DM produced during the inflation. The connection between the parameters of the inflation and those of the reheating can be obtained by the following relation 

    ρ_re/ρ_e=e^-3N_re(1+ω_re),

where ρ_re and ρ_e are the energy density at the end of reheating (re) and the end of inflation, respectively, and N_re is the number of e-folds during the reheating corresponding to the equation-of-state parameter denoted by ω_re. The energy density at the end of reheating is determined in terms of the reheating temperature T_re and the effective number of relativistic degrees of freedom g_re at the reheating epoch as ρ_re=π^2g_reT^4_re/30. With the energy density at the end of inflation given by the inflation potential at the end of the inflation V_e as ρ_e≃3V_e/2, we express the reheating temperature as follows

    T_re   =   [(45/π^2g_re)^3π^2A_s/2]^1/12(M^2_Pm_X)^1/3(50.55 GeV/m_X)^7/48e^-3/4N_re(1+ω_re)
          ×[n(n-2β)-1+√(1-n^2+4n(1+n)β+4n^2β^2)/n(n+1)]^n/2.


In order to determine N_re, we use the relation k/(a_0H_0)=a_kH_k/(a_0H_0) with k=0.002 Mpc^-1 being the pivot scale and a_k (H_k) is the scale factor (the Hubble parameter) at the moment which the mode k crosses the horizon, which leads to <cit.>

    k/a_0H_0=e^-(N_k+N_re)a_re/a_mrea_mreH_mre/a_0H_0H_k/H_mre.

From the conservation of entropy, we can find the ratio of a_re to a_0 as follows

    a_re/a_0=(43/11g_s,re)^1/3T_0/T_re,

where g_s,re is the effective number of relativistic degrees of freedom that contribute to the entropy and T_0 is the present temperature of the universe. Then, the number of e-folds at the end of reheating is given by

    N_re   =   4/1-3ω_re[67.38-N_k-ln( k/a_0H_0)-1/12ln g_re +1/3lng_re/g_s,re+1/4ln(π^2A_s/3).
          .+7/16ln(50.55 GeV/m_X)+n/2ln(n(n+1)/n(n-2β)-1+√(1-n^2+4n(1+n)β+4n^2β^2))].


From Eqs. (<ref>) and (<ref>), we can find the relation between the reheating temperature T_re and the mass of the brane-vector DM. In Fig. <ref>, we depict this relation for n=1 (the left panel) and n=10 (the right panel). The region is bounded by the solid and dashed purple curve represents our prediction with the 68% CL value of the scalar spectral index as n_s=0.965±0.004.

We see that the present observation of n_s leads to the large uncertainty for the prediction of the reheating temperature in terms of the mass of the brane-vector DM. The CMB experiments in the future which will be able to decrease the uncertainty of the n_s measurement will provide a more accurate prediction for this relation.


The spectrum of inflationary GWs <cit.>, which is a function of the frequency f, is connected to the brane-vector DM as

    Ω_GWh^2(f)   ≃   A_s/3(πΩ_mhf/a_0H_0)^2(50.55 GeV/m_X)^7/4exp{-1/8(50.55 GeV/m_X)^7/4log(f/f_0)}
          ×g_in/3.36[3.9/g_s,in]^4/3[3j_1(z_k)/z_k]^2T^2_1(f/f_eq)T^2_2(f/f_re),

where “in" denotes the moment that the mode k crosses the horizon, j_1(z) is the spherical Bessel function of the first kind, z_k≡4π f/(a_0H_0), the functions T_1,2(x) describe the change of the primordial GW spectrum at the frequency f_re≃0.26Hz× T_re/(10^7GeV) <cit.> due to the presence of the reheating and are given as follows

    T_1(x)   =   (1+1.57x+3.42x^2)^1/2,
    
    T_2(x)   =   (1-0.32x+0.99x^2)^-1/2.

In Fig. <ref>, we show the prediction of the primordial GW spectrum for various values of the brane-vector mass and the parameter n. In addition, we include the projected sensitivities of the different GW observations, represented by the colored regions, which are the future SKA <cit.>, the space-based LISA <cit.>, BBO <cit.>, DECIGO <cit.>, and Ultimate-DECIGO <cit.>.

We observe that Ultimate-DECIGO can probe the reheating temperature and the brane-vector DM in its allowed mass region if the parameter n in the inflation potential is large enough. On the contrary, if n is low, Ultimate-DECIGO only allows us to probe the low mass region of the brane-vector DM. In this sense, the observation of the primordial GWs produced during the inflation provides a potential way to probe the extra-dimensions and branes: the reheating temperature and the mass of the brane-vector DM which are determined by the spectrum of the primordial GWs are compared to the prediction which is given in Eqs. (<ref>) and (<ref>).




 §.§ Natural potential

We study the connection of the brane-vector DM to the inflation with the natural potential or the axion potential <cit.> which appears naturally in the compactifications of M/string theory <cit.> and provides a plausible solution to the strong CP problem <cit.>. The inflation potential in the natural inflation is given as follows

    V(ϕ)=Λ^4[1-cos(ϕ/f)],

where f is a parameter. Using Eq. (<ref>) with the inflation potential (<ref>), we find the slow-roll parameters as follows

    ϵ = γ^21+cos(ϕ/f)/[1-cos(ϕ/f)]^2
    η = γ^2cos(ϕ/f)/[1-cos(ϕ/f)]^2

where

    γ≡M_P^2m_X/fΛ^2.

At the horizon crossing of the mode k, corresponding to ϵ≪ 1, we find the value of the inflaton field in terms of the spectral index n_s and the brane-vector mass as follows

    cos(ϕ_k/f)≃ -1 + 4/25γ^2[5(1-n_s)-γ(γ+√((5-5n_s)/2+γ^2))]≡ X_k,

where the parameter γ can be expressed as

    γ/√(2)≃1-n_s-300(GeV/m_X)^7/4/√(1-n_s-240(GeV/m_X)^7/4).


The number of e-folds is

    N_k=1/γ^2[X_k-X_f - 2log( 1+X_k/1+X_f)  ],

where X_f≡cos(ϕ_f/f) with ϕ_f being the value of the inflaton field at the end of inflation which is determined by the condition ϵ≃ 1 leading to

    X_f≃ 1+γ/2(γ-√(γ^2+8)).


In Fig. <ref>, we show the dependence of the scalar spectral index n_s on 
the number of e-folds N_k.

We can see that the brane-vector DM is only consistent with the natural inflation in the 2σ region for the value of n_s and its low mass region around 280 GeV. However, as we see above, the low mass region of the brane-vector DM is hard to detect in the spectrum of the primordial GWs in the near future.



§ CONCLUSION


In the present paper, we have explored dark matter and its connection to the inflationary observations and the spectrum of the primordial gravitational waves in a scenario in which our observable universe is realized as a flexible 3-brane that could oscillate in a five-dimensional bulk spacetime compactified on a circle. (Such a scenario can be considered as a low-energy limit of ten-dimensional string theory which is reduced on a five-dimensional internal manifold.) The off-diagonal components of the bulk metric that is the gauge field corresponding to the U(1) isometry of the internal space would absorb the scalar mode describing the fluctuation of the 3-brane in the bulk. As a result, this leads to a massive vector field living on the 3-brane, so-called the brane-vector. Because the coupling of the brane-vector is strongly suppressed by the observed Planck scale, the observed relic abundance of the brane-vector dark matter is essentially due to the production of its longitudinal mode by the quantum fluctuations in the inflationary stage. In this way, the tensor-to-scalar ratio is determined in terms of the brane-vector mass or the brane tension. In addition, the mass of the brane-vector enters into the Friedmann equation describing the evolution of the brane universe, which is very sensitive with respect to the region of the high energy density such as the inflation scale. Therefore, we have studied the imprints of the brane-vector dark matter in the inflationary observables and the spectrum of the primordial gravitational waves. This can open a promising observation window to detect the extra-dimensions and branes which play a fundamental role in string/M theory.


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 Paper Title: Quasilocal Newtonian limit of general relativity and galactic dynamics.


apsrev4-1



marco.galoppo@pg.canterbury.ac.nz
School of Physical & Chemical Sciences, University of Canterbury, 
 Private Bag 4800, Christchurch 8140, New Zealand


federico.re@unimib.it
Dipartimento di Fisica Giuseppe Occhialini, Università di Milano Bicocca,
 Piazza dell'Ateneo Nuovo 1, 20126, Milano, Italy
 &
INFN, sezione di Milano, Via Celoria 16, 20133, Milano, Italy


david.wiltshire@canterbury.ac.nz
School of Physical & Chemical Sciences, University of Canterbury, 
 Private Bag 4800, Christchurch 8140, New Zealand

§ ABSTRACT 
A new Newtonian limit of general relativity is established for stationary axisymmetric gravitationally bound differentially rotating matter distributions with internal pressure. The self-consistent coupling of quasilocal gravitational energy and angular momentum leads to a modified Poisson equation. The coupled equations of motion of the effective fluid elements are also modified, with quasilocal angular momentum and frame–dragging leading to novel dynamics. The solutions of the full system reproduce the phenonomenology of collisionless dark matter for disc galaxies, offering an explanation for their observed rotation curves. Halos of abundant cold dark matter particles are not required. 


Quasilocal Newtonian limit of general relativity and galactic dynamics
    David L. Wiltshire
    August 5, 2024
======================================================================



The Newtonian limit of general relativity is crucially important to both fundamental physics and astrophysical phenomenology. In cosmology two key features consistent with observation are: (i) far from any localised sources the Universe is not empty; (ii) the Universe is not globally rotating on the largest scales. 

Conventionally, one invokes an asymptotic Minkowski space, with respect to which additive Newtonian potentials are defined, despite observation (i). Moreover, even though known rotating asymptotically flat solutions of Einstein's equations—the Kerr and Kerr-Newman geometries—possess global non-zero angular momenta at spatial infinity, it is implicitly assumed that superposition of the effectively quasilocal angular momenta of such pointlike sources can be reconciled with observation (ii).

In this Letter we show that such implicit assumptions are misguided, even at nonrelativistic orbital speeds. The conventional limit neglects an important coupling of the quasilocal energy and angular momentum defined by the regional time–averaged motion of matter sources. We set out a new, self-consistent quasilocal Newtonian limit of general relativity, which does not assume a Minkwoskian spacetime background a priori. It introduces novel, first-order features that fundamentally modify the dynamics considered. A direct consequence is the elimination of the need for abundant cold dark matter particles in the modelling of galactic dynamics.

The prescription of a fixed, global spacetime background is not a requirement of general relativity. Indeed, the nonlinearity of Einstein's field equations (EFE), which arises from the self–interaction of matter and geometry, results in dynamical, regional backgrounds in which spacetime itself carries its own energy and angular momentum <cit.>. Moreover, the highly nonlinear nature of general relativity gives rise to a noncommutativity of averaging and limiting procedures in spacetime <cit.>. This has fundamental consequences for the dynamics of non-pointlike sources of large spatial extent, such as disc galaxies <cit.>.

The crucial ingredient of the differentially rotating models <cit.>, and in particular of the exact solutions of Ref. <cit.>, is that essential nonlinearity is retained before any low energy limit is considered. No global background is assumed. This contrasts with: (a) approaches that investigate frame–dragging on asymptotically flat <cit.> or asymptotically Friedmann <cit.> backgrounds, necessitating dark matter halos; (b) dark matter alternatives which modify the laws of gravitation, e.g., modified Newtonian dynamics (MOND) <cit.>.

In this Letter we demonstrate how insights from the conventional Newtonian framework may nonetheless be embedded in a quasilocal framework with an effective fluid. Furthermore, Einstein was never fully satisfied by the extent to which the EFE embody Mach's principle, viz., “Local inertial frames are determined through the distributions of energy and momentum in the universe by some weighted average of the apparent motions” <cit.>. The quasilocal Newtonian limit exhibits such a suitable weighted average, which we apply to disc galaxies showing that the EFE may actually contain an understanding of inertia consistent with MOND phenomenology.

The spacetime metrics can be written in the generalized Lewis–Papapetrou–Weyl form <cit.>,

    
    s^2     =-c^2 e^2Φ(r, z)/c^2( t+A(r, z)  ϕ)^2 +e^-2Φ(r, z)/c^2[W(r,z)^2 ϕ^2 +e^2k(r, z)/c^2( r^2+ z^2)]  ,



The energy-momentum tensor takes the form

    T^μν=((r, z)+p(r,z)/c^2)   U^μU^ν  + p(r,z) g^μν,

where (r,z) is the local matter density, p(r,z) is the effective pressure, and each element of the fluid possesses a 4–velocity U^μ, given by

    U^μ∂_μ=(-H(r,z))^-1/2(∂_t +Ω(r,z)  ∂_ϕ)  .

Here dϕ/dt=Ω(r, z) uniquely defines the angular speed of rotation at any point, and H(r, z) is a normalization factor. Since U^μU_μ=-c^2, it follows that

    H=-e^2Φ/c^2(1+A  Ω)^2+e^-2Φ/c^2W^2   Ω^2/c^2  .

Relevant physical velocities have to be identified to correctly implement any low-velocity limit. In the present case, we define the kinetic and dragging velocities

    := r Ω  , 
        := r χ  ,

where χ := -g_tϕ/g_ϕϕ is the frame–dragging term. For systems in the low-energy regime—i.e., nonrelativistic relative local velocities v ≪ c, weak pseudo-Newtonian potential Φ v^2, nonrelativistic frame–dragging, and small pressure p  v^2—then to leading order  coincides with the special relativistic interpretation of the redshift, whilst  follows by analogy <cit.>.

Let us then write the EFE as

    R_μν = 8 π G/c^4(T_μν-1/2T g_μν)  ,

where T=T^μ_μ = 3p- c^2. With (<ref>)–(<ref>) this yields the partial differential equations (PDEs) <cit.>

   
    Φ^,a_,a + W^,aΦ_,a/W + c^4A^,aA_,a/2W^2e^4Φ/c^2 = 4 π G e^2(k-2Φ)/c^2[(+p/c^2)(1+AΩ)^2e^2Φ/c^2+c^-2W^2Ω^2e^-2Φ/c^2/-H+2p/c^2]   , 
       A^,a_,a - W^,aA_,a/W + 4/c^2Φ^,aA_,a = 16 π G/c^4 W^2Ω1+AΩ/H(+p/c^2)e^2(k-2Φ)/c^2  , 
       W^,a_,a = 16 π G/c^4p   , 
        W_,rr-W_,zz+2/c^2(k_,zW_,z-k_,rW_,r)+2W/c^4(Φ_,r^2 - Φ_,z^2) + c^2/2We^4Φ/c^2(A_,z^2 - A_,r^2) = 0   , 
       c^2/2We^4Φ/c^2A_,z A_,r-W_,rz-2W/c^4Φ_,rΦ_,z+1/c^2(k_,zW_,r-k_,rW_,z) = 0   , 
       1/c^2(Φ^,a_,a-k^,a_,a)-1/2WW^,a_,a + 1/c^2W^,aΦ_,a/W - 1/c^4Φ^,aΦ_,a+ c^2/4W^2e^4Φ/c^2(A_,z^2 + A_,r^2) = 4 π G/c^2e^2(k-2Φ)/c^2(-p/c^2)  ,
 

where a∈{r,z}. Eq. (<ref>) and (<ref>) are the tt and rzcomponents of (<ref>) respectively, (<ref>) and (<ref>) are obtained from the R_zz± R_rr equations, (<ref>) is found by taking the combination W^-1c^-2(g_ϕϕR_tt-2 g_tϕR_tϕ+g_ttR_ϕϕ), and finally (<ref>) follows from the combination 2c^-2(R_tϕ-A R_tt). 

The system of PDEs is completed by the perfect fluid elements' equations of motion, which give 

    
    Hp_,a/e^2(k-Φ)/c^2=[Φ_,a+(c^2 A_,a+2AΦ_,a)Ω+(c^2AA_,a+A^2Φ_,a)Ω^2]e^2Φ/c^2+Ω^2(W^2Φ_,a/c^2-WW_,a)e^-2Φ/c^2  ,


We now take an expansion in powers of v/c to implement the nonrelativistic limit, where v is any relevant local velocity. We apply the self-consistent ansatz 

    W(r,z) = r + 𝒪(v^4/c^4)   ,

to solve (<ref>) up to fourth order. From (<ref>), and the definition of χ, we find A = r  /c^2+𝒪(v^3/c^3). Thus, by (<ref>) it follows that 

    H = -1 + ^2/c^2 - 2/c/c + 2 Φ/c^2 + 𝒪(v^4/c^4) .

Hence, by direct substitution of (<ref>) into (<ref>)–(<ref>) we find

    ΔΦ + 1/2r^2||∇⃗Ł_D||^2 = 4 π G  + 𝒪(v^2/c^2)  , 
       Δ̂Ł_D = 0 + 𝒪(v^2/c^2)   , 
        k_,r = 1/4r(Ł_D,z^2-Ł_D,r^2)+ 𝒪(v^2/c^2)    , 
        k_,z = -1/2r(Ł_D,z Ł_D,r) + 𝒪(v^2/c^2)   ,

where Ł_D := r is the quasilocal angular momentum per unit of mass associated with spacetime rotation, Δ is the standard laplacian and Δ̂ := ∂^2_r -(1/r)∂_r + ∂^2_z, is the Grad–Shafranov laplacian <cit.>. The number of equations reduces by one, as (<ref>) is equivalent to (<ref>) to the order considered. Furthermore, we see that the fluid effective pressure does not enter the EFE as a source term at this order of approximation. 

Eq. (<ref>) is a general relativistic generalization of the Poisson equation, and coincides with it in the absence of spacetime rotation. The extra term in (<ref>) can be interpreted as the rotational energy associated with the quasilocal angular momentum of the averaged background spacetime. This term is absent in the conventional Newtonian approximation in which the background is taken to be Minkowski space a priori. Furthermore, when moved to the r.h.s. of (<ref>), it can be identified as an effective density, ρ𝒬=-||∇⃗Ł_D||^2/(8π G  r^2).  This represents a negative gravitational binding energy relative to the background <cit.>.

Even for nonrelativistic dragging velocities spacetime rotation gives first-order corrections to the field equations. For fixed , Eq. (<ref>) shows that the strength of the pseudo-Newtonian potential needed to sustain equilibrium decreases as the local rotational energy of the background geometry increases. Furthermore, the pseudo-Newtonian potential in (<ref>) is essential to accommodate nonrigid rotation of the perfect fluid source. Indeed, if we set Φ=0 then (<ref>)–(<ref>) are the same equations that characterize a system of rigidly rotating dust <cit.>. As such this choice is unphysical, requiring an identification of  purely with rotational energy.

At the same order, equations (<ref>) reduce to 

    -p_,r/=Φ_,r+Ω Ł_D,r - ^2/r + 𝒪(v^2/c^2)   , 
       -p_,z/=Φ_,z+ΩŁ_D,z+𝒪(v^2/c^2)  .

Hence, the effective pressure of the fluid still plays a crucial role in determining the physics of the system, as expected. In Eq. (<ref>) and (<ref>), we recognize the gradient of the pseudo-Newtonian potential and the centrifugal acceleration in the term ^2/r. The conventional Newtonian equations follow in the case of negligible dragging (≪). However, more generally Eqs. (<ref>), (<ref>) once again show the crucial role that the quasilocal angular momentum of the regional background plays in determining the dynamics. For fixed  and p, the magnitude of the pseudo-Newtonian potential needed to match the l.h.s. of (<ref>) and (<ref>) decreases linearly with the quasilocal angular momentum of the underlying averaged geometry.

The quasilocal Newtonian limit defined here is thus realized by Eqs. (<ref>)-(<ref>). As a first application of the quasilocal Newtonian limit, we derive the dragging velocity profile for a disc galaxy rotation curve supported exclusively by baryonic matter. The baryonic mass distribution in a disc galaxy can be written as =  +, where  and  are the densities of the bulge and the disc, respectively. 

We assume a spherical bulge component given by the Plummer density profile <cit.>

    (R) = 3^2/4π(R^2+^2)^5/2 ,

where R = √(r^2+z^2),  is the scale parameter of the bulge and  is the total bulge mass. The resulting circular velocity in the conventional Newtonian limit is

    (R) = √(G R^2(R^2+^2)^-3/2)  .


The baryonic disc matter distribution is usually taken as (r, z)=Σ(r)Z(z). The surface density of the disc is well approximated by the exponential profile

    Σ(r)=/2π^2e^-r/  ,

where  is the total disc mass and  is the scale length in the radial direction for the galactic disc. The distribution along z, with ∫ Z(z)dz=1, is assumed to be

    Z(z)=1/2e^-|z|/  ,

where  is the thickness scale length and ≪. When the thickness  tends to zero, we apply the conventional infinitely thin disc approximation, i.e., Z(z)=δ(z)<cit.>. The Newtonian circular velocity field in the thin disc approximation, , is given by <cit.>

    
    (r) = √(G/2^3r^2[I_0(r/2)K_0(r/2) -I_1(r/2)K_1(r/2)])  ,


where I_{0,1} and K_{0,1} are modified Bessel functions of the first and second kind.

To explain the observed rotation curves in the conventional Newtonian framework, a spherical cold dark matter (DM) halo must be included, so that the matter density is =  +, where  is typically given by the isothermal profile <cit.>

    (R)=ρDM0/1+(R/)^2  .

Here  is the halo scale parameter, and ρDM0 is the maximum DM density. The DM halo contribution to the rotation curve, , is then <cit.>

    (R)^2=4π G ^2 ρDM0(1- arctan(R/)/R/)   .

Thus the circular velocity of the matter in Newtonian modelling of the disc galaxy, , on the equatorial plane, is given by

    (r,0)=√((r,0)^2+(r)^2+(r,0)^2)  .

In Fig. <ref>, we consider a Milky Way-like galaxy with = 0.8 · 10^10M_⊙, = 0.8 kpc, = 8.1 · 10^10M_⊙, = 2.1 kpc, = 5.69 kpc and ρDM0= 1.88·10^-21 kg/m^3. We plot the predicted rotation curve (r,0), and the respective (r,0), (r) and (r,0). 


To apply the general relativistic solution, let us now assume that =, i.e., that the inferred Newtonian rotation curve corresponds to that of a disc galaxy, and that =. We will demonstrate that the quasilocal energy and angular momentum of the galaxy produce observations equivalent to those of a conventional cold dark matter halo.

We begin by writing the pseudo-Newtonian potential as Φ=-, so that  is given by the Newtonian solution for the chosen , while  is defined as the solution of the Poisson equation sourced purely by the quasilocal rotational energy of the spacetime 

    Δ=||∇⃗Ł_D||^2/2r^2  .


We now solve (<ref>), (<ref>), and (<ref>) for Ł_D by noting that in the thin disc approximation the pressure term is exactly the same for both the Newtonian and the general relativistic cases. Moreover, a similar ansatz can be applied to the spherical bulge, where dark matter is not required to explain the observed dynamics, i.e., it is well captured by purely Newtonian models with purely baryonic matter. Thus we can take p(r,z):=p N (r,z) for the whole system, where p N (r,z) is the expected effective Newtonian pressure. We then use Eqs. (<ref>),(<ref>) to deduce

    = ΩŁ_D,r -(^2-^2)/r   , 
        = ΩŁ_D,z ,

where := √(^2+^2) is the Newtonian contribution to the observed rotation curve from the baryonic mass.

Substituting (<ref>), (<ref>) in (<ref>) it follows that

    Δ =Ω_,rŁ_D,r +2ΩŁ_D,r/r +Ω_,zŁ_D,z -(^2-^2)_,r/r  ,

where we have also used the quasilocal Newtonian limit of the (tϕ) Einstein Eq. (<ref>). Eqs. (<ref>), (<ref>) then give a first-order nonlinear PDE for Ł_D

    ∇⃗Ł_D·[∇⃗Ł_D - 2∇⃗(r)] + 2r(^2-^2)_,r = 0  .

We evaluate Eq. (<ref>) on the galactic plane, z=0, where thanks to the plane symmetry we have Ł_D,z(r,0)=vK,z(r,0) = 0. Thus, the nonlinear PDE (<ref>) is reduced to an algebraic equation for Ł_D,r on the galactic plane, whose solutions are given by 

    Ł_D,r=(r)_,r±√([(r)_,r]^2 - 2r(^2-^2)_,r)  .


We choose the minus sign in (<ref>) so that it reduces to the expected Newtonian case for =. Furthermore, given the baryonic density distribution and the conventional Newtonian profile for (r,0), we can numerically solve (<ref>) for (r,0). In Fig. <ref> we show the resulting frame–dragging velocity profile generating the simulated galaxy rotation curve, . The frame–dragging speed reaches a maximum 42.2km/s at r=24.4kpc, well within our nonrelativistic approximation. Furthermore, as we would expect: (i)  decreases with increasing radius, tending to zero at infinity; and (ii)  is negligible in the bulge, showing the self-consistency of the pressure ansatz used in the calculations. 


The new quasilocal Newtonian limit applies to stationary axisymmetric matter distributions with geometry (<ref>). These symmetry assumptions mean that for a Milky Way-like disc galaxy the approximations here, along with the exact solutions <cit.> hold on time scales, 10^7 t10^9yr. Eqs. (<ref>)–(<ref>) will not apply during galaxy formation, and therefore do not explain how disc galaxies form in a cold dark matter free environment. While such complex problems must be left to future work, the precise definition of finite infinity <cit.> provides a mathematical framework for further exploring the quasilocal Newtonian limit, in settings from the early Universe through to highly dynamical astrophysical systems. The interplay of quasilocal gravitational energy and angular momentum must play a key crucial role.

In summary, we have shown that via a novel self-consistent quasilocal Newtonian limit, general relativity can explain the observed rotation curves of disc galaxies without the need for halos of cold dark matter particles. The phenonomenology of collisionless dark matter is reproduced by the regional gravitational energy and angular momentum of the average spacetime geometry.

This discovery has far-reaching consequences for cosmology, astrophysics and particle physics. Indeed, many interesting questions are now opened up. Within the given realization of the quasilocal Newtonian limit, disc galaxy stability, gravitational lensing and fits to observational datasets must be addressed to definitively test the new paradigm. Comparisons with MOND phenomenology potentially opens the prospect not only of placing MOND within the theoretical framework of general relativity, but also of providing insights into the development of this important new physical limit of Einstein's theory. Finally, in light of our results, the phenomenology of dark matter in the early universe, in the dynamics of galaxy clusters, and in other astrophysical environments should now be reconsidered and further investigated.

Acknowledgments  DLW is supported by Marsden Fund grant M1271 administered by the Royal Society of New Zealand, Te Apārangi. We thank Davide Astesiano, Luca Ciotti, Sergio Cacciatori, John Forbes, Vittorio Gorini, Christopher Harvey-Hawes, Frederic Hessman, Morag Hills, Emma Johnson, Zachary Lane, Ryan Ridden-Harper, Antonia Seifert, Chris Stevens, Shreyas Tiruvaskar and Michael Williams for useful discussions. 






 Paper Title: Information Scrambling at Quantum Hall Interfaces and Their Analog to Black Hole Event Horizon.









Ken K. W. Ma Hometown: Orlando, Florida 
 
kwokwai_ma@alumni.brown.edu
Kun Yang () Department of Physics, Florida State University and the National High Magnetic Field Laboratory, 

 Address: 1800 E Paul Dirac Dr, Tallahassee, Florida 32310 

kunyang@magnet.fsu.edu





Information Scrambling at Quantum Hall Interfaces and Their Analog to Black Hole Event Horizon
    Ken K. W. Ma and Kun Yang
    August 5, 2024
==============================================================================================



The black hole information paradox has been hotly debated for the last few decades without a full resolution. This makes it desirable to find analogues of this paradox in simple and experimentally accessible systems, whose resolutions may shed light on this longstanding and fundamental problem. Here, we review and resolve the apparent “information paradox" in two different interfaces separating Abelian and non-Abelian quantum Hall states. In both cases, the information carried by the pseudospin degree of freedom of the Abelian anyons get scrambled when they cross the interface and enter the non-Abelian quantum Hall liquid. Nevertheless, it is found that the scrambling mechanism depends on the nature of the interface. The corresponding analogues of different concepts in black hole physics such as event horizon, black hole interior, Hawking radiation, and Page curve will also be discussed.





§ INTRODUCTION


The existence of black holes has received strong support from recent observations <cit.>. Instead of being a region which nothing can escape from, Hawking predicted that a black hole emits radiation and evaporates slowly <cit.>. He also concluded that the radiation carries no information except mass, angular momentum, and charge of the black hole <cit.>. This result points to possible loss of information in black holes. On one hand, it is consistent with the no-hair theorem <cit.>. On the other hand, quantum mechanics forbids information loss due to the unitarity of time evolution. This apparent contradiction leads to the black hole information paradox <cit.>. It is believed that the resolution of this paradox may provide important clues on how to combine quantum mechanics and general relativity.

Various approaches have been proposed to resolve this black hole information paradox. Among them, the holographic principle <cit.> supports the preservation of unitarity and information. In particular, information can be encoded holographically on surfaces, such as the event horizon. This belief is substantiated by the discovery of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence <cit.>. Recently, the firewall scenario <cit.> was proposed to resolve the conflict between black hole complementarity <cit.> and monogamy of entanglement <cit.>. If this conjecture is correct, the firewall at the event horizon (or black hole's boundary) may also break the entanglement between the outgoing and the in-falling particles. Thus, the boundary can be as important as, or even more important than, the interior of a black hole.

Suppose black hole evaporation is a unitary process. Then how is information hidden in the black hole released from Hawking radiation? Page argued that the release of information starts slowly at the beginning, but becomes faster in the later stage of the evaporation <cit.>. If the system was initially in a pure state, entropy of the radiation (coming from its entanglement with the remainder of black hole) would first increase from zero but eventually decrease back to zero when the black hole evaporates completely, thus recovering the pure state nature of the system and all the (quantum) information it carries. This feature is now known as the Page curve. Based on quantum information theory, the thought experiment by Hayden and Preskill (Hayden-Preskill protocol) has provided further insight on retrieving information from Hawking radiation <cit.>. Suppose the black hole has already passed its Page time and become maximally entangled with its previously emitted Hawking radiation. If the internal dynamics of black hole can be described by an instantaneous random unitary transformation, then any additional information entering the black hole can be recovered from Hawking radiation almost immediately (a very short time compared to the lifetime of the black hole) <cit.>. The protocol has postulated the existence of information scrambling <cit.>, which describes the dispersal of local information into entanglement and correlation throughout the entire system. Thus, the original information is stored nonlocally and cannot be recovered via local measurement. This concept has attracted considerable attention in the context of many-body dynamics <cit.> and quantum neural network <cit.>, which has been verified in recent experiments <cit.>. In addition, recent studies have recovered the Page curve for AdS black holes <cit.>. However, a full resolution of the paradox remains an open problem <cit.>. It is thus desirable to mimic the information paradox in simple and experimentally accessible systems, that allows for a complete understanding of this process.

Somewhat similar to the holographic principle, the bulk-edge correspondence relates the topologically protected edge modes and bulk topological orders in fractional quantum Hall (FQH) systems <cit.>. This allows us to learn about the bulk by probing the edge of the system <cit.>. Comparatively speaking, interfaces between a pair of FQH states are explored much less <cit.>. In fact, the physics and dynamics of interfaces are much richer than simple edges <cit.>. For example, as we demonstrate below, certain interfaces allow quasiparticle tunneling between two different FQH states, even if the quasiparticles are of very different nature. If they have different internal degrees of freedom, (local) information carried by them needs to be transmuted (or scrambled in a specific way) to prevent information loss. This motivates us to explore analogues of black hole information paradox in quantum Hall interfaces.

In this chapter, we review and resolve a condensed matter analogue of information paradox in two different quantum Hall (QH) interfaces <cit.>. We first consider the interface between Halperin-331 <cit.> and Pfaffian (Moore-Read) <cit.> QH states. The latter is a famous non-Abelian state which hosts non-Abelian anyons that may be useful in topological quantum computation <cit.>. Meanwhile, both 331 and Pfaffian states may be realized in bilayer systems or wide quantum wells at filling factor ν=1/2 <cit.>. Due to the competition between interlayer tunneling and intralayer Coulomb interaction, a phase transition between the 331 and Pfaffian states was predicted <cit.>. This suggests the possibility of creating a 331-Pfaffian interface by controlling the tunneling strengths in different regions of the bilayer system<cit.>. Interestingly, the Abelian 331 state has a pseudospin (layer) degree of freedom, which is absent in the Pfaffian state. If the original information carried by the pseudospin degree of freedom becomes irrecoverable after quasiparticles cross the interface and enter the Pfaffian liquid, it leads to an “information paradox". We demonstrate that the information is scrambled and stored nonlocally in the Pfaffian liquid and the interface. We also mimic black hole evaporation in the same system, and find it satisfies the Page curve naturally. In other words, the original pseudospin information is recovered and the “information paradox" in our model is resolved. Following the similar logic, we consider the “information paradox" in another QH interface formed by Halperin 330 state and Read Rezayi state at level four (abbreviated as RR_4 state) <cit.>. On one hand, the setting closely resembles the 331-Pfaffian interface by having a pseudospin degree of freedom in the 330 state that is absent in the RR_4 state. Both states may be realized in bilayer systems at ν=2/3 <cit.>. On the other hand, we will show that the nature of 330-RR_4 interface is different from the 331-Pfaffian interface. This leads to different mechanisms for scrambling the original pseudospin degree of freedom carried by the Abelian anyons in the two QH interfaces. Here, we must emphasize that we are not aiming at a resolution of the original information paradox in astrophysical black holes. This is clearly unachievable by proposing a simple analogy. Instead, we want to simulate some important concepts in resolving the original paradox in a simple and accessible manner.



§ BASICS OF QUANTUM HALL EFFECT


To set the stage for later discussion, we first review briefly some basic concepts in quantum Hall (QH) physics <cit.>. Electrons moving in two dimensions (x-y plane) and a perpendicular magnetic field (in the z direction) have their energy levels being quantized in Landau levels. A very important parameter characterizing the system is the Landau level filling factor ν, which is the ratio between the number of electrons and the number of magnetic flux quanta enclosed by the system. Of particular interest is the case ν<1, where only the lowest Landau level is partially filled by electrons at low temperature, known as the fractional QH (FQH) regime. Since the kinetic energy of these electrons is quenched due to Landau quantization, the interaction between them dominates the properties of the system. Various FQH states, which possess numerous fascinating properties, are realized in this strongly correlated electronic system. The exotic properties of FQH states are associated with the topological order they possess <cit.>. Most prominent among them is the existence of low-energy excitations (quasiparticles) that have fractional charges and obey fractional statistics (between bosonic and fermionic statistics) <cit.>. A famous example is the Laughlin state at ν=1/3 <cit.>, in which quasiparticle with a fractional charge e/3 <cit.> and a fractional statistics 2π/3 can exist <cit.>. Note that both fractional charge <cit.> and fractional statistics were observed experimentally <cit.>. Such exotic quasiparticles are called anyons, and the possible types of anyons are associated with the specific topological order.



 §.§ Bulk-edge correspondence and conformal field theory


The bulk-edge correspondence, another consequence of the topological order, relates the edge structure and the bulk topological order in FQH systems. In particular, it predicts the existence of gapless edge modes described by conformal field theories (CFTs), and there is a one-to-one correspondences between the bulk topological order and edge CFT <cit.>. In our previous example, the edge of the Laughlin state at ν=1/3 has a single chiral bosonic edge mode ϕ, which can be described by the Lagrangian density,

    ℒ_1/3
    =-3/4π∂_xϕ(∂_t-v∂_x)ϕ.

Here, v is the speed of the edge mode, and ϕ is a (chiral) bosonic field. In general, the edge of a FQH liquid can have more than one edge mode. For Abelian FQH states, the corresponding edge theory is described by <cit.>

    ℒ_ edge
    =-1/4π∑_i,jK_ij∂_tϕ_i∂_xϕ_j
    -1/4π∑_i,jV_ij∂_xϕ_i∂_xϕ_j.

Importantly, the K matrix encodes all information of the topological order. For a FQH state in a bilayer system, it may (but not always) be described by a two-component topological order which has a 2× 2 K matrix. In this situation, two different edge modes exist. Furthermore, the possible type of edge modes is not limited to bosonic mode. Other types of modes such as Majorana fermion modes exist when the topological orders are non-Abelian <cit.>.

With the knowledge of the edge structure in hand, different low-energy excitations in the FQH system can be described or created by suitable CFT operators <cit.>. For example, a charge-e/3 quasiparticle and an electron in the ν=1/3 Laughlin state are created by the operators :exp(iϕ): and :exp(3iϕ):, respectively. Here, :𝒱: denotes the normal ordering of the vertex operator 𝒱. When there is no confusion, this normal ordering notation will be dropped in the later discussion. For a FQH state being described by a multicomponent topological order, there are multiple types of anyons (described by different CFT operators) that have the same electric charge. In other words, the anyons have an additional degree of freedom. This point will become clear when we discuss our setup.



§ INFORMATION PARADOX IN 331-PFAFFIAN INTERFACE



Both Halperin-331 and Pfaffian quantum Hall liquids have the same Landau level filling factor ν=1/2, which can be realized in a bilayer system. For the 331 liquid, it is described by a two-component topological order with the K matrix <cit.>,

    K=[ 3 1; 1 3 ].

The two different edge modes are denoted as ϕ_↑ and ϕ_↓. The two most relevant operators creating an electron are exp(3iϕ_↑+iϕ_↓) and exp(iϕ_↑+3iϕ_↓). On the other hand, the Pfaffian liquid is described by a single-component non-Abelian order with K=2. Its edge has a bosonic mode ϕ and a Majorana fermion mode ψ <cit.>. The corresponding electron operator is ψexp(2iϕ). Since the Halperin-331 and Pfaffian edges have opposite chiralities at the interface, the interface is described by the Lagrangian density <cit.>,

    ℒ
    =   -1/4π∑_i,jK_ij∂_tϕ_i∂_xϕ_j
    +2/4π∂_tϕ_l∂_xϕ_l
    -iψ_l∂_tψ_l
    
       -ℋ(ϕ, ψ).

Here, the indices i,j=↑, ↓ denote the layer or the pseudospin.
All ϕ_↑, ϕ_↓, and ϕ_l are charge modes. The first two are right-moving along the edge of the 331 liquid, whereas the last one is left-moving along the edge of the Pfaffian liquid. The Pfaffian liquid also has a left-moving neutral Majorana fermion mode ψ_l along the edge. The edge structures of both quantum Hall states are illustrated in Fig. <ref>(a). As shown by one of us, a relevant random electron tunneling between the Pfaffian and 331 edges can lead to a phase transition at the interface <cit.>.

Now, we follow Ref. <cit.> and briefly summarize how different modes get localized at the interface. This also allows us to introduce useful notations for later discussion. One can define a charge mode ϕ_r=ϕ_↑+ϕ_↓ and a neutral spin mode ϕ_n=ϕ_↑-ϕ_↓ in the 331 liquid. Using this new set of modes, the topological term of the 331 edge becomes

    ℒ_331
    =-2/4π∂_tϕ_r∂_xϕ_r
    -1/4π∂_tϕ_n∂_xϕ_n.

The overall charge density at the interface is given by

    ρ(x)
    =1/2π∂_x(ϕ_↑+ϕ_↓+ϕ_l)
    =1/2π∂_xϕ_c.

The random electron tunneling between the Pfaffian and 331 edges is described by

    H_ T
    =   ∫ξ(x)ψ_l
    (e^3iϕ_↑+iϕ_↓+2iϕ_l
    +e^iϕ_↑+3iϕ_↓+2iϕ_l) dx
    +H.c.
    
    =   ∫ |ξ(x)|ψ_l(x) ψ_r(x)cos[2ϕ_c(x)+φ(x)] dx.

