Paper Title: Decoupled Anisotropic Buchdahl's Relativistic Models in $f(\mathbb{R},\mathbb{T})$ Theory.







Decoupled Anisotropic Buchdahl's Relativistic Models in f(ℝ,𝕋) Theory
    Tayyab Naseer tayyabnaseer48@yahoo.com;
tayyab.naseer@math.uol.edu.pk  and M. Sharif msharif.math@pu.edu.pk 

Department of Mathematics and Statistics, The University of Lahore,

1-KM Defence Road Lahore-54000, Pakistan.
    
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§ ABSTRACT 
This paper constructs three different anisotropic extensions of the
existing isotropic solution to the modified field equations through
the gravitational decoupling in f(ℝ,𝕋) theory.
For this, we take a static sphere that is initially filled with the
isotropic fluid and then add a new gravitational source producing
anisotropy in the system. The field equations now correspond to the
total matter configuration. We transform the radial metric component
to split these equations into two sets characterizing their parent
sources. The unknowns comprising in the first set are determined by
considering the Buchdahl isotropic solution. On the other hand, we
employ different constraints related to the additional gravitational
source and make the second system solvable. Further, the constant
triplet in Buchdahl solution is calculated by means of matching
criteria between the interior and exterior geometries at the
spherical boundary. The mass and radius of a compact star LMC X-4
are used to analyze the physical relevancy of the developed models.
We conclude that our resulting models II and III are in
well-agreement with acceptability conditions for the considered
values of the parameters.

Keywords: Modified theory;
Anisotropy; Gravitational decoupling. 

PACS: 04.40.Dg; 04.50.Kd; 04.40.-b.



§ INTRODUCTION


Cosmologists have recently made significant discoveries that
challenge the perception of the arrangement of astrophysical
structures in our universe. Rather than a random distribution, these
structures appear to be systematically organized, sparking
considerable interest among researchers. The detailed study of such
interstellar bodies has become a focal point for scientists seeking
to figure out the mystery of the accelerated expansion of the
cosmos. It has become evident through various experiments that our
universe contains abundance of a force countering the pull of
gravity, and thus driving such expansion. This mysterious force is
referred to the dark energy due to its obscure nature that continues
to confuse researchers. While general relativity (𝔾ℝ)
explains this expansion up to some extent, it grapples with issues
related to the cosmological constant. Therefore, some extensions of
this theory need to be proposed.

A straightforward generalization to 𝔾ℝ is the
f(ℝ) theory that represents a significant advancement in
the field of theoretical physics. It introduces a modification to
the Einstein-Hilbert action, where the roles of the Ricci scalar
ℝ and its generic function are interchanged. This theory
has provided promising results and has been employed to investigate
self-gravitating systems <cit.>-<cit.>. Astashenok et al.
<cit.> investigated the causal limit of maximum mass for stars in
this modified framework. Their findings led to the conclusion that
the secondary component of the compact binary GW190814 is likely to
be a neutron star, a black hole, or possibly a rapidly rotating
neutron star, ruling out the possibility of it being a strange star.
There is a body of literature, pointing out such important works by
various researchers <cit.>-<cit.>. Bertolami et al. <cit.>
initially presented the idea of studying the fluid-geometry
interaction in f(ℝ) framework and executed this by
combining the matter Lagrangian density and ℝ as a single
function. This notion prompted astronomers to put their focus on the
discussion of the rapid expansion of the universe <cit.>.

Soon after this, Harko et al. <cit.> generalized this concept at
the action level, and pioneered a new gravitational theory. This was
named as f(ℝ,𝕋) theory where the effects of
geometry and matter configuration are coupled through ℝ
and trace of the energy-momentum tensor (𝔼𝕄𝕋)
𝕋. This generalized function results in the non-conserved
system, hence, an extra force (that depends on physical parameters
like pressure and density <cit.>) appears in the gravitational
field which forces the test particles to move in a non-geodesic
path. Houndjo <cit.> employed a minimal model of this modified
theory to explain the conversion of the matter-dominated phase into
late-time acceleration era successfully. Among several
f(ℝ,𝕋) models, ℝ+2ϖ𝕋
gained much importance in the literature that produces physically
acceptable compact interiors. This model is adopted by Das et al.
<cit.> to establish a gravastar-like model consisting of three
layers, each of them is represented by different equations of state.
Multiple stellar interiors have been discussed by using various
approaches in the context of this modified model
<cit.>-<cit.>. One significant notion of
f(ℝ,𝕋) theory is that it considers the quantum
effect, leading the possibilities of particle production. This
aspect holds much importance in astrophysical research as it posits
a connection between quantum theory and f(ℝ,𝕋)
gravity. Some notewrothy applications of this theory in both
astrophysics  and cosmology can be seen in <cit.>-<cit.>.
Recently, Zaregonbadi et al. <cit.> have examined the viability
of this modified theory to explore the effects of dark matter on the
galactic scale.

The standard model of the cosmos is believed to be mainly based on
the homogeneity and isotropy at a large-scale, however, there exists
pressure anisotropy at some small-scales <cit.>-<cit.>.
Further, the geometry of compact objects guarantees the presence of
anisotropic pressure and inhomogeneous fluid distribution. The
former factor appears when there is a difference between the
pressure in both radial and transverse directions. There is a class
of components that produces anisotropy in the interior configuration
such as phase transitions <cit.>, pion condensation
<cit.>, neutron stars surrounded by a strong electromagnetic
field <cit.>, and some other factors <cit.>. Another
factor that causes the anisotropy is the gravitational effects
produced by the tidal forces <cit.>. The isotropy of our
universe has recently been examined by analyzing X-rays coming from
galactic clusters <cit.>. The same strategy has then been
carried out on massive structures, and it was concluded that the
nature of cosmos is anisotropic <cit.>. The pressure anisotropy
is thus much significant to be studied and affects several physical
characteristics like the energy density, gravitational redshift and
mass, etc., of a self-gravitating system.

The self-gravitating celestial objects represented by non-linear
gravitational field equations in 𝔾ℝ or other modified
theories prompted astrophysicists to obtain their exact/numerical
solutions. The compact interiors could be of physical interest only
if the corresponding formulated solution fulfills the required
conditions. Multiple techniques, in this regard, have recently been
suggested, one of them is the gravitational decoupling that is
employed to model a compact interior possessing multiple sources
including anisotropy, heat dissipation, shear, etc. The initial idea
of this technique was based on the fact that the field equations
comprising different sources can be decoupled into multiple sets,
and thus it becomes an easy task to solve each set individually.
Ovalle <cit.> recently pioneered the minimal geometric
deformation (MGD) scheme which provides some enticing ingredients to
formulate physically acceptable stellar solutions in the braneworld.
Following this, an isotropic spherically symmetric matter
distribution was discussed by Ovalle and Linares <cit.> who
formulated an analytical solution in the braneworld and found it
consistent with Tolman-IV ansatz. Casadio et al. <cit.> obtained
the Schwarzschild geometry in the context of Randall-Sundrum
braneworld theory by extending the above strategy.

Ovalle and his collaborators <cit.> adopted a spherically
symmetric isotropic interior and developed its physically feasible
anisotropic version using the MGD scheme. Sharif and Sadiq <cit.>
introduced the straightforward extension of this technique to the
charged case where they constructed two different anisotropic
counterparts of Krori-Barua metric potentials and discussed their
stability. Multiple metric ansatz have been extended to obtain their
corresponding anisotropic analogs through MGD in modified
f(ℝ) theory <cit.>. The Durgapal-Fuloria metric
coefficients have been taken as a seed isotropic source through
which several physically relevant anisotropic solutions were
obtained <cit.>. Different researchers proposed the anisotropic
extensions of isotropic Heintzmann as well as Tolman VII solutions
and found them stable in the considered range <cit.>. Sharif
and Ama-Tul-Mughani <cit.> took the axial spacetime into account
and formulated their corresponding well-behaved solutions. We have
also obtained such charged/uncharged anisotropic analogs of
Krori-Barua ansatz in a strong non-minimally coupled gravitational
theory <cit.>-<cit.>.

This article formulates three different anisotropic solutions that
are, in fact, the extensions of an isotropic interior to the
additional matter source coupled gravitationally to the seed source
in f(ℝ,𝕋) framework. The following lines present
how this paper is organized. The fundamentals of the modified
f(ℝ,𝕋) theory and its relevant field equations
corresponding to the total (seed and newly added) matter source are
formulated in the next section. Section 3 introduces the
MGD transformation that helps to decouple the field equations into
two sets. We then adopt the Buchdahl's solution in section
4 and calculate the unknown constants through boundary
conditions. Section 5 presents some conditions whose
fulfillment leads to physically acceptable model. Three newly
developed anisotropic solutions and their graphical interpretation
are provided in section 6. Lastly, we summarize our
outcomes in the last section.



§ F(ℝ,𝕋) GRAVITY


The Einstein-Hilbert action for the modified
f(ℝ,𝕋) theory becomes (with κ=8π) after
the inclusion of an additional field as <cit.>

    S=∫√(-g)[f(ℝ,𝕋)/16π
    +_m+η_𝔇]d^4x,

where _m is the fluid's Lagrangian density. We suppose an
extra source to be gravitationally coupled with the parent matter
configuration whose corresponding Lagrangian is denoted by
_𝔇. The decoupling parameter η explains
how much an extra source influences the physical properties
characterizing a self-gravitating system. Also, the metric tensor
g_σω in this case provides the determinant indicated
by g. The least action principle is applied on the action
(<ref>), leading to the field equations given by

    𝔾_σω=8π𝕋_σω^(tot),

where 𝔾_σω and
𝕋_σω^(tot) describe the geometric
sector and the interior fluid distribution, respectively. The later
term is further classified as

    𝕋_σω^(tot)=𝕋_σω^(eff)+η𝔇_σω=1/f_ℝ𝕋_σω+𝕋_σω^(C)+η𝔇_σω.

Here 𝔇_σω is the additional fluid source.
Moreover, the effective matter sector
𝕋_σω^(eff) is divided into the
usual and modified 𝔼𝕄𝕋s. The later term
𝕋_σω^(C) takes the form

    𝕋_σω^(C)   =   1/8π
    f_ℝ[f_𝕋𝕋_σω+{ℝ/2(f/ℝ
    -f_ℝ)-_mf_𝕋}g_σω
       -   (g_σω-∇_σ∇_ω)f_ℝ+2f_𝕋g^ζβ∂^2
    _m/∂ g^σω∂
    g^ζβ],

where f_𝕋 and f_ℝ mean ∂
f(ℝ,𝕋)/∂𝕋 and ∂
f(ℝ,𝕋)/∂ℝ, respectively.
Furthermore,
≡1/√(-g)∂_σ(√(-g)g^σω∂_ω)
is the D'Alembert operator and ∇_σ is the covariant
divergence. Also, we adopt _m=P (P is an isotropic
pressure), leading to ∂^2 _m/∂
g^σω∂ g^ζβ=0.

We consider that the geometrical structure is initially filled with
an isotropic fluid which can be expressed through the following
𝔼𝕄𝕋

    𝕋_σω=(μ+P)𝒦_σ𝒦_ω+P
    g_σω,

where μ and 𝒦_σ are the energy density and
four-velocity, respectively. We obtain the trace of Eq.(<ref>) as
follows

    2f+𝕋(f_𝕋+1)-ℝf_ℝ-3∇^σ∇_σf_ℝ-4f_𝕋_m
    +2f_𝕋g^ζβg^σω∂^2_m/∂
    g^ζβ∂ g^σω=0.

Since the action of this theory involves matter-geometry coupled
functional, the disappearance of the matter term, i.e.,
𝕋=0 (or vacuum case) reduces all our results in
f(ℝ) gravity. Moreover, the non-zero divergence of the
usual 𝔼𝕄𝕋 is observed in this theory due to such strong
interaction. Resultantly, an additional force appears in the field
of self-gravitating object that alters the geodesic path of the
moving test particles. Its mathematical expression is given by

    ∇^σ𝕋_σω   =f_𝕋/8π-f_𝕋[(𝕋_σω
    +Υ_σω)∇^σlnf_𝕋+∇^σΥ_σω
       -8πη/f_𝕋∇^σ𝔇_σω
    -1/2g_ζβ∇_ω𝕋^ζβ],

where
Υ_σω=g_σω_m-2𝕋_σω-2g^ζβ∂^2
_m/∂ g^σω∂ g^ζβ.

We consider a spherical interior geometry that is distinguished from
the exterior region at the hypersurface Σ given by the
following line element

    ds^2=-e^ν_1 dt^2+e^ν_2 dr^2+r^2(dθ^2+sin^2θ
    dϑ^2),

where ν_1=ν_1(r) and ν_2=ν_2(r). The corresponding
four-velocity now takes the form in terms of temporal metric
component as

    𝒦_σ=-δ^0_σ
    e^ν_1/2=(-e^ν_1/2,0,0,0).

In order to have some meaningful results, we adopt a standard
modified model. Although the literature presents several
matter-geometry coupled (minimal as well as non-minimal)
f(ℝ,𝕋) models, however, we adopt a minimal one
which is given by

    f(ℝ,𝕋)=f_1(ℝ)+
    f_2(𝕋)=ℝ+2ν_3𝕋,

where 𝕋=-μ+3P and ν_3 is a real-valued constant.
There are two main reasons behind this choice highlighted as


  * For complicated f(ℝ,𝕋) functionals containing exponential, logarithmic,
or polynomials terms of 𝕋, or higher-order curvature
terms such as ℝ^2, we will obtain a very complicated form
of the field equations. So, the problem appears when it comes to
split those equations into two sets using MGD technique (following
section elaborates MGD methodology thoroughly). In such cases, we
may be unable to obtain two new sectors characterizing their parent
fluid sources.


  * On the other hand, in the case of complex
f(ℝ,𝕋) functionals, an issue appears when we
deal with matching conditions across the spherical interface.
Therefore, in order to adjust these issues, we choose the functional
given in Eq.(<ref>).

Moreover, Ashmita et al. <cit.> derived potential slow-roll
parameters by adopting multiple inflation potentials in this
gravitational theory and found their results to be consistent with
the experimental data only when -0.37<ν_3<1.483. This model has
also been used to formulate physically relevant anisotropic version
of the interior Tolman-Kuchowicz spacetime <cit.>. We have
developed acceptable decoupled solutions corresponding to
anisotropic configuration in this theory <cit.>.

The spherical spacetime (<ref>) produces the field equations
corresponding to the modified model (<ref>) as

    e^-ν_2(ν_2'/r-1/r^2)
    +1/r^2=8π(μ-η𝔇_0^0)+ν_3(3μ-P),
       e^-ν_2(1/r^2+ν_1'/r)
    -1/r^2=8π(P+η𝔇_1^1)-ν_3(μ-3P),
    
       e^-ν_2/4[ν_1'^2-ν_2'ν_1'+2ν_1”-2ν_2'/r+2ν_1'/r]
    =8π(P+η𝔇_2^2)-ν_3(μ-3P),

where the entities along with ν_3 appear as the
f(ℝ,𝕋) corrections and prime means
∂/∂ r. Moreover, we obtain the generalized
form of Tolman-Opphenheimer-Volkoff equation from (<ref>)
corresponding to the model (<ref>) as

    dP/dr+ν_1'/2(μ+P)+ην_1'/2(𝔇_1^1-𝔇_0^0)+ηd𝔇_1^1/dr
       +2η/r(𝔇_1^1-𝔇_2^2)=-ν_3/4π-ν_3(μ'-P'),

which verifies the nature of this extended theory of gravity to be
non-conserved. Since Eq.(<ref>) is a combination of different
forces that maintain the hydrostatic equilibrium inside a celestial
object, it plays an important role to study the interior's
structural changes. It is observed that the extraction of the
solution of a system (<ref>)-(<ref>) becomes obscure due to
the entanglement of a large number of unknowns, i.e.,
(ν_1,ν_2,μ,P,𝔇_0^0,𝔇_1^1,𝔇_2^2).
Therefore, it becomes necessary to adopt some constraints,
otherwise, the system cannot be solved uniquely. In this context, a
systematic scheme <cit.> is adopted to fulfill our requirement.



§ MINIMAL GRAVITATIONAL DECOUPLING


Gravitational decoupling is an efficient approach that transforms
the metric potentials in a new reference frame and makes it easy to
construct the solution of highly non-linear field equations
representing compact systems. For this, a new line element is
considered as a solution to the field equations
(<ref>)-(<ref>) given by

    ds^2=-e^ν_4(r)dt^2+1/ν_5(r)dr^2+r^2(dθ^2+sin^2θ
    dϑ^2).

Following equations transform the metric potentials and decouple the
field equations as

    ν_4→ν_1=ν_4+ηt_1,   ν_5→
    e^-ν_2=ν_5+ηt_2,

where t_1 and t_2 correspond to g_tt and
g_rr components, respectively. Gravitational decoupling offers
two different techniques, i.e., minimal and extended deformations.
The main difference between minimal and extended decoupling schemes
is that the geometric deformation is applied only on the radial
metric component in the former case, leaving temporal coefficient as
an invariant quantity <cit.>. However, in the later scenario,
both temporal as well as radial potentials are deformed to covert
the field equations in a new reference frame <cit.>. Moreover,
the MGD technique works as long as the interaction between the
matter sources is purely gravitational, implying that each fluid
source must be conserved individually. This is in contrast with the
extended decoupling approach in which the total fluid configuration
is conserved but the non-conservation phenomenon occurs when it
comes to individual systems of equations. The transfer of the energy
between different source is also allowed in this scenario. Further,
the deformation function plays a crucial role in the process of
gravitational decoupling. Its selection is based on the specific
characteristics of the problem and the intended simplification, with
the additional requirement of ensuring the spherical symmetry of the
solution. In the current setup, we have t_1→
0, t_2→T. Hence, Eq.(<ref>)
switches into

    ν_4→ν_1=ν_4,   ν_5→
    e^-ν_2=ν_5+ηT,

where T=T(r). It must be kept in mind that such
a linear mapping does not bother the considered spherical symmetric.
We apply the transformation (<ref>) on the system
(<ref>)-(<ref>) to divide into two different sets
corresponding to η=0 and 1. The first set corresponds to the
initial (isotropic) source and is given by

    e^-ν_2(ν_2'/r-1/r^2)
    +1/r^2=8πμ+ν_3(3μ-P),
       e^-ν_2(1/r^2+ν_1'/r)
    -1/r^2=8π P-ν_3(μ-3P),
       e^-ν_2/4[ν_1'^2-ν_2'ν_1'+2ν_1”-2ν_2'/r+2ν_1'/r]
    =8π P-ν_3(μ-3P).

The simultaneous solution of Eqs.(<ref>) and (<ref>) results
in the state variables as

    μ   =e^-ν_2/8r^2(ν_3^2+6πν_3+8π^2)[ν_3
    r ν_1 '+(3 ν_3 +8 π ) r ν_2 '+2 (ν_3 +4 π )
    (e^ν_2 -1)],
    
    P   =e^-ν_2/8r^2(ν_3^2+6πν_3+8π^2)[(3
    ν_3 +8 π ) r ν_1 '+ν_3  r ν_2 '-2 (ν_3 +4 π )
    (e^ν_2 -1)].

Contrariwise, the field equations representing the additional matter
distribution (𝔇^σ_ω) are obtained as

    8π𝔇_0^0=T'/r+T/r^2,
       8π𝔇_1^1=T(ν_1'/r+1/r^2),
       8π𝔇_2^2=T/4(2ν_1”+ν_1'^2+2ν_1'/r)
    +T'(ν_1'/4+1/2r).


It is interesting to note that both (seed and new) sources are
individually conserved, and hence, the energy's exchange between
them is not allowed in this scheme. Further, we successfully
decouple the set of equations (<ref>)-(<ref>) that makes
easier to solve both sets independently. There appear four unknowns
(μ,P,ν_1,ν_2) in Eqs.(<ref>) and (<ref>), thus we
must require a well-behaved metric ansatz to equal the number of
unknowns to that of equations. Also, four unknowns
(T,𝔇_0^0,𝔇_1^1,𝔇_2^2)
are appeared in the system (<ref>)-(<ref>), a unique
solution shall be obtained by adopting a constraint on
𝔇-sector. Since the additional source makes the initial
matter source anisotropic, we identify the matter determinants as
follows

    μ̃=μ-η𝔇_0^0,  P̃_r=P+η𝔇_1^1,   P̃_=P+η𝔇_2^2,

which help to define the total anisotropic factor as

    Π̃=P̃_-P̃_r=η(𝔇_2^2-𝔇_1^1),

verifying its disappearance for the case when η=0, i.e., the
effect of a new matter source is removed.



§ BUCHDAHL SOLUTION AND BOUNDARY CONDITIONS


This section is devoted to the formulation of a solution
corresponding to the first set given by (<ref>) and
(<ref>). We have already discussed that a particular form of
metric potentials is required to solve the system. In this context,
Buchdahal ansatz <cit.> has attracted significant attention
among researchers. This assumption has proven valuable in
investigating almost all physically viable known super-dense star
models. Vaidya and Tikekar <cit.> further refined the Buchdahl
ansatz and discussed spheroidal geometries for the 4-dimensional
hypersurface. Such a particular spheroidal condition has been proved
to be highly effective in obtaining exact solutions to the field
equations, a task that proves challenging in numerous other
scenarios. Kumar et al. <cit.> extensively explored this
particular spacetime, focusing on charged compact objects coupled
with the isotropic fluid distribution. Additionally, Sharma et al.
<cit.> determined the maximum attainable masses and radii for
various values of the surface density within the Vaidya-Tikekar
spacetime. Maurya et al. <cit.> adopted this metric and discussed
eight different cases for Buchdahl's dimensionless parameter and
found all of them valid at every point within the interior geometry.
Various authors used this metric to produce viable as well stable
charged/uncharged interiors in the contexts of 𝔾ℝ and
f(ℝ,𝕋) theory <cit.>-<cit.>. This metric is
adopted as follows

    e^ν_1(r)   =   C_1[(1+C_3r^2)^3/2+C_2(5+2C_3r^2)
    √(2-C_3r^2)]^2,
    
    e^ν_2(r)   =   2(1+C_3r^2)/2-C_3r^2,
    μ   =   3C_3/8(ν_3+2π)(ν_3+4π)(C_3r^2+1)^2{(C_3r^2+1)√(2-C_3^2r^4+C_3r^2)
       -   C_2(C_3r^2-2)(2C_3r^2+5)}^-1[2 (C_3 r^2+1) √(2-C_3^2 r^4+c
    r^2)
       ×   {3 ν_3 +2 π(C_3
    r^2+3)}-C_2(C_3 r^2-2)
    {3 ν_3 (4C_3 r^2+7)
       +   4 π(C_3 r^2+3) (2 C_3
    r^2+5)}],
    
    P   =   3C_3/8(ν_3+2π)(ν_3+4π)(C_3r^2+1)^2{C_2(C_3r^2-2)(2C_3r^2+5)
       -   (C_3r^2+1)√(2-C_3^2r^4+C_3r^2)}^-1[2 (C_3r^2+1) √(2-C_3^2
    r^4+C_3r^2)
       ×   {ν_3(2C_3r^2-3)+6π(C_3r^2-1)}-C_2(C_3
    r^2-2) {4 (C_3r^2+1)
       ×   (2ν_3C_3r^2+π(6C_3r^2+3))-3
    ν_3 }],

where C_1, C_2 and C_3 are
unknown quantities that must be determined to perform the graphical
analysis.

In this perspective, the junction (or matching) conditions play a
vital role in the study of structural characteristics of a compact
object at some boundary surface (Σ: r=R). The
interior geometry now becomes in terms of ansatz (<ref>) and
(<ref>) as

    ds_-^2   =-C_1[(1+C_3r^2)^3/2+C_2(5+2C_3r^2)
    √(2-C_3r^2)]^2dt^2
       +2(1+C_3r^2)/2-C_3r^2
    dr^2+r^2dθ^2+r^2sin^2θ dϑ^2,

whose corresponding exterior spacetime is described by the
Schwarzschild metric. This is given as follows

    ds_+^2=-r-2M/rdt^2+r/r-2Mdr^2+
    r^2dθ^2+r^2sin^2θ dϑ^2,

where M shows the total mass. The first fundamental form
of these constraints provide continuity of the metric potentials at
the hypersurface as

    g_tt^- _=^Σ g_tt^+,    g_rr^- _=^Σ g_rr^+,

providing the following expressions by employing the metrics
(<ref>) and (<ref>) as

    R-2M/R   =   C_1[(1+C_3R^2)^3/2
    +C_2(5+2C_3R^2)√(2-C_3R^2)]^2,
    R/R-2M   =   2(1+C_3R^2)/2-C_3R^2.



Since we have two equations and three unknowns, we need one more
constraint. For this, we use the second fundamental form that claims
that the isotropic pressure disappears at the boundary, i.e.,
P _=^Σ 0. Thus, Eq.(<ref>) provides

    2 (C_3r^2+1) √(2-C_3^2
    r^4+C_3r^2){ν_3(2C_3r^2-3)+6π(C_3r^2-1)}
       -C_2(C_3 r^2-2) {4
    (C_3r^2+1)(2ν_3C_3r^2+π(6C_3r^2+3))-3
    ν_3 }=0.

By solving Eqs.(<ref>)-(<ref>) simultaneously, we get
the triplet (C_1,C_2,C_3) as

    C_1   =   1/3F^2(R-2M/R),
    C_2   =   R(3R-4M)√(2R(R-2M)/(4M-3R)^2){4M(5ν_3+12π)-9R(ν_3+2π)}/(2M-R){8MR(7ν_3+6π)
    -16ν_3M^2-9R^2(ν_3-4π)},
    C_3   =   -4M/R^2(4M-3R),

where F is provided by

    F   =3(R/3R-4M)^3/2-6R{4M(5ν_3+12π)
    -9R(ν_3+2π)}/8MR(7ν_3+6π)-16ν_3M^2-9R^2(ν_3-4π)
       ×[(4M-5R)/(2M-R)√(2M-R/4M-3R)√(R(R-2M)/(4M-3R)^2)].

We use the calculated values of the mass and radius as
M=1.04 ± 0.09 M_ and R=8.301 ±
0.2 km, respectively, of a compact star candidate LMC X-4
<cit.> to evaluate the above constant triplet. Table 1
provides these values corresponding to different choices of the
model parameter. It is observed that the constant C_1
(C_2) is in inverse (direct) relation with the
parameter ν_3, however, the unknown C_3 is not
dependent on ν_3.



§ PHYSICAL ACCEPTABILITY CONDITIONS OF A COMPACT MODEL


There are several criteria in the literature to check physical
acceptability of compact structures <cit.>-<cit.>. Some other
works are <cit.>-<cit.>. They are highlighted as follows.


  * One of the fundamental aspects is the existence of geometric singularities inside the
star. The behavior of the metric components should be positively
increasing and free from singularity in an acceptable
self-gravitating interior. The metric coefficients (<ref>) and
(<ref>) in the core of star become

    e^ν_1(r)|_r=0=C_1[1+5√(2)C_2]^2,  
    e^ν_2(r)|_r=0=1,

and their first derivatives are

    (e^ν_1(r))'   =6C_1C_3rf(r){(1+C_3r^2)^3/2+C_2(5+2C_3r^2)
    √(2-C_3r^2)}/√(2-C_3r^2),
    
    (e^ν_2(r))'   =12C_3r/(2-C_3r^2)^2,

where
f(r)=C_2-2C_2C_3r^2+√(2+C_3r^2-C_3^2r^4).
We notice that the above both derivatives disappear at the center
implying the regularity of these potentials. Figure 1
reveals that they are minimum at r=0 and increasing outwards to
reach their maximum at the spherical surface.


  * The profile of matter determinants such as energy density and
pressure components should be finite and positive everywhere.
Further, they must reach their maximum (minimum) at r=0
(Σ: r=R). Likewise, the decreasing trend of these
variables towards the boundary can also be assured if their first
derivative disappears at r=0 and negative outwards.


  * The mass of the spherical stellar structure is defined by

    m(r)=1/2∫_0^Rw^2μ
    dw.

The mass-radius ratio, known as the compactness, measures that how
tightly the particles are bound with each other in a
self-gravitating system. Mathematically, we have

    ζ(r)=m(r)/r.

Its value must be less than 4/9 everywhere in a spherical
interior <cit.>. We can also describe the surface redshift in
terms of mass or compactness of a star. This is given as

    z(r)=1-√(1-2ζ(r))/√(1-2ζ(r)).

Since we are discussing anisotropic matter distribution, the
redshift reaches its maximum by 5.211 at the boundary surface to
get a feasible model <cit.>.


  * Another key feature to check feasibility of the stellar model
is the energy conditions which are in fact linear combinations of
the matter determinants. These are given as follows

    μ≥ 0,   μ+P_≥ 0,
          μ+P_r≥
    0,   μ-P_≥ 0,
          μ-P_r≥ 0,   μ+2P_+P_r≥ 0.

Among these conditions, the dominant energy bounds (i.e.,
μ-P_≥ 0 and μ-P_r≥ 0) are the most important
because they demand μ≥ P_ and μ≥ P_r
everywhere.


  * Different techniques have been suggested in the
literature to check the stability of the compact stars, one of them
is based on the sound speed whose radial and tangential components
are v_sr^2=dP_r/dμ and
v_s^2=dP_/dμ, respectively. Abreu et al.
<cit.> suggested that if the speed of light is greater than
that of sound (i.e., 0 < v_sr^2,  v_s^2 < 1), the
causality would be preserved in a system. Likewise, Herrera
<cit.> proposed that if the total radial force changes its sign
throughout the evolution, then the idea of cracking would occur in
the interior fluid. It must be avoided to obtain a stable model. He
provided that the cracking cannot be occurred only if -1 <
v_s^2-v_sr^2 < 0 holds.





§ FORMATION OF DIFFERENT ANISOTROPIC MODELS


Here we consider two sources characterizing different matter
distributions, the unknowns are increased in the field equations.
Therefore, we choose three different constraints to make such
differential equations solvable. They are given by


  * Model I: Density-like constraint

  * Model II: Pressure-like constraint

  * Model III: A linear equation of state




 §.§ Model I


We consider a constraint depending on the energy density of the
original and anisotropic source to obtain solution to the total
interior fluid. This is given as follows <cit.>

    μ=𝔇_0^0.

We use Eqs.(<ref>) and (<ref>) in (<ref>), and obtain
the differential equation as

    1/8π{T'(r)/r+T(r)/r^2}
    -e^-ν_2/8r^2(ν_3^2+6πν_3+8π^2)
       ×[ν_3 r ν_1 '+(3 ν_3 +8 π ) r ν_2 '+2 (ν_3 +4
    π ) (e^ν_2 -1)]=0.

In terms of Buchdahl's ansatz (<ref>) and (<ref>), the above
equation takes the form

    1/8π{T'(r)/r+T(r)/r^2}-3
    C_3[8 (ν _3+2 π) (ν _3+4 π)
    (C_3 r^2+1)^2 {(C_3
    r^2+1)
       ×√(C_3 r^2+2-C_3^2
    r^4)-C_2 (C_3 r^2-2) (2 C_3
    r^2+5)}]^-1[2 (C_3
    r^2+1)
       √(C_3 r^2+2-C_3^2 r^4)(2 π(C_3 r^2+3)+3 ν _3)-C_2
    (C_3 r^2-2) {3 ν _3(4 C_3
    r^2+7)
        +4 π(C_3 r^2+3) (2
    C_3 r^2+5)}]=0.

The above equation contains one unknown (i.e., the deformation
function T(r)). However, the exact solution of this
equation is not possible because of the terms appearing in the
square root. Therefore, we use the numerical integration to make
this function known. For this purpose, we use the initial condition
T(0)=0 by providing the range of the radius of the
considered star. The corresponding deformed g_rr^-1 component
can be calculated by

    e^-ν_2(r)=2-C_3r^2+2ηT(1+C_3r^2)/2(1+C_3r^2).


We perform the graphical analysis of the obtained deformation
function and its corresponding matter determinants, anisotropy,
energy conditions and some other parameters for the considered
modified model (<ref>). Moreover, we choose η=0.1, 0.3 and
ν_3=0.25, 0.5, 0.75, 1 to study the effect of decoupling
strategy and the modified gravity on the interior of a compact star.
The deformation function obtained from Eq.(<ref>) does not
depend on the decoupling parameter, thus its variation with respect
to different values of the model parameter is plotted in Figure
2. It is observed that this function possesses increasing
trend outwards, however, it takes smaller values for increasing
ν_3.

The corresponding matter variables such as effective energy density,
effective radial/tangential pressures and anisotropy are calculated
through Eqs.(<ref>) and (<ref>). The trend of these
variables is plotted in Figure 3, indicating an acceptable
behavior. We notice that the energy density (pressure components)
decreases (increase) with the increment in η. However, they
decrease by increasing the model parameter. Such a behavior can be
observed by the numerical values provided in Tables 2 and
3. Further, the zero radial pressure at the spherical
boundary is observed for each parametric value (upper right plot).
The anisotropic factor, in this case, becomes null in the core and
increases, otherwise. The parameters in Figure 4 possess
increasing and acceptable profile throughout. The energy density in
Eq.(<ref>) corresponds to the total matter source involving
modified corrections, thus the mass function is dependent on the
decoupling and model parameters. The smaller values of both
parameters ν_3 and η provide more massive interiors (upper
left plot). Figure 5 shows the dominant energy bounds,
displaying positive profile and thus leading to a viable solution.
Figure 6 (lower plot) declares that our resulting model I
is physically unstable because there occurs a cracking in the
interior for every parametric choice.






 §.§ Model II






In this subsection, we construct our second model by choosing the
following constraint <cit.>

    P=𝔇_1^1.

Using Eqs.(<ref>) and (<ref>) in the above equation, we
have

    T(r)/8π{ν_1'/r+1/r^2}
    -e^-ν_2/8r^2(ν_3^2+6πν_3+8π^2)
       ×[(3 ν_3 +8 π ) r ν_1 '+ν_3  r ν_2 '-2 (ν_3 +4
    π ) (e^ν_2 -1)]=0,

that, in terms of the ansatz (<ref>) and (<ref>), leading to

    T(r)/8π{6 C_3
    (C_2-2 C_2 C_3
    r^2+√(C_3r^2+2-C_3^2
    r^4))/(C_3 r^2+1) √(C_3
    r^2+2-C_3^2 r^4)-C_2 (C_3 r^2-2)
    (2C_3 r^2+5)+1/r^2}
       -3
    C_3[8 (ν _3+2 π) (ν _3+4 π)
    (C_3 r^2+1)^2 {(C_3
    r^2+1)√(C_3 r^2+2-C_3^2 r^4)
       -C_2 (C_3 r^2-2) (2 C_3
    r^2+5)}]^-1[2 (C_3
    r^2+1)√(C_3 r^2+2-C_3^2 r^4){ν _3
    
       ×(2 C_3 r^2-3)+6 π(C_3 r^2-1)}+C_2 (C_3
    r^2-2) {3 ν _3-4 (C_3
    r^2+1)
       ×(2 C_3 ν _3 r^2+π(6 C_3 r^2+3))}]=0.

The above equation provides the deformation function as

    T(r)=3πC_3r^2/(ν_3+2π)(ν_3+4π)(C_3r^2+1)^2[C_2 {C_3 r^2 (14 C_3
    r^2-5)-10}
       +√(C_3
    r^2+2-C_3^2 r^4)(-7 C_3
    r^2-1)]^-1[2 (C_3 r^2+1)
    √(C_3 r^2+2-C_3^2 r^4)
       ×{ν _3 (2 C_3 r^2-3)+6 π(C_3
    r^2-1)}-C_2 (C_3 r^2-2) {4
    (C_3 r^2+1)
       ×(2
    C_3 ν _3 r^2+π(6 C_3 r^2+3))-3
    ν _3}].


Figure 7 exhibits the plot of the above radial deformation
function for the considered pressure-like constraint with respect to
the chosen values of parameters. We observe that this function
initially increases for all values of model parameter, i.e., ν_3
∈ (0,1), and then decreases to reach its minimum at the
hypersurface. The effective matter determinants along with
anisotropic factor corresponding to the above deformation function
can be obtained by making use of Eqs.(<ref>) and (<ref>). We
also examine the profile of such effective quantities in Figure
8. We find from the upper left plot that the energy density
possesses the same (opposite) behavior as that of the first model
near the center (spherical boundary). Tables 4 and
5 provide the numerical values that confirm such profile of
these matter determinants. Further, both pressure ingredients and
anisotropy show a consistent behavior. The radial pressure and
anisotropic factor are found to be null at the boundary and center,
respectively (right plots).



Figure 9 displays the variation in multiple factors with
respect to r and parametric values, and we find them consistent
with observed data. Since we observe the matter triplet
(μ̃,P̃_r,P̃_̃) to be positive
everywhere, the dominant energy bounds are only needed to check. We
plot them in Figure 10 and deduce that the corresponding
solution is viable. We also check the stability of the resulting
solution in Figure 11, showing that our model II becomes
stable through both criteria (sound speed and cracking approach).








 §.§ Model III


Here, we assume a linear equation that connects different components
of 𝔇-sector given as follows <cit.>

    𝔇_1^1(r)=τ_1𝔇_0^0(r)+τ_2,

where τ_1 and τ_2 are real-valued constants whose
different values can be taken to check how the corresponding
solution is influenced by these unknowns. Making use of
Eqs.(<ref>) and (<ref>) in the above linear equation yields

    T(r)(ν_1'/r+1/r^2-τ_1/r^2)-τ_1T'(r)/r-8πτ_2=0.

One can involve 𝔇_2^2(r) component in
Eq.(<ref>), however, its presence will make the above equation
more complicated due to the appearance of second order derivatives
of the metric potentials. Joining the metric ansatz (<ref>) and
(<ref>) with (<ref>), we get

    T(r){6 C_3 (√(C_3
    r^2+2-C_3^2 r^4)+C_2-2 C_2 C_3
    r^2)/(C_3 r^2+1) √(C_3
    r^2+2-C_3^2 r^4)-C_2 (C_3 r^2-2)
    (2C_3r^2+5)
       +1-τ_1/r^2}-τ_1T'(r)/r-8πτ_2=0.

This is first order differential equation in T(r) whose
analytical solution is not possible due to the appearance of a term
in the square root. Therefore, we employ a numerical integration
with an initial condition T(0)=0 to obtain the
deformation function for τ_1=1.6 and τ_2=-0.001.



Figure 12 shows increasing trend outwards and in an inverse
relation with the model parameter. We utilize Eqs.(<ref>) and
(<ref>) to construct the corresponding matter triplet
(μ̃,P̃_r,P̃_) and anisotropy. The
variation of these variables with respect to r, η and ν_3
is shown in Figure 13, indicating an acceptable profile
(see Tables 6 and 7 for numerical values). They
show the same behavior as we have found for the first and second
models. Figure 14 presents the plots of the mass function,
redshift and compactness that are consistent with their acceptable
limits. Further, this solution produces less massive interior in
comparison with models I and II. The variation of the energy
conditions μ̃-P̃_r ≥ 0 and
μ̃-P̃_≥ 0 is shown in Figure 15,
demonstrating that our model III is viable. Both criteria of
stability (Figure 16) indicate that the developed model is
stable for every parametric choice.