Here, ξ(x) denotes the random tunneling amplitude. In the second line, |ξ(x)| and
φ(x) are the magnitude and the phase of ξ(x), respectively. We have also fermionized exp[iϕ_n(x)]=ψ_r(x)+iψ_R(x). The resulting edge modes are shown in Fig. <ref>(b). If H_ T is relevant in the renormalization group sense, then both charge modes, ψ_l, and ψ_r are localized at the interface. After the localization, only a single right-moving Majorana fermion mode remains gapless and propagates freely at long distance, as shown in Fig. <ref>(c). Notice that this gapless mode is neither an original edge mode of the 331 nor the Pfaffian state.



Since Pfaffian and 331 states are quantum Hall states formed by superconducting pairing between composite fermions <cit.>, both of these two states have quasiparticles with the smallest possible charge of e/4 <cit.>. However, there is a fundamental difference between these quasiparticles. For the 331 state, there are two different types of Abelian e/4 quasiparticles created by the vertex operators, e^iϕ_↑ and e^iϕ_↓. One may view them as quasiparticles with different pseudospins. When we formulate the information paradox in the following discussion, this pseudospin will be regarded as the degree of freedom of the Abelian quasiparticles. For the Pfaffian state, there is only one type of e/4 quasiparticle created by σ e^-iϕ_l/2 <cit.>. Here, σ is the twist field with a scaling dimension 1/16 in the chiral Ising CFT <cit.>. We summarize the three primary fields and their properties in Table <ref>. In particular, σ satisfies the fusion rule σ×σ=ψ+I. Note that we have omitted the subscript l for the Majorana field to make the discussion of Ising CFT general. Its proper meaning should be clear from context. The fusion rule indicates that the quasiparticle is non-Abelian. An interesting question is what happens if a quasiparticle is dragged from the 331 liquid in to the Pfaffian liquid? It seems that the pseudospin information would be lost. In this sense, we can define the interface between the two different QH liquids as the “event horizon with a firewall" in our setup. This definition or analogy makes sense since the interface plays the role of a one-way surface of information in our setup, and the “destruction" of pseudospin information at the interface resembles a firewall conjectured in Ref. <cit.>. In addition, the Pfaffian liquid can be viewed as the interior of a “black hole". Suppose this analogous black hole can evaporate (discussed in Sec. <ref>) and the original pseudospin information cannot be recovered at the end of the evaporation. Then, the lost of information contradicts to the fact that quasiparticle tunneling is a unitary process. Thus, we have identified an apparent “information paradox".





 §.§ 331-Pfaffian interface from anyon condensation



Before resolving the paradox, we reformulate the above discussion in the framework of anyon condensation <cit.>. This technique has been commonly applied to study possible transitions between topologically ordered phases. In the context of quantum Hall physics, it was used to study the interface between Pfaffian and non-Abelian spin-singlet (NASS) quantum Hall states <cit.>. In this section, we first use anyon condensation to deduce the CFT description of the 331-Pfaffian interface. In the next section, we apply the same technique to resolve the paradox. Along the way, we adopt a pedagogical approach and aim at relating the rather abstract technique to the more physical picture in Sec. <ref>. It will allow us to highlight the advantages of applying anyon condensation in studying quantum Hall interfaces.

From Sec. <ref>, we know that the edges of the Halperin-331 and Pfaffian liquids are described by CFTs with central charges 2 and 3/2, respectively. These two edges are counterpropagating at the interface. Hence, we expect the resulting CFT describing the 331-Pfaffian interface has a net central charge of 2-3/2=1/2. To deduce exactly what the CFT is, it is first necessary to separate the charge and neutral sectors for both Halperin-331 and Pfaffian liquids. It is because a charge mode cannot be gapped out by coupling to a neutral mode in a usual situation. Equivalently, we do not consider the possibility of condensing charge bosons, which will break the U(1) gauge symmetry. The separation was already achieved in Sec. <ref>. In particular, the combination of charge modes ϕ_c=ϕ_r+ϕ_l was shown to be gapped out (more precisely, localized) by H_ T <cit.>. Therefore, we can focus our discussion on the neutral sectors.

As stated previously, the neutral sector of the Pfaffian state is described by a chiral Ising CFT. For the Halperin-331 state, its neutral sector is governed by the spin mode ϕ_n, which is described by the U(1)_4 CFT. Different primary fields in this Abelian CFT are summarized in Table <ref>. Note that any two vertex operators in the form
e^iαϕ_n/2 and e^i(α+4ℤ)ϕ_n/2 are identified.



The structure (remaining gapless modes) of the 331-Pfaffian interface is solely determined by anyon condensation in the neutral sectors. This condensation occurs in the
U(1)_4 ×Ising CFT. We emphasize again that the bar denotes conjugation of the Ising CFT due to the opposite chiralities between the 331 and Pfaffian edges at the interface.  Compared to the original CFT, anyons in the conjugate CFT have the same fusion rules, but complex conjugated topological spins and braiding phases. Alternatively, one may interpret the condensation as a coset construction <cit.>. We label a generic anyon as (e^iαϕ_n/2, t̅). Here, the parameter
α=0,1,2,3 determines the corresponding primary fields in the U(1)_4 CFT. Meanwhile, t̅={I̅, ψ̅, σ̅} denotes the primary fields in the conjugate Ising CFT. In the present case, there is only one condensable boson,
B=(e^iϕ_n, ψ̅). The condensation of B leads to confinement of some of the anyons in the condensed phase. An anyon remains unconfined if and only if it has a trivial mutual statistics with B. This condition ensures that an unconfined anyon has a consistent topological spin in the condensed phase. Furthermore, two anyons are identified when they differ from each other by a multiple of B. Using operator product expansion, it is straightforward to deduce the six (or three after identification) deconfined anyons in the condensed phase. They are listed in Table <ref>. Their corresponding topological sectors, namely Ĩ, ψ̃, and σ̃ are defined according to their conformal spins. From the table, we conclude that the 331-Pfaffian interface is described by a chiral Ising CFT. Note that this Ising CFT has an opposite chirality to the one describing the Pfaffian edge at the interface.



Now, one may wonder why going through such abstract and seemingly redundant procedures to find out the CFT describing the interface. Doesn't the net central charge c=1/2 directly indicate that it should be an Ising CFT? There are two reasons for analyzing this simple system by anyon condensation. First of all, it is fortunate that for the 331-Pfaffian interface, the mechanism and consequences of anyon condensation can be visualized in a very transparent and physical manner, but this is a very special case. In Eq. (<ref>), the electron tunneling between counterpropagating edges at the interface couples ψ_l and e^iϕ_n. This leads to a mass term and eventually gaps out the counterpropagating ψ_l and ψ_r. Only ψ_R remains gapless at the interface. This was demonstrated by fermionizing e^iϕ_n=ψ_r+iψ_R. This type of arguments does not always generalize to more complicated interfaces. On the other hand, the condensation of B systematically captures the gaping process and leads to a correct CFT description of the 331-Pfaffian interface. More importantly, anyon condensation relates every primary field in the original and condensed phases. These relations cannot be obtained from the argument in Sec. <ref>.



 §.§ Transmutation of pseudospin information
 


We now discuss the transmutation of pseudospin information when Abelian quasiparticles cross the interface. Let us first comment on the charge sectors. As one will see, they basically play no role in the resolution of the paradox. Since charge-e/4 quasiparticles are allowed in both Halperin-331 and Pfaffian liquids, dragging quasiparticles across the interface does not require any absorption of net charge at the interface. Furthermore, the gapping of ϕ_c indicates that the dragging will not create any low-energy charge excitation at the interface <cit.>.

Thus, we focus on the neutral sectors. An Abelian quasiparticle in the Halperin-331 liquid has its pseudospin degree of freedom carried solely by the neutral mode ϕ_n. This is observed by writing

    e^iϕ_↑   =e^iϕ_r/2e^iϕ_n/2,
    
    
    e^iϕ_↓   =e^iϕ_r/2e^-iϕ_n/2.

Hence, the vertex operators 𝒱_1 and 𝒱_3 encode the spin-up and spin-down states of an Abelian charge-e/4 quasiparticle, respectively. These two operators are not defined in the Pfaffian liquid. To understand the transmutation of quasiparticles when they cross the interface, we need to represent the four primary fields in the U(1)_4 CFT as different products between two Ising CFTs. One of them describes the interface, whereas the other describes the Pfaffian order. Both Ising CFTs now have the same chirality to match the central charges, 1=1/2+1/2. From Table <ref>, one can obtain the inverted expressions:

    𝒱_0
       ≡ I_1
    =I_1/2×Ĩ+ψ×ψ̃,
    
    𝒱_1
       ≡ e^iϕ_n/2
    =σ×σ̃,
    
    𝒱_2
       ≡ e^iϕ_n
    =ψ×Ĩ+I_1/2×ψ̃,
    
    𝒱_3
       ≡ e^-iϕ_n/2
    =σ×σ̃.

The tilded and untilded fields are in the CFTs describing the interface and the Pfaffian liquid, respectively. Also, the subscripts 1 and 1/2 in the identity fields denote the central charges of the corresponding CFTs. When there is no confusion, these subscripts will be skipped.

Eq. (<ref>) suggests that the U(1)_4 CFT is obtained from condensing the boson b=(ψ,ψ̃) <cit.>. This result is consistent with the orbifold construction <cit.>. After the condensation, one of the unconfined particles is (σ, σ̃). We should state clearly that these two twist fields describe excitations (anyons) at different regions of the system, so it is meaningless to consider their fusion. In other words, the present situation is different from the case of a Pfaffian liquid, in which σ in the bulk and σ at the edge created from vacuum must fuse to I for conserving fermion parity. Importantly,

    (σ, σ̃)×(σ, σ̃)
    =(ψ, ψ̃)+(I, Ĩ)+(ψ, Ĩ)+(I, ψ̃).

The first two terms on the right hand side show that two orthogonal copies of vacuum exist, so (σ, σ̃) needs to split into two inequivalent types of anyons in the resulting U(1)_4 CFT <cit.>. We denote them as (σ, σ̃)_1 and (σ, σ̃)_2. Both of them have conformal spins 1/8, which are identified as the vertex operators 𝒱_1 and 𝒱_3 in the U(1)_4 CFT  (see Table <ref>). The fusion rules are consistent by imposing the conditions
(σ, σ̃)_1×(σ, σ̃)_1=
(σ, σ̃)_2×(σ, σ̃)_2
=𝒱_2
and (σ, σ̃)_1×(σ, σ̃)_2=𝒱_0. Following Ref. <cit.>, we interpret the above result as an incoming pseudospin-up quasiparticle transmutes into a neutral anyon σ̃ at the interface, and another anyon σ free to move in the Pfaffian liquid. To be more precise, the last anyon actually carries charge e/4, but we skip displaying its charge sector e^-iϕ_l/2 explicitly. The same conclusion holds for an incoming quasiparticle with pseudospin down. We illustrate the results in Fig. <ref>.





  §.§.§ Matching of Hilbert spaces and analogue of information scrambling



It is obvious that the total quantum dimension of (σ, σ̃)_1 and (σ, σ̃)_2 is two. It matches the two-dimensional Hilbert space spanned by the pseudospin degree of freedom of an Abelian charge-e/4 quasiparticle. This matching is guaranteed mathematically by the commutativity between fusion and restriction in anyon condensation <cit.>. Interestingly, the information of pseudospin is being stored nonlocally at the interface and in the interior of Pfaffian liquid. There is no local measurement that can distinguish between (σ, σ̃)_1 and (σ, σ̃)_2. Hence, it is impossible to recover the original information from any local measurement. This feature resembles a quantum information scrambling, which can be defined as the spreading of local information into many-body entanglement and correlation in the whole system <cit.>.

The situation becomes more interesting when we keep dragging more Abelian quasiparticles across the interface. Suppose N-1 charge-e/4 quasiparticles were already dragged. We assume the corresponding N-1 anyons σ̃ created at the interface are well separated from each other, so that no fusion occurs between them. We also pose the same assumption for the N-1 anyons created in the Pfaffian liquid. Consider dragging an additional charge-e/4 quasiparticle across the interface. This process increases both numbers of σ̃ and σ by one. As a result, there are N neutral anyons
σ̃ at the interface, and N non-Abelian anyons in the interior of the Pfaffian liquid. The dimension of the corresponding topological Hilbert space is then increased by a factor of two, which is consistent with the one bit of information carried by the additional Abelian quasiparticle from the Halperin-331 liquid. We illustrate the example of N=6 in Figs. <ref>(a) and <ref>(b). Now, we relax the confining potential, and allow the anyons to move and braid <cit.>. The braiding can further scramble the original information <cit.>. The N anyons at the interface are indistinguishable, so are those N anyons in the Pfaffian liquid. Meanwhile, the information carried by pseudospins of the original N Abelian quasiparticles is still preserved. Both Hilbert spaces for indistinguishable anyons at the interface and indistinguishable anyons in the Pfaffian liquid have dimensions 2^N/2. It is intriguing that the interface and the Pfaffian liquid store the same amount of information. This does not hold in the Pfaffian-NASS interface <cit.>.

The above discussion also suggests another important feature. In addition to being stored nonlocally, the original pseudospin information is actually “hidden" in the fusion channels of the non-Abelian anyons. Hence, the scrambled information is protected topologically and will not be destroyed by any local perturbation. This property is essential in topological quantum computation (TQC) <cit.>.



 §.§ Upper bound of information storage and holographic principle



Our previous discussion assumed that local anyons in the system can be well separated to prevent fusion. This assumption leads to a natural question. How much information can be stored nonlocally with the topological protection that has been described?

Recall that the minimum separation between two anyons is in the order of the magnetic length ℓ_B, so that they are well defined individually and do not fuse. From this, one may naively think that the maximum amount of information can be stored is N_A∼ A/πℓ_B^2, where A denotes the area of the Pfaffian liquid. This argument is valid if the information is carried solely by anyons in the Pfaffian liquid. However, this is not true in the present case. We have assumed both Pfaffian liquid and 331-Pfaffian interface were initially in the ground state with no excitations. As we discussed, the nonlocal storage of pseudospin information of the Abelian quasiparticles from the Halperin-331 liquid requires both anyons at the interface and in the Pfaffian liquid.

For an interface with a length (perimeter) L, it can only accommodate N_L∼ L/ℓ_B neutral anyons σ̃. Since the radius R of a circular quantum Hall droplet satisfies R≫ℓ_B, one has N_L≪ N_A. When the number of σ̃ gets close to or exceeds N_L, different σ̃ anyons start to fuse. The resulting particles will be either a fermion or a boson that can propagate back to the Halperin-331 liquid. More explicitly, one has

    (I,σ̃)×(I,σ̃)
    =(I,Ĩ)+(I,ψ̃).

For the first fusion outcome, the two σ̃ anyons at the interface can fuse to a neutral boson with its spin part described by 𝒱_0 [see Eq. (<ref>)]. This neutral boson can then split to a pair of quasihole and quasiparticle with opposite charges but the same pseudospin, and propagate in the 331 liquid. For the second fusion outcome, the neutral fermion can split to a pair of quasihole and quasiparticle with opposite pseudospins propagating in the 331 liquid. Consequently, some of the hidden information is released and being accessible by local probes. Therefore, the “black hole” is no longer completely black. Note that the released information is not protected topologically and can suffer from quantum decoherence. The discussion shows that the length of the interface sets an upper bound of storing information nonlocally and topologically via (σ,σ̃) pairs. Furthermore, the magnetic length ℓ_B=√(1/eB) (in the unit of h/2π=c=1) plays the role of Planck length in the present system. Here, B denotes the magnetic field.

The above observation actually resembles the argument from holographic principle in black holes. Based on this principle, the maximum amount of information can be stored in a black hole is not determined by its volume, but bounded by its area <cit.>. This is because the Bekenstein entropy of the black hole is proportional to its area <cit.>, which limits the number of degree of freedoms the black hole can have. In contrast to the quantum Hall interface, the black hole can always store and “hide" more information by increasing its area. Since the length of the quantum Hall interface is assumed to be fixed, the analogous (a weaker version of) holographic principle there implies that Hawking radiation in the form of Abelian quasiparticles and quasiholes will be released when the bound N_L∼ L/ℓ_B is reached. Roughly speaking, any additional incoming information is thus reflected by the interface (event horizon).



§ INFORMATION PARADOX IN 330-RR_4 INTERFACE


In the previous section, we demonstrated that an original Abelian anyon from the Halperin-331 QH liquid would transmute into a pair of anyons, when it crosses the interface and enters the non-Abelian Pfaffian QH liquid. More specifically, one of the resulting anyons is neutral and being created at the interface, and the remaining one is created in the Pfaffian liquid. By proximitizing different pairs of QH liquids, a wide variety of interfaces can be formed. The corresponding quasiparticle transmutation (if any) will depend on the properties of the interface. Hence, different ways of scrambling the original information may be realized in different QH interfaces. Recall that the 331-Pfaffian interface has a chiral central charge of c=2-3/2=1/2. This indicates that there must be a gapless chiral Majorana fermion mode on the interface <cit.>. As a result, some of the scrambled information can be stored at the interface. Meanwhile, an interface with a zero chiral central charge can be created by proximitizing a pair of different QH states with identical central charges for their edges. Interfaces of this kind have been studied extensively, and importantly, can be gapped or gapless. When both QH phases are Abelian, they are described by the K-matrix formalism <cit.>. This enables one to study the gapness of the interface through the concepts of Lagrangian subgroups <cit.> or the null vector criteria <cit.>. For an interface involving non-Abelian QH state(s), its gapness and the corresponding gapped phases can be explored by the anyon condensation approach <cit.>. A rigorous mathematical formalism was developed in <cit.>. We note that gapped interfaces between different non-Abelian topologically ordered states is still an ongoing research topic.

Motivated by the above discussion, we consider in this section the information paradox and the corresponding quasiparticle transmutation in the interface between the Abelian Halperin-330 state <cit.> and the non-Abelian Read-Rezayi state at level four <cit.>. Both states or phases may describe the fractional QH (FQH) state in a bilayer system at the total Landau-level filling factor ν=2/3. Whether these phases are favorable or not in a realistic sample depends on the actual microscopic details of the system. Here, we assume both states and the interface between them can be realized, and study the possible consequences. Previous theoretical work suggested that a continuous phase transition between the Halperin-330 state and the RR_4 state might be triggered by tuning the interlayer tunneling strength in a bilayer FQH system <cit.>. This transition and similar phase transitions in bilayer systems at other filling factors can be studied systematically through the anyon condensation approach <cit.>. In particular, the condensation of a specific type of anyon in the RR_4 state leads to the Halperin-330 state. Based on the “folding trick" <cit.>, it is expected that a gapped boundary between these two FQH states can form. Nevertheless, the precise gapping mechanism and the possibility of having different phases for the gapped interface have not been addressed. Suppose the interface can be gapped, the scrambled pseudospin information may be entirely stored by anyons in the non-Abelian RR_4 FQH liquid with no information stored at the interface. This feature is completely different from the one in the aforementioned 331-Pfaffian interface <cit.>.



 §.§ Review of Halperin-330 and Read-Rezayi states



The Halperin-330 state is a two-component Abelian topological order, which is characterized by the matrix <cit.>,

    K=[ 3 0; 0 3 ],

and the associated t=(1,1)^T. Clearly, one has ν=t^T K^-1t=2/3. In a more explicit form, the corresponding topological term for the Halperin-330 edge is

    L_0
    =-3/4π(∂_tϕ_↑∂_xϕ_↑
    +∂_tϕ_↓∂_xϕ_↓).

Here, the subscripts ↑, ↓ denote the layer or the pseudospin index. Roughly speaking, each layer of the Halperin-330 state is a simple Laughlin state at 
ν=1/3  <cit.>. Since ϕ_↑ and ϕ_↓ have the same chirality, the edge is maximally chiral and has a central charge c=2. 

The two most relevant vertex operators for electrons are exp(3iϕ_↑) and exp(3iϕ_↓). These two different operators create respectively an electron in the upper and the lower layer. We call them as electrons with pseudospin up and pseudospin down. Both electrons operators have scaling dimensions 3/2, indicating that they are indeed fermionic as required. The most fundamental anyon that the Halperin-330 state can host has charge e/3. Depending on its pseudospin, it is created by the operator exp(iϕ_↑) or exp(iϕ_↓). The OPE between any one of them and each of the electron operators is singlevalued. For example, one has the following OPEs:

    lim_z→ w
    e^3iϕ_↑(z)× e^iϕ_↑(w)   ∼ (z-w)e^4iϕ_↑(w),
    
    lim_z→ w
    e^3iϕ_↓(z)× e^iϕ_↑(w)   ∼ e^i[3ϕ_↓(w)+ϕ_↑(w)].

Notice that |α|=1 is the smallest possible nonzero value for the generic operator exp(iαϕ_↑) to have singlevalued OPEs with both electron operators. This fact verifies that the e/3 anyon is the smallest-charge anyon that can be host by the Halperin-330 state.



  §.§.§ Parafermion conformal field theory and Read-Rezayi states


Different from the Halperin-330 state, the Read-Rezayi states are a series of non-Abelian topological orders introduced from the CFT approach <cit.>. Each RR state consists of two different types of CFTs. First, it has a compactified U(1) holomorphic boson ϕ. On the edge of the system, ϕ corresponds to the gapless charge mode described by

    L_ϕ
    =-1/4πν∂_xϕ(∂_t+v∂_x)ϕ.

This mode fixes the charge density ρ(x)=∂_xϕ/2π and the quantized electrical Hall conductance σ_xy=ν e^2/h. The possible values of the filling factor ν depend on the second type of CFT in the RR state, which is the chiral ℤ_k parafermion CFT with the central charge <cit.>,

    c=2(k-1)/k+2.
 

The above parafermion CFT can be obtained from the SU(2)_k/U(1) coset construction <cit.>. Here, k∈ℕ is the level of the corresponding Wess-Zumino-Witten model. Following the notations in Ref. <cit.> (which are more convenient for the general discussion), each Virasoro primary field in the parafermion CFT is labeled as Φ^ℓ_m with ℓ+m≡ 0 (mod  2). The index ℓ=0,1,⋯, k. By imposing the field identification, Φ^ℓ_m=Φ^ℓ_m+2k=Φ^k-ℓ_m-k, the index m can be restricted to the range -ℓ<m≤ℓ, and ℓ>0. Hence, there are in total k(k+1)/2 Virasoro primary fields. Their scaling dimensions are given by

    h^ℓ_m=ℓ(ℓ+2)/4(k+2)-m^2/4k.

Since the CFT is chiral or holomorphic, the scaling dimension of the field is also its conformal spin. The fusion rule between different fields is

    Φ^ℓ_1_m_1×Φ^ℓ_2_m_2
    =∑^min(ℓ_1+ℓ_2, 2k-ℓ_1-ℓ_2)_ℓ=|ℓ_1-ℓ_2|Φ^ℓ_m_1+m_2.

When k≥ 2, the ℤ_k parafermion CFT is non-Abelian. A famous example is the Ising CFT at k=2, which has a non-Abelian anyon σ corresponding to the field Φ^1_1. The Ising CFT is relevant in the description of FQH state at ν=5/2 <cit.>. The parafermion CFTs and their related FQH states have attracted considerable amount of attention as the non-Abelian anyons there are useful in topological quantum computation <cit.>. 

An anyon (including the electron) in the RR state is created by the operator, 
ηexp(iωϕ). Here, η is a Virasoro primary field in the parafermion CFT. It is customary to choose η=Φ^k_k-2 for constructing the electron operator. The conformal spin for Φ^k_k-2e^iϕ/ν is

    h=k-1/k+1/2ν.

By requiring ν>0 and h being an half-integer, the possible filling factor is fixed at ν=k/(Mk+2), where M is a positive odd integer. For k=4 and M=1, the Read-Rezayi state may describe a FQH state of electrons at ν=2/3. This matches the filling factor of the Halperin-330 state. In particular, the Lagrangian density describing the chiral Bose mode ϕ_l on the RR_4 edge is

    L_ϕ_l=-3/8π∂_xϕ_l(-∂_t+v∂_x)ϕ_l.

Since we will eventually study the interface between two FQH liquids, we define the chirality of ϕ_l as opposite to the chiralities of the two Bose modes on the Halperin-330 edge. The subscript l indicates that ϕ_l is a left-moving mode. As a reminder, the complete edge theory for the RR_4 state also involves the neutral parafermionic sector (see the particular discussion in Ref. <cit.>). 

For later reference, the ten different Virasoro primary fields in the ℤ_4 CFT and their fusion rules are listed in Table <ref>. Note that we have switched to another set of notations (somehow more convenient for this particular discussion) for the primary fields. The identification with the notations in the previous discussion:

    I   =Φ^4_4, ψ_1=Φ^4_2, ψ_2=Φ^4_0, ψ_3=Φ^4_-2, 
    σ_+=Φ^3_3,
    
    σ_-   =Φ^1_1, ϵ=Φ^2_0, ρ=Φ^2_2, 
    χ_-=Φ^3_1, χ_+=Φ^3_-1.

After determining the electron operator, all other operators for quasiparticles can be deduced in a “brute-force" manner by requiring the OPE between ηexp(iωϕ) and the electron operator is single-valued. This restricts the possible values of ω in the vertex operator for each separate η as listed in Table <ref>. From the result, the charge of the corresponding quasiparticles 2ω/3 (in unit of e) and the whole spectrum of anyons that the Read-Rezayi state can host is determined. Notice that the quantum dimension of the anyon a is determined by the largest eigenvalue of the fusion matrix N_a with matrix elements (N_a)_bc=N^c_ab. Here, N^c_ab is defined from the fusion between the anyons a and b, and the resulting anyon c:  

    a× b=∑_c N^c_ab c.

The maximum eigenvalue is positive and nondegenerate, as guaranteed by the Perron-Frobenius theorem. For an Abelian anyon, the quantum dimension is one. Otherwise, the anyon is non-Abelian.







 §.§ Mathematical details of anyon condensation



Here, we discuss the interface between the Halperin-330 RR_4 states (abbreviated as 330-RR_4 interface) from the anyon condensation perspective. Readers who are uninterested in the mathematical details may skip this subsection. The corresponding physical picture for the fully gapped interface and mechanism of scrambling the pseudospin degree of freedom in the system are reviewed in Secs. <ref> and <ref>.

Since both edges of the QH states have the same central charge but with opposite chiralities at the interface, the interface has a zero chiral central charge. In principle, this kind of interface can be (but not guaranteed to be) fully gapped. For example, the edge of the ℤ_2 toric code (i.e., the boundary between it and the vacuum) remains gapless if the chosen boundary does not break the translational symmetry <cit.>. However, the edge in a generic situation (with the so-called smooth or rough boundary) is gapped due to the condensation of either one of the self-bosons, e or m on the edge <cit.>. The discussion suggests that there are two different possible gapped boundaries. When the gapped boundary is obtained by condensing e, then m is confined and becomes a boundary excitation, or vice versa. Can the 330-RR_4 interface be fully gapped? If the answer is affirmative, there will be no gapless excitations on the interface. Then, it may be possible to transmute the original information carried by certain types of the Abelian anyons completely into topological information stored by anyons in the non-Abelian RR_4 liquid.

In the following discussion, we first employ the anyon condensation approach to explore the gapness of the 330-RR_4 interface. We use 𝒜 and ℬ to denote the sets of anyons for the Halperin-330 state and RR_4 state, respectively. The possible phases of the interface are determined by the resulting phases from condensing different possible anyons (if it occurs) in 𝒜×ℬ̅. Here, ℬ̅ indicates the conjugation of ℬ. In the present case, we have

    𝒜   ={e^iαϕ_↑e^iβϕ_↓},
    
    ℬ̅   ={η e^iω_ηϕ_l}.

Here, α, β are integers, and η denotes the Virasoro primary field in the ℤ_4 parafermion CFT. For each η, the possible values of ω_η are given by the last row in Table <ref> with m∈ℤ. Since both electrons in the Halperin-330 state and RR_4 state are fermions that have trivial mutual statistics with every anyon in 𝒜×ℬ̅, 𝒜×ℬ̅ is a fermionic topological order. Specifically, we introduce the symbol Ψ_e=e^3iϕ_↑ to denote the electron with pseudospin up for the Halperin-330 state. It is a trivial fermion in 𝒜×ℬ̅.



  §.§.§ Determination of Lagrangian subset


The interface can be fully gapped only if there exists a Lagrangian subset ℒ⊂𝒜×ℬ̅, in which the condensed anyons are the only deconfined anyons in the condensed phase. To identify some possible ℒ, we follow the strategy in Ref. <cit.> by first condensing Abelian anyons. After that, we will analyze the resulting phase and check for the necessity of condensing more anyons in the system to obtain a Lagrangian subset. Finally, we discuss the underlying reason that gives rise to the anyon condensation. It is vital to clarify that one can actually condense non-Abelian anyons, but the strategy below greatly simplifies the analysis. When the Lagrangian subset ℒ only consists of Abelian bosons, then the explicit conditions for having a fully gapped interface are:


  * trivial monodromy for all a∈ℒ: M^aa̅_I=1;

  * trivial mutual statistics for all a, b∈ℒ: M^ab_a× b=1;

  * confinement for all a∉ℒ: there exists at least one b∈ℒ that has a non-trivial braiding phase with a.

The condition (i) ensures that a is a boson. If a is its own antiparticle (i.e., a=a̅ and a×a̅=I), then a should have an integer conformal spin. When a is not its own antiparticle, then it further requires that b=a× a also has an integer conformal spin <cit.>. For condition (ii), it implies that the anyon c=a× b also needs to be a boson, otherwise it needs to be condensed as well <cit.>. Note that one can also condense fermions (by fusing it with the trivial fermion) in a fermionic topological order <cit.>, but this turns out unnecessary in our discussion.



  §.§.§ Separation of charge and neutral sectors


From Table <ref>, it is clear that the four Abelian anyons in the ℤ_4 parafermion CFT are I, ψ_1, ψ_2, and ψ_3. To deduce the set of condensable Abelian bosons, one can in principle analyze all anyons with η={I, ψ_1, ψ_2, ψ_3} and generic values for α, β, and ω_η in the vertex operators. However, this general treatment turns out to be unphysical. As we will discuss in Sec. <ref>, the gapped interface originates from the gapping or localization of counterpropagating edge modes due to electron and quasiparticle tunneling process. Therefore, the charge modes and neutral modes should obtain expectation values independently. This suggests us to separate the charge and neutral sectors in the problem, which is achieved by introducing a new set of modes for the Halperin-330 edge:

    ϕ_r=ϕ_↑+ϕ_↓,
    
    ϕ_n=ϕ_↑-ϕ_↓.

They correspond to the overall charge mode and neutral spin mode for the Halperin-330 edge, respectively. Both modes are right-moving. Using the new set of modes, the topological term in Eq. (<ref>) becomes

    L_0=-3/8π(∂_tϕ_r∂_xϕ_r
    +∂_tϕ_n∂_xϕ_n).
 
The t vector becomes t=(1,0)^T, which verifies that ϕ_n is a neutral mode. Furthermore, we have

    𝒜×ℬ̅
    ={e^iQϕ_r e^iSϕ_n}×{η e^iω_ηϕ_l}.

Here, Q=(α+β)/2 and S=(α-β)/2. Since α∈ℤ and β∈ℤ, both Q and S can only take half-integer or integer values. The corresponding anyon in 𝒜 has charge 2Q/3 (in the unit of e). Furthermore, the conformal spins for the two vertex operators exp(iQϕ_r) and 
exp(iSϕ_n) are

    s_Q=h_Q=Q^2/3 , 
    s_S=h_S=S^2/3.

Each of them will change by an integer value whenever Q or S is changed by 3ℤ. Physically, this corresponds to the fusion between the anyon with a trivial boson in the Halperin-330 state. Thus, we can identify a pair of anyons with 
(Q, S) and (Q+3ℤ, S+3ℤ). More explicitly, one can rescale the fields φ_r=ϕ_r/2 and φ_n=ϕ_n/2 and rewrite Eq. (<ref>) as

    L_0=-6/4π(∂_tφ_r∂_xφ_r
    +∂_tφ_n∂_xφ_n).
 
The possible vertex operators take the form exp(ipφ_r) and 
exp(iqφ_n), where both p and q can only take integer values now. Thus, both φ_r and φ_n (and hence the original fields, ϕ_r and ϕ_n) are compactified bosons in the U(1)_6 CFT which has exp(6iφ) as the trivial boson. For later discussion, the six primary fields involving the neutral spin mode in the U(1)_6 CFT  are summarized in Table <ref>. Moreover, a pair of anyons with (Q,S) and [Q+3(ℤ+1/2), S+3(ℤ+1/2)] differ by the fusion of an odd multiples of electrons in the Halperin-330 state. 





  §.§.§ Condensation of Abelian bosons


Following the above discussion, we look for possible values of Q, ω_η, and S such that both anyons with operators exp(iQϕ_r)exp(iω_ηϕ_l) and ηexp(iSϕ_n) are bosonic. The conformal spin for exp(iQϕ_r)exp(iω_ηϕ_l) is

    s(Q,ω_η)=Q^2-ω_η^2/3.

Therefore, a possible solution for s(Q,ω_η)=0 (or alternatively, the null vector for the K matrix describing the charge sector) is (Q,ω_η)=(1,1). Hence, we first condense the following four bosons:

    b_0
       =e^im_0(ϕ_r+ϕ_l),
    
    
    b_1
       =ψ_1 e^3iϕ_n/2e^i(m_1+1/2)(ϕ_r+ϕ_l),
    
    
    b_2
       =ψ_2 e^im_2(ϕ_r+ϕ_l),
    
    
    b_3
       =ψ_3 e^3iϕ_n/2e^i(m_3+1/2)(ϕ_r+ϕ_l).

Here, all m_i∈ℤ. It is easy to check that all of the above bosons satisfy the conditions (i) and (ii) in the previous discussion. This verifies that all of them can be condensed simultaneously <cit.>. It is worthwhile for clarifying that being a boson does not mean that it must be condensable. For example, the condensation of non-Abelian bosons can be obstructed by a no-go theorem <cit.>. Also, a self-dual boson having Frobenius-Schur indicator ϰ=-1 is not condensable <cit.>. Nevertheless, none of these limitations applies here.

Since a deconfined anyon must have trivial mutual statistics with all the condensed bosons, we first put m_2=0 to eliminate a large set of possibilities. The corresponding monodromy is given by

    M^ηψ_2_η×ψ_2
    =exp[-(2π i)(h_η×ψ_2-h_η-h_ψ_2)].

Hence, all anyons with η={σ_+, σ_-, χ_+, χ_-} are confined. This result was used by Barkeshli and Wen in showing that the phase transition from the RR_4 state to the Halperin-330 state could be described by the anyon condensation of ψ_2 <cit.>. By setting m_2≠ 0, the monodromy between Y=ηexp(iSϕ_n)exp(iQϕ_r)exp(iω_ηϕ_l) and b_2 is given by

    M^Yb_2_Y× b_2
    =exp[4m_2 π i/3(Q-ω_η)] , 
    m_2∈ℤ.

Here, η={I, ψ_1, ψ_2, ψ_3, ϵ, ρ}. Thus, the set of possible deconfined anyons is further reduced to
e^iω_η(ϕ_l+ϕ_r)
e^3iγϕ_r/2
e^iSϕ_n×{I, ψ_1, ψ_2, ψ_3, ϵ, ρ}, with γ∈ℤ. This set of anyons have trivial mutual statistics with b_0 also. By requiring them to have trivial mutual statistics with b_1 and b_3, it further reduces the possible set of deconfined anyons to four apparently different classes. For the first class, we have

    𝒯_1
       =e^im(ϕ_l+ϕ_r)
    e^3i(p+1/2)ϕ_r
    e^i(q+1/2)ϕ_n×{I, ψ_2, ϵ}
       ∼ e^im(ϕ_l+ϕ_r)[e^3i(ϕ_r+ϕ_n)/2]
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ψ_2, ϵ}
       ∼ e^im(ϕ_l+ϕ_r)Ψ_e
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ψ_2, ϵ}
       ∼Ψ_e
    {𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}.

All m, p, and q in the above calculation are integers. It is recalled that 
Ψ_e=exp(3iϕ_↑)=exp[3i(ϕ_r+ϕ_n)/2] is the electron in the Halperin-330 state. In the second line, we used the fact that ϕ_n is a compactified boson in the U(1)_6 CFT, and labeled the corresponding vertex operators by the symbols in Table <ref>. Moreover, we used the fact that exp(3ipϕ_r) is a trivial boson to make the identification denoted by ∼. In the last line, the identification is made by fusing the set of anyons in the third line with the condensed boson b_2 with m_2=-m. 

Using similar procedures, we determine the second class of deconfined anyons as

    𝒯_2
       =e^im(ϕ_l+ϕ_r)
    e^3ipϕ_r
    e^iqϕ_n×{I, ψ_2, ϵ}
       ∼{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}.

By setting η={ψ_1, ψ_3, ρ}, we have the remaining two classes of deconfined anyons,

    𝒯_3
       =e^i(m+1/2)(ϕ_l+ϕ_r)
    e^3ipϕ_r
    e^i(q+1/2)ϕ_n×{ψ_1, ψ_3, ρ}
       ∼ e ^i(ϕ_l+ϕ_r)/2{𝒱_1, 𝒱_3, 𝒱_5}×{ψ_1, ψ_3, ρ},

and

    𝒯_4
       =e^i(m+1/2)(ϕ_l+ϕ_r)
    e^3i(p+1/2)ϕ_r
    e^iqϕ_n×{ψ_1, ψ_3, ρ}
       ∼Ψ_e  e ^i(ϕ_l+ϕ_r)/2{𝒱_1, 𝒱_3, 𝒱_5}×{ψ_1, ψ_3, ρ}.

Meanwhile, they are actually equivalent to 𝒯_2 and 𝒯_1, respectively. This is observed by fusing 𝒯_3 and 𝒯_4 with either b_1 or b_3 to obtain

    𝒯_3    ∼{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}∼𝒯_2,
    
    𝒯_4    ∼Ψ_e{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}∼𝒯_1.

Therefore, the above four classes of deconfined anyons can be combined into

    𝒯   ={1, Ψ_e}×{𝒱_0, 𝒱_2, 𝒱_4}×{I, ϵ}
       ={1, Ψ_e}×𝒯_B.

Since 𝒱_2 and 𝒱_4 cannot be obtained from any fusion between the bosons in the set {b_0, b_1, b_2, b_3}, this set is not a Lagrangian subset. In the second line of Eq. (<ref>), 
𝒯_B is a bosonic order with 𝒱_0 I being the trivial boson. The separable form in Eq. (<ref>) is guaranteed for any Abelian fermionic order <cit.>. The Abelianity of 𝒯 becomes transparent after splitting the non-Abelian anyons in 𝒯_B. Notice that (Ψ_e)^2=exp(3iϕ_r)exp(3iϕ_n)∼ 1. 



  §.§.§ Splitting of non-Abelian anyons and the ℤ_3 toric code


Since some of the anyons in the original phase have been identified as the trivial vacuum after the anyon condensation, the unconfined non-Abelian anyons in 𝒯 may need to split. Consider the fusion:

    ϵ×ϵ
    =I+ ψ_2+ϵ∼ 2I + ϵ.

Due to the identification ψ_2∼ I, the vacuum appears twice. Consequently,
ϵ needs to split into two Abelian anyons, ϵ=ϵ_1+ϵ_2. This matches the quantum dimension as 2=1+1. Furthermore, the splitting and the fusion rules are consistent only if

    ϵ_1×ϵ_1=ϵ_2,
    
       ϵ_2×ϵ_2=ϵ_1,
    
       ϵ_1×ϵ_2
    =ϵ_2×ϵ_1=I.

Then, the fusion rule ρ×ρ=I+ψ_2+ϵ implies that the non-Abelian anyon ρ also needs to split, ρ=ρ_1+ρ_2. After the splitting, a possible (but not unique) consistent set of fusion rules are

    ρ_1×ρ_1=ϵ_1 , ρ_2×ρ_2=ϵ_2 , ρ_1×ρ_2=I,
    
       ϵ_1×ψ_1=ρ_2
     , ϵ_2×ψ_1=ρ_1,
    
       ρ_1×ψ_1=ϵ_2
     , ρ_2×ψ_1=ϵ_1,
    
       ρ_1×ϵ_1=ψ_1,
     , ρ_1×ϵ_2=ρ_2,
    
       ρ_2×ϵ_1=ρ_1,
     , ρ_2×ϵ_2=ψ_3.

Note that ψ_3∼ψ_1 in the condensed phase since they differ by a fusion with ψ_2 (i.e, b_2 with m_2=0). Therefore, the fusion rules involving ψ_3 are the same as those involving ψ_1. By making the change(s), ϵ_1↔ϵ_2, or/and ρ_1↔ρ_2, one can still obtain a consistent set of fusion rules. This is reasonable as it is impossible to uniquely define or fix the anyons resulting from the splitting. On the other hand, they must be inequivalent. Without loss of generality, we will stick with the set of fusion rules in Eqs. (<ref>) – (<ref>) in the following discussion.

The above discussion shows that 𝒯_B actually consists of nine different Abelian anyons, so 𝒯 is an Abelian fermionic topological order. Then, what is the topological order 𝒯? By labeling the nine anyons in 𝒯_B as shown in Table <ref>, it is observed that these anyons take the form of e^p m^q, where p, q∈ℤ. Moreover, one has e^3=m^3=𝕀. Hence,𝒯_B is the ℤ_3 toric code. This topological order is a generalization of the more famous ℤ_2 toric code that only has four anyons, {1, e, m, f≡ em} <cit.>. Since it is the topological order built from (mathematically, the modular tensor category constructed over) the quantum double model with the finite group ℤ_3 <cit.>, the ℤ_3 toric code is also denoted as 𝔇(ℤ_3) <cit.>. It was suggested that this special topological order can be realized by proximitizing a bilayer system of electrons and holes in separate Laughlin states with respective filling factors ± 1/3 to a superconductor <cit.>. It was also pointed out that the more general ℤ_p toric code appears as the symmetry-enriched neutral sector of non-diagonal quantum Hall states <cit.>. Here, we have shown that the interface between the Halperin-330 and RR_4 states with suitable anyon condensation also leads to the ℤ_3 toric code. The physical origin that triggers such an anyon condensation will be discussed in Sec. <ref>. On the application side, it was suggested that the ℤ_3 toric code could be used in implementing universal topological quantum computation <cit.>. To summarize, the condensation of b_0, b_1, b_2, and b_3 leads to the condensed phase with deconfined anyons,

    𝒯={1, Ψ_e}×𝔇(ℤ_3).

Note that 𝒯 does not describe a fully gapped interface. This is because 
𝔇(ℤ_3) is not a topologically trivial order. Alternatively, the edge of 𝔇(ℤ_3) is gappable but the edge remains gapless unless a further anyon condensation occurs, which we are going to discuss below.






  §.§.§ Two different phases of gapped interfaces


As we have shown, the set {b_0, b_1, b_2, b_3} is not a Lagrangian subset, and the condensation of this set of bosons does not lead to a fully gapped interface. On the other hand, a fully gapped interface can be achieved by further condensing some bosonic particles in 𝔇(ℤ_3). By recycling the results in Refs. <cit.>, one can immediately conclude that there are two different types of fully gapped interfaces between the Halperin-330 FQH state and the RR_4 FQH state.

The first possible kind of fully gapped interface is obtained by a further condensation of e and e^2 in 𝔇(ℤ_3). It is straightforward to verify that both of them satisfy conditions (i) and (ii) in the previous discussion. This implies that they can be condensed simultaneously. Moreover, their condensation leads to the confinement of the remaining anyons (except the identity 𝕀) in 
𝔇(ℤ_3). Therefore, we determine the first Lagrangian subgroup <cit.> for 𝒜×ℬ̅ as

    ℒ_e
    ={b_0, b_1, b_2, b_3}×{𝕀, e, e^2}.

The corresponding fully gapped interface is usually known as the e-boundary. Notice that the symbol × actually means picking an element from each set on the right hand side of Eq (<ref>) and then fuse them (see Eq. (<ref>) for example). The product structure of ℒ_e ensures that it is a maximal set of condensable bosons.

Another possible type of fully gapped interface is obtained by condensing m and m^2 in 𝔇(ℤ_3) instead. This leads to the second Lagrangian subgroup,

    ℒ_m
    ={b_0, b_1, b_2, b_3}×{𝕀, m, m^2}.

The corresponding fully gapped interface is known as the m-boundary. For both e- and m-boundaries, the remaining deconfined anyons outside the Lagrangian subgroups are ℱ_0={1, Ψ_e}. This is the trivial fermionic topological order, which indicates that the condensation of anyons in ℒ_e or ℒ_m leads to a fully gapped interface <cit.>.



 §.§ Physical picture of the gapped interface
  


While anyon condensation has provided a systematic and mathematical approach in studying the gapness of the interface, it will be also desirable to understand the gapping of the interface in a more physical picture. We claim that the anyon condensation may originate from the electron and quasiparticle tunneling processes at the interface in two different steps.

At the beginning, the Halperin-330 and Read Rezayi states are two topologically distinct phases. Therefore, only electrons can tunnel across the two different FQH liquids. In general, the electron tunneling process is described by the following Lagrangian density,

    L_el, tun
    =    ξ_1,a(x)(ψ_1 e^3iϕ_l/2e^3iϕ_↑)^a
    
       +ξ_2,a(x)(ψ_1 e^3iϕ_l/2e^3iϕ_↓)^a
    +H.c.
    
    =    ξ_1,a(x)[ψ_1 e^3iϕ_n/2 e^3i(ϕ_l+ϕ_r)/2]^a
    
       +ξ_2,a(x)[ψ_1 e^-3iϕ_n/2 e^3i(ϕ_l+ϕ_r)/2]^a
    +H.c.

Note that H.c. stands for the Hermitian conjugation. We include the exponent a>1 to describe multi-electron tunneling processes. As the electron operators ψ_1 e^3iϕ_l/2 and e^3iϕ_σ (where σ=↑, ↓) enter the same number of times, L_el, tun conserves the total electric charge. Furthermore, the tunneling amplitudes ξ_1,a(x) and ξ_2,a(x) are random functions in x. This is because the electron tunneling generally does not conserve momentum, and disorder needs to be involved. Due to the random nature of the tunneling, some of the processes are actually irrelevant in the renormalization group sense <cit.>. In this situation, we will need to assume the tunneling strength W_a, defined as ξ_a(x)ξ_a(x')=W_aδ(x-x'), is sufficiently large so that the charge modes 3a[ϕ_l(x)+ϕ_r(x)]/2 can still be pinned. Then, this combination of counterpropagating charge modes obtains a nonzero expectation value, and indicates the localization (analogous to mode gapping in nonrandom tunneling) of charge modes. For the neutral sector, it is expceted that the combination ψ_1 e^± 3iϕ_n/2 will also be gapped by ℒ_el, tun in the strong coupling regime. By defining (ψ_1)^2∼ψ_2, (ψ_1)^3∼ψ_3, and (ψ_1)^4∼ I, the tunneling processes described by ℒ_el, tun lead to the condensation of b_0, b_1, b_2, and b_3 in Eqs. (<ref>) – (<ref>). 

After the above condensation or charge-mode localization, the interface remains gapless. Meanwhile, quasiparticles with charges e/3 and 2e/3 can tunnel across this “new" gapless interface. Let us specifically consider the tunneling of charge e/3 anyon with the topological sector ρ in the Read-Rezayi liquid. This process can be described by the following Lagrangian density,

    L_e/3, tun
    
    =    ζ_1(x)(ρ e^iϕ_l/2e^iϕ_↑)
    +ζ_2(x)(ρ e^iϕ_l/2e^iϕ_↓)
    +H.c.
    
    =    ζ_1(x)[ρ𝒱_1 e^i(ϕ_l+ϕ_r)/2]
    +ζ_2(x)[ρ𝒱_5 e^i(ϕ_l+ϕ_r)/2]
    +H.c.

Since the charge modes have been localized, we focus on the neutral sector. Now, both combinations ρ𝒱_1 and ρ𝒱_5 have zero conformal spins (see Tables <ref> and <ref> for reference), which indicates that they are bosonic. In principle, they may be condensed or gapped as well. Consider the set of anyons generated from any fusion between ρ𝒱_1 and 
ρ𝒱_5, we have

    {ρ𝒱_1, ρ𝒱_5}×{ρ𝒱_1, ρ𝒱_5}
    ={I, ψ_2, ϵ}×{𝒱_0, 𝒱_2, 𝒱_4}

A further fusion with {ρ𝒱_1, ρ𝒱_5} gives

    {ψ_1, ψ_3, ρ}×{𝒱_1, 𝒱_3, 𝒱_5}.

Hence, the fusion between ρ𝒱_1 and ρ𝒱_5 generates the list of anyons (more precisely, their neutral sectors) in ℒ_e and ℒ_m. In this sense, the condensation described by the Lagrangian subgroup 
ℒ_e or ℒ_m may be understood as originating from the gapping of modes due to e/3 quasiparticle tunneling at the interface. However, this is only possible if we can treat the whole system as a single topological phase. Otherwise, only electrons can tunnel across two topologically distinct phases (for example, between a FQH liquid and a normal metal) <cit.>.



 §.§ Transmutation of pseudospin information



Previously, we found that it is possible (at least in principle) to form a gapped interface between the Halperin-330 state and the Read-Rezayi state at level four. Although both states may describe the FQH state in a bilayer system at total filling factor 2/3, they host different sets of anyons. Specifically, the Halperin-330 state is a two-component state, in which the anyons possess the layer or pseudospin degree of freedom that is absent in the RR_4 state. Meanwhile, the RR_4 state supports non-Abelian anyons that have more complicated fusion rules than the usual Abelian quasiparticles. This observation leads to a natural question: What happens when an Abelian quasiparticle from the Halperin-330 liquid crosses the gapped interface and enters the non-Abelian RR_4 FQH liquid? In the opposite direction, what happens when we drag a non-Abelian quasiparticle from the RR_4 liquid to the Halperin-330 liquid? The answers to these questions lead to the ideas of topological quantum information scrambling and Andreev-like reflection of non-Abelian anyons. 

An anyon a originally in the topological phase 𝒜 can pass through the interface (described by the condensed phase of 𝒜×ℬ̅) and transmutes into an anyon b in ℬ if and only if ab is a deconfined anyon in the condensed phase. This idea was introduced to study the transmutation between anyons in the interface between the Pfaffian and non-Abelian spin-singlet FQH states <cit.>. Now, we employ the same kind of argument to study the transmutation of Abelian anyons in the Halperin-330 FQH liquid when they cross the interface and enter the RR_4 liquid. 



  §.§.§ Transmutation of Abelian charge e/3 anyon


First, an Abelian charge e/3 anyon in the Halperin-330 state can have pseudospin up or pseudospin down. We denote them as (e/3, ↑) and (e/3, ↓), respectively. Their associated CFT operators are

    (e/3, ↑)≡exp(iϕ_↑)
    =𝒱_1exp(iϕ_r/2),
    
    
    (e/3, ↓)≡exp(iϕ_↓)
    =𝒱_5exp(iϕ_r/2).

When they cross the fully gapped 330-RR_4 interface, their transmutation depends on the type of the gapped interface (i.e., whether it is an e- or m-boundary). 

Suppose the fully gapped interface is described by the e-boundary. Then, the deconfined anyons are those in ℒ_e in Eq. (<ref>). In particular, we have the following pair of deconfined anyons:

    e^2× b_1
       = [ρ_1 exp(iϕ_l/2)] [𝒱_1exp(iϕ_r/2)],
    
    
    e× b_1
       = [ρ_2 exp(iϕ_l/2)] [𝒱_5exp(iϕ_r/2)].

These are the only two deconfined anyons that involve (e/3,↑) and (e/3,↓). Since e^2× b_1 and e× b_1 are condensed bosons in 
ℒ_e, they are identified as the vacuum sector in the condensed phase. Based on the interpretation in Ref. <cit.>, (e/3, ↑) and (e/3, ↓) will transmute into ρ_1 exp(iϕ_l/2) and ρ_2 exp(iϕ_l/2) in the RR_4 FQH state, respectively. It is important to clarify that these transmuted anyons are now in ℬ, but not ℬ̅. Since ρ_1≠ρ_2, the resulting Abelian anyons are different.



Consider the case when the fully gapped interface is described by the m-boundary that has the Lagrangian subset ℒ_m in Eq. (<ref>). Now, we have the following pair of deconfined anyons:

    m× b_1
       = [ρ_2 exp(iϕ_l/2)] [𝒱_1exp(iϕ_r/2)],
    
    
    m^2× b_1
       = [ρ_1 exp(iϕ_l/2)] [𝒱_5exp(iϕ_r/2)].

Thus, (e/3, ↑) and (e/3, ↓) will transmute respectively into 
ρ_2 exp(iϕ_l/2) and ρ_1 exp(iϕ_l/2). Notice that the result is different from the one in the e-boundary. We summarize the above results in Fig. <ref>.





  §.§.§ Transmutation of Abelian charge 2e/3 anyon


Similarly, there will be a transmutation of the pseudospin information carried by a charge 2e/3 anyon when it crosses the interface and enters the RR_4 FQH liquid. There are three different pseudospin states for the charge 2e/3 anyon in the Halperin-330 FQH liquid. They are spin up, spin zero, and spin down. Physically, they can be viewed as the combination of two spin-up, one spin-up and one spin-down, and two spin-down e/3 anyons. The corresponding CFT operators are

    (2e/3, ↑)
       ≡exp(2iϕ_↑)
    =𝒱_2exp(iϕ_r),
    
    
    (2e/3, 0)
       ≡exp[i(ϕ_↑+ϕ_↓)]
    =𝒱_0exp(iϕ_r).
    
    
    (2e/3, ↓)
       ≡exp(2iϕ_↓)
    =𝒱_4exp(iϕ_r).


Following the previous discussion, we first discuss their transmutation when the fully gapped interface is described by the e-boundary. In this case, one has the following three deconfined anyons,

    e× b_0
       = [ϵ_1 exp(iϕ_l)] [𝒱_2exp(iϕ_r)],
    
    𝕀× b_0
       = [I exp(iϕ_l)] [𝒱_0exp(iϕ_r)],
    
    
    e^2× b_0
       = [ϵ_2 exp(iϕ_l)] [𝒱_4exp(iϕ_r)],

Thus, (2e/3, ↑) transmutes into ϵ_1 exp(iϕ_l); (2e/3, 0) becomes I exp(iϕ_l); (2e/3, ↓) becomes ϵ_2 exp(iϕ_l). For the case of having the m-boundary, one has

    m^2× b_0
       = [ϵ_2 exp(iϕ_l)] [𝒱_2exp(iϕ_r)],
    
    
    m× b_0
       = [ϵ_1 exp(iϕ_l)] [𝒱_4exp(iϕ_r)].

In this scenario, (2e/3, ↑) transmutes into ϵ_2 exp(iϕ_l), whereas (2e/3, ↓) becomes ϵ_1 exp(iϕ_l). The results are illustrated in Fig. <ref>.



The original pseudospin information of a charge e/3 or a charge 2e/3 anyon from the Halperin-330 state is entirely encoded in the neutral spin mode ϕ_n. Clearly, this is a local information which can be accessed via local measurement. More specifically, we know in which layer the Abelian anyon was created. Depending on both the original pseudospin state and the boundary type of the fully gapped interface, the pseudospin information will be transmuted into ρ_1 or ρ_2 (for the charge e/3 anyon), or ϵ_1, ϵ_2, or I (for the charge 2e/3 anyon). Although one knows that the split anyons, say ρ_1 and ρ_2, are different, it is impossible to distinguish between them via any local measurement. In fact, their inequivalence is only manifested topologically in their braiding and fusion rules with other anyons (including themselves) as summarized in Eqs. (<ref>) – (<ref>). In this sense, we claim that the original local pseudospin information is completely scrambled into a form of nonlocal information being stored and protected topologically by the anyons in the RR_4 FQH liquid. No information is stored on the interface. It is reasonable as the interface is fully gapped, and does not support any low-energy gapless excitations there. This feature is different from the situation of having a gapless QH interface. As a concrete example, our recent work <cit.> showed that the pseudospin information for an Abelian charge e/4 anyon from the Halperin-331 state will be scrambled and stored nonlocally by a pair of vortices (one on the interface, and another one in the Pfaffian FQH liquid) when the Abelian anyon crosses the gapless 331-Pfaffian interface. The comparison here demonstrates the dependence of the scrambling mechanism on the gapness of the QH interface.



 §.§ Some remarks


One may realize that the transmuted particle from the charge 2e/3 anyon can be obtained directly by fusing a pair of transmuted particles from the charge e/3 anyons. This is not a coincidence. It is guaranteed from the commutativity between restriction and fusion in anyon condensation <cit.>. This important property also ensures that the total quantum dimension of the split anyons is identical to the quantum dimension of the original non-Abelian anyon. Moreover, the total conformal spin for the transmuted anyon(s) matches the one for the original anyon. In the present case, we have a fully gapepd interface, and the vertex operators describing the charge sector of the deconfined anyons always have zero conformal spins. Thus, the conformal spin for the parafermionic field η from the ℤ_4 CFT must match with the conformal spin for the vertex operator describing the neutral spin mode ϕ_n. This holds in all our results, and further highlights the advantage of separating the charge and neutral sectors as we did in our analysis.

In principle, one can drag more quasiparticles from the Halperin-330 liquid into the RR_4 liquid. Suppose the resulting anyons are well separated from each other (i.e., with a separation much larger than the magnetic length). Then, the braiding between the transmuted anyons will lead to a further scrambling of the original pseudospin information <cit.>. In order to recover the original pseudospin information, one basically needs to “pull" all the anyons back to the Halperin-330 liquid. However, the recovered information will be in a highly entangled form, which again resembles the definition of quantum information scrambling.



§ SIMULATION OF BLACK HOLE EVAPORATION
 

In order to resolve the information paradox in our model, it is necessary to mimic a black hole evaporation in the quantum Hall interface. To simplify our discussion, we only consider the 331-Pfaffian interface here. As we will show, the process recovers the original information carried by the pseudospin degree of freedom naturally. To simplify the discussion, we assume only charge-e/4 Abelian quasiparticles were dragged across the interface before the “evaporation". In general, one can also drag quasiholes and quasiparticles with other charges. Under the above assumption, we argued in Sec. <ref> that neutral anyons σ̃ and non-Abelian charge-e/4 quasiparticles carrying σ are created. This is illustrated in Figs. <ref>(a) and <ref> (b).



The “black hole evaporation" is simulated by shrinking the Pfaffian region. Experimentally, it may be achieved by reducing the interlayer tunneling in the bilayer system <cit.>. When a non-Abelian quasiparticle reaches the shrinking interface, it is released back to the Halperin-331 liquid. This process plays the role of Hawking radiation in the present setup. There are two different mechanisms for the conversion from a non-Abelian quasiparticle into an Abelian quasiparticle. First, the former may encounter an existing σ̃ at the interface. In this case, they recombine and transmute back to an Abelian quasiparticle [see Eqs. (<ref>) and (<ref>)]. This special scenario is shown in Fig. <ref>(c). In order for this recombination to occur, it requires a highly delicate control of the shrinking process. Thus, it is unlikely to recombine all the existing σ̃ and σ (with charge sector skipped) in this way. On the other hand, it is likely that a non-Abelian anyon reaches the interface at a position with no neutral anyon σ̃. Being already outside the Pfaffian liquid, the non-Abelian anyon still needs to transmute into an Abelian quasiparticle. In this case, a pair of σ̃ needs to be created at the interface. One of them combines with the non-Abelian anyon to covert into an Abelian quasiparticle in the Halperin-331 liquid. The remaining one is left at the interface. This mechanism is shown in Fig. <ref>(d). Since the additional pair of σ̃ are created from the vacuum, they must have their fusion channel in the trivial topological sector. Hence, they do not carry additional information. All above processes are unitary, so information should be preserved.



 §.§ Recovery of Page curve


Now, we show that the above “black hole evaporation" recovers all original information and satisfies the Page curve. Here, the first subsystem consists of anyons remaining in the Pfaffian liquid and the interface. Another subsystem consists of Abelian quasiparticles in the Halperin-331 liquid. For simplicity, we call these two subsystems as (I) and (II), respectively. Since we treat the Pfaffian liquid as the black hole and the interface as an event horizon, reduced density matrices at different stages are obtained by partial tracing out (I). At the beginning of the evaporation, (II) is in a vacuum state with no quasiparticles, so its entropy is zero. When the Pfaffian liquid starts to shrink, the entropy of Abelian quasiparticles originating from their entanglement with (I) increases. However, the increase in entanglement entropy will not continue forever. By keep shrinking the Pfaffian region, the number of non-Abelian anyons and the dimension of the corresponding Hilbert space decrease. Hence, the dimensions of Hilbert spaces of (I) and (II) will become comparable and eventually equal to each other. The entanglement entropy reaches its maximum at this moment <cit.>, which is known as the Page time. The Page time depends on the actual shrinking process. After passing the Page time, the entanglement entropy starts to decrease. 

In the most idealistic (yet most unlikely) situation which one can recombine all N non-Abelian anyons with the originally existing σ̃ (exist before the evaporation) at the interface, the Page time occurs when N/2 quasiparticles are released to the 331 region. This feature does not hold in a generic situation. One can actually deduce the average entanglement entropy of (II) in the most idealistic case. We assume the initial state of the total system (before shrinking the Pfaffian region) is a random pure state |Ψ⟩ in the 2^N-dimensional Hilbert space. The entanglement entropy is averaged with respect to the unitary invariant Haar measure on the space of unitary vectors |Ψ⟩ in the 2^N-dimensional Hilbert space <cit.>. Suppose j non-Abelian anyons have been dragged out from the Pfaffian region and transmuted back to Abelian quasiparticles in the Halperin-331 liquid. The corresponding Hilbert space dimensions of (I) and (II) are given by n=2^N-j and m=2^j, respectively. When m≤ n, the conjecture by Page (later proved by Sen <cit.>) suggests that the average entanglement entropy of (II) takes the form <cit.>,

    ⟨ S_ (II)⟩≡ S_m,n
    =(∑_k=n+1^mn1/k)-m-1/2n.

For m>n, one obtains ⟨ S_ (II)⟩ by interchanging m and n in Eq. (<ref>). By plotting ⟨ S_ (II)⟩ versus lnm, one concludes that it is identical to the one in Fig. 1 of Ref. <cit.>.

From our previous discussion, it is very likely that the number of σ̃ anyons increases during the “black hole evaporation". This leads to two consequences. First, it is possible that all non-Abelian anyons in the Pfaffian liquid have been released to the 331 liquid, but some σ̃ anyons still remain at the interface. These anyons are entangled with the Abelian anyons in the Halperin-331 region, so the entanglement entropy is still nonzero. Second, the bound N_L∼ L/ℓ_B can be satisfied easily during the shrinking process. The discussion in Sec. <ref> showed that pairs of Abelian quasiparticles and quasiholes with opposite charges will be released to the 331 region. After eliminating the Pfaffian region completely, the total number of anyons in the 331 region needs not be equal to the number in the original configuration (before sending the anyons in the Pfaffian liquid). Only in the most idealistic situation that we mentioned previously, these two numbers are equal as shown in Fig. <ref>(a) and (e). In general, the final state of the system will have a superposition of different total numbers of anyons in the 331 liquid. It is reasonable since the total charge in the system is still conserved. This idea is illustrated in Fig. <ref>(f). In particular, Fig. <ref>(f) denotes a superposition state of 6+M charge-e/4 quasiparticles and M charge--e/4 quasiholes, where M is a non-negative integer. This kind of superposition state is actually a closer analogue of the actual Hawking radiation emitted from a black hole, which consists of different types of particles or excitations.

Independent of the actual shrinking process, the system must return to a pure state when the Pfaffian region is eliminated completely. Then, the entanglement entropy goes back to zero and resembles the Page curve. The original pseudospin information is recovered but in a highly entangled form. Thus, the paradox in our model is resolved. Our above discussion suggests that the Page curve in the present system should be more complicated than the one in Fig. 1 of Ref. <cit.>.



§ SUMMARY AND DISCUSSION


To conclude, we have identified and resolved an “information paradox" in the 331-Pfaffian and the 330-RR_4 quantum Hall interface respectively <cit.>. In both cases, the paradox originates from an apparent inability to recover the original pseudospin information carried by Abelian quasiparticles after they cross the interface and enter the non-Abelian single-component QH liquid. 

Employing the technique of anyon condensation, we found that each incoming Abelian anyon needs to be transmuted into new types of anyons when the former crosses the interface. Moreover, the mechanism of transmutation and its corresponding scrambling of pseudospin information depend on the nature of quantum Hall interfaces. For the 331-Pfaffian interface, an incoming Abelian anyon is transmuted into a pair of non-Abelian anyons. One of them is created in the Pfaffian liquid, whereas the other is created on the interface. This is expected because the interface remains gapless and being described by a conformal field theory with central charge c=1/2. On the other hand, the 330-RR_4 interface can be fully gapped. In this case, the incoming Abelian anyon is completely transmuted into a non-Abelian anyon in the Read-Rezayi quantum Hall liquid. Abelian anyons with opposite pseudospins would transmute into non-Abelian anyons satisfying different fusion channels.

Regardless of the transmutation mechanism, the original pseudospin information is stored nonlocally in the system. As a result it cannot be recovered by any local measurement. We believe this is a fair analogy to an object falling into a real black hole, in the sense that while the information it carries is not lost, it does become inaccessible to an (outside) observer. This resembles the idea of quantum information scrambling, which is consistent with the modern viewpoint that black holes are fast (perhaps the fastest) information scramblers <cit.>. The matching between the dimensions of Hilbert spaces for the Abelian quasiparticle and non-Abelian anyons further verifies the preservation of information.

In addition, we considered the case when more quasiparticles are dragged across the interface. Focusing on the 331-Pfaffian interface, we argued that the maximum amount of information the system can store in a topologically protected way is bounded by the length of the interface. This feature is reminiscent of a similar bound in black hole set by its area due to the holographic principle and the Bekenstein entropy. Finally, we discussed the simulation of black hole evaporation by shrinking the Pfaffian region which releases quasiparticles back to the 331 liquid. We demonstrate explicitly that the corresponding entanglement entropy would follow the Page curve. Hence, the original pseudospin information is recovered and the “information paradox" in our model is resolved. Note that remnants may be left at the end of evaporation in actual astrophysical black holes <cit.>. This may provide an alternative resolution of the information paradox, which is not addressed in the present work.

It is quite surprising that the seemingly simple quantum Hall interface has a rich analogy with black hole physics. Nevertheless, we need to point out some potential differences between our model and real astrophysical black holes. For a (semi)classical black hole, the horizon is not expected to have an effect on an infalling object (the so-called “no drama scenario”), including the information carried by it. Whether this remains to be the case or not for a fully quantum-mechanical black hole is unclear. A firewall at the horizon is a possible scenario that is currently under investigation and debate <cit.>. In our model, the Abelian quasiparticles must be transmuted when they cross the interface. This is inevitable as the Abelian and non-Abelian quantum Hall liquids support different degrees of freedom. Thus, the interface in our model behaves like a firewall. In our opinion, this interface may be a very simple and accessible “black hole firewall", which deserves more attention. In future work, it will be tempting to examine possible analogy of black hole thermodynamics in quantum Hall interfaces. It is also interesting to examine whether quantum Hall interfaces can provide an easy simulation of (a topological version of) the Hayden-Preskill protocol. 

The black-hole information paradox is arguably one of the most fundamental problems in physics, which involves gravitation, quantum field theory, and in particular, quantum information science. This long-standing problem is currently being actively studied by physicists in many different areas, and from very different perspectives (but so far only theoretically). Its resolution may well pave the way for the quantum theory of gravity, the holy grail of theoretical physics. While there is a lack of complete parallel between our model and certain “believed” processes in actual black holes (especially in the description of black hole evaporation which should be spontaneous), the analogy presented here provides a simple and accessible platform to simulate (i) apparent information loss, (ii) information scrambling, and (iii) information recovery. We believe these are arguably the most important and central concepts in understanding and resolving the original information paradox. Furthermore, our work may open a new research direction of studying how local information can be transmuted and stored nonlocally in an actual black hole. Since the concept of firewall and many other aspects in the paradox are still under intense debate, it is worthwhile to have simple analogies that capture some of the relevant concepts (but not necessarily all details precisely) in the original problem. In addition, our results have established a connection between quantum information, black hole physics and quantum Hall physics, and may bring experimentalists into this exciting research area.  

Last but not least, it is worthwhile to mention that a deep connection between quantum Hall effect and gravitational physics has been revealed in previous work <cit.>, leading to the recent experimental discovery of chiral graviton<cit.>. In particular, Refs. <cit.> have suggested a possible simulation of Hawking-Unruh effect by scattering quasiparticles in quantum Hall systems. We are optimistic that more connections between black hole physics and quantum Hall physics may be discovered in the future.



§ ACKNOWLEDGMENT


We thank N. E. Bonesteel for a useful discussion in our original work. This work was supported by the National Science Foundation Grants No. DMR-1932796 and DMR-2315954, and performed at the National High Magnetic Field Laboratory which is supported by National Science Foundation Cooperative Agreement No. DMR-2128556, and the State of Florida.

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The gapping of modes is realized when the tunneling amplitude in H_T is a constant instead of a random variable in Eq. (<ref>). Strictly speaking, this kind of nonrandom tunneling is unphysical as there is a momentum mismatch between different modes. However, gapped modes and localized modes discussed in Sec. <ref> have the same physical consequences in the present discussion.

footnote0
In our opinion, this is a big advantage over the Pfaffian-NASS interface <cit.>. In that case, the NASS quantum Hall liquid at ν=4/7 has quasiparticle with the smallest charge of e/7. Hence, a net charge of e/4-e/7=3e/28 is absorbed, which leads to an accumulation of charges at the interface. Furthermore, the difference between filling factors of the two quantum Hall liquids may complicate the gapping of counterpropagating charge modes at the interface. This difference also introduces additional challenges in realizing the interface experimentally.

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Although Fig. <ref> may imply the interface is a one-dimensional object, it is not. The interface actually penetrates in both quantum Hall liquids and has a finite width (but much shorter than its length), similar to the interface between Pfaffian and anti-Pfaffian liquids <cit.>. Therefore, braiding of anyons can occur at the interface.

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footnote2
In the Pfaffian-NASS interface, one may consider a similar “information paradox" by dragging a charge-e/7 non-Abelian quasiparticle from the NASS liquid to the Pfaffian liquid. Depending on the species, the neutral sector is given by σ_↑ or σ_↓ in the c=6/5 CFT. Both transmute into (σ̅, σ), where σ̅ is a primary field in the ℳ(5,4) minimal model describing the interface <cit.>. Since σ̅ has a quantum dimension d=(1+√(5))/√(2)>√(2), the Hilbert space of anyons at the interface has a larger dimension.

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In the general case, the maximal set of condensable bosons may not be closed under fusion due to the splitting of non-Abelian anyons. In this case, a more appropriate terminology should be Lagrangian subset <cit.>. Meanwhile, ℒ_e and ℒ_m are indeed closed under fusion in our present case.

footnote-random
Since the tunneling process is random, the renormalization group (RG) equation reads 
dW/d(ln s)=(3-2Δ)W. Here, s is the scaling factor of time and space, and 
ξ(x)ξ(x')=Wδ(x-x'). For the single electron tunneling process (i.e., a=1), the scaling dimension for the operator involving the neutral modes ψ_1exp(iϕ_n) is already h_n=3/4+3/4=3/2. The scaling dimension of the operator involving charge modes exp[i(ϕ_l+ϕ_r)/2] is non-universal but must be positive. Therefore, the process is irrelevant in the RG sense. On the other hand, the corresponding RG equation for nonrandom tunneling is dW'/d(ln s)=(2-Δ)W', where W' stands for the tunneling strength. Depending on the interaction between 
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 Paper Title: Embedding Warm Natural Inflation in $f(φ)T$ gravity.



 


Embedding Warm Natural Inflation in f(ϕ)T gravity
     Sabina Yeasmin [Email:sabyes22@gmail.com] , Atri Deshamukhya  [Email: atri.deshamukhya@aus.ac.in]

Department of Physics, Assam University, Silchar, Assam 788011, India
    
=============================================================================================================================================================================



  

  0.85
  
 We study warm inflation in the framework of f(ϕ)T gravity, where ϕ is the inflaton and T is the trace of the energy-momentum tensor. The inflaton field is assumed to roll on the natural potential and the result is analyzed in light of Planck 2018 and BICEP/Keck 2021 data. We start our work by obtaining the field equations under slow-roll approximations. We then evaluate the scalar and tensor power spectra and their corresponding spectral index and tensor-to-scalar ratio with a temperature-dependent form of the dissipation coefficient during the inflationary era. We find that the warm inflation model in f(ϕ)T gravity is compatible with observational bands.  
    
            
    
 
    
  

 

§ INTRODUCTION

According to the Standard Model of Cosmology and observational data, the universe has undergone two phases of accelerated expansion throughout its evolution, the inflationary phase at its early stage and a second phase of accelerated expansion in recent times. This second phase of accelerated cosmic expansion is believed to be a consequence of the presence of dark energy exerting negative pressure. General Relativity is not ideal for explaining the existence of the dark sector i.e. dark matter and dark energy of the universe <cit.>. To overcome such shortcomings researchers started studying the different higher-order extensions to General Relativity and exploring alternative theories of gravity <cit.>. Modified theories of gravity not only address the late-time acceleration but these theories are also applied to explain the early Universe, e.g., the inflationary era. 

In modified gravity theories, the Einstein-Hilbert action is modified either by extending the geometric or the matter part or both. Unlike Einstein's theory, whose field equations contain only up to second-order derivatives, the modified theories with higher derivative Ricci/Riemann tensor gravity models include higher derivatives <cit.>. A significant difference is therefore expected between general relativity and modified gravity predictions.

The f(R) gravity model is the first proposed modification of the Einstein gravity where the curvature scalar in the Einstein-Hilbert action has been replaced by a general function of Ricci scalar R. This theory describes both inflation and late-time expansion of the universe. The first viable f(R) model to study inflation was proposed by Nojiri and his collaborators <cit.>. T. Harko et al. <cit.> introduced another modified gravity theory called f(R, T) theory where the gravitational field is assumed to be coupled to the trace T of the energy-momentum tensor of the matter in the action. Later several inflationary models have been studied within the context of the  f(R, T) gravity theory <cit.>. There are many other technically allowed modified gravity theories that have been investigated in literature in the context of inflationary cosmology like f(𝒯) <cit.>, f(R, G) <cit.>, Gauss-Bonnet (GB) theory (f(G) gravity) <cit.>, Brans-Dicke theory <cit.>, etc. where 𝒯 denotes the Torsion scalar and G is Gauss-Bonnet scalar.

Another recent modification is the f(ϕ)T theory of gravity, proposed by Zhang et al. <cit.>, which is an extension of the simplest f(R,T) gravity theory. This modified theory of gravity has been investigated by adding a coupling term between the scalar field ϕ and the trace of the energy-momentum tensor into the action in general relativity. They studied cold inflationary dynamics in this framework by considering three well-motivated potentials: chaotic, natural, and Starobinsky, and found that chaotic and natural inflationary models are in better agreement with the observational constraints. However, a larger value of the tensor-to-scalar ratio was obtained for the Starobinsky model. Although natural inflation in f(ϕ)T gravity is consistent with the observational data, a major problem with that is a larger axion decay constant is required to agree with the observational constraints. So, it is difficult to embed this model in fundamental theories like string theory <cit.>. In this study, we show that it is possible to lower the value of the axion decay constant below the Planck scale if the natural inflationary model in f(ϕ)T theory of gravity is studied in the context of the warm inflationary scenario.  






In 1995, Berera <cit.> introduced the idea of warm inflation as an alternative approach to resolving the graceful exit problem of the standard inflation dynamics.
 The basic concept of warm inflation is that during inflationary period, the inflaton field does interact with other existing fields <cit.> which leads to vacuum energy decay to the other lighter fields, thereby creating particles simultaneously with the process of inflation. Thus in warm inflation, radiation is produced simultaneously with the expansion of the universe and there can be a smooth transition from the inflationary phase to the radiation-dominated phase without the requirement of invoking any separate reheating phase. In this scenario, dissipative effects play an important role during inflation.

 


  In warm inflation scenario, energy dissipates from the inflaton field into radiation during the inflationary phase and this mechanism is parameterized by a term Γ called dissipation coefficient <cit.>. Because of this dissipative coupling between inflaton and radiation field inflation can last long even if the potential is not very flat. In this work, we use cubic temperature-dependent dissipation coefficient which affects the prediction of cosmological observables like it suppresses the value of the tensor-to-scalar ratio. The latest Cosmic Microwave Background (CMB) observations of BICEP/Keck <cit.> combined with the Planck 2018 data <cit.> sets an  bound on the tensor-to-scalar ratio of r<0.036 at the 95% confidence level. We show that the warm natural inflation in f(ϕ)T gravity model is consistent with this observational bound on r.
  
 

The paper is organized as follows: In section 2, we present a brief overview of f(ϕ)T gravity in the FLRW background. In section 3, we investigate warm inflation in f(ϕ)T gravity and study the cosmological perturbations originating from f(ϕ)T gravity during warm inflation and also formulate the corresponding power spectrum, the tilt of spectral index, and tensor-to-scalar ratio. In section 4, we study warm inflation with  Natural potential in weak and strong dissipative regimes for variable dissipation coefficient. In section 5, we present our conclusions. 




§ OVERVIEW OF F(Φ)T GRAVITY
    
 
 The action in f(ϕ)T gravity theory as proposed in <cit.> is given by,


    S = ∫[R/2κ + λ f(ϕ) T + L_m ] √(-g) d^4x

 where a coupling term (coupling between inflaton (ϕ) and trace of the energy-momentum tensor (T) is added to the Hilbert-Einstein action. Here, κ=8π G=1/ M_P^2, M_P is the reduced Planck mass, R stands for Ricci scalar, L_m is the matter Lagrangian density, g is the determinant of the metric tensor, λ is the model parameter, and f(ϕ) is a dimensionless function of the inflaton ϕ which satisfies the condition f(0)=0. This indicates that when inflaton field decays and its energy no longer dominates the universe, Einstein gravity is recovered. Also in the limit λ→ 0, the action in equation (<ref>) returns to the action in the Einstein theory.  
 