§ CONCLUSIONS


In this paper, we have formulated different anisotropic extensions
of the known isotropic solution through gravitational decoupling for
the model f(ℝ,𝕋)=ℝ+2ν_3𝕋.
Initially, we have assumed that a sphere is filled with an isotropic
fluid that becomes anisotropic after inserting an additional source
at the action level. The modified form of the action (<ref>) has
then triggered the field equations involving the effects of both
seed isotropic as well as additional sources and modified theory. We
have then split these equations into two sets through MGD strategy.
We have dealt with the unknowns of the first set by choosing
Buchdahl anstaz given by

    e^ν_1(r)   =C_1[(1+C_3r^2)^3/2+C_2(5+2C_3r^2)
    √(2-C_3r^2)]^2,
    
    e^ν_2(r)   =2(1+C_3r^2)/2-C_3r^2,

and calculated the triplet
(C_1,C_2,C_3) through junction
conditions at Σ: r=R. Moreover, the
𝔇-sector (<ref>)-(<ref>) comprised four
unknowns that needed to be determined. Therefore, we have imposed
different constraints on 𝔇_σω, resulting in
three different solutions. We have then added these solutions
corresponding to both fluid sources through the parameter η and
obtained new anisotropic extensions.

We have discussed acceptability conditions such that the resulting
solution must be in agrement with them. Such physical
characteristics of all the developed models have been interpreted
graphically for ν_3=0.25,0.5,0.75,1 and η=0.1,0.3. We have
observed an acceptable behavior of the matter determinants and
anisotropy in each case for all parametric values. The numerical
values of the mass function indicates that the second model produces
massive interior in comparison to the others. The redshift and
compactness factors have also been shown compatible with the
experimental data. The dominant energy bounds are satisfied in the
region 0<r<R, ensuring the viability of the obtained
solutions. Finally, we have noticed that the cracking occurs only in
the first model, hence, models II and III are stable.

Morales and Tello-Ortiz <cit.> extended the Durgapal's fifth
model to the anisotropic domain through the MGD approach and
obtained the more dense structure in comparison with our developed
solutions. Andrade and Contreras <cit.> used Tolman IV,
Heintzmann IIa and Durgapal IV as seed solutions, and proposed
different anisotropic extensions by using the definition of the
complexity factor. The anisotropy was also discussed in the interior
of SMC X-1 and Cen X-3 stars from which we found this factor to be
much greater as compared to that obtained in the current setup.
Maurya et al. <cit.> investigated the possible existence of
compact stars influenced by the electromagnetic field in the
framework of Gauss-Bonnet gravity and determined the exact solutions
to the corresponding field equations in contrast with our work. It
is important to stress here that our models I, II and III are
consistent with <cit.>, <cit.> and <cit.>,
respectively. All our results reduce to 𝔾ℝ for
ν_3=0.


Data Availability Statement: This manuscript has no
associated data.




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 Paper Title: Holographic thermodynamics of a five-dimensional neutral Gauss-Bonnet AdS black hole.


 Enabling Next-Generation V2X Perception: 
 Wireless Rigid Body Localization and Tracking

    Niclas Führling0000-0003-1942-8691, Hyeon Seok Rou0000-0003-3483-7629, Graduate Student Members, IEEE,

Giuseppe Thadeu Freitas de Abreu0000-0002-5018-8174, David González G.0000-0003-2090-8481, Senior Members, IEEE, and Osvaldo Gonsa0000-0001-5452-8159




N. Führling, H. S. Rou and G. T. F. Abreu are with the School of Computer Science and Engineering, Constructor University, Campus Ring 1, 28759, Bremen, Germany (emails: [nfuehrling, hrou, gabreu]@constructor.university).

D. González G. and O. Gonsa are with the Wireless Communications Technologies Group, Continental Automotive Technologies GmbH, Guerickestrasse 7, 60488, Frankfurt am Main, Germany (emails: david.gonzalez.g@ieee.org, osvaldo.gonsa@continental-corporation.com).

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








§ INTRODUCTION


Black hole thermodynamics in asymptotically anti-de Sitter (AdS) spacetime has advanced to the forefront of current research activities in theoretical high energy physics. One of the intriguing aspects of theoretical physics is duality which has revolutionized perspectives of physics at a fundamental level. Dualities relate two different facets of nature and thus they lead to an enhanced understanding of physics. An implication of these dualities is that, while the mathematical structures of the two theories may be identical, there may exist differences from a physical standpoint. One of the interesting dualities is the anti-de Sitter/conformal field theory (AdS/CFT) correspondence <cit.> which connects the CFT on a flat spacetime to string theory. This is interesting because string theory is a theoretical framework for quantum gravity, whereas quantum field theory is not. 

Now, even though these two theories are different, the AdS/CFT correspondence assumes the existence of a correspondence between the two theories and this is one of the most attractive features of this duality, which pushes research for its deeper understanding and possible applications. A renowned example of a phase transition in black hole spacetimes is the radiation/black hole first-order phase transition of Hawking and Page, observed for Schwarzschild-AdS black holes immersed in a bath of radiation <cit.>. Such a phenomenon has a dual interpretation for a boundary quantum field theory via the AdS/CFT correspondence and is related to a confinement/deconfinement phase transition in the dual quark gluon plasma <cit.>. For charged black hole in AdS spacetime one likewise observes a small/large black hole first-order phase transition reminiscent of the liquid/gas phase transition of the Van der Waals fluid <cit.>. But when the cosmological constant is interpreted as the thermodynamic pressure and the black hole mass is interpreted as the thermodynamic enthalpy <cit.>, the phase behavior of the charged AdS black hole is exactly the same with that of the Van der Waals fluid and they have the same critical exponents <cit.>. This version of the black hole thermodynamics in the presence of negative cosmological constant is interpreted as the extended phase space thermodynamics or generically speaking, the black hole chemistry <cit.>. The phase structures in the extended phase space is much richer, including reentrant  phase transition <cit.>, superfluid-like phase transition <cit.>, triple points <cit.> etc. Recent advancements have yielded numerous successful interpretations of the origins of the extended phase space thermodynamics. Notable among these are the extended Iyer-Wald formalism <cit.>, the coupling constant of gauge field perspective <cit.>. Additionally, a mechanism for the higher-dimensional origin of a dynamical cosmological constant was proposed <cit.>.

Despite significant advancements have been achieved in the field of thermodynamics for black holes in AdS spacetime, the holographic interpretation of black hole chemistry has remained an enigma <cit.>. From the viewpoint of the AdS/CFT correspondence, the negative cosmological constant Λ sets the asymptotic structure of the bulk spacetime and is related to the number of colours N in the dual gauge field theory <cit.>

    kL^d-2/16π G=N^2,

where L is the AdS radius and the numerical factor k depends on the details of the particular holographic system. Hence, variation of the negative cosmological constant Λ corresponds to changing the number of colors N and hence varying the dual gauge field theories in the boundary <cit.>.

Recently there have been considerable progress in developing an exact dictionary between the black hole chemistry and the thermodynamics of the dual gauge field theory. Considering the viewpoint that Newton's constant is treated as a variable <cit.>, and extended to the more general CFT metric with boundary radius different from the AdS radius, Visser argued that holographic thermodynamics requires a chemical potential for color and obtained the thermodynamic Euler equation for high energy states of large-N gauge theories <cit.>. Systematic works have been done based on this viewpoint <cit.>. 
Building on Visser’s viewpoint that the radius of the CFT metric need not coincide with the AdS radius, and allowing the conformal factor to vary while keeping Newton’s constant fixed, thereby resulting in the boundary volume 𝒱 and the central charge C being completely independent,  Ahmed et al. constructed an exact dictionary between the thermodynamics of the charged AdS black hole and the dual conformal field theory <cit.>. Based on this perspective, a lot of work has been done for the holographic thermodynamics for AdS black holes, including the charged AdS black hole <cit.>, Kerr AdS black hole <cit.>, Kerr-Newman AdS black hole <cit.>. Inspired by this viewpoint, we investigate the holographic CFT thermodynamics for the five-dimensional neutral Gauss-Bonnet AdS black hole. Different from the recent work of Cui et al. <cit.>, where the Newton's constant is regarded as a dynamical variable, we take the viewpoint of Ahmed et al. in which the Newton's constant is treated as a fixed quantity but the conformal factor is supposed to vary. 

From the perspective of the gauge/gravity correspondence, the thermodynamics of AdS black holes is completely equivalent to the thermodynamics of the dual gauge fields. Through the extended bulk thermodynamics for the five-dimensional Gauss-Bonnet AdS black hole, we derive the first law of the CFT thermodynamics which is obtained by directly translating the arbitrary conformal factors in the dual CFT and investigate the phase behavior and criticality of the CFT. We find that the free energy exhibits a characteristic swallowtail behavior and the critical points and critical exponents are the same as that of the five-dimensional Gauss-Bonnet AdS black hole. 


The outline of the paper is as follows. In Sec. <ref>, we derive the AdS/CFT dictionary and the laws for the CFT residing in the boundary dual to the five-dimensional Gauss-Bonnet AdS black hole. In Sec. <ref>, we investigate the phase behavior and criticality of the CFT. Finally, in the last section we discuss the results and conclude them. 




§ HOLOGRAPHIC THERMODYNAMICS OF FIVE-DIMENSIONAL GAUSS-BONNET ADS BLACK HOLE



In this section, we tentatively establish a connection between the extended thermodynamics of the five-dimensional neutral Gauss-Bonnet AdS black hole and the corresponding extended thermodynamics of dual CFT. By exploring the relationship between these two realms, we aim to shed light on the profound similarities and underlying principles that govern their thermodynamic properties.




 §.§ Extended bulk thermodynamics



In this subsection, we briefly review the main points of the extended black hole thermodynamics for the five-dimensional neutral Gauss-Bonnet AdS black hole. A d-dimensional neutral Gauss-Bonnet AdS black hole is a solution for the gravitational field equation derived by the action <cit.>

    S=∫ d^dx√(-g)1/16π G_N(ℛ-2Λ+α_GBℒ_𝒢ℬ)

with the Gauss-Bonnet term

    ℒ_𝒢ℬ=ℛ_μναβℛ^μναβ -4ℛ_μνℛ^μν+ℛ^2,

where G_N is the Newton constant, α_GB is the Gauss-Bonnet coupling constant, and the cosmological constant is given by

    Λ=-(d-1)(d-2)/2L^2

with L the AdS radius.






Within the context of the low energy effective action formalism in heterotic string theory, the Gauss–Bonnet term is the leading order quantum correction to gravity <cit.>. The Gauss–Bonnet coupling constant is related to the string scale and can be identified with the inverse of the string tension <cit.>. Hence, we only consider positive Gauss-Bonnet coupling constant in this paper.

The theory admits an exact neutral black hole solution in five dimensions. The metric for the five-dimensional neutral Gauss-Bonnet AdS black hole is <cit.>

    ds^2=-f(r)dt^2+f^-1(r)dr^2+r^2[dθ^2+sin^2θ (dϕ^2+sin^2ϕ dψ^2)]

with the metric function given by

    f(r)=1+r^2/2α(1-√(1+32α G_NM/3π r^4+2αΛ/3)),

where α=2α_GB.

The solution is a steady spherical solution with the event horizon r_h determined by f(r_h)=0. From this equation, we can express the mass of the black hole as

    M=π/8G_N(-1/2Λ r_h^4+3r_h^2+3α).

From the extended black hole thermodynamic perspective, the mass of the black hole is interpreted as the enthalpy H=M rather than the internal energy U of the gravitational system.

By requiring the absence of a conical singularity at the event horizon in the Euclidean action, we can get the Hawking temperature of the black hole, which is given by

    T_H=f'(r_h)/4π=3r_h-Λ r_h^3/6π r_h^2+12πα.

From the viewpoint of the extended black hole thermodynamics, the cosmological constant Λ is interpreted as the thermodynamic pressure and the black hole mass is regarded as the thermodynamic enthalpy H <cit.>. The thermodynamic pressure is

    P=-Λ/8π G_N=3/4π G_N1/L^2,

and its conjugate thermodynamic volume is

    V=π^2 r_h^4/2.

The Gauss-Bonnet coupling constant α is a thermodynamic variable, with the corresponding conjugate thermodynamic variable given by

    𝒜=(∂ M/∂α)_S,P=-π/8G_N-4Λ r_h^4+9r_h^2-6α/r_h^2+2α.

The entropy of the black hole is

    S=∫_0^r_h(∂ M/∂ r_h)_P,αdr =π^2 r_h/2G_N(r_h^2+6α).

Evidently, the presence of the Gauss-Bonnet correction to Einstein's theory of gravity leads to a notable revision of the Bekenstein-Hawking area law when considering the black hole's temperature as its surface gravity, as emphasized in Ref. <cit.>.

The first law of black hole thermodynamics and the Smarr relation for the five-dimensional Gauss-Bonnet AdS black hole are <cit.>

    dM    =    TdS+VdP+ 𝒜dα,
    
      2M    =    3TS-2PV+2𝒜α.


Using the first law of the bulk thermodynamics, we proceed to derive the boundary thermodynamics in the next subsection. This derivation establishes a tentative relationship between the thermodynamic properties of the bulk system and its boundary counterpart, allowing for a comprehensive analysis of their interconnected behavior. The insights gained from this derivation might enhance our understanding on the overall thermodynamic framework.





 §.§ Extended boundary thermodynamics



From the perspective of the AdS/CFT correspondence, the dual CFT lives on the conformal boundary of the asymptotically AdS spacetime. According to the recent development of the holographic chemistry <cit.>, the boundary metric of the CFT dual to the five-dimensional Gauss-Bonnet AdS spacetime is <cit.>:

    ds^2=ω^2(-dt^2+L^2dΩ_3^2 ),

where dΩ_3^2 is the metric on the three-dimensional unit sphere:

    dΩ_3^2=dθ^2+sin^2θ (dϕ^2+sin^2ϕ dψ^2),

and ω=R/L is the dimensionless conformal factor that is free to vary <cit.>. It reflects the conformal symmetry of the boundary theory. R is the radius of the boundary.

In the gauge/gravity duality, the central charge of the CFT is related to the AdS radius L and the gravitational coupling constant G_N in the bulk theory. For Einstein gravity the central charge usually reads <cit.>

    C=Ω_d-1 L^d-1/16π G_N,

where Ω_d-1 is the volume of a (d-1)-dimensional unit sphere. Since the precise coefficient of the central charge is irrelevant for our purpose and only the combination dC/C appears in the first law of the CFT, the dimensional dependent coefficient is normalized such that it agrees with the coefficient A of the Euler density in the trace anomaly <cit.>.

From the first law of black hole thermodynamics and the Smarr relation for the five-dimensional Gauss-Bonnet AdS black hole, we can express the first law in terms of the boundary thermodynamics

    δ(M/ω)=(T/ω)δ S+M-TS-𝒜α/ωδ(L^3/G_N)/(L^3/G_N)+√(G_N)𝒜/ω Lδ(α L/√(G_N))-(Ẽ/3𝒱)δ𝒱,

where 𝒱=2π^2 R^3 is the spatial volume of the boundary on which the CFT resides.
In the derivation of the first law in terms of the boundary thermodynamics, the Newtown constant G_N is regarded as a fixed constant, but the conformal factor ω is free to vary.
Employing the following holographic dictionary:

    Ẽ   =M/ω,                 T̃=T/ω,               S=S, 
    𝒜̃   =√(G_N)𝒜/ω L,
                    α̃=α L/√(G_N),        𝒫=Ẽ/3𝒱, 
    
          C   =π L^3/8G_N,              μ=1/C(Ẽ- T̃ S-𝒜̃α̃),

the extended first law of thermodynamics for the five-dimensional Gauss-Bonnet AdS black hole is dual to the following thermodynamic first law in the CFT:

    δẼ=T̃δ S+𝒜̃δα̃+μδ C-𝒫δ𝒱,

where 𝒫 is the field theory pressure, and μ is the chemical potential associated with the central charge C.  The generalized Smarr relation (<ref>) for the AdS black hole is dual to an Euler equation in a large N gauge field theory, which takes the following form:

    Ẽ=T̃ S+𝒜̃α̃+μ C.

One noticeable point is that the Euler equation does not include a 𝒫𝒱 term.












From the above AdS/CFT dictionary, the central charge C in the extended boundary thermodynamics is dual to the pressure P of the extended bulk thermodynamics. Hence it is natural to take the viewpoint that the conjugate thermodynamics of the central charge—the chemical potential μ in the extended boundary thermodynamic is dual to the thermodynamical volume V of the extended black hole thermodynamics. 

We define the dimensionless parameters x and y as

    x=/L,      y=α/L^2.

Then, in terms of the parameters x and y, the CFT thermodynamic quantities are given by

  
  * energy:
  
    Ẽ  =3C/R(x^4+x^2+y),

  
  * temperature:
  
    T̃=1/2π R2x^3+x/x^2+2y,

  
  * entropy:
  
    S  =4π Cx(x^2+6y),

  
  * pressure:
  
    𝒫 =C/2π^2R^4(x^4+x^2+y),

  
  * chemical potential:
  
    μ=1/Rx^2[-x^2(x^2-1)+6y(x^2+1)]/x^2+2y,

  
  * the Gauss coupling constant α̃ and its conjugate CFT thermodynamic quantity 𝒜̃:
  
    α̃=√(8CL^3/π)y,     𝒜̃ =-1/R√(π C/8L^3)24x^4+9x^2-6y/x^2+2y.








Having the thermodynamics for the CFT dual to the five-dimensional Gauss-Bonnet AdS black hole, we delve into the exploration of the phase transition  and critical phenomena for the CFT.





§ THERMODYNAMIC ENSEMBLES IN THE DUAL CFT



The extended thermodynamics and phase transition for CFT states that are dual to the singly-spinning asymptotically AdS black hole were investigated by Ahmed et al., and various phase transitions and critical behavior in the CFT were uncovered <cit.>. Gong et al. found that the critical exponents for the CFT states in the vicinity of the critical point are the same with that of the Kerr-Newman AdS black hole <cit.>. In this section, we investigate the phase behavior and criticality exhibited by the canonical ensemble within the CFT

framework.





 §.§ Canonical ensemble: swallowtail behavior and critical point



The canonical ensemble for five-dimensional Gauss-Bonnet AdS black hole has been well study in the framework of extended black hole thermodynamics, i.e. black hole chemistry <cit.>. In these studies it was found that the five-dimensional Gauss-Bonnet AdS black hole displays the swallowtail behavior, which indicates the small/large black hole phase transition reminiscent of the Van der Waals like phase transition. Besides, it was shown that the critical point for the five-dimensional Gauss-Bonnet AdS black hole phase transition is <cit.>

    r_hc=√(6α),      P_c=1/48πα,       T_c=1/2√(6α)π.

In this subsection, we investigate the phase behavior of the CFT states dual to the five-dimensional neutral Gauss-Bonnet AdS spacetime.

From the previous subsection for the study of the boundary thermodynamics, we can get the relevant Helmholtz free energy in the fixed (C, 𝒱, α̃) canonical ensemble:

    F=Ẽ-T̃S=-C/Rx^6+18x^4y-x^4+3x^2y-6y^2/x^2+2y.


To illustrate the phase behavior of the CFT states, we draw the diagram for the Helmholtz free energy F vs the temperature T in Fig. <ref>. 
Though we have chosen the boundary radius R=10^4 and the thermodynamic variable of the CFT α̃=1 in the figures, the scale of the boundary radius R and α̃ do not affect the qualitative thermodynamic behavior of the system. The figures show the characteristic swallowtail behavior which is similar to the liquid/gas phase transition for the Van der Waals fluid. The brown dashed and blue curves in Fig. <ref> resemble the shape of a swallowtail. The swallowtail behavior for the CFT states demonstrates the existence of a first-order phase transition. When the central charge C equals the critical value C_c, it becomes a second-order phase transition. However, the swallowtail disappears when the central charge C is smaller than the critical value C_c.

As depicted in Fig. <ref>, each swallowtail consists of three distinct piecewise smooth branches. The parameter x increases along the curve, as evidenced by the direction of the arrows traced along the curve in Fig. <ref>. Since the entropy of the CFT states is a monotonic increasing function of x, as indicated by eq. (<ref>), the entropy increases along the direction of the arrows depicted on the curve. The black line starting from zero temperature is the low entropy branch, since the value of x is the smallest on this branch. The entropy increases along the read dashed curve, which is the intermediate entropy branch. The branch that the Helmholtz free energy F extends to minus infinite is the largest entropy branch. For any swallowtail curve in the figure, starting at low temperature T, the branch that minimizes the Helmholtz free energy is the thermodynamically favored state. Hence the low entropy branch is thermodynamically stable for low temperature. As the self-intersection point for the low entropy branch and high entropy branch has the same Helmholtz free energy, it is the phase transition point. The low and high entropy phases can exist together. After crossing the self-intersection point, the higher entropy branch is more stable. 




The stability of the CFT states can also be captured by the heat capacity. For positive heat capacity, the system is locally thermodynamically stable; while negative heat capacity indicates thermodynamically unstable. For the canonical ensemble with fixed C, 𝒱 and α̃, the heat capacity is

    𝒞_C, 𝒱, α̃=T(∂ S /∂T)_C, 𝒱, α̃= 12 πC(2 x^3+x) (x^2+2 y)^2/x^2 (12 y-1)+2 x^4+2 y.



The heat capacities for the low and high entropy branches of the Helmholtz free energy in Fig. <ref> are both positive, hence they are thermodynamically stable. While the heat capacity for the intermediate entropy branch is negative, hence it is thermodynamically unstable. We also draw the heat capacity for the CFT states as in Fig. <ref>. For the brown dashed and blue line, the central charge C is larger than the critical value C_c=9πα̃/(2√(G_N)). For the red large dashing line, the central charge is equal to the critical value C = C_c. For the black dotted line, the central charge C is smaller than the critical value C_c. As we can see from the figure, the heat capacity is continues and finite for small central charge C< C_c. However, there are points where the heat capacity is divergent for large central charge C ≥  C_c. The divergent points of the heat capacity correspond to the intersection points between the small/large entropy branches and intermediate entropy branches in Fig. <ref>.





Besides the Helmholtz free energy, the oscillatory behavior of the temperature T against the entropy S can also depict the phase transition of the system. 
The oscillatory behavior for curves in the T̃-S diagram in Fig. <ref> shows the phase transition of the boundary CFT. For large central charge C>C_c, the curves (the brown dashed and blue lines) in the T̃-S plane display the oscillatory behavior. The oscillatory behavior of the T̃-S diagram indicates there exists a first-order phase transition. For small central charge C≤ C_c, the oscillatory behavior disappears and the temperature T̃ is a monotonic increasing function of the entropy S, as can be seen in the figure for the red dashing and black dotted lines.




From the T̃-S oscillatory curve in Fig. <ref>, we can get the critical point for the phase transition. The minimal and maximal points of the T̃-S oscillatory curve coincide and exhibit a horizontal tangent at the critical point. Mathematically, the critical point satisfies the following equations:

    (∂T̃/∂ S)_C, 𝒱, α̃=0,     (∂^2T̃/∂ S^2)_C, 𝒱, α̃=0.

From the above two equations, we can get the critical point, which is determined by the following two parameters

    x_c=1/√(6),      y_c=1/36.

Some straightforward calculations reveal that the critical point for the boundary CFT satisfies

    r_hc=√(6α).

Evidently, the critical point of the thermodynamics for the boundary CFT is the same as the critical point for the bulk phase transition as indicated in eq. (<ref>). 
At the critical point, the other thermodynamic quantities are

    T̃_c   =√(6)/2π R,       𝒜̃_c=-9 π/8√(G_N) R,
         𝒫_c= α̃/2 π√(G_N)R^4, 
     
    S_c   =√(6)/√(G_N)π^2α̃,       
    C_c=9π/2√(G_N)α̃,
        μ_c=1/4 R.


From the critical point, we can easily find an interesting relation among the critical pressure 𝒫_c, critical volume 𝒱_c, critical temperature T̃_c, and the critical entropy S_c:

    𝒫_c𝒱_c/T̃_c S_c=1/3.






 §.§ Maxwell's equal area law and coexistence curve




The physical processes of crossing the coexistence phase often unveil intriguing phenomena, such as latent heat <cit.>. As shown by previous work <cit.>, both the swallowtail behavior of the free energy and Maxwell's equal area law can reveal phase transition of a thermodynamic system and they are consistent with each other. In this subsection,  we derive Maxwell's equal area law from the change in the Helmholtz free energy during the phase transition between low and high entropy states. Then we employ Maxwell's equal area law to construct an analytical formula for the coexistence curve in the T̃-S plane.

To solve the coexistence curve for the low and high entropy states of the boundary CFT, we need to solve the phase transition points in the F-T̃ plane in Fig. <ref>. We focus on the self-intersection points in Fig. <ref>. These two points are low and high entropy phases. These two coexistence phases correspond to point A and point C in the T-S plane, respectively. A phase of the boundary CFT thermodynamic system in the fixed (C, 𝒱, α̃) canonical ensemble can be determined by the parameter x. We denote the low entropy phase A as (x_L,y,R,C) and the high entropy phase C as (x_H,y,R,C).

For the phase transition between the low entropy state A and high entropy state C in Fig. <ref>, the Helmholtz free energies of the two phases are the same and the difference is zero:

    Δ F=0.

 From the Helmholtz free energy

    F=Ẽ-T̃S,

and the first law (<ref>), we have

    dF=SdT̃-𝒫d𝒱+𝒜dα+μ dC.

Integrating it and referring to eq. (<ref>), we obtain

    ∫^C_A dF=∫^C_A SdT-∫^C_A Pd 𝒱+∫^C_A𝒜̃dα+∫^C_Aμ dC=0.

Evidently, the exactly result of each integral in eq. (<ref>) depends on the physical process.
For the fixed C, 𝒱, and α canonical ensemble, eq. (<ref>) becomes

    ∫^C_A SdT=0.

The result shows that the areas in the region 1 and region 2 in Fig. <ref> are the same. It suggests that Maxwell's equal area law holds for the phase transition of the boundary CFT in the canonical ensemble. As we can see from Fig. <ref>,

it is more convenient and direct to formulate Maxwell's equal area law (<ref>) in the following form

    ∫^C_ATdS=T(A)[S(C)-S(A)].

From the temperature in eq. (<ref>) and the entropy in eq. (<ref>) of the CFT, we can get the integration, which is

    ∫^C_ATdS=∫^x_H_x_LT(∂ S/∂ x)dx=3C/R(x_L^2+x_H^2+1)(x_L+x_H)(x_H-x_L).

Besides, the right-hand side of eq. (<ref>) is

    T̃(A)[S(C)-S(A)]
           =2C/R2x_L^3+x_L/x_L^2+2y(x_L^2+x_Lx_H+x_H^2+6y)(x_H-x_L).

Then, Maxwell's equal area law (<ref>) indicates the following equation

    3(x_L+x_H)(x_L^2+2y)(x_L^2+x_H^2+1)
            =2(2x_L^3+x_L)(x_L^2+x_Lx_H+x_H^2+6y).

On the other hand, the temperatures of the phase A and phase C are the same:

    T̃(A)= T̃(C).



From Maxwell's equal area law in eq. (<ref>) and eq. (<ref>), we can get the parameters characterizing the low entropy phase A and high entropy phase C:

    x_L   =√(1-24 y-√(48 y (9 y-1)+1))/√(2), 
    
            x_H   =6 √(2) y/√(1-24 y-√(48 y (9 y-1)+1)).

The isothermal temperature for phases A and C, representing the temperature of coexistence, can be determined. The temperature indicates the precise conditions at which coexistence takes place. The isothermal temperature is

    T̃=√(1-24 y-√(48 y (9 y-1)+1))(2-24 y-√(48 y (9 y-1)+1))/√(2)π  R (1-20 y-√(48 y (9 y-1)+1)).

Since we are considering the fixed (C, 𝒱, α̃) canonical ensemble, the function for the coexistence curve for the canonical ensemble is given by eq. (<ref>) and

    C=π/8√(G_N)α̃/y.

The above equations (<ref>) and (<ref>) are the parameter equations for the coexistence curve. The temperature for the coexistence of the two phases is a function of the parameter y and boundary volume 𝒱. To write the equation in a more concise manner, we define

    τ=T̃/T̃_c,      c=C/C_c.

The coexistence curve equation can be written as

    τ= √(-√(3)√(1/c^2-4/c+3)-2/c+3)(c (√(3)√(1/c^2-4/c+3)-6)+2)/c (3 √(3)√(1/c^2-4/c+3)-9)+5.

One of the most remarkable properties of the above equation for the coexistence curve is its independence of the CFT thermodynamic variable α̃, as well as the boundary volume 𝒱. From the coexistence curve in Fig. <ref>, the temperature for the low and high entropy coexistence phases decreases with the increasing of the inverse of the central charge 1/C.

In the coexistence curve, a first-order phase transition occurs between the low and high entropy states for C> C_c. At the critical point, a second-order phase transition occurs. However, there is no phase transition for C<C_c.





 §.§ Critical exponents



Physics near the critical point often exhibits fascinating phenomena. Critical exponents characterize the behavior of physical quantities in the vicinity of a critical point. These critical exponents are considered universal, as they remain independent of the specific characteristics of the physical system. In this subsection, we study the critical behavior of some physical quantities and calculate the corresponding critical exponents for the thermodynamics of the CFT residing on the boundary.

In the vicinity of the critical point, the critical behavior of the CFT dual to the five-dimensional neutral Gauss-Bonnet AdS black hole can be precisely characterized by the critical exponents denoted as α', β, γ, and δ. These critical exponents are explicitly defined as follows:


  * The critical exponent α' governs the behavior of the heat capacity of the CFT at constant chemical potential as compared to the heat capacity of an AdS black hole at constant volume

    C_μ=T̃(∂ S/∂T̃)_μ∼(-T-T_c/T_c)^-α'.


  * The critical exponent β determines the behavior of the order parameter η. Since the entropy increases along the arrows in the swallowtail curve in Fig. <ref>, and the difference of the entropies S_H-S_L changes discontinuously in the first-order phase transition but continuously at the critical point, we choose the entropy difference between high and low entropy phases as the order parameter. Then the critical exponent β is defined as

    η=  S_H-S_L/S_c∼(-T̃-T_c/T_c)^β.













  * The critical exponent δ determines the behavior of the pressure on the critical point for extended black hole thermodynamics. Through the dual relation, the critical exponent δ for the CFT is defined as

    C-C_c∼ (  μ-μ_c)^δ.



These critical exponents characterize the critical phenomena for the boundary CFT. To calculate the critical exponents for the CFT, we define

    t=T̃/T̃_c-1,     
        s=S/S_c-1,     χ=μ/μ_c-1.

At the critical point, the heat capacity for the fixed chemical potential is

    C_μ,𝒱,α̃=T̃(∂ S/∂T̃)_μ,𝒱,α̃=4 √(6)/√(GN)π^2 α̃,

which is not divergent.  Hence the critical exponent  governing the behavior of the heat capacity for the CFT is α'=0.

To calculate the critical exponent governing the behavior of the order parameter, we start at the entropy of the CFT at the phase transition point. Together with the entropy (<ref>), and the parameter labeling the low and high entropy states (<ref>), the entropy of the phases A and C in Fig. <ref> are

    S_L   =√(2)π  C √(1-24 y-√(48 y (9 y-1)+1))(1-12 y-√(48 y (9 y-1)+1)),
    
        S_H   =144 √(2)π  C y^2 (1-12 y-√(48 y (9 y-1)+1))/(1-24 y-√(48 y (9 y-1)+1))^3/2.

In terms of the parameter y, the order parameter can be expressed as

    η   =S_H-S_L/S_c
       =(12 y+√(48 y (9 y-1)+1)-1)^2 (36 y+√(48 y (9 y-1)+1)-1)/8 √(3) y (-24 y-√(48 y (9 y-1)+1)+1)^3/2.

From the temperature T̃ in eq. (<ref>),  we have

    t   =T̃/T̃_c-1
       =√(1-24 y-√(48 y (9 y-1)+1))(24 y+√(48 y (9 y-1)+1)-2)/√(3)(20 y+√(48 y (9 y-1)+1)-1)-1.

Solving eq. (<ref>), we can get a simple result for the parameter y in terms of t:

    y= 1/36(-2 t^2-4 t+1).

Together with eqs. (<ref>) and (<ref>), the order parameter can be formulated in terms of the temperature parameter t:

    η   =4 (t (t+2)-√(3)√(t (t+1)^2 (t+2))+1)^2 (√(3)√(t (t+1)^2 (t+2))-3 t (t+2))/(1-2 t (t+2)) (4 t (t+2)-2 √(3)√(t (t+1)^2 (t+2))+1)^3/2
       =4√(6)t^1/2+25√(6)t^3/2+𝒪(t^5/2).

Hence, the critical exponent is given by β=1/2.

To calculate the critical exponent δ, through the definition of t and the equation for the temperature (<ref>), we can solve the third-order equation for x in terms of the parameters t and y. Then the chemical potential χ can be formulated as a function of the parameters t and y. 
 







Taking y=y_c=1/36 and expand χ near the critical point, we obtain

    χ(1+3c)(c-1)^1/3/3c^4/3∼(c-1)^1/3,

where c is the reduced central charge as defined in eq. (<ref>). Then we have

    c-1 ∼χ^3=(μ/μ_c-1)^3.

Hence, the critical exponent is δ=3.

Thus, we have obtained the critical exponents α', β and δ associated with the T̃-S criticality of the CFT,  where the conformal factor ω is free to vary but with fixed Newton's constant. Our results suggest that these critical exponents associated with the T̃-S criticality of the CFT are the same as those of the five-dimensional Gauss-Bonnet AdS black hole <cit.>.




§ CONCLUSION AND DISCUSSION



In the AdS/CFT correspondence, black holes in AdS spacetime correspond to thermal states of the boundary field theories. Based on the recently proposed viewpoint of the holographic thermodynamics, which treats the conformal factor ω as a variable, we derived the first law and Euler equation for the thermodynamics of the CFT, and established a holographic dictionary between the thermodynamics of the five-dimensional  neutral Gauss-Bonnet AdS black hole and the dual boundary thermodynamics. Having the thermodynamics for the dual CFT, we investigated the phase transition and criticality of the CFT in the fixed (C, 𝒱, α̃)  canonical ensemble. Just like the free energy for the five-dimensional neutral Gauss-Bonnet AdS black hole, the free energy for the CFT displays a characteristic swallowtail behavior. This indicates a first-order phase transition below the critical point and a second-order phase transition at the critical point. Through the T̃-S oscillatory behavior, we obtained the critical point for the thermodynamics of the CFT in the  T̃-S plane. The critical point for the CFT is exact the same with that of the five-dimensional neutral Gauss-Bonnet AdS black hole. Using Maxwell's equal area law, we got the coexistence curve for the high and low entropy phases of the CFT. Besides, we got the critical exponents for the CFT, and found that these critical exponents associated with the T̃-S criticality are the same as those of the five-dimensional Gauss-Bonnet AdS black hole.

Critical exponents describe the singularity behavior of thermodynamic variables such as the heat capacity, order parameter etc. They are universal and satisfy the scaling hypothesis. The critical exponent γ governing the isothermal compressibility of black hole thermodynamics is defined as

    κ_T=-1/v(∂ v/∂ P)_T∼(-T-T_c/T_c)^-γ.

As we argued in Sec. <ref>, the black hole thermodynamic pressure P is dual to the CFT central charge C, and the black hole thermodynamic volume V is dual to the CFT chemical potential μ.
In our investigation of the critical exponents for the CFT thermodynamics in the T̃-S plane, we found that if we define the critical exponent γ just replacing the pressure P and the  specific volume v by the central charge C and the chemical potential μ respectively

    κ_T̃=-1/μ(∂μ/∂ C)_T̃,𝒱,α̃∼(-T-T_c/T_c)^-γ,

then, the critical exponent is given by γ=2/3, which is different from the prediction of the scaling hypothesis. Perhaps this is due to the fact that it is the black hole thermodynamic volume V instead of the specific volume v that is dual to the CFT chemical potential μ. This leads to the result that the isothermal compressibility is not defined just by replacing the pressure P and the  specific volume v by the central charge C and the chemical potential μ in the T̃-S plane.

Criticality is a very important phenomena for thermodynamic systems and critical exponents are universal. Our investigation of the thermodynamics and criticality from the gravity/gauge duality viewpoint might lead to new insight to the understanding of the Van der Waals like phase behavior of the five-dimensional Gauss-Bonnet AdS black hole.







We are grateful to Chuan Chen, Shan-Ping Wu, and Yu-Peng Zhang for useful discussions. This work was supported by National Key Research and Development Program of China (Grant No. 2021YFC2203003), the National Natural Science Foundation of China (Grants No. 12305065, No. 12247178, No. 12347177,  No. 12075103 and No. 12247101), the China Postdoctoral Science Foundation (Grant No. 2023M731468), the 111 Project (Grant No. B20063), and Lanzhou City's scientific research funding subsidy to Lanzhou University.


















JHEP




 Paper Title: Positive Mass in General Relativity Without Energy Conditions.













alves.nickolas@alumni.usp.br    
Center for Natural and Human Sciences, Federal University of ABC,
Av. dos Estados, 5001, 09210-580, Santo André, SP, Brazil

andre.landulfo@ufabc.edu.br
Center for Natural and Human Sciences, Federal University of ABC,
Av. dos Estados, 5001, 09210-580, Santo André, SP, Brazil


bcosta@troy.edu
Center for Relativity and Cosmology, Troy University,
 Troy, AL, 36082, USA







§ ABSTRACT 
A long-standing problem in physics is why observed masses are always positive. While energy conditions in quantum field theory can partly answer this problem, in this paper we find evidence that classical general relativity abhors negative masses, without the need for quantum theory or energy conditions. This is done by considering many different models of negative-mass “stars” and showing they are dynamically unstable. A fortiori, we show that any barotropic negative-mass star whose pressure is not a monotonically increasing function of the energy density must be dynamically unstable. Furthermore, we argue that all acceptable barotropic models of negative-mass stars must be in this class, and are thus unstable.




Positive Mass in General Relativity Without Energy Conditions
    Bruno Arderucio Costa 0000-0001-5182-2010
    July 29, 2024
=============================================================





§ INTRODUCTION

A simple, yet profound, question one can ask about gravity is: “why do things fall down?” This is a conspicuous experimental fact about gravitational physics, and it is certainly fundamental for the formation of structures in the Universe and the existence of life itself. The answer, however, may dive deep into the foundations of physics. 

From a purely quantum field theoretical point of view, one could answer that things fall down because gravity is mediated by a spin 2 field—the graviton. As is well-known <cit.>, even-spin mediators lead to attractive forces for particles with charges with the same sign. Hence, positive masses must attract each other. This is in contrast to the situation with electromagnetism, in which the spin 1 photon makes opposite charges attract each other, but similar charges repel each other. 

A follow-up question arises: why do we only observe “gravitational charges” (, masses) of the same sign? The absence of negative masses is a prominent feature of nature which can be deemed curious since one at first would expect the situation for positive and negative masses to be symmetric.

The fact is that assuming that the equivalence principle holds in such cases (i.e., inertial and gravitational masses are equal), the situation is not symmetric. The reason is the curious gravitational behavior of masses with different signs that can be understood in Newtonian gravity, for simplicity. Consider a pair of masses. If both the masses are positive, then the situation is standard and the masses will be attracted to each other. In particular, the gravitational force on each particle and their acceleration point in the same direction (see figure <ref>). If the masses have opposite signs, the forces are repulsive, but the net effect is that the negative mass pursues the positive mass, while the latter runs from the former. This is because the force and acceleration on the negative mass point to different directions (F = - ma). Finally, for a similar reason, the gravitational force between two negative masses is attractive, but the effect is repulsive. This is pictured in figure <ref>. These examples make it clear that the gravitational dynamics of negative masses is fairly different from that of positive masses. While negative and positive masses can lead to similar orbits in Newtonian mechanics <cit.>, we know of no experimental evidence supporting the existence of negative masses.