 In this work, we have considered the form of f(ϕ) as f(ϕ)=√(κ)ϕ. Also, we have used the natural units such that c=ℏ=1 and used the (-,+,+,+) sign convention for the metric tensor throughout this manuscript. With the form of f(ϕ)=√(κ)ϕ, the action reads

    S = ∫[R/2κ +  λ√(κ)ϕ T + L_m ] √(-g) d^4x

 By varying the action (<ref>) with respect to the metric, we get the modified Einstein field equation in the following form,

    R_αβ - 1/2g_αβR = κ T_αβ^(eff)

where T_αβ^(eff) is the effective stress-energy tensor given by,

    T_αβ^(eff)= T_αβ - 2 λ√(κ)ϕ( T_αβ - 1/2T g _αβ + Θ_αβ)

where T is the trace of the energy-momentum tensor T_αβ in Einstein gravity. In flat FLRW background T_αβ is given by


    T_αβ = ∂_αϕ∂_βϕ + g_αβ(1/2ϕ̇^2 - V(ϕ))

where V(ϕ) is the potential of the inflaton. For inflaton field, Θ_αβ reads as

    Θ_αβ=-∂_αϕ∂_βϕ-T_αβ

From equations (<ref>),(<ref>) and (<ref>), the effective energy density and pressure for inflaton field are obtained as

    T_00^(eff)= ρ_ϕ^(eff)=(1+2λ√(κ)ϕ)ϕ̇^2/2+(1+4λ√(κ)ϕ)V


    T_ij^(eff)=  P_ϕ^(eff) g_ij=[(1+2λ√(κ)ϕ)ϕ̇^2/2-(1+4λ√(κ)ϕ)V]g_ij

 
Substituting equation (<ref>) into the 00 component of the modified Einstein field equation (<ref>), we get

    H^2 = κ/3[ϕ̇^2/2 (1+ 2λ√(κ)ϕ) + V (1+ 4λ√(κ)ϕ)]

which is known as the modified Friedmann equation. Here we have defined the Hubble parameter as H=ȧ/a. Similarly, substituting equation (<ref>) into the ij component of the modified Einstein field equation (<ref>), we get

    ä/a = - κ/3[ϕ̇^2 (1+ 2λ√(κ)ϕ) - V (1+ 4λ√(κ)ϕ)]

This is the modified acceleration equation. We can also obtain the expression for Ḣ as,

    Ḣ = ä/a - H^2 = - κ/2 (p^eff + ρ^eff)= -κ/2[ϕ̇^2(1+ 2λ√(κ)ϕ)]

 Equations (<ref>) and (<ref>) are known as the modified Friedmann second equation. In the limit λ=0, these equations reduce to those in Einstein gravity.
 
Also the continuity equation or the modified Klein-Gordon equation can be written as,

    ϕ̈(1+ 2λ√(κ)ϕ) + 3Hϕ̇(1+ 2λ√(κ)ϕ) +λ√(κ)ϕ̇^2 +   (1+ 4λ√(κ)ϕ)  V^'+4λ√(κ) V = 0

 where the prime denotes derivative with respect to the inflaton field.
  



 





























































 


























  

















 




  




 












  




 




  





















 

§ WARM INFLATION IN  F(Φ)T GRAVITY

  In the warm inflationary paradigm, the inflaton field interacts with other fields during the inflationary period, leading to the decay of the inflaton field into radiation. To capture this feature, a dissipation co-efficient Γ is added to the equation of motion describing the warm inflationary scenario. The equations that completely describe the dynamics of the warm inflation scenario in f(ϕ)T gravity can be described as  

    H^2=κ/3(ρ_ϕ^(eff)+ρ_γ)


    ρ̇_ϕ^(eff)+3H(ρ_ϕ^(eff)+P_ϕ^(eff))=-Γϕ̇^2


    ρ̇_γ+4Hρ_γ=Γϕ̇^2

Here ρ_ϕ^(eff), ρ_γ and P_ϕ^(eff) are the energy density of the scalar field, energy density of the radiation field and pressure of the scalar field respectively. Γ is the dissipation coefficient which describes the decay of inflaton into radiation during inflationary phase.

Modified equation of motion of the inflaton field in warm inflation can be derived by substituting equations (<ref>) and (<ref>) into equation (<ref>), 
 
    ϕ̈(1+ 2λ√(κ)ϕ) + 3Hϕ̇(1+ 2λ√(κ)ϕ+Q) +λ√(κ)ϕ̇^2 + (1+ 4λ√(κ)ϕ) V^'+4λ√(κ) V = 0

 where Q=Γ/3H is the dissipation rate which describes the effectiveness at which the inflaton energy converts into radiation. The Q≫1 corresponds to strong dissipation regime, and the Q≪1 corresponds to weak dissipation regime <cit.>. The presence of extra friction term in the equation of motion of the inflaton suggests that radiation will not be redshifted during inflation, because, inflaton field continuously converts into radiation through dissipation

 For inflation to occur and last long enough, the potential energy of the inflaton field must be dominated over both the kinetic energy of the inflaton field and the energy density of the radiation field, however, the radiation energy density still satisfies the condition ρ_γ^1/4>H. That means T>H during warm inflation and this condition is considered as the condition for warm inflation. It is also assumed that the production of radiation is quasi-stable. These approximations are called slow-roll approximations which can be quantified as 

     
 
    (1+2λ√(κ)ϕ)/2ϕ̇^2+ρ_γ<<(1+4λ√(κ)ϕ)V

 
    (1+2λ√(κ)ϕ)ϕ̈<< 3Hϕ̇(1+ 2λ√(κ)ϕ+Q)

 
    ρ̇_γ<< 4Hρ_γ


    √(κ)ϕ̇^2<<Hϕ̇

 Under these considerations, ρ_ϕ≃ V(ϕ), and the dynamical equations (<ref>), (<ref>) and (<ref>) read as
 
    H^2=κ/3(1+4λ√(κ)ϕ)V

 
    3Hϕ̇(1+ 2λ√(κ)ϕ+Q) + (1+ 4λ√(κ)ϕ)V^'+4λ√(κ) V = 0

 
    ρ_γ=Γϕ̇^2/4H =Cτ^4

 where C=g_*π^2/30 is the Stefan-Boltzmann constant and g_* is the number of degrees of freedom for the radiation at temperature τ. (For calculation we will take C = 70 for g_*= 200)

The slow-roll approximation can be parameterized by several slow-roll parameters namely ϵ, η, and β. Using equations (<ref>), (<ref>) and (<ref>), we can derive the modified slow-roll parameters in the framework of f(ϕ)T gravity in terms of the inflaton potential V(ϕ) as

 
    ϵ   =   -Ḣ/H^2
       =   1/2κ(1+2λ√(κ)ϕ+Q)(V^'/V+4λ√(κ)/1+4λ√(κ)ϕ)^2

  
  
    η   =   -ϕ̈/Hϕ̇
       =   1/κ(1+2λ√(κ)ϕ+Q)(V^''/V+λ√(κ)(6+8λ√(κ)ϕ+8Q)/(1+4λ√(κ)ϕ)(1+2λ√(κ)ϕ+Q)V^'/V.
       -   .8λ^2κ/(1+2λ√(κ)ϕ+Q)(1+4λ√(κ)ϕ))

 and
  
    β   =   -ρ̇_γ/Hρ_γ
       =   1/κ(1+2λ√(κ)ϕ+Q)Γ^'/Γ(V^'/V+4λ√(κ)/1+4λ√(κ)ϕ)

  The conditions satisfied by the slow-roll parameters for warm inflation to occur are ϵ≪ 1, |η| ≪ 1 and |β| ≪ 1 <cit.>. Violation of the slow-roll conditions marks the end of inflation.

 Another significant parameter during inflation is the number of e-folding. This parameter quantifies the extent of expansion of the universe throughout the inflationary epoch. The number of e-foldings when the inflation field ϕ rolls from its value ϕ_i to ϕ_f is estimated as:

 
    N   =   ∫^t_2_t_1 H dt=∫^ϕ_f_ϕ_iH/ϕ̇dϕ
       =    -κ∫^ϕ_f_ϕ_i(1+2λ√(κ)ϕ+Q)/V^'/V+4λ√(κ)/1+4λ√(κ)ϕdϕ



 
 


 




 §.§ Perturbation Spectra

In this section, we develop the theory of cosmological perturbations for warm inflation in the f(ϕ)T theory of gravity. In warm inflationary models T>H, the fluctuations of the inflaton field will be produced by thermal effects due to radiation. Consequently, the source of density fluctuations in warm inflationary models is the thermal fluctuations in the radiation field rather than quantum fluctuations. These thermal fluctuations in the radiation field produce fluctuations in the inflaton field. These fluctuations lead to perturbations in the metric which in turn perturb the equation of motion of inflaton field. In the following, we will calculate the fluctuations of the inflaton field. To compute the inflaton fluctuation we start with the perturbations of the FRW metric in the spatially flat gauge
which is given by

    ds^2=-(1+2A)dt^2+2a(t)∂_iBdtdx^i+a^2(t)δ_ijdx^idx^j

 Where A(x,t) and B(x,t) are spece-time dependent scalar perturbations of the metric. Under this perturbation, we expand the inflaton field ϕ(x,t)= ϕ(t)+δϕ(x,t), where δϕ(x,t) is the linear response due to the thermal stochastic noise. Using the slow-roll conditions, we find the perturbed equation of the inflaton field in the momentum space given by,
 
    (1+2 λ√(κ)ϕ)δ̈ϕ̈_k+3H(1+2λ√(κ)ϕ+Q)δ̇ϕ̇_k+k^2/a^2(1+2λ√(κ)ϕ)δϕ_k=(1+2λ√(κ)ϕ)ϕ̇Ȧ
       +   k^2/a(1+2λ√(κ)ϕ)ϕ̇B-(2(1+4λ√(κ)ϕ)V^'+8λ√(κ) V+Γϕ̇)A

 And the perturbed Einstein equation becomes
 
    3H^2A+k^2/aHB=-4π Gδρ

  
    HA=4π G (ρ+p)δ u

  In the above equations, ρ and p denote the total energy density and pressure respectively, δρ denotes the perturbations at the linear order of the total energy density, and δ u is the scalar part of the linear perturbation of the 4- velocity. Substituting the expressions of A and B from equations (<ref>) and (<ref>) into equation (<ref>), we get the perturbed equation of motion of inflation field as

    (1+2λ√(κ)ϕ)δ̈ϕ̈_k(t)+3H(1+2λ√(κ)ϕ+Q)δ̇ϕ̇_k(t)+k^2/a^2(1+2λ√(κ)ϕ)δϕ_k(t)=ξ_k(t)

where the thermal stochastic noise source ξ_k(t) is introduced to describe thermal fluctuations. In the slow-roll regime, the term δ̈ϕ̈_k in equation (<ref>) can be neglected. Thus equation (<ref>) becomes

    3H(1+2λ√(κ)ϕ+Q)δ̇ϕ̇_k(t)+k^2/a^2(1+2λ√(κ)ϕ)δϕ_k(t)=ξ_k(t)

The solution of equation (<ref>) is
  
 
    δϕ_k(t)=1/3H(1+2λ√(κ)ϕ+Q)exp[-t/𝒯]∫_t_0^texp[t^'/𝒯]ξ_k(t^')dt^'+ δϕ_k(t_0)exp[-t-t_0/𝒯]

 where,
  𝒯(ϕ)=3H (1+2λ√(κ)ϕ+Q)/(1+2λ√(κ)ϕ) k^2/a^2= 3H (1+2λ√(κ)ϕ+Q)/(1+2λ√(κ)ϕ)k_p^2 and k_p is the physical wave number. The first term in the right-hand side of the solution (<ref>) is the noise contribution which acts to thermalize δϕ. whereas the second term is the memory term for the initial value of δϕ which is exponentially damping i.e. it becomes negligible with time.  If k_p of one mode of δϕ_k  is smaller than the freeze-out physical wave number k_F, then the mode will not thermalize during the Hubble time, and for k_p≳ k_F, the mode will be thermalized and the memory term will vanish within Hubble time. So, the freeze-out wave number k_F is   
 
    k_F=√(3H^2(1+2λ√(κ)ϕ+Q)/1+2λ√(κ)ϕ)

The power spectrum for the scalar fluctuations in warm inflation has the form 
 
    P_R=(H/ϕ̇)^2δϕ^2

where the fluctuations of scalar field can be obtained through the relation


    δϕ^2   =   k_F τ/2π^2

Combining equations (<ref>),(<ref>) and (<ref>), the expression for scalar power spectrum P_R for warm inflation in f(ϕ)T gravity leads to

    P_R    =   H^3 τ/2π^2 ϕ̇^2√(3(1+2λ√(κ)ϕ+Q)/1+2λ√(κ)ϕ)

In the limit λ→ 0, the above expression goes back to the form of P_R for warm inflation in Einstein gravity.


The power spectrum for the tensor perturbation is

    P_T   =   16 H^2/π M_P^2
       =   16 (1+4λ√(κ)ϕ)V/3π M_P^4

The spectral index n_s and tensor to scalar ratio r are defined as <cit.>

    n_s-1=d ln P_R/d ln k


    r= P_T/P_R

where P_R  is the scalar power spectrum and P_T is the tensor power spectrum <cit.>.

 



§ WARM NATURAL INFLATION IN F(Φ)T GRAVITY WITH TEMPERATURE-DEPENDENT DISSIPATION COEFFICIENT


  A successful inflationary model must generate sufficient expansion to solve the horizon problem. To satisfy this constraint, the slope of the inflaton potential must be very flat. This imposes a restriction on the choice of potentials. Moreover, it is expected that such a potential should be motivated by a fundamental theory. The natural inflation model is a well-motivated model where axion plays the role of inflaton field and the shift symmetry present in the axionic theory assures the flat potential required for inflation. We, therefore, attempt to study 
 natural warm inflation in the framework of f(ϕ, T) gravity. The form of the natural potential we are considering is given by <cit.> 

    V(ϕ)= μ^4(1+cos(ϕ/𝔣))

where 𝔣 is the decay constant and μ is the mass scale of the axion. We assume that the inflaton couples with light gauge fields and produces thermal friction which is given by the dissipation coefficient of the  form <cit.>

    Γ(τ)=C_Γτ^3/𝔣^2










where C_Γ is a dimensionless factor proportional to the coupling constant between the inflaton field ϕ and gauge field.
  
With this form of dissipation coefficient, we carry out our further analysis in two dissipative regimes viz weak Q<<1 and strong Q>>1.
  
  
  

 §.§ Weak Dissipative Regime

   
  In weak dissipative regime, Q<<1 and  the slow-roll parameters ϵ, η and β take the forms,
 

    ϵ=1/2 κ (1+2λ√(κ)ϕ)(4 λ√(κ)/4 λ√(κ)ϕ +1-sin(ϕ/𝔣)/𝔣(cos(ϕ/𝔣)+1))^2


    η   =   1/κ (1+2λ√(κ)ϕ)(-8 λ ^2 κ/(2 λ√(κ)ϕ +1) (4 λ√(κ)ϕ +1)-cos(ϕ/𝔣)/𝔣^2 (cos(ϕ/𝔣)+1).
       -   .λ√(κ) (8 λ√(κ)ϕ +6) sin(ϕ/𝔣)/𝔣 (2 λ√(κ)ϕ +1) (4 λ√(κ)ϕ +1)(cos(ϕ/𝔣)+1) )

 
    β= 1/κ(1+2λ√(κ)ϕ)Γ^'/Γ( 4λ√(κ)/1+4λ√(κ)ϕ-sin(ϕ/𝔣)/𝔣(cos(ϕ/𝔣)+1))

  
 and the equation of motion of the inflaton field reads

    3Hϕ̇(1+ 2λ√(κ)ϕ) + (1+ 4λ√(κ)ϕ)V^'+4λ√(κ) V = 0
 
 Using equations (<ref>) and (<ref>), we obtain

    ϕ̇=-√(μ ^4 (4 λ√(κ)ϕ +1) (cos(ϕ/𝔣)+1))(4 λ√(κ)/4 λ√(κ)ϕ +1-sin(ϕ/𝔣)/𝔣(cos(ϕ/𝔣)+1))/√(3 κ) (2 λ√(κ)ϕ +1)

 The temperature of the thermal bath can be obtained by using equation (<ref>) as
 
    τ=(  3 Q ϕ̇^2/4C)^1/4

 Also, the relation (<ref>), reads
 
    τ=(3Q H 𝔣^2/C_Γ)^1/3

Equating equations (<ref>) and (<ref>), we get

    Q= (1/4 C)^3 (C_Γ^4 μ ^4 ) ((4 λ√(κ)ϕ +1) (cos(ϕ/𝔣)+1)) (4 λ√(κ)𝔣/4 λ√(κ)ϕ +1-sin(ϕ/𝔣)/cos(ϕ/𝔣)+1)^6/(9 κ^5 𝔣^14) (2 λ√(κ)ϕ +1)^6


The violation of the slow-roll conditions governs the end of inflation. It is numerically checked that ϵ violates the slow-roll condition first. So, by using the condition ϵ(ϕ_f)=1, we determine the value of the scalar field at the end of inflation. The number of e-foldings (equation (<ref>)) in this regime becomes,  


    N   =    - κ∫^ϕ_f_ϕ_i(1+2λ√(κ)ϕ)/V^'/V+4λ√(κ)/1+4λ√(κ)ϕdϕ


Using equation (<ref>), the initial field value (ϕ_i) is calculated for N=60. The temperature of the thermal bath can be obtained by inserting the equations (<ref>) and (<ref>) into equations (<ref>),


    τ= 0.144 √(C_Γ^4 μ ^8 (4 λ√(κ)ϕ +1)^2 (cos(ϕ/𝔣)+1)^2      (4 λ√(κ)𝔣/4 λ√(κ)ϕ +1-sin(ϕ/𝔣)/cos(ϕ/𝔣)+1)^6     (4 λ√(κ)/4 λ√(κ)ϕ +1-sin(ϕ/𝔣)/𝔣(cos(ϕ/𝔣)+1))^2/κ^5 C^4 𝔣^14(2 λ√(κ)ϕ +1)^8)

  From the above relation, we can obtain the temperature of the thermal bath for any value of ϕ. Also, by substituting the relations (<ref>), (<ref>), (<ref>) and (<ref>) into equations (<ref>) and (<ref>), we obtain the spectral index and tensor-to-scalar ratio at initial field value ϕ_i.

  To evaluate the viability of any inflationary model, it is necessary to check whether the values of cosmological observables are in good agreement with the observational data. So, in order to constrain different parameters used in our model, we consider the latest limits set by Planck 2018 and BICEP/Keck data on n_s and r.

  The predictions of the spectral tilt and tensor-to-scalar ratio for three representative values of dissipation parameters are shown in  Fig. <ref>. The plots are obtained by varying λ and fixing the values of the mass scale μ=0.001 M_P and axion decay constant 𝔣=0.4M_P for 60 e-foldings. It is seen that variation of tensor-to-scalar ratio is very slow and with the increasing dissipation parameter (C_Γ), its values go on decreasing. The ranges of model parameter λ are found to be [93.6, 97.3] for C_Γ=1.1×10^5, [93.57, 97.27] for C_Γ=5×10^5 and [93.47, 97.17] for C_Γ=7.38×10^5. The plot shows that the warm natural inflationary scenario in the f(ϕ)T gravity framework is consistent with observational constraints on r and n_s in weak dissipative regime. 
 



   






 




 §.§ Strong Dissipative Regime

   
  In strong dissipative regime, Q>>1. So, the slow-roll parameters ϵ, η and β in this regime become,
 

    ϵ=1/2 κ (2λ√(κ)ϕ +Q)(4 λ√(κ)/4 λ√(κ)ϕ +1-sin(ϕ/𝔣)/𝔣(cos(ϕ/𝔣)+1))^2


    η   =   1/κ (2λ√(κ)ϕ+Q)(-8 λ ^2 κ/(2 λ√(κ)ϕ +Q) (4 λ√(κ)ϕ +1)-cos(ϕ/𝔣)/𝔣^2 (cos(ϕ/𝔣)+1).
       -   .λ√(κ) (8 λ√(κ)ϕ +6+8Q) sin(ϕ/𝔣)/𝔣 (2 λ√(κ)ϕ +Q) (4 λ√(κ)ϕ +1)(cos(ϕ/𝔣)+1) )

 
    β= 1/κ(2λ√(κ)ϕ+Q)Γ^'/Γ( 4λ√(κ)/1+4λ√(κ)ϕ-sin(ϕ/𝔣)/𝔣(cos(ϕ/𝔣)+1))

 and the equation of motion of the inflaton field becomes

    3Hϕ̇( 2λ√(κ)ϕ+Q) + (1+ 4λ√(κ)ϕ) V^'+4λ√(κ) V = 0
 
  From Γ(τ)=C_Γτ^3/𝔣^2, we have τ=(3Q 𝔣^2 H/C_Γ)^1/3. Equating this with τ=(  3 Q ϕ̇^2/4C)^1/4, we get
 
    ϕ̇^6=3 Q (4 C)^3 (𝔣^2 H/C_Γ)^4

 Expressions of ϕ̇ and Q can be obtained by solving equations (<ref>) and (<ref>). These expressions are substituted in equations (<ref>), (<ref>) and (<ref>) to obtain the slow-roll parameters ϵ, η and β in terms of model parameters. In this case, we have numerically checked that inflation ends when ϵ reaches unity and found the corresponding value of inflaton field (ϕ_f). The number of e-foldings (equation (<ref>)) in this regime becomes,  


    N   =    -κ∫^ϕ_f_ϕ_i(2λ√(κ)ϕ+Q)/V^'/V+4λ√(κ)/1+4λ√(κ)ϕdϕ

 
 Using the above definition of N, we obtain the initial field value (ϕ_i) for 60 e-foldings. From the expressions of the power spectrum P_R (equation (<ref>)) and P_T (equation (<ref>)), the spectral index n_s and tensor-to-scalar ratio r are computed at ϕ_i for different values of model parameters.
 
The depictions of the spectral tilt and tensor-to-scalar ratio are shown in  Fig. <ref>. Here also we consider three representative values of C_Γ. In each case, the mass scale and the axion decay constant are set at μ=0.001 M_P, 𝔣=0.4M_P. The range of model parameter λ is found to be [88.3, 95.1] for C_Γ=10^6, [95.75, 104.3] for C_Γ=1.1 × 10^6 and [103.2, 114] for C_Γ=1.2 × 10^6. The values of n_s and r are in good agreement with Planck 2018 <cit.> and BICEP <cit.> data at 95% CL.



 
 

 



§ CONCLUSION

Over the past few decades, extensive investigations have been conducted to understand and model the dynamics of the universe. The standard model of cosmology successfully describes the formation and evolution of the universe. However, this model has some problems like horizon and flatness problems which remain as open issues. In order to overcome these shortcomings, the idea of inflationary cosmology appears necessary in the early era of the universe. Even though general relativity provides remarkably accurate predictions for describing cosmological phenomena, it can not explain the dark sector of the universe. For this reason, investigating alternative theories of gravity arises as a promising path. 

In this work, we explore natural warm inflation in the framework of f(ϕ)T gravity theory. In warm inflation, the interactions between scalar and other fields during inflation are taken into account which provides the dissipation term. Here we consider a cubic temperature-dependent dissipation coefficient.

We have investigated this model in two separate dissipative regimes viz weak and strong and in each case, we have evaluated the cosmological observables for certain ranges of model parameters.

In the weak dissipative regime, the cosmological observables are functions of model parameter λ, axion decay constant 𝔣, mass scale μ, and dissipation coefficient. We have presented the predictions of the spectral tilt and tensor-to-scalar ratio for dissipation parameter C_Γ=1.1×10^5, C_Γ=5×10^5 and C_Γ=7.38×10^5 by varying λ and keeping the other parameters fixed.  The obtained value for the spectral index and the tensor-to-scalar ratio in the selected range of the model parameter λ  falls in the 1σ confidence level of the Planck 2018 data and the joint Planck and BICEP results. Hence, we claim that this model in weak dissipative regime is capable of making predictions consistent with the observational data.

In the strong dissipative regime also, the cosmological observables are the function of model parameter λ, axion decay constant 𝔣, mass scale μ, and dissipation coefficient as expected. Here, we have plotted the predictions of the spectral tilt and tensor-to-scalar ratio for dissipation parameter C_Γ=10^6, C_Γ=1.1×10^6 and C_Γ=1.2×10^6 by varying λ and keeping the other parameters fixed. The result shows that for some selected ranges of the model parameter λ, values of the spectral index, and the tensor-to-scalar ratio match in the 1σ confidence level of the Planck 2018 data and their combination with BICEP/Keck data. Hence, this model is consistent with the observational data in strong disspative regime also.

It is found that the model parameter λ has a minimal impact on the tensor-to-scalar ratio, as its values exhibit slow variation. So, it can be concluded that the tensor-to-scalar ratio is less sensitive to the change in the model parameter. The model produces smaller values of tensor-to-scalar ratio in strong dissipative regime. Because, the scalar power spectrum is completely dominated by the dissipation, which strongly suppresses the tensor-to-scalar ratio. Additionally, the value of the axion decay constant can be lowered below the Planck scale which is 0.4 M_P in this model. We can realize the natural warm inflation scenario in f(ϕ)T gravity for much smaller values of 𝔣 (sub-Planckian scale) compared to the corresponding cold inflation case, which makes it theoretically more sound.

 

§ ACKNOWLEDGEMENT

 We are thankful to Prof. Sudhakar Panda for fruitful insights.

 
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 Paper Title: Transmission of quantum information through quantum fields in curved spacetimes.

 







mk7592@princeton.edu
Department of Physics, Princeton University, Princeton, New Jersey 08544, USA


erickson.tjoa@mpq.mpg.de
Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, D-85748 Garching, Germany


§ ABSTRACT 

We construct a relativistic quantum communication channel between two localized qubit systems, mediated by a relativistic quantum field, that can achieve the theoretical maximum for the quantum capacity in arbitrary curved spacetimes using the Unruh-DeWitt detector formalism. Using techniques from algebraic quantum field theory, we express the quantum capacity of the quantum communication channel purely in terms of the correlation functions of the field and the causal propagator for the wave equation. Consequently, the resulting quantum channel, and hence the quantum capacity, are by construction manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and the choice of Hilbert space (quasifree) representations of the field.  



Transmission of quantum information through quantum fields in curved spacetimes
    Erickson Tjoa
    August 5, 2024
===============================================================================







§ INTRODUCTION


In the spirit of it-from-(qu)bit, quantum information theory has been extensively used to understand generic physical phenomena from quantum many-body physics to high-energy physics, from applied to foundational physics. Relativistic quantum information (RQI) is one of the many attempts to realize this spirit, which thus far focuses on the role of relativity in quantum information-processing tasks, as well as a deeper understanding of fundamental theories such as relativistic quantum field theories (QFT). The role of quantum information theory in relativistic QFTs cannot be understated: a non-exhaustive list includes rigorous study of the entanglement structure, complexity, and measurement theory for QFTs <cit.>; operational approach to particle production due to non-inertial motion and gravitational fields <cit.>; connections between relativistic causality, information-theoretic causality and reference frames <cit.>;  quantum information protocols mediated by the relativistic quantum field <cit.>; etc. 

This work is motivated by a somewhat vague, ambitious, and fundamental question: 

    Problem: What is the information-carrying capacity of quantum fields?

One can think of several ways to answer this question by making the problem statement more precise. For our purposes, we will adopt an operational approach, namely, whether two parties Alice and Bob, each in possession of some quantum-mechanical probes (atoms, qudits, oscillators, etc.) are able to transmit (quantum) information using the relativistic quantum field as a medium — that is, as a relativistic quantum channel mediating the communication between Alice and Bob. The paradigmatic model of such a communication channel is through the Unruh-DeWitt (UDW) particle detector model and its generalizations <cit.>. Indeed, transmission of classical information has been analyzed by many authors both perturbatively and non-perturbatively and in flat and curved spacetimes <cit.>. Transmission of quantum information in the non-perturbative regime has been studied only in flat spacetimes in very specific examples <cit.>. Note that in all these cases, the communication is encoded through the field amplitude and not, say, through the polarization degrees of freedom of the electromagnetic field, since on its own the polarization is blind to the relativistic features and constraints. 

Here we present the most general formulation for the transmission of quantum information between Alice and Bob, each in possession of a two-level system (a `qubit detector'), by suitable coupling to a relativistic scalar field in arbitrary curved spacetime. Clearly, our work constitutes a natural extension of the flat spacetime result in <cit.>. However, we will do so by exploiting techniques from algebraic quantum field theory (AQFT) <cit.>. Crucially, what we gain from this will not be a simple rewriting of all the formulae in the literature: we will see that the quality of the relativistic quantum channel mediated by the quantum scalar field can be formulated purely in terms of smeared correlation functions of the field and the detector parameters. Our results make clear what features of the field-mediated communication channel are necessary for perfect transmission of quantum information, and they are by construction manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and (to some extent) the Hilbert space representations of the field. In particular, it automatically works for all quasifree representations that give rise to Fock representations <cit.>).

This paper is organized as follows. In Section <ref> we briefly review the scalar field theory in curved spacetimes using the AQFT framework. In Section <ref> we describe our UDW detector setup and review the quantum capacity of a quantum channel. In Section <ref> we give the protocol that allows the transmission of quantum information through the quantum field. In Section <ref> we discuss the validity of the channel and study the regimes in which it fails. In Section <ref> we conclude with some discussions of our results and outlook. We set c=ħ=1 and adopt mostly-plus signature for the metric.




§ SCALAR FIELD THEORY IN CURVED SPACETIME




In this section we review the scalar field theory in curved spacetimes within the AQFT framework, adapting the convention given in <cit.> (for more details and reviews, see also <cit.>). 



 §.§ AQFT for scalar field


Consider a Klein-Gordon (KG) scalar field ϕ in an (n+1)-dimensional globally hyperbolic Lorentzian spacetime ℳ with metric tensor g_ab, whose equation of motion is given by the KG equation: 

    (-∇_a∇^a + m^2 +ξ R)ϕ = 0 ,

where ∇_a is the covariant derivative with respect to the Levi-Civita connection, m is the mass parameter, R is the Ricci scalar curvature, and ξ≥ 0 controls the non-minimal coupling of ϕ to R. Global hyperbolicity implies that we have a foliation of spacetime Σ and the KG equation is well-posed with initial data given on Cauchy surfaces. 

The KG operator P̂ -∇_a∇^a + m^2 +ξ R admits a unique advanced and retarded Green's functions G_A(,),G_R(,) = G_A(,) which satisfies

    P̂_ G_R/A(,) = δ^n+1(-) ,

where the subscript on P̂_ means the KG operator is w.r.t. the variable . We can use these Green's functions to construct the antisymmetric causal propagator (also known as the Pauli-Jordan distribution)

    E(,)  G_R(,) - G_a(,) .

Canonical quantization of the field ϕ gives rise to a field operator ϕ̂ obeying the canonical commutation relations (CCR) 

    [ϕ̂(),ϕ̂()] = i E(,) .

As ϕ̂ is an operator-valued distribution, in the algebraic approach to QFT we consider smeared field operators, i.e., by viewing the field operator as an -linear map from the test space of smooth compactly supported functions  to a *-algebra (), i.e., as a map ϕ̂: C^∞_0 ()→𝒜() with

    ϕ̂(f)∫ V  f()ϕ̂() .

The algebra of observables () is defined to be the collection of sums of products of ϕ̂(f) over all functions in  obeying the smeared CCR relation (<ref>), i.e., a CCR algebra <cit.>. The KG equation is encoded in the fact that we identify ϕ̂(f) ≡ 0 if f=P̂h for some h∈. 



The dynamical content of the field theory is reflected in terms of the solution space of the KG equation. The solution space _() can be equipped with a symplectic form σ:_()×_()→, defined as

    σ(ϕ_1,ϕ_2) ∫_Σ_tΣ^a [ϕ_1∇_aϕ_2 - ϕ_2∇_aϕ_1] ,

where Σ^a = -t^a Σ, -t^a is the inward-directed unit normal to the Cauchy surface Σ_t, and Σ = √(h) ^n is the induced volume form on Σ_t <cit.>. As is well-known, this definition is independent of the choice of Cauchy surface. The field operator ϕ̂(f) can be expressed as symplectically smeared field operator  <cit.> 

    ϕ̂(f) ≡σ(Ef,ϕ̂) ,

and the CCR algebra can be written as 

    [σ(Ef,ϕ̂),σ(Eg,ϕ̂)] = iσ(Ef,Eg) = i E(f,g) ,

where σ(Ef,Eg) = E(f,g) in the second equality follows from Eq. (<ref>) and (<ref>). 

Since the smeared field operators are unbounded operators, for free fields it is more convenient technically to work the Weyl algebra (), whose elements are bounded operators constructed by “exponentiating” the field operators. The Weyl algebra () is a unital C^*-algebra generated by elements that formally take the form 

    W(Ef) ≡e^iϕ̂(f) ,   f∈ .

These elements satisfy Weyl relations:

    W(Ef)^†   = W(-Ef) , 
       
        W(E (P̂f) )    =  ,
    
        W(Ef)W(Eg)    = e^-i/2E(f,g) W(E(f+g))

where f,g∈. The third relation enforces relativistic causality (or microcausality).


In AQFT, a quantum state is defined as a -linear functional ω:()→ (similarly for ()) such that 

    ω() = 1 ,  ω(A^† A)≥ 0  ∀ A∈() .

The state ω is pure if it cannot be written as ω= αω_1 + (1-α)ω_2 for any α∈ (0,1) and any two algebraic states ω_1,ω_2; otherwise we say that the state is mixed. 


The way to pass from AQFT to the Hilbert space approach is through Gelfand-Naimark-Segal (GNS) reconstruction theorem <cit.>. The GNS theorem gives us, for a given algebra of observables () and a state ω, a GNS triple (ℋ_ω, π_ω,|Ω_ω⟩), where π_ω: ()→ℬ(ℋ_ω) is a Hilbert space representation with respect to state ω. In its GNS representation, any state ω is realized as a vector state |Ω_ω⟩∈ℋ_ω and  A∈() are represented as bounded operators Âπ_ω(A)∈ℬ(ℋ_ω), and we write ω(A) = ⟨Ω_ω|Â|Ω_ω|$⟩. Since there exists infinitely many unitarily inequivalent representations of the CCR algebra, the algebraic approach allows us work with all representations at once and only pick the physically relevant representation at the very end. 

For a fixed stateω, we can construct the Wightmann-point functions, defined by

    𝖶(f_1,...,f_n)ω(ϕ̂(f_1)...ϕ̂(f_n))

wheref_j∈and the RHS is computed within some GNS representation of(). The GNS representation of the Weyl algebra()allows us to calculate Eq. (<ref>) by differentiation[Strictly speaking, the state on () and on () are different, but they are in one-to-one correspondence whenever they are related by Eq. (<ref>) (see <cit.> for good exposition on this). ]: for example,  the smeared Wightman two-point function reads

    𝖶(f,g) ≡ -∂^2/∂ s∂ t|_s,t=0ω(e^iϕ̂(sf)e^iϕ̂(tg))

where the RHS is calculated in the GNS representation of()<cit.>. 

The consensus within the AQFT community is that physically reasonable states should be Hadamard states<cit.>. Very roughly speaking, these states respect local flatness and the expectation values of all observables (in particular, the renormalized stress-energy tensor) are finite <cit.>. A particularly nice subclass of Hadamard states is the family of quasifree states: for these states, all odd-point functions in the sense of (<ref>) vanish and all higher even-point functions can be written as in terms of just two-point functions. In modern terminology, the term quasifree state is synonymous with Gaussian state, where the one-point functions need not vanish and higher-point functions only depend on one- and two-point functions <cit.>.

The relevance of quasifree states lies in the fact that they are completely specified once we know the Wightman two-point functions. More precisely, any given quasifree stateωis associated with a real-bilinear inner productμ:_()×_()→that satisfies <cit.>
    |σ(Ef,Eg)|^2 ≤μ(Ef,Ef)μ(Eg,Eg) ,

where we recall thatEf,Eg∈_()for allf,g∈. The inequality is saturated ifω_μis a pure quasifree state. Any quasifree state can then defined by those that satisfy

    ω(W(Ef))  e^-μ(Ef,Ef)/2 .

However, this definition is not useful unless we can compute the induced norm||Ef|| √(μ(Ef, Ef)). 

It turns out that||Ef||is related to the Wightman two-point function as follows <cit.>: we complexify_()into the space of complex solutions_()and define the KG bilinear product by 

    ⟨φ_1,φ_2|_⟩ iσ(φ_1^*,φ_2)

forφ_1,φ_2∈_(). It can be shown that for any quasifree state, there exists a subspaceℋ⊂_()such that(ℋ,⟨·,·|_⟩)is a Hilbert space and an-linear mapK:_()→ℋsuch that for allφ_1,φ_2∈_()* K_()+i K_() is dense in ℋ;
    
  * μ(φ_1,φ_2) = ⟨Kφ_1,Kφ_2|_⟩;
    
  * σ(φ_1,φ_2) = 2⟨Kφ_1,Kφ_2|_⟩;
    
  * _()≅ℋ⊕ℋ, where ℋ is the complex-conjugate Hilbert space of ℋ and ⟨u,v|_⟩=0 for all u∈ℋ and v∈ℋ.

In a more familiar language of canonical quantization, the mapKprojects to the “positive-frequency part” of a real solution to the KG equation. The Wightman two-point function is then given by <cit.>
    𝖶(f,g)    = ⟨KEf,KEg|_⟩
       = μ(Ef,Eg) + i/2E(f,g) ,

where we have used the fact thatσ(Ef,Eg) = E(f,g). SinceE(f,g)is antisymmetric, it follows that||Ef||^2 = μ(Ef,Ef) ≡𝖶(f,f)and hence

    ω(W(Ef)) =   e^-1/2𝖶(f,f) .

Indeed, we may very well take Eq. (<ref>) as the definition of quasifree states. 




The most important example of a quasifree state is the “vacuum state”ω, where we can write the (unsmeared) vacuum Wightman function as 

    ω(ϕ̂()ϕ̂()) ≡𝖶(,)    = ∫^n  u^*_() u^*_() ,

whereu_()are called the positive-frequency modes of KG operator with respect to the KG inner product

    (ϕ_1,ϕ_2)_kg iσ(ϕ_1^*,ϕ_2) ,

whereϕ_j∈_()are complexified solutions to Eq. (<ref>). The vacuum stateωis not unique and there is a sense in which every GNS vector is a “vacuum state” in that particular representation. However, once a particular basis of modes{u_}are chosen, it fixes the vacuum state as one whose Wightman function satisfies (<ref>). In flat spacetime, the standard plane-wave basis in the inertial coordinates defines the Minkowski vacuum, which is the only state that respects the full Poincaré symmetry and minimizes the expectation value of the (renormalized) stress-energy tensor <cit.>. In curved spacetimes, there are multiple inequivalent states that qualify as a vacuum state <cit.>.




 §.§ Relationship with canonical quantization
 

The usual Fock space in canonical quantization arises from the GNS reconstruction applied to()with respect to some vacuum stateω. Technically speaking, any pure quasifree (Hadamard) state qualifies as a vacuum state, as their GNS representation gives rise to the Fock vacuum in the standard sense <cit.> (see also <cit.> for a different definition in Minkowski spacetime). This is the best one can do in generic curved spacetimes as there is no preferred vacua without further assumptions such as time translation symmetry. 

In more detail, the GNS Hilbert spaceℋ_ωfor the pure quasifree representation is the Fock space over the one-particle Hilbert spaceℋℋ_ω≡ = ⊕_n=0^∞ℋ^⊙ n ,

whereℋ^⊙0≅andℋ^⊙nmeans symmetrized direct sum (then-particle sector of the Fock space). In this representation, we can write

    π_ω(ϕ̂(f)) = â((KEf)^*) + â^†(KEf) .

In what follows we dropπ_ωif it is clear from the context that we are using a Fock representation. Note that if we consider complex smearing functionsf:→, we can write

    ϕ̂(f) ≡ϕ̂( f) + i ϕ̂( f) .

The operatorsâ(u^*),â^†(v)are smeared ladder operators[There are various conventions on how to label the smeared ladder operators (see <cit.>). We follow <cit.> in that it views â,â^† as being linear in their arguments. The convention in <cit.> does not include complex conjugation in the argument for â(·) and views â as an anti-linear map while â^† is linear.] obeying the CCR

    [â(u^*),â^†(v)] = ⟨u,v|_⟩

on a suitable dense domain of the Fock space.