Within classical general relativity, this makes sense. One has a limited amount of matter types (, of stress-energy-momentum tensors) that can be considered in the theory. These correspond to the fields of the standard model of particle physics and their emergent consequences. It turns out that all known classical forms of matter satisfy certain energy conditions, which are impositions made on the stress tensor to restrict the allowed behavior of the matter in a certain spacetime. Within classical general relativity, it is well-known that the so-called dominant energy condition (DEC), together with other reasonable assumptions, is sufficient to ensure the positivity of the total mass in an asymptotically flat spacetime at a given time <cit.>. Hence, classical general relativity provides an answer to why we do not observe negative masses: as long as the matter content obeys the classical energy conditions, it follows that the total mass must be positive.

It happens, however, that quantum mechanical systems can easily violate the classical energy conditions, including the DEC. This means that quantum mechanical systems can have regions with negative energy density. For example, the Casimir effect <cit.> leads to a negative energy density between two conducting plates. It is then natural to wonder whether one could produce a macroscopic object with negative mass by exploiting quantum effects. 

With this in mind, Costa and Matsas recently analyzed whether quantum mechanical effects could lead to macroscopic negative masses by considering the Casimir system <cit.>. They noticed that to keep the Casimir plates separated, it is necessary to hold them apart with “struts.” Assuming these struts are classical, they must satisfy the DEC, the authors showed that the positive mass due to the struts is sufficient for the complete system to have a positive mass.

While Costa and Matsas analyzed the Casimir effect in Minkowski spacetime, their conclusions suggested the conjecture (already stated at the end of Ref. <cit.>) that a “cosmic-weight watcher must rule out from nature regular asymptotically flat stationary solutions of Einstein’s equations with M < 0”. This statement is vague regarding what could count as a “cosmic-weight watcher,” but some other conditions such as restrictions on the matter content or stability are known to be needed. For example, in Ref. <cit.>, Novikov, Bisnovatyi-Kogan, and Novikov considered a number of possible stellar solutions with negative energy density. Their interest was mostly in the fact that a star with negative energy density tends to expand due to gravitational effects and contract due to the star's pressure—which is the exact opposite of the roles pressure and gravity play in regular stars. With this goal, the authors studied a few different models that show how general relativity allows these solutions, but do not comment on whether the matter necessary to form these stars exists or whether these solutions are stable. 

If one decides to take input from the matter theory, then there is evidence that negative masses should be forbidden. While quantum fields do not obey the classical energy conditions, there is evidence that they obey weakened energy conditions, which hold only on average. In rough terms, while quantum field theory allows negative energy densities in a given region, these negative energy densities must be balanced by positive energy densities somewhere else. With this assumption, and assuming that the Einstein field equations hold semiclassically, one can conclude under some geometric assumptions on the underlying spacetime that mass must always be positive. This conclusion is mostly supported by a theorem due to Penrose, Sorkin, and Woolgar <cit.>, which must be complemented by a theorem due to Borde <cit.> to translate the result into a condition about the stress tensor.

It is interesting, however, to return attention to the original cosmic-weight watcher conjecture and ask: can general relativity discard negative mass solutions without the aid of quantum field theory? This would provide an alternative mechanism for the nonoccurrence of negative masses that does not rely on quantum theory at all. Hence, if true, it would show that gravity itself abhors negative masses, regardless of the matter content.

A natural candidate for the role of a cosmic-weight watcher is the stability of solutions. More specifically, we conjecture that there are no stable regular asymptotically flat stationary solutions of Einstein’s equations with M < 0. This time, we consider the possibility that equilibrium solutions are allowed but discard their relevance based on whether or not they are dynamically stable. Our findings provide strong evidence for this refined conjecture. Namely, all models of negative-mass stars we considered turned out to be unstable under linear perturbations or presented other problems. A fortiori, we establish that any stars with (*Pρ)_s < 0 at any point must be dynamically unstable, and all acceptable negative-mass stars must satisfy this condition somewhere (and therefore are unstable). This analysis does not require the use of energy conditions, and thus the conclusion is very different from the earlier results on positive mass theorems.

The structure of the paper is as follows. Section <ref> reviews the Tolman–Oppenheimer–Volkoff equation, which is the basic equation for the hydrostatic equilibrium of a static and spherically symmetric star in general relativity. Section <ref> reviews, for completeness, the role of energy conditions in ruling out negative masses, with particular emphasis on the semiclassical scenario, in which the averaged null energy condition (ANEC) is relevant. Section <ref> discusses how to study the stability of static, spherically symmetric stars in general relativity and how to adapt that formalism to stars with negative masses. Section <ref> then provides concrete examples of negative-mass stars. Section <ref> gives our main results on how negative-mass stars are hydrodynamically unstable. We conclude in section <ref>. Appendix <ref> exhibits how classical energy conditions can be violated in quantum field theory. Appendix <ref> expresses the ANEC integral (which is essential for verifying whether a given spacetime satisfies the ANEC) in a simpler form for a TOV-like star. Appendix <ref> computes a pair of integrals involving bump functions that are relevant for the stability analysis of a constant density star, and later for general stars.

We employ the same conventions used in Ref. <cit.>, which includes abstract index notation and has the sign convention +++ in the Misner–Thorne–Wheeler classification <cit.>. Latin indices stand for abstract indices. We also use geometric units with G=c=1. 



§ THE TOLMAN–OPPENHEIMER–VOLKOFF SOLUTION

To study negative-mass stars, we work on the framework of standard general relativity with some simplifying symmetry assumptions about the source T_a_b and metric. At this point we refrain from interpreting its origin.

The first symmetry we demand is that the spacetime is stationary, which is motivated by the fact we would like to describe a negative-mass star that can retain equilibrium. Then we ask for spherical symmetry to keep the calculations simple and manageable. Finally, we impose that the material composing the star is isotropic, meaning all of the three principal pressures of the stress tensor coincide. This simplifies the analysis, but this condition could likely be lifted at the expense of the calculations becoming more complex. 

Within these assumptions, there is a spherical coordinate system in which the line element can be written in the form

    s^2 = - e^2ϕ(r)t^2 + e^2ψ(r)r^2 + r^2 Ω^2,

where Ω^2 is the line element for the round metric in the unit two-sphere. r is such that the area of the spheres with constant coordinate radius r is 4 π r^2.

The symmetry assumptions also restrict the stress tensor to have the form

    T_a_b = ρ(r) u_au_b + P(r) (g_a_b + u_au_b),

where

    u^a = e^-ϕ(r)(t)^a.

Notice that u^a is a normalized timelike vector which is everywhere parallel to the stationary Killing vector field. 

The problem of solving the Einstein field equations with these ansätze was originally considered in these coordinates by Tolman, Oppenheimer, and Volkoff <cit.> and is reviewed in standard textbooks in general relativity <cit.> and stellar structure <cit.>. It can be shown that the solution inside the star is given by the line element

    s^2_< = - e^2ϕ(r)t^2 + (1 - 2m(r)/r)^-1r^2 + r^2 Ω^2,

where the subscript < in s^2_< indicates this is the interior solution. The functions ϕ and m are determined by means of the differential equations

    mr   = 4 πρ(r) r^2, 
    ϕr   = 4 π P(r) r^3 + m(r)/r[r - 2m(r)], 
    Pr   = -(ρ + P) 4 π P(r) r^3 + m(r)/r[r - 2m(r)].

Equation (<ref>) is known as the Tolman–Oppenheimer–Volkoff (TOV) equation.

The system of differential equations given by Eqs. (<ref>) to (<ref>) is underdetermined, since it involves four unknowns (ρ, P, m, and ϕ), but only three equations. In most applications, the remaining equation is supplied in the form of a thermodynamical equation of state relating P and ρ. The procedure to solve the TOV system is then to integrate Eqs. (<ref>) and (<ref>) from the center to the border of the star with the aid of the equation of state. This is often done numerically by providing the initial conditions m(0) = 0 and P(0) = P_0 [It is also possible to provide ρ(0) = ρ_0 instead of P(0) = P_0. The condition m(0) = 0 is used to avoid the presence of a physical singularity at the origin (this singularity would be analogous to the singularity at the origin of Schwarzschild spacetime). Notice also that in practical numerical computations, one does not give the initial condition precisely at r = 0, where the right-hand side of the TOV equation is indeterminate at best. Rather, one introduces a small cutoff r_0 > 0 and provides initial conditions at r_0. Here, we use this method with initial conditions of the form ρ(r_0) = ρ_0 and m(r_0) = 4/3πρ_0 r_0^3.]. One then proceeds with the integration until the boundary of the star is reached at r=R, defined by the condition that 

    P(R) = 0.

One then usually assumes that the star ends at R and ρ and P are understood to vanish for r > R. In this outer region, the metric is given by the standard Schwarzschild metric in vacuum with mass parameter

    M = m(R).


Once ρ, P, and m are known, one can solve Eq. (<ref>) to obtain ϕ. The boundary condition is then that the metric of the spacetime is continuous across the stellar boundary at r = R and translates into the condition [In numerical computations, it may be more efficient to solve for ϕ while solving for the remaining variables. This can be done by imposing any initial condition for ϕ at the center of the star, and then using the fact that Eq. (<ref>) is linear in ϕ to subtract the value obtained at the boundary and add the value desired at the boundary, hence fixing the appropriate boundary condition.]

    e^2ϕ(R) = (1 - 2M/R).


The boundary conditions at R, Eqs. (<ref>) to (<ref>), can be deduced from the Israel junction conditions <cit.> with the additional assumption that the stress-energy tensor is not singular at the stellar surface. Stars with discontinuous pressure can be described by allowing a thin matter shell at r=R, in which case the boundary conditions are relaxed as well.

In most practical applications in astrophysics, one is interested in a particular model for a star and wants to understand how this model gravitates. For example, one can consider an equation of state modeling a neutron star and use it to solve the TOV system. Nevertheless, this approach is restrictive in the study of negative-mass stars, whose equation of state is not predetermined. Instead of using only equations of state to fix a stellar solution, one can impose that the star should have a particular energy density profile. In other words, one imposes by hand that the function ρ(r) is given by an ansatz. With this function fixed, one then solves the TOV system with the condition that the star ends at some predefined radius r=R. This method was employed in Ref. <cit.> and seems more convenient for generating negative-mass stars since one can fine-tune the behavior of the stellar mass. 



 §.§ Example: The Schwarzschild Star

    A first example of a negative-mass star is based on a star of constant density profile ρ(r) = ρ_0. This is known as a Schwarzschild star <cit.>. Under this ansatz, the TOV system can be handled analytically. The mass parameter is given by
    
    m(r) = 4 πρ_0 r^3/3.

    The boundary condition m(R) = M fixes the value of M according to
    
    ρ_0 = 3 M/4 π R^3.

    We are interested in the case with M < 0 (and hence ρ_0 < 0).

    The solution for the pressure is the same as in the positive-mass case. It is given by 
    
    P(r) = ρ_0 [√(1 - 2M/R) - √(1 - 2M r^2/R^3)/√(1 - 2M r^2/R^3) - 3 √(1 - 2M/R)].

    The pressure is everywhere non-negative and finite if M/R < 4/9, which holds for a wide class of equations of state and is known as the Buchdahl limit <cit.>. In particular, notice the pressure is everywhere non-negative and finite when the star has negative mass. 

    Having found a star with a negative mass, there are two important questions we would like to address. 
    
        
  i. Is the matter composing this star allowed by quantum field theory?
        
  ii. Is this configuration stable?
    
    We consider them in the next two sections.



§ ENERGY CONDITIONS

It is not surprising that the Einstein equations allow for negative-mass stars. After all, any Lorentzian geometry is a solution to the Einstein equations as long as one chooses the right stress-energy-momentum tensor. 

To put constraints on the physical reasonableness of a given solution, one often imposes energy conditions <cit.>. These are restrictions on the stress tensor that enforce it to have some interesting properties, typically understood as energy being positive in a suitable sense. The weakest of the classical energy conditions is the null energy condition (NEC), reviewed for example in Refs. <cit.>. It states that for all null vectors k^a the stress tensor satisfies the bound

    T_a_bk^ak^b≥ 0.

In the particular case of interest, a perfect fluid, this condition states that ρ + P ≥ 0. 

The NEC is not the only energy condition of classical interest. It is important to point out the dominant energy condition (DEC), which states that for all future-directed causal vectors ξ^a it holds that

    -T^a_bξ^b

is causal and future-directed. This condition essentially imposes that all observers see causal, future-directed energy fluxes. For a perfect fluid, the DEC states that ρ≥P. Notice that the DEC implies the NEC. The interest in this condition is that it was used by Schoen, Yau, and Witten to obtain the first positive-mass theorems in general relativity <cit.>. Hence, within suitable assumptions, the dominant energy condition ensures that the spacetime has a positive mass.

All known forms of classical matter satisfy the NEC [It is known that some forms of classical matter, such as a non-minimally coupled scalar field, violate the NEC—see, , Refs. <cit.>. Nevertheless, we are restricting our focus to the classical fields associated with the standard model and we assume general relativity to be the theory describing gravitational interactions. In particular, once we assume gravity to be described by general relativity, we are also assuming that all standard model fields are minimally coupled. In any case, this does not affect the main points of our discussion.], and in fact even the DEC. Does our star?

The positive mass theorems due to Schoen, Yau, and Witten already imply that the star cannot satisfy the DEC. Using Eq. (<ref>), one can promptly show that

    ρ(r) + P(r) = ρ_0 [2 √(1 - 2M/R)/3 √(1 - 2M/R) - √(1 - 2Mr^2/R^3)].

By close inspection, one can tell that the term in brackets is always positive for 0 ≤ r ≤ R. Since ρ_0 < 0, we conclude that ρ(r) + P(r) < 0. Hence, the star cannot be built from any known form of classical matter.

This is not surprising. After all, we started this discussion by considering the Casimir effect, which is inherently quantum. There is no a priori reason to expect that classical configurations of matter could lead to negative energy solutions, but, in principle, one may question whether quantum configurations can lead to macroscopic negative masses since they are known to allow local violations of the positivity of energy—in fact, all classical energy conditions, including the NEC, can be easily violated by exploiting quantum effects, as shown in appendix <ref>. Is there a quantum analog of the classical energy conditions? 

There is. The averaged null energy condition (ANEC), in the conventions of Ref.  <cit.>, states that, for any inextendible null geodesic γ with affine parameter λ, it holds that

    ∫_-∞^+∞T_a_b(γ(λ)) γ̇^a(λ) γ̇^b(λ) λ≥ 0

whenever the integral is absolutely convergent. T_a_b is now understood as the renormalized expectation value of the stress tensor. Notice the idea is then that the NEC can be locally violated, but it still holds on average—an idea originally due to Tipler <cit.>. An even weaker condition is the achronal ANEC (AANEC), which states that the ANEC must hold only for achronal geodesics, , for null geodesics such that no pair of its points can be connected by a timelike curve. The restriction of the ANEC to achronal geodesics was originally considered by Wald and Yurtsever <cit.>, and it is interesting because there is physical reason to believe that the AANEC is true in all physically reasonable circumstances and that it could be implied by a fundamental principle of full quantum gravity—see Refs. <cit.>. Furthermore, the ANEC is equivalent to the quantum null energy condition <cit.>, which is interesting in its own right and is implied in turn by an interesting conjecture pertaining to the quantum focusing of congruences of geodesics <cit.>.

The fact that the ANEC demands that energy be positive on average is remarkably similar to how Costa and Matsas dealt with the Casimir system: while the energy density could be negative within the plates, it was positive somewhere else, and the net energy was positive. Hence, the ANEC appears to provide an interesting criterion to establish the net positivity of mass. Instead of focusing on forcing the energy density to be positive everywhere, we simply require that negative energy densities be compensated somewhere else. Furthermore, it should be mentioned that the ANEC does hold in the Casimir system <cit.>.

Assuming that every AANEC integral is absolutely convergent, this intuition is correct. This is a corollary of works by Borde <cit.> and Penrose, Sorkin, and Woolgar <cit.>, which we briefly review below. 

Borde's theorem is a statement about the occurrence of conjugate points in geodesics satisfying a weaker form of an averaged energy condition. His original result, focusing theorem 2 in Ref. <cit.>, applies to any causal geodesic, but we will focus on the case of null geodesics. Borde's theorem is the following. 


    Consider a complete null geodesic with tangent vector k^a and affine parameter λ. For each ϵ > 0, assume there is some b > 0 such that for any λ_1 < λ_2 there is a pair of intervals I^- < λ_1 and I^+ > λ_2—each with length larger than or equal to b—such that
    
    ∫_λ'^λ”R_a_bk^ak^bλ≥ - ϵ∀λ' ∈ I^-, ∀λ”∈ I^+.

    If k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 at some point on γ, then γ contains a pair of conjugate points.


This is a generalization of an earlier result due to Hawking and Penrose that establishes the occurrence of conjugate points based on the validity of the null convergence condition, which states that R_a_bk^ak^b≥ 0 for all null vectors k^a <cit.>. The basic idea of both proofs is to use the curvature condition to ensure that the geodesics do not defocus, while the condition k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 ensures that the geodesic interacts with curvature at some point for the focusing to start. The main technique in the proof consists in manipulating the Raychaudhuri equation <cit.>. 

Under the hypothesis of absolute convergence of the integral ∫_-∞^∞ R_abk^ak^bλ, we can show the following intuitive corollary.


    Consider a complete null geodesic with tangent vector k^a and affine parameter λ. Suppose that
    
    ∫_-∞^+∞R_a_bk^ak^bλ≥ 0

    and that the integral converges absolutely. If it holds that k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 at some point on γ, then γ must contain a pair of conjugate points.


The assumption of absolute convergence is used to ensure that the “tails” of the integrand as λ→ +∞ are negligible compared to the “bulk” contributions. Notice that if the Einstein field equations hold (as we assume they do), then the condition on the integral of R_a_bk^ak^b is equivalent to the ANEC integral being non-negative. Furthermore, it is known that achronal null geodesics cannot have conjugate points (Proposition 4.5.12 in Ref. <cit.>). Hence, we get the following result.


    Consider a complete null geodesic with tangent vector k^a and affine parameter λ. Suppose that the Einstein equations and the ANEC hold. If k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0 at some point on γ, then γ is not achronal.


This was known to Penrose, Sorkin, and Woolgar at the time of writing of Ref. <cit.>. The result they established is then a result concerning the positivity of mass based on the existence of achronal geodesics in the spacetime. 

The basic idea behind the argument runs as follows. The Shapiro time delay <cit.> shows that null geodesics that pass closer to a positive mass are delayed (in coordinate time) relative to geodesics that pass farther away. Analogously, one concludes that geodesics passing close to a negative mass will be “faster” (in coordinate time) than geodesics that pass far from the negative mass. With this in mind, Penrose, Sorkin, and Woolgar constructed the fastest causal curve from the past null infinity to the future null infinity of the conformal completion of an asymptotically flat spacetime and showed this is an achronal geodesic. This geodesic can pass either through the interior of the spacetime or avoid it completely by travelling only through the conformal boundaries of the spacetime. Due to the Shapiro time delay, if the spacetime has negative mass, this geodesic must enter the spacetime. They then conclude that if the spacetime has no complete achronal null geodesics, it must follow that the spacetime mass must be non-negative. 

The precise statement is as follows <cit.>:


    Let M ∈ℝ and let B ⊆ℝ^3 be a ball with radius R > 2 M. Consider the metric *g^M_a_b on ℝ× (ℝ^3 ∖ B) with line element
    
    s_M^2 = - (1 - 2M/r) t^2 + (1 - 2M/r)^-1r^2 + r^2 Ω^2.

    A metric g_a_b on ℝ× (ℝ^3 ∖ B) is said to be uniformly Schwarzschildean if there is an M ∈ℝ such that, in the coordinates of Eq. (<ref>), it holds that
    
    g_μ_ν - *g^M_μ_ν = o(M r^-1), 
    ∂_μ(g_ν_ρ - *g^M_ν_ρ) = o(M r^-2).

    In the case M = 0, the above notation is understood to mean the metric is flat for r > R, where R ≥ 0 is some constant.



    Consider a spacetime (M,g) taken to be asymptotically flat at null and spatial infinity and uniformly Schwarzschildean. Let 𝒟 = I^-(ℐ^+) ∩ I^+(ℐ^-) be the domain of outer communications of the spacetime. Assume that 𝒟∪ℐ^+ ∪ℐ^- is globally hyperbolic as a subset of the conformal extension of M. If there are no achronal null geodesics connecting ℐ^- to ℐ^+ and passing through 𝒟, then it follows that the ADM four-momentum of M is future-causal.  


Recall that the ADM four-momentum (after Arnowitt, Deser, and Misner <cit.>) is a notion of four-momentum of the whole spacetime measured at a single instant of time (which is to be understood as a spacelike hypersurface of the spacetime). 

The Penrose–Sorkin–Woolgar theorem can then be combined with corollary <ref> to yield the following result. 


    Consider a spacetime (M,g) taken to be asymptotically flat at null and spatial infinity and uniformly Schwarzschildean. Let 𝒟 = I^-(ℐ^+) ∩ I^+(ℐ^-) be the domain of outer communications of the spacetime. Assume that 𝒟∪ℐ^+ ∪ℐ^- is globally hyperbolic as a subset of the conformal extension of M. Suppose the Einstein field equations hold, that the AANEC integrals are always absolutely convergent and non-negative, and that the null generic condition holds—, that for all null geodesics, there is some point at which k^ck^dk_[_aR_b_]_c_d_[_ek_f_]≠ 0. Then the ADM four-momentum of the spacetime is future-causal.


In the above result, the AANEC is sufficient (as opposed to the ANEC), because we only need to rule out the achronal geodesics. If a geodesic is already known to be chronal, it can be ignored. Notice then that the AANEC will typically be a vacuous truth. 

The exterior solution for a spherically symmetric star in general relativity is always the Schwarzschild solution due to Birkhoff's theorem. Hence, all spherically symmetric stellar spacetimes must be uniformly Schwarzschildean, and hence satisfy the conditions for the Penrose–Sorkin–Woolgar theorem. It follows that a star satisfying the AANEC and the null generic condition can never have negative mass in general relativity. Due to the strong evidence supporting the AANEC <cit.>, it seems unlikely that one can build negative-mass stars within the domain of the semiclassical Einstein equations. 



§ DYNAMICAL STABILITY OF STARS

From the perspective of averaged energy conditions, it seems unlikely that one can build negative-mass stars. However, this requires a lot of input from quantum theory, and it would be interesting to try to discard negative masses based purely on classical physics. With this in mind, in this section, we study this problem from the perspective of stability analysis. 

There are three obvious ways in which a star can be spontaneously destroyed: through hydrodynamic instabilities, thermodynamic instabilities, and the consumption of nuclear fuel. In regular stars, the time scales associated with hydrodynamic processes are much shorter than those associated with thermal processes, which, in turn, are much shorter than those associated with nuclear processes. Ref. <cit.>, for example, considers the Sun and estimates the time scales for hydrodynamic processes at around t_H ∼e3, for thermal processes at t_T ∼3e7, and t_N ∼e10 for the burning of nuclear fuel. Hence, hydrodynamic stability is the most important one in regular stars.

While the estimates for the Sun cannot be trusted when dealing with negative-mass stars, it still seems natural to focus on the hydrodynamic stability. While this is partially motivated by the time scale estimates we just mentioned, there is also the important reason that a thermodynamic or nuclear analysis would require a precise model of the matter constituting the star, for example in the form of an equation of state. This model, however, is not available, since we are precisely avoiding making too many assumptions on the composition of matter. Hence, we have insufficient information for discussing the thermal or nuclear stabilities of negative-mass stars. Furthermore, since they would necessarily be made of unconventional matter (or purely of quantum effects), one lacks a good justification for the use of traditional techniques developed for regular matter. Thus, we focus on the hydrodynamic stability.

The simplest way of discussing the hydrodynamic stability of a star is to consider linear perturbations of the stellar parameters and metric coefficients about the background spacetime provided by the TOV solution. One then writes the metric and the stellar parameters as the equilibrium solutions to the TOV equations plus small time-dependent perturbations. The Einstein equations for these perturbations are linearized. The end goal is to see whether the perturbations grow in time or remain bounded. If they grow in time, the star is deemed unstable. 

This problem was originally considered by Chandrasekhar <cit.> (see also the review in Chap. 26 of Ref. <cit.>), who simplified the problem to the study of solutions of the so-called Chandrasekhar pulsation equation. This is an equation that describes the Lagrangian displacement of the perturbation ξ. This is a measure of how much each fluid element in the star is dislocated by the perturbation. If the fluid element was originally at r before the perturbation, it is taken to r+ ξ(t,r) by the perturbation. The pulsation equation is given by

    r[p(r) χr] + q(r) χ(t,r) = w(r) [2]χt,

where χ is a “renormalized Lagrangian displacement”, from which all remaining perturbations can be promptly calculated. It is given in terms of the “true Lagrangian displacement” ξ by 

    χ(t,r) = r^2 e^-ϕ(r)ξ(t,r).

The coefficient functions p (not to be mistaken for the pressure P), q, and r are determined by the background stellar spacetime according to


    p(r)    = e^ψ + 3 ϕγ P/r^2, 
    
        q(r)    = e^ψ + 3 ϕ[1/r^2(P + ρ)(Pr)^2 - 4/r^3Pr - 8πP/r^2  e^2 ψ (P+ρ)], 
    
        w(r)    = (ρ + P)/r^2 e^3ψ + ϕ.


The coefficient p in the Chandrasekhar pulsation equation involves the “effective polytropic index”

    γ = 1/P(Pn)_ρ[n - (ρ + P) (nρ)_P],

where n is the baryon number density (with antibaryons counted negatively).

Typically, the boundary conditions for the problem are that <cit.> 

    lim_r → 0χ(t,r)/r^3 < ∞and[γ P e^ϕ/r^2χr]_R = 0.

The first one ensures χ and its derivative stay finite at the center of the star, while the second one enforces that fluid elements on the boundary of the star stay at the boundary of the star [This is done by imposing that the Lagrangian perturbation of the pressure vanishes at the boundary.]. Additional conditions may be imposed to keep ξ finite at the boundary <cit.>.

The solution of Eq. (<ref>) is carried out by the separation of variables χ(t,r) = τ(t)ζ(r). The resulting equations are 

    r[p(r) ζr] + q(r) ζ(r) + σ^2 w(r) ζ(r) = 0,

also known as the Chandrasekhar pulsation equation, and

    [2]τt + σ^2 τ = 0.


The boundary conditions take the form

    lim_r → 0ζ(r)/r^3 < ∞and[γ P e^ϕ/r^2ζr]_R = 0.


The solution to Eq. (<ref>) is 

    τ(t) = τ(0) cos(σ t) + τ̇(0)/σsin(σ t),

where the dot denotes a time derivative. Notice, in particular, that if σ^2 < 0 (, if σ is imaginary), then τ(t) grows exponentially with time for a generic initial condition. In this case, the perturbation is unstable, and so is the star. The main question for stability with the Chandrasekhar pulsation equation is then to study whether σ^2 > 0 always holds or whether there are exceptions. If there is a single admissible value of σ for which σ^2 < 0, then the star is unstable, because τ(t) will grow exponentially in time for this mode.

Eq. (<ref>), when combined with the boundary conditions of Eq. (<ref>), corresponds to a Sturm–Liouville problem <cit.> with eigenvalues σ^2. This determines the allowed values of σ. For example, in standard astrophysical applications, it holds that p(r) > 0 and w(r) > 0 in (0,R). This ensures the problem is a self-adjoint eigenvalue problem in a Hilbert space, and thus σ^2 is always real.

The standard methods for studying stability with the Chandrasekhar pulsation equation involve techniques from Sturm–Liouville theory. For example, method 2-C in Ref. <cit.> obtains the number of unstable modes by solving numerically the Chandrasekhar equation with σ^2 = 0 and counting the number of zeros in the solution. By the Sturm Comparison Theorem <cit.>, this number of zeros can be understood as the number of eigenvalues smaller than zero, , the number of eigenvalues with σ^2 < 0. This is only possible because, for usual stars, p(r) > 0 and w(r) > 0 for r ∈ (0,R), which means the problem is sufficiently regular to use standard results from Sturm–Liouville theory (although the endpoints r=0 and r=R are often singular). Alternatively, method 2-D in Ref. <cit.> uses a variational technique to obtain the sign of the smallest eigenvalue, which is a technique based on the so-called min-max principle of functional analysis <cit.>.

There are a few reasons, however, that keep the standard Chandrasekhar pulsation equation from being straightforward when dealing with negative-mass stars. The first is that the NEC violations which are required in negative-mass stars (otherwise the Penrose–Sorkin–Woolgar theorem would imply the star has positive mass) lead to sign flips in the coefficients p and w. This means we are apparently outside the domain of the regular Sturm–Liouville problem, and there is no a priori reason to believe that results such as the Sturm Comparison theorem still hold. Hence, at first glance, it seems challenging to extract the sign of σ^2 in a NEC-violating star.

The second reason is also practical in nature and involves the difficulty in knowing the equation of state for the star. Recall that the Chandrasekhar pulsation equation involves the effective polytropic index γ. For an adiabatic perturbation, one can rewrite this expression as <cit.>

    γ = ρ + P/P(Pρ)_s,

where s is the entropy per baryon in the star. In the case of a barotropic equation of state (, an equation of state of the form P = P(ρ)), we can write

    (Pρ)_s = Pr(ρr)^-1

and compute γ as a function of r without using further information about the equation of state. However, for an equation of state that is not barotropic, this ratio of radial derivatives could correspond to the partial derivative with different quantities being held constant because (*Pρ)_T, (*Pρ)_s, and (*Pρ)_n, for example, need not coincide. 

If the star in question was obtained from an equation of state, this is not a problem. However, stars obtained from energy density profiles provide us more freedom to exploit when building examples. For these stars, there is no a priori reason to believe that its constituent matter is barotropic. Hence, the need to know γ is particularly inconvenient.

Finally, a third objection to the Chandrasekhar pulsation equation is conceptual. One of the assumptions that enter Chandrasekhar's derivation is baryon number conservation. Some authors go as far as saying that “The most fundamental law of thermodynamics—even more fundamental than the `first' and `second' laws—is baryon conservation” <cit.>. Nevertheless, it is not clear whether this is applicable in the case of a negative-mass star. In the negative-mass case, it is natural to begin by generalizing “baryon number” to “particle number”, where particle refers to the particles of whichever field composes the star. Nevertheless, even with this generalization the conservation of particle number is still unclear. For example, the quantum state violating the NEC shown in appendix <ref> is not an eigenstate of the particle number operator, and as a consequence, the number of particles is not defined. Hence, the number of particles in a negative mass star is, in general, ill-defined. Hence, we prefer to avoid techniques which rely on baryon number conservation.

Luckily, within suitable assumptions, these difficulties can be overcome. The regularity issue can be solved by noticing that all coefficients flip signs precisely when P+ρ vanishes (p and q can undergo additional sign flips, but these are unimportant). Due to this, the problem can be multiplied by the sign of P+ρ to yield a sufficiently regular problem. This allows the use of some of the techniques in Ref. <cit.>. The main available technique is method 2-D, which is numerically costly, but manageable for our purposes. 

The difficulty with γ can be overcome in the particular case of a barotropic equation of state, and the validity of this hypothesis can be verified by making a parametric plot of P(r) against ρ(r) and checking whether the resulting graph yields a function. While this is not proof that the equation of state is barotropic, it gives evidence that the star can be well-approximated by a barotropic equation of state. In practice, however, it may not be possible to obtain this equation of state.

The difficulty concerning baryon number conservation can be solved in the particular case of an adiabatic perturbation. The hypothesis of baryon number conservation enters Chandrasekhar's derivation in order to express the Eulerian pressure perturbation P as a function of the Lagrangian displacement ξ. For a general perturbation, this method seems adequate. For an adiabatic perturbation, however, we can impose simply that the Lagrangian pressure perturbation Δ P satisfies

    Δ P = (Pρ)_s Δρ.

With the aid of Eq. (<ref>), one can show that this prescription yields the same result obtained by Chandrasekhar <cit.> using baryon number conservation. Recall that the Lagrangian perturbation Δα of a quantity α is the perturbative change in a quantity as one follows a fluid element, whereas the Eulerian perturbation α is the change as one considers a fixed point in space <cit.>. They are related, to linear order, by

    Δα(t,r) = α(t,r) + ξ(t,r) αr(r),

where *αr refers to the background (unperturbed) value of α.

We notice that an incorrect approach one could consider to avoid these difficulties is to impose that the perturbations satisfy a simple equation of state that is not necessarily the same as the one of the underlying star. For example, one could impose that the Lagrangian perturbation of the pressure of the star always vanishes. This would correspond to a dust-like perturbation. However, in this case, the equations describe the evolution of the dust particles on a background describing a static star, rather than the motion of the star itself. Hence, one cannot conclude anything about stability in such an approach, but rather only about the behavior of the new matter in the star. For example, one can conclude that grains of dust in the Sun move either toward the surface or the center, not that the Sun is unstable.

With all of these considerations, the remaining mathematical difficulty in the problem is that both the coefficients p and w in the Sturm–Liouville problem change signs. In particular, the fact that w changes sign means we are not considering an eigenvalue problem in a Hilbert space, but rather in a more general vector space with an indefinite inner product. This keeps us from using standard results from functional analysis on Hilbert spaces. However, we can manipulate the Chandrasekhar pulsation equation into a form corresponding to a self-adjoint problem in a Hilbert space. This is because all of the coefficient functions in the Sturm–Liouville problem change sign simultaneously, precisely at the point in which the NEC is violated. We can then remove the sign of the “NEC factor” P + ρ and get to a problem that is better behaved. In more detail,



 §.§ Alternative Form of the Chandrasekhar Pulsation Equation

    We assume the perturbations to be adiabatic, so that Eq. (<ref>) holds. Recall the Chandrasekhar equation in the Sturm–Liouville form of Eq. (<ref>), with the relevant functions being given in Eqs. (<ref>) to (<ref>) and (<ref>). Recalling that the TOV equations yield 
    
    Pr = - (P+ρ) ϕr,

    we find that the Sturm–Liouville coefficient functions become 
    
    
    p(r)    = e^ψ + 3 ϕP+ρ/r^2(Pρ)_s, 
    
            q(r)    = e^ψ + 3 ϕ[(P + ρ)/r^2(ϕr)^2 + 4(P+ρ)/r^3ϕr - 8πP/r^2  e^2 ψ (P+ρ)], 
    
            w(r)    = (ρ + P)/r^2 e^3ψ + ϕ,

    
    which used Eq. (<ref>). 
    
    We notice that all three functions above are proportional to P+ρ, which can flip its sign inside the star due to NEC violations and cause w to reverse its sign as well. Suppose we eliminate this sign term from the equation. In that case, we will get to a more standard Sturm–Liouville problem, which in the worst-case scenario will still correspond to a self-adjoint eigenvalue problem in a Hilbert space (rather than an indefinite inner product space).

    This can be easily done by defining the function 
    
    s(r) = sign(P(r)+ρ(r)).

    Where the sign function is defined by
    
    sign(x) = 
                +1,    if  x > 0, 
    
                0,    if  x = 0, 
    
                -1,    if  x < 0.


    We can now write the Chandrasekhar pulsation equation as
    
    r[s(r)p(r) ζr] + s(r)q(r) ζ(r) + σ^2 s(r)w(r) ζ(r) = 0,

    which is possible because p, q, and w all vanish at the point in which s(r) does. In particular, the derivative of s(r) does not lead to any problems because p(r) vanishes at the point in which the derivative does not. By inspecting Eq. (<ref>) one can tell that s(r)w(r) is manifestly non-negative, and this ensures the Sturm–Liouville problem takes place in a Hilbert space. 

    For future convenience, we introduce the shorthand notation
    
    p_⋆(r) = s(r)p(r)

    and similarly for q_⋆ and w_⋆.



 §.§ Variational Technique for Stability Analysis

    To see whether a Sturm–Liouville problem admits negative eigenvalues, we can use the so-called min-max principle (Theorem XIII.1 in Ref. <cit.>). For our purposes, it states that given a self-adjoint operator L with spectrum spec(L) on a Hilbert space, it holds that
    
    infspec(L) ≤⟨ψ|Lψ⟩/⟨ψ|⟩

    for all ψ∈Dom(L) with ψ≠ 0. Ref. <cit.> makes the additional assumption that L is bounded from below, but since we are only interested in knowing whether spec (L) extends to negative values, we do not need this additional assumption (having a spectrum which extends to minus infinity means there is a negative value in the spectrum). This is a costly albeit standard technique in the study of stellar stability used by Chandrasekhar in the same references in which the pulsation equation was introduced <cit.>. It corresponds to method 2-D in Ref. <cit.>. This method is also widely used in textbook quantum mechanics to find upper bounds on the ground state energy of a system <cit.>.
    
    In the Sturm–Liouville problem with coefficient functions p_⋆, q_⋆, and w_⋆ the inner product is given by 
    
    ⟨φ|ψ⟩ = ∫φ(r) ψ(r) w_⋆(r) r,

    where we assume the functions to be real. We can then cast the problem in the form
    
    L ζ = σ^2 ζ,

    with 
    
    L = - 1/w_⋆(r)[r[p_⋆(r) r] + q_⋆(r)].

    
    L is self-adjoint [The standard references <cit.> do not pay close attention to the difference between symmetric and self-adjoint operators, so neither will we. Details of this type within Sturm–Liouville theory can be found, for example, in Ref. <cit.>.] when we assume homogeneous boundary conditions (our case of interest). Since w_⋆ can only be zero at points in which p_⋆ and q_⋆ vanish (which turns out to be a point in which p_⋆' also vanishes) and all of these functions vanish in precisely the same way, the operator L is well-defined.
    
    Using integration by parts one can write
    
    ⟨ψ|Lψ⟩ = p_⋆(0)ψ'(0)ψ(0) - p_⋆(R)ψ'(R)ψ(R) 
     + ∫_0^R p_⋆(r) ψ'(r)^2 - q_⋆(r)ψ(r)^2 r.

    The boundary conditions we are interested in, given in Eq. (<ref>), are such that we can assume the boundary contributions to vanish. Hence, we can write that 
    
    infspec(L) ≤∫_0^R p_⋆(r) ψ'(r)^2 - q_⋆(r)ψ(r)^2 r/∫_0^R w_⋆(r) ψ(r)^2 r≡ S[ψ],

    where ψ is assumed to satisfy the boundary conditions in Eq. (<ref>). Since the denominator on the right-hand side is always positive, the question of stability is reduced to whether S[ψ] admits negative values for some ψ satisfying the boundary conditions.



 §.§ Bardeen's Technique for Stability Analysis

    An alternative method of studying the stability of a star using the Chandrasekhar pulsation equation is due to Bardeen <cit.> (see also Method 2-C in Ref. <cit.>). It uses the Sturm Comparison Theorem <cit.> and the fact that the n-th eigenfunction of a Sturm–Liouville problem has n roots to establish the number of unstable modes in the star. The basic idea is that increasing the eigenvalue in the Sturm–Liouville equation increases the number of roots a solution with fixed initial (not boundary) conditions has in the interval (this follows from the Sturm Comparison Theorem). As a consequence, one can count how many roots the numerical solution with σ^2=0 and this will correspond to the number of eigenvalues with σ^2 < 0. This number therefore corresponds to the number of unstable modes in the star.

    The basic process is the following. One numerically solves the pulsation equation with the ansatz σ^2 = 0. If the stellar surface is a regular point for the Sturm–Liouville problem, then one starts from there. Otherwise, one starts from the center. One counts how many roots the numerical solution has inside the star and this gives the number of unstable modes. If the numerical solution reaches the other end of the star (either the center or the boundary, depending on where integration started) and satisfies the boundary condition there, then σ^2 = 0 is one of the eigenvalues of the problem, meaning there is a mode with neutral stability.

    This method is much more efficient than the variational technique but has some drawbacks. The variational technique relies on a very general result from functional analysis (the min-max principle), while Bardeen's method depends on the Sturm Comparison Theorem, which is a result about a certain class of Sturm–Liouville problems. In particular, the theorem requires that p_⋆ be positive on (0,R), and hence Bardeen's technique cannot be applied to stars in which P+ρ flips signs or in which (*Pρ)_s < 0. This excludes all acceptable models of negative-mass stars (see section <ref>), but the method is still convenient to illustrate how the variational technique works.



 §.§ Example: Relativistic Polytropes

    As a warm-up, let us use these stability criteria against some simple examples. Namely, we consider relativistic polytropes in the sense originally considered by Tooper <cit.>. These models were also studied by Chandrasekhar in Ref. <cit.> as an example application of the original pulsation equation.

    The equation of state we will consider is
    
    P = ρ^Γ,

    where Γ is a constant. Since we will do the computations numerically, from this point onward we also set R_0 = 1, where R_0 is some arbitrarily chosen length scale. This length scale is to be understood as the typical length scale of the negative mass objects under consideration. Different choices for R_0 affect the quantitative results in the sense that the equations of state and density profiles considered are different depending on the value of R_0. For example, Eq. (<ref>) would typically involve a dimensional constant in front of ρ^Γ, which we are setting to 1. This constant depends on R_0. Due to the absence of a preferred length scale to be considered, we choose to keep R_0 unspecified. R_0 cannot be too small to the point of the classical description of spacetime to be untrustworthy, but very large values of R_0 would likely be uninteresting on experimental grounds.
    
    The mass-radius diagrams and mass per central density diagrams for some values of Γ are shown in figures <ref>, <ref>, and <ref>. In the Newtonian case, Γ = 4/3 and Γ = 5/3 are common stellar models <cit.>.

    

    

    

    The spiral structures in the mass-radius diagrams for these three different sequences of stars are a very generic feature of the TOV equation, as per Theorem 6.4 on Ref. <cit.>. The most important feature for stability analyses is the sign of 
    
    Mρ_0,

    where M is the equilibrium value of the mass, for it provides a back-of-the-envelope stability criterion (see, , Ref. <cit.>). Indeed, consider a sequence of stars and pick a star with mass M̅ > 0 and Mρ_0 > 0. Perturb this star by increasing the central density according to
    
    ρ_0 →ρ_0 + ρ_0

    (supposing ρ_0, ρ_0 > 0), assuming that M̅ is kept constant during this process. Since *Mρ_0 > 0, M̅ is smaller than the mass M of the equilibrium star with central density ρ_0 + ρ_0. This means the gravitational field of the perturbed star is weaker than it should be to retain equilibrium. Consequently, the increased pressure forces the star to expand, thereby decreasing the central pressure and density, and bringing the perturbed star back to the original equilibrium state. Notice that if we had *Mρ_0 < 0 instead, the perturbation would grow and the star would be unstable. It should be pointed out that this is a necessary, but not sufficient, criterion for stability. 

    To perform more detailed stability analyses, we need to fix the boundary conditions to be considered in the Chandrasekhar pulsation equation. Following Eq. (<ref>), we take 
    
    ζ(r) ∼ r^3  at  r ≈ 0

    and
    
    ζr = 0  at  r = R.

    
    We will first use the variational technique to obtain the sign of the lowest mode of oscillation. We pick a trial function and compute S[ψ] to check its sign. If it is negative, the star is unstable. To abide by the boundary conditions, we will use trial functions of the form 
    
    ψ_n(r) = r^3 - 3 R^-n/3+n r^3+n

    where the coefficient was adjusted so that ψ_n'(R) = 0. The constant n must be positive to obey the boundary conditions, but it is otherwise arbitrary. These types of trial functions were considered in Ref. <cit.>, where they were used with more elaborate techniques to estimate the first few eigenvalues of the Chandrasekhar pulsation equation. In scenarios in which the star is very inhomogeneous these trial functions can lead to misleading estimates. Yet, since we are only interested in the sign of the smallest eigenvalue we do not need to worry about precise approximations.

    Using these trial functions, we can compute S[ψ_n] and check its sign. The results are shown in table <ref>. We see that the variational criterion reproduces the back-of-the-envelope expectations we had for the sequences with Γ = 2 and Γ = 5/3, but not for Γ = 4/3. For the sequence with Γ = 4/3, the star with ρ_0 = 5.623e-1 passes the necessary condition for stability from the back-of-the-envelope criterion, but more careful calculations show it is not stable. In fact, it is well known <cit.> that in Newtonian theory a polytrope is only stable if Γ > 4/3, and the bound gets tighter in relativity. This is consistent with the fact that the back-of-the-envelope criterion is only a necessary, not sufficient, condition for stability.
    
    

    Of course, the positive values of S[ψ_n] in table <ref> do not imply stability, they simply fail to falsify it. In the absence of other methods, we could repeat the computation for other values of n and choices of test functions.

    In these examples, the pulsation equation is sufficiently well-behaved to employ Bardeen's technique. We report the results in table <ref>, all of which agree with the results obtained through variational methods. Notice that Bardeen's technique can establish stability and confirms that the positive values of table <ref> indeed correspond to stable stars.

    



§ EXAMPLES OF NEGATIVE-MASS STARS

We are now ready to show different stellar models with negative masses and study their stability. 

The two main methods for generating negative-mass stellar models are explored in Ref. <cit.>: proposing an equation of state, or giving an energy density profile. The latter is more adequate for generating unusual stars that partially satisfy the ANEC since it allows fine-tuning the stellar energy. We will focus on this method first. 



 §.§ Solutions with Density Profiles

    Generating solutions with a density profile involves different techniques than solving the TOV system with an equation of state. With an equation of state one integrates from the center outward to ensure the condition that m(0) = 0, which is necessary to avoid a Schwarzschild-like singularity at the origin. With a density profile, however, this condition is already ensured because one can define m directly as 
    
    m(r) = ∫_0^r 4 πρ(r') r'^2 r',

    where ρ(r) is a previously chosen function. Hence, it is secured that m(0) = 0. Hence, the initial condition we use instead is that P(R) = 0 for some previously chosen R. For simplicity, we take R=1. This immediately determines that the star's mass is M = m(1), which can be fine-tuned by carefully selecting ρ(r). We now solve the TOV system for the pressure, as the remaining variables ρ and m are already known. 
    
    It will also be interesting to consider the value of the quantity 
    
    F[ρ] = ∫_0^R (P+ρ) e^-ϕ(r)/√(1 - 2m(r)/r)r

    which, on account of Eq. (<ref>), is proportional to the value of the ANEC integral along a radial null geodesic. The sign of F[ρ] for a given density profile ρ decides whether the ANEC is satisfied for radial null geodesics. We know that, due to the Penrose–Sorkin–Woolgar theorem and Borde's theorem, the AANEC will fail in all of the negative-mass stellar spacetimes we are about to consider. However, F[ρ] allows us to understand this failure in more detail. 
    
    
    
    Table <ref> exhibits the masses and values of F[ρ] for different choices of profiles. All models have positive pressures in the interior of the star. Models <ref>, <ref>, and <ref> are particular cases of the ones considered in Ref. <cit.>. Models <ref>, <ref>, and <ref> to <ref> are natural generalizations of those models. All of these models have a large amount of negative energy density in the center of the star, and the absolute value of the energy density decreases from the center to the border. This is similar to a regular star, apart, of course, from the sign. Models <ref> to <ref> consistently violate the NEC in the sense that P(r) + ρ(r) < 0 for all r < R (some of them have P(R)+ρ(R) = 0). In this sense, they are completely made of negative energy.

    Models <ref> to <ref> are less usual. They start with a small amount of negative energy density at the center (often taken to be zero for simplicity) and the absolute value of the energy density grows as one gets farther away from the center. This is an uncommon situation and has the peculiar feature that the equation of state yields a non-vanishing pressure for a vanishing energy density, but we include this case for completeness because it is still easy enough to analyze. These models partially satisfy the NEC because in the deep interior of the star, the pressure is positive, but the energy density is very small. The NEC is still violated on the outer layers (as the Penrose–Sorkin–Woolgar theorem demands).

    Models <ref> to <ref> are the most interesting examples. Their density profiles are plotted in figure <ref>. These density profiles flip signs inside the star. In these two particular cases, the stars have a positive mass core, but their outer layers are made of negative mass densities. Since the pressure is everywhere positive inside the star, the core satisfies the NEC, but the outer layers do not. As one can tell from table <ref>, this can be fine-tuned in such a way that the ANEC is respected by radial geodesics (even though the Penrose–Sorkin–Woolgar theorem implies it must be violated for other geodesics). The reason is that radial null geodesics will cross a region with a substantial amount of positive energy, which suffices to balance out the negative amounts obtained in the outer shells. Nevertheless, since the outer shells have negative energy densities, a “glancing” null geodesic, , one that barely penetrates the stellar surface can cross a region of strictly negative energy and violate the ANEC. This is illustrated in figure <ref>. Incidentally, the Penrose–Sorkin–Woolgar theorem entails that at least one of these glancing geodesics must be inextendible and achronal.
    
    

    
    
    Once we consider the gravitational dynamics of particles with different mass signs as discussed in the introduction, configurations with sign flips appear to be unstable. The positive core is repelled by the outer layers but has nowhere to go. The perfect spherical symmetry of the star keeps the core in what seems to be an unstable equilibrium. Meanwhile, the outer layers fall toward the core. This argument suggests the impossibility of the reversed situation: a negative core would be attracted by the positive layers, which would be repelled by the core. It seems one would need, at the very least, negative pressures in the positive-energy region to keep the system together. Indeed, suppose the total mass M is negative and that ρ(R) > 0. Since P(R) = 0, we see that the TOV equation leads to
    
    Pr_R = - ρ(R) M/R(R-2M) > 0.

    Hence, P(r) must be negative for r < R sufficiently close to R. Negative pressures have been considered elsewhere in the study of ultra-compact objects, such as in the Mazur–Mottola gravastar <cit.>. We shall not focus on this direction in this work due to the reasons discussed at the end of section <ref>. 
    
    The above argument suggests that negative-mass stars with a sign flip should be unstable against perturbations that break spherical symmetry. It should be mentioned, however, that negative-mass stars without any sign flips also have curious equilibrium dynamics <cit.>. Since the gravitational interaction between negative masses is repulsive, the gravitational force tries to make a negative-mass star explode. Meanwhile, the pressure gradient applies an outward-pointing force on the fluid elements, which are then accelerated inward due to the negative mass sign. Hence, in a negative-mass star, the pressure pulls the star inward while gravity tries to make the star explode (rather than collapse). Thus, the roles of pressure and gravity in the stellar equilibrium dynamics are reversed.



 §.§ Solutions with Equations of State

    Next, we consider examples of stars with a negative mass arising from a (barotropic) equation of state. 

    First, we should mention it seems to be particularly difficult to use this method to generate finite stars with a sign flip, such as models <ref> and <ref> of table <ref>. The reason is as follows. Assuming the star has positive pressure, there is a point inside the star in which the NEC holds (because the pressure is always positive and the sign flip implies there is some point with positive energy density). If the NEC held everywhere, the star would have positive mass due to the Penrose–Sorkin–Woolgar theorem. Hence, the NEC must be violated somewhere. By continuity, this implies that there is a point with P+ρ = 0. Since the pressure obeys the TOV equation and we are assuming the equation of state to be barotropic, we find that
    
    ρr = - (P+ρ) 4π P r^3 + m/r[r-2m]ρP.

    We see then that, at the point with P+ρ = 0, both the derivatives of P and ρ typically vanish on account of Eqs. (<ref>) and (<ref>). Since the differential equations are of first order, this means P and ρ become constant from that point onward, which implies the star will be infinite.

    This behavior can be bypassed if the equation of state is fine-tuned so that *ρP diverges precisely when P+ρ vanishes. This makes P+ρ traverse the point where it is zero, and the TOV equation allows the pressure to continue evolving. 

    Such fine-tuning can be easily enforced when creating an equation of state, so it is not problematic as long as one is aware of it. There is, however, a second concern. Both the TOV equation and Eq. (<ref>) also have a factor of the form
    
    4 π P r^3 + m.

    Assuming the NEC holds at the center of the star and that the pressure is positive, one can conclude that the strong energy condition (SEC) holds at the center of the star: ρ(0) + 3P(0) ≥ 0 and ρ(0) + P(0) ≥ 0 <cit.>. In fact, we have ρ(0) + 3P(0) > 0, since P(0) > 0. This implies that, for sufficiently small r,
    
    4 π P r^3 + m = 4π/3∫_0^r (3P(0) + ρ(r')) r'^2 r' > 0,

    which follows from the fact that ρ(0) + 3P(0) > 0 implies ρ(r) + 3P(0) > 0 for sufficiently small r, by continuity. Hence, 4 π P r^3 + m is positive near the star's center. Meanwhile, it is negative at its boundary because there we have P(R) = 0 and m(R) < 0. Hence, it must flip sign at least once due to the intermediate value theorem. At the point in which 4 π P r^3 + m = 0, we run into a new problem as before: the evolution of P and ρ reaches an extremum. Since the derivative of m does not vanish, the solution manages to traverse this point and keep evolving. However, unless *ρP diverges to keep *ρr nonzero, ρ will “turn around”: the extremum point r_0 is reached and, if ρ decreased for r<r_0, it starts to grow for r>r_0. This means that if ρ approaches the value ρ^* in which P(ρ^*) = 0 for r<r_0 (signaling the end of the star), it departs from ρ^* for r>r_0.

    Counteracting this effect requires an even finer tuning of the equation of state to ensure that *ρP diverges exactly at the point in which 4 π P r^3 + m = 0. This difficulty is illustrated for model <ref> of table <ref> in figures <ref> and <ref>.

    

    

    With this in mind, we focus on stars with negative-semidefinite energy density. Novikov, Bisnovatyi-Kogan, and Novikov previously considered two examples of this kind <cit.>, so we consider one of their models (their other model, with a linear equation of state, does not lead to finite-radius stars) and a few generalizations. More specifically, we will consider stars with an equation of state with the form
    
    P = (-1)^Γρ^Γ,

    where Γ is a positive integer for simplicity[In astrophysics, it is more common to drop the signal in Eq. (<ref>) and consider Γ = 1 + 1/n. This would be a relativistic polytrope, such as the ones considered in section <ref>. However, since ρ is now negative, we keep Γ an integer so that the equation of state is differentiable.]. Ref. <cit.> considered this equation of state with Γ=1—which leads to an infinite star—and with Γ=2. Notice this model is essentially a relativistic polytrope with negative mass. 

    As noted in Ref. <cit.>, there is a bound on the parameters of the equation of state for the star to have finite size. If the (negative) central density ρ_0 is such that
    
    ρ_0 ≤ρ_c ≡ - 3^1/1-Γ,

    then the star is infinite. The reason is that for ρ_0 = ρ_c the star admits a solution with a constant density (and hence constant pressure). Notice that in the Γ=1 case, the bound diverges, and all central densities lead to infinite stars. 

    Figure <ref> exhibits the mass-radius diagrams and mass per central density diagrams for a few stars satisfying the equation of state (<ref>). It is easy to see how the masses (and thus the radii) diverge as ρ_0 approaches the critical central density ρ_c = -3^1/1-Γ. 

    

    It is interesting to point out that the models with the equation of state (<ref>) consistently violate the NEC.



§ STABILITY OF NEGATIVE-MASS STARS

The steps needed to perform the stability analysis of negative-mass stars are now the following:

    
  * choose a density profile or an equation of state (in which case a value for the central density will also be needed);
    
  * solve the TOV equation with this information and obtain the functions p_⋆, q_⋆, and w_⋆ of section <ref>;
    
  * if the model came from a density profile, verify that the parametric plot of (ρ(r),P(r)) is the graph of a function to ensure the model can be assumed to be barotropic;
    
  * employ the variational or the Bardeen technique to determine if the star is stable.


For some of the models we want to consider, the trial functions of Eq. (<ref>) will not be sufficient to establish instability. Hence, we define a new family of trial functions. To do so, we will first define the auxiliary function

    θ_0(x) = 
            0,    if  x ≤ 0, 
    exp-1/x,    if  x >0.

Here, θ_0 is a common example of a function that is everywhere smooth but fails to be analytic at x = 0. Using θ_0 we can construct the functions

    θ_a,b(x) = θ_0(a^2 - (x-b)^2)/θ_0(a^2).

These are standard examples of smooth bump functions. These functions are everywhere smooth but have compact support. Namely, θ_a,b(x) = 0 for any x such that x-b > a. For our purposes, we will consider bump functions with 0 < b-a < b+a < R, which ensures they are non-vanishing only in the interior of the star under consideration. Since the function and all of its derivatives vanish at the origin and the stellar surface, the boundary conditions for the pulsation equation are satisfied.



 §.§ Models with Density Profiles

Variational stability tests for the models arising from density profiles are shown in table <ref>. It turns out that, for many models considered in table <ref>, the trial function (<ref>) with n=2 was sufficient to establish instability. Recall that all of these models violate the NEC consistently inside the star: they are such that P(r) + ρ(r) < 0 for all r < R (but possibly not at r=R). Nevertheless, all of these models have *Pρ < 0, which renders p_⋆ negative. As a result, we cannot apply Bardeen's technique.



The trial function (<ref>) does not yield a good bound for the models that partially satisfy the NEC (models <ref> to <ref> in table <ref>). For them, a better result is obtained by using the bump function (<ref>). The results are shown in table <ref>. All models are unstable. Since p_⋆ changes its sign inside the star, we cannot apply Bardeen's technique.






 §.§ Schwarzschild Star

Tables <ref> and <ref> address almost all models of table <ref>, the exception being model <ref>—the Schwarzschild star. A naive calculation for model <ref> would lead to infinite *Pρ and to an ill-definition of p_⋆. This is because *ρr is zero since the density is constant. Nevertheless, one can treat (*Pρ)_s as finite by considering it corresponds to the value of the perturbations and that these perturbations do not have constant energy density. In Refs. <cit.>, Chandrasekhar does exactly this by treating γ (as defined on Eq. (<ref>)) as being constant. He then finds there is a critical value for γ which determines the onset of stability. We can establish a similar result. Define p̅_⋆ through p_⋆(r) = γp̅_⋆(r). For the Schwarzschild star, one can compute p̅_⋆, q_⋆, and w_⋆ exactly. The important properties about them is that p̅_⋆ and q_⋆ are negative throughout the interior of the star, p̅_⋆ only vanishes at the boundary, and q_⋆ never vanishes. The variational criterion establishes then that the star will be unstable if and only if,

    ∫_0^R p̅_⋆(r) ψ'(r)^2 rγ - ∫_0^R q_⋆(r)ψ(r)^2 r/∫_0^R w_⋆(r) ψ(r)^2 r < 0

for some ψ≠ 0 satisfying the boundary conditions. This means the necessary and sufficient condition for stability is 

    γ > ∫_0^R q_⋆(r)ψ(r)^2 r/∫_0^R p̅_⋆(r) ψ'(r)^2 r

for some ψ≠ 0 satisfying the boundary conditions. The expression on the right-hand side can never vanish. Hence, we can define a critical value for γ by

    γ_c = inf_ψ∫_0^R q_⋆(r)ψ(r)^2 r/∫_0^R p̅_⋆(r) ψ'(r)^2 r

where in the infimum it is understood that ψ≠ 0 and that the boundary conditions are satisfied. 

By considering the trial functions (<ref>), one can show that 

    γ_c ≤inf_n ∫_0^R q_⋆(r)ψ_n(r)^2 r/∫_0^R p̅_⋆(r) ψ_n'(r)^2 r≈0.777774

The actual infimum over n>0 can also be found analytically and occurs in the limit n → 0.

The bump functions (<ref>) generate much better estimates. Fix ϵ > 0 and denote I_ϵ = (ϵ,R-ϵ). Assume ϵ < b-a < b+a < 1 - ϵ, which means θ_a,b is compactly supported in I_ϵ. In this case, we have

    
    γ_c    ≤∫_0^R q_⋆(r)θ_a,b(r)^2 r/∫_0^R p̅_⋆(r) θ_a,b'(r)^2 r, 
       ≤max_I_ϵ q_⋆(r) ∫_0^R θ_a,b(r)^2 r/min_I_ϵp̅_⋆(r) ∫_0^R θ_a,b'(r)^2 r, 
       = c ∫_0^R θ_a,b(r)^2 r/∫_0^R θ_a,b'(r)^2 r,


where c is the finite and positive constant obtained from the ratio between the minimum and maximum. The minimum and maximum are finite and non-vanishing because we assumed only a compact region of the integral contributes. Notice that the value of b does not change the integral as long as we integrate over the whole support of θ_a,b. This leads to 

    γ_c ≤ c ∫_-a^+aθ_a,0(r)^2 r/∫_-a^+aθ_a,0'(r)^2 r≤ c I(a),

where I(a) is defined as

    I(a) = ∫_-a^+aθ_a,0(r)^2 r/∫_-a^+aθ_a,0'(r)^2 r.


These integrals can be evaluated analytically (see appendix <ref>). One finds that, for a ≪ 1, I(a) behaves as I(a) = a^4 + a^6 (Eq. (<ref>)). Hence, by taking a sequence of bump functions with smaller values of a we can bound γ_c by increasingly smaller values. In the limit when we take the infimum, we find that γ_c = 0.

Since γ_c vanishes, then the fact that the Schwarzschild star consistently violates the NEC together with Eq. (<ref>) implies that stability requires

    (Pρ)_s > 0.

We will comment on this later in this section.



 §.§ Models with an Equation of State

Stability tests for the models arising from the equation of state (<ref>) are shown in table <ref>. All tests indicated instability. For fixed Γ and n, the value of S[ψ_n] appeared to increase in absolute value as ρ_0/ρ_c increased. This suggests that more negative mass leads to less instability since σ^2 is bounded farther away from zero. Larger values of σ in absolute value are associated with faster exponential growth of the perturbations.



All of these models have negative values for p_⋆ due to the fact that (*Pρ)_s < 0. Hence, we cannot apply Bardeen's method.



 §.§ Missing Examples

When dealing with the Schwarzschild star, we found that it is possible to have stability if (*Pρ)_s > 0. If one pays close attention to the models we considered so far—both coming from profiles and from equations of state—one would notice that they all violate (*Pρ)_s > 0 at some point inside the star. Therefore, it is necessary to discuss this condition further.

Firstly, we notice that in a negative-mass star, it is expected that (*Pρ)_s < 0, which is the opposite of usual thermodynamics. This is motivated by the fact that regions with very large negative values of ρ will tend to induce stronger gravitational effects, and thus are more susceptible to “exploding” (recall that in a negative-mass star, the equilibrium dynamics are reversed). To counter this effect, large pressures are expected to be necessary to balance the forces and maintain equilibrium. One could argue, however, that this argument relies too heavily on intuition coming from positive energy physics, and therefore we will not depend solely on it.

Let us find a stronger argument then. Under the assumption of the absence of singularities and a barotropic equation of state, (*Pρ)_s > 0 implies negative energies are only possible if there are also negative pressures. The reason is as follows. First, notice this condition implies the pressure is a monotonically increasing function of the energy density. Next, notice that the TOV equation with vanishing energy density (, in the vacuum) leads to

    Pr = - 4 π P^2 r,

where we used the absence of singularities to ensure that m = 0. Since the equation of state is barotropic, we can write

    ρr = - 4 π P^2 r (ρP)_s = 0,

where the expression vanishes by the assumption that ρ = 0. Since (*ρP)_s ≠ 0, we conclude P = 0. Hence, the pressure must vanish when the energy density vanishes. Since P(ρ) is increasing, we conclude that (*Pρ)_s > 0 demands negative pressures for negative energy densities. Notice also that this means the NEC must be violated at all points with negative energy densities. In the slightly more general case with (*ρP)_s ≥ 0 we can conclude that the pressure is non-positive for negative energy densities (and thus the NEC is still violated).

Can a star with (*ρP)_s > 0 have negative mass and negative pressure? Consider a star with negative pressure throughout the interior: P(r) < 0 for all r ∈ [0,R). At the center, we have P(0) = P_0 < 0 and ρ(0) = ρ_0 < 0. Then the TOV equation tells us that, close to the center,

    Pr = - (P + ρ) 4 π P r^3 + m/r [r- 2m]≤ 0,

because P, ρ, and m are all negative there. We also have (*ρP)_s > 0, *ρr≤ 0 as well. Hence, P and ρ can only become more negative as one gets farther from the center. Hence, there is no point R with P(R) = 0 and the star is infinite. If we only assume (*Pρ)_s ≥ 0, we can adapt the argument by noticing that continuity impedes the positivity of *ρr at the points in which (*ρP)_s diverges.

This is expected. In a negative-mass star, gravity is responsible for expanding the star, while pressure is responsible for contracting it. For pressure to contract the star, the pressure gradient must point inward. Hence, if pressure is negative, the star cannot be finite because the pressure would need to grow from a negative value up to zero close to the border, and hence the pressure gradient would point in the wrong direction close to the border.

This can be made rigorous using the TOV equation. 


        Consider a spherically symmetric static star in general relativity with radius R > 0 defined by the condition that P(R) = 0, with P(r) < 0 for all R - ϵ < r < R, for some ϵ > 0. Assume that the star's pressure P is differentiable in [R-ϵ,R] and that the energy density ρ is continuous in this interval. If the star has a negative total mass, the NEC must be satisfied at a subinterval of [R-ϵ,R].
    

    Since the star has a negative total mass, we know that m(R) < 0. We also know that P(R) = 0 by definition. Notice then that
    
    4π P(R) R^3 + m(R) < 0

    and 
    
    R - 2m(R) > 0.


    Since P(r) < 0 for all R - ϵ < r < R, we know that *Pr≥ 0 at R. Hence, the TOV equation reads
    
    Pr_R = - (P(R)+ρ(R)) 4π P(R) R^3 + m(R)/R [R - 2m(R)]≥ 0.

    We see then that P(R) + ρ(R) ≥ 0.  

    If P(R) + ρ(R) > 0, the result follows from continuity. Assume next that P(R) + ρ(R) = 0. Since m(R) < 0, continuity implies there is some δ > 0 (δ < ϵ) such that m(r) < 0 for r ∈ (R-δ,R]. For r ∈ (R-δ,R) ⊊ (R-ϵ,R) we know that P(r) < 0. In particular,
    
    P(R - δ/2) < 0.

    Since P(R) = 0, the mean value theorem ensures that there is some point r^* ∈(R - δ/2,R) with P(r^*) > 0. At this point, we know that 4π P(r^*) (r^*)^3 + m(r^*) < 0. The TOV equation then enforces that P(r^*) + ρ(r^*) > 0. Continuity ensures positivity holds in an interval.


Hence, we learn that a negative mass star with negative pressure near the border must satisfy the NEC near the boundary. This contradicts our desires that (*Pρ)_s ≥ 0 with P(ρ=0) = 0. After all, there is at least one interval inside the star in which the pressure is negative, but the energy density is positive. We conclude that the scenarios with negative pressure and (*Pρ)_s ≥ 0 cannot lead to negative-mass stars. Since the star ends when the pressure changes sign, we conclude (*Pρ)_s < 0 somewhere in the star.

It should be noted that if (*Pρ)_s < 0 anywhere inside the star, then there is a region with p_⋆ < 0, as one can tell from Eq. (<ref>). This leads to two important observations.

    
  * Bardeen's technique is generally not applicable to negative mass stars, as it relies on the assumption that p_⋆ > 0 throughout the interior of the star (this is a condition for the Sturm Comparison Theorem).
    
  * One will often be able to use a small bump function with large enough derivatives to make S[θ_a,b] negative by restricting the support of θ_a,b to the region in which p_⋆ is negative.


This second observation corroborates that negative mass stars cannot be stable in general relativity. The variational criterion establishes that the star will be unstable if there is any ζ satisfying the boundary conditions (<ref>) such that

    ∫_0^R p_⋆(r) (ζ'(r))^2 r - ∫_0^R q_⋆(r) (ζ(r))^2 r < 0.

Suppose now that p_⋆ is negative and non-vanishing in some interval (c,d) ⊆ (0,R) (if p_⋆ is negative at any point, it is negative in an interval by continuity). Choose ζ to be a bump function of the form (<ref>) with support in (c,d). Then Eq. (<ref>) is equivalent to

    1 > ∫_c^d q_⋆(r) (ζ(r))^2 r/∫_c^d p_⋆(r) (ζ'(r))^2 r.

Notice that a sufficient condition for instability is that 

    1 > max q_⋆∫_c^d (ζ(r))^2 r/min p_⋆∫_c^d (ζ'(r))^2 r,

where the maximum and minimum are taken over the interval (c,d). We have now reduced the problem to something similar to the analysis of the Schwarzschild star. Namely, by taking ζ to be a sufficiently sharp bump function we can make the right-hand side arbitrarily small, which entails instability.

Hence, stars with (*Pρ)_s < 0 at any point inside the star are unstable. This does not require any assumption on the sign of the energy density [The argument fails in the particular scenario in which (*Pρ)_s < 0 only at points with P+ρ = 0, since then p_⋆ vanishes. Nevertheless, P+ρ = 0 means the pressure is constant by the TOV equation. Furthermore, since (*Pρ)_s < 0, applying the chain rule to the TOV equation also leads to the conclusion that *ρr = 0. Hence, if (*Pρ)_s < 0 at a point with P+ρ = 0, then the star will be infinite and the solution is not relevant for our purposes.]. 



 §.§ Instability Timescales

    We finally estimate the timescales associated with the hydrodynamical stabilities we considered. For this, we must recover the units.

    Let N(σ) denote the absolute value of the numerical value obtained for σ in the system of units we have been using so far, with G = c = R_0 = 1, as discussed at the end of section <ref>. Then the value of σ in an arbitrary system of units will be given by
    
    σ = N(σ) c/R_0.

    Accordingly, the timescale in which the instability happens is 
    
    T = R_0/N(σ) c.

    Notice then that the timescale is the time it takes for light to cross a length R_0 corrected by the numerical factor associated with the instability. 

    According to tables <ref>, <ref>, and <ref>, the smallest value we found for σ^2 was of order unity (Γ = 3, ρ_0 = 0.1ρ_c on table <ref>). Hence, the smallest value we found for N(σ) was at order N(σ) ∼ 1. Larger values would lead to smaller instability timescales, so this is the “most stable scenario”. 

    We thus find that the instability timescales for the models we considered are such that
    
    T ≲R_0/c.

    It would be incorrect to conclude from this expression that the instability is faster than light. One should recall that the instabilities we are considering are local in nature. They are derived from the pulsation equation, which describes how each fluid element moves around its equilibrium position. A way to picture this is to remember that Sturm–Liouville problems traditionally have an infinite and unbounded sequence of eigenvalues. Hence, even in a stable star, there are oscillation modes with arbitrarily large frequencies, which are thus associated with arbitrarily small oscillation periods. The phenomenon we are seeing here is the same. T does not measure the time it takes for the star to undergo gravitational collapse, for example, but rather the time it takes for the perturbations to grow significantly.

    For R_0 ∼7e8 (about the size of the solar radius), we get T ∼2e2, indicating that only larger structures with negative masses can live longer than a couple of minutes, posing the astrophysical question of their formation.



§ CONCLUSIONS

We have discussed the properties of negative-mass relativistic stars. In particular, we reviewed how quantum theory, grounded on Borde's and Penrose–Sorkin–Woolgar's theorems, hampers the existence of negative-mass stars. Furthermore, we found that classical general relativity abhors negative masses in the sense that all interesting models of negative mass stars considered in this work—arising both from an equation of state or an energy density profile—turned out to be unstable. A fortiori, we showed that any barotropic star with (*Pρ)_s < 0 at some point must be dynamically unstable, and also that all interesting barotropic negative-mass stars satisfy this condition. None of our stability analyses depend on the validity of energy conditions.

It is interesting to notice the implications of these results to the cosmic-weight watcher conjecture about negative masses. While general relativity cannot forbid an equilibrium solution with negative mass—after all any Lorentzian geometry is a solution to the Einstein field equations for the appropriate stress tensor—it may forbid stable solutions with negative mass. This happens in the cases considered in this work. While one can construct a negative-mass star, one cannot expect it to be stable under small hydrodynamical perturbations. 

Notice this gives a classical answer to a problem that may appear to be quantum in nature. One would typically expect that the energy conditions of quantum theory would be necessary, not only sufficient, to forbid negative-mass relativistic stars. The argument would be essentially that quantum field theory gives the input on the sorts of matter available to general relativity and thus forbids otherwise valid solutions. However, our results indicate that this is not the only mechanism used by the cosmic weight-watcher of Costa and Matsas to forbid negative masses. Stability is also an important mechanism that seems to rule out negative masses even at a classical level. 

We thank Daniel A. T. Vanzella, and George E. A. Matsas for illuminating discussions that initiated and fuelled this project. Part of the calculations in this work were carried out with the aid of  <cit.> (and in particular the  package <cit.>) in a license to the University of São Paulo (N. A. A.'s previous institution). N. A. A. was supported by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001.




§ NEC VIOLATIONS IN QFT

    For completeness, in this appendix, we give a short, but standard, argument showing that one can violate the null energy condition in quantum field theory. The argument is similar to the one given in Refs. <cit.>. Shortly, we will find a state for which the expectation value of T̂_a_bk^ak^b— where T̂_a_b is the stress tensor operator, k^a is a null vector, and the colons denote normal ordering—becomes negative. This appendix uses Planck units G = c = ħ = 1.

    We consider a minimally coupled massive scalar field in Minkowski spacetime. The stress-energy tensor for such a field is known to be 
    
    T̂_a_b = ∇_aφ̂∇_bφ̂ - 1/2η_a_b[∇_cφ̂∇^cφ̂ + m^2 φ̂^2],

    where the hats remind us that we should consider these objects as operators. We will soon need to renormalize the stress tensor, which can be implemented by the normal-ordering prescription.

    For any null vector field k^a,
    
    T̂_a_bk^ak^b = k^a∇_aφ̂k^b∇_bφ̂≡ℋ̂,

    where we defined the “null Hamiltonian density” ℋ̂ with the sole intention of simplifying the notation. As a side note, if we were in the classical theory Eq. (<ref>) would be manifestly non-negative since we would have
    
    T_a_bk^ak^b = (k^a∇_aφ)^2 ≥ 0,

    showing that a minimally coupled classical scalar field always obeys the NEC.

    Our next step is to express the normal-ordered operator ℋ̂ in terms of creation and annihilation operators. For this, we decompose the quantum field φ̂ in creation and annihilation operators by writing
    
    φ̂(x) = 1/(2π)^d/2∫[â_p e^i p · x + â_p^† e^-i p · x] [d]p/√(2 ω_p),

    where p · x = p_μx^μ,
    
    p^ap_a = - (p^0)^2 + p^2 = -m^2

    is understood, and we denote ω_p = √(p^2 + m^2). For generality, we are assuming at this stage a d+1 dimensional spacetime.

    The decomposition in Eq. (<ref>) leads to the canonical commutation relations
    
    â_pâ_q^† = δ^(d)(p-q).


    From Eq. (<ref>) we extract
    
    k^b∇_bφ̂ = i/(2π)^d/2∫[â_p e^i p · x - â_p^† e^-i p · x] k^bp_b[d]p/√(2 ω_p).


    From Eqs. (<ref>) and (<ref>) we find that 
    
    
    ℋ̂ = -1/(2π)^d∫[â_pâ_q e^i (p + q) · x - â_p^†â_q e^-i (p - q) · x - â_q^†â_p e^-i (q-p) · x + â_p^†â_q^† e^-i (p + q) · x] (k^bp_b)(k^cq_c) [d]p[d]q/2 √(ω_pω_q).

    

    The next step is to find a state |ψ⟩ which yields ℋ̂(x)ψ < 0 in some region. We take
    
    |ψ⟩ = cosβ|0⟩ + sinβ|fg⟩,

    where β is a real parameter which we will fix later, |0⟩ is the (Minkowski) vacuum, and 
    
    |fg⟩ = (∫√(2 ω_p) f(p)/k^bp_bâ^†_p[d]p∫√(2 ω_q) g(q)/k^cq_câ^†_q[d]q) |0⟩

    is a smeared two-particle state. The smearing is necessary to later keep the expectation value of ℋ̂ well-defined. The factors of ω and k^ap_a are chosen for later convenience. The normalization of this state is given by
    
    ⟨f_2 g_2||f_1 g_1⟩ = (f_2,f_1) (g_2,g_1) + (f_2,g_1) (g_2,f_1),

    where 
    
    (f, g) = ∫ f^*(p) g(p) 2 ω_p[d]p/(k^bp_b)^2.

    For simplicity, we will take f = g. We obtain ⟨ff|=⟩ 1 imposing
    
    (f,f) = 1/√(2).

    
    We recall that the Minkowski vacuum is annihilated by all annihilation operators, and this implies
    
    ℋ̂(x)0 = 0.

    We also have the normalization condition ⟨0|=⟩ 1.

    We notice then that
    
    ℋ̂ψ = sin(2β) [0ℋ̂ff] 
     + sin^2βℋ̂ff.


    The matrix elements are given by
    
    0ℋ̂ff = - 2/(2π)^d∫ f(p) f(q) e^i (p+q) · x[d]p[d]q.

    and
    
    ℋ̂ff = 4/√(2) (2π)^d∫ f^*(p)f(q) e^- i (p-q)· x[d]p[d]q.


    In total, we find that
    
        
    ℋ̂(x)ψ = - 2 sin(2 β)/(2π)^d[∫ f(p) f(q) e^i (p+q) · x[d]p[d]q] + 4 sin^2β/√(2) (2π)^d∫ f^*(p)f(q) e^- i (p-q)· x[d]p[d]q.

    Once x is fixed, one can then choose β arbitrarily small so that the first term is negative. In the small β limit, the second term becomes negligible, exhibiting local violations of the NEC. 

    Let us consider now what happens when we integrate along a null geodesic parallel to k^a, which will allow us to consider the ANEC. For simplicity, we focus on a massive field m>0 in a two-dimensional Minkowski spacetime (d = 1). A curve parallel to k^a is described in coordinates by x^μ = λk^μ, where λ is an affine parameter. 

    With this in mind, we see that Eq. (<ref>) leads to the expression 
        
    ∫ℋ̂ψλ = - 2 sin(2 β)/(2π)^d-1[∫ f(p) f(q) δ((p+q) · k) [d]p[d]q] 
     + 4 sin^2β/√(2) (2π)^d-1∫ f^*(p)f(q) δ((p-q) · k) [d]p[d]q.

    
    The first integral vanishes because the delta's argument,
    
    (p+q)· k = (p + q) k - (ω_p + ω_q) k^0,

    is strictly positive since k^0 = k and ω_p > p. Notice the main reason this quantity cannot vanish is the fact that ω_p > 0, so we cannot have p^a = -q^a because ω_p = - ω_q is impossible.

    To deal with the second term, we use the simplifying assumption that d=1. Then the ANEC integral becomes
    
    ∫ℋ̂ψλ = 4 sin^2βω_p/√(2)∫f^*(p)f(p)/ω_p - pp,

    which is manifestly positive. Therefore, we see that the ANEC holds for the state |ψ⟩ of a massive scalar field in a two-dimensional Minkowski spacetime.