The standard canonical approach is recovered by working with the unsmeared field operatorϕ̂()and considering the Fourier mode decomposition

    ϕ̂() = ∫^n â_ u_() + â_^† u_^*() ,

whereu_()andu_^*()are positive- and negative-frequency modes normalized to Dirac delta functions via the KG inner product:

    ⟨u_,u_'|_⟩   = δ^n(-') ,
    ⟨u^*_,u^*_'|_⟩   = -δ^n(-') ,
    ⟨u_,u^*_'|_⟩   = 0 .

These modes are not proper elements of the one-particle Hilbert space but they are convenient to work with. In particular, we see thatâ^*_≡â(u_^*)andâ_^†≡â^†(u_). If we were to pick any other quasifree state, the resulting one-particle structure will differ and this can give rise to Hilbert space representations that are unitarily inequivalent, e.g., by considering Kubo-Martin-Schwinger (KMS) thermal states <cit.>. 

Using these modes, we can decompose the solutionEfin the “eigenmode basis”{u_,u_^*}Ef ≡∫^n ⟨u_,Ef|_⟩u_ + ⟨u^*_,Ef|_⟩u_^* .

From the CCR and the definition of the Wightman function, we see that

    iE(,')    = ∫^n u_()u_^*(') - u_^*()u_(') ,

the positive-frequency partKEfofEfreads

    KEf    = ∫^n ⟨u_,Ef|_⟩u_≡∫^n f_ u_

where

    f_-i∫ V f()u_^*() .
 
Furthermore, sinceKEf lies inℋ, it is convenient to use the notation|KEf⟩∈ℋand|u_⟩for the (improper) basisu_. In this notation, we write

    |KEf⟩ = ∫^n f_|u_⟩ ,

This suggests that we can label the ladder operators using the “momentum space smearing function”f_rather than the “real space positive-frequency solution”KEf, i.e., we can write

    ϕ̂(f)    = â(f_^*) + â^†(f_) ≡∫^n â_^†f^*_ + â^†_f_
       ≡ϕ̂(f_) .

Essentially, we are using the elements of the one-particle Hilbert spaceℋto label the field observables rather than using the spacetime smearing functions. This notation is particularly convenient if the focus is more about which modes of the field that the detectors are coupled to <cit.>.



§ SETUP


In this section, we introduce the UDW detector model that we will use to couple Alice's and Bob's qubits to the field and briefly review the notion of quantum capacity for a quantum channel and coherent information. These are relevant measures of how much quantum information can be transmitted (on average) through the quantum channel between Alice and Bob.



 §.§ UDW detector model


In the UDW framework, Alice and Bob carry their own qubit (`detector') with a free HamiltonianĤ_ν= 1/2Ω_νσ̂_z, whereν∈A, B,Ω_νthe energy gap andσ̂_zthe Pauli-Zoperator. We denote the eigenstates ofσ̂_sby|±_s⟩withs ∈x,y,z, where we do not label the Pauli matrices withνto reduce notational clutter and the context will make it clear.  The excited and ground states of the free HamiltonianĤ_νare given by|±_z⟩with eigenvalues±Ω_ν/2respectively.

The UDW interaction between Alice's and Bob's detectors and the field is assumed to be linear in the field as it represents a simplified model of dipole interaction𝐝̂·𝐄̂in quantum optics, i.e., as a spin-boson type interaction <cit.>. The general form of such an interaction is given by a linear combination of the form[This is a slight modification of the usual spin-boson model since typically spin-boson models are assumed to have time-independent interactions <cit.> and to account for the covariant formulation of the UDW model <cit.>.]
    ĥ_I,ν() ∑_iλ_ν,i f_ν,i() m̂_ν,i(τ()) ⊗ϕ̂() ,

whereλ_ν,iis the coupling strength,m̂_ν,i(τ)is a Hermitian operator acting on the qubit, andf_ν,i()is a spacetime smearing function defining the duration of the interaction and the spatial profile of the detector. The most commonly used UDW model involves only one term inĥ_I,ν() = λ_νf_ν()m̂_ν(τ())⊗ϕ̂()<cit.>, but we will need the more general version in (<ref>) for our purposes. We drop the tensor product symbol if the context is clear. 


The time evolution operator generated by the interaction Hamiltonian (<ref>) reads

    Û = 𝒯_t exp[-i ∫_V∑_νĥ_I,ν]

whereV [n] √(-g)is the invariant volume element and the time-ordering is assumed to be given by some global time function (see <cit.> for discussions about some subtleties of time-ordering for the spatially smeared UDW model). In general, it is not possible to solve the the time evolution analytically due to the time-dependence of the interaction, thus many analyses involving the UDW detectors consider perturbative methods in the weak coupling regimes. However, our goal in this paper is to establish a perfect quantum channel between Alice and Bob and as was explained in <cit.>, this requires our model to be in the non-perturbative regime.

In order to construct the perfect quantum channel that can transmit quantum information perfectly, we need to be able to calculate the interaction unitaryÛnon-perturbatively (in the sense of not truncating the Dyson series in the weak coupling regime). For this purpose, we will need to use the so-called delta-coupling regime <cit.>, where the interaction timescale is assumed to be much faster than all the relevant timescales of the problem so that the interaction can be taken to occur at a single instant in time. Delta-coupled detectors have been used in many contexts such as entanglement extraction, relativistic communication, work extraction, and many others (see, e.g., <cit.>). As each detector has its own rest frame, we need to consider the Fermi normal coordinates (FNC) associated with each observer  <cit.>, whose center-of-mass (COM) trajectory is parametrized by their respective proper times <cit.>.

The physically reasonable assumption we make is that in the FNC, ≡(τ,), the spacetime smearing functions can be decomposed as

    f(()) = χ(τ)F() ,

where(τ,=0)labels the COM trajectory. In other words, Alice (and Bob) can in their rest frames distinguish the spatial profile (which is assumed to satisfy `Born rigidity', i.e., time-independent with respect toτ) from the switching function that controls the duration and strength of the interactions. The Hamiltonian density in this case reads


    ĥ_I,ν() ∑_iλ_ν,iχ_ν,i(τ)F_ν,i() m̂_ν,i(τ ) ϕ̂(())

and the delta coupling interaction corresponds to the case when

    χ_ν,i(τ) = δ(τ-τ_ν,i)

for some constantsτ_ν,i∈. 

For our purposes, we consider the scenario where Alice and Bob each interact with the field via delta coupling twice, so that we need to consider multiple interaction timesτ_ν,iwithτ_ν,1 < τ_ν,2. More generally, if Alice (or Bob) performs a sequence ofNdelta interactions withτ_ν,i<τ_ν,i+1, the corresponding unitary time evolutionÛ_νcan be decomposed into a sequence of simple-generated unitaries<cit.> (see also <cit.>)

    Û_ν   = ∏_j=1^NÛ_ν,j ,  Û_j = exp[-i λ_νm̂_ν(τ_ν,j)  ϕ̂(F_ν,j)]

The time-ordering𝒯is taken care of since the unitariesÛ_iare written in time-increasing order. When two or more observers are involved, we can use the global time coordinates to orderÛ_aandÛ_b: for example, if Alice and Bob are spacelike separated then the ordering does not matter, while Alice unitary acts before BobÛ_bif(f_b)is in the causal future of(f_a). The unitaryÛ_νcan be computed non-perturbatively sinceÛ_jis a controlled unitary: for any two Hermitian operatorsÂ, B̂such thatÂ=∑_i=1^ra_j|a_j⟩⟨a_j|is finite-dimensional operator in its spectral decomposition, we can write

    exp[i Â⊗B̂] = ∑_j=1^r|a_j⟩⟨a_j|⊗exp[ia_j B̂] .

As we will see, for our setup the full unitary time evolutionÛwill consist of products of Alice's two simple-generated unitaries and also Bob's two simple-generated unitaries.

Last but not least, let us recall the unitaries found in <cit.> for Minkowski spacetime. There Alice performs two simple-generated unitaries, but one of them involves coupling to a conjugate momentum.

    Û_a = e^iσ̂_x π̂(F_a)e^iσ̂_z ϕ̂(F_a) ,

where the spatially smeared operators are

    ϕ̂(F_a)    = λ_1∫^3 F_a()ϕ̂(t_a,) ,
    π̂(F_a)    = λ_2∫^3 F_a()∂_tϕ̂(t_a,) .

Note that here they taket_at_A,1andt_A,2 = t_a+ϵand consider the limit asϵ→0^+so that the unitary generated by the conjugate momentum acts later. Bob's unitary is then constructed as a “suitable inverse” of Alice's unitary to faithfully extract Alice's qubit. 

Our construction requires us to make two modifications to the above unitaries. First, we can generalize the conjugate momentum operator in curved spacetime, i.e., 

    π̂() ∂_tϕ̂()≡√(h) t^a∇_a ϕ̂() ,

wheret^ais the forward pointing unit normal, orthogonal toΣ_twith an induced metrich. Second, it is straightforward to see from integration by parts that 

    π̂(f)    = ∫ V f()∂_tϕ̂() = ϕ̂(-∂_tf) .

Thus we can always view the smeared conjugate momentum operator as a smeared field operator with a different smearing function (namely, its time derivative). In Minkowski spacetime, using the spacetime smearing notation we can thus write

    π̂(F_a)≡π̂(δ· F_a) = ϕ̂(-δ'· F_a)  ,

whereδ'(t)is the distributional derivative of the Dirac delta functionδ(t). This second step might appear unconventional sinceπ̂is not usually regarded as a spacetime-smeared operator, but this provides a simple way to recast all basic observables in terms of a single field operatorϕ̂. The connection between the modern AQFT approach based on spacetime-smeared operators (cf. Section <ref>) and the “old” (non-covariant) AQFT formulations based on two spatially smeared operators{ϕ̂|_Σ,π̂|_Σ}at a fixed Cauchy sliceΣwith equal-time CCR[ϕ̂(t,),π̂(t,)]=iδ^(n)(-)can be made more precise and rigorous  (see, e.g., <cit.>, and also <cit.> for details). 

The two modifications above suggest that what we really need for the generalization to curved spacetime is that Alice couples to two smeared field operatorsϕ̂(f_1),ϕ̂(f_2)that do not commute, i.e.,E(f_1,f_2)≠0, and not necessarily to the field and its conjugate momentum. Indeed, in the modern AQFT framework the coupling to momentum is somewhat awkward because the perfect quantum channel in <cit.> requires that Alice couples toϕ̂(F_a)more strongly thanπ̂(F_a), but the coupling strengthsλ_1,λ_2would need to have different units, so one needs an additional length scale to compare their strengths. By considering different smeared field operatorsϕ̂(f_i), we can haveλ_iwith the same units and we could use other length scales (e.g., from the effective size of the spatial smearing) if we insist on interpreting one of them as conjugate momentum. In what follows we assume thatf_i(and henceλ_i) have the same units for simplicity.




 §.§ Quantum capacity and coherent information


Here we briefly review the notion of quantum capacity of a quantum channel and quantum coherent information, following closely the exposition in <cit.> (see <cit.> for more in-depth details). We denote by𝒟(ℋ)the space of density operators associated with some Hilbert spaceℋ. LetΦ:𝒟(ℋ_a)→𝒟(ℋ_b)be a quantum channel — a completely-positive trace-preserving (CPTP) map — between Alice's and Bob's density operators. 

An ideal quantum communication channel is one that is able to send any quantum state from Alice to Bob, or equivalently, one that preserves entanglement reliably since Alice's state can always be viewed as an entangled state with some environment that purifies it.  Schematically the idea is the following. First, Alice prepares a purification of the stateρ_a∈𝒟(ℋ_a), namely,ρ_ea|ψ_ea⟩⟨ψ_ea|∈ℋ_e⊗ℋ_awithℋ_a≅ℋ_e. Then Alice applies an encoding channelℰ:𝒟(ℋ_e⊗ℋ_a)→𝒟(ℋ_e⊗ℋ_a^⊗N)to map her share of pure state intoNquantum systems, i.e.,

    ρ_ea,N (_e⊗ℰ)(ρ_ea) .

After encoding, Alice sends each quantum system throughNindependent uses of the communication channelΦ, i.e.,

    ρ_eb,NΦ^⊗ N(ρ_ea,N)∈𝒟(ℋ_e⊗ℋ_b^⊗ N) .

Bob's task is to use a decoding channel𝒟:𝒟(ℋ_e⊗ℋ_b^⊗N)→𝒟(ℋ_e⊗ℋ_b), such that

    ρ_eb (_e⊗𝒟)(ρ_eb,N) .

This is a good encoding-decoding procedure if for a givenϵwe can have

    ||ρ̃_eb-ρ_eb||_1≤ϵ

whereρ̃_ebis the state obtained ifΦis replaced with a noiseless channel_a→b:(ρ∈𝒟(ℋ_a)) ↦(ρ∈𝒟(ℋ_b)). 

The rate of the communication channel is the number of qubits transmitted per use of the channel, i.e.,

    R 1/nlog_2(ℋ_a)

and this rate is said to be achievable if for anyδ,ϵ>0, there exists an(n,R-δ,ϵ)for sufficiently largen. The quantum capacity𝒬[Φ]of the channelΦis defined by the supremum of all achievable rates (see <cit.> for more details). 

For practical calculations that we need, we consider a useful form of the quantum capacity in terms of an information-theoretic measure called coherent information. Given a quantum channelΦ:𝒟(ℋ_a)→𝒟(ℋ_b), let|ψ_ea⟩be a purification of Alice's input stateρ_aand define

    ρ_eb    (⊗Φ)(|ψ_ea⟩⟨ψ_ea|) .

Usingρ_b _e(ρ_eb) =  Φ(ρ_a), the coherent informationI_c(ρ_a,Φ)is defined to be <cit.>
    I_c(ρ_a,Φ)     S(Φ(ρ_a)) - S(ρ_eb) .

We may sometimes writeI_c(ψ_ea,Φ)to make clear the joint state onEA. The coherent information depends on the input stateρ_aand the channelΦbut is independent of its purification. From this, we can then define the coherent information of a quantum channelI_c(Φ)by maximizing over all input states:

    I_c(Φ)   max_ρ_aI_c(ρ_a,Φ) .

Note that for qubit channels we have that-1≤I_c(ρ,Φ)≤1and0≤I_c(Φ)≤1<cit.>. We can then rewrite the quantum capacity using the Lloyd-Shor-Devetak (LSD) formula <cit.>, given by <cit.>
    𝒬[Φ]lim_n→∞1/nI_c(Φ^⊗ n) ,

where the RHS is also known as regularized coherent information.  

Despite the somewhat physically intuitive expression for𝒬[Φ], the asymptotic nature of Eq. (<ref>) makes quantum capacity very difficult to compute in most cases unless the quantum channel has very specific properties (e.g., when the quantum channel is additive, namely𝒬[Φ^⊗n] = n𝒬[Φ], which is false in general <cit.>). However, for us we do not actually need this: we follow the strategy adopted in <cit.>, which is to construct a quantum channel for qubits that hasI_c(Φ) →1by appropriately adjusting the setup parameters. This works because the quantum coherent information is superadditive<cit.>:

    I_c(Φ^⊗ N) ≥ NI_c(Φ)

which immediately implies that

    𝒬[Φ] ≥ I_c(Φ) ≥ I_c(ρ_a,Φ) .

For qubit channels, we have0≤𝒬[Φ]≤1, thus we can construct an essentially perfect quantum channel if we can show thatI_c[Φ]can be made arbitrarily close to 1. 

The interpretation of the above analysis is that in the limit where𝒬[Φ]→1, the quantum channel can effectively transmit quantum information (qubits) reliably in one use of the channel. It is perhaps not surprising that this is possible if and only if the whole procedure is essentially a “swapping operation”, i.e., Alice swaps her qubit `into the field', and then Bob swaps out the (approximate) embedded qubit in the field state to recover Alice's state. In effect, this suggests that the core of the protocol comprises of two components: (1) Bob being positioned correctly in spacetime in order to recover Alice's qubit, and (2) Alice and Bob being able to implement as high quality `SWAP gates' as possible[ In <cit.>, Alice's unitary is called ENCODING gate, and Bob's unitary is called DECODING gate. While the terminology has physical justification, it does not coincide with the encoding and decoding channels ℰ,𝒟 even in the case of a single use of the communication channel Φ (N=1), as they are not part of Φ whose capacity is being computed. The ENCODING and DECODING gates were part of the definition of the communication channel.].

As a final remark, we would like to point out that the existence of many superadditive quantum channels are in some sense a `curse' to finding a good characterization of quantum capacity — we refer the readers to <cit.> for one of the more recent summary and progress on this front. From Eq. (<ref>) we see that we need0<I_c(Φ) ≤1for the channel to be able to transmit some quantum information and our construction capitalizes on this. However, the case forI_c(Φ) = 0is much more subtle, as many quantum channels exhibit strict superadditivity in both weak and strong sense <cit.>. Weak superadditivity means that there existsNsuch thatI_c(Φ^⊗N) > N I_c(Φ), and strong superadditivity meansI_c(Φ_1⊗Φ_2) > I_c(Φ_1)+I_c(Φ_2)for two channelsΦ_1,Φ_2. In both cases, the `single-letter' formula for coherent informationI_c(Φ)is not sufficient to calculate the quantum capacity,  although some exceptions are known. For instance, all degradable channels (such as some regimes of the dephasing channels) are known to be weakly additive, in which case𝒬[Φ] = I_c(Φ)and hence non-positive channel coherent information would imply zero quantum capacity <cit.>. All anti-degradable channels (such as all entanglement-breaking channels and certain regimes of quantum erasure channels) are known to have zero quantum capacity, though the argument does not rely on additivity and more towards no-cloning type arguments <cit.>. Our construction (similarly for <cit.>) exploits the structure of the detector-field coupling to make the channel coherent information tractable.





§ PROTOCOL FOR THE PERFECT RELATIVISTIC QUANTUM CHANNEL


In this section, we present the protocol for Alice and Bob to implement a perfect quantum channel through the field. While our protocol closely follows <cit.>, our approach naturally extends to arbitrary background spacetimes and choices of vacuum states of the field by recasting the protocol in the AQFT framework. 

A brief visual of the quantum channel is depicted in Fig. <ref>. Mathematically, the quantum channel from Alice to Bob generically takes the form

    Φ(ρ_a) = _a, ϕ[Û_b^†Û_a^†( ρ_a⊗0⊗ρ_b ) Û_a^†Û_b^†]  ,

where|0⟩is the field vacuum[The state |0⟩ can be replaced with any GNS vector state |Ω_ω⟩ associated with any of the quasifree representations, which accounts for the existence of many unitarily inequivalent ground states in QFT <cit.>.],ρ_bis Bob's initial state, andÛ_aandÛ_bare the unitaries implemented by Alice and Bob through the UDW interaction with the field. The task is to construct a perfect quantum channel that can faithfully transmit quantum information to arbitrary accuracy.

Essentially, the basic idea is for Alice to (approximately) perform a `SWAP' gate[Technically speaking, the qubit and the quantum field have different Hilbert space dimensions but we will call them `SWAP' gates anyway since it would be the usual SWAP gate if we were to replace the quantum field with a third qubit. The fact that we are “embedding” the qubit state into the (much higher-dimensional) field can be viewed as a form of encoding operation, which justifies calling Alice's unitary ENCODING gate in <cit.>, and similarly Bob's unitary can be viewed as decoding operation from the field to his qubit system.] on her qubit state to the field, effectively encoding it into some superposition of approximately orthogonal coherent states, so that Bob can extract the qubit state by performing another swap operation. The entire problem thus reduces to analyzing under what circumstances the UDW interaction allows for nearly perfect `SWAP' operations on both parties which will depend on both the coupling strengths and the supports of the interactions. In particular, we will see that Bob can only perform a perfect `SWAP' gate if he is causally connected to Alice and the details of the interactions determine the quality of the swap operation. 

More precisely, Alice starts with an arbitrary state|ψ_a⟩c_1 |+_z⟩ + c_2 |-_z⟩, equivalentlyρ_a=|ψ_a⟩⟨ψ_a|, and encodes that state into the field using a `SWAP' gate. Bob should then be able to decode the state from the quantum field into his own Hilbert space. If the `SWAP' gatesÛ_aandÛ_bare implemented correctly, then we should have a perfect quantum channelΦ(ρ_a) = ρ_ain the idealized limit. We start by discussing how Alice encodes her state into the field:


    Let  |ψ_a⟩ = c_1 |+_z⟩ + c_2 |-_z⟩ be Alice's initial state and let |0⟩ be the initial vacuum state of the field. Consider Alice's unitary operation at time t=t_a given by[The usual convention would be Û_a =  e^-i σ̂_x ϕ̂(f_2) e^-i σ̂_z ϕ̂(f_1), but we opt for the positive sign for the exponent for convenience as many expressions appear more symmetric (this was also implicitly done in <cit.>). This amounts to the replacement ĥ_I→ -ĥ_I, or equivalently f_i→ -f_i. ]
    Û_a = e^i σ̂_x ϕ̂(f_2) e^i σ̂_z ϕ̂(f_1) .

    Then Alice can approximately swap the state |ψ_a⟩ into the field provided the following conditions are satisfied 
    
    E(f_1, f_2)    = π/4 mod  2π ,
    
            E(f_1, f_2)^2    ≫𝖶(f_2, f_2) .

    The resulting state after Alice's operation reads 
    
    Û_a (c_1 |+_z⟩   + c_2 |-_z⟩) |0⟩
       = |+_y⟩ (c_1 |+α⟩ - i c_2 |-α⟩) ,

    where |±α⟩ are coherent states of the quantum field defined by
    
    |±α⟩ e^± i ϕ̂(f_1)|0⟩.

    In order to prove this statement, we will explicitly apply Eq. (<ref>) to Alice's qubit and the field and show that the qubit state is encoded under the conditions in Eqs. (<ref>) and (<ref>).

    The first unitary in Eq. (<ref>) gives 
    
    e^i σ̂_z ϕ̂(f_1)|ψ_a⟩|0⟩   = c_1 |+_z⟩|+α⟩  + c_2 |-_z⟩|-α⟩.

    Next, we need to apply the second unitary e^i σ̂_x ϕ̂(f_2) but we first note that 
    
    ϕ̂(f_2) |±α⟩ = ± E(f_1, f_2) |±α⟩ + e^± i ϕ̂(f_1)ϕ̂(f_2) |0⟩ ,

    which follows directly from the Baker-Campbell-Haussdorff (BCH) formula and the CCR. Now we see that if 
    
    E(f_1, f_2)^2 ≫0ϕ̂(f_2) ϕ̂(f_2)0 = 𝖶(f_2, f_2)

    then |±α⟩ become approximate eigenvectors of ϕ̂(f_2)
    ϕ̂(f_2) |±α⟩≈± E(f_1, f_2) |±α⟩.

    If we pick the eigenvalues to be E(f_1, f_2) = π/4 mod 2π, then the action of Alice's full unitary Û_a gives
    
    Û_a    (c_1 |+_z⟩ + c_2 |-_z⟩) |0⟩
       = e^i σ̂_x ϕ̂(f_2)(c_1 |+_z⟩|+α⟩  + c_2 |-_z⟩|-α⟩)
       ≈ c_1 e^i π/4σ̂_x |+_z⟩|+α⟩  
            + c_2 e^-i π/4σ̂_x |-_z⟩|-α⟩
       = |+_y⟩(c_1 |+α⟩ - i c_2 |-α⟩) ,

    where we used e^i π/4σ̂_x |+_z⟩ = |+_y⟩ and e^-i σ̂_x π/4|-_z⟩ = -i|+_y⟩ in the last equality.



The second of Alice's conditions in Eq. (<ref>) can be interpreted as a “strong coupling condition” that requires Alice's coupling through the field via the interaction profilef_1be much stronger than throughf_2. In <cit.>, this corresponds to choosingf_2such thatϕ̂_2corresponds to conjugate momentumπ̂(F_a)and the condition translates to Alice coupling to the fieldϕ̂(F_a)much more strongly than toπ̂(F_a)(cf. Eq. (<ref>)). We can interpret Eq. (<ref>) as a “fine-tuning condition” that performs near exactπ/4rotations in the Bloch sphere so that Alice's second unitary decouples her detector from the quantum field.

Now we move on to discussing how at a later time, Bob can decode Alice's state from the quantum field into his own qubit detector. We notice that the protocol in Proposition <ref> leaves Alice's system in the state|+_y⟩, therefore if Bob's qubit starts in that initial state, it seems that Bob should simply apply the inverse ofÛ_ato decode the state from the field, i.e.,

    Û_b^†Û_a^† =  e^-i σ̂_z ϕ̂(f_1) e^-i σ̂_x ϕ̂(f_2) ,

where the qubit operatorsσ̂_sare now understood to act on Bob's Hilbert space. If Bob were able to implement this unitary then we have (almost trivially) achieved our goal of creating a perfect quantum channel from Alice to Bob. The issue is that Alice's operations defined throughϕ̂(f_i)are defined at timet=t_a, while we want Bob to decode from the field using only field operatorsϕ̂(g_i)localized at timet=t_b. We need a way to write the operatorsϕ̂(f_i)using different spacetime smearing functionsg_idefined at a different Cauchy slice. To achieve this we use the following theorem:

    Suppose we have two smeared field operators ϕ̂(f) and ϕ̂(g), then the two operators are equal ϕ̂(f) = ϕ̂(g) if the smearing functions satisfy
    
    Ef = Eg.

    The proof is almost immediate from the symplectic definition of smeared operators in Eq. (<ref>). If we assume Ef = Eg[The converse statement is false. From the axioms of AQFT, we also have that ϕ̂(P̂f)=0 which is the KG equation. Thus ϕ̂(g) is only unique up to homogeneous solution of the KG equation, i.e., g∼ g+P̂f.], then 
    
    ϕ̂(f) = σ(Ef, ϕ̂) = σ(Eg, ϕ̂) = ϕ̂(g) .


Some remarks are in order in view of the above Proposition <ref>. First, intuitively Alice's state is encoded into the field amplitude that propagates into the causal future of Alice's interaction region 𝒥^+(f_i), where𝒥^+((f_i))is the causal future of(f_i); for Bob to decode that state, his qubit has to be coupled everywhere where the initial data has propagated at a given timet=t_b, which in turn forces Bob's detector to interact with the field with the correct support and coupling strengths. In particular,(g_i)should essentially beΣ_t_b ∩(Ef_i), where(Ef_i)⊆𝒥^+(f_i). For simplicity, we assume that(f_1)=(f_2)and hence(g_1)=(g_2), which in the delta-coupling scenario is easily enforced as we are fixing the spatial profile of the interaction on Alice's side — this, in turn, fixes the needed spatial profile for Bob's coupling. 

Second, despite its simplicity and simple interpretation, Eq. (<ref>) is largely impractical to use for actually calculating Bob's spatial smearing functionsG_i(x)for a given timet = t_b. Using the eigenmode basis provided in Eq. (<ref>) and the momentum space smearing functions in Eq. (<ref>), the condition in Eq. (<ref>) can be equivalently written in `momentum space' as

    f_ k = g_ k.

These equations in momentum space are often easier to work with since in many simple cases (such as all FRW spacetimes) the spatial part of the mode functionu_k() ∝e^i k ·xand we can use Fourier transforms, which is similar to what was originally done in <cit.> where Bob's spatial smearing functions were related to Alice's in momentum space. More generally, in generic curved spacetimes, we do not have a simple plane-wave basis, so it is essential that all the conditions we demand for the perfect quantum channel are given only in terms of the correlation functions of the field without reference to Fourier transforms. 

Last but not least, note that there is an asymmetry in how we quantify the quality of Alice's and Bob's respective `SWAP' gates, i.e., how close their unitaries are in achieving the ideal `SWAP' operation. On the one hand, Alice needs two smearing functionsf_1andf_2that satisfy the conditions in Proposition <ref> to approximate the ideal `SWAP'. On the other hand, Bob needs two smearing functionsg_1andg_2that are related to Alice's via Proposition <ref> and hence automatically satisfy the conditions in Proposition <ref>. However, Proposition <ref> alone is not enough for the perfect quantum channel: ifEf_i ≠Eg_i, Bob will not be able to decode Alice's state even though by construction Bob's smearing functionsg_1andg_2satisfy Eqs. (<ref>) and (<ref>). In a sense, to decode the state using the `SWAP' operation with the field Bob needs an extra condition, namely that Bob needs to interact “at the correct spacetime regions”. The reason for this asymmetry is that the Hilbert space dimension of the fieldϕ̂is (much) larger than that of the detectors: it is much simpler for Alice to encode her state in a larger system but Bob will need additional information to decode it. 

In summary, we have constructed a perfect relativistic quantum channel that allows Alice to send her qubit state to Bob through the quantum field. The quantum channel consists of two steps:

  *  At time τ= τ_a, Alice encodes her qubit into the field using the unitary Û_a such that the conditions in Proposition <ref> are satisfied.
    
    
  *  At a later time τ = τ_b, Bob decodes the state from the field using the unitary Û_b defined in Eq. (<ref>). However, to implement this protocol, Bob's detector must interact across a spacetime region that satisfies the conditions specified in Proposition <ref>.

As the proper times are not the same in both trajectories, in practice one can use the ambient global coordinate systems and foliation to help with the time-ordering of these operations. Once timesτ_aandτ_bare specified, we demand that Alice's spatial smearing functionsF_i(x)fulfill conditions (<ref>) and (<ref>), and Bob's functionsG_i(x)fulfill the condition (<ref>)[In principle, Alice or Bob could each need up to four spatial smearing functions since f_i or g_i could include both δ(t) and δ'(t) terms. For example, in flat spacetime, we can have f_1 = δ(t) F_11( x) + δ'(t)F_12( x) (similarly for f_2), but for simplicity these can be chosen to be equal. Both g_1 and g_2 are fixed by f_1 and f_2 in the ideal case.]. 

 In order to simplify our subsequent discussions, we make a physically reasonable assumption that Alice's interaction profilesf_iare completely fixed except their coupling constants, i.e., in FNC we have

    f_i = λ_i χ_i(τ)F()

whereχ_iare either the Dirac delta functionδor its distributional derivativeδ'and the spatial profile is the same for bothf_1,f_2(hence they have the same supports(f_i) = (F)⊆). In other words, Alice only varies her interaction profilef_iby tuning the coupling strengthsλ_iwithout changing the `shape' of the interaction profile, and this is sufficient to satisfy the perfect `SWAP' conditions (<ref>)-(<ref>) in Proposition <ref>.




§ ANALYTIC TESTS OF THE QUANTUM CHANNEL


In this section, we will prove that the protocol given in Section <ref> produces a perfect quantum channel in the correct limit without any approximations like Eq. (<ref>). This proves that entanglement can always be perfectly transmitted between two localized qubits through a quantum field regardless of the background spacetime geometry. We also discuss the case when Alice and Bob can in principle perfectly implement the SWAP gates with the field but are spacelike separated from each other. For this channel, the quantum and classical channel capacity are both zero, as we expect with relativistic causality. 




 §.§ Ideal case: perfect quantum channel


Recall that our task is to show that the channel coherent informationI_c(Φ)can be made arbitrarily close to 1, thus proving that the (relativistic) quantum channel can transmit quantum information perfectly. This means computing the coherent informationI_c(ρ_a,0,Φ)and maximizing over all possible input states on Alice's side. Using the definition of coherent information (<ref>), it turns out that it is sufficient to consider the case when Alice's initial state is the maximally mixed stateρ_A, 0 = 1/2 and Bob initial state is|+_y⟩state in order to show that we can attain maximum channel coherent informationI_c(Φ)→1and hence maximal quantum capacity. We stress that this does not mean Alice should send a maximally mixed state to Bob, which is “garbage” as a message and is even classical, since in practice Alice wants to be able to send pure states properly. Also, in general, it is not true thatI_c(Φ)is maximized by the maximally mixed state: for example, the completely depolarizing channelΦ(ρ) = /2hasI_c(/2,Φ) = -1but the channel coherent informationI_c(Φ)=0is attained by any pure state.


In order to show thatI_c(Φ)→1is attainable, we need to introduce a qubit of the environmentEthat purifies Alice's maximally mixed state/2, such that the joint state of Alice and the environment is the maximally entangled state|ψ_ea⟩ = 1/√(2) (|-_z⟩ |+_z⟩ + |+_z⟩ |-_z⟩ )_ea. As discussed in Section <ref>, a perfect quantum channel will preserve all the entanglement between Alice and the environmentEAto Bob and the same environmentEB. To computeI_c (/2, Φ), we must evaluate the stateρ_ebin Eq. (<ref>). 

Writing out the expression forρ_ebin terms of the unitariesÛ_aandÛ_b, we obtain:

    ρ_eb=_Aϕ [Û_bÛ_a( ψ_ea⊗0⊗+_y ) Û_a^†Û_b^†].

The simplest way of continuing is to decompose the unitaries using projections onto the eigenvectors of the qubit operators. That is, forÛ_awe write

    Û_a   = ∑_x,z ∈{±}P̂_x P̂_z ⊗ e^i x ϕ̂(f_2) e^i z ϕ̂(f_1)
       = ∑_x,z ∈{±}P̂_x P̂_z ⊗ W(xEf_2) W(zEf_1)

whereP̂_iare the projectors onto the eigenstates ofσ̂_iand in the last line we used the Weyl algebra notation introduced in Eq. (<ref>). For Bob's unitary, we write

    Û_b = ∑_x,z ∈{±}P̂_z P̂_x ⊗ W(-zEf_1) W(-xEf_2)

where the projection operators are understood to act on Bob's system. We have written Bob's unitary as the inverse of Alice's, the imperfect case whereÛ_b ≠Û_a^-1is discussed in the following sections. We also expand the Bell state as|ψ_ea⟩ = 1/√(2) ∑_j |-j_z⟩ |j_z⟩withj ∈{ ±}to get the expression

    ρ_eb   = 1/2∑_j,k,x_i, z_iω(Ô) k_zP̂_z_1P̂_x_1P̂_x_4P̂_z_4j_z
       ×|-j_z⟩_e⟨-k_z|⊗P̂_z_3P̂_x_3|+_y⟩_b⟨+_y|P̂_x_2P̂_z_2

where we write ω(Ô) ≡⟨0|Ô|0|$⟩ and

    Ô   = W(-z_1Ef_1) W(-x_1Ef_2) W(x_2Ef_2) W(z_2Ef_1) 
       × W(-z_3Ef_1) W(-x_3Ef_2) W(x_4Ef_2) W(z_4Ef_1),

x_i stands for x_1, x_2, x_3, x_4, similarly for z_i, and all sums are taken over the set {±}. Provided that we can obtain an expression for ω(Ô), a computer algebra system can straightforwardly sum over the 2^10 terms in the sum. We start by redefining summation indices x_1 ↦ -x_1, z_1 ↦ -z_1, x_3 ↦ -x_3, and z_3 ↦ -z_3 such that  

    ρ_eb   = 1/2∑_j,k,x_i, z_iω(Ô) k_zP̂_-z_1P̂_-x_1P̂_x_4P̂_z_4j_z
       ×|-j_z⟩_e⟨-k_z|⊗P̂_-z_3P̂_-x_3|+_y⟩_b⟨+_y|P̂_x_2P̂_z_2

and Ô only contains terms without minus signs

    Ô   = W(z_1Ef_1) W(x_1Ef_2) W(x_2Ef_2) W(z_2Ef_1) 
       × W(z_3Ef_1) W(x_3Ef_2) W(x_4Ef_2) W(z_4Ef_1).

In <cit.>, 0Ô0 is evaluated using the Baker-Campbell-Hausdorff (BCH) formula and Wick's theorem. In this paper, we adopt a much simpler approach using the AQFT framework. It is straightforward (albeit somewhat tedious) to apply the third Weyl condition W(Ef) W(Eg) = e^- i/2E(f,g) W(E(f+g)) seven times to obtain

    Ô   = W(E((z_1+z_2+z_3+z_4)f_1+(x_1+x_2+x_3+x_4)f_2)) 
       × e^-i/2 (x_1+x_2)(z_1-z_2-z_3-z_4) E(f_1, f_2)
       × e^-i/2 (x_3+x_4)(z_1+z_2+z_3-z_4) E(f_1, f_2)

where only one Weyl operator remains multiplied by a phase factor. 

Before we evaluate the expectation value of the remaining Weyl operator, it will be convenient to consider the decomposition of a Wightman two-point function into its real (symmetric) and imaginary (anti-symmetric) parts, i.e.,

    𝖶(f,g)    = 1/2ω({ϕ̂(f),ϕ̂(g)})+1/2ω([ϕ̂(f),ϕ̂(g)])
       = 1/2𝖧(f,g) + i/2E(f,g) ,

where we used the CCR to obtain the second term and 𝖧(f,g) ω({ϕ̂(f),ϕ̂(g)}) ≡1/2μ(Ef,Eg) is the smeared Hadamard two-point distribution of the field (cf. Eq. (<ref>)). This decomposition is important because only the symmetric part 𝖧(f,g) is state-dependent, while E(f, g) is independent of the state. This is useful as it tells us which parts of our analysis depend on the field state. Furthermore, while we have assumed that the field starts in some vacuum state for simplicity, but the protocol in Section <ref> automatically extends to all quasi-free states such as thermal states <cit.>, squeezed vacuum state <cit.>, and with slight modifications to more general Gaussian states such as coherent states (see, e.g., <cit.>). 

We can now use the definition of quasifree state ω in Eq. (<ref>) for the vacuum state to evaluate the expectation value ω(Ô) in terms of Wightman and Hadamard functions giving us

    ω(Ô)    = e^-1/2 (z_1+z_2+z_3+z_4)^2 𝖶(f_1, f_1)-1/2 (x_1+x_2+x_3+x_4)^2 𝖶(f_2, f_2)
       × e^-1/2 (x_1+x_2+x_3+x_4)(z_1+z_2+z_3+z_4)𝖧(f_1, f_2)
       × e^-i/2 (x_1+x_2)(z_1-z_2-z_3-z_4) E(f_1, f_2)
       × e^-i/2 (x_3+x_4)(z_1+z_2+z_3-z_4) E(f_1, f_2) .

Note that this expression is valid automatically for any quasifree states such as squeezed vacuum state and (squeezed) thermal state <cit.>.  The density matrix in the ordered basis ρ_eb in the |+_z+_z⟩_be, |+_z-_z⟩_be, |-_z+_z⟩_be, |-_z-_z⟩_be now reads

    ρ_eb = (P_-     0     A     C
     
        0     P_+     X     B
    
        A^*     X^*     P_+     0
    
        C^*     B^*     0     P_-)

where

    P_±   = 1/4(1 ± e^-2 𝖶(f_1, f_1) sin(2E(f_1, f_2)) )
    
        X    = 1/4sin(2E(f_1, f_2)) ( e^- 2 𝖶(f_2, f_2) +  sin(2E(f_1, f_2)) )
    
        A    = - i/4 e^ -2 𝖶(f_1, f_1) cos(2E(f_1, f_2))
       ×( sin(2E(f_1, f_2)) -e^ -2W(f_2, f_2)sinh(4𝖶(f_2, f_1)) )
    
        B    = -i/4 e^ - 2 𝖶(f_1, f_1) cos(2E(f_1, f_2)) 
       ×(sin(2E(f_1, f_2)) + e^-2𝖶(f_2, f_2)cosh(4 𝖶(f_1, f_2)))
    
        C    = 1/4 e^ - 8 𝖶(f_1, f_1)sin(2E(f_1, f_2)) 
       ×( sin(2E(f_1, f_2)) - e^-2 𝖶(f_2, f_2)cosh(4𝖧(f_1, f_2)) ) .



Now that we have an analytic expression for ρ_eb, we can use our two conditions in Proposition <ref> and verify that the coherent information can be made arbitrarily close to 1. First we use the fine-tuning condition E(f_1, f_2) = π/4 which simplifies ρ_eb  considerably to

    ρ_eb = (P̃_-     0     0    C̃
     
        0    P̃_+    P̃_+     0
     
        0    P̃_+    P̃_+     0
    C̃    0    0    P̃_- )

where

    P̃_±   = 1/4(1 ± e^-2𝖶(f_2, f_2))
    C̃   = 1/4 e^-8 𝖶(f_1, f_1)(e^-2𝖶(f_2, f_2)cosh(4𝖧(f_1, f_2)) +1).