§ ANEC IN A STELLAR SPACETIME

In this appendix, we rewrite the ANEC integral in a way that is easier to compute in a spacetime representing a star. To do so, we begin by noticing that Eqs. (<ref>) and (<ref>) imply that 

    ∫T_a_bγ̇^aγ̇^bλ = ∫ (P+ρ) (u_aγ̇^a)^2 λ.


Notice that

    γ̇^a(λ) = tλ(t)^a + rλ(r)^a + φλ(φ)^a,

where we are considering an arbitrary null geodesic and will impose the radial condition only later. Notice that spherical symmetry allowed us to make the simplifying assumption that θ = π/2 without any loss of generality. 

The spacetime has a timelike Killing vector field and an axial Killing vector field (among others). This allows us to identify the conserved quantities 

    ε = - (t)^aγ̇^bg_a_b = e^2ϕ(r)tλ
    andℓ = (φ)^aγ̇^bg_a_b = r^2 φλ,

which are interpreted as energy and angular momentum.

It follows then that

    γ̇^a(λ) = ε e^-2ϕ(r)(t)^a + rλ(r)^a + ℓ/r^2(φ)^a,

and hence, using Eq. (<ref>),

    u_aγ̇^a = - e^-ϕ(r)ε.


The condition γ̇^aγ̇_a = 0 implies that 

    (rλ)^2 = (1 - 2m(r)/r)(ε e^-2ϕ(r) - ℓ^2/r^2).

Notice that (assuming 2m(r) < r for all r, which holds in our cases of interest) the point of closest approach, r_min, is characterized by

    r_min^2 e^-2ϕ(r_min) = ℓ^2/ε^2,

and if this equation admits multiple solutions, then r_min is the largest solution with r_min < R.

Let us focus on the time interval in which the coordinate r grows along the geodesic. We choose λ such that r(0) = r_min, so that this corresponds to computing the integral

    ∫_0^+∞T_a_bγ̇^aγ̇^bλ = ∫_0^+∞ (P+ρ) (u_aγ̇^a)^2 λ,

which we already know will yield 

    ∫_0^+∞T_a_bγ̇^aγ̇^bλ = ε^2 ∫_0^+∞ (P+ρ) e^-2ϕλ.

We can change the integration parameter from λ to r by using Eq. (<ref>) with the additional assumption that *rλ > 0. This yields, upon simplification,

    ∫_0^+∞T_a_bγ̇^aγ̇^bλ
     = ε∫_r_min^R(P(r)+ρ(r)) e^-ϕ(r)r/√((1 - 2 m(r)/r)(1 - ℓ^2/ε^2e^2ϕ(r)/r^2)).


To get the full ANEC integral, we notice that the expression is symmetric on whether r is increasing or decreasing. Hence, we find that

    ∫_-∞^+∞T_a_bγ̇^aγ̇^bλ
     = 2 ε∫_r_min^R(P(r)+ρ(r)) e^-ϕ(r)r/√((1 - 2 m(r)/r)(1 - ℓ^2/ε^2e^2ϕ(r)/r^2)).


In the particular case with ℓ = 0, corresponding to a radial geodesic, we get r_min = 0 and 

    ∫_-∞^+∞T_a_bγ̇^aγ̇^bλ = 2 ε∫_0^R (P+ρ) e^-ϕ(r)/√(1 - 2m(r)/r)r.




§ BUMP FUNCTION INTEGRALS

While dealing with the stability of the Schwarzschild star, we faced the integrals

    ∫_-a^aθ_a,0(r)^2 rand∫_-a^aθ'_a,0(r)^2 r.

In this appendix, we evaluate them exactly. We begin by defining

    I_1(a) = ∫_-a^aθ_a,0(r)^2 r

and

    I_2(a) = ∫_-a^aθ'_a,0(r)^2 r.


From the definition of θ_a,0 (Eq. <ref>), it is straightforward to show that

    I_1(a) = e^2/a^2∫_-a^a e^2/r^2-a^2r

and

    I_2(a) = 4 e^2/a^2∫_-a^ar^2 e^2/r^2-a^2/(r^2-a^2)^4r.


Using the fact that the integrands are even and performing the substitution r = ax we get that

    I_1(a) = 2 a e^2/a^2∫_0^1 e^2/a^2(x^2-1)x

and

    I_2(a) = 8 a^-5 e^2/a^2∫_0^1x^2 e^2/a^2(x^2-1)/(x^2-1)^4x.


Next, we make the substitutions 

    u = - 1/x^2-1,

which involve

    x = u/2 u √(u^2 - u).

This leads to 

    I_1(a) = a e^2/a^2∫_1^+∞e^-2u/a^2/u √(u^2 - u)u

and

    I_2(a) = 4 a^-5 e^2/a^2∫_1^+∞ e^-2u/a^2 u√(u^2 - u)u.


The problem of finding both I_1(a) and I_2(a) has now been reduced to computing integrals of the form

    J(z;α,β) ≡∫_1^+∞ e^- z u u^-α (u-1)^β-1u.

For I_1 we have z = 2/a^2, α = 3/2, and β = 1/2. For I_2, z = 2/a^2, α=-3/2, and β = 3/2. The trick to solve this integrals is the same one often used in symbolic integration: we express the integrals in terms of Meijer G-functions <cit.> and then apply an integration theorem. 

We thus begin by noticing that <cit.>

    e^-z u = G^1,0_0,1(z u| 0pt0),

which leads us to 

    J(z;α,β) = ∫_1^+∞ G^1,0_0,1(z u| 0pt0) u^β (u-1)^γu.

The result of this integral is known and given by <cit.>

    J(z;α,β) = Γ(β) G^2,0_1,2(z | 0ptαα-β,0),

where Γ is the gamma function. Therefore, we conclude that

    ∫_1^+∞e^-2u/a^2/u √(u^2 - u)u = √(π) G^2,0_1,2(2/a^2| 0pt3/20,1)

and

    ∫_1^+∞ e^-2u/a^2 u √(u^2 - u)u = √(π)/2 G^2,0_1,2(2/a^2| 0pt-3/2-3,0).


Bringing everything together, we find that

    I_1(a) = a √(π) e^2/a^2 G^2,0_1,2(2/a^2| 0pt3/20,1)

and

    I_2(a) = 2 √(π) a^-5 e^2/a^2 G^2,0_1,2(2/a^2| 0pt-3/2-3,0).


Ultimately, the quantity we are interested in is the function

    
    I(a)    = I_1(a)/I_2(a), 
       = a^6/2G^2,0_1,2(2/a^2| 0pt3/20,1)/G^2,0_1,2(2/a^2| 0pt-3/2-3,0).


More specifically, we are interested in the behavior of I(a) for small a. To that end, it is useful to use an asymptotic expansion for the Meijer G-functions. For large z ∈ℝ, z>0, it holds that <cit.>

    G^2,0_1,2(z| 0ptαα-β,0) ∼ e^-z z^-β(1 + ∑_k=1^+∞ M_k z^-k),

where M_k are constants determined by the specific values of α and β. We thus find that 

    G^2,0_1,2(z| 0pt3/20,1)/G^2,0_1,2(z| 0pt-3/2-3,0) = z + 1.


Using this asymptotic expression we conclude that, in the limit with a ≪ 1, it holds that

    I(a) = a^4 + a^6.

Hence, I(a) tends to zero as a → 0.






 Paper Title: Energy-time uncertainty relation from entropy measures.




An inhomogeneous p-laplacian equation with a Hardy potential
    Genival da Silva[email: gdasilva@tamusa.edu, website: <www.gdasilvajr.com>]
    August 5, 2024
===============================================================================





§ INTRODUCTION



Position (q) and momentum (p) are two complementary quantities of a quantum particle, satisfying an uncertainty relation first derived by H. Kennard <cit.> in 1927, and H. Weyl <cit.> in 1929, mathematically represented by

    Δ q Δ p ≥ħ/2,

where Δ q and Δ p represent the standard deviation of the position and momentum, respectively, and ħ is the reduced Planck constant. This equation expresses Heisenberg's Uncertainty Principle (HUP), which states that it is not possible to simultaneously determine the position and momentum of a quantum particle. The same expression is obtained by considering position and momentum as operators that satisfy the commutation relation

    [q̂, p̂] = iħ.

Substituting this into Robertson's uncertainty relation (1929) <cit.>,

    Δ A Δ B ≥1/2| ⟨ψ | [Â, B̂] | ψ⟩|,

where |ψ⟩ is the physical state of the system, we obtain (<ref>).

Heisenberg also postulated the energy-time uncertainty relation, expressed mathematically as

    Δ t Δ E ≥ħ/2,

where Δ E represents the energy dispersion and Δ t is defined as a specific time interval. The inequality (<ref>) cannot be interpreted in the same way as the well-known Heisenberg relation between position and momentum (<ref>). This distinction arises because in quantum mechanics, time is regarded as an evolution parameter rather than as an observable. Hence, it cannot be represented by a self-adjoint operator (conjugated to the Hamiltonian) as argued by Pauli due to the bounded nature of the continuous energy spectrum <cit.>.

The time-energy uncertainty relation was initially deduced in the work by Mandelstam and Tamm <cit.>, and subsequently discussed by various authors <cit.>.

There exist several methods to derive the energy-time uncertainty relation, (<ref>). Particularly noteworthy is the approach presented in <cit.>, which provides a derivation from first principles involving a canonical transformation in classical mechanics. This transformation yields a new canonical momentum (the energy E) and a new canonical coordinate T (referred to as tempus), which is conjugate to the energy. It is crucial to emphasize that tempus, as a new canonical coordinate, is conceptually distinct from the time t that denotes the system's evolution. The canonical invariance of the Poisson bracket ensures that energy and tempus follow the same algebraic relationships as momentum and position. Upon quantization of the system, the uncertainty relation (<ref>) emerges.



To introduce the quantum effects of gravity, the uncertainty relation for position and momentum, (<ref>), is extended to a Generalized Uncertainty Principle (GUP), which ensures a nonzero minimal uncertainty in position. In <cit.>, it is demonstrated that the GUP arises from the consideration of non-extensive entropies, denoted as S_±. The deformation parameters, which determine the magnitude of minimal uncertainty effects, are associated with S_+ and S_- entropies, and are negative and positive, respectively. This enables exploration of various possibilities in the quest for physical implications. Therefore, non-extensive statistics can be interpreted as a signature of quantum gravity.

In recent years some works have explored the relation between quantum gravity and non-extensivity as well from different perspectives<cit.>.

GUP is also derived from different proposals: in <cit.>, the scattering of strings at ultra-high energies is considered to analyze the divergences of quantum gravity at the Planck scale; in <cit.>, a gedanken experiment is proposed to measure the area of the apparent horizon of black holes in the context of quantum gravity; and <cit.> explores the idea that spacetime in the Planck region fluctuates, leading to the possibility of virtual micro-black holes affecting the measurement process.



The non-extensive entropies S_±, first introduced in <cit.>, were derived within the framework of Superstatistics <cit.>. These entropies depend only on the probabilities associated with the microscopic configuration of the system and enable exploration of non-equilibrium phenomena. These generalized entropies represent the entire family of entropies dependent solely on probability <cit.>. The statistics associated with S_+ and S_- exhibit distinct characteristics: the former gives rise to an effective potential related to an effective repulsive contribution term, while the latter contributes effectively attractively <cit.>.

In this work, we expand upon the results obtained in <cit.>. As observed, the generalized entropies ensuring quantum gravity effects lead to modifications in the uncertainty relation and, consequently, in the algebra between position and momentum. Similarly, these entropies are expected to modify the uncertainty relation between time and energy, (<ref>). 


The structure of the paper is as follows: In section <ref>, we present the derivation of the uncertainty relation between energy and time by introducing a canonical variable called tempus, which is conjugate to energy. In section <ref>, we discuss the origin of non-extensive entropies S_± in the domain of Superstatistics and how GUP emerges from them. In section <ref>, we derive the generalized energy-time uncertainty relation by examining the modified commutation relation between the tempus operator and the relativistic effective energy. Conclusions are given in section <ref>.




§ DERIVATION OF THE ENERGY-TIME UNCERTAINTY RELATION


A rigorous derivation of the uncertainty relation between time and energy was first provided by <cit.>. In this work, we will follow a different approach, as presented by <cit.>, where a new canonical variable, called tempus T, conjugate to energy E is defined through a canonical transformation. This approach ensures that they satisfy the classical Poisson algebra

    {T,E}=1,

similar to the position q and momentum p

    {q,p}=1.

According to <cit.>, tempus is related to time t in a conservative system as follows

    T=t-t_0,

here, t_0 is an arbitrary constant. Without loss of generality, t_0=0 can be set, making tempus equivalent to the time 
t in the classical regime.

































































































 
 
As we see from (<ref>) the tempus, and the energy are canonical conjugates. Therefore, when the system is quantized, the operators T̂ and Ê satisfy the commutation relation

    [T̂,Ê]=iħ,

and considering the Robertson relation (<ref>),



one obtains the uncertainty relation

    Δ TΔ E≥ħ/2.

The relationship between the standard deviation of the tempus operator, represented by Δ T, and the time interval Δ t is given by

    Δ T=|d⟨ T⟩/dt|Δ t,

and replacing in the tempus-energy uncertainty relation (<ref>), is obtained

    Δ tΔ E≥ħ/2/|d⟨ T⟩/dt|.

Now, with the help of Ehrenfest's Theorem, we can find the time derivative of the expected value of the operator T̂, considering that the operator itself is not explicitly time-dependent, i.e., ∂T̂/∂ t=0. Then,

    d⟨ T⟩/dt=1/iħ⟨[T̂,Ĥ]⟩,

and for a system with a time-independent Hamiltonian (conservative system) Ĥ=Ê, thus, from (<ref>) one arrives at energy-time uncertainty relation (<ref>). Reference <cit.> shows that the expectation value of the tempus operator is directly related to the time t of the system's evolution, which directly leads, from (<ref>), to (<ref>).






Another alternative way to obtain (<ref>) is from the uncertainty relationship for position and momentum (<ref>), considering Δ E=v_GΔ p and Δ q=v_GΔ t, where v_G is the group velocity of the wave packet. Considering these factors, an alternative derivation of the uncertainty relation between energy and time is obtained.




§ MODIFIED ENTROPIES AND THE GENERALIZED UNCERTAINTY PRINCIPLE


In the Heisenberg uncertainty relation, the gravitational interaction between particles was completely neglected, although this was somehow justified by the considerable weakness of gravity when compared with other fundamental interactions.

Incorporating gravity into quantum measurement processes led to the generalization of the uncertainty relation (<ref>). This modification is known as the Generalized Uncertainty Principle (GUP), which introduces additional terms that account for gravitational effects at very small scales, typically at the Planck scale.

GUP can alternatively be derived from modified entropies <cit.>, in particular of the ones that depend only on the probability <cit.>. These entropies are obtained in the Superstatistics formalism <cit.>, mathematically expressed as

    S_+   =   k_β∑_l^Ω(1-p_l^p_l),
    
        S_-   =   k_β∑_l^Ω(p_l^-p_l-1),

and in terms of the Boltzmann-Gibbs (BG) (Shannon) entropy S_BG=k_βlnΩ, for p_l=1/Ω, can be written as (considering the expansion)

    S_±=S_BG∓1/2!S_BG^2 e^-S_BG+⋯,

The associated probabilities and, consequently, the exponential and logarithms are also generalizations of the usual ones; the exponentials result in
   
    exp_±(-x)=exp(-x)∑_j=0^∞a^±_jx^j,

   where the dimensionless coefficients a^±_j are fixed <cit.>. One can then define in space of phase a generalized probability
   
    𝒫_±=exp(-H̅)∑_j=0^∞a̅^±_jH̅^j,

being H̅ the dimensionless Hamiltonian, and by these means generalized Hamiltonians <cit.>

    H_±=1/2m(p^2+α_±/3p^4+⋯),









in terms of low-energy momentum p. Then, it is possible to define a generalized momenta p_± in terms of p, as follows

    p_±=p+α_±/3 p^3+⋯,

where, α_±=α^0_±/M_x^2 (c=1 is assumed) is the deformed parameter, M_x is a certain energy scale, and α^0_± is a dimensionless constant

    α_±^0=3 ((a^±_1)^2-2a^±_2)/8 (1-a^±_1).

The commutation relation between p̂_± and q̂ can be calculated using equation (<ref>), and the usual commutation relation [q̂,p̂]=iħ

    [q̂,p̂_±]=iħ(1+α_±p̂^2+⋯),

taking into account that α_±p̂_±^2=α_±p̂^2+⋯, it allows us to express (<ref>) as

    [q̂,p̂_±]=iħ(1+α_± p̂_±^2).

A compact way to express the modified momentum p_± in (<ref>), which satisfies the commutation relation (<ref>), is

    p_±=1/√(α_±)tan(√(α_±)p),

where the domain of p is restricted to -π/2√(α_±)<p<π/2√(α_±). It is essential to mention that the coordinate q and the effective momentum p_± are no longer canonical conjugates <cit.>.

The generalized uncertainty relation associated with the modified algebra in (<ref>) is

    Δ qΔ p_±≥ħ/2(1+α_±(Δ p_±)^2),

which is a modification of the uncertainty relation in (<ref>), taking into account quantum corrections of gravity. The parameter α_±^0 can have positive and negative values, depending on the considered statistics, i.e.,  α^0_+=-0.560565 and α^0_-=0.361022 <cit.>.  When the positive value is set, (<ref>) introduce a minimal length in the position, (Δ q)_min=ħ√(α_-). If we choose the Planck scale M_x=M_Pl the parameter α^0_- is in order of unity and the minimal uncertainty in the position becomes the Planck length l_Pl.  On the other hand, when the negative value is chosen, a maximum uncertainty in the momentum is obtained, (Δ p)_max∼ 1/√(|α_+|).




§ GENERALIZED ENERGY-TIME UNCERTAINTY RELATION


It has been demonstrated that starting from non-extensive entropies, it is possible to obtain a generalization to the uncertainty relation (<ref>) to obtain (<ref>). Using the same arguments, we can find a modified uncertainty relation between time and energy, which should be a generalization of (<ref>) to include quantum gravity corrections.



Before finding the uncertainty relation between time and energy, we must express the relativistic version of the effective Hamiltonian H_± in (<ref>). In the relativistic regime, energy E and momentum p are written as the components of a four-momentum vector p_0=(E,p). Similarly, we should be able to write a modified four-momentum with components P_±=(E_±,p_±), where p_± is defined in (<ref>). On the other hand, E_± must be the relativistic energy for the Hamiltonian H_± in (<ref>). We define this as

    E_±=E+α_±/3 E^3+⋯.

It can be easily verified that this expression reduces to the Hamiltonian (<ref>) in the non-relativistic limit. With this purpose, we rewrite (<ref>) as follows

    E_±=E(1+α_±/3 E^2+⋯),

and considering the relativistic energy relation express as E= m√(1+p^2/m^2) in non-relativistic limit, we have

    E_±=m(1+p^2/2m^2)[1+α_±/3p^2+α_±/3m^2+⋯],

which reduces to

    E_±=m(1+α_±/3m^2)+(1+α_± m^2)p^2/2m+α_±/3p^4/2m+⋯.

As we have seen already, the deformation parameters α_± are expressed in terms of the Planck energy M_Pl through the relation α_±=α^0_±/M_Pl^2. This allows us to rewrite (<ref>) as

    E_±=m(1+α^0_±/3(m/M_Pl)^2)+(1+α^0_±(m/M_Pl)^2)p^2/2m+α_±/3p^4/2m+⋯,

where α^0_± is a dimensionless parameter defined in the previous section. On the other hand, m represents the energy at low scales, implying that M_Pl≫ m, and therefore m/M_Pl≪ 1.  Notice that the conditions m ≪ M_pl have to be stronger than the condition p ≪ m, i.e. m/M_Pl≪ p/m. This means that the mass of the particle has to be below the Planck scale, and therefore it can not be an excited state of string theory (for example). Considering this fact, we can write the effective energy (<ref>) in a familiar form

    E_±≃ m+1/2m(p^2+α_±/3p^4+⋯).

The first term represents the rest energy of the particle, and the second one represents the effective Hamiltonian (<ref>). With this, we demonstrate that (<ref>) is the relativistic expression of (<ref>). The same expression can be obtained from the effective momenta (<ref>) when a relatively high energy is applied to a particle with relatively low rest mass, it can be approximated as E≃ p in high-energy physics.

So, from (<ref>) we have

    E_±=p+α_±/3 p^3+⋯.

This expression is interpreted as a modified dispersion relation (MDR) derived from entropic principles. Building on other proposals in quantum gravity, similar MDRs to (<ref>) have been derived; for example, we cite <cit.>. In the non-relativistic approximation, the MDR in (<ref>) reduces, using (<ref>), to

    ω_±(k)=m/ħ+ħ k^2/2m(1+α_±ħ^2k^2/2m)+⋯,

where the de Broglie relations for energy and momentum for matter waves E = ħω and p = ħ k, respectively, have been utilized. Clearly, this expression reduces to the usual dispersion relation ω(k) ≈m/ħ + ħ k^2/2m when α_±→ 0. 

In <cit.>, it is argued that the emergence of MDR can be attributed to a fluctuating background dominated by quantum effects. This is because the dispersion relation of a particle propagating through a fluctuating spacetime differs from that of a particle moving in a smooth background. Furthermore, we can argue that the dispersion relation is modified due to non-extensivity, which is a signature of quantum gravity as it was discussed in <cit.>.









Starting from (<ref>), similar to what was done in (<ref>), we can calculate the commutation relation between the operator T̂ and the effective energy Ê_±, obtained from entropic principles. This yields

    [T̂,Ê_±]=iħ(1+α_±Ê^2+⋯),

which is a generalization of (<ref>), taking into account the quantum gravity effects associated with the non-extensivity of the entropies considered in (<ref>) and (<ref>). Unlike the standard case, tempus and energy aren't longer canonical conjugates. Replacing the deformed Heisenberg algebra (<ref>) in the general uncertainty relation (<ref>), we obtain

    Δ TΔ E_±≥ħ/2(1+α_± (Δ E)^2+⋯).

Similarly to the usual case, we can find the relationship between Δ T and Δ t by considering the fact that the operator T̂ has not been modified and satisfies the usual algebra with Ê. Therefore, for a conservative system, from (<ref>) we have that Δ T = Δ t. So, (<ref>) become in

    Δ tΔ E_±≥ħ/2(1+α_± (Δ E)^2+⋯).

This is the modified uncertainty relation for time and energy, derived from non-extensive entropies. Equation (<ref>) is a generalization of (<ref>) that incorporates high-energy corrections encoded in the non-extensive deformation parameter α_±. Additionally, if we consider the de Broglie relation for matter waves, then in (<ref>) we obtain the modified time-frequency uncertainty relation

    Δ tΔω_±≥1/2(1+α_±ħ^2 (Δω)^2+⋯),

for which we have considered Δ E_±=ħΔω_±. For example, in <cit.>, a Gaussian signal is considered, resulting in Δ t=1/√(2) and Δω=√((π-2)/2π), and therefore the usual uncertainty relation gives Δ tΔω=0.301. In this case, from (<ref>), we have Δ tΔω_+=0.301-2.38× 10^-54 and Δ tΔω_-=0.301+1.53× 10^-54. Clearly, in both cases, it can be seen that the first correction term is very small compared to 0.301. This leads us to interpret that the corrections introduced by the non-extensive entropies S_+ and S_- only play an important role at Planck scales (frequencies), which corroborates the results obtained in <cit.>.

Considering the usually relation Δ t Δ E ≥ħ/2, it can be concluded from (<ref>) that

    Δ E_±=Δ E(1+α_± (Δ E)^2+⋯).

This result can be interpreted in two ways. If the positive deformation parameter α_- is considered, the energy fluctuation increases in the region where the contribution of gravity is significant, that is, at Planck scales, which implies the appearance of an unobservable region Δ x that cannot be probed <cit.>. On the other hand, if the negative parameter α_+ is considered, the energy fluctuation decreases. Furthermore, as α_+ (Δ E)^2<1 must be satisfied in (<ref>), the energy fluctuation reaches its maximum in this scenario. This behavior has been observed in models that consider a crystal-like universe, whose lattice spacing is of the order of the Planck length <cit.>.

Now, we intend to calculate the quantum-corrected radiation temperature of a black hole. From (<ref>), we have

    Δ E_±ħ/2Δ t+α_±ħ/2Δ t(Δ E)^2+⋯.

In <cit.>, a heuristic derivation of the radiation temperature of a black hole is presented by considering particles as real gas at the horizon. Since these particles are moving in a space slice of thickness Δ x, each of them has an uncertainty in the kinetic energy equal to Δ E=3k_βΘ/2, where Θ is the temperature of this gas of real particles. This provides an estimate of the temperature. We will follow a similar procedure to calculate the contributions of quantum gravity effects to Hawking radiation. For a black hole, we have Δ t=2 R^2/GM and Δ E=ħ/16GM, where R=2GM is the Schwarzschild radius, G is the gravitational constant, and M is the mass of the black hole. Thus, in (<ref>), we have

    Δ E_±ħ/16 GM+α_±(ħ/16 GM)^3,

and the corresponding modified Hawking radiation temperature associated with the different statistics imposed by the entropies S_+ and S_- are

    Θ_+=   ħ/24k_β GM[1-9|α_+|k_β^2/4(ħ/24 k_β G M)^2],
    Θ_-=   ħ/24k_β GM[1+9α_-k_β^2/4(ħ/24 k_β G M)^2],

where k_β is the Boltzmann constant. The first term in (<ref>) and (<ref>) is the usual Hawking radiation temperature Θ=ħ/24k_β GM <cit.>, and the second term contains quantum gravity effects. In the classical limit, where the mass M of the black hole is sufficiently large, the correction terms in (<ref>) and (<ref>) are too small to be considered compared to the first term. For example, suppose M=M_sun=2× 10^30kg; the contribution of quantum gravity effects is on the order of 10^-96 compared to 1 for the first term. On the other hand, if we consider a micro black hole with a mass on the order of the Planck mass M∼ M_Pl, quantum gravity effects will be significant. This coincides with the observation in <cit.> that the modifications are relevant in the quantum gravity realm.  


From (<ref>), it can be seen that the radiation temperature for micro black holes, denoted as Θ_+, is lower than the Hawking radiation temperature Θ. This result implies that quantum gravity effects related to the non-extensivity associated with entropy S_+ decrease the black hole's radiation temperature. Conversely, quantum effects due to S_- increase the temperature of the quantum micro black hole, as shown in (<ref>), compared to the usual Hawking temperature. Several authors by other means have also obtained corrected black hole temperatures in different contexts <cit.>.



We can extend the commutation relation (<ref>) to a form analogous to (<ref>) considering all corrections terms in α_±, or equivalently, by setting α_±Ê_±^2=α_±Ê^2+⋯. Thus, we have

    [T̂,Ê_±]=iħ(1+α_±Ê_±^2),

and the corresponding uncertainty relation

    Δ TΔ E_±≥ħ/2(1+α_±(Δ E_±)^2).

Using the same argument as in (<ref>), we can express the modified uncertainty relation between time and energy from (<ref>), which is as follows

    Δ tΔ E_±≥ħ/2(1+α_±(Δ E_±)^2),

that is the generalized version of (<ref>), the energy-time Generalized Uncertainty Relation. As mentioned earlier, the deformation parameter α_± has two possible values, one positive, α_-, and the other negative, α_+, depending on the statistics considered. Choosing the positive parameter in (<ref>), associated with the entropy S_-, leads to a minimum time interval

    (Δ t)_min=t_min=ħ√(α_-).

On the contrary, if one chooses the negative value of the deformation parameter denoted by α_+, associated with the entropy S_+,  a maximum energy width is found

    (Δ E_+)_max=E_max=1/√(|α_+|).

Taking into account that in the Planck regimen α_±=α_±^0/M_Pl^2, then the minimum time in (<ref>) is on the order of the Planck time t_Pl, t_min=t_Pl√(α_-^0). In (<ref>), this implies that the maximum energy is on the order of the Planck energy M_Pl, i.e., E_max=M_Pl/√(|α_+^0|). Notice first that the extreme magnitudes are of the order of the Planck scale. Also, is important to see that from the different statistics we obtain minimum time or maximum energy. This is given by the signs of the α_±, meaning that one class of entropies corresponds to one case and the other entropy to the other one.


 Non-extensive statistics characterized by entropies S_± lead to distinct physical behaviors. For example, in (<ref>) and (<ref>), S_- leads to a minimum time and an increase in the radiation temperature of a mini black hole, while in (<ref>) and (<ref>), S_+ leads to maximum energy and a decrease in the Hawking temperature. This fact was already noted in <cit.> because the statistics associated with S_+ and S_- have different characteristics. The first entropy gives rise to an effective potential related to an effective repulsive contribution term, while the second one gives rise to an effective attractive contribution <cit.>.

 In <cit.>, the decay laws are discussed based on their relationship with the energy spectrum of the system <cit.>. Due to (<ref>), the representation of the energy eigenstates must change, which would result in a modified non-decay amplitude χ(t) <cit.>, and consequently, the decay law Q(t). In future work, we will analyze these and other related issues more carefully.




§ CONCLUSIONS


With the goal of introducing the quantum effects of gravity into measurement processes, in an effective manner, the uncertainty relation between position and momentum is modified (<ref>). This modification is named the Generalized Uncertainty Principle (GUP). Various approaches have demonstrated the emergence of GUP <cit.>. Notably, an intriguing connection has been established between non-extensive entropies and GUP, as S_± in (<ref>)-(<ref>), depending only on the probability, <cit.> (see (<ref>)). This leads us to the conclusion that non-extensivity can be interpreted as a consequence of quantum gravity. The statistics associated with S_+ and S_- yields distinct interpretations. For instance, those linked to S_+ result in a maximum uncertainty for the momentum, while those associated with S_- lead to a minimum uncertainty for the position. Furthermore, S_+ gives rise to an effective potential correlated with a repulsive interaction, whereas S_- gives rise to an effective attractive interaction <cit.>.

Using the probability distributions derived from the generalized entropies S_±, (<ref>), we obtain two effective Hamiltonians, labeled in (<ref>), which include correction terms at the Planck scale. At high-energy and high-velocity regimes, the energy and the momentum exhibit similar characteristics; therefore we express the effective Hamiltonian in its relativistic form (<ref>), in which the concepts of relativity and quantum theory are combined. As a result, from the generalized relativistic energy, the Modified Dispersion Relation (MDR) (<ref>) is derived. Unlike other proposals, such as <cit.>, this relation is obtained starting from entropic principles. This leads us to conclude that the dispersion relation is modified as a consequence of the non-extensivity, with the latter being a signature of quantum gravity <cit.>.

We employ the definition of the tempus operator <cit.>, computing the commutator with the relativistic effective energy. This yields a modified commutation relations (<ref>), leading to the generalized energy-time uncertainty relation (<ref>). Consistently with previous findings, the corrections arising from S_+ and S_- exhibit differences. The deformation parameter of the commutator α_+ derived from S_+ results in a decrease in energy fluctuation and the radiation temperature of a micro black hole, see (<ref>) and (<ref>), as well as in maximum uncertainty in energy (<ref>). Conversely, the deformation parameter α_- derived from S_- leads to an increase in the energy fluctuation and radiation temperature, as indicated in (<ref>) and (<ref>), as well as to the emergence of a minimum time interval (<ref>). In the Planck regime, the minimum time interval is on the order of the Planck time, and the maximum uncertainty in energy is of the order of the Planck energy.







The outcomes presented in this work strengthen the proposition that non-extensivity in entropy is a manifestation of quantum gravity <cit.>. The quantum corrections originating from the generalized entropies S_± prove to be substantial on both Planck scales (GUP). Grasping these findings unveils considerable potential for applying non-extensive entropies to understand phenomena entailing quantum gravity, a subject that will be explored further in forthcoming research.



N. C. B and W. Y thank the support of the University of Guanajuato grants:  CIIC 251/2024 ”Teorías efectivas de cuerdas y exploraciones de aprendizaje de máquina”,  CIIC 264/2022
"Geometría de dimensiones extras en teoría de cuerdas y sus aplicaciones físicas". NCB would like to thank the Grant CONAHCyT A-1-S-37752, “Teorías efectivas de cuerdas y sus aplicaciones
a la física de partículas y cosmología” and  University of Guanajuato Grant CIIC
224/2023 "Conjeturas de gravedad cuántica y el paisaje de la teoría de cuerdas". NCB would like to acknowledge 
support from the ICTP through the Associates Programme (2023-2029), and the Isaac Newton Institute for Mathematical Sciences, Cambridge, for support and hospitality during the programme "Black holes: bridges between number theory and holographic quantum information" where work on this paper was undertaken. This work was supported by EPSRC grant no EP/R014604/1.  O. O and W. Y thanks to the grant by the University of Guanajuato CIIC 168/2023 "Non-extensive Entropies Independent of Parameters" and CIIC 156/2024 "Generalized Uncertainty Principle, Non-extensive Entropies, and General Relativity", as well as the CONAHCyT Grant CBF2023-2024-2923 "Implications of the Generalized Uncertainty Principle (GUP) in Quantum Cosmology, Gravitation, and its Connection with Non-extensive Entropies.
















JHEP




 Paper Title: Finite distance effects on the Hellings-Downs curve in modified gravity.





    guillem.domenech@itp.uni-hannover.de

    apostolos.tsabodimos@stud.uni-hannover.de



^aInstitute for Theoretical Physics, Leibniz University Hannover, Appelstraße 2, 30167 Hannover, Germany.
^b Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut, 30167 Hannover, Germany


§ ABSTRACT 
There is growing interest in the overlap reduction function in pulsar timing array observations as a probe of modified gravity. However, current approximations to the Hellings-Downs curve for subluminal gravitational wave propagation, say v<1, diverge at small angular pulsar separation. In this paper, we find that the ORF for the v<1 case is sensitive to finite distance effects. First, we show that finite distance effects introduce an effective cut-off in the spherical harmonics decomposition at ℓ∼√(1-v^2)  kL, where ℓ is the multipole number, k the wavenumber of the gravitational wave and L the distance to the pulsars. Then, we find that the overlap reduction function in the small angle limit approaches a value given by π kL v^2 (1-v^2)^2 times a normalization factor, exactly matching the value for the autocorrelation recently derived. Although we focus on the v<1 case, our formulation is valid for any value of v.


Finite distance effects on the Hellings-Downs curve in modified gravity
    Apostolos Tsabodimos^a
    August 5, 2024
=======================================================================






§ INTRODUCTION 


The unique nature of pulsars, characterized by their remarkably stable rotational periods, allows for their utilization in a wide range of applications. These applications extend from the creation of Galactic maps to the exploration of Gravitational Waves (GWs) in the frequency range of 1-100 nHz. Of particular interest is the latter, an exciting prospect pioneered by Sazhin <cit.>, Detweiler <cit.>, Hellings and Downs <cit.>, and its subsequent analysis. Since then, substantial progress has been made, encompassing both phenomenological studies <cit.> and efforts to constrain theoretical models and test the predictions of General Relativity <cit.>.

With the rapid evolution of technology and expertise in recent years <cit.>, the prospect of reliably measuring a stochastic GW background in the aforementioned frequencies has become increasingly feasible. This advancement is driven by techniques in the field of Pulsar Timing Array (PTA) <cit.>. In fact, PTA collaborations recently reported 3-4σ evidence of a GW background <cit.>. A key aspect of PTA measurements is the identification of the Overlap Reduction Function (ORF) <cit.>, commonly denoted as Γ_ab(ξ_ab), where ξ_ab is the angle between pulsars “a” and “b”. In General Relativity (GR), the ORF is commonly known as the Hellings-Downs curve <cit.>.

Interestingly, it has been shown that the ORF can also be used to test the phase velocity of the GWs <cit.>. A phase velocity of the GWs with v≠ 1 is a signature of modifications of gravity. For example, it is possible to have v<1 or v>1 in scalar-tensor theories of gravity <cit.> (see Refs. <cit.> for recent reviews and Refs. <cit.> for an Effective Field Theory (EFT) perspective). In terms of metric transformations, v≠ 1 could also be motivated by disformal couplings to a scalar field <cit.>. In massive gravity one finds that the phase velocity is frequency dependent <cit.>. One can also use general EFTs to study modifications to the propagation speed and polarization of GWs <cit.>. It is important to note that although the LIGO/VIRGO/KAGRA collaboration placed stringent constraints on the propagation speed of GWs <cit.> at frequencies of around 100  Hz, it does not exclude departures from v=1 at lower frequencies, see, e.g., the arguments provided by Ref. <cit.>. Thus, PTAs provide a complementary probe to modified gravity. In general, one finds a different shape and frequency dependence of the ORF <cit.>, namely Γ_ab(kL,ξ_ab), where k is the wavenumber of the GW and L the distance to the pulsars. 

However, current analytical approximations to ORF in the v<1 case diverge at the small pulsar separation <cit.>. Recently, Ref. <cit.> found an exact formula for the value of the autocorrelation, i.e., the correlation of timing residuals of a single pulsar, say Γ_aa, for general values of v. In particular, they provide the autocorrelation value for the v<1 case in the large kL limit. However, there is the open question of how the ORF for v<1 approaches the autocorrelation value and why and when the analytical approximation breaks down.


The goal of this paper is to investigate the finite distance effects on the ORF with a special focus on the v<1 case. To do so, we work in the spherical harmonic decomposition and express the Legendre functions of the harmonic coefficients in terms of spherical Bessel functions. As the spherical Bessel functions extend over the whole real line, they have, in our opinion, a more manageable dependence on the multipole number ℓ. With our alternative formulation, we discover that the approximations derived in Refs. <cit.> are valid up to a cut-off at ℓ∼√(1-v^2) kL. Naively, a cut-off for ℓ≳ kL is to be expected as the angular separation corresponding to the multipole number (roughly ξ∼ 1/ℓ) is smaller than the typical features in the antenna response pattern (which is of the order of 1/kL). Thus, multipoles higher than the cut-off do not contribute to the description of the antenna response. in Fig. <ref>, we show the antenna pattern for v={0.8,1,1.2}.[We follow the procedure described in Ref. <cit.> with the antenna response modified to

    R^A(k,Ω̂)=(1-e^-ikL(v+Ω̂·p̂))v/2e^A_ij(Ω̂)p_ip_j/v+Ω̂·p̂ .

The first term inside the parenthesis is the so-called redshift transfer function and the rest is called geometrical projection factors. The subscript A refers to plus + or cross × polarization.
 We thank B. Allen for the suggestion to draw the antenna pattern.] At the end of the paper, we show that such cut-off reconciles the ORF of Refs. <cit.> with the autocorrelation value of Ref. <cit.>.





This paper is organized as follows. In Sec. <ref>, we present an alternative formulation to the harmonic coefficients in terms of one integral of spherical Bessel functions. We dedicate Sec. <ref> to study the behavior of the integral for ℓ<kL and ℓ> kL. We show an exponential suppression in the latter case for ℓ> √(1-v^2) kL. In Sec. <ref>, we compute the ORF for v<1 and show that it is finite in the small angle limit. We conclude our work in Sec. <ref> with discussions and future directions.






§ REVIEW AND RE-FORMULATION IN TERMS OF BESSEL FUNCTIONS 


Our starting point is the expression of the Hellings-Downs curve in the spherical harmonic expansion, which following the notation of Ref. <cit.> reads

    Γ(kL,ξ)= C∑_ℓ=2^∞ a_ℓ P_ℓ(cosξ) ,

where C is a normalization constant, which we will fix to C=3/64 to compare with Ref. <cit.>, 

    a_ℓ=(2ℓ+1)2(ℓ-2)!/(ℓ+2)!|c_ℓ|^2 ,

and

    c_ℓ=∫_-1^1dxv/v+x(1-e^-ikL(v+x))(1-x^2)^2d^2/dx^2P_ℓ(x) .

We refer the reader to Refs. <cit.> for more details on the derivation of the above formulas. In Eq. (<ref>), we denote with v=ω(k)/k the phase velocity of the GWs (where ω(k) is the dispersion relation), which in principle can have any positive value as well as k dependence.[A k dependence in the phase velocity appears in massive gravity <cit.> where ω^2=k^2+m_g^2 with m_g being the graviton’s mass.] It should be noted that we also assumed pulsars at equal distances for simplicity. The main goal of this section is to recast c_ℓ in a more suitable form for integration and analytical approximations. 

We start noting that the terms containing v+x may be written in an integral form, namely

    v/v+x(1-e^-ikL(v+x))=i∫_0^kLdq  e^-i(v+x)q .

We note that similar replacements are often referred to as “Feynman's trick”.
With the identity (<ref>), Eq. (<ref>) becomes[We also used that

    d/dxP_ℓ(x)=ℓ+1/1-x^2(xP_ℓ(x)-P_ℓ+1(x))   and   (ℓ+2)P_ℓ+1(x)=(2ℓ+3)xP_ℓ+1(x)-(ℓ+1)P_ℓ(x) .

See Sec. 14.10 of Ref. <cit.> for more details.
]

    c_ℓ=iv(ℓ+1)∫_0^kLdq  e^-ivq∫_-1^1dx  e^-ixq{((ℓ+2)x^2-ℓ)P_ℓ(x)-2xP_ℓ+1(x)} .

Now we can perform the x integral by noting that (see Eq.18.17.19 in Ref. <cit.>)

    ∫_-1^1dx e^-ixq P_ℓ(x)=2 (-i)^ℓ j_ℓ(q) ,

where j_ℓ(q) is the spherical Bessel of ℓ-th order. Then, integrating Eq. (<ref>) over x and after some integration by parts,[We used that 
    ∫_-1^1 dx  x^n e^-ixq P_ℓ(x)=2(-1)^ℓ i^ℓ+n j^(n)_ℓ(t) ,

where we indicate the n-th derivative with a superscript parenthesis.
] we obtain our main formula, which is given by

    c_ℓ=2(-1)^ℓ    i^ℓ+1v(ℓ+1){((ℓ+2)v^2-ℓ) I_ℓ(kL,v)-2iv  I_ℓ+1(kL,v)
       
    -e^-ivkL[ℓ+ivkL/kL(ℓ+2)j_ℓ(kL)-(ℓ+4) j_ℓ+1(kL)]} ,

where we used that j_ℓ(0)=0 for ℓ>0 and, for convenience, we introduced

    I_ℓ(kL,v)=∫_0^kLdq  j_ℓ(q)e^-ivq .

With the above procedure, we have reduced the calculation of c_ℓ given by Eq. (<ref>) to a definite integral of a spherical Bessel function. Now, it all boils down to calculating I_ℓ(kL,v), which unfortunately does not have an analytical solution in terms of known functions, as far as we are aware. It is interesting to note that the spherical Bessel functions in Eq. (<ref>) can have an amplitude of O(ℓ^2). Thus, for finite kL, cross terms in |c_ℓ|^2 that include Bessel functions can introduce O(1) modulations on the amplitude of a_ℓ. In the calculations of the ORF though, they mostly average out.




§ ANALYTICAL APPROXIMATIONS TO THE INTEGRAL  


Before going into the detailed analytical approximations, it is instructive to look at Eq. (<ref>) qualitatively. First, we note that in PTAs, we have GW frequencies between f∼ 10^-9-10^-8  Hz and distances to pulsars of about L∼ 1000  lyr. It follows that typical values for kL∼ 2π f L are of the order of 10^3-10^4. We will then always consider that kL≫1 in Eq. (<ref>). However, this is not enough to only consider the kL→∞ limit of the integral due to the behavior of the spherical Bessel function. This is because the spherical Bessel function, j_ℓ(q), first grows as q^ℓ, peaks at around q∼ℓ, and continues to infinity with damped oscillations. This difference between q>ℓ and q<ℓ tells us that there will be a different behavior for ℓ≪ kL and ℓ≫ kL.

To understand the change in behaviour of I_ℓ(kL,v) (<ref>) in terms of kL, ℓ and v, it is instructive to take the large ℓ limit approximation of the spherical Bessel function given by

    j_ℓ(μβ)≈1/μ√(cosβ/tanβ)cos(μ(β-tanβ)+π4) ,

where μ=ℓ+1/2. The above equation is valid for 0<β<π/2 and follows from the approximation to J_μ(x) derived by Rayleigh (see Sec. 8.22 of Ref. <cit.>). Within such approximation we have that Eq. (<ref>), for ℓ<kL, reads

    I_ℓ(ℓ<kL,v)≈     I_ℓ(ℓ,v)
       +∫_0^β_kLdβ/2μ√(tanβ/cosβ)(e^iμ(β-tanβ)+iπ4+e^-iμ(β-tanβ)-iπ4)e^-ivμβ ,

where β_kL=arccos(ℓ/kL). In this form, it is clear that the integrand of Eq. (<ref>) contains two highly oscillating functions with frequency proportional to μ (or ℓ). 

We can gain insight into the behavior of Eq. (<ref>) by looking for saddle points of the imaginary part of the integrand. For v<1, we find that the saddle point of the product of the oscillations lies at sinβ=v. This means that the integrand contributes most around q∼μ/√(1-v^2), which for small values v is, unfortunately, in the regime where Eq. (<ref>) is not very reliable, at least regarding the amplitude. Nevertheless, approximating the integral by the integrand at that point captures the essence of the integral. Namely, it will be an oscillating function of ℓ for a given v<1 (actually, the integrand resembles a Hankle function of zeroth order as it goes as e^-iμ(arcsin v+π/4); something close to what we shall find later). In the opposite regime, namely v>1, the saddle point is complex. If we say that β=a+ib, the saddle point is located at a=π/2 and cosh b=v (at the other extreme of the real β line). This means that the integral is exponentially suppressed with ℓ, roughly as e^-μarccosh v. Both behaviors, for v<1 and v>1 are consistent with the kL→∞ solutions to I_ℓ (<ref>), given by Eq. (<ref>). We thus conclude that the saddle point argument, though not completely rigorous, implies that for v<1, no “constructive interference” occurs when

    ℓ≳ℓ_ cut=√(1-v^2) kL .

This is related to the argument in Ref. <cit.> that c_ℓ for ℓ≫ kL is exponentially suppressed. For v>1 the behavior change might happen at lower ℓ as the saddle point corresponded to larger values of q. However, this will not be relevant for the discussion of the ORF as, in this case, only the lower multipoles contribute. We also note that for q<ℓ, the spherical Bessel function oscillates no more, and both cases should have a similar behavior. We proceed to study the integral in two different limits and derive the precise scaling of the integral (<ref>) with ℓ and kL.




 §.§ Analytical approximation in the  regime 


We start by looking at the ℓ≪ kL limit to Eq. (<ref>), which corresponds to the case when there are many GW oscillations within the angular scale corresponding to ℓ. So, effectively we may send kL→∞. In this limit, there are closed formulas in terms of Legendre polynomials.[These formulas can be derived using the formulas in Secs. 10.22 and 14.3 of Ref. <cit.>, in particular using Eq. 10.22.56 and the hypergeometric function representation of the Legendre polynomials. Alternatively, one can ask a software like Mathematica.] Explicitly, we find that

    I_ℓ(kL→∞,v)={   π/2(-i)^ℓ P_ℓ(v)+(-i)^ℓ+1Q_ℓ(v)   (v<1)
       (-i)^ℓ+1 Q_ℓ(v)    (v>1)
    . ,

where Q_ℓ(v) is the Legendre polynomial of the second kind for v<1, also known as Ferrers function, and Q_ℓ(v) is the Legendre function for v>1. It is interesting to note that, in our formulation, the difference between v>1 and v<1 comes from whether the frequency of the oscillations of the exponential is fast enough to interfere with the oscillations of the Bessel function (see also the discussion on the saddle point approximation at the beginning of the section). In the original formulation, the difference between the v>1 and v<1 could be associated with the seeming pole at x=-v <cit.>. Regarding the Hypergeometric representation of Legendre functions, the values v<1 and v>1 correspond to a different choice of branch cuts. Also note that, for the same reason, the integral Eq. (<ref>) is not well-defined for v=1 in the kL→∞ limit, though c_ℓ in (<ref>) is. Now, with Eq. (<ref>), we find that Eq. (<ref>), in the ℓ≪ kL regime, is approximated by

    c_ℓ   (ℓ≪ kL)≈ 4(-1)^ℓ
       ×{   v(ℓ+1)/2[((ℓ+2)v^2-ℓ)(Q_ℓ(v)+iπ2P_ℓ(v))-2v(Q_ℓ+1(v)+iπ2P_ℓ+1(v))]    (v<1)
       1    (v=1)
       v(ℓ+1)/2[((ℓ+2)v^2-ℓ) Q_ℓ(v)-2v Q_ℓ+1(v)]    (v>1)
    . ,

which coincides with the results of Ref. <cit.>.[After using that P_ℓ(-x)=(-1)^ℓP_ℓ(x) and Q_ℓ(-x)=(-1)^ℓ+1Q_ℓ(x).] For the limit of v→1 we took the total expression in terms of Legendre polynomials and used the asymptotic formulas near the singularity.[For example, we have that P_ℓ(x→ 1^-)→ 1 and Q_ℓ(x→ 1^-)→1/2ln(2/1-x)-γ_E-ψ(ℓ+1), where ψ(ℓ) is the digamma function and γ_E≈ 0.577 Euler’s constant.] All divergences cancel, and we obtain the well-known result for v=1.






 §.§ Analytical approximation in the  regime 


We now turn to the ℓ>kL regime, which corresponds to the case when the angular resolution associated with the multipole number is finer than the number of GW oscillations. This is the regime where we expect finite distance effects to take place. Noting that the ℓ>kL regime roughly corresponds to the regime where the spherical Bessel function does not oscillate, we use the power series expression for j_ℓ(q), namely

    j_ℓ(q)=√(π)/2∑_m=0^∞(-1)^m/m! Γ[ℓ+m+3/2](
    q/2)^ℓ+2m .

Using the expansion Eq. (<ref>), we can formally write Eq. (<ref>) as 

    I_ℓ(kL,v)=√(π)   ∑_m=0^∞(-1)^m(
    2v)^-1-ℓ-2m/m! Γ[ℓ+m+3/2]( Ci_ℓ+2m+1[vkL]-i  Si_ℓ+2m+1[vkL]) ,

where Ci_n[z]=∫_0^zt^n-1cos t dt and Si_n[z]=∫_0^zt^n-1sin t dt respectively are the generalized sine and cosine integrals.[See Sec. 8.21 of Ref. <cit.> for more details.] In the current form, there is no resummation formula for Eq. (<ref>), to the best of our knowledge. However, we can resum Eq. (<ref>) under a series of approximations.

We first note that, both Ci_n[z] and Si_n[z], where in our set up z=vkL, decrease similarly with decreasing z for z≪ n, namely Ci_n[z]∝ z^n/n and Si_n[z]∝ z^n/(1+n).[For z≪ 1 we have that Si_n[z]∝ z^n+1. However, this corresponds to the case where vkL≪1, where a good approximation is to neglect the exponential in Eq. (<ref>) as it barely oscillates. Then Eq. (<ref>) is well approximated by 
    ∫_0^kLdq j_ℓ(q)=√(π)/(l+1) Γ[ℓ+3/2](kL/2)^l+1 
       _1F_2(l+12;l+32,l+32;-(kL2)^2) .
] we also notice that the remaining factor is an oscillatory function slowly decaying with z. For the Ci_n[z] it is initially very similar to cos(z) and for the Si_n[z] to sin(z). We then use the approximation that

    Ci_n[z≪ n]≈z^n/ncos(z)   and   Si_n[z≪ n]≈z^n/1+nsin(z) .

Then, we find that

    I_ℓ(kL,v)   ≈√(π)∑_m=0^∞(-1)^m(
    2kL)^1-ℓ-2m/m! Γ[ℓ+m+3/2](cos(vkL)/ℓ+2m+1-isin(vkL)/ℓ+2m+2)
       
    =√(π)/(l+1) Γ[ℓ+3/2](kL/2)^l+1{ 
       _1F_2(l+12;l+32,l+32;-(kL2)^2)cos(vkL)
                       -isin(vkL) (l+1)/l+2 
       _1F_2(l2+1;l2+2,l+32;-(kL2)^2)} ,

    where _1F_2(a;b,c;-x^2) is the generalized Hypergeometric function. We note that this approximation is close to considering a constant phase in the integral Eq. (<ref>) and integrating the spherical Bessel function only. In fact, since v<1, we expect that the approximation should still be valid for ℓ>vkL, which goes below the cut-off at ℓ∼ kL. Thus, this approximation is particularly good for the case with v≪1.


In the current form, Eq. (<ref>) is not very enlightening. So, let us discuss some interesting limits. First, keeping the leading order term in kL/ℓ in a Taylor series expansion yields

    I_ℓ(kL,v)   ≈√(π)/(ℓ+1) Γ[ℓ+3/2](kL/2)^ℓ+1(cos(vkL)-i (l+1)/l+2sin(vkL))
       ≈e^ℓ/√(2)ℓ(kL/2ℓ)^ℓ+1e^-ivkL .

Note that, despite the exponential dependence on ℓ, Eq. (<ref>) quickly decays for ℓ≫ kL.
In the transitional regime, where ℓ≳ kL, the kL/ℓ dependence can be inferred from the following. First, we note that for ℓ≳ kL, the integrand in Eq. (<ref>) peaks at the upper integration limit, that is, at q=kL. We then approximate the integral by the value of the integrand at the maximum, which gives I_ℓ(kL≲ℓ,v)∼ j_ℓ(kL)e^-ivkL up to O(1) errors. We then use the approximation formula in transitional regions to j_ℓ(kL) given by Nicholson (see Sec. 8.43 of Ref. <cit.>), which reads

    j_ℓ(kL≲ℓ)   ≈√(2/3π)√(ν-kL)/kLK_1/3[2 √(2)
       (ν-kL)^3/2/3 √(kL)]
       ≈1/2^3/4ℓ(1-kL/ℓ)^1/4(ℓ/kL)^3/4e^-ℓ√(ℓ/kL)2 √(2)(1-kL/ℓ)^3/2/3 ,

where ν=ℓ+1/2, K_1/3[x] is the modified Bessel function of the second kind and in the last step we took the large ℓ limit and we approximated K_1/3[x]∼√(π2x)e^-x. Thus, for ℓ≳ kL, the initial decay of Eq. (<ref>) is proportional to e^-ℓ√(ℓ/kL), which is close to the behaviour speculated in Ref. <cit.>. For ℓ≫ kL then it falls off as e^ℓ(kL/ℓ)^ℓ+1. In fact,  for practical purposes, we find that the approximation given by

    I_ℓ(kL≲ℓ,v)   ≈e^-ivkL/2^3/4ℓ(1-kL/ℓ)^1/4(ℓ/kL)^3/4e^-ℓ√(ℓ/kL)2 √(2)(1-kL/ℓ)^3/2/3 ,

already gives a good approximation to the integral in the relevant range of ℓ≳ kL, and it is much less numerically expensive than Eq. (<ref>). Though the approximation is not good for ℓ≫ kL, the integral is already several orders of magnitude smaller and does not contribute to the Hellings-Downs curve (<ref>).  




We show in Fig. <ref> the values of the integral I_ℓ(kL,v) for v=1/2 and v=10^-3 with kL=100. We choose a relatively low value of kL for illustration purposes. Later, we will fix kL=1000 to estimate the ORF numerically. In Fig. <ref> we show the numerical results together with the approximation (<ref>) for ℓ<kL and both Eqs. (<ref>) and (<ref>) for ℓ>kL. See how the qualitative approximation given by the modified Bessel function, Eq. (<ref>), describes well the exponential decay of the integral for ℓ>kL. We also checked numerically that the Taylor series expansion of Eq. (<ref>), namely Eq. (<ref>), describes well the decay but only for ℓ≫ kL. It gives a bad approximation for ℓ near kL. We conclude that the integral I_ℓ(kL,v) is well approximated by the large kL approximation (<ref>) and a cut-off at ℓ∼√(1-v^2)kL (see the discussion around Eq. (<ref>)). We also checked numerically that the value of ℓ_ cut describes well the start of the decay.

We end this section by noting that the calculations in this subsection are valid for any value of v. Nevertheless, for v≥ 1, the summation in the Hellings-Downs curve, Eq. (<ref>), converges quickly enough so that the effects are negligible.




§ FINITE DISTANCE EFFECTS TO THE ORF 


Here we compute the ORF (<ref>) and focus on the coincident pulsar limit, where previous approximations gave a divergent result. We start by computing a_ℓ given by (<ref>), which contains |c_ℓ|^2. For numerical calculations, we use the full c_ℓ given by (<ref>). For the analytical approximations we neglect the Bessel functions in (<ref>) and take

    |c_ℓ|^2≈ 4v^2(ℓ+1)^2 |((ℓ+2)v^2-ℓ) I_ℓ(kL,v)-2iv  I_ℓ+1(kL,v)|^2 .
 
Though we neglect the Bessel functions, it is interesting to note that for ℓ≫ kL, a Taylor series expansion of the Bessel term in (<ref>) yields

    ℓ+ivkL/kL(ℓ+2)j_ℓ(kL)-(ℓ+4) j_ℓ+1(kL)≈√(π) (l+2) (kL)^ℓ-1ℓ/ 2^ℓ+1Γ(l+3/2)≈e^ℓ/2^ℓ+3/2(kL/ℓ)^ℓ-1 ,

where in the last step, we took the large ℓ limit. It is interesting to note that in this limit, the contribution from the spherical Bessel functions, Eq. (<ref>) decays with two powers of (kL/ℓ) less than Eq. (<ref>). We confirm numerically that the Bessel contribution to c_ℓ quickly dominates for ℓ> kL. This is nevertheless unimportant as the suppression is huge, and it does not contribute to the ORF.

Because in Sec. (<ref>) we showed that c_ℓ is exponentially suppressed for ℓ>√(1-v^2) kL, we consider for analytical purposes that the ORF (<ref>) for v<1 is well approximated by

    Γ(kL,ξ)= C∑_ℓ=2^ℓ_ cut a_ℓ P_ℓ(cosξ) ,
 
where we introduced a cut-off to the sum at ℓ_ cut=√(1-v^2) kL.[Strictly speaking the sum is from ℓ=2 until the integer number closer to ℓ_ cut. For large ℓ this does not make a difference.] Thus, we only need to know the value of a_ℓ in the ℓ≪ kL regime. As shown in Ref. <cit.>, the large ℓ limit of c_ℓ in the case of v<1 yields[We used that P_ℓ(cosθ)≈√(θ/sinθ)J_0((ℓ+12)θ) and Q_ℓ(cosθ)≈ -π2√(θ/sinθ)Y_0((ℓ+12)θ). ]

    c_ℓ(ℓ≪ kL;ℓ≫1)≈ (-1)^ℓ iπ   
    v(ℓ+1)√(θ/sinθ)
       ×[((ℓ+2)v^2-ℓ)H_0^(1)((ℓ+12)θ)-2vH_0^(1)((ℓ+32)θ)] ,

where cosθ=v and H_0^(1)(x) is the Hankle function of the first kind. Further using that in the ℓ≫1 limit H_0^(1)(x≫1)≈√(2π x)e^i(x-π/4) and that the prefactor in a_ℓ (<ref>) goes as 4/ℓ^3, one obtains that

    a_ℓ(ℓ≪ kL;ℓ≫1)≈ 8π v^2(1-v^2)^3/2 .

Namely, for large enough ℓ the coefficients a_ℓ are constant. 



We show in Fig. <ref> the a_ℓ for the case of kL=100 and kL=1000, respectively, on the left and right panels. The former is for illustration purposes; the latter is of the order of current observations.  In the left panel, we also fixed v=10^-2 as a small v example, but the approximations work well in other cases, too. The exception is v≳ 1/3 where the cut-off in ℓ∼√(1-v^2)kL is noticeably different than kL. In that case, the ℓ>kL approximation (<ref>) gives the right exponential decay but shifted to ℓ=kL. For smaller values of v, the approximations get better, and Eq. (<ref>) is relatively good for some ℓ inside the ℓ<kL regime. In the right panel, we show the case of kL=1000 for v={9/10,1/2,10^-1,10^-2}. There it is clear that there is a very sharp cut-off roughly at ℓ∼√(1-v^2) kL. Let us remind the reader that value for ℓ_ cut is analytically derived in the beginning of Sec. <ref>. Though one could try to numerically find an effective cut-off, we will see shortly that ℓ_ cut given by Eq. (<ref>) recovers the value of the autocorrelation function.

Now, let us discuss the impact of Eq. (<ref>) on the ORF. In Ref. <cit.>, the constant approximation is taken up to ℓ→∞, which, using the resummation formulas of the Legendre polynomials, yields a factor 1/√(1-cosξ). However, this is divergent in the ξ→ 0 limit, as noted in Refs. <cit.>. Physically speaking, divergence is due to the fact in the strict kL→∞ limit, the pulsars are infinitely far away, and the “surfing” <cit.> on top of the GW takes place infinitely. In our terminology, the cut-off is sent to ℓ_ cut→∞, and the sum diverges proportionally to ℓ_ cut. However, for finite values of kL, we have that the value of ORF (<ref>) in the ξ→0 limit is finite and given by 

    Γ(kL,ξ=0)=3/64∑_ℓ=2^ℓ_ cut a_ℓ ,

where we set the normalization constant to 3/64 for comparison with Ref. <cit.>.
We give an analytical estimate as follows. Using that kL≫1 and so that ℓ_ cut≫1, we note that most of the contribution to the coincident limit comes from the large ℓ sum. We then assume that a_ℓ is constant for all ℓ and given by Eq. (<ref>), which yields that

    Γ(kL,ξ→0)≈3/64a_ℓℓ_ cut=3π/8 kL  v^2(1-v^2)^2 .

We numerically checked for kL=1000 that this limit is accurate within 0.1%. This further supports our analytical estimation.[It is interesting that Γ(kL,ξ→0) is not necessarily large despite the fast growth for small but not too small ξ. In fact, for v<1/√(kL), it achieves values less than in the GR case. This limit is, unfortunately, not so interesting from the observational point of view, as in PTAs the significant for v<0.4 the statistical significants drops below 3σ as show in Ref. <cit.>using the CPTA data.] We also note that our limit for ξ→0 of the ORF matches exactly the value of the auto-correlation obtained in Ref. <cit.>; see their Eq. (28)  with the identification of vkL=2π fL. 



We show in Fig. (<ref>) the ORF (<ref>) for kL=100 (top panels) and kL=1000 (bottom panels) and v={9/10,1/2,10^-1,10^-2}. We also include the GR case (v=1) for comparison. In solid lines, we show the numerical result. The nearby dashed lines show the analytical approximation given by Eq. (<ref>) with c_ℓ from Eq. (<ref>) together with Eq. (<ref>). First, look at the top and bottom panels on the right. Note how the analytical approximation agrees well with the numerical one in the small ξ limit. We also see that the ORF approaches a constant for values of ξ below ξ∼ 2/ℓ_ cut. We estimated such value of ξ by requiring that the next order correction in a Taylor series expansion of the Legendre polynomials for ξ≪1 becomes of O(1). This estimate yields log_10ξ≈ -1.7 and log_10ξ≈ -2.6 respectively for kL=100 and kL=1000, which is a reasonable estimate compared with the right panel of Fig. (<ref>). We also see that the asymptotic value for ξ→0 matches the analytical approximation given by Eq. (<ref>) in the large kL limit.

Now, looking at the left top and bottom panels of Fig. (<ref>) we see that the numerical and analytical approximation are similar, though they do not overlap as much as in the small ξ regime. We see that for kL=100, there are more wiggles than kL=1000 in the numerical and analytical cases due to finite distance effects. Increasing the value of kL smooths the curve. Besides the wiggles, we still see some differences between the numerical and analytical results, even for kL=1000. There it is likely that some component is due to numerical errors as the I_ℓ (<ref>) is the integral of highly oscillatory functions. Nevertheless, it is plausible that finite distance effects remain even for relatively large kL. From Fig. <ref>, it is clear that the analytical approximation given by Eq. (<ref>) does not capture the behavior of a_ℓ near ℓ_ cut. It would be interesting to carry out a more detailed and precise numerical study and, perhaps, refine the analytical approximations to the ORF for v<1.




§ CONCLUSIONS 


The ORF in PTAs provides a complementary way to test gravity by measuring the phase velocity v of the GWs. In general, modified gravity predicts a phase velocity with v≠ 1, such as in scalar-tensor theories (see Ref. <cit.> for a review), and possible with a k dependence, as in massive gravity <cit.>. Thus, departures of v=1 in the ORF would hint at modifications of gravity.

Motivated by recent studies <cit.>, that encountered a divergent ORF in the small angle limit when kL→∞ for the v<1 case, we investigated finite distance effects on the ORF. To do that, we reformulated the calculation of the harmonic coefficients c_ℓ in terms of a definite integral of spherical Bessel functions given by Eq. (<ref>). As we argued at the beginning of Sec. <ref>, the approximation derived by Refs. <cit.> is valid up to ℓ_ cut∼√(1-v^2) kL. For ℓ≳ℓ_ cut the integral (and c_ℓ) decays exponentially, roughly as (ℓ/kL)^3/4× e^-ℓ√(ℓ/kL), see Eq. (<ref>) and the discussion around it. For ℓ≫ kL it then decays as e^ℓ/ℓ×(kL/ℓ)^ℓ+1, see Eq. (<ref>).

The physical interpretation of our results, based on the nice physical explanations of Ref. <cit.>, is as follows. First, the factor kL is the ratio of the “arm-length” (the distance to the pulsar) to the wavelength of the GW. Namely, kL tells us how many GW oscillations undergoes the pulse from the pulsar until Earth. More precisely, the factor kL enters in the redshift transfer transfer function (roughly the term inside parenthesis in Eq. (<ref>)) in the antenna pattern of PTAs, yielding an angular pattern where points with vanishing redshift are separated by an angle v+cosξ=2n/kL where n∈ℤ^+ (see also Fig. 5 and 6 of Ref. <cit.> and Fig. <ref>). Thus, the typical angular separation of these features is ξ∼ 1/kL. Then, in the spherical harmonic decomposition, where the multipole number is roughly related to the inverse angular separation (i.e. ℓ∼π/ξ), it is expected that multipole numbers with ℓ≳ kL contain no more information of the antenna response and their coefficients are, therefore, suppressed. 

Focusing on the small angle limit of the ORF, we found that using the large ℓ approximation for a_ℓ (<ref>), first derived by Refs. <cit.>, up to the cut-off ℓ_ cut (<ref>) exactly recovers the value of the autocorrelation derived in Ref. <cit.>. We also checked our analytical estimates for the ORF numerically for kL=1000 and found a good agreement in the small angle regime. We thus showed that the ORF for v<1 is finite and that it correctly approaches the limiting value for ξ→ 0. Outside the small ξ regime, we find that the numerical and analytical approximations are similar, although they do not match as well as in the small ξ limit. This difference could be partly due to numerical errors, but it is possible that finite distance effects on the ORF will remain for relatively large values of kL. Basically, we find that the a_ℓ≈ constant approximation (<ref>) for ℓ≫ 1 is not good for values close to ℓ∼ℓ_ cut. It is therefore important to carry out more precise numerical studies and perhaps refine the analytical templates of the ORF for v<1.

We also note that while we focused on the case v<1 our approach is also valid for any value of v. However, for v≥ 1 the numerical integration of Eq. (<ref>) might be more challenging in the regime ℓ≪ kL, with kL≫1, as the oscillations from the exponential have a higher frequency and there are non-trivial cancellations. Nevertheless, for v>1 there are already analytical expressions for c_ℓ after dropping the exponential in (<ref>), see, e.g., Refs. <cit.>.  Thus, our approach is most suitable for v<1 as it is sensitive to finite distance effects. For instance, we could numerically compute c_ℓ up to ℓ=1100 for kL=1000, which is challenging to do in the formulation of Legendre polynomials (<ref>). Interestingly, Ref. <cit.> tested the value of v using the ORF of CPTA data. They found that while not statistically significant there are two preferred values of v=0.98 and v=1.04 in the data. For v<0.4 though, the statistical significance dropped below 3σ. It would be interesting to derive more precise templates with our formulation for v≲ 1 and test them against the data.

Lastly, we note that throughout the paper, we assumed equal distance pulsars for simplicity. It would be interesting to redo the calculations when the distance to the pulsars is different from each other, which is the more realistic situation. We leave this issue for future work, as it is out of the scope of this paper. We end by speculating that it may be possible to derive an exact formula for the coefficients a_ℓ without assuming the limit kL≫ 1. Looking at the exact expression for the autocorrelation given by Eq. (23) of Ref. <cit.>, we notice that all terms in that formula can be expressed as an infinite sum of the product of spherical Bessel functions and Legendre polynomials (see, e.g., Secs. 6.10 and 10.6 of Ref. <cit.>). It is thus plausible that a_ℓ may also be written as a series expansion in terms of such functions. In the worst-case scenario, it may only be possible in the exact ξ→ 0 limit. We leave this exercise for future work.





§ ACKNOWLEDGMENTS
 
G.D. would like to thank B. Allen, M. Sasaki, K. Schmitz and J. Tränkle for their helpful discussions. G.D. is supported by the DFG under the Emmy-Noether program grant no. DO 2574/1-1, project number 496592360.



 



 Paper Title: Prospects for Cosmological Research with the FAST Array: 21-cm Intensity Mapping Survey Observation Strategies.


	
	
	
 
	
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	
	
	
	
	
	Corresponding author.
zhangxin@mail.neu.edu.cn
	
	
	Key Laboratory of Cosmology and Astrophysics (Liaoning) & College of Sciences, Northeastern University, Shenyang 110819, China
	Key Laboratory of Data Analytics and Optimization for Smart Industry (Ministry of Education), Northeastern University, Shenyang 110819, China
	National Frontiers Science Center for Industrial Intelligence and Systems Optimization, Northeastern University, Shenyang 110819, China
	
	
		
	
	
	
	
	
	§ ABSTRACT 
	Precise cosmological measurements are essential for understanding the evolution of the universe and the nature of dark energy. The Five-hundred-meter Aperture Spherical Telescope (FAST), the most sensitive single-dish radio telescope, has the potential to provide the precise cosmological measurements through neutral hydrogen 21-cm intensity mapping sky survey. This paper primarily explores the potential of technological upgrades for FAST in cosmology. The most crucial upgrade begins with equipping FAST with a wide-band receiver (0 < z < 2.5). This upgrade can enable FAST to achieve higher precision in cosmological parameter estimation than the Square Kilometre Array Phase-1 Mid frequency. On this basis, expanding to a FAST array (FASTA) consisting of six identical FAST would offer significant improvements in precision compared to FAST. Additionally, compared with the current results from the data combination of cosmic microwave background, baryon acoustic oscillations (optical galaxy surveys), and type Ia supernovae, FASTA can provide comparable constraints. Specifically, for the dark-energy equation-of-state parameters, FASTA can achieve σ(w_0) = 0.09 and σ(w_a) = 0.33.

		
	
	
	
	Prospects for Cosmological Research with the FAST Array: 21-cm Intensity Mapping Survey Observation Strategies
    Xin Zhang
    
==============================================================================================================



§ INTRODUCTION

 
    Dark energy is a component of the universe that causes the late-time cosmic acceleration. Although it is known to comprise 68% of the total energy density of the universe, the nature of dark energy remains unknown. The Λ cold dark matter (ΛCDM) model, the standard model of cosmology, treats dark energy as a constant energy density filling space homogeneously. However, the latest baryon acoustic oscillation (BAO) observation from the Dark Energy Spectroscopic Instrument (DESI) suggests that dark energy may potentially be a dynamical energy density evolving with time <cit.>. This finding indicates the possibility of dark-energy dynamical evolution, but more observations and advanced instruments are needed to fully confirm this and to understand the evolutionary dynamics of dark energy.

BAO from the large-scale structure (LSS) is a late-time cosmological probe that can be used as a standard ruler to study the expansion history of the universe <cit.>. Currently, BAO studies primarily rely on optical galaxy surveys <cit.>. While effective, these surveys are limited in terms of efficiency and coverage.  To overcome these limitations and achieve more precise cosmological measurements, advancements in radio astronomy have developed a promising method known as 21 cm intensity mapping (21 cm IM). After reionization, neutral hydrogen (H  I) mainly exists within galaxies. This technique measures the intensity of the characteristic 21 cm emission line from H  I to study the LSS <cit.>. This method does not require resolving individual galaxies, thus it has high survey efficiency. The characteristics of the 21 cm signal allow us to probe deeper into the universe and directly obtain redshift information. Recent forecasts and observational studies have confirmed the potential of 21 cm IM in cosmology <cit.>.





	
	To achieve more precise cosmological measurements, advanced radio telescopes are utilized in 21 cm IM. These telescopes include those currently in use, such as the Five-hundred-meter Aperture Spherical Telescope (FAST) <cit.>, the MeerKAT <cit.>, the Tianlai <cit.>, the Canadian Hydrogen Intensity Mapping Experiment (CHIME) <cit.>, and those planned for future use, such as the Square Kilometre Array (SKA) <cit.>, the Baryon Acoustic Oscillations from Integrated Neutral Gas Observations (BINGO) <cit.>, and the Hydrogen Intensity and Real-time Analysis eXperiment (HIRAX) <cit.>.

    
    
    
    Among these, FAST is particularly significant due to its large aperture and sensitivity. The current FAST has achieved substantial success in many areas, particularly in fast radio burst and gravitational wave research <cit.>. However, to harness its potential for 21 cm IM and to extend its capability for cosmological studies, several upgrades are planned for FAST  <cit.>. 
    
    This paper aims to explore how potential future upgrades to FAST could promote cosmological research by using 21 cm IM. For the expansion of FAST itself, equipping it with a wide-band receiver aims to extend its 21 cm signal detection capability, enabling broader redshift coverage <cit.>. This will allow us to observe a larger volume of the universe, aiding in constructing a more precise three-dimensional map of the LSS. Moreover, a phased array feed (PAF) is planned to enhance its survey efficiency <cit.>.
    Additionally, the expansion of the construction around FAST is also considered. The plan involves building 40-meter diameter spherical radio telescopes (40 m dishes) in two phases around FAST: the first phase involves constructing six 40 m dishes, and the second phase will increase this number to twenty. 
	The most significant upgrade is the FAST Array (FASTA), which plans to add five more identical 500-meter diameter telescopes to join the current FAST <cit.>. This network will significantly improve sensitivity and survey efficiency, enhancing the ability to explore LSS and providing higher precision in cosmological parameter estimation.
	
    In this paper, we present a forecast for estimating cosmological parameters using simulated 21 cm IM data and discuss how the various extensions of FAST will improve the precision of these constraints.
    
    In Section <ref>, we introduce the 21 cm IM simulation and the methods used to constrain cosmological parameters. In Section <ref>, we discuss the performance of various configurations for 21 cm IM and the resulting cosmological constraints. In Section <ref>, we conclude with perspectives for future research.
	
	In our analysis, we adopt the Planck best-fit ΛCDM model as fiducial cosmology  <cit.>. The parameters are H_0 = 67.3 km s^-1 Mpc^-1, Ω_m = 0.317, Ω_b = 0.0495, Ω_k = 0, σ_8 = 0.812, and n_s = 0.965.
	
	



§ 21 CM INTENSITY MAPPING

   

 §.§ Signal

   21 cm IM measures the intensity of 21 cm signals and converts it into brightness temperature.  The mean temperature of H  I can be expressed as <cit.>
   
    T_b = 3/32 π𝒽 c^3 A_10/k_B m_pν_21^2(1+z)^2/H(z)Ω_H   I(z) ρ_c, 0 ,

   where 𝒽 is the Planck constant, c is the speed of light, A_10 is the Einstein coefficient for spontaneous emission <cit.>, k_B is the Boltzmann constant, m_p is the proton mass, ν_21 is the H  I line frequency, H(z) is the Hubble parameter, Ω_H  I(z) is the fractional density of H  I, and ρ_c, 0 is the current critical density of the universe.
   
    The signal covariance is expressed in terms of the power spectrum
    <cit.>
		
    C^S(q, y) = T_b^2(z_i)/r^2 r_ν (b_H   I + fμ^2)^2 exp(-k^2μ^2σ_NL^2)  P(k, z),

	where q = k_⊥ r and y = k_ r_ν. r denotes the comoving distance to z, and r_ν corresponds to the comoving distance for the frequency interval. b_H   I represents the bias factor of H  I <cit.>. f is the linear growth rate of the structure. μ = k_ / k is the cosine of the angle between the wavevector k and the line of sight. σ_NL=7 Mpc represents the non-linear dispersion scale <cit.>. P(k, z) is the matter power spectrum at wavenumber k and z, which can be calculated by CAMB <cit.>.
	
		
	

 §.§ Noise

	The noise covariance can be written as <cit.>
	
    C^N(q, y)=T_sys ^2/t_tot Δν N_b N_d U_bin B_⊥^-2 B_^-1,

	where T_sys includes the instrumental temperature T_inst and the sky background radiation T_sky≈ 60  K× (ν / 300 MHz)^-2.5.  t_tot represents the total integration time, N_b is the number of beams, N_d is the number of dishes in the array, and Δν is the total bandwidth. U_bin = S_areaΔν̃ represents the survey volume element at a given redshift, where S_area is the survey area and Δν̃ = Δν/ν_21 is the dimensionless bandwidth for a given redshift bin. B_ is the radial beam response
	
    B_(y)=exp(-(y δν / ν_21)^2/16 ln 2),

	where δν is the bandwidth of an individual frequency channel. B_⊥ is the transverse beam response in single dish mode
	
    B_⊥(q)=exp(-(q λ/D_dish)^2/16 ln 2),

	where D_dish is the diameter of the dish.

	

 §.§ Foreground

	The foreground is an important component of the signal received by the telescope. It primarily originates from the galaxy and extragalactic point sources. At this stage, the effectiveness of methods for subtracting foregrounds from observational data has been very limited. The mainstream methods mainly include Polynomial Fitting, Principal Component Analysis, Independent Component Analysis, and Artificial Intelligence <cit.>. We assume a method that can effectively subtract the foreground, and the residual foreground model can be written as <cit.>
	
    C^F(q, y)=ε_FG^2 ∑_X A_X(l_p/2 π q)^n_X(ν_p/ν_i)^m_X,

	where ε_FG is a factor that describes the efficiency of foreground subtraction. ε_FG = 1 represents no foreground subtraction, while ε_FG = 0 represents perfect foreground subtraction. We assume ε_FG=10^-6, as this level is effective for extracting cosmological signals. X is the different foreground component shown in Table <ref>. 




    






 §.§  Instrument parameters

    

    
    
In this paper, we consider several different configurations of FAST and Square Kilometre Array Phase 1 - Mid frequency (SKA1-Mid) for simulating the 21 cm IM. The specific instrument parameters are listed in Table <ref>.

The current capability of FAST to detect 21 cm signals covers the redshift range 0 < z < 0.35. A new wide-band receiver under test extends this range to 0 < z < 1.84 <cit.>. Previous galaxy surveys, however, have performed precise BAO measurements within z < 2.5, achieving 1%-2% accuracy <cit.>. This paper assumes the upgraded new wide-band receiver on FAST can detect 21 cm signals up to 0 < z < 2.5. 