Next, we use the strong coupling condition E(f_1, f_2)^2 ≫𝖶(f_2, f_2). Recall that we are assuming for simplicity that on Alice's side, her interaction profile will be kept fixed except for the coupling strengths, thus only λ_1,λ_2 are adjustable parameters. In this case, the strong-coupling condition essentially means that λ_1≫λ_2 and hence we have the hierarchy of scales

        
    𝖶(f_1,f_1) ∼𝒪(λ^2_1)    ≫ E(f_1,f_2)^2∼𝒪(λ^2_1λ^2_2) 
    
            E(f_1,f_2)^2∼𝒪(λ^2_1λ^2_2)    ≫𝖶(f_2,f_2)∼𝒪(λ^2_2)
    𝖶(f_1,f_1)∼𝒪(λ^2_1)    ≫𝖧(f_1,f_2) ∼𝒪(λ_1λ_2)


With this hierarchy, we can express λ_1 in units of λ_2 so that  λ_1 cλ_2 for some c≥ 0, which is also convenient since in general λ_i has physical units[The unit of λ_i is [Length]^n-3/2, and we have assumed for simplicity that λ_i have the same units since f_i are chosen to be (cf. the end of Section <ref>)]. We can thus write


    𝖶(f_1,f_1)    = c^2λ_2^2 𝗐_1 ,  𝖶(f_2,f_2) = λ_2^2 𝗐_2 ,
    
        E(f_1,f_2)    = cλ^2_2𝖾 ,    𝖧(f_1,f_2) = cλ_2^2𝗁 ,


where 𝗐_1,𝗐_2,𝖾,𝗁 are fixed by the spacetime smearings. Proposition <ref> then translates to

    λ_2^2(c^2𝖾^2)≫𝗐_2 ,  λ_2^2(c𝖾) = π/4 2π .

Note that the fine-tuning condition can still be satisfied: for example, for a fixed coupling λ_2 we find c such that 

    λ_2^2(c𝖾) = π/4 2π ,

and we can always ensure that c is sufficiently large so that

    λ^2_2(c^2𝖾^2) = c·𝖾π/4+2π n≫𝗐_2

in order to satisfy both conditions in (<ref>). 

In terms of the rescaled coupling constants, the strong-coupling condition implies that for fixed λ_2 and c≫ 1, we get

    ρ_eb(P̃_-     0     0     0
     
        0    P̃_+    P̃_+     0
     
        0    P̃_+    P̃_+     0
    
        0     0    0    P̃_- ) .

Using negativity 𝒩 as an entanglement measure <cit.>, 

    𝒩[ρ_eb] = ||ρ_eb^Γ_e||_1-1/2 = 1/2e^-2λ_2^2𝗐_2

where ρ_eb^Γ_e is the partial transpose of ρ_eb and ||·||_1 is the trace norm. This means that ρ_eb is always an entangled state for any finite coupling λ_2≪λ_1 and the channel is able to preserve some quantum entanglement. 

Finally, we obtain the perfect quantum channel in the idealized limit where c→∞ and λ_2→ 0 such that λ_1 is finite and still satisfies (<ref>), i.e., 

    ρ_eb = 1/2[ 0 0 0 0; 0 1 1 0; 0 1 1 0; 0 0 0 0 ] ,

i.e., a Bell state with maximum negativity 𝒩[ρ_eb] = 1/2. Therefore, in this limit the coherent information I_c(Φ)→ 1, which corresponds to a perfect quantum channel that can transmit a maximally entangled state perfectly. 




 §.§ Alice and Bob are spacelike separated



The preceding analysis shows that the ability to implement the approximate `SWAP' gate depends on how closely one is able to satisfy the fine-tuning and strong-coupling conditions in Proposition <ref>. While this is straightforward for Alice, Proposition <ref> tells us that for Bob to emulate ϕ̂(f_i) defined at some (global coordinate) time t(τ_a),  Bob's spacetime smearing functions g_i at time t(τ_b) has to be somewhat delocalized over some region of spacetime to catch all the signals from Alice. Since the quantum channel Φ is built out of a relativistic quantum field, we ought to be able to show that there can be no transmission of both classical and quantum information through Φ.

Formally, we start from the CCR which tells us that 

    E(f_i,g_j) = 0    ∀ i,j=1,2

when Alice and Bob are spacelike-separated, i.e., (g_i)∩𝒥^±((f_i)) = ∅. Note that the Hadamard function 𝖧(f_i,g_j) is generically nonzero for spacelike-separated regions due to entanglement contained in the field state <cit.>. The calculations proceeds similarly as in the previous section, except that Bob will have his own arbitrary smearing functions and their respective UDW interaction unitaries are



    Û_A   = ∑_x,z ∈{±}P̂_x P̂_z ⊗ W(xEf_2) W(zEf_1)
    Û_b   = ∑_x,z ∈{±}P̂_z P̂_x ⊗ W(-zEg_1) W(-xEg_2).


The condition for spacelike separation in Eq. (<ref>) will be enforced later in the calculation once everything is written in terms of two-point functions. 

We can analyse this scenario very generally by allowing Alice and the environment to start in any arbitrary state ρ_ea instead of a maximally entangled state or purifications of any initial state ρ_a,0. The expression for the density matrix ρ_eb now reads


    ρ_eb   = ∑_x_i, z_iω(Ô)  _a[ P̂_x_4P̂_z_4ρ_aeP̂_-z_1P̂_-x_1 ]
       ⊗P̂_-z_3P̂_-x_3|+_y⟩⟨+_y|_bP̂_x_2P̂_z_2

where the projection operators act on Alice or Bob and Ô is now defined to be

    Ô   = W(z_1Ef_1) W(x_1Ef_2) W(x_2Eg_2) W(z_2Eg_1)  
       × W(z_3Eg_1) W(x_3Eg_2) W(x_4Ef_2) W(z_4Ef_1).

Applying the third Weyl relation and using Eq. (<ref>), this evaluates to 

    
    ω(Ô) 
           = 
        e^-i/2 (z_1 - z_4)(x_1 + x_4) E(f_1, f_2)  
        e^-i/2 (z_2 + z_3)(x_3 - x_2) E(g_1, g_2)
        e^-i/2 (z_2 + z_3)(z_1 - z_4) E(f_1, g_1)
       × 
        e^-i/2 (z_1 - z_4)(x_2 + x_3) E(f_1, g_2)
        e^-i/2 (z_2 + z_3)(x_1 - x_4) E(f_2, g_1)
        e^-i/2 (x_2 + x_3)(x_1 - x_4) E(f_2, g_2)
       ×  e^-1/2(
          (z_1+z_4) (z_1+z_4)  𝖶(f_1, f_1)
        + (x_1+x_4) (x_1+x_4)  𝖶(f_2, f_2)
        + (z_2+z_3) (z_2+z_3)  𝖶(g_1, g_1)
        + (x_2+x_3) (x_2+x_3)  𝖶(g_2, g_2) )
       × e^-1/2(
          (x_1+x_4) (z_1+z_4)  𝖧(f_1, f_2)
        + (z_2+z_3) (z_1+z_4)  𝖧(f_1, g_1)
        + (x_2+x_3) (z_1+z_4)  𝖧(f_1, g_2) )
       × e^-1/2(
          (z_2+z_3) (x_1+x_4)  𝖧(f_2, g_1)
        + (x_2+x_3) (x_1+x_4)  𝖧(f_2, g_2)
        + (x_2+x_3) (z_2+z_3)  𝖧(g_1, g_2) ) .

This is the most general form that ω(Ô) can take in terms of Alice and Bob's two-point functions. We now enforce spacelike-separation using Eq. (<ref>), giving us

    ω(Ô) 
           = 
        e^-i/2 (z_1 - z_4)(x_1 + x_4) E(f_1, f_2)  
        e^-i/2 (z_2 + z_3)(x_3 - x_2) E(g_1, g_2)
       ×  e^-1/2(
          (z_1+z_4) (z_1+z_4)  𝖶(f_1, f_1)
        + (x_1+x_4) (x_1+x_4)  𝖶(f_2, f_2)
        + (z_2+z_3) (z_2+z_3)  𝖶(g_1, g_1)
        + (x_2+x_3) (x_2+x_3)  𝖶(g_2, g_2) )
       × e^-1/2(
          (x_1+x_4) (z_1+z_4)  𝖧(f_1, f_2)
        + (z_2+z_3) (z_1+z_4)  𝖧(f_1, g_1)
        + (x_2+x_3) (z_1+z_4)  𝖧(f_1, g_2) )
       × e^-1/2(
          (z_2+z_3) (x_1+x_4)  𝖧(f_2, g_1)
        + (x_2+x_3) (x_1+x_4)  𝖧(f_2, g_2)
        + (x_2+x_3) (z_2+z_3)  𝖧(g_1, g_2) ) .


The density matrix ρ_eb can now be calculated 

    ρ_eb = ρ_e⊗(1/2    X
     X^*    1/2)

with ρ_e = _a [ρ_ea] and 

    X    = -i/2 e^-2 𝖶(g_1, g_1)
       [e^-2 𝖶(g_2, g_2)cosh(2𝖧(g_1, g_2)) + sin(2E(g_1, g_2))].


Observe that the density matrix ρ_eb depends only on the initial condition of the environment and Bob's local two point functions, i.e. not involving Alice's smearing functions at all, and is clearly separable across B and E. 
Consequently, the quantum channel preserves no entanglement at all from the joint system EA (viewing it as ⊗Φ). In fact, since we start from any arbitrary state ρ_ea, we have effectively shown that Φ is a PPT channel <cit.>, and hence also an entanglement-breaking channel (because PPT is equivalent to separability for two qubits), and these are known to fall under the class of anti-degradable channels with zero quantum capacity 𝒬[Φ] = 0  <cit.>. 

From an information-theoretic viewpoint, zero quantum capacity says nothing about the ability of the quantum channel to transmit classical information, since the classical channel capacity 𝒞[Φ] is an upper bound for quantum capacity, i.e., 𝒞[Φ] ≥𝒬[Φ]. Quantum channels can also transmit classical bits by, for instance, considering only diagonal density matrix

    ρ_a,0 = (α    0
     0     1-α) ,  α∈ [0,1] .

and indeed it has been shown that for certain delta-coupled UDW models,  the classical capacity 𝒞[Φ] can be computed exactly and is zero if Alice and Bob are spacelike <cit.>. Our quantum channel is more complicated as Alice and Bob each perform a sequence of two simple-generated unitaries, but it is not necessary for us to find the explicit formula for 𝒞[Φ] to show that it is zero. This follows directly from the fact that the density matrix in Eq. (<ref>) does not depend on ρ_a,0 at all, therefore the quantum channel carries no information whatsoever about Alice's initial state, hence 𝒞[Φ] = 0. 

One of the main features of a relativistic quantum channel is precisely that one should be able to show that it respects the causal structure of spacetime by construction, and this requires one to be able to show, for instance, that no information transmission is allowed in spacelike separation. In fact we can say more: our formalism does not require us to specify anything about the (violation of) strong Huygens' principle <cit.>, since everything is encoded in the causal propagator. For example, in (3+1)-dimensional Minkowski spacetime, Bob can be timelike-separated from Alice but for a massless scalar field, the causal propagator has zero support in the interior of the light cone (see, e.g., <cit.>). In this case, if Bob also coupling functions to the field  g_i are strictly in the timelike interior and  (g_i)∩𝒥^+( f_j) = ∅ for any i,j=1,2, then 𝒬[Φ]=𝒞[Φ]=0.




 §.§ Intermediate scenarios


In the two preceding subsections, we have analyzed two opposite regimes: the ideal case with Ef_i = Eg_i and the worst case with Alice and Bob spacelike-separated. Let us make some comments on intermediate regimes. 

When the coherent information of a quantum channel is not close to 1, e.g., when 0 ≤ I_c(Φ) ≪ 1, it is in general very difficult to make quantitative claims about the quantum channel capacity for the reasons like superadditivity described in Section <ref>. In fact, even for quantum channels with I_c(Φ) = 0, only small subsets of these are known to have zero quantum capacity, namely the anti-degradable channels <cit.> and the PPT channels <cit.> (see also <cit.>). For intermediate scenarios, such as when Bob is only partially covering  𝒥^±((f_i)) or implementing faulty SWAP gates we are not able to give generic quantitative statements about 𝒬[Φ], as is the case for generic quantum communication channels. 

However, in some situations we may try to use no-cloning type argument to conclude that 𝒬[Φ]=0 in some scenarios as was done in <cit.> (see also <cit.>). Suppose Bob is allowed to couple only to at most half of Alice's signal at some future time t that defines the Cauchy slice for Bob's interaction. Consider another observer Charlie who now chooses to couple to the other half of the region at the same time slice so that jointly the interaction support is the full region needed for the perfect quantum channel. Since Charlie's interaction is by construction spacelike separated from Bob, Charlie's interaction is invisible to Bob (and vice versa). If Bob and Charlie were able to individually decode Alice's qubit, this would violate the no-cloning theorem, so we can conclude that the quantum capacity must be zero if the UDW interactions are set up this way[It is worth emphasizing that the protocol we have (and also in <cit.>) are generically not ideal, since for any finite interaction strengths the `SWAP' gates are never perfect and the `logical qubit' in the field is only approximate — consequently one has to be careful about using exact cloning argument. In practice, it would be better to think of the scenario in terms of approximate cloning. Indeed, in Figure 8 of <cit.> the coherent information vanishes smoothly as one varies Bob's interaction support when analyzing the cloning-type argument.]. 

The above discussion is not necessarily the only (or natural) way to view this. One way to think about this from the perspective of the quantum field is this: if Bob manages to decode Alice's state at time t, the `logical qubit' state c_1|+α⟩-ic_2|-α⟩ in Eq. (<ref>) would have been taken out by Bob, and the field would now change into a new state that has no logical qubit in it (essentially because Bob swapped out the state into his physical detector). Charlie, who couples to the field at any time t+ϵ for any arbitrary ϵ>0, would no longer have any logical qubit to extract from the field, so Charlie cannot extract another qubit for free. Yet another way to see this is that Bob's interaction is designed to implement the inverse unitary Û_a^-1, and this requires one to find g_i in 𝒥^+((f_j)) that can undo Alice's operation. Therefore, if we choose to pick g_i that does not cover the full region needed for the decoding, then Bob fails to implement Û_a^-1 and instead performs a different unitary operation. In this interpretation, we would say that Bob is unable to perform the required `SWAP' gate to decode Alice's state. Showing that Bob has zero quantum capacity would mean showing that Bob's operations somehow completely depolarize the qubit state and lose all the quantum information in it — we will not pursue this further in this paper and leave this for future work.




§ DISCUSSION AND OUTLOOK



We motivated our work through the question: What is the information-carrying capacity of quantum fields? The UDW framework allows us to formulate a precise statement of this question, in terms of quantum capacity of the relativistic quantum channel between Alice and Bob who try to transmit quantum information between their qubit detectors through the quantum field.  In this work, we have successfully generalized the results of <cit.>, in particular <cit.>, where we construct a quantum channel that can transmit quantum information arbitrarily well through a relativistic quantum scalar field in arbitrary curved spacetimes. 

   

The features of our construction is that the resulting quantum channel, and hence the quantum capacity, are manifestly diffeomorphism-invariant, respect the causal structure of spacetime, and are independent of the details of the background geometry, topology, and the choice of Hilbert space (quasifree) representations of the CCR algebra. Furthermore, in this generalization we no longer need to deal with (violation of) the strong Huygens' principle that may occur in curved spacetimes for non-conformally invariant fields (see, e.g.,<cit.>), since this is fully encoded in the causal propagator. We also showed explicitly that the quantum and classical capacity is zero for spacelike separated observers, as we expect from quantum communication channel built from a relativistic quantum field.

Crucially, in this generalized formulation it becomes clear that there are two essential features of a near-perfect quantum channel that can transmit quantum information arbitrarily well: (1) Alice and Bob's interactions need to satisfy the so-called fine-tuning and strong coupling conditions in Proposition <ref> in order to approximate an ideal `SWAP' gates; (2) Bob needs to couple to the field at appropriate spacetime regions, fixed by Proposition <ref>, so that he could execute the approximately ideal `SWAP' gates. Indeed, where Bob should be is fixed by the causal propagator that determines where Alice's information propagates in spacetime. The asymmetry between Alice and Bob can be viewed as the consequence of the asymmetry between encoding the qubit state into the field, with the logical basis being the coherent states |±α⟩, and decoding the state while taking into account relativistic causality. In this sense, our construction constitutes the most general relativistic qubit channel within the UDW framework that can transmit quantum information arbitrarily well in appropriate regimes and we also provided analyses when the quantum channel is sub-optimal.


There are several future directions that merit investigation, at least from a more fundamental standpoint:

    
  * Is it possible to construct a perfect relativistic quantum channel using higher-dimensional detector models (using qudit detector models <cit.> rather than qubits)? A priori it is not obvious how to construct logical qudits in the field using only controlled displacement operators and what kind of interaction Hamiltonians are needed to do this, if it exists at all. It is also natural to consider several emitters and receivers, exploiting the fact that one can interpret displacement operations can be viewed as Weyl generators with respect to some spacetime smearing functions <cit.>.

    
  * As mentioned in <cit.>, one may be interested in more directional coupling, which can be envisaged in (1+1)D models (see, e.g., <cit.>). Our protocol works automatically simply because only the two-point functions change to reflect that the left- and right- movers of the field modes are independent. A more interesting setup would be to study how realistic settings involving waveguides and non-trivial dispersion relations can be adapted for this purpose. 

    
  * Is the information-carrying capacity different between different fields with different spins and statistics? For example, how does one compare the scalar field with a fermionic field? The issue one has to deal with is that we have to separate the effects coming from the model dependence of the coupling (e.g., we cannot couple σ̂_x⊗ψ̂ where ψ is the fermionic field) from truly field-theoretic physics that contribute to the quantum capacity, if we wish to do it in the same way as we have done in this paper (see, e.g., some proposed fermionic couplings in RQI literature <cit.>).

    
  * Our construction relies on the fact that the quantum field is non-interacting (Gaussian). There are very few studies involving the UDW model with interacting QFTs such as the ϕ^4 theory, even perturbatively and for a single qubit (see, e.g., <cit.> for one example). The effects of interactions are unclear for (long-distance) quantum communication, both perturbatively and non-perturbatively. A possibly relevant testbed would be to consider known rigorous constructions in (1+1)D interacting QFTs (see, e.g., <cit.>). 

    
  * Is it possible to systematically study quantum capacity in the non-relativistic quantum many-body settings? Somewhat related studies to our setup are, for example, <cit.>. It would be interesting to see the similarities and differences when they are compared with setups involving relativistic quantum fields.

We leave these problems open for the future.




§ ACKNOWLEDGMENT



M. K. acknowledges support from the Princeton International Internship Program (IIP). E. T. acknowledges funding from the Munich Center for Quantum Science and Technology (MCQST), funded by the Deutsche Forschungsgemeinschaft (DFG) under Germany’s Excellence Strategy (EXC2111 - 390814868). 


apsrev4-2


 Paper Title: Comparative of light propagation in Born-Infeld, Euler-Heisenberg and ModMax nonlinear electrodynamics.


Fontaine-Laffaille Theory over Power Series Rings
    Christian Hokaj
    August 5, 2024
=================================================





§ INTRODUCTION


Nonlinear electromagnetic effects become significant when the electromagnetic field strengths approach the critical fields E_cr≈ m^2_ec^3/(eħ)≈ 10^18 Volt/m or B_cr≈ 10^9 Tesla; B_cr represents the field at which the cyclotron energy equals m_ec^2, where m_e is the mass of the electron, and it defines the field scale at which the external field becomes significant on quantum processes; it is also known as Schwinger critical field and it marks out the threshold of nonlinear QED scale <cit.>. Moreover, anyone familiar with quantum field theory is led to the  conclusion that a classical phenomenological theory that pretends a description of polarization effects (light-light interaction or pair production in vacuum excited by an electric field) has to be nonlinear.

The nonlinear electromagnetic (NLEM) interactions can be described in a classical way using an effective Lagrangian that  depends nonlinearly on the two Lorentz and gauge invariants, F and G, of the Faraday tensor F_μλ=∂_μA_λ-∂_λA_μ,


    F=F^μλF_μλ= 2(B^2-E^2),    G=F^*μλF_μλ= -4 B⃗·E⃗,
 
where F^*μλ=1/2ϵ^μλαβF_αβ.  

Maxwell's electrodynamics cannot describe nonlinear interactions between two electromagnetic waves, for example, the interaction light-by-light; these phenomena are well known in QED, and considering classical nonlinear electrodynamics (NLED) it is possible to describe them <cit.>. 
Several NLED proposals exist to describe such nonlinear effects, but three of them stand out:  Born-Infeld (BI), Euler-Heisenberg (EH), and ModMax theories. 

  
In 1934, Born and Infeld <cit.> presented a theory with nonlinear corrections to Maxwell electrodynamics with the aim of solving the singularity of the electric field self-energy of charged particles;  they propose the existence of a maximum attainable electromagnetic field, denoted by the BI parameter b, with a magnitude b= e / r_0^2 = 10^20 Volt/m, where r_0 is the classical electron radius. The interpretation of this absolute field is also that the classical self-energy of the electron is equal to its mass-energy at rest <cit.>. 
The Lagrangian for this NLED is established as 

    L_BI=-4b^2 (1-√(1+F/2b^2-G^2/16b^4)).


This classical theory effectively models vacuum polarization as a material medium.  
From the viewpoint of QED, the BI modifications of the Maxwell theory result in a change in the photon-photon scattering signal. This is the reason why it has been proposed that experiments that measure the QED process could constrain the maximum field b of the BI theory <cit.>. Another interesting feature is that it presents neither birefringence nor shock waves; the absence of birefringence leads to the idea that vacuum birefringence experiments could set bounds on the BI parameter b  <cit.>; also experimental constrains have been proposed through interferometry <cit.>.
It is also worth noting that there is a connection of BI-NLED with string theory; in bosonic open string theory
the BI action arises as an effective low-energy action <cit.>.


On the other hand, the EH theory was derived from QED principles by W. Heisenberg and H. Euler in 1936 <cit.>  from the perspective of QED  <cit.>. It considers vacuum polarization to one loop and is valid for electromagnetic fields that change slowly compared to the inverse of the electron mass.  By treating the vacuum as a medium, EH effective action predicts rates of nonlinear light interaction processes.
The Lagrangian for EH NLED is 

    L_EH=-F/4+α^2/90 m_e^4(F^2+7/4G^2),

where α is the fine structure constant and m_e the electron mass.

An interesting aspect of NLED are the symmetries that are preserved.
The BI NLED is not conformally invariant while the EH NLED is neither conformal invariant or dual invariant. The question then arose if a NLED fulfilling the same symmetries as Maxwell's existed; i.e. a NLED with conformal and electric-magnetic duality invariances. It indeed exists. Recently  was derived a NLED theory endowed with the two symmetries <cit.>.  It is characterized by a dimensionless parameter γ and it reduces to Maxwell theory if γ=0. This theory, known as Modified Maxwell (ModMax, MM) NLED, has stimulated research in several aspects, from classical solutions <cit.> to supersymmetric analysis <cit.>. The Lagrangian for the ModMax NLED is 

    L_MM=-F/4coshγ +sinhγ/4√(F^2+G^2).
    

In this work, we discuss the phase velocities of light moving in the three previously described NLEDs; although the parameter of each NLED is different we would like to know which one is the most effective in slowing down light velocity in a magnetic or electric background. Additionally, 
there is not a criterion to constrain the values of the ModMax parameter γ, apart from being positive, as energy conditions of a MM black hole demand.   We use BI and EH theories to set bounds on γ through the comparison of the phase velocities of light propagating in a magnetic (or electric) background.  In a second part of the paper, regarding NLED coupled to gravity, we compare the shadows of the respective black hole solutions; confronting their shadows with the observational evidences from Sagittarius A^* and the M87^* galaxy
allows to delimit the ranges of the nonlinear  parameters (μ for EH, b for BI, and γ for ModMax) that would fit with charged astrophysical supermassive black holes.


The paper is organized as follows: in Section <ref> it is presented the effective or optical metric, that is the background metric modified by the nonlinear electromagnetic effects; the null geodesics of the effective metric are the actual trajectories of light propagation; then the phase velocity general expression is derived. In Section <ref>, the effective metric and phase velocity of light propagation in an intense magnetic  background are given for each one of the NLEDs under study. Through phase velocities for the BI and EH NLEDs we set bounds on the  ModMax nonlinear parameter γ.
In Section <ref>, we consider the static spherically black hole solutions of the three analyzed NLEDs and calculate the  corresponding shadows that are then confronted versus the Sagittarius A^* and the M87 galaxy observed shadows. These tests set bounds on the possibility of a charged supermassive black hole. Finally, the conclusions are given in the last section.





§ NLED EFFECTIVE METRIC AND PHASE VELOCITY



It is well known that intense EM fields, where the Maxwell theory is no longer valid, can resemble a curved spacetime, in the sense that light trajectories are not straight lines but undergo deflection. Deviations from the straight trajectories in vacuum are described in NLED by the null trajectories of an effective or optical metric. The concept of effective metric in nonlinear electrodynamics has its origin in the works by  G. Boillat <cit.>, Bialynicki-Birula <cit.>,   Plebański <cit.> and more recently by Novello <cit.>. In this work, we take the approach developed by Plebański and later on by Novello and collaborators. 

The equations of the propagation of the field discontinuities in nonlinear electrodynamics by a Lagrangian L(F, G)  can be derived by analyzing the propagation of linear waves associated with the discontinuity of the field, f_αβ,  in the limit of geometrical optics <cit.>.

 Considering a null vector k_μ=(ω,k⃗) that is normal to the characteristic surfaces or wavefronts, the effective  metric g_ eff^μν must be the metric in which the wave vector is null, k^μk_μ=0,


    g_ eff^(a) μνk_μk_ν=0,    a=1,2,

where the (a) superscript corresponds to the two metrics that can arise in NLED when birefringence occurs.
The propagation vector can be expressed as  



    k_(a)^2= - λ_aF^τμF^ν_τk_μk_ν,   a=1,2,

where  λ_a can be interpreted as the  birefringence or refractive index of a material medium and is  given in terms of the NLED Lagrangian and its derivatives by,


    λ_a=-4L_FF+Ω_(a)L_FG/L_F+G(L_FG+Ω_(a) L_GG),   a=1,2,

where  L_X=d L/dX, X=F, G, and Ω_(a) in Eq. (<ref>), depends on the derivatives of the Lagrangian with respect to the invariants <cit.>.  The expression for the effective metric can be identified from the dispersion relation, Eq. (<ref>) with Eq. (<ref>) is, 


    { g^μν + λ_at^μν} k_μk_ν= 
     g_ eff  (a)^μνk_μk_ν=0,   a=1,2,

where t^μν=F^μλF_λ^ν, is defined by the particular (intensive) fields F_μν of the system, and g^μν  is the background metric that can be the Minkowski metric or a general curved spacetime, for instance, a black hole or a cosmological spacetime. 
Then  the effective metrics for general nonlinear electrodynamics with Lagrangian L(F,G) are given by,

    g_ eff (a)^μν =  g^μν+ λ_a t^μν,    a=1,2.




If the Lagrangian is such that L_FG=0, the effective metrics become


    g_ eff (1)^μν   =   (L_F-2L_GG F )g^μν-4L_GGt^μν, 
    
    g_ eff (2)^μν   =     L_Fg^μν-4L_FFt^μν,


Additionally, when we turn off either the electric or the magnetic field, i.e., E⃗=0 or B⃗=0, then G=4E⃗·B⃗=0, and the effective metrics become


    g_ eff (1)^μν   =     L_F g^μν, 
    
    g_ eff (2)^μν   =     L_Fg^μν-4L_FFt^μν,

one of  the effective metrics, g_ eff (1), is conformal to the background metric; in this case, one of the optical paths is the null geodesic of the background metric and the other one is the null geodesic of the effective metric g_ eff (2). 

In this work, we analyze three different NLEDs: BI, EH and ModMax, and here we present their respective effective metrics, as listed below. 

Note that the BI effective metric does not present birefringence, then there is only one,

    g_ eff, BI^μν=(b^2+F/2) g^μν+F^μ_λF^λν.


On the other hand, the EH NLED has two effective metrics, given by Eqs. (<ref>) and (<ref>), explicitly, 


    g_ eff(1), EH^μν = - 1/4( 1 + μ F ( 5/2 F^2 - 7/4 G^2)/(F^2 + 7/4 G^2)^3/2) g^μν
     - 7/4μ F^2/(F^2 + 7/4 G^2)^3/2 F^μ_λF^λν,

and


    g_ eff(2), EH^μν= - 1/4( 1 - μ F /(F^2 + 7/4 G^2)^1/2) g^μν
     - 7/4μ G^2/(F^2 + 7/4 G^2)^3/2 F^μ_λF^λν,

where  we  denote μ = 2 α^2/(45 m_e^4).

Finally, ModMax NLED does present birefringence but one of the two effective metrics is conformal to the background metric; such that the effective metrics are given by


    g_ eff(1), MM^μν=g^μν+4tanhγ/√(F^2+G^2)+Ftanhγt^μν,



    g_ eff(2), MM^μν=g^μν.

Note that the electromagnetic tensor F_μν and its invariants that appear in the previous equations correspond to the intense electromagnetic background field that the light wave traverses with the phase velocity derived from the corresponding g_ eff^μν.
In the Maxwell case  L=-F/4,   L_F=-1/4,    L_FF= 0 and L_G= 0, then both effective metrics become conformal to the background metric, g_ eff^(1) μν = g_ eff^(2) μν= g^μν, and the null geodesics coincide with the light trajectories in the background spacetime.

See <cit.> for a study on the Fresnel equation in nonlinear electrodynamics and <cit.> for a classification of the effective metrics.




 §.§ Phase velocity of light propagation in the Effective Metric


Here we explain how the phase velocity of light can be calculated
from the dispersion relation.  Let us consider a  null vector k_μ={ω, k⃗}. If k_i is the light propagation  direction, from the dispersion relation,  Eq. (<ref>), 


    g_ eff(a)^ttω^2+2g_ eff(a)^itω k_i+g_ eff(a)^ij k_i k_j=0,

where i,j subscripts (or superscripts) denote the spatial coordinates and t denotes the time coordinate. Defining the normalized wave vector k̃_i=k_i/|k⃗|, Eq. (<ref>) can be written  as 


    g_ eff(a)^ttω^2/|k⃗|^2+2g_ eff(a)^itω/|k⃗|k̃_i+g_ eff(a)^ijk̃_ik̃_j =0.


Then the light's phase velocity,  v=ω/|k⃗|, for propagation in the direction k̃_i,  v^i is given by 

    (v^i)_a= ω/|k⃗|k̃_i=g_ eff(a)^itk̃_i/g_ eff(a)^tt±√((g_ eff(a)^itk̃_i/g_ eff(a)^tt)^2-g_ eff(a)^ijk̃_ik̃_j/g_ eff(a)^tt),      a=1,2.


The two possible effective metrics g_ eff(a)^μν, a=1,2, render two
dispersion relations that correspond to two modes of polarization, this is known as the birefringence effect <cit.>. Note that these velocities differ from the phase velocities of a massless test particle that follows the null geodesics of the background metric g_μν and  is given by analogous expressions making g_ eff(a)^μν↦ g^μν, 


    v_ massless = ±√(-g^ijk̃_ik̃_j/g^tt).




§ COMPARATIVE OF THREE NONLINEAR ELECTRODYNAMICS: LIGHT PROPAGATION



In this Section, our goal is to study light propagation in the three NLED theories and to evaluate how strong electromagnetic fields impact its phase velocity. We will compare the phase velocities of light propagating in a magnetic  background in a Minkowski spacetime, and then in an electric background. 



 §.§ NLED phase velocities in a magnetic background 

We examine the phase velocities of light propagating in the z direction, while the magnetic background field is oriented making an angle with the propagation direction; we consider the magnetic field with one component along the propagation direction and a second component as transversal or perpendicular,  B⃗=  B_z ẑ + B_xx̂ + B_yŷ =B_z ẑ + B_⊥ (x̂ + ŷ). We calculate the phase velocities for the case of a static observer or as measured in the Laboratory.
The phase velocities corresponding to the three NLEDs are derived from Eq. (<ref>),  considering a Minkowski background,
g^μν= η^μν =  diag [-1,1,1,1]; while the uniform magnetic field components are F^xy=-B_z,   F^xz=B_y,    F^yz=-B_x, and the invariants, F= 2B^2,    G=0.

For the EH NLED, the expressions for the phase velocities are

    v_EH1^2=1-14 μ  B_⊥^2/10 B^2 μ +1,    v_EH2^2=1-8 μ  B_⊥^2/1-4 B^2 μ.

Considering terms up to μ ( α^2), the EH phase velocities are given by,


    v_EH1,B^2=1- 14 μ B_⊥^2,     v_EH2,B^2=1- 8μ B_⊥^2.

And performing an average over polarization and direction,

    v_EH,B^2=1- 11 α/270 π( B_⊥/B_cr)^2,

where we substituted μ in terms of the critical field  μ= α/(90 π B_cr^2); that gives a light velocity decreasing respect the one in vacuum c=1 of

    v_EH,B^2=1- (9.5 × 10^-5) ( B_⊥/B_cr)^2.



For  the BI NLED, there is no birefringence, and the phase velocity is 

    v_BI^2=1-B_⊥^2/b^2+B^2,

with b being the maximum attainable electric or magnetic field.
Up to first order in 1/b^2, neglecting terms 1/b^4 and higher, 


    v_BI,B^2=1-B_⊥^2/b^2.

Such that light velocity is reduced the most, for fields with a component perpendicular to the light propagation (B_⊥) of intensities of the order of b= 10^11 Tesla.

For the ModMax NLED, there is birefringence and one of the light trajectories is 
the one in vaccum, with phase velocity v^2=c^2=1. While the other one is

    v_MM^2=1-2 B_⊥^2/B^2( tanh(γ)/1 + tanh(γ )).


The phase velocities as  functions of the corresponding adimensional parameters: μ B^2 for EH; B^2/b^2 for BI are illustrated in Fig. <ref>. For the ModMax NLED the phase velocity depends on B_⊥/B; in Fig. <ref>  is shown how the phase velocity decreases in ModMax case for B_⊥/B =0.1 .    The differences in the decreasing of the phase velocities  is qualitatively the same for BI and EH, while  the Mod Max case is illustrated as a function of B_⊥/B in Fig. <ref>. 





 §.§ NLED phase velocities in an electric background 

The phase velocities when considering an electric background field in a Minkowski spacetime are presented. The setting is analogous to the one for the magnetic background: light propagating in the ẑ direction and the electric background of the form E⃗= E_z ẑ + E_⊥ (x̂ + ŷ). 

The two possible phase velocities for the EH NLED are given by 

    v_EH1,E^2=1-14 E_⊥^2 μ/4 E^2 μ +1,    v_EH2,E^2=1-8 E_⊥^2 μ/12 E^2 μ +1.

For the BI NLED, since there is no birefringence, the phase velocity reads

    v_BI,E^2=1-E_⊥^2/b^2.

While  phase velocity in a ModMax NLED is 

    v_MM,E^2=1-2 E_⊥^2/E^2( tanh(γ)/1 + tanh(γ )).

The behavior at the order μ E^2, E^2/b^2∼ 10^-2 is very similar the one for the magnetic background field. Moreover, the phase velocities for the wave in the presence of ModMax magnetic and electric fields in Eqs. (<ref>) and (<ref>) have the same dependence on the magnitude of the fields, a consequence of the duality invariance required by ModMax theory.  




 §.§ Bounds on the ModMax parameter γ from BI and EH


The phase velocity of light propagation in a strong magnetic field background obtained from the ModMax theory is given by,

    v_MM^2=1-2 B_⊥^2/B^2 ( tanh (γ )/1 + tanh (γ )).

As a consequence that there are no bounds for the allowed values of γ, and that for large values of γ, tanh(γ) ↦ 1, in the expression for the phase velocity, the factor with γ tends to 1/2 and then v_MM can be made arbitrarily small, even for small values of the magnetic field if the component that is perpendicular to the wave propagation is of the order of the total magnetic field, B_⊥≈ B. 
Since EH describes the nonlinear electromagnetic effect of vacuum polarization and BI the effect of photon splitting, the phase velocity of light propagation derived from these effects compared with the one from the ModMax theory  can give  insight of the ranges of γ. For this reason we search for bounds on γ from the phase velocities derived for BI and EH.



  §.§.§ Bounds on the NLED ModMax parameter  from BI


Comparing the phase velocity for light propagation that arises in the BI theory  in a magnetic background, Eq. (<ref>) with the phase velocity  from the ModMax theory, Eq. (<ref>),  it is obtained for the parameter γ


    tanhγ = B^2/2 b^2- B^2≈B^2/2 b^2,

if we consider a magnetic field background of the order of the maximum attainable magnetic field b, i.e. B/b ≈ 1, we obtain the bound 
γ= 0.549306. While if  B is a tenth of b, B/b ≈ 0.1 then the bound on the ModMax parameter is γ= 0.005. Smaller magnetic fields give smaller bounds for γ.
Then BI constrains γ to the interval 0 < γ≤ 0.549306.





  §.§.§ Bounds on the NLED ModMax parameter  from EH


The two phase velocities for light propagation in a strong magnetic background that arise in the EH theory, Eqs. (<ref>), are compared with the MM phase velocity.  From the first one, v_EH1,B  compared with the phase velocity for the ModMax theory, Eq. (<ref>), we obtain for the parameter γ the constraint


    tanhγ = 7 α/90 π(B/B_cr)^2 = 1.8 × 10^-4(B/B_cr)^2,

Such that for fields near the critical one, B ≈ B_cr
the nonlinear MM parameter is γ = 1.8 × 10^-4.

Following the same procedure for the second velocity  v_EH2,B in Eqs.  (<ref>) we obtain a second bound of γ = 10^-4,
for fields B ≈ B_cr = 10^9 Tesla.
Such that the EH theory sets 
γ in the interval 0.0001 < γ < 0.00018.




Therefore the  EH theory delimits  the nonlinear ModMax parameter to the interval 0.0001 < γ < 0.00018, while BI places  γ in a wider interval, 0 ≤γ≤ 0.549306. We should take into account that EH and BI apply to different nonlinear electromagnetic effects that arise at different orders of magnitude of the background fields
( B_cr≈ 10^9 Tesla for EH and b ≈ 10^11 Tesla for BI), 
but both serve to the purpose of delimiting the value of γ, i.e. the effect of the ModMax electrodynamics in light propagation 
should resemble in a certain way the ones of the BI or EH electrodynamics. The BI constrain 0 < γ < 0.549306  includes the constraints derived from EH,  therefore the greater field intensity allows a larger γ in the ModMax theory.




§ COMPARATIVE OF THREE NONLINEAR ELECTRODYNAMICS: BLACK HOLE SHADOWS



It is well known that the coupling of Nonlinear Electrodynamics (NLED) with Einstein's gravity can yield black hole solutions. In this section, we focus on the static black holes corresponding to the three NLEDs discussed above and we compare the metric functions for these BHs with the Reissner-Nordstrom one, that is the linear limit; in the comparison is also included the Schwarzschild BH as a reference. Additionally, we compare the resulting shadow radii  with the observed shadows of Sagittarius A^* and M87^* black holes <cit.>. 