Additionally, constructing 6×40 m dishes or 20×40 m dishes around FAST is considered. These dishes will use the same receiver and feed system as FAST, so their frequency bands and N_b will be identical. Given the outstanding performance of FAST, we also consider constructing five additional identical FAST telescopes to form FASTA, significantly improving survey efficiency. Furthermore, we consider equipping FAST with a PAF to increase the value of N_b to 100.

Regarding instrument temperature, in Ref. <cit.> the author measured the temperature of FAST near boresight to be 20 K in the L-band. We assume that the frequency band can be broadened while maintaining the instrument temperature at 20 K. For the PAF, we assume the temperature to be approximately 30 K due to the increase in the number of beams. The total observation time is assumed to be 10000 hours. The calculation for the survey area is as follows

	
    S_area = η∫_ϕ∫_θsinθdθdϕ ,

	where η represents the fraction of the initial survey area that remains usable for scientific observation after excluding the Milky Way. For FAST/FASTA, we considered η = 0.7 and the range of declination from -14^∘ to +66^∘, yielding an area of approximately 8100 deg^2.

For SKA1-Mid, we considered band I (0.35<z<3) planned for 21 cm IM, with the survey area of 20000 deg^2 and the total observation time of 10000 hours. The instrument temperature is as follows <cit.>

    T_inst=15  K+30  K(f/GHz-0.75)^2 .

	

 §.§ The Fisher matrix

	
	
	
	
    
	
	
	
	
	
	
	
	
	
	To achieve optimal observational results, different telescopes require different survey strategies, so the methods for calculating C^tot also vary. For a single dish or identical dishes observing the same area of the sky (such as FAST or FASTA), the total covariance matrix is expressed as

    C^tot = C^S + C^N + C^F.


For the combination of different telescopes, we consider two scenarios. The first scenario involves two telescopes in close geographic proximity observing the same area of the sky, such as combining 40 m dishes and FAST. This approach maximizes the reduction of thermal noise impact on observations. Since C^S and C^F are only related to the observed sky area, they remain fixed. Different telescopes have varying C^N due to their differing thermal noise integration effects. Therefore, the total covariance matrix can be expressed as

    C^tot = C^S + ((C_1^N)^-1 + (C_2^N)^-1)^-1 + C^F,

where C_1^N and C_2^N are the noise covariance matrices of the different dishes, with (C^N)^-1 denoting their inverse matrices. The second scenario involves two telescopes with significant differences in latitude, resulting in a very small common observable sky area. Therefore, they must observe different areas of the sky, which also maximizes the survey volume. Since both the observed sky areas and the telescopes are different, the total covariance matrix is given by

    C^tot = ((C_1^tot)^-1 + (C_2^tot)^-1)^-1,

where C_1^tot and C_2^tot are the total covariance matrices of the different dishes.

	After obtaining the total Fisher matrix C^tot, we can transform it into the Fisher matrix of observations {p_i} <cit.>
	
    F_i j = 1/2 U_bin∫d^2 q  d y/(2 π)^3[∂ln C^tot(q, y)/∂ p_i∂ln C^tot(q, y)/∂ p_j].

	In this paper, we mainly calculate the Fisher matrix for several key observables to constain the evolution of LSS, as well as the growth and geometry of the universe. These include the transverse distance indicator D_A(z), radial distance indicator H(z) and the parameter combination fσ_8(z) from redshift space distortions (RSDs), where f(z) is growth rate and σ_8 is the amplitude of matter density fluctuations on a scale of 8 h^-1 Mpc (with h the dimensionless Hubble constant).
    
   
    
    
	

 §.§ Cosmological parameter constraints

    We use the Fisher matrix to propagate errors from observational errors to the BAO parameters D_A(z), H(z), and fσ_8(z). Then, we use the errors of BAO parameters to form the likelihood function for cosmological models. Finally, we employ Markov Chain Monte Carlo (MCMC) analysis to constrain the cosmological parameters <cit.>.
 
 To compare with the mainstream observations, we calculated the constraints from the latest observations. The data used include: CMB – Planck 2018 high-ℓ TT, TE, EE, and low-ℓ TT, EE, and PR4 lensing data <cit.>; BAO – DESI data <cit.>; and supernova (SN) – Pantheon+ data <cit.>. The data combination of CMB, BAO, and SN in this paper is also abbreviated as CBS.

	The equation of state (EoS) of dark energy is given by w = p/ρ, where p is the pressure and ρ is the energy density. We employ different models to describe its dynamics. (i) ΛCDM: w = -1. (ii) wCDM: w(z) = w, where w is a constant EoS for dark energy. (iii) w_0w_aCDM: w varies with redshift according to the parameterization w(z) = w_0 + w_az / (1 + z), where w_0 is the present-day value of the equation of state parameter and w_a describes its evolutionary behavior.
	


§ RESULTS AND DISCUSSION

	In this section, we  first discuss the capabilities of various instruments and their combinations in 21 cm IM using the relative errors of BAO parameters {ε(D_A(z)), ε(H(z)), ε(fσ_8(z))}, where ε(ξ) = σ(ξ)/ξ. We then present the cosmological parameter constraints employing FAST, SKA1-Mid, FASTA, and CBS, followed by comparison and discussion of the results.
    
    
    
    
    
	
	

 §.§ Combination of FAST and SKA-Mid

 
 In this subsection, we discuss the capabilities of SKA1-Mid and FAST with a wide-band receiver, as well as their combination in 21 cm IM. Due to geographical factors, the overlapping observable sky area for FAST and SKA1-Mid is very limited, which necessitates a strategy where each telescope observes different sky regions.

From Figure <ref>, it is evident that FAST and SKA1-Mid have distinct advantages across different redshift ranges. SKA1-Mid achieves tight constraints within the range of 0.35 < z < 0.8 due to its high survey efficiency. In this range, the relative errors in BAO parameters are typically below 3%. On the other hand, FAST attains robust constraints within the range of 0.8 < z < 2.5 because of its high transverse resolution and sensitivity. In this range, the relative errors in BAO parameters are typically around or below 4%.


Their combined observations enable precise measurement of BAO parameters across the redshift range of 0 < z < 2.5, achieving constraints on D_A(z), H(z), and fσ_8(z) to below 2% for most of this range, and to as low as 1% at z ≈ 1.

Upgrading the existing receiver to detect 21 cm signals up to 0 < z < 2.5 will enable FAST to utilize its high-redshift advantage in 21 cm IM. Combined with SKA1-Mid, this complementary strategy will improve measurements of BAO parameters across a wide redshift range.

    

 §.§ Combination of FAST and 40 m dishes

	In this subsection, we discuss the extent to which incorporating 40 m dishes with FAST enhances the constraints on BAO parameters in 21 cm IM. An array of 40 m dishes, located in close geographical proximity to FAST, enhances the survey efficiency by enabling observations of the same sky regions.

Figure <ref> illustrates the fractional constraints achievable on D_A(z), H(z), and fσ_8(z) using 21 cm IM with an array of 40 m dishes, FAST, and their combined configurations. The 40 m dishes perform better at low redshifts because they have higher survey efficiency. FAST is more effective at higher redshifts due to its greater sensitivity and resolution. The combination of FAST and 40 m dishes significantly reduces observational errors, particularly at high redshifts. The relative errors in BAO parameters are typically below 4%, with errors reaching as low as 2% at z ≈ 1.

For the configurations of 6× 40 m dishes and 20× 40 m dishes combined with FAST, Figure <ref> shows that the larger array provides only marginal improvements. This result suggests that the combination of FAST with 6× 40 m dishes offers a more optimal configuration for the 21 cm IM survey.

     
    

 §.§ FASTA

	In this subsection, we present the cosmological parameters constraints by using simulations of 21 cm IM data with different instruments and current observational data from CBS. Figure <ref> shows the  parameter constraints under the 1σ and 2σ confidence intervals for the ΛCDM, wCDM, and w_0w_aCDM models. The specific values are listed in Table <ref>.

Compared to the results in Ref. <cit.> for FAST (0 < z < 0.35), a wide-band receiver proposed in this paper utilizes the resolution advantage of the FAST at high redshifts, significantly enhancing the constrains on cosmological parameters. The main specific parameter constraints are as follows: for the ΛCDM model, FAST achieves σ(Ω_m) = 0.0076 and σ(H_0) = 0.51  km s^-1 Mpc^-1. For the wCDM model, the constraint is σ(w) = 0.036. For the w_0w_aCDM model, the constraints are σ(w_0) = 0.14, and σ(w_a) = 0.57. Compared to SKA1-Mid, FAST also has higher sensitivity and lower receiver noise, which enables it to achieve better parameter constraints. For the dark-energy EoS parameters, there is an improvement of approximately 20% for w, 12.5% for w_0, and 21.9% for w_a.

	Further advances are observed with FASTA. From Figure <ref>, FASTA performs well in observing BAO parameters across the entire band, achieving an observational error as low as 1%. This precise observation of BAO parameters translates into tight constraints on cosmological parameters. 
 
In the ΛCDM model, FASTA provides best constraints, σ(Ω_m) = 3.6 × 10^-3 and σ(H_0) = 0.26  km s^-1 Mpc^-1, which surpasses other instruments and CBS.
In the wCDM model, FASTA also provides best constraints, σ(Ω_m) = 0.0037, σ(H_0) = 0.39  km s^-1 Mpc^-1, and σ(w) = 0.02. For the parameter w, FASTA represents an improvement of approximately 55.56% compared to SKA1-Mid, 44.44% compared to FAST, and 16.67% compared to CBS.
In the w_0w_aCDM model, FASTA achieves highly precise parameter constraints, with σ(Ω_m) = 0.012, σ(H_0) = 0.89  km s^-1 Mpc^-1, σ(w_0) = 0.09, and σ(w_a) = 0.33. For w_0 and w_a, FASTA represents an improvement of approximately 43.75% and 54.79% compared to SKA1-Mid, and 35.71% and 42.11% compared to FAST, while achieving constraints comparable to CBS using 21 cm IM  alone.

In summary, FASTA significantly improves cosmological parameter constraints across various models, surpassing other instruments and achieving results comparable to CBS using 21 cm IM  alone.



 
 
 
	
        
	

 §.§ Phased array feed

	In this subsection, we discuss the enhancements and challenges of upgrading the feed of FAST to PAF for 21 cm IM. PAF can achieve various beamforming modalities by adjusting the phase and amplitude of each element in the array, thereby generating multiple beams simultaneously. This capability covers a larger field of view (FOV) and enables rapidly beam steering and scanning <cit.>.  At the same time, different beamforming strategies significantly impact noise characteristics, and data calibration and processing.  We considered two beamforming strategies in terms of survey efficiency.
 
    In the first scenario, we consider a fixed FOV and maximize the N_b within this field. The size of a beam is given by θ = c/(ν D_dish). If there is no overlap between the beams, the maximum N_b can be written as
    
    N_b = FOV·(ν D/c)^2 .


    In this case, C^N follows the same pattern as described in Ref. <cit.>
    
    C^N(q, y) → C^N(q, y) ×1,    ν>ν_crit 
    (ν_crit  / ν)^2,    ν⩽ν_crit  .

	We assume ν_crit=1420 MHz. As shown in Figure <ref>, the errors increase significantly at low frequencies. This is because at low frequencies, the beam size increases, resulting in a decrease in N_b.
 
	Another scenario is a fixed N_b. We assume the case of 100 beams. As shown in Figure <ref>, the increase in the N_b significantly enhances survey efficiency, achieving best constraint result of 2% around z = 1. However, in practice, the FOV does not increase indefinitely as the beam size increases at lower frequencies. As the FOV certainly deviates from the boresight, the survey efficiency decreases. This requires actual measurements to PAF, but here we make an ideal estimate. The real situation is expected to lie between the fixed FOV and the fixed N_b scenarios.


	

§ CONCLUSION

	In this paper, we explore several potential upgrades to FAST using 21 cm IM to enhance cosmological research. Those upgrades include FAST with a wide-band receiver, the combination of 40 m dishes and FAST, FASTA, and FAST with a PAF. We simulated the full 21 cm IM power spectrum to constrain the BAO and RSD parameters D_A(z), H(z), and fσ_8(z), and used these results to evaluate the capabilities of these different configurations. Finally, we used the simulated BAO and RSD data to constrain the cosmological parameters of the ΛCDM, wCDM, and w_0w_aCDM models.

	Firstly, the wide-band receiver utilizes the capabilities of FAST in the redshift range 0.8 < z < 2.5, achieving better cosmological parameter constraints than SKA1-Mid. It also enables FAST to form a complementary observation with SKA1-Mid, which has strengths in the redshift range of 0 < z < 0.8, providing high-precision observations across a wider redshift range.
 
	Secondly, we discuss the combination of the 40 m dishes and FAST. Incorporating 40 m dishes with FAST significantly enhances survey efficiency, thereby reducing BAO parameter observational errors to typically below 2%. For the combination with FAST, 6×40 m dishes are a more balanced configuration compared to 20×40 m dishes for 21 cm IM.
	
	Next, we discuss the cosmological parameter constraints for FASTA. Through BAO parameter measurements approaching 1% accuracy, it shows excellent performance in constraining cosmological parameters. Specifically, in the w_0w_aCDM model, FASTA achieves highly precise parameter constraints, with improvements of 43.75% for w_0 and 54.79% for w_a over SKA1-Mid, and 35.71% for w_0 and 42.11% for w_a over FAST, achieving results comparable to CBS using 21 cm IM  alone.


	Lastly, we analyze FAST with a PAF. We consider two beamforming strategies, a fixed FOV and a fixed N_b. The real situation is likely to fall between these two cases. This provides a basis for understanding the advantages and limitations of different beamforming strategies for 21 cm IM. A balanced beamforming strategy will be beneficial for effectively utilizing PAF.
	
	In summary, these upgrades to FAST have enormous potential to improve the precision of cosmological measurements. Despite facing some technical challenges, they can greatly enhance our understanding of the evolution of the universe and the nature of dark energy.
	
	
	
	We thank Guang-Chen Sun, Ji-Guo Zhang,  Zi-Qiang Zhao, and Lu Feng for helpful discussions and suggestions. We are grateful for the support from the National SKA Program of China (grant Nos. 2022SKA0110200 and 2022SKA0110203), the National Natural Science Foundation of China (grant Nos. 11975072, 11875102, and 11835009), and the National 111 Project (Grant No. B16009).
		
	
	

	


 Paper Title: gwforge: A user-friendly package to generate gravitational-wave mock data.





kbc595@psu.edu


0000-0003-4750-5551]Koustav Chandra
Institute for Gravitation & the Cosmos and Department of Physics, The Pennsylvania State University, University Park PA 16802, USA
















§ ABSTRACT 

The xg gw detectors, with their improved sensitivity and wider frequency bandwidth, will be capable of observing almost every cbc signal from epochs before the first stars began to form, increasing the number of detectable binaries to hundreds of thousands annually. This will enable us to observe compact objects through cosmic time, probe extreme matter phenomena, do precision cosmology, study gravity in strong field dynamical regimes and potentially allow observation of fundamental physics beyond the standard model. However, the richer data sets produced by these detectors will pose new computational, physical and astrophysical challenges, necessitating the development of novel algorithms and data analysis strategies. To aid in these efforts, this paper introduces , a user-friendly, lightweight Python package, to generate mock data for xg detectors. We demonstrate the package's capabilities through data simulation examples and highlight a few potential applications: performance loss due to foreground noise, bright-siren cosmology and impact of waveform systematics on binary pe.






§ INTRODUCTION
 




    
    
    



The xg ground-based gw detectors, such as the ce and the et, promise to significantly improve both the detection horizon and the frequency bandwidth of observable cbc signals <cit.>. This expanded detection horizon will allow us to observe cbc signals from greater distances, thereby increasing the number of detectable binaries by several orders of magnitude. Additionally, the increased bandwidth of these detectors could allow for the observation of a cbc signal over durations spanning minutes to even hours <cit.>. 

Given this, the xg gw detectors will have far-reaching scientific potential. However, the “richer” gw detector data will present computational, physical and astrophysical challenges. Existing gw analysis tools, waveform models and detector calibration techniques will be insufficient in xg era <cit.>. Also, the sheer volume of observed signals will necessitate the development of novel algorithms for detecting the signals and extracting the science out of them. It is, therefore, imperative to quantify the limitations of current tools when employed in xg data analysis. 


To support this effort and ensure that xg detectors will achieve their scientific potential, it is crucial to generate playground datasets to allow the gw community to develop and tune their algorithms. Such an effort will drive the development of new data analysis strategies, such as faster search and pe techniques, improved waveform modelling, dealing with non-stationary noise and methods to identify subtle physical effects, such as finite size effects in ns. 

This paper introduces , a user-friendly Python package to simulate the mock data for the proposed xg detectors based on our understanding of our universe. 

One primary reason for developing is the anticipated abundance of signals in xg data, necessitating systematic data generation that accounts for realistic population models and the complexities of overlapping signals. This code is built on existing gw packages, such as , ,  and .



The subsequent sections of this paper are structured to highlight the user-friendly nature of the code. Sec <ref> provides an overview of the code, including installation instructions in Sec. <ref>. Sec. <ref> provides the necessary theoretical background and an example to generate a day's worth of coloured Gaussian xg network data with cbc signals. Sec. <ref> is devoted to a few studies one can do using the prepared dataset in the previous section, and we conclude in Sec. <ref>.




    



§ CODE OVERVIEW






 §.§ Installation


The source code is available on the repository <cit.>, which includes instructions for contributing to code development. Comprehensive documentation regarding installation, functionality, and user syntax is provided on the project’s website <cit.>. Example setup scripts discussed in this work are available in the repository. 




 §.§ Packages



At the top level, comprises three main packages: , and . The package provides functions essential to generate individual compact binary properties θ based on hyper-parameters Λ that describe the population characteristics. These characteristics are defined by phenomenological population models or detailed physical simulations, such as population synthesis or N-body dynamical simulations generated using population synthesis codes such as Compas <cit.>. integrates all phenomenological population models from  and includes additional distribution functions such as a  and . Users can also utilise outputs from population synthesis simulations to generate their own populations and define cosmological parameters, such as the Hubble constant H_0, the CMB temperature T_CMB, etc., to specify their Universe.

The package provides core functionality for generating either zero-noise or coloured Gaussian noise given a noise psd. It implements current and future ground-based gw detectors, detailing their locations, orientations, and various noise PSDs of existing and future instruments. Users can also implement custom gw detectors akin to  <cit.>. Additionally, this package allows users to introduce data gaps, enabling the simulation of detector downtimes. This is important, as it creates a more realistic scenario, reflecting the potential for missing signals or having signals that are only partially present. By accounting for detector downtimes and data gaps, one can better understand the impact of these factors on gravitational wave detection and analysis, thereby improving the robustness of signal detection and parameter estimation methodologies. 



Finally, the  module provides essential functionalities for injecting or adding simulated gw signals into the data. It supports all standard waveform approximants available via the  package <cit.> and/or those made available via 's waveform plugin.

Future updates will expand this module to include the capability to add non-Gaussian transients, SNe signals, and other gw transients, thereby making the network output more closely “mock” the real data.




 §.§ Executables



To facilitate usability, we developed three key executables using the aforementioned packages: , , and . Each executable operates based on a configuration file tailored to its specific function. Additionally, the  executable integrates all three tools and utilises the high-throughput computing framework HTCondor. This integration significantly accelerates the data generation process and simplifies user interaction by managing all ancillary tasks automatically. 


The following section provides an explicit example of using these executables.





§ SIMULATING A DAY'S WORTH OF MOCK DATA








 §.§ Background






The observed gw strain s(t |θ, I) at a detector I can be expressed as:

    s(t |θ, I) = 1/D_Lℝ[F^I h]

where θ is a vector representing the source's properties,  D_L is the luminosity distance to the source, ℝ denotes the real part, and

    F = F_+^I + iF_×^I

is the antenna response function that depends on the sky location (α, δ) and the polarization angle (ψ). The term h represents the complex binary waveform on a sphere. This waveform can be decomposed as a sum of spin-weighted spherical harmonic modes h_ℓ m, so that the waveform along any direction (ι, φ) in the binary's source frame is given by:

    h = ∑_ℓ≥ 2∑_m=-ℓ^ℓ^-2Y_ℓ,m (ι, φ) h_ℓ m(t |Ξ)

Here, ^-2Y_ℓ,m (ι, φ) are the spin -2 weighted spherical harmonics, and (ι, φ) represent the binary's inclination and azimuth, respectively. The vector Ξ denotes the binary's intrinsic parameters whose dimensionality depends on the binary type. For example, in quasi-spherical bbh, Ξ includes the two component masses m_i and the six spin degrees of freedom χ_i. In contrast, for quasi-circular or non-precessing bns, Ξ represents the component masses m_i,  the tidal parameters Λ_i and the aligned spin components χ_i ·L̂, with L being the binary's orbital angular momentum.

Since the source's total merger rate and redshift distribution determine the number of signals in a given simulated dataset, begins by first estimating the average time interval ⟨Δ t ⟩ between two adjacent signals of the same type given the source population and the cosmological parameters. It assumes that all cbc systems are formed through isolated formation channels involving a common envelope phase. This allows it to derive the redshift distribution of cbc in the source frame as follows <cit.>:

    ℛ(z)    ∝∫_τ_min^∞ℛ_∗ [z_f(z, τ)] P(τ)dτ
       ∝∫_z^∞ℛ_∗ (z_f) P[τ(z, z_f)] dt(z_f)/dz_f  dz_f .

Here, ℛ_∗ is the cosmological star formation rate density, P(τ) is the probability density function of the delay time, z_f is the redshift at which the stellar binary forms, z is the redshift at which the compact binary merges, τ=t(z)-t(z_f) is the delay time and t(z) is the lookback time at redshift z. The proportionality constant is defined such that ℛ(z=0) = ℛ_0 where ℛ_0 is the local merger rate density of the cbc source.

The lookback time,

    t(z) = 1/H_0∫_z^∞dz/(1+z) √(Ω_Λ + Ω_m(1+z)^3),

depends on the Hubble constant, H_0, the dark matter, Ω_Λ and matter density, Ω_m. All these cosmological parameters, along with the star formation rate, are adjustable by the user, allowing them to simulate their own Universe. However, currently only implements P(τ) ∝ 1/τ.

Given this, the merger rate per redshift bin in the detector frame is then given by <cit.>:

    dR/dz = ℛ/1+zdV(z)/dz,

where the comoving volume element is given as:


    dV(z)/dz = c/H_04 π D_L^2/(1+z)^2 √(Ω_Λ + Ω_m(1+z)^3) .



Here, c is the light speed, and D_L is the luminosity distance to the source defined as:

    D_L =  c(1+z)/H_0∫_0^z dz'/√(Ω_Λ + Ω_m(1+z')^3)

The 1/(1+z) factor in Eq. (<ref>) accounts for time dilation.

obtains ⟨Δ t ⟩ as follows:

    ⟨Δ t ⟩ = 1 / ( ∫^z_max_0 dR/dz dz ) .

where z_max is the maximum redshift used in the simulation. Based on ⟨Δ t ⟩ and the total analysis time, T_a of the network, determines the expected number of signals, N = T_a / ⟨Δ t ⟩, and correspondingly draws the binary parameters from the provided population models.


Below, we discuss an example to build a mock catalogue based on our current understanding of bbh binary population distribution. 




 §.§ Simulating the mock GW catalogue


To create a mock bbh catalogue using , one must first define a configuration file like the one provided below:

    [style=bash]
[Redshift]
redshift-model = MadauDickinson
redshift-parameters = `gamma': 2.7, `kappa': 5.6, `z_peak': 1.9
local-merger-rate-density = 22
maximum-redshift = 30
; custom cosmology
cosmology = custom 
H0 = 67.7
Om0 = 0.31
Ode0 = 0.69
Tcmb0 = 2.735
; analysis start time
gps-start-time = 1893024018

[Mass]
mass-model = PowerLaw+Peak
mass-parameters = `alpha':3.37, `beta': 0.76, `delta_m':5.23,  `mmin':4.89, `mmax':88.81, `lam':0.04, `mpp': 33.60, `sigpp':4.59

[Spin]
spin-model = default
spin-parameters = `mu_chi':0.26, `sigma_squared_chi':0.02, `sigma_t':0.87, `xi_spin':0.76

[Extrinsic]
    



Within the  section, users must specify the redshift evolution model, local merger rate density in , analysis start time (in GPS time), and the maximum redshift. Currently, supports two different redshift evolution models: the power-law redshift evolution model and the other that convolves the Madau-Dickinson star-formation rate with an inverse time-delay model to determine the cbc's redshift evolution model <cit.>. Users can define custom cosmological and redshift parameters; otherwise, default values from <cit.> and Madau-Dickinson parameters are used <cit.>.

Similarly, the  and  sections provide options for various phenomenological mass and spin distribution models, as detailed in <cit.>. Besides predefined models, can utilize outputs from population-synthesis codes, creating interpolated distribution functions from variable and associated probability arrays, enabling sampling from simulated distributions.

Similarly, the  and  sections offer a selection of phenomenological mass and spin distribution models, as documented in <cit.>. 

The  section deals with the other extrinsic parameters. Unless otherwise specified, employs the following default distribution choices for these parameters (See Table <ref>) for definitions of the symbols):





Finally, if the binary contains a ns or a pair of ns, can set the tidal parameters Λ_i based on an eos. This necessitates a pre-shipped  table, examples of which can be found at <cit.>. Additional configuration file examples are also provided at the same location.

Once the configuration file is defined, the next step is to execute:

[style=bash]
    gwforge_population –config-file bbh.ini –output-file bbh.h5


to create the mock bbh catalogue and store as in an  file.





 §.§ Mock noise generation



After generating the mock gw catalogue, the next step is to define the detector network and simulate their noise. This can be achieved by creating a configuration file, for example, , as shown below:

    [style=bash]
[IFOS]
detectors = [`CE20', `CE40', `ET']
sampling-frequency = 8192
noise = gaussian
    

and executing:


[style=bash]
    gwforge_noise –config-file xg.ini –output-directory output/data –gps-start-time 1895616018 –gps-end-time 1895656978


This initialises a three-detector network consisting of two L-shaped ce observatories at two fiducial locations: ce40 off the coast of Washington State and ce20 off the coast of Texas, along with an underground triangular et located in Sardinia, Italy <cit.>. The simulated noise thus produced will be Gaussian coloured by the detector noise psd, with data sampled at 8192 Hz over approximately one day. The data will be stored as  files in the  directory, named in the format , with channel names . To simulate zero-noise realisations, the user has to set the flag .

also includes the two LIGO detectors in the US <cit.>, Virgo in PISA <cit.>, LIGO-Aundha in India <cit.> and the KAGRA detector in Japan <cit.>. It is also possible to define new detectors characterised by their geometry, location and frequency response. This feature is particularly useful for developing the science case for new proposals and optimising the design and placement of new detectors, such as placing the et observatory in the Netherlands or building a pair of et observatories at two different locations <cit.>.




 §.§ Mock data generation






The final step involves injecting the simulated gw signals into the network data. Similar to the previous steps, this requires a configuration file, for example, , as shown below: 


[style=bash]
[IFOS]
detectors = [`CE20', `CE40', `ET']
channel-dict = `CE20':`CE20:INJ', `CE40':`CE40:INJ', `ET':`ET:INJ'
sampling-frequency = 8192
minimum-frequency = 6

[Injections]
injection-file = bbh.h5
injection-type = bbh
waveform-approximant = IMRPhenomXO4a



As with the noise generation step, the  section requires specification of the detector network and sampling frequency. Additionally, this section must include the minimum frequency cut-off for signal generation and the channel names for the frame files. If not specified, assumes that the reference frequency for measuring the spin angles is the same as the minimum frequency used for signal generation.

The  section should contain the following keys: , which points to the file containing the mock bbh catalogue (e.g., ); , which specifies the type of injection being made; and , which indicates the waveform model to use for signal generation, with  being an example <cit.>. Currently, supports all waveform approximants available via the  package <cit.>. To use waveforms beyond those available in , a new plugin package must be created to advertise the new waveform model to for waveform generation <cit.>.




To inject the signal waveforms into the generated data over the specified GPS time interval, the user needs to execute the following command:

[style=bash]
gwforge_inject –config-file forged-data/inject-bbh.ini –data-directory /output/data –gps-start-time 1895616018 –gps-end-time 1895656978        



This command injects the signal waveforms into the generated noise over the specified GPS time interval, creating the mock data for further study.




 §.§ Using 




Since the data generation process can be computationally expensive and time-consuming, users are encouraged to take advantage of , which leverages the high-throughput computing framework HTCondor. This significantly accelerates the data generation process by abstracting all other housekeeping logic away from the user. Once the mock catalogues are generated and the different configuration files are created, the following commands can be executed:

[style=bash]
# Activate Conda environment
conda activate gwforge-test ||  echo "Failed to activate Conda environment." >   2; exit 1; 

# Set output directory
output_directory=forged-data

# Workflow submission script
gwforge_workflow     –gps-start-time 1895616018     –gps-end-time  1895702418     –output-directory "output_directory/output" –noise-configuration-file "output_directory/xg.ini"     –bbh-configuration-file "output_directory/bbh-inject.ini" –bns-configuration-file "output_directory/bns.ini"     –nsbh-configuration-file "output_directory/nsbh.ini"  –workflow-name work –submit-now


This will create an HTCondor meta-scheduler to manage dependencies between jobs and systematically parallelize the tasks, accelerating the mock data generation process. Fig <ref> shows 1024s of simulated data when assuming Gaussian noise, including signals from Population-A (See Sec. <ref> for details). As can be seen, the data will never be “signal-free”. 





§ EXAMPLE USAGE OF THE MOCK DATA




 §.§ Foreground Noise


The increased bandwidth of xg detectors will allow low-mass cbc signals to remain within the detector's sensitive bandwidth for hours. Consequently, multiple signals will simultaneously appear in the data stream, the number of which will depend on the sources' astrophysical merger rate density. Characterised by their rapid frequency evolution, these signals will predominantly overlap at low frequencies and “separate” as they approach merger. Therefore, current modelled detection techniques can identify and measure their merger time to𝒪(10ms)accuracy <cit.>. As a result, unlike in LISA, where overlapping signals lead to source confusion, these signals will be resolvable and only contribute to foreground noise.



However, as the number of overlapping signals per time-frequency bin is small, their combined contribution won't obey the central limit theorem. Therefore, the data cannot be assumed to follow a wide-sense stationary Gaussian process — an assumption critical to the currently used noise psd estimation technique such as the Welch averaging <cit.> and the BayesLine method <cit.>. As a result, using these techniques without modification can result in biased psd estimates. 
Such a deviation can introduce systematic biases in parameter inference and reduce cbc detectability.

To demonstrate this, we simulate multiple realisations of 512s mock ce40 data using the  package and estimate the noise psd using the Welch method. Mathematically, this method assumes the data to be wide-sense stationary and divides the discrete time domain data d[j] into K overlapping segments d_k[j] of length L, each windowed by w[j] which for our purpose, is the Hann truncation function. The discrete Fourier transform d̃_k[i] of each windowed segment is computed using an FFT routine, and the periodogram P_k[i] is obtained as

    P_k[i] = 1/U |d̃_k[i]|^2,

where U = ∑_n=0^L-1 w[j]^2. We, following <cit.>, use mean-median averaging of the periodograms over all segments to obtain the final psd estimate. This approach significantly reduces the variance of the spectral estimate due to glitches, thereby providing a more accurate representation of the data's power distribution across frequencies.

We assume two different source populations — population-A quantified by bns and bhns merger rate density ofℛ_BNS^A=320/Gpc^3/yr,  ℛ_BHNS^A=90/Gpc^3/yrrespectively and population-B with merger-rate density ofℛ_BNS^B=1700/Gpc^3/yr,  ℛ_BHNS^B=200/Gpc^3/yr<cit.>. While overlapping bbh signals will lead to larger foreground noise, their lower local merger rate and shorter duration in the detector band suggest that they will rarely contribute to the foreground noise. We therefore useℛ_BBH=30/Gpc^3/yrin both the cases.

Fig. <ref> shows the simulated and the estimated ce40 asd, with the latter obtained using the Welch method. It is estimated by dividing multiple-realisation of 512s of data into 16s subsegments and 50% overlap, akin to that used in a PyCBC search. Consistent with previous works <cit.>, we find that the deviation is mainly concentrated between 6-30Hz, reaching≳10%at 15Hz for the upper merger rate. This can reduce the detectability of bbh with a high detector frame (redshifted) total mass, as shown in Fig. <ref>. In particular, depending on the binary's detector frame mass, the horizon distance [The horizon distance, D_H, for our case, is the farthest luminosity distance to which we can confidently detect an optimally oriented, located, non-spinning symmetric mass bbh assuming an optimal snr threshold of 8.] loss to systems withM_T(1+z) ≳1000M_⊙can be≳15%. Similar to the Welch estimate, the median BayesLine output, which is widely used in gwpe, will also show a deviation, unless a joint spectral and signal estimate is performed with precision (see Appendix <ref> for a discussion).



    

 §.§ Bright-siren cosmology
gw signals from cbc sources can be used as probes for cosmic expansion <cit.>. The luminosity distance,D_Lto a cbc can be directly estimated using the gw waveforms(t|θ) ∝1/D_Lwithout needing external distance calibrators <cit.>. If the cbc's cosmological redshift,z, can be estimated by other means, then one can infer the local expansion rate,H_0and other cosmological parameters <cit.>, such as the fractional energy density of dark energy,Ω_Λand dark matter,Ω_m. For example, observing the em counterparts from GW170817 allowed us to identify the source's host galaxy, NGC 4993, enabling the first direct measurement ofH_0using gw <cit.>.

Another promising em-bright cbc source is bbh mergers occurring in gas-rich environments such as agn disks <cit.>. bbh in such environments can efficiently merge due to gas torques and dynamical encounters, potentially producing detectable em radiation during or after the merger. Such detection is feasible even against bright agn disks, provided the disks are thin and relatively low in luminosity or if the merger remnant is ejected from the optically thick midplane and undergoes super-Eddington accretion.

xg detectors promise to play a pivotal role in inferring cosmological parameters <cit.>. To demonstrate how the mock data can be used to perform bright-siren cosmology, we analyse 8s of xg detector data segment containing a GW190521-like binary that coincides with an em counterpart with{α_em, δ_em, z_em}same as the simulated signal. The data includes Gaussian noise and Population-A binaries, as introduced in the previous subsection. The binary and cosmological parameters used for the analysis are tabulated in Table <ref>. We use the Bayesian pe library  <cit.>, waveform model  <cit.> and off-the-shelf dynamic nested sampler  <cit.>. We use the same waveform model for injection and recovery to prevent waveform systematics. Consistent with the mock data generation set-up, we use a minimum-frequency cutoff off_min=6Hz and perform a flare-unrestricted pe with standard priors <cit.> and psd calculated using the Welch method. 


Fig. <ref> shows the posterior probability distributions for the sky-location of simulated binary. As can be seen, the sky location of the flare and the gw signal are consistent. In fact, the odds of a common source hypothesisC, against a random coincidence,𝒪_C/R > 2 ×10^7, if we assume the prior odds,π_C/R,  to be unity.



Having confirmed such an association, we use the obtained posterior samples to inferH_0<cit.> given the gw and em datasets,d_gwandd_em:

    p(H_0     | d_em, d_gw) ∝ π(H_0) 
       ×ℒ( d_gw|α_em, δ_em, D_L (z_em| H_0) ) 
       ×π_pop(z_em| H_0)/β(H_0) .

Hereπ(H_0)is the prior onH_0,ℒis the marginalised gw likelihood function evaluated at the distance and sky location of the electromagnetic source,π_pop(z_em |H_0)is the assumed redshift distribution of the binary population andβ(H_0)is the fraction of binary population that will be simultaneously detected in gw and em window. 




Fig <ref> shows our results, assuming a flatΛCDM cosmology and matter constraints from <cit.>. We assumeπ(H_0)=𝒰(35, 140)km/s/Mpcand consider two redshift distribution models: one that assumes that bbh merger rate is constant up to redshiftz=10and the other that follows the Madau-Dickinson star-formation rate used to simulate the binary population. For both cases, we first estimateβ(H_0)following <cit.> and then inferH_0to be67.7^+0.36_-0.36km/s/Mpcat 68% CI, thereby being consistent with the value used in simulating the binary population. It shows that if the uncertainty in luminosity distance and the counterpart redshift measurement is small, xgem-bright sources can precisely measureH_0, irrespective of the underlying population distribution assumption.


In Fig <ref>, we present the measurements ofΩ_mandH_0, when using flat priors in the rangeH_0=[35,140]km/s/Mpc,Ω_m=[0,1]and assumingΛCDM cosmology. For this, we obtain the joint posterior on(H_0, Ω_m)by modifying Eq. (<ref>) to the following:

    p(H_0, Ω_m     | d_em, d_gw) ∝ π(H_0, Ω_m) 
       ×ℒ( d_gw|α_em, δ_em, D_L (z_em| H_0, Ω_m) ) 
       ×π_pop(z_em| H_0, Ω_m)/β(H_0, Ω_m) .

Consistent with Fig. <ref>, the maximum likelihood point is close to the true value. However, as expected, the posterior forH_0is broader.


    

 §.§ PE biases due to waveform systematics

gw from cbc carry characteristic information about the astrophysical properties of compact objects, such as their masses and spins. A significant challenge in inferring these parameters is the systematic error introduced by the waveform approximants. If two waveform models,𝒲_1and𝒲_2do not agree for a particular choice ofθ, i.e.δh = h_1(t |θ, 𝒲_1) - h_2(t |θ, 𝒲_2) ↛0, then it can significantly skew pe outcomes. This is especially problematic if the difference exceeds the detector noise level.


More rigorously, if the two waveform models are not faithful up to a particular level determined by the signal's snr, then the  disagreement between the waveform models will lead to differences in pe outputs.  This issue is particularly critical in xg era, where we anticipate observing over10^5compact binary signals per year, few of which can have a snr greater than 100. Since the statistical uncertainty of parameter estimates decreases with an increase in snr, the measurability of signal parameters can be significantly affected if the current waveform models are used, particularly when analysing high snr events <cit.>. Consequently, it is crucial to understand how waveform systematics can lead to incongruous results.

While a detailed study is deferred to future work, we provide an illustrative example of using the simulated mock data to identify biases in pe due to waveform systematics. We note that these studies must be performed in zero-noise so that no systematics are introduced due to noise. Here, we give an example where the noise is Gaussian and includes population-A signals.

To this end, we re-analyse the mock data segment containing the GW190521-like  signal with parameters tabulated in Table <ref> with three different state-of-the-art waveform models, namely ,  and  <cit.>. For each of these analyses, we use , employ standard priors and the standard likelihood function for gw transients <cit.> and evaluate it from a fixed minimum frequency off_min=6Hz identical to the reference frequency choice.  

Fig. <ref> shows a corner plot comparing a few binary parameters. As can be seen, depending on the waveform approximant, the recovered parameters can be significantly different from the injected value. This is because these waveform models differ in their construction; therefore, measurements relying on them can have systematic uncertainties. This has not been an issue for almost all second-generation gw candidates, as differences caused by different approximant choices have been lower than those induced by noise fluctuations. But for loud xg events, this can introduce significant biases as the statistical uncertainty of parameter estimates decreases with an increase in snr <cit.>. Standard approaches to account for modelling errors, such as combining posterior for analysis performed using different approximants to yield a joint posterior that is effectively marginalised over the uncertainty inherent in the waveform models, won't alone be able to account for the errors and biases that occur during the recovery of source parameters due to waveform modelling choices.