 §.§  NLED BH Metric functions

We consider the corresponding BH metrics as the background metrics. It is interesting to compare the metric functions for the BHs of the three NLEDs in consideration with the one of  Reissner-Nordstrom, the static spherically symmetric solution of the coupled Einstein-Maxwell equations. For the static spherically symmetric line element, that is the background metric g_μν,

    ds^2= - f(r) dt^2 + dr^2/f(r) + r^2 d Ω,

the corresponding metric functions f(r) for Schwarzschild (Schw), Reissner-Nordstrom (RN), Euler-Heisenberg (EH),  Born-Infeld (BI), and ModMax (MM), are, respectively, 

    f_Schw.(r)    =    1-2M/r,
    
    f_RN(r)    =    1-2 M/r+Q^2/r^2, 
    
    f_EH(r)    =    1-2 M/r+Q^2/r^2 - μ  Q^4/20 r^6,


    f_BI(r) = 1-2 M/r+2/3 b^2 r^2 (1-√(Q^2/b^2 r^4+1))+2 Q^2 /3r√(b/Q) F[ ArcCos(b r^2/Q-1/b r^2/Q+1),1/√(2)],


    f_MM(r) = 1-2 M/r+e^- γ Q^2/r^2.
 

In Fig. <ref>, we plot the metric functions for the Schwarzschild, RN, EH, BI and ModMax black holes, for a fixed  BH charge Q=0.9. In the BI case  for b>0.7, f_BI(r) behaves similarly near the origin than the one for RN, while if b<0.7, the metric function f_BI(r) resembles the  Schwarzschild behavior near r=0 <cit.>.





 §.§ The shadow of the NLE black holes


Considering the equations of motion of a test particle in the external field of a static BH, the radial motion is governed by

    ṙ^2 
       + L^2/g_rr( g_ϕϕ/r^4
       +g_tt/f^2 (r)1/b^2 
       - δ/L^2)
       = 0,

where δ= 1,0,-1 for space-like, null, or time-like geodesics, respectively; the test particle angular momentum is L,  and  the impact parameter b= L/E, where  E is the particle energy.
In NLED photons travel through null geodesics of the effective metric, then  in Eq. (<ref>) δ=0 and g ↦ g^ eff(a), such that the radial motion of the light ray obeys,

    ṙ^2 
       + L^2/g^ eff(a)_rr( g^ eff(a)_ϕϕ/r^4
       +g^ eff(a)_tt/f^2 (r)1/b^2)
       = 0,    a=1,2,

where a=1,2 denote the two effective metrics, given by  Eq. (<ref>) for BI, Eqs. (<ref>)- (<ref>) for EH, and Eqs.   (<ref>)- (<ref>) for ModMax. The electromagnetic tensor components  F_μν for the construction of the effective metric  correspond to the ones of  a charged static BH, that in case of both charges, electric Q_e and magnetic Q_m  are given by 


    F^tr = Q_e/r^2 L_F,    F_θϕ = Q_m sinθ,

we shall consider only electric charged BHs, Q_e=Q.  

From 
ṙ^2 + V^(a)_ eff =0, we identify the effective potential as 

    V^(a)_ eff=L^2/g^ eff(a)_rr( g^ eff(a)_ϕϕ/r^4
       +g^ eff(a)_tt/f^2 (r)1/b_a^2),    a=1,2.

The radius of the  circular orbits r_c 
corresponds to an extreme of the effective potential,

    V^(a)_ eff |_r_c =0,   dV^(a)_ eff/dr |_r_c  =0,   a=1,2.



The critical impact parameters can be obtained in terms of the effective metrics as 

    (b^2_c)_a= ( -r_c^4 g_tt^ eff (a)/f^2 (r_c)g_ϕϕ^ eff (a))_r_c,   a=1,2.

The NLED theories that present birefringence have two values for the critical impact parameter, corresponding to the two effective metrics;  for the NLED-BHs of our study, EH, BI and ModMax, respectively, are


    b_c,EH1^2=r_c^2/f_EH(r_c)( 1-10μQ^2/r_c^4/1+4μQ^2/r_c^4),    b_c,EH2^2=r_c^2/f_EH(r_c)( 1+4μQ^2/r_c^4/1+12μQ^2/r_c^4),



    b_c,BI^2=r_c^2/f_BI(r_c)( 1-Q^2/b^2 r_c^4),



    b_c,MM1^2= e^-2γr_c^2/f_MM(r_c),    b_c,MM2^2= r_c^2/f_MM(r_c).


We calculate the radii of the shadows for the three NLED-BHs and compare them with the RN BH. The shadow can be determined by considering a light ray from the observer position r_o to the unstable circular orbit (UCO) making an angle ψ_sh with respect to the radial direction. The expression for the deviation angle associated with the shadow is <cit.>

    sin^2(ψ_sh)_a=-b^2_ca(g_ϕϕ^ eff (a)/g_tt^ eff (a)f^2 (r_o)/r_o^4)_r_o,    a=1,2.

where  b_c is the impact parameter of the UCO with radius r_c.

The NLED theories that present birefringence have two values for the shadow radius  r_sh, corresponding to the two effective metrics. The radius of the BH shadow is approximated in terms of the observer position r_o and the angle ψ_sh as <cit.>

    r_sh=r_otanψ_sh≈ r_osinψ_sh,

then, we obtain that for the EH BH there are two radii, then two shadows, 

    r_sh,EH1^2=r_c^2 (f_EH(r_o)/f_EH(r_c)) ( r_c^4 -10 Q^2 μ/r_c^4 +4Q^2 μ) ( r_o^4 +4 Q^2 μ/r_o^4 -10Q^2 μ),



    r_sh,EH2^2=r_c^2 (f_EH(r_o)/f_EH(r_c)) (r_c^4 +4 Q^2 μ/r_c^4 +12Q^2 μ) ( r_o^4 +12 Q^2 μ/r_o^4 +4 Q^2 μ).

The static BI BH has one shadow radius, 

    r_sh,BI^2 = r_c^2 ( f_BI(r_o)/f_BI(r_c)) ( 1 - Q^2/b^2 r_c^4)/(1-  Q^2/ b^2 r_o^4).

And although there are two critical impact parameters for the ModMax BH, an observer located at infinity observes only one shadow,

    r_sh,MM^2= r_c^2 ( f_MM (r_o)/f_MM (r_c)).

Considering that the observer is at infinity r_o↦∞, then f(r_o ↦∞)=1 and at first order in the nonlinear parameters, the radii of the shadows are, for EH BH up to first order in μ,


    r_sh,EH1^2= (r_c^2/f_EH(r_c)) (1 - 14 Q^2 μ/r_c^4),    r_sh,EH2^2= (r_c^2/f_EH(r_c)) (1 - 8Q^2 μ/r_c^4).

For the BI BH up to first order in 1/b^2,

    r_sh,BI^2= (r_c^2/f_BI(r_c)) (1 - Q^2 /b^2 r_c^4).

While for the ModMax BH the shadow radius is

    r_sh,MM^2= (r_c^2/f_MM(r_c)).

It is worth to note that r_c is different for each BH; r_c is the radius of the photosphere,  precisely the radius at which is maximum  the effective potential for the null geodesics of the effective metric;
i.e. r_c is calculated from the conditions V_ eff^' =0 and  V_ eff^''  < 0. For RN the radius of the UCO is given by


    r_c,RN= 3M/2( 1 + √(1- 8 Q^2/9M^2)),

Moreover, the impact parameter and shadow radius at infinity of the RN BH can be derived from any of the above expression in the limit of the vanishing nonlinear parameter, and are given by,

    b_c,RN^2=r_c^2/f_RN(r_c) =  r_sh,RN^2= 3^3 M^2  ( 1+ √(1- 8Q^2/9M^2))^4/8 ( 1 - 2Q^2/3M^2 +  √(1- 8Q^2/9M^2)),

while for Schwarzschild it is well known that the radius of the photosphere r_c = 3 M, while the shadow radius is r_sh,Schw= 5.196 M. 




 §.§ Confronting the NLED BHs shadow with observations


In this section we test the NLED BH shadows with the observations of the shadow reported by the EHT for the supermassive BHs in the center of the Milky Way (Sgr A^*) and the M87^* galaxy. This gives insight about is a supermassive BH can bear electric or magnetic fields of high intensities.


Fig. <ref> shows the shadow radii for the BHs in consideration as a function of the charge Q, contrasted with the σ intervals given by the EHT for Sgr A^* <cit.>. As the charge increases the radius of the shadow decreases. 

The radius of the shadow for BI and the second solution of EH behave similarly for values of the nonlinear parameters b=0.7, μ=0.5; for Q/M ≤ 0.2 both BHs are in excellent agreement with the shadow of Sgr A^* and up to Q/M ≤ 0.7 their shadows remain in the 1 σ band of Sgr A^*. The ModMax BH is the one that allows  the  largest charge/mass ratio Q/M keeping its shadow radius in the 1σ region for γ = 0.5.




In Fig. <ref> the shadow radii for the NLED BHs  is contrasted with the observations of  M87^* <cit.>. Note that the 1σ constraint reaches values above the radius for the Schwarzschild BH, nonetheless the BHs in consideration do not exceed the Schwarzschild radius. 
The confidence intervals of M87^*  are less restrictive than the ones for Sgr A^*; that allows larger values of Q/M BH in the 1 σ band.
For Q/M ≤ 0.35 EH and BI BHs are in excellent agreement with the shadow of M87^*and up to Q/M ≤ 0.8 their shadows remain in the 1 σ band. While the ModMax BH remains in the 1 σ band for  Q/M ≤ 1 for γ = 0.5.




In Fig. <ref> we compare the 1σ bands for the SgrA^* and M87^* being the light gray the 1σ band  for M87^* and the dark gray the one for SgrA^*. As noted before, the confidence intervals for M87^* include the ones for SgrA^*.





Since tighter confidence intervals come from SgrA^*, 
in Fig. <ref> we confront  the shadow radii of EH1, BI and MM BHs for different values of their respective nonlinear parameters μ, b and γ. For the EH BH, we only plot the first solution as the behaviors are similar; as the value of μ increases, the allowed values of the charge that fall in the 1σ and 2σ regions, are more restricted, however μ=0.3 admits charge-mass ratios of 0.8 ≤ Q/M in the 1 σ interval and 0.9 ≤ Q/M in the 2 σ interval. 

For the BI BH increasing b the radius of the shadow tends to the one of the RN BH, as expected, since in the limit b→∞ Maxwell electrodynamics is recovered. It is important to highlight that Figure <ref> shows that BI and EH nonlinear theories do not allow higher values of the charge  in agreement with the observations; then, if one looks for the existence of a charged black hole, neither BI or EH NLEDs are the theories that would help to achieve that, at least in the range of supermassive black holes; however, up to a relation charge mass Q/M ≈ 0.3 charged BHs are not ruled out for any of the examined NLEDs.

In contrast, for the MM BH, if γ→ 0, we recover the radius of the shadow of the RN BH, and  as γ increases higher values of the BH charge are admitted; for γ =0.5 the whole range of Q/M ≤ 1 is in the 1 σ band.
Moreover, the MM BH fits better the shadow observations than RN.





 
  
   
    
     
 






§ CONCLUSIONS



We considered three nonlinear electrodynamics (NLED) theories: Born-Infeld (BI), Euler-Heisenberg (EH), and modified Maxwell (ModMax, MM) to study light propagation in a strong magnetic (electric)  background; the effect of these intense fields in light propagation is the slowing down of the phase velocity. Another effect that arises due to the high intensities of the fields is birefringence in vacuum. For the NLED theories in consideration, only the BI theory does not present birefringence; and in the case of the ModMax theory, as a consequence of the conformal symmetry, one of the light trajectories coincides with the null geodesics of the Minkowski space.  The NLED theory that diminishes the phase velocity the most is the EH theory and the effect is enhanced in proportion to the magnetic (electric) component that is orthogonal to the propagation direction. 
 Our treatment is valid for very strong fields; if we consider, for instance, that B^2/b^2 ≈ 10^-2, then the BI field is of the order of 10^11 Tesla that is one hundred times the critical Schwinger field or  B_ cr≈ 10^9 Tesla. It is important to note that for fields of the orders B^2/b^2≈10^-2 and μ B^2≈ 10^-2 the diminishing in the velocities for BI and EH is qualitatively similar, as illustrated in  Figure <ref>. The plots that serve the comparison are in terms of dimensionless parameters μ B^2 for EH and B^2/b^2 for BI. Still, we should keep in mind that the critical fields of both nonlinear theories are different by two orders of magnitude, being the BI theory the one with the larger critical field. 
 
 Additionally, the comparison between the phase velocities is useful to set bounds on the ModMax nonlinear parameter γ since so far it is not constrained apart from the restriction of being positive that is derived from the energy conditions in the ModMax BH. BI constrains γ to the interval  0 < γ < 0.549306; the larger bound for BI fields B ≈ b with b = 10^11Tesla  (b = 10^20 Volt/m)  being the maximum attainable electromagnetic BI field. Then
we derive that the EH theory is more stringent constraining  γ to the interval 0.0001 <γ < 0.00018 for fields of the order of B_cr∼ 10^9 Tesla  ( E_cr=10^18 Volt/m ).  

In the second part of the paper we contemplate light propagation in the vicinity of NLED BHs as the background metric and calculate the corresponding shadows 
to compare the NLED effects with the ones for the linear solution of the Einstein-Maxwell equations, the Reissner-Nordstrom (RN) metric and, as a reference, we include in the comparison the Schwarzschild BH. Then these shadows are
confronted  with the observational shadows from Sagittarius A^∗ and M87^*.  The situation is different for each electrodynamics.  Due to birefringence, EH presents two shadows and BI only one. For ModMax one of the effective metrics is the background metric as a consequence of the conformal invariance.

We can determine ranges for the nonlinear parameters μ for EH, b for BI, and γ for ModMax, for their shadows to be in agreement with the evidence of Sagittarius A^* and M87^*.
 If one is looking for a charged BH, the most effective theory is ModMax, which coupled with Einstein equations results in a metric function similar to the one for RN but with a screened charge; the ModMax BH allows  values of the BH charge-mass Q/M ≤ 1 still in the 1 σ confidence interval with the observations of Sgr. A^* and M87^*. Meanwhile, by increasing the BI and EH nonlinear parameters, we are not able to reach higher values of the BH charge that keep the shadow in the 1σ interval of Sgr.A^* that sets the most stringent astrophysical limits for the NLED parameters. 

 In summary, up to a relation charge mass Q/M ≤ 0.3  astrophysical charged BH are not ruled out for any of the examined NLEDs; their shadows remain in the 1σ interval of Sgr.A^*.  However neither BI-BH or EH-BH point to a possible astrophysical supermassive charged BH with Q/M > 0.3. In contrast, the Mod Max BH shadow remains in the 1 σ interval for BH charge-mass  0  ≤ Q/M ≤ 1, fitting data even better than the Einstein-Maxwell solution RN.




EGH acknowledges financial support by Conahcyt-Mexico through the Ph. D.  scholarship No. 761878.
The work of AM has been sponsored by Conahcyt-Mexico through the Posdoc Project I1200/311/2023. 
NB acknowledges partial support by Conahcyt-Mexico 
project CBF2023-2024-811. 






JHEP























 Paper Title: Road to Singularity Theorems with Torsion.






venn@fias.uni-frankfurt.de




ujjwal.agarwal@mff.cuni.cz



vasak@fias.uni-frankfurt.de



§ ABSTRACT 
This study examines the formulation of a singularity theorem for timelike curves including torsion, and establishes the foundational framework necessary for its derivation.
We begin by deriving the relative acceleration for an arbitrary congruence of timelike curves. The resulting “deviation equation” offers an alternative pathway to the well-known Raychaudhuri equation with torsion. Conjugate points are then introduced and analyzed in relation to the behavior of the scalar expansion. Together with the sensible requirement of hypersurface orthogonality, the Raychaudhuri equation is examined for several specific cases of torsion that are prominent in the literature.
Our findings indicate that a totally antisymmetric torsion tensor does not influence the behavior of the congruence of timelike curves. Finally, we formulate a singularity theorem for timelike curves and highlight the critical requirement of non-autoparallel curves.



Road to Singularity Theorems with Torsion
    D. Vasak
    
=========================================







§ INTRODUCTION
 
Despite its tremendous success, Einstein's General Relativity (GR) is plagued by the appearance of singularities, as 
elucidated by the renowned Penrose-Hawking singularity
theorems <cit.>.
Singularities are physically undesirable as they often signify
divergences in measurable quantities and the violation of
physical laws.
While there are ways to avoid, e.g., the Big Bang singularity
within the framework of GR <cit.>,
we are going to adopt a different approach here.

In Metric-Affine Gravity (MAG) the linear connection is
promoted to an independent dynamic variable and provides,
besides the pseudo-Riemannian metric, additional degrees of 
freedom. These additional degrees of freedom are referred to
as “torsion” and “nonmetricity”.
Geometrically, torsion measures the non-closure of a parallelogram formed by parallel transporting two infinitesimal vectors along each other <cit.>. Physically, torsion is believed
to be closely related to the spin-density of matter <cit.> and is able to account for the
effects of Dark Energy in the late Universe <cit.>.
Nonmetricity on the other hand causes lengths of vectors and
angles between vectors to change when performing a parallel 
transport. This geometrical picture is however not 
complemented by any promising physical significance yet.
Therefore we are going to restrict ourselves to the study
of a connection including torsion while neglecting any
nonmetricity contributions, i.e., we assume a metric
compatible connection.

The framework for establishing and proving singularity theorems is based on the Raychaudhuri equation. It 
determines how a ball of events within the congruence of 
curves contracts/expands as we move along the flow of the curves. A crucial aspect of the Raychaudhuri equation involves the imposition of energy conditions (ECs). These ECs are ad-hoc conditions based on our current physical interpretation of gravitational forces.
By applying these ECs to the conventional Raychaudhuri equation, it can be concluded that a collection of vorticity-free geodesics will converge to a focal point, ultimately leading to the formation of a spacetime singularity. However, the presence of torsion modifies the kinematics, potentially leading to different outcomes. Therefore, the study of kinematics in spacetimes with torsion could provide insights into modified ECs, which may enhance our understanding of gravity and potentially avoid the formation of singularities.


This paper is organised as follows: <Ref>
introduces the basic definitions of our framework and establishes the conventions used throughout the paper.
The deviation equation for timelike curves is derived in
<Ref> and culminates in the Raychaudhuri eq.
with torsion in <Ref>. We discuss the relation
between the conventional kinematic variables and their geometric counterparts in <Ref>. 
The notion of conjugate points is generalised to the case
of arbitrary timelike curves in <Ref>
and allows us, together with hypersurface orthogonality
(<Ref>), to 
investigate particular torsion forms in <Ref>.
Finally, we present and prove a singularity theorem for arbitrary timelike curves with torsion in <Ref>, and conclude the paper with a summary of our findings in <Ref>.

Throughout this paper we employ natural units, in which
ħ=c=1.
Furthermore, the metric is assumed to have signature
(-+++), following the standard convention in <cit.>.






§ THEORETICAL FRAMEWORK


In metric compatible MAG the connection is given by

    Γ^λ_μν = Γ^λ_μν + K^λ_μν,

where 

    Γ^λ_μν =
        1/2g^λρ(∂_νg_μρ + ∂_μg_νρ
        - ∂_ρg_μν)

are the Christoffel symbols with g_μν being the metric, and 

    K^λ_μν 
        g^λρ(S_ρμν - S_μρν - S_νρμ)

is the contortion tensor defined in terms of the torsion
tensor S^λ_μνΓ^λ_[μν].

Curvature is encoded within the Riemann-Cartan tensor

    R^λ_σμν∂_μΓ^λ_σν - ∂_νΓ^λ_σμ + Γ^λ_ρμΓ^ρ_σν - Γ^λ_ρνΓ^ρ_σμ,

which may be split into its Levi-Civita contribution, 
denoted by a ring, and its torsional part using (<ref>):

    R^λ_σμν =    R^λ_σμν
        + ∇_μK^λ_σν - ∇_νK^λ_σμ
       + K^λ_ρμK^ρ_σν
        - K^λ_ρνK^ρ_σμ.


Covariant derivatives act on an arbitrary (1,1) tensor T as

    ∇_μT^α_β = 
        ∂_μT^α_β + 
        Γ^α_ρμT^ρ_β
        - Γ^ρ_βμT^α_ρ,

and accordingly on higher/lower rank tensors. As before, this 
expression may be split into

    ∇_μT^α_β = 
        ∇_μT^α_β + 
        K^α_ρμT^ρ_β
        - K^ρ_βμT^α_ρ.


With torsion present, the notions of autoparallel and geodesic
curves differ in general. A geodesic is a curve that 
minimises/maximises the path length

    S = ∫√(-g_μνdx^μ/dλdx^ν/dλ)dλ,

where x^μ(λ) denotes the curve in local 
coordinates and λ is the curve parameter. The path
length S is extremal if δ S = 0, which leads to the
geodesic equation

    d^2 x^μ/dλ^2
        + Γ^μ_αβdx^α/dλdx^β/dλ = 0.

Notice that only the Levi-Civita part of the connection contributes to the geodesic equation.
An autoparallel curve on the other hand generalises the notion
of a straight line to curved space. Let X^μ
denote the tangent vector field to a curve, then the condition
for this curve to be an autoparallel reads X^α∇_αX^μ = 0. In local coordinates with X^μ = dx^μ/dλ, this condition
is equivalent to

    d^2 x^μ/dλ^2
        + Γ^μ_αβdx^α/dλdx^β/dλ = 0.

Evidently now the full connection, more specifically the symmetric part Γ^μ_(αβ), has
to be taken into account. Thus torsion enters the autoparallel equation in general.
Only for a totally antisymmetric torsion tensor (or a vanishing torsion) do the notions of geodesics and 
autoparallels coincide.








§ DEVIATION EQUATION


Let ℳ be an n dimensional Lorentzian manifold
with an independent metric compatible connection Γ^λ_μν
and γ(s) be a curve in ℳ with tangent vectors

    V̂_γ(s)^μdγ^μ/d s(s),

defined at every point along the curve.
Let further U^μ be a normalised timelike vector field,
i.e., g(U,U) = U^μU_μ = -1 at every point where U^μ is defined.
We may create a one-parameter family of non-intersecting curves by moving each point on
γ(s) along the integral curves of U^μ.
This family shall be denoted by γ_s (τ)
<cit.>. In other words, for fixed s, the quantity 
γ_s (τ) constitutes a timelike curve parametrised by the 
affine parameter τ which coincides with the proper time.
We thus have

    U_γ_s(τ)^μ = ∂γ_s^μ/∂τ(τ)

at any point within the family of curves,
where γ_s^μ(τ) are the coordinates
of the curves in a local frame. We can extend the 
vector field V̂^μ across the whole family 
γ_s (τ) by pushing all V̂^μ along the integral curves to an arbitrary
parameter value of τ.
For that matter we define the flow ϕ_τ of U^μ as usual by

    ϕ_τ(γ_s(0)) γ_s(τ),

where γ_s(0) = γ(s).
It is well known that  the ϕ_τ constitute a 
one-parameter family of diffeomorphisms. We then extend
V̂^μ to

    Ṽ_γ_s (τ) (ϕ_τ)_* (V̂_γ(s)),

where (ϕ_τ)_* denotes the pushforward of the flow.
In other words

    Ṽ_γ_s (τ)^μ = ∂γ_s^μ/∂ s (τ),

see <Ref> for details.
Our goal is to determine how the spacelike separation between neighbouring timelike curves changes as we move along them.
This is essentially described by the change of Ṽ^μ along the curves. In general
however, Ṽ^μ may have components
parallel to U^μ which we do not care about.
A remedy is to project Ṽ^μ onto 
a vector subspace orthogonal to U^μ first
and consider the change of this orthogonal component only.
The projection to the orthogonal subspace is achieved with
the projection tensor

    P^μ_νδ^μ_ν
        + U^μU_ν.

We thus define the new deviation vector field,

    V^μ P^μ_νṼ^ν,

orthogonal to U^μ, see Fig. <ref>.




To determine how V^μ changes along the curves, we first note that 
the Lie derivative of Ṽ along U vanishes

    ℒ_U Ṽdef=.d/dτ((ϕ_τ)^* (Ṽ))|_τ=0 = 0,

where (ϕ_τ)^* = (ϕ_τ^-1)_* denotes the pullback,
since 

    (ϕ_τ)^* (Ṽ) = V̂

is independent of τ.
But ℒ_U Ṽ = [U,Ṽ] and
therefore we have that 

    0    = [U,Ṽ]^μ
       = U^α∂_αṼ^μ - 
        Ṽ^α∂_α U^μ
       = U^α∇_αṼ^μ - 
        Ṽ^α∇_α U^μ + 2 U^αṼ^βΓ^μ_
        [αβ]
       = U^α∇_αṼ^μ - 
        Ṽ^α∇_α U^μ + 2 U^αṼ^β S^μ_αβ,

and hence

    U^α∇_αṼ^μ = 
        Ṽ^α∇_α U^μ + 2 U^αṼ^β S^μ_βα.

To proceed, note that 

    U^β∇_β
        P^ρ_λ=
        A^ρU_λ + A_λ
        U^ρ,

where A^μ U^β∇_β U^μ
is the autoparallel acceleration.
Now multiplying (<ref>) with P^ν_μ
and summing over μ, we find, after some algebra and 
with (<ref>), that

    U^α∇_α V^μ =    
        A^μU_αṼ^α
        + U^μA_αṼ^α+
        Ṽ^α∇_α U^μ
       + 2 U^αṼ^β 
        P^μ_ν
        S^ν_βα.

After projecting and using
A^λU_αṼ^α + Ṽ^α∇_α U^λ = 
    V^α∇_α U^λ
this reads

    P^λ_μU^α∇_α V^μ =
        V^α∇_α U^λ + 2 U^αṼ^β 
        P^λ_ν
        S^ν_βα

and tells us how the spacelike separation of neighbouring curves changes along the curves.

We may go one step further and determine how 
P^λ_μU^α∇_α V^μ
changes along the curves, which leads us to the 
acceleration of nearby timelike curves

    a^μ P^μ_ρU^β∇_β(P^ρ_λU^α∇_α V^λ).

Plugging in (<ref>), employing
(<ref>), and using

    ∇_α∇_β C^μ = ∇_β∇_α C^μ + R^μ_λαβ
        C^λ + 2S^λ_αβ∇_λC^μ

for any vector C^μ,
we finally find the deviation equation
for timelike curves

we find the intermediate result

    a^μ =   (U^β∇_βV^α)∇_αU^μ + 
        V^αP^μ_ρU^β∇_β∇_αU^ρ
       + 2[A^αṼ^γ
        P^μ_λS^λ_γα
        + U^α(Ṽ^β∇_βU^γ)P^μ_λS^λ_γα
       + 2U^αU^ξṼ^ρ
        S^γ_ρξP^μ_λS^λ_γα + U^αṼ^γA^μ
        U_λS^λ_γα
       + U^αṼ^γP^μ_λ
        U^β∇_βS^λ_γα],

where we further used P^μ_ρA^ρ = A^μ, P^μ_ρU^ρ = 0,
and P^μ_ρP^ρ_λ = P^μ_λ.


    a^μ =    R^μ_λαβU^λ
        U^αV^β + 
        4 S^λ_βαU_λ
        U^αṼ^β
        A^μ
       +
        A^μA_αV^α
        + V^αP^μ_ρ∇_αA^ρ
       +2P^μ_λ[U^αṼ^ρ
        U^β∇_β
        S^λ_ρα
        + U^αS^λ_ραṼ^β∇_βU^ρ
       + 2U^αU^βṼ^ξS^λ_ρα
        S^ρ_ξβ
        + S^λ_ραṼ^ρ
        A^α].







For a totally antisymmetric torsion tensor,
the deviation equation admits the form

    a^μ =    R^μ_λαβU^λ
        U^αV^β + 
        A^μA_αV^α
        +V^αP^μ_ρ∇_αA^ρ
       +  
        2[U^αṼ^ρU^β∇_β
        S^μ_ρα + 
        U^αS^μ_ραṼ^β∇_βU^ρ
       + 2U^αU^βṼ^ξS^μ_ρα
        S^ρ_ξβ
        + P^μ_λS^λ_ραṼ^ρ
        A^α].


In the case of a vanishing torsion and autoparallel curves (A^μ = 0), we recover the known geodesic deviation
equation a^μ = R^μ_λαβU^λ
U^αV^β from standard GR.
















§ RAYCHAUDHURI EQUATION WITH TORSION


We can make use of (<ref>) to get an evolution equation
for ∇_ν U^μ. This will naturally lead
us to the Raychaudhuri equation.

Compare (<ref>) with an intermediate result of the deviation equation where the definition of the Riemann-Cartan tensor has not been employed yet, and solve for the covariant derivative of 
∇_ν U^μ along the curves.
One finds

    P^μ_λ V^α U^β∇_β(∇_α
        U^λ) =    - Ṽ^β(∇_βU^α)(
        ∇_αU^μ)
       - 2 U^βṼ^λS^α_λβ∇_αU^μ
       + R^μ_αβλU^α
        U^βV^λ
        + P^μ_ρV^α∇_αA^ρ
       - U_βṼ^β
        A^α∇_αU^μ.

After using (<ref>) and realising that there is an overall
arbitrary contraction with Ṽ, we arrive, after some manipulations, at 
the evolution equation for ∇_ν U^μ

    D/Dτ(∇_ν U^μ) =    -(∇_ν U^α)
        (∇_α U^μ)
        - 2U^αS^β_να∇_β U^μ
       + R^μ_αβνU^α
        U^β + ∇_νA^μ,

where D/Dτ U^β∇_β. We may further express 

    ∇_ν U_μ≡D_νU_μ
        + (δ_ν^αδ_μ^β -
        P_ν^αP_μ^β)
        ∇_α U_β,

where we defined 

    D_νC_μ
    P_ν^αP_μ^β∇_α C_β

for an arbitrary covector C_μ. It turns out that
(δ_ν^αδ_μ^β -
P_ν^αP_μ^β)
∇_α U_β = -U_ν
A_μ and thus 

    ∇_ν U_μ = D_νU_μ
        -U_νA_μ.

We choose to split 

    D_νU_μ   = 
        D_(νU_μ) +
        D_[νU_μ]
       = D_(νU_μ) -1/n-1θ P_μν + 1/n-1θ P_μν +
        D_[νU_μ],

where 

    θD^μU_μ = ∇^μU_μ
 
is the volume scalar/scalar expansion describing the mean separation between curves 
and n is the dimension of our manifold. Furthermore, we define
the symmetric and trace-free shear

    σ_μνD_(νU_μ) -1/n-1θ P_μν,

which measures any anisotropic evolution of curves, and the 
antisymmetric vorticity

    ω_μνD_[νU_μ],

describing rotational behaviour. Thereby we obtain the following split:

    ∇_ν U_μ = 1/n-1θ P_μν + σ_μν +
        ω_μν
        -U_νA_μ,

which further implies that 

    ∇_ν A_μ   =
        D_νA_μ + 1/n-1θ
        U_μA_ν + U_μ(
        σ_να - ω_να)
        A^α
       - U^α∇_α(
        U_νA_μ) + A_ν
        A_μ.

Therefore we may reframe the evolution equation (<ref>) as

    D/Dτ(∇_ν U_μ) =    - 1/(n-1)^2θ^2P_μν + R_μ_αβνU^α
        U^β
       - 2/n-1θ(σ_μν + ω_μν) - σ_αμσ_ν^α - ω_αμω_ν^α
       + 2σ_α[μω_ν]^α
        + D_νA_μ + 
        2/n-1θ U_(μA_ν)
       + 2 U_(μσ_ν)α
        A^α - 2U_[μω_ν]α
        A^α + A_μA_ν
       - U^α∇_α(A_μU_ν) - 2/n-1θ S_μναU^α
       + 2(1/n-1θ U_μ - A_μ)U^αU^β
        S_αβν
       + 2(σ_μ^β + ω_μ^β)
        U^αS_βαν.

Equation (<ref>) encodes the full information
of the evolution of ∇_ν U_μ
along the flow of the curves. We can now choose to split 
this information into several parts based on
(<ref>). This provides us with three evolution
equations for each of the variables θ, ω_μν and σ_μν.

The evolution equation for the scalar expansion θ is obtained after taking the trace of (<ref>). This is the well-known Raychaudhuri
equation in n dimensions with torsion for general timelike curves

    d/dτθ =    -1/n-1θ^2
        -R_(μν)U^μU^ν
        -σ_μνσ^μν
        +ω_μνω^μν
       + D_μA^μ + 
        A_μA^μ + 2/n-1θ 
        S_μU^μ - 2 S_μνρ
        U^μU^νA^ρ
       - 2 S_μνρσ^μνU^ρ +
        2 S_μνρω^μνU^ρ,

where S_μ S^α_μα and d/dτ U^β∂_β,
in accordance with <cit.>. Note that the covariant derivative 
D/Dτ
reduced to the ordinary derivative d/dτ
since their action on a scalar field, here θ, is 
equivalent.

The evolution equation for the vorticity is found by antisymmetrising
(<ref>):

    D/Dτω_μν = 
            R_[μ|αβ|ν]U^α
        U^β - 2/n-1θω_μν
        + 2σ_α[μω_ν]^α
       + D_[νA_μ]
        - 2U_[μω_ν]α
        A^α - 2/n-1θ S_[μν]αU^α
       + 2(1/n-1θ U_[μ - A_[μ)U^αU^β
        S_|αβ|ν]
       + 2(σ_[μ^β + ω_[μ^β)
        U^αS_|βα|ν].


We define the symmetric trace-free operation “⟨⟩” on any index pair
of an arbitrary tensor C of rank greater or equal 2 as

    C_⟨μν⟩ C_(μν) - 
        1/n-1P_μνC^α_α.

Then 

    ∇_⟨νU_μ⟩ = 
        σ_μν - U_⟨ν
        A_μ⟩.

Acting with this operation on (<ref>) results
in the evolution equation for the shear

    D/Dτσ_μν = 
            R_⟨μ|αβ|ν⟩U^α
        U^β - 2/n-1θσ_μν
        - σ_α⟨μσ_ν⟩^α
       - ω_α⟨μω_ν⟩^α + D_⟨νA_μ⟩
        + 2/n-1θ U_(μA_ν)
       + 2 U_(μσ_ν)α
        A^α + A_⟨μA_ν⟩
        - 2/n-1θ S_⟨μν⟩αU^α
       + 2(1/n-1θ U_⟨μ - A_⟨μ)U^αU^β
        S_|αβ|ν⟩
       + 2(σ_⟨μ^β + ω_⟨μ^β)
        U^αS_|βα|ν⟩.


















§ GEOMETRIC KINEMATIC VARIABLES


The previous sections are based on a particular definition
of the kinematic variables θ, σ_μν, and ω_μν
in terms of a decomposition of ∇_ν
U_μ. This choice is used in a number of references such as <cit.> and roots in the direct analogy to
the torsionless case. However, as evident from (<ref>), the complete change of the separation
vector along the curves is not fully determined by
∇_νU_μ but there are 
further components such as torsion and the autoparallel acceleration
A^μ. Hence there is a point to be made for 
decomposing not just ∇_νU_μ
but the whole r.h.s. of (<ref>) into its 
trace, symmetric tracefree and antisymmetric part. The resulting geometric variables then include the complete information about the
evolution of the separation vector and describe the relative motion of nearby
curves.
This path is taken in, e.g., <cit.>.
Here we shall introduce these geometric kinematic variables,
henceforth labelled by a “(g)” superscript,
and relate them to the ones used before.

We can express (<ref>) as

    U^β∇_β V^μ =
        V^β∇_β U^μ
        + U^μA_βV^β
        + 2 U^αV^β 
        P^μ_ν
        S^ν_βα,

or, after projecting, as

    P^μ_λU^β∇_β V^λ = B_β^μ
        V^β

with

    B_β^μ   
        P^μ_λP_β^ρ(
        ∇_ρ U^λ
        + U^λA_ρ    
        + 2 U^α
        P^λ_νS^ν_ρα)
       = D_βU^μ
        + 2U^αP^μ_ν
        S^ν_βα.

Now instead of just decomposing D_βU^μ, we decompose the 
whole tensor B_β^μ as

    B_νμ = 1/n-1θ^(g)
        P_μν + σ^(g)_μν
        + ω^(g)_μν,

where

    θ^(g)    B_α^α
    σ^(g)_μν   
        B_(νμ) - 1/n-1θ^(g)
        P_μν
    ω^(g)_μν   
        B_[νμ].

These are related to the previously introduced variables by direct computation

    θ^(g)   = θ - 2U^αS_α
    σ^(g)_μν   = σ_μν + 2U^αP_⟨μ|β
        S^β_|ν⟩α
    ω^(g)_μν   =
        ω_μν +
        2U^αP_[μ|β
        S^β_|ν]α.

Writing the evolution equations in terms of the geometric
variables or in the old variables is a matter of taste.
The torsional terms which are absorbed in the geometric
variables in (<ref>) appear explicitly on the 
r.h.s. of the evolution equations in the previous section,
while in terms of geometric variables only appear implicitly
within them. The evolution equations in the geometric variables therefore look different. However they include 
exactly the same physical information.

Nevertheless, our later investigations of the Raychaudhuri equation must be conducted in terms of the geometric variables. The explicit form of the equation is crucial for establishing appropriate ECs and ultimately determining the formation of focal points.
Therefore, in the following we shall proceed to work
with the Raychaudhuri equation in geometric variables

    d/dτθ^(g) =    -1/n-1θ^(g)^2
        -R_(μν)U^μU^ν
        -σ^(g)_μνσ^(g)^μν
       +ω^(g)_μνω^(g)^μν
        + D_μA^μ + 
        A_μA^μ
       - 2/n-1θ^(g) 
        S_μU^μ
        - 2 S_μνρ
        U^μU^νA^ρ
       + 2 S_μνρσ^(g)^μνU^ρ
        -
        2 S_μνρω^(g)^μνU^ρ
       -2D/Dτ(
        U^αS_α),

which is in agreement with <cit.>. Note that different index placements in the definitions of the covariant derivative and the vorticity tensor result in the fact that our torsion and vorticity tensors admit a relative minus sign compared to the torsion and vorticity defined in <cit.>.


The vorticity equation reads

    D/Dτω^(g)_μν = 
            R_[μ|αβ|ν]U^α
        U^β - 2/n-1θ^(g)ω^(g)_μν
       + 2σ^(g)_α[μω^(g)_ν]^α
        + D_[νA_μ]
        - 2U_[μω^(g)_ν]α
        A^α
       + 2/n-1θ^(g) S_[μν]αU^α
       - 2(1/n-1θ^(g) U_[μ + A_[μ)U^αU^β
        S_|αβ|ν]
       + 2(ω^(g)_[μ^β-σ^(g)_[μ^β)
        U^αS_ν]βα
       + 2(ω^(g)_[μ^β-σ^(g)_[μ^β)
        U_ν]U^αU^ρ
        S_ρβα
       + 2U^λ(U_μ
        S_[να]λ - 
        U_ν
        S_[μα]λ)
        A^α
       + 2 D/Dτ(
        U^αP_[μ|β
        S^β_|ν]α).



















§ CONJUGATE POINTS


The notion of conjugate points, to be defined below,
is a vital component in establishing and proving the singularity theorem later on. To discuss them, we have to 
revisit the deviation equation (<ref>):

    a^μ =    R^μ_λαβU^λ
        U^αV^β
        + 4S^λ_βαU_λ
        U^αV^βA^μ
       + A^μA_βV^β
        +P^μ_λV^β∇_βA^λ
       +  
        2P^μ_λ[U^α
        V^βU^ρ∇_ρ
        S^λ_βα
        + U^αS^λ_ρα
        V^β∇_βU^ρ
       + 2U^αU^ξ
        V^βS^λ_ρα
        S^ρ_βξ + 
        S^λ_βα
        V^βA^α].

Notice that here we expressed (<ref>) solely
in terms of V^β, instead of the 
unprojected Ṽ^β. The reason
for this will become clear soon.
We further have the definition (<ref>), which may be reframed as

    a^μ =    D^2/Dτ^2V^μ + 2U^μ
        U^λV^β∇_β
        A_λ
        -4U^μU^αA_λ
        S^λ_βαV^β
       - U^μV^βU^α∇_αA_β
        -A^μV^βA_β.

Together with (<ref>) we thus find

    D^2/Dτ^2V^μ = 
           [R^μ_λαβU^λ
        U^α
        + 4S^λ_βαU^α(U_λ
        A^μ + A_λU^μ)
       + 2A^μA_β
        + (∇_βA^μ)
       + U^μ(U^α{∇_αA_β} - U^α{∇_βA_α})
        
       +  
        2P^μ_λ(U^α
        U^ρ{∇_ρ
        S^λ_βα}
        + U^αS^λ_ρα{∇_βU^ρ}
       + 2U^αU^ξ
        S^λ_ρα
        S^ρ_βξ
        + S^λ_βαA^α)]
        V^β.

By writing out the covariant derivatives on the l.h.s. explicitly, we may express this deviation equation
as an ordinary, linear, second order differential equation[
   It is possible to choose a tetrad frame, that is parallely transported along our curve of interest through
    the Fermi derivative,
    in which the spin-connection components vanish <cit.>. 
    In this case, the connection reduces to the Weitzenböck connection
    Γ^μ_αβ = e_c^μ∂_βe^c_α.
    On a mathematical level, this fact 
    ensures the validity of the strong Equivalence Principle in presence of torsion. Compared to the standard GR case, this 
    does not simplify our equation substantially, hence we choose
    to continue with the general connection.]

    d^2/dτ^2V^μ = 
           [R^μ_λαβU^λ
        U^α
        + 4S^λ_βαU^α(U_λ
        A^μ + A_λU^μ)
       + 2A^μA_β
        + (∇_βA^μ)
       + U^μ(U^α{∇_αA_β} - U^α{∇_βA_α})
        
       +  
        2P^μ_λ(U^α
        U^ρ{∇_ρ
        S^λ_βα}
        + U^αS^λ_ρα{∇_βU^ρ}
       + 2U^αU^ξ
        S^λ_ρα
        S^ρ_βξ
        + S^λ_βαA^α)
       -d/dτ(
        U^γΓ^μ_βγ)
        + U^αU^γΓ^μ_λαΓ^λ_βγ
       -2U^γΓ^μ_λγ(U^λA_β + 
        {∇_βU^λ}
       +2U^αP^μ_ν
        S^ν_βα)
        ]
        V^β.


Generalising the 
definition of a Jacobi field <cit.> to the case
including torsion, we call a solution V^μ of the
deviation equation (<ref>)
a Jacobi field on γ_s'(τ), where s' has some fixed but arbitrary value.
As in the standard case, we call two points p,q ∈γ_s'(τ) conjugate if there exists a Jacobi field that
is not identically zero and vanishes at p and q, see Fig. <ref>.









Choose the parametrisation of the curves such that p
lies at τ = 0. Following <cit.>, a general solution of the linear, second order ordinary
differential equation (<ref>) then admits the form

    V^μ(τ) = E^μ_ν(τ)dV^ν/dτ(0)

for some matrix E^μ_ν(τ),
since V^μ(0) = 0 per assumption. Plugging this ansatz back into (<ref>) yields the corresponding
differential equation for the matrix E.

At the point q we are required to have V^μ(τ_q) = 0. Then the homogeneous linear system

    0 = E^μ_ν(τ_q)dV^ν/dτ(0)

has a non-trivial solution only if det(E) = 0 at q.
From (<ref>) it follows that

    D/DτV^μ
        =[U^μA_β + ∇_βU^μ + 
        2U^αP^μ_νS^ν_βα]V^β

and with (<ref>) therefore

    d/dτE^μ_ν
        =   [U^μA_β + ∇_βU^μ + 
        2U^αP^μ_νS^ν_βα
       -U^αΓ^μ_βα]E^β_ν.

In matrix notation this reads

    C = ( d/dτE)E^-1,

where

    C^μ_β 
        U^μA_β + ∇_βU^μ + 
        2U^αP^μ_νS^ν_βα
        -U^αΓ^μ_βα.

On the one hand taking the trace of (<ref>) yields

    tr(C) = tr(( d/dτE)E^-1) = 
        1/det(E)d/dτdet(E),

whereas from (<ref>) we get

    tr(C) = θ - U^α[2S_α+ Γ^μ_μα].

With θ^(g) = θ - 2U^αS_α and 
Γ^μ_μα = Γ^μ_μα
we conclude

    θ^(g) = d/dτlndet(E) + 
        U^αΓ^μ_μα.


As the timelike curves approach the point q, det(E) tends to 0. Thus
d/dτlndet(E)→ -∞ and hence also 
θ^(g)→ -∞. The converse also holds since the connection is assumed to admit only finite values.
Therefore, even with torsion, we retain the fact that the 
points p and q are conjugate if and only if 
lim_τ→τ_qθ^(g) = -∞, as in standard GR.
However, the question where E becomes singular highly depends on the
torsional contributions present in the respective differential equation.

The standard GR form of (<ref>) is recovered
by setting torsion to zero and by going to a local
inertial frame where the connection vanishes.







Now let us restrict ourselves to the study of autoparallel
curves, i.e., A^μ = 0. Then, generalising the 
definition of a Jacobi field <cit.> to the case
including torsion, we call a solution V^μ of the autoparallel
deviation equation, i.e., (<ref>) with A^μ = 0:

    P^μ_ρU^β∇_β(P^ρ_λU^α∇_α V^λ) =    R^μ_λαβU^λ
        U^αV^β
       +  
        2P^μ_λ[U^αṼ^ρU^β∇_β
        S^λ_ρα
       + U^αS^λ_ραṼ^β∇_βU^ρ
       + 2U^αU^βṼ^ξS^λ_ρα
        S^ρ_ξβ]

a Jacobi field on γ_s'(τ), where s' has some fixed but arbitrary value.
For A^μ = 0 we have that U^β∇_βP^ρ_λ = 0 and Ṽ^α∇_α U^λ = 
    V^α∇_α U^λ.
With  P^μ_ρP^ρ_λ = P^μ_λ, P^μ_λV^λ = V^μ,
the definition

    S̅^μ_ρα 
        P^μ_λS^λ_ρα,

and the antisymmetry of the torsion tensor in its last two 
indices,
we may rewrite (<ref>) more neatly as

    D^2/Dτ^2V^μ = 
           [R^μ_λαβU^λ
        U^α
        +  
        2(U^α
        U^λ∇_λS̅^μ_βα
       + U^αS̅^μ_λα∇_βU^λ
        + 2U^αU^λS̅^μ_ρα
        S^ρ_βλ)]V^β.

By writing out the covariant derivatives on the l.h.s. explicitly, we may express this deviation equation
as an ordinary second order differential equation

    d^2/dτ^2V^μ = 
           [R^μ_λαβU^λ
        U^α
        + U^λU^ρ(Γ^α_ρλΓ^μ_βα - Γ^α_βλΓ^μ_αρ) 
       - U^αd/dτΓ^μ_βα
        +
        2(U^α
        U^λ∇_λS̅^μ_βα
       + U^αS̅^μ_λα∇_βU^λ
        + 2U^αU^λS̅^μ_ρα
        S^ρ_βλ
       - U^αΓ^μ_λα∂_β
        U^λ - 2 U^αΓ^μ_λα
        U^ρ(S^λ_ρβ - S̅^λ_ρβ))]V^β,

where d/dτ U^α∂_α.
It is possible to choose a tetrad frame, that is parallely transported along our curve of interest through
the Fermi derivative,
in which the spin-connection components vanish <cit.>. 
In this case, the connection reduces to the Weitzenböck connection

    Γ^μ_αβ = e_c^μ∂_βe^c_α.

On a mathematical level, this fact 
ensures the validity of the strong Equivalence Principle in presence of torsion.

As in the standard case, we call two points p,q ∈γ_s'(τ) conjugate if there exists a Jacobi field that
is not identically zero and vanishes at p and q, see Fig. <ref>.









Choose the parametrisation of the curves such that p
lies at τ = 0. Following <cit.>, a general solution of the linear, second order ordinary
differential equation (<ref>) then admits the form

    V^μ(τ) = A^μ_ν(τ)dV^ν/dτ(0)

for some matrix A^μ_ν(τ),
since V^μ(0) = 0 per assumption. Plugging this ansatz back into (<ref>) we find that
the matrix A correspondingly satisfies 

    d^2/dτ^2A^μ_ν = 
           [R^μ_λαβU^λ
        U^α
        + U^λU^ρ(Γ^α_ρλΓ^μ_βα - Γ^α_βλΓ^μ_αρ) 
       - U^αd/dτΓ^μ_βα
        +
        2(U^α
        U^λ∇_λS̅^μ_βα
       + U^αS̅^μ_λα∇_βU^λ
        + 2U^αU^λS̅^μ_ρα
        S^ρ_βλ
       - U^αΓ^μ_λα∂_β
        U^λ - 2 U^αΓ^μ_λα
        U^ρ(S^λ_ρβ - S̅^λ_ρβ))]A^β_ν.

We used here that dV^ν/dτ(0) is an arbitrary initial condition
not equal to zero. If it was zero, p and q would not be conjugate points.

At the point q we are required to have V^μ(τ_q) = 0. Then the homogeneous linear system

    0 = A^μ_ν(τ_q)dV^ν/dτ(0)

has a non-trivial solution only if det(A) = 0 at q.

From (<ref>) it follows that

    D/DτV^μ
        =[∇_βU^μ + 
        2U^αS̅^μ_βα]V^β

and with (<ref>) therefore

    d/dτA^μ_ν
        =[∇_βU^μ + 
        2U^αS̅^μ_βα
        -U^αΓ^μ_βα]A^β_ν.

In matrix notation this reads

    C = ( d/dτA)A^-1,

where

    C^μ_β∇_βU^μ + 
        2U^αS̅^μ_βα
        -U^αΓ^μ_βα.

On the one hand

    tr(C) = tr(( d/dτA)A^-1) = 
        1/det(A)d/dτ
        (det(A)),

whereas from (<ref>) we get

    tr(C) = θ - U^α[2S_α+ Γ^μ_μα].

With θ^(g) = θ - 2U^αS_α and 
Γ^μ_μα = Γ^μ_μα, where the circle
denotes the Levi-Civita connection,
we conclude

    θ^(g) = d/dτ(lndet(A)) + 
        U^αΓ^μ_μα.


As the autoparallel curves approach the point q, det(A) tends to 0. Thus
d/dτ(lndet(A)) → -∞ and hence also 
θ^(g)→ -∞. The converse also holds since the connection is assumed to admit only finite values.
Therefore, even with torsion, we retain the fact that the 
points p and q are conjugate if and only if 
θ^(g)→ -∞, as in standard GR.
However, where A becomes singular highly depends on 
torsional contributions, as evident from the 
differential equation (<ref>).

The standard GR form of (<ref>) is recovered
by setting torsion to zero and by going to a local
inertial frame where the connection vanishes.











§ HYPERSURFACE ORTHOGONALITY


Our congruence of timelike curves is not completely free
but has to obey certain conditions in order to be
physically feasible. More specifically, we require the 
vector field U to be hypersurface orthogonal. This means 
that at every point where U is defined, there exists locally
an embedded submanifold which is orthogonal to U,
see Fig <ref>.

If this were not the case then it would be impossible to 
foliate spacetime into equal-time 3-surfaces and hence
the standard ΛCDM treatment of the Universe would
collapse.

Mathematically, hypersurface orthogonality is realised 
with Frobenius' theorem. As in <cit.>,
let p∈ℳ and consider subspaces W_p⊆ T_pℳ and U_p^*⊆ T_p^*ℳ with
dim(W_p) = m ≥ 1, m < n and dim(U_p^*) = n-m such that for all u ∈ U_p^* and all
q ∈ W_p

    u(q) = 0.

This is possible since any subspace of an ambient vector space equipped with a non-degenerate symmetric bilinear form,
together with its orthogonal complement, have complementary dimension.
Here, U_p is the orthogonal complement of W_p.
Let O_p be some open neighbourhood of p and define

    W    _q∈ O_p W_q ⊆ Tℳ
    
        U^*    _q∈ O_p U_q^* ⊆ Tℳ^*,

where Tℳ is the tangent bundle and “⊔” denotes the
disjoint union. Frobenius' theorem states that W admits
integral submanifolds if and only if for all X,Y ∈Γ(W) and for all u∈Γ(U^*) 

    u(ℒ_X Y) = 0,

where Γ(W) and Γ(U^*) denote the set of all sections of the bundle W and the cotangent bundle U^* respectively. But ℒ_X Y = [X,Y] and hence
(<ref>) reads u_α[X,Y]^α = 0.
Writing the commutator as in (<ref>), pulling
u_α into the covariant derivative and using the 
requirement that u_α Y^α = 0 = u_α X^α,
we arrive at

    X^α Y^β(∇_[βu_α]+ S^γ_αβu_γ)
        = 0.

Since X and Y are arbitrary, we find equivalently that

    ∇_[αu_β]-S^γ_αβu_γ
        = ∑_i=1^n-m(u_i)_[α(v_i)_β],

where the u_i are n-m linearly independent elements 
of Γ(U^*) (thus in particular (u_i)_α Y^α = 0 = (u_i)_α X^α for all i) and (v_i) are arbitrary
elements in Γ(U^*).

Let us consider the special case where dim(U^*) = 1,
which is the case relevant for us. Then we may choose u itself as (u_1) and we get 

    ∇_[αu_β]-S^γ_αβu_γ
        = u_[αv_β],

where v ∈Γ(U^*) arbitrary. Choose v such that
P_ν^βv_β = 0 and contract
the whole equation above with P_μ^αP_ν^β.
With ω_νμ = D_[μu_ν] we finally get

    ω_μν = S^α_νμu_α
        + 2S^α_[ν|β|u_μ]
        u_αu^β.

This holds for an arbitrary u∈Γ(U^*). For our specific 
case we choose to consider the vector field U^α tangent 
to the timelike curves and its associated 1-form defined via
the musical isomorphism U_α g_αβU^β.
Then the condition for U^α to be hypersurface
orthogonal reads

    ω_μν = S^α_νμU_α
        + 2S^α_[ν|β|U_μ]
        U_αU^β,

with ω_νμ =
D_[μU_ν] the vorticity
encountered previously,
or 

    ω^(g)_μν = (S_ανμ + 2S_[μν]α)U^α
        + 4U_[μS^α_ν]β
        U_αU^β

in geometric variables, in agreement with <cit.>.
For a totally antisymmetric torsion, these conditions reduce
to

    ω_μν = S^α_νμU_α

and

    ω^(g)_μν = S_μν^αU_α.


Therefore, in the presence of torsion, we can no longer assume that vorticity vanishes, as is typically done in the standard GR treatment. It must be emphasized that torsion, and torsion alone, is responsible for the generation of vorticity.
















§ CASE STUDIES


The Raychaudhuri equation allows us to predict the appearances
of conjugate points (also referred to as focal points) provided that we assume an energy condition.
For the case of timelike curves, as studied here, the relevant
condition is the strong energy condition (SEC).

In this section we are going to investigate different specific
torsion forms and examine how a modified SEC with torsion
leads to the appearance of focal points.
Torsion may be split, with respect to the action of the Lorentz group, into three irreducible tensors <cit.>: A totally antisymmetric part,
a vectorial contribution and the rest which is traceless
but not totally antisymmetric.
We are going to investigate the former two parts in detail and 
consider the special case of a Weyssenhoff fluid
relevant most prominently in Einstein-Cartan-Sciama-Kibble (ECSK) theory. For a recap of the most well-known energy
conditions, see <Ref>.





 §.§ Totally Antisymmetric Torsion

In the case of a totally antisymmetric torsion tensor
S_μνλ = S_[μνλ]
the geometric expansion scalar coincides with the conventional one θ^(g) = θ.
For autoparallel curves, and for n=4,
the Raychaudhuri equation (<ref>) reads

    d/dτθ =    -1/3θ^2
        -R_μνU^μU^ν
        -σ_μνσ^μν
        +ω_μνω^μν
        + 2 S_μνρω^μνU^ρ.

Requiring hypersurface orthogonality, this can be reframed
into

    d/dτθ =    -1/3θ^2
        -R_μνU^μU^ν
        -σ_μνσ^μν
       -S^α_μνS^β^μνU_αU_β.

One can show that in the case of a totally antisymmetric
torsion tensor,

    R_μνU^μU^ν = 
        R_μν
        U^μU^ν - 
        S^α_μνS^β^μν
        U_αU_β.

Therefore we conclude that in this case, the Raychaudhuri equation does
not contain any torsional contributions and reduces to its standard form

    d/dτθ = -1/3θ^2
        -R_μνU^μU^ν
        -σ_μνσ^μν.

Additionally one may verify that θ = θ
and σ_μν = σ_μν.










We may further simplify the torsion tensor by using the
Hodge star operator to relate the 3-form 
S_α_μν to a 1-form

    (⋆ S)_μ = 1/3!ϵ^αβγ_μS_αβγ,

where ϵ_μαβγ denotes
the Levi-Civita tensor. We call s_μ 
(⋆ S)_μ and reverse the above equality,
with conventions as in <cit.>,
to find

    S_μνλ = 
        ϵ_αμνλs^α.

The Raychaudhuri equation then admits the form

    d/dτθ =    -1/3θ^2
        -R_(μν)U^μU^ν
        -σ_μνσ^μν
       -2[s_μs^μ + 
        (s_μU^μ)^2].


Since σ_μν and 
ω_μν have both their indices projected 
orthogonal to U^α, they are both spacelike
in the sense that
σ_μνσ^μν≥ 0
and ω_μνω^μν≥ 0.

But ω_μνω^μν = 2[s_μs^μ + 
    (s_μU^μ)^2] and therefore

    s_μs^μ + 
        (s_μU^μ)^2 ≥ 0.


Hence we can generalise Wald's proposition <cit.> to:



Let (ℳ,g) be a spacetime, equipped with an independent
metric compatible affine connection Γ. Let the Ricci tensor satisfy R_μνξ^μξ^ν≥ 0
for all timelike ξ^μ and let the torsion tensor S_μνλ be totally antisymmetric. Consider a hypersurface orthogonal congruence
of timelike autoparallels emanating from p ∈ℳ.
If θ_0θ(τ_0) < 0 at some 
point r∈ℳ at parameter value τ_0 of a member of the congruence in the future 
of p, then there exists a point q conjugate to p within
proper time τ≤ -3/θ_0 from r (assuming the congruence extends that far).


The proof follows quickly based on the discussion before:
Note that 

    d/dτθ≤ -1/3θ^2

supposed that all assumptions in the proposition hold.
Upon choosing τ_0 = 0, we can solve this differential inequality to find

    θ(τ)^-1≥θ_0^-1 + 1/3τ.

Now if θ_0 < 0 at some point r, then θ^-1
must cross zero within the time interval τ∈ [0,-3/θ_0]. Since θ is negative before the crossing, this means that θ→ -∞ within this 
time interval. But θ→ -∞ is equivalent 
to the existence of a conjugate point.


Note that a “spacetime” is defined in the sense of Wald,
namely a smooth manifold equipped with a Lorentzian metric.
Further, due to the fact that torsion completely cancels out here,
the SEC R_μνξ^μξ^ν≥ 0 is not modified at all.

It is likewise not difficult to show that 
σ^(g)_μν = σ_μν and that
the shear equation
(<ref>) also reduces to its standard form without torsional contributions:


    D/Dτσ_μν = 
           R_⟨μ|αβ|ν⟩U^α
        U^β - 2/3θσ_μν
        - σ_α⟨μσ_ν⟩^α.

Hence there are also no residual
direction-dependent torsional effects to be expected.

Lastly, the evolution equation for the vorticity
(<ref>) becomes

    U^α
        U^β(R_[μ|αβ|ν]
        + 2S_β^γ_[μ
        S_|γα|ν]) = 0.

But R_[μ|αβ|ν]
U^αU^β = 0, due to the symmetries
of the Riemann-tensor, and U^αU^β
S_β^γ_[μ
S_|γα|ν] = 0 hold separately. Therefore
the evolution equation for the vorticity is merely an identity
and does not provide us with additional constraints.













 §.§ Vectorial Torsion

The second irreducible component of torsion is given by

    S_μνλ = -2/3g_μ[νS_λ].


Hypersurface orthogonality demands here that
ω^(g)_μν = 0 and we find the Raychaudhuri equation for autoparallels
to admit the form

    d/dτθ^(g) =    -1/3θ^(g)^2
        -R_μνU^μU^ν
        -σ^(g)_μνσ^(g)^μν
       - 2/3θ^(g)S_μU^μ
        -2D/Dτ(S_μU^μ).

In order to render dθ^(g)/dτ
negative and hence establish the formation of conjugate points, 
the modified SEC

    R_μνξ^μξ^ν≥ - 2/3θ^(g)S_μξ^μ
        -2D/Dτ(S_μξ^μ)

for any timelike ξ^μ has to be assumed.
Then indeed, since σ^(g)_μνσ^(g)^μν≥ 0 due to σ^(g)_μν being spacelike,
we have that dθ^(g)/dτ≤ -θ^(g)^2/3. Therefore we have:


Let (ℳ,g) be a spacetime, equipped with an independent
metric compatible affine connection Γ. Let the Ricci tensor satisfy (<ref>)
for all timelike ξ^μ and let the torsion tensor S_μνλ be of form (<ref>). Consider a hypersurface orthogonal congruence
of timelike autoparallels emanating from p ∈ℳ.
If θ_0θ(τ_0) < 0 at some 
point r∈ℳ at parameter value τ_0 of a member of the congruence in the future 
of p, then there exists a point q conjugate to p within
proper time τ≤ -3/θ_0 from r (assuming the congruence extends that far).


The proof proceeds exactly as in the previous subsection.


In <cit.>, a particular realisation of this vectorial torsion is used.
More specifically

    S_μνλ = 2ϕ g_μ[νU_λ],

where ϕ = ϕ(t) is a scalar function that depends only
on cosmic time and g_μν is the Friedmann-Lemaître-Robertson-Walker (FLRW) metric.
Therefore S_μ = -3ϕ U_μ. Since <cit.> studies
behaviours of the background cosmology, the Cosmological Principle is required to hold.
The most general allowed values of the
torsion tensor in a spacetime compatible with the Cosmological
Principle are provided in <cit.>. Indeed, (<ref>) is a subset of these
allowed values if U_i = 0 for i=1,2,3.
This ensures that S_00i = 0 for i=1,2,3, as
desired. Since the vector field U^μ is assumed
to be normalised, we conclude that
U^μ = (1,0,0,0)^⊺. Then indeed
U^μU_μ=-1 and one may verify that
U^α∇_αU^μ = 0,
which means that U^μ corresponds to an autoparallel.

The Raychaudhuri equation reduces to

    d/dtθ^(g) =    -1/3θ^(g)^2
        -R_00
        -σ^(g)_μνσ^(g)^μν
       - 2θ^(g)ϕ(t)
        -6d/dtϕ(t),

where the derivative is now with respect to cosmic time
since U^μ = (1,0,0,0)^⊺ and thus
d/dτ=d/dt.
Convergence of the congruence is hence ensured if 

    R_00≥ - 2θ^(g)ϕ(t)
        -6d/dtϕ(t).

Decomposing 

    R_00 = R_00 -
        6d/dtϕ

with R_00 = -3ä/a 
and using θ^(g) = θ - 2U^αS_α = θ - 6ϕ
with θ = 3ȧ/a + 6ϕ,
we find θ^(g) = 3ȧ/a and
restriction on the Universe's expansion

    ä≤  2ȧϕ(t).

Implications of this restriction would be interesting
to explore in future work.











 §.§ Rest


    d/dτθ^(g) =    -1/n-1θ^(g)^2
        -R_μνU^μU^ν
        -σ^(g)_μνσ^(g)^μν
       +ω^(g)_μνω^(g)^μν
        + 2 S_μνρσ^(g)^μνU^ρ
       -
        2 S_μνρω^(g)^μνU^ρ,












 §.§ Weyssenhoff fluid

The Weyssenhoff fluid is a perfect fluid with spin. 
Macroscopically it is a continuous medium, however, 
microscopically the Weyssenhoff fluid is characterised by the spin of matter fields <cit.>.
It is mainly studied within the context of ECSK
theory, where it serves as the source of torsion <cit.>. In ECSK, torsion is related algebraically to the
spin of matter via

    S^λ_μν + 2δ^λ_[μS_ν] = 
        κΣ^λ_μν,

where κ 8π G/c^4 and Σ^λ_μν is the spin-density tensor.

It was postulated in <cit.> that for a 
Weyssenhoff fluid, the source Σ^λ_μν admits the form

    Σ^λ_μν = U^λΣ_μν,

where U^λ denotes the 4-velocity of the
fluid and Σ_μν is an antisymmetric
tensor describing the spin-density of matter. 
Furthermore, it was shown in <cit.> that
the Frenkel condition 

    U^μΣ_μν = 0

necessarily arises, meaning that the spin of the matter fields
is spacelike, hence Σ_μνΣ^μν≥ 0.
Inserting the ansatz (<ref>) into (<ref>),
and contracting over λ and ν, we find that
the torsion vector vanishes S_μ = 0.
Therefore the torsion tensor finally reads

    S^λ_μν = κ 
        U^λΣ_μν

for a Weyssenhoff fluid in ECSK.
For simplicity we set κ = 1 in the following (or
likewise we could absorb κ into Σ_μν).

For a Weyssenhoff fluid we find that θ^(g) = 
θ, ω^(g)_μν = ω_μν, and σ^(g)_μν = σ_μν.
Then the Raychaudhuri equation for autoparallels becomes

    d/dτθ = -1/3θ^2
        -R_μνU^μU^ν
        -σ_μνσ^μν
        +ω_μνω^μν,

where the vorticity, assuming hypersurface orthogonality,
can be shown to exactly match the spin-density
ω_μν = Σ_μν.
In order to establish a similar proposition as in the previous sections we would now have to require the modified SEC, i.e., that

    R_μνξ^μξ^ν≥Σ_μνΣ^μν

holds for all timelike vectors ξ^μ.

Explicitly separating Levi-Civita and torsional contributions,
(<ref>) becomes

    d/dτθ =    -1/3θ^2
        -R_μν
        U^μU^ν
        -σ_μνσ^μν
       -2U^μ(∂_αΣ^α_μ + 
        Γ^λ_μαΣ_λ^α),

with modified SEC

    R_μνξ^μξ^ν≥ 
        -2ξ^μ(∂_αΣ^α_μ + 
        Γ^λ_μαΣ_λ^α)

for all timelike ξ^μ. As in the previous section, for the Weyssenhoff fluid we have that 
θ = θ.



Let (ℳ,g) be a spacetime, equipped with an independent
metric compatible affine connection Γ.
Let matter be described by a Weyssenhoff fluid in EC theory,
i.e., torsion admits the form S^λ_μν = κ U^λΣ_μν, where 
κ 8π G/c^4, U^λ is the 4-velocity of the fluid and
Σ_μν an antisymmetric tensor describing
the spin-density of matter.
Let the Ricci tensor satisfy (<ref>)
for all timelike ξ^μ. Consider a hypersurface orthogonal congruence
of timelike autoparallels emanating from p ∈ℳ.
If θ_0θ(τ_0) < 0 at some 
point r∈ℳ at parameter value τ_0 of a member of the congruence in the future 
of p, then there exists a point q conjugate to p within
proper time τ≤ -3/θ_0 from r (assuming the congruence extends that far).



The shear equation is found to be 

    D/Dτσ_μν = 
            R_⟨μ|αβ|ν⟩U^α
        U^β - 2/3θσ_μν
        - σ_α⟨μσ_ν⟩^α
       - Σ_α⟨μΣ_ν⟩^α

or likewise 

    D/Dτσ_μν = 
           R_⟨μ|αβ|ν⟩U^α
        U^β - 2/3θσ_μν
        - σ_α⟨μσ_ν⟩^α
       - 2U^αU^βU_(μ|∂_βΣ_|ν)α
        + 2U^αU^βΓ^λ_αβU_(μΣ_ν)λ
       - 2U^α∂_⟨μΣ_ν⟩α
        +2U^αΓ^λ
        _α⟨νΣ_μ⟩λ
        +2Σ^λ_(μσ_ν)λ

after splitting the Levi-Civita contributions and torsional parts,
with σ_μν = σ_μν.
Torsion is therefore anticipated to induce direction-dependent effects on the congruence of curves.
The vorticity equation
on the other hand reduces to an algebraic equation for the
spin-density of matter

    2Σ_[ν|α∇_|μ]U^α=
        -Σ^λ_[μΣ_ν]λ - 2/3θΣ_μν
        + 2σ_α[μΣ_ν]^α,

or equivalently

    3 A_ρΣ_[μ^ρU_ν] =
        Σ_μ^ρ∇_[νU_ρ]
        - Σ_ν^ρ∇_[μU_ρ].

At first glance, (<ref>) appears to impose a different constraint than (<ref>). Investigating the consistency of these two equations is necessary, though it falls outside the scope of this work.


























§ SINGULARITY THEOREM WITH TORSION


In order to be able to prove the generalised Singularity
Theorem below for timelike curves with torsion, we need 
to state Wald's proposition <cit.>
in the most general case.
Define the scalar 

    C(U^μ)     ω^(g)_μνω^(g)^μν
        + D_μA^μ + 
        A_μA^μ
       - 2/3θ^(g) 
        S_μU^μ
        - 2 S_μνρ
        U^μU^νA^ρ
       + 2 S_μνρσ^(g)^μνU^ρ
        -
        2 S_μνρω^(g)^μνU^ρ
       -2D/Dτ(
        U^αS_α),

then the Raychaudhuri eq. (<ref>)
takes the more tractable form

    d/dτθ^(g) = 
            -1/3θ^(g)^2
            -R_μνU^μU^ν
            -σ^(g)_μνσ^(g)^μν + C(U^μ).

In this general formulation we have:

Let (ℳ,g) be a spacetime, equipped with an independent
metric compatible affine connection Γ. Let the Ricci tensor satisfy R_μνξ^μξ^ν≥ C(ξ^μ)
for all timelike ξ^μ. Consider a hypersurface orthogonal congruence
of timelike curves emanating from p ∈ℳ.
If θ_0θ^(g)(τ_0) < 0 at some 
point r∈ℳ at parameter value τ_0 of a member of the congruence in the future 
of p, then there exists a point q conjugate to p within
proper time τ≤ -3/θ_0 from r (assuming the congruence extends that far).


Note that we are not restricting ourselves to 
autoparallel curves here. 

Another crucial point is that singularity theorems usually have a requirement on causality. In our case
that will be the condition of spacetime being globally hyperbolic. This means that spacetime has a Cauchy surface, which per definition is a closed achronal set Σ
for which D(Σ) = ℳ. Achronal means that 
no point in Σ belongs to the chronological future of any other point in Σ, while D(Σ) is the domain of dependence of Σ which contains roughly speaking all
points in spacetime that are causally connected to Σ.





The Singularity Theorem for timelike
curves with torsion thus reads:




Let (ℳ,g) be a globally hyperbolic spacetime, equipped with an independent
metric compatible affine connection Γ. Let the Ricci tensor satisfy R_μνξ^μξ^ν≥ C(ξ^μ)
for all timelike ξ^μ.
Assume there exists a smooth spacelike Cauchy surface Σ
on which θ^(g)≤ K < 0 everywhere for a past directed normal timelike congruence, where K is a constant. Then no past directed timelike curve
from Σ can have length greater that 3/K.



    The proof proceeds as in <cit.>, however with some subtleties. Assume there exists a
    past-directed timelike curve γ from Σ
    with length greater than 3/K. Let p denote
    a point on γ lying at a distance beyond 
    3/K from Σ. Then by <cit.> there
    exists a maximum length curve γ̃ from 
    Σ to p. Certainly γ̃ must also
    have length greater than 3/K and further be timelike (since only timelike curves can be homotopic to other timelike curves). But then we know <cit.> that
    γ̃ must be a timelike geodesic with no
    conjugate point between Σ and p. This however
    contradicts Proposition  <ref>.
    Hence the assumption is wrong and γ cannot exist.


Two remarks are in order.
First, in the presence of a general torsion tensor, we are compelled to consider non-autoparallel curves, i.e., U^α∇_α
U^μ≠ 0, during the step of connecting
Σ and p with a maximal length curve. The connecting
curve is a geodesic but should in general not be considered 
an autoparallel, since that would imply torsion
to be totally antisymmetric (recall that the totally antisymmetric
torsion case has been shown to be equivalent to the standard
case with no torsional contributions).
Second, the deviation of the case above to the standard case
without torsion is essentially encoded within C(ξ^μ) (apart from the hidden dependence
of the Ricci tensor on torsion). 










§ CONCLUSION


This work established the foundation for formulating and ultimately proving a singularity theorem for timelike curves with torsion. Although the Raychaudhuri equation incorporating torsion has been already obtained in the literature, we presented an alternative approach by initially deriving the general deviation equation for timelike curves.
The deviation equation furthermore allowed us to generalise the concept of conjugate points, and to connect the appearance of conjugate points to the behaviour of the scalar 
expansion. 
Based on compatibility with standard cosmology, we opted to impose the condition of hypersurface orthogonality.
We observed that this condition necessitates the presence of a non-vanishing vorticity tensor, which is in stark contrast to the standard case without torsion.
For several specific torsion forms, we subsequently established modified SECs under which the emergence of conjugate points is assured. In particular we found that 
in the case of a totally antisymmetric torsion tensor, all
torsional contributions cancel out.
For the case of a Weyssenhoff fluid we discovered that 
vorticity is exactly equal to the spin-density of matter, and consequently to torsion, which facilitates the general consensus that torsion 
constitutes a twisting of spacetime.
Finally we formulated and proved a singularity theorem for 
timelike curves with torsion. We observed that in the case where torsion is not totally antisymmetric, the inclusion of non-autoparallel curves is essential for incorporating the contributions of torsion. In other words, the autoparallel acceleration terms involving A^μ must be integrated into the Raychaudhuri equation.

While the formulation of singularity theorems represents an essential initial step, the ultimate objective is to circumvent the conditions imposed by these theorems to prevent the occurrence of singularities. The scalar C introduced in the previous section fundamentally indicates how torsion can violate the modified SECs and, consequently, which torsional forms are physically feasible.
We anticipate a more thorough investigation of this matter in future research.

Furthermore, we would like to emphasise that the status of ECs remains uncertain. While they serve as useful guidelines for physically reasonable scenarios, it is evidently necessary to violate them in certain cases to prevent unphysical behavior. Thus, a reevaluation of the standing and interpretation of these ECs is warranted, as highlighted in 
<cit.>.










Since this paper is a formative paper on the mathematics of the subject and a tentative analysis for the various torsion cases, the following comments can be discussed in next paper. Hence they are just for perspective to ensure a coherent final draft of paper instead of suggesting further additions.
1) The discussion on energy condition can be argued in 2 different ways. 
1.1) Firstly, we can motivate them from understanding of Physics. This is to say for example that nearby geodesics must be attractive, and so Ric(ξ,ξ) must be constrained so that (̇θ) is negative. However, one must still see that some of the energy conditions can be discussed in such a manner without Einstein equations. As in, if like WEC one has to motivate the energy conditions through the stress energy tensor then its relation to raychaudhuri equation is not self-evident. but rather implied through the energy condition. As such whether such a condition imply collapse of nearby geodesics is uncertain.
1.2) We can keep them as they are and find out novel behaviour which go against physically acceptable behaviour. For example, if the current form of SEC implies that for torsion spacetimes that a 
2) Other aspect, and this is for the next paper but still, has to do with the source of the torsion. For example, whether the spin density increases for collpasing matter. Indeed for the wessenhof fluid we exxpect that a negative theta will give postive time derivative of spin density. Hence eventually the collapse will stop.








§ ACKNOWLEDGEMENTS

The authors are indebted to the “Walter Greiner-Gesell­schaft zur Förderung
der physi­ka­lischen Grundlagenforschung e.V.” (WGG) in Frankfurt for their support. 
AvdV and DV especially thank the Fueck Stiftung for support. 

The authors also wish to thank Jürgen Struckmeier, Johannes Kirsch, Vladimir Denk, Marcelo Netz-Marzola, David Benisty and Tomoi Koide for valuable discussions.












§ EXTENDING THE DEVIATION VECTOR FIELD


This section elaborates on the intermediate steps between  
(<ref>) and (<ref>). First act with (<ref>)
on an arbitrary smooth function f on ℳ.
Then, per definition of the pushforward, the r.h.s. yields

    ((ϕ_τ)_* (V̂_γ(s)))(f) = 
        V̂_γ(s)(f∘ϕ_τ).

Choose any local frame x, then we may write

    V̂_γ(s)(f∘ϕ_τ) = 
        V̂_γ(s)^μ(∂/∂ x^μ)_γ(s)(f∘ϕ_τ),

where (∂/∂ x^μ)_γ(s) denotes the coordinate basis of the
tangent space at γ(s). The action of the coordinate
basis is given by

    (∂/∂ x^μ)_γ(s)(f∘ϕ_τ) = ∂/∂ x^μ.((f∘ϕ_τ)∘ x^-1)|_x(γ(s)).

Now choose another frame y, rewrite 
(f∘ϕ_τ)∘ x^-1 = f∘ y^-1∘( y∘ϕ_τ∘ x^-1) and use the chain rule to find
that the r.h.s. of (<ref>) becomes

    .∂ (f∘ y^-1)/∂ y^ν|_y(γ_s(τ)).∂(y^ν∘ϕ_τ∘ x^-1)/∂ x^μ|_x(γ(s)),

which is equal to 

    .∂(y^ν∘ϕ_τ∘ x^-1)/∂ x^μ|_x(γ(s))(∂/∂ y^ν)_γ_s(τ)(f).

The contraction of (<ref>) with

    V̂_γ(s)^μdef=dγ^μ/d s(s) = d(x^μ∘γ)/d s(s),

and the use of the chain rule yield

    d/ds.(y^ν∘ϕ_τ∘ x^-1∘ x ∘γ)|_s(∂/∂ y^ν)_γ_s(τ)(f),

or 

    ∂/∂ s.(y^ν∘γ_s (τ))|_s(∂/∂ y^ν)_γ_s(τ)(f),

which is equivalent to

    ∂γ_s^ν(τ)/∂ s(∂/∂ y^ν)_γ_s(τ)(f).

Comparing (<ref>) with the l.h.s. of (<ref>):

    Ṽ_γ_s (τ)^ν(∂/∂ y^ν)_γ_s(τ)(f),

we finally find

    Ṽ_γ_s (τ)^μ = 
        ∂γ_s^μ(τ)/∂ s.






§ ENERGY CONDITIONS


Energy conditions are generically posed, independent of details of matter source, conditions on the behaviour of matter on a dynamic curved space. Since these conditions are independent of the idiosyncrasies of matter fields, these conditions typically, in conjunction with Einstein's field equations, lead to far-reaching consequences in determining the global structure of spacetime manifolds and the nature of the gravitational field, for example, through singularity theorems. If one assumes the energy-momentum tensor of the matter fields to be of Hawking Ellis type I <cit.> and, further, utilises the Einstein field equations, then it is possible to propose these conditions in various forms, see Table <ref>. 

One can propose a form involving the curvature of spacetime (geometric form), energy-momentum tensor (physical form), or the energy density and pressure of the matter fields (effective form) <cit.>. However, it is difficult to suggest one form or the other to be fundamental. For example, the weak energy condition only has a clear physical justification through its physical form. On the other hand, one may suggest that only covariant (observer-independent) formulations of the conditions should be considered fundamental.

