 §.§ Other possible challenges


The data generated using  can be used to address several other challenges, some of which are summarised below:

  * Testing faster and more robust search and pe algorithms: The xg detectors are expected to observe hundreds of thousands of binary signals annually. Developing optimal signal identification methods and rapid parameter estimation algorithms tailored to compact binary analysis will be essential for extracting science from the signal-rich dataset. In this context, the traditional requirement of achieving a ≳ 5σ confidence level may no longer be necessary. Instead, search design requirements can be adapted to enable faster detection and pe algorithms can be developed and/or tuned for faster and more precise estimation of signal properties.

    Moreover, non-stationary noise due to glitches and foreground noise will necessitate a joint analysis of the signal of interest and the remaining data. In fact, current search techniques can be replaced by coherent Bayesian search algorithms that will allow for the identification of potential candidates, give rapid, fully coherent parameter estimates of the signal source, and assign astrophysical probabilities of the candidate events <cit.>. The simulated datasets will allow the gw community to develop, test, and refine algorithms to meet these scientific objectives.


    
  * Measurement of nseos:gw from binaries consisting of a pair of ns, or a ns and a bh can be used to measure the tidal deformability of ns and constrain the nature of matter at supernuclear densities <cit.>. Given that these effects appear in the fifth pn order, this will be best measured for signals with snr of order thousands, akin to those expected to be observed in xg era. The  dataset will contain such signals depending on the initial conditions and can be used to verify how precisely and accurately one can measure the nseos. Further, it can be used to quantify any observed bias due to the foreground noise. Also, depending on the waveform model used, the post-inspiral portion of the signals may also be present in the simulated dataset which can be used to study how well we can constrain the presence of phase transitions at several times the nuclear densities <cit.>.        

    
  * Testing rapid pre-merger sky localisation algorithms: Low latency detection and pre-merger localisation of potential bns signals, especially those within the reach of electromagnetic observatories, will be an important scientific goal for xgcbc algorithms as they can allow for a multiband understanding of bns. The  dataset can be used to mimic online analysis. It will allow the community to combine multi-messenger observation of future mergers to more precisely determine the nseos and learn about the engine powering gamma-ray bursts, kilonovae and the formation of heavy elements.     
    
    
  * Estimate the strength of the stochastic background due to compact binary population: The aggregate of all sources, particularly binary neutron stars will contribute to foreground noise which could interfere with the study or the establishment of upper limits on the cosmological gw background. Therefore, it is crucial to either remove or account for this foreground to measure the properties of a primordial background accurately.  datasets will facilitate extensive testing of xg algorithms on large and realistic samples.

    
  * Investigating Deviations from gr and False Violations: allows injecting non-GR signals through a  plugin, enabling users to introduce and analyze these signals. This feature facilitates studies on the efficacy of conventional tests in detecting deviations from GR. Additionally, users can inject GR signals to investigate scenarios where tests of GR may fail. Such failures could arise due to waveform systematics, sub-optimal detector noise modelling, and various astrophysical aspects. By examining both non-GR and GR signals,  dataset will allow us to perform mock tests of GR in xg era.

    
  * Determining Merger Rates and Reconstructing the Underlying Distribution of Source Populations: Given the anticipated high number of detections, utilizing current hierarchical Bayesian inference techniques to determine the population properties of a cbc type may become impractical. However, the sheer volume of signals allows for an alternative approach: multiplying the posteriors of individual events of a particular source type to infer population properties. The  dataset will enable the validation of this hypothesis and facilitate the testing of faster and more accurate Bayesian inference techniques.

    



    




§ CONCLUSION


In the upcoming decades, xg detectors are expected to replace the current gw detection facilities. With longer baselines and new infrastructures, these detectors can make transformative discoveries across astronomy, physics and cosmology. However, the “richer”gw detector data will pose new computational, physical and astrophysical challenges, which may not be straightforward to mitigate. 

For example, the sheer volume of signals will require developing novel signal identification and extraction algorithms. Non-stationarity due to foreground noise will pose additional challenges, especially in noise modelling, complicating subsequent analysis.  Furthermore, detecting loud signals will necessitate the development of improved waveform models to achieve unprecedented precision in parameter estimations and avoid false indications of deviations from gr. 

To simulate these challenges, we introduce , a user-friendly and lightweight Python package designed to generate mock data for xg detectors. Through various data simulation examples, we demonstrate the capabilities of . These examples include examining foreground noise from overlapping signals with intersecting time-frequency tracks, demonstrating the potential of bright sirens forH_0measurement, and discussing parameter estimation biases due to waveform systematics. We also highlight a few potential challenges that can be addressed using the simulated dataset. These demonstrations underscore the potential applications of  in supporting the development of new analysis methods and preparing the scientific community for the next era of gravitational wave astronomy.




§ ACKNOWLEDGEMENTS

The author thanks B.S. Sathyaprakash, Ian Harry, Rossella Gamba and  Ish Gupta for their insightful comments and valuable suggestions. He extends special thanks to Sophie Hourihane for her assistance with BayesLine. Heartfelt appreciation is also given to Lorine Talhaoui, Tobias Forge, Papa Nihil, and N. Ghouls for their continuous inspiration throughout the development of this work. 

The author acknowledges the generous support of the National Science Foundation (NSF) through grant PHY-2207638. He is also grateful for the computational resources provided by the Gwave (PSU) cluster, which were essential for the numerical work conducted in this research.

The  package extensively utilises the following  packages:  <cit.>,  <cit.>,  <cit.>, and  <cit.>. Furthermore, various Python packages such as  <cit.>,  <cit.>,  <cit.>, and  <cit.> are used in this work.




§ BAYESLINE PSD ESTIMATE



The Welch method for estimating noise power spectral density (PSD) generates an “off-source” noise PSD by median-averaging the spectral estimates of neighbouring data segments. However, it is not widely used in gwpe due to its sensitivity to non-Gaussian transients or “glitches”. Moreover, even for wide-sense stationary Gaussian noise, the data whitened by the Welch method doesn't produce a𝒩(0,1)distribution, leading to biased pe estimates <cit.>.

Hence, current lvk analyses use the “on-source” Bayesian spectral estimates, such as those provided by the BayesLine algorithm <cit.>. This method approximates the noise psd by generating a parameterised fit involving a combination of splines and Lorentzians. This has yielded unbiased and statistically consistent results, with whitened residuals that follow a𝒩(0,1)distribution. 

Furthermore, this methodology, originally developed for joint spectral and signal inference, will be indispensable in the upcoming xg era, where no signal-free off-source data will be available. However, current lvk analysis uses sequential modelling, which may lead to notable discrepancies from joint analyses as detector sensitivities improve. This discrepancy becomes more pronounced, particularly for ground-based xg detectors, as the sum of foreground noise won't follow a Gaussian distribution.


To illustrate that, we re-compute the noise psd of a 512s segment of data as discussed in Sec. <ref> using the BayesLine algorithm. Fig. <ref> shows our result. Like the Welch estimate, the foreground noise significantly affects noise characterisation at lower frequencies, with a more pronounced impact observed for population-B data, which assumes upper merger rates for bns and bhns systems. This can affect pe results, potentially broadening constraints on parameters like chirp mass (ℳ_c) and/or symmetric mass ratio (η) for longer signals while not significantly affecting the measurability of parameters appearing at higher pn orders (see <cit.> for a possible way to mitigate this). However, this may substantially impact binary parameter measurements for short-lived signals, such as imbh binaries. A detailed investigation is deferred to future work.







aasjournal

 Paper Title: Spinning waveforms in cubic effective field theories of gravity.
































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        ^aCentre for Theoretical Physics, Department of Physics and Astronomy, 
        
        Queen Mary University of London, Mile End Road, London E1 4NS, United Kingdom^bNiels Bohr International Academy,
        Niels Bohr Institute, University of Copenhagen,
        
        Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmarka.brandhuber@qmul.ac.ukgraham.brown@qmul.ac.ukgang.chen@nbi.ku.dkg.travaglini@qmul.ac.ukp.vivesmatasan@qmul.ac.uk
        
        	QMUL-PH-24-11
        
        
        We derive analytic all-order-in-spin expressions  for  the leading-order time-domain waveforms  generated in the scattering of two Kerr black holes with arbitrary masses and spin vectors in the presence of all independent cubic deformations of Einstein-Hilbert gravity. These are  the  two parity-even interactions I_1 and G_3, and  the parity-odd ones 
        Ĩ_1 and G̃_3. 
        Our results are obtained using three independent methods: a particularly efficient  direct integration and tensor reduction approach; integration by parts combined with  the method of differential equations; and finally a residue computation. 
        For the case of the G_3 and G̃_3 deformations we can express the spinning waveform  in terms of the scalar waveform  with appropriately shifted impact parameters, which are reminiscent of Newman-Janis shifts. For I_1 and Ĩ_1 similar shifts occur, but  are   accompanied by additional contributions that cannot be captured by simply shifting the scalar I_1 and Ĩ_1 waveforms. 
        We also show the absence of leading-order corrections to gravitational memory.  
         Our analytic results are  notably compact,  and we compare the effectiveness of the three methods used to obtain them.  We also briefly comment on the magnitude of the corrections to observables due to cubic deformations.  
        
        
        
        
        
        Spinning  waveforms in cubic effective field theories of gravity 
        
            and Pablo Vives Matasan^a
            Submitted August 5, 2024
        ==================================================================
        
        
        
        § INTRODUCTION
        
        
        
        The high accuracy achieved in current observations of gravitational waves by the LIGO-Virgo-KAGRA collaboration, along with the potential of further  improvements  in upcoming experiments such as LISA, strongly motivates the pursuit of 
        increasing precision in theoretical calculations.
        One is then faced with the need to perform higher-loop calculations in Newton's constant 
        G, incorporating  spin effects, and also considering    potential modifications to the Einstein-Hilbert theory arising from as yet undiscovered higher-dimensional interactions. 
        
        
        
        Quadratic corrections in the curvatures are known to leave scattering amplitudes invariant <cit.>, hence the first  deformations to be studied appear at dimension six, that is they are cubic in the curvatures (see <cit.> for work on dimension-eight operators). 
        Specifically, we consider the deformations 
        
            I_1    R^αβ_μνR^μν_ρσR^ρσ_αβ ,      
            I_2 R^μνα_βR^βγ_νσR^σ_μγα .
        
        However,  instead of    I_2,  we  prefer working with  the combination 
        
            G_3  I_1 - 2 I_2  ,
        
        which is a topological term in six dimensions and has vanishing four-dimensional graviton amplitudes. 
        We will also study the effect of  the parity-odd couplings Ĩ_1 and G̃_̃3̃, which are obtained from the parity-even ones by replacing one of the Riemann curvatures R^μναβ by the dual R̃^μναβ=  
        (1/2)  ϵ^μνρσR_ρσ^αβ.
        Summarising, the effective action we will work with is 
        
            S = ∫d^4x  √(-g) (    -2/κ^2R      +  β_1 I_1 + β_2 G_3 + β̃_1 Ĩ_1 + β̃_2 G̃_3)  .
        
        Particular choices of the coefficients β_1, β_2, β̃_1, β̃_2  correspond to specific theories
        [For instance β_1 = -2/κ^2 α^'  2/48,  β_2 = -2/κ^2 α^'  2/24, β̃_1=β̃_2=0 reproduces the low-energy effective action of bosonic string theory.], but we will be agnostic and treat each deformation independently.  
        
        Cubic  deformations  have been studied  
        both using  general relativity approaches <cit.>, as well as 
        amplitude methods applied to the case of spinless binaries <cit.>, or systems where only one black hole is spinning and has a  much larger mass than the other black hole  <cit.>.
        Related work on other modified theories of gravity include <cit.>. We also note that in <cit.> a bound of ℓ_ EFT≤ 38.2 km was determined for the fundamental length scale of cubic theories, specifically for the  case where β_1 =β̃_1, β_2 =β̃_2=0. Furthermore,   <cit.> investigated the potential  detectability of cubic deformations of gravity in a gravitational-wave event from the merger of two stellar-mass black holes. 
        
        In a typical encounter, we expect both celestial bodies to have non-zero spin vectors, along with arbitrary masses, and this is the scenario we considered in <cit.>. 
        In that paper we focused   on  elastic processes, related to the two-to-two scattering amplitude of Kerr black holes,  and   using the KMOC formalism <cit.> we computed the leading-order  impulse (or momentum kick) and spin kick in a generic   hyperbolic encounter. These observables  vanish at tree level, and hence a one-loop calculation was required. 
        For the sake of gravitational-wave observations, the most relevant quantity to compute is the waveform, and this is what we  focus on in this paper. 
        
        
        Waveforms for hyperbolic spinning encounters in general relativity were derived to leading order using a worldline quantum field theory approach in <cit.> for the spinless case, reproducing the classic results of Kovacs and Thorne <cit.>. This was later extended to include spinning bodies in   <cit.>, with results valid  up to quadratic order in the spin. Results for the scattering waveform of spinless bodies at next-to-leading order were later derived using amplitude approaches in <cit.>,  while for spinning objects, 
        leading-order expressions 
        valid to high order in the spins were obtained in <cit.>, also using amplitude methods,   which could readily be extended to any spin order once the relevant Compton amplitudes are available. 
        
        
        In order to derive the waveforms  we will follow the adaptation of the KMOC approach presented in <cit.>. In that paper it was shown that, to leading order in the coupling, gravitational  waveforms can be  expressed as a Fourier transform to impact parameter space of the five-point amplitude describing the scattering of two spinning celestial objects with the emission of a graviton. In fact, only the physical singularities of this amplitude contribute to the waveform; as we approach the poles, the five-point amplitude factorises onto a four-point Compton amplitude and a three-point amplitude of two spinning objects with one graviton. 
        Compact expressions for  the Compton amplitude in the presence of cubic deformations were derived  in <cit.>, and we will use them to derive all-order in spin expressions for the tree-level waveforms of two Kerr black holes in a cubic background.
        
        
        After constructing the integrand from factorisation, we are left with the task of taking its Fourier transform. We will do this using three different approaches, which will give us the possibility of comparing their own merits. The first approach  is that of direct integration, building on  the seminal work of <cit.>.
        In this method, it is possible to write down ansätze for certain tensor (i.e. not necessarily scalar) integrals which are invariant under rescaling of the integration variable. Such integrals often contain poles on the integration contour and, for the rescaling argument to be valid, they  must  be regularised with a principal value prescription. This is possible  since  additional contributions arising from the   i ε prescriptions 
         are free of physical poles and hence cancel, as we have checked explicitly.  
         Higher-rank tensor integrals can then be produced efficiently with a generating function technique which is applicable thanks to the presence of a Fourier transform to impact parameter space. Effectively, integrals with  (loop-momentum dependent) numerators can be obtained by differentiating a master integral with a shifted impact parameter with respect to certain scalar auxiliary variables. This approach avoids  differentiation with respect to   four-momenta as well as lengthy Lorentz contractions, thereby leading to very compact results. 
        
         A second independent approach is based on integration by parts (IBP) reductions and the method of differential equations <cit.>. The waveforms corresponding to I_1 and its parity-odd version Ĩ_1
        contain spurious poles of the form (sinh x)/x, which we avoid by rewriting this function in an integral representation that is well suited for IBP reductions.  As a consequence, the Fourier transforms in the I_1 or Ĩ_1 cases are reduced to a single  simple master integral. We also note that this approach is systematic and  can be extended to one-loop waveforms with or without spin.  Finally, a third approach we  pursued employs Cauchy's residue theorem. This is straightforward to implement and highly  efficient, though it generally produces less compact  analytic results  compared to  the   other two methods, with  direct integration   yielding  the most compact results. 
        The three approaches give results in 
        perfect agreement. 
        
        Remarkably, we are able to write the spinning waveforms in terms of  Fourier transforms to impact parameter space where most (and, for the G_3 and G̃_3 cases, all)
        of the spin dependence is encoded in spin-dependent shifts of the impact parameter. These  shifts are reminiscent of the Newman-Janis shift,  which made an appearance  in amplitude contexts in <cit.>,  
        and more recently in the computation of spinning waveforms in <cit.>. 
        For the G_3 and G̃_3 waveforms, we are able to write the spinning waveforms entirely in terms of non-spinning ones (with shifted impact parameter). 
        
        The case of waveforms in the presence of parity-odd deformations is very easy to discuss and in fact does not require new calculations: parity-odd waveforms can be obtained from parity-even ones by simply swapping the “plus” and “cross” polarisations.  
        Investigating parity-odd deformations of gravity  is not merely an academic exercise, given the early 
         indications of parity violation in  the Cosmic Microwave Background and  the  
        large-scale structure of galaxies <cit.>.  
        
        A feature of all the leading waveforms in the presence of cubic deformations is that they do not modify the gravitational memory, which is easily proven using the connnection between the memory and soft theorems <cit.>. 
        
        
        Our  waveform results can be found in our   
        https://github.com/QMULAmplitudes/Cubic-Corrections-to-Spinning-Observables-from-Amplitudes Cubic Corrections to Spinning Observables from Amplitudes GitHub repository.  
        
        The rest of the paper is organised as follows. 
        In Section <ref> we summarise the expressions for the classical Compton amplitudes derived in <cit.>. 
        In Section <ref> we present the relevant formulae to derive the waveforms from  the factorisation diagrams of the five-point amplitudes. We also construct these factorisations using the classical Compton amplitudes and the classical three-point amplitudes of two spinning particles and one graviton. 
        In Section <ref> we present the direct integration method of the waveform, inspired by the work of <cit.>. The method is illustrated in great detail in the Appendices, while in this section  we use the results of the integrals to derive the  final expressions for the  waveforms with  G_3 and I_1 deformations. 
        In Section <ref> we present our second alternative derivation of the waveforms using a  systematic method proposed in <cit.> for the tensor integral generating functions <cit.>.    
        A third derivation of the waveforms is then briefly shown  in 
        Section <ref> using residues, much in line with our previous work <cit.>. Section <ref> illustrates the waveforms for various values of the relative velocities of the black holes and their spins, for the G_3 and I_1 deformations. These waveforms  clearly  show the absence of a  contribution to the gravitational memory, which we   demonstrate  in 
        Section <ref>. 
        Finally, in 
        Section <ref>  
        we conclude by comparing the size of cubic corrections to that of Post-Minkowskian (PM) corrections. 
        
        
        Two Appendices deal with the evaluation of  the  integrals needed to compute our waveforms. Specifically, in 
        Appendix <ref> we evaluate the necessary master integrals using a combination of explicit evaluations and educated guesses; and in 
        Appendix <ref> we present the generating function technique described earlier, which allows to compute with great economy all the necessary higher-tensor integrals. 
        
        
        
        
        § SUMMARY OF CLASSICAL SPINNING COMPTON AMPLITUDES WITH CUBIC DEFORMATIONS
        
        
        
        In this section we briefly review the Compton amplitudes in the presence of parity-even and parity-odd cubic deformations derived in <cit.>. 
        We denote by k_i and ε_k_i, i=1,2, the momenta and polarisations of the emitted gravitons, while m and a are the mass and ring radius of the black hole, with p being the classical momentum.  
        
        
        
         §.§ Parity-even deformations
        
        For the parity-even cubic deformations, the classical Compton amplitudes 
        in the gauge k_1 ε_k_2 = k_2 ε_k_1 =0 
        are found to be
        [From now on,   we will omit the coefficients of the cubic interactions β_1, β_2, β̃_1, β̃_̃2̃, which  can  be reintroduced at the end of the calculations. Furthermore, we  note that in our normalisations Newton's  constant is defined as   G κ^2 /(32π).]<cit.>
            M_I_1(p, k_1, k_2)    = i ( κ/2)^4   24 (k_1  k_2)^2 (ε_k_1ε_k_2) 
                {cosh ( a q ) [ 2 (pε_k_1) (pε_k_2) - m^2 (ε_k_1ε_k_2)  ]
                -i sinh (a q)/a  q[ 
            (pε_k_1 ) ϵ( ε_k_2 p a q) +(pε_k_2 ) ϵ( ε_k_1 p a q) 
            ] }  ,
        
            and
        
            M_I_2(p, k_1, k_2)    =
                i   ( κ/2)^4 6   (k_1  k_2)  (ε_k_1ε_k_2)^2  
                {cosh ( a q ) [ ( p (k_1 - k_2))^2 + m^2 (k_1  k_2)   ]
                -2i  sinh (a q)/a  q p (k_1 - k_2) ϵ( k_1 k_2  p a)  
            ] }  ,
        
            while 
            
            M_I_1(p, k_1^±±, k_2^∓∓)= M_I_2(p, k_1^±±, k_2^∓∓)=0 
                     ,
        
                where 
                q=k_1 + k_2.
                
        The amplitudes  (<ref>) and (<ref>) can also be rewritten in spinor-helicity variables:
        [Our spinor conventions are the same as in  <cit.>.]
            M_I_1(p, k_1^++, k_2^++)    = 
                i( κ/2)^4 3! [1  2]^4/q^2{ - 4 cosh(a q)   (p k_1)(p k_2) 
                +1/2p(k_1-k_2) sinh (a q)/a  q( [1|p|2⟩ [2|a|1⟩ - [2|p|1⟩ [1|a|2⟩) 
             }  ,
        
            and 
            
            M_I_2(p, k_1^++, k_2^++)    = i/2( κ/2)^4   3! [1  2]^4/q^2{cosh (a q) [ ( p (k_1 - k_2))^2 + m^2 (k_1  k_2)   ]
                        
               +1/2p(k_1-k_2) sinh (a q)/a  q( [1|p|2⟩ [2|a|1⟩ - [2|p|1⟩ [1|a|2⟩)  } .
        
        In the following we will focus on the G_3 deformations instead of I_2. The corresponding classical Compton amplitude in spinor-helicity variables is  
        
            M_G_3(p, k_1^++, k_2^++) = -3   i m^2  ( κ/2)^4 cosh (a q) [1  2]^4   
                .
        
        Note that this amplitude has no term proportional to (sinh a q) /(a q).
        
        
        
         §.§ Parity-odd deformations
        
        
        Finally, as discussed in <cit.>
        the amplitudes with parity-odd deformations can be obtained easily from the parity-even  ones:  for the case of positive (negative)  helicity, they are equal to those for the corresponding parity-even case multiplied by a factor of  +i (-i), and we recall that the only non-vanishing amplitudes are those where the two gravitons have the same helicity.  
        
        
        
        
        § CONSTRUCTING THE TIME-DOMAIN WAVEFORMS
        
        
         §.§ General results
        
        
        The derivation of the time-domain waveforms using the KMOC formalism was presented  in <cit.>.  
        We will work in the  far-field limit, that is at large observer distance r|x⃗|  and time t with fixed retarded time u t-r.  
        The quantity of interest  is then 
        
            ⟨ h_+± i h_×⟩   ⟨ h_μν^ out⟩^μν_(±±)1/4π r (h_+^∞± i h_×^∞)   ,
        
        for which one finds that
        [See for example Section 5 of <cit.> for a detailed derivation.]
            h^∞_+± i  h^∞_×  =    
              
              κ∫_-∞^+∞dω/2π e^-i ω u 
             [ 	 
            		 
               θ(ω) .W(b, k^±)  |_k= ω (1, 𝐱̂)
             +   
             θ(-ω) . W^∗(b,k^∓) |_k= -ω (1, 𝐱̂)]
            		,
        
        where  W=W(b⃗, k; h)  is the spectral waveform for the emission of a graviton of momentum k and helicity h, 
        which satisfies,  at tree level, 
            
            .    W^∗ (b, k^h)|_k=-ω(1, 𝐱̂) = 
            .    W(b, k^-h)|_k=ω(1, 𝐱̂)  .
        
        Using this result, one can easily combine  the two terms in (<ref>) into 
        
            h^∞_+± i  h^∞_× = 
            κ∫_-∞^+∞dω/2πe^- i ω u. 
            W(b, k^±)|_k=ω(1, 𝐱̂)  .
        
        At leading order (tree level) this can be neatly obtained from the  five-point classical bremsstrahlung amplitude ℳ_5(q_1, q_2, a_1, a_2; h) describing the scattering of two objects with ring radii a_1 and a_2 with the emission of a graviton with momentum k= q_1 + q_2 and helicity h: 
        
            W(b, k^h )   -i ∫dμ^(4)  e^i(q_1 b_1 + i q_2  b_2)  ℳ_5(q_1, q_2, a_1, a_2; h)   ,
        
        where  
        
            dμ^(D)d^Dq_1/(2π)^D-1d^Dq_2/(2π)^D-1  (2π)^D  δ^(D) (q_1 + q_2 - k) δ(2  _1 q_1 ) δ(2  _2 q_2 )  ,
        
        and   q_1,2= p_1,2- p_1,2^' are the momentum transfers. The barred variables 
        <cit.>  are defined as 
        
            p_1    = p̅_1 + q_1/2  ,      p_1^' = p̅_1 - q_1/2  , 
            
            		p_2    = p̅_2 +  q_2/2  ,      p_2^' = p̅_2 - q_2/2  ,
        
        and satisfy  
            p̅_1 q_1 =p̅_2 q_2 =0   .
        
        However, at leading order we can actually drop the distinction between p̅_i and p_i, hence we will henceforth drop bars. Furthermore, as in <cit.>,
        in order to simplify the notation  we will  work with the quantity 
        
            h^∞(u)    κ∫_-∞^+∞dω/2πe^- i ω u. 
            W(b, k)|_k=ω(1, 𝐱̂)
               = -iκ∫_-∞^+∞dω/2πe^- i ω u∫d^4q_1/(2π)^2δ(2  p_1 q_1 ) δ(2  p_2 (k-q_1) )  e^i(q_1 b_1 + q_2  b_2) ℳ_5   ,
        
        where we omitted the helicity dependence, 
        performed the q_2 integration using momentum conservation from (<ref>)
        and defined
        
            kωk̂ =ω(1, 𝐱̂)  .
        
        This is the main formula we will use. We also note that b b_1 - b_2 is the asymptotic impact parameter, which  satisfies 
        
            b v_1 = b v_2 = 0  .
        
        
         §.§ Kinematics of the five-point scattering
        
        
        The time-domain waveform (<ref>) requires knowledge of the classical part of the five-point bremsstrahlung amplitude, that is the amplitude of two spinning particles with  one emitted graviton. 
        The kinematics of the process is shown below: 
        
            [ [scale=12,baseline=([yshift=-1mm]centro.base)]
            			00 at (0+,0+) (centro) ;
            			 at (-3pt+,-3pt+) (uno) p_1;
            			 at (-3pt+,3pt+) (due) p_2;
            			 at (3pt+,3pt+) (tre) p_2^';
            			 at (3pt+,-3pt+) (quattro) p_1^';
            			 at (5.55pt+,) (cinque)
            			
            			k= q_1 + q_2;
            
            			 [thick,double,orange] (uno) – (centro);
            			 [thick,double,amber] (due) – (centro);
            			 [thick,double,amber] (tre) – (centro);
            			 [thick,double,orange] (quattro) – (centro);
            			 [vector,double] (cinque) – (centro);
            
            			 [->] (-2.8pt+,-2pt+) – (-1.8pt+,-1pt+);
            			 [<-] (2.8pt+,-2pt+) – (1.8pt+,-1pt+);
            			 [<-] (-1.8pt+,1pt+) – (-2.8pt+,2pt+);
            			 [->] (1.8pt+,1pt+) – (2.8pt+,2pt+);
            			 [->] (2.3pt+,-0.6pt+) – (3.75pt+,-0.6pt+);
            
            			 [shading=radial] (centro) circle (1.5pt);                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   ]
        
        The  two Kerr black holes  have  masses m_1 and m_2  with 
        
            p_1^2 = (p_1^')^2 = m_1^2  ,       p_2^2 = (p_2^')^2 = m_2^2  .
        
        It is also convenient to  introduce four-velocities v_1 and v_2 as
        
            p_1  m_1 v_1  ,       p_2  m_2 v_2  ,
        
        so that  v_1^2 = v_2^2 = 1. The two black holes have ring radii a_1 and a_2, where for a single black hole we recall that the ring radius a^μ is related to the spin vector as a^μ S^μ/ m <cit.>, where 
        
            S^μ = 1/2 mϵ^μναβ p_ν S_αβ ,
        
        with S^μν being the spin tensor. 
        The ring radius satisfies the covariant spin supplementary condition    p a(p) =0. 
         Finally, we also define the Lorentz factor
        
            σ v_1  v_2  ≥ 1 .
        
        Note that σ=1/√(1-ẋ⃗̇^2), with  ẋ⃗̇ being the relative velocity of one of the two black holes  in the rest frame of the other. For instance, in  the rest frame of particle 1 we can write  v_1^μ=(1,0,0,0) and   v_2^μ=σ(1, ẋ⃗̇). 
        
        
        
         §.§ Waveform integrands
        
        
        An important simplification  in the calculation of the waveforms  consists in the fact that only the residues on the physical factorisation channels  are needed in order to obtain  the waveform. These can  be computed from the two factorisation diagrams 
        
            [baseline=([yshift=-0.4ex]current bounding box.center)]every node=[font=]
            		 (p1) 1;
            			 [above=1.5cm of p1](p2)2;
            			 [right=1.5cm of p2] (u1) [HV]H;
            			 [right=1.2cm of u1] (p3)2^';
            			 [right=1.5cm of p1] (b1) [dot];
            			 [right=1.2cm of b1] (p4) []1^';
            			 [above=0.4cm of p1] (cutLt);
                         [right=0.6cm of cutLt] (cutL);
            			 [right=1.8cm of cutL] (cutR)k;
            			 [right=0.5cm of b1] (cut1);
            			 [above=0.2cm of cut1] (cut1u);
            			 [below=0.2cm of cut1] (cut1b);
                        [vector, double] (u1) – (b1); 
            			* 
            			(p2) – [thick, amber, double] (u1) – [thick, double, amber] (p3),
            			(b1)–[opacity=0,momentum=q_1](u1),  (u1)– [vector, double] (cutR), (p1) – [thick, orange, double] (b1)– [thick, orange, double] (p4), (cutL)–[dashed, red,thick] (cutR),
            			;
            		      [baseline=([yshift=-0.4ex]current bounding box.center)]every node=[font=]
            		 (p1) 1;
            			 [above=1.5cm of p1](p2)2;
            			 [right=1.5cm of p2] (u1) [dot];
            			 [right=1.2cm of u1] (p3)2^';
            			 [right=1.5cm of p1] (b1) [HV]H;
            			 [right=1.2cm of b1] (p4) []1^';
            			 [above=1.1cm of p1] (cutLt);
                         [right=0.6cm of cutLt] (cutL);
            			 [right=1.8cm of cutL] (cutR)k;
            			 [right=0.5cm of b1] (cut1);
            			 [above=0.2cm of cut1] (cut1u);
            			 [below=0.2cm of cut1] (cut1b);
                        [vector, double] (b1) – (u1); 
            			* 
            			(p2) – [thick, amber, double] (u1) – [thick, amber, double] (p3),
            			(u1)–[opacity=0,momentum'=q_2](b1), (b1)– [vector, double] (cutR), (p1) – [thick, orange, double] (b1)– [thick, double, orange] (p4), (cutL)–[dashed, red,thick] (cutR),
            			;
        
        which  correspond to the two possible factorisations as q_1^2→ 0 or q_2^2→ 0, with q_1 + q_2 = k. 
        In each diagram, the key ingredients are the Compton amplitudes for two spinning objects, which were computed for the parity-even and parity-odd cubic deformations  in <cit.> and are quoted in Section <ref>. Using these, we now construct the integrands for the  parity-even and parity-odd cubic deformations of gravity. 
        
        For simplicity, from now on we focus on the case of an emitted graviton with positive helicity. The two factorisation diagrams are then given by 
        
            _I, q_1^2    = i/q_1^2 M_3(-p_1, q_1^–) M_I(-p_2, - q_1^++, k^++)    , 
            _I, q_2^2    = i/q_2^2 M_3(-p_2, q_2^–) M_I(-p_1, - q_2^++, k^++)   ,
        
        with  
        
            _5, I  = _I, q_1^2+ _I, q_1^2 +  contact  terms  .
        
        Here  M_I denotes  the Compton amplitude in the presence of a deformation I,  with I∈ (I_1, G_3, Ĩ_1, G̃_3), as given in Section <ref>. 
        M_3 is  
        a classical three-point amplitude with one graviton of momentum q and two massive 
        spinning particles with momenta p and -(p+q) and ring radius a. For the two possible helicities of the emitted gravitons one has <cit.>
            M_3 (p, q^++) = -i ( κ/2) e^a q(
                ⟨ξ | p | q]/⟨ξ q⟩)^2  ,      
                M_3 (p, q^–) = -i ( κ/2) e^-a q(
                ⟨ q  | p | ξ ]/[ q ξ] )^2  ,
        
        where ξ and ξ̃ are reference spinors. 
        Finally, we observe that the second factorisation diagram in (<ref>) can be obtained from the first as 
        
            _I, q_2^2 = ._I, q_1^2|_1↔2  ,
        
        where 1↔2 means swapping  the masses, spin vectors and momenta, in addition to also exchanging q_1 and q_2.
        
        
        With these ingredients it is very easy to construct the integrand of the waveforms. For the case of the G_3 deformation,  and choosing as reference spinor of the three-point amplitude the spinor of the external graviton, we get 
        
            _G_3, q_1^2 = - i (κ/2)^53  m_1^2 m_2^2/q_1^2  [k| v_1 q_1 | k]^2  e^- a_1  q_1cosh( a_2 (q_1 - k) )  .
        
        For I_1, the result is
        
            _I_1, q_1^2   =  -i (κ/2)^53  m_1^2 m_2^2/q_1^2[k| v_1 q_1 | k]^2/ q_1 ke^- a_1  q_1×
               [ - 4 cosh [ a_2 (k-q_1)] (-v_2 q_1) (v_2  k) - 2 i  v_2  (k+q_1) sinh a_2 (k-q_1)/a_2 (k-q_1)ϵ ( k v_2 q_1 a_2)]  .
        
        Note that because of the first  δ-function in (<ref>)
        we can simplify the above expression using v_1 q_1=0 into 
        
            _I_1, q_1^2   = -  i (κ/2)^512    m_1^2 m_2^2/q_1^2  (q_1 k)  
              [k| v_1 q_1 | k]^2(v_2  k )  e^- a_1  q_1×
               [   cosh [ a_2 (k-q_1)]  (v_2  k) -  i sinh a_2 (k-q_1)/a_2 (k-q_1)ϵ ( k v_2 q_1 a_2)]  .
        
        
        § TIME-DOMAIN WAVEFORMS FROM DIRECT INTEGRATION 
        
        
        There are several approaches to explicitly perform the integrations in (<ref>). 
        In this section  we follow  the direct integration approach of <cit.>, while in Section <ref> we present  an alternative derivation  based on Cauchy's  residue theorem. 
        
        
        The first step in the direct integration method consists in 
        performing the ω integration
        in (<ref>) using
        the delta function δ (2 p_2  (k-q_1)). This localises the graviton energy ω to the value ω^∗ given by  
        
            ω^∗ = q_1 v_2/k̂ v_2  ,
        
        where we recall from (<ref>) that 
           kωk̂.
        One then quickly arrives at <cit.>
            h^∞(u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._5|_k = ω^∗k̂  ,
        
        with the following definitions: 
        
            b̃   b̃_1 - b̃_2  ,  
                
            b̃_i     b_i + u_i v_i  ,      
               u_i  u - k̂ b_i/k̂ v_i  ,      i=1,2.
        
        We note that the modified (shifted) impact parameter b̃ has the useful  property that 
        
            k̂b̃=0  ,
        
        since k̂b̃_i =u. We are then left to perform integrations of the form (<ref>). For clarity we have derived and collected all required  master integrals in Appendix <ref> 
        and <ref>, where a detailed description of the direct integration method is included. 
        
        
        
        
        
        
        
         
        
        
        
         §.§ The G_3 waveform
        
        We now discuss the G_3 waveform. 
        The relevant integral is (<ref>), where we can replace the five-point amplitude with the contributions of the two poles,  _G_3, q_1^2, _G_3, q_2^2 and discarding contact terms (see (<ref>)).   
        To proceed, we first rewrite  
        
            |k] √(ω) |k̂]  ,
        
        and recall that ω is localised at the q_1-dependent value ω^∗ given in (<ref>), and where | k̂] is ω-independent. 
        We also rewrite the cosh term in M_G_3(-p_2, - q_1^++, k^++) in terms of exponentials, with  the net effect of shifting the impact parameters. Doing so, we obtain for  
        the contribution to the time-domain waveform  arising from the first diagram in (<ref>), 
        
            h^∞, (1)_G_3 (u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._G_3, q_1^2|_k = ω^∗k̂
               =- 3/2(κ/2)^6
                m_1 m_2/(k̂ v_2)^3∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^2/q_1^2[ e^i q_1  [  b̃ +  i (ã_1 - ã_2) ] + e^i q_1  [  b̃ + i (ã_1 + ã_2)]] 
                = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3 X_μ X_ν v_2 α v_2β[ 
                .^μναβ|_b̃→b̃ + i(ã_1 - ã_2) + .^μναβ|_b̃→b̃+ i(ã_1 + ã_2)] 
                = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3[ 
                .^(1)|_b̃→b̃ + i(ã_1 - ã_2) + .^(1)|_b̃→b̃+ i(ã_1 + ã_2)]  ,
        
        where we used the expression for _G_3, q_1^2 derived earlier in (<ref>). We also  defined
        
            X_μ = [k̂| v_1 σ_μ |k̂]   ,
        
        and
        
            ã_i   a_i - v_i a_ik̂/v_ik̂ ,
        
        which allows us, using (<ref>), to rewrite
        [In writing (<ref>) we have also used that q_1 v_1=0 and replaced a_1 q_1 with ã_1 q_1. This is useful in order to maintain the symmetry between the two factorisation diagrams in (<ref>) so that the second can easily be obtained from the first upon performing suitable replacements, see (<ref>). 
        ]
            a_2  (k-q_1) = -ã_2  q_1  .
        
        The definitions of the tensor integrals ^μ_1 …μ_n can be found in (<ref>). 
         
        The integrals in the second line of (<ref>) can be computed elegantly using the method of generating functions introduced in Appendix <ref>. In particular  (<ref>) and (<ref>) are relevant for this case, with the result
        
            ^(1)   =∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^2/q_1^2 e^i q_1 b̃
               = -1/8 π(∂^4/∂^2 t_1 ∂^2 t_21/|b̃_(1)+t_1 v_2(1) + t_2 X_(1)|)_t_1=t_2=0 .
        
        Here the subscript (1)  indicates the projection imposed by the delta function and implemented by the projector 
        
            P_1^μν=η^μν-v_1^μ v_1^ν ,
        
        which allows us to define
        
            V^μ_(1) = P_1^μν V_ν ,
        
        and
        
            |V_(1)| = √(-V_(1) V_(1))
            = √(-V  P_1  V) .
        
        We also note that X^μ_(1) = X^μ.
        
        With these definitions and performing the derivatives in (<ref>) we find
        
            ^(1)     = -3/8 π|b̃_(1)|^5[ 
            2 (X  v_2 (1))^2
                
            + 5/|b̃_(1)|^2(v_2(1)^2 (X b̃_(1))^2 
            + 4 (X  v_2 (1)) 
            (X b̃_(1))(b̃_(1) v_2(1))
            +7 (b̃_(1) v_2)^2/|b̃_(1)|^2 (X b̃_(1))^2)
            ]  ,
        
        which can be simplified further into
        [We have also rewritten b̃_(1) v_2(1) as b̃_(1) v_2, since for any two vectors m^μ and n^μ we have m_(1) n_(1) =m n_(1) = m_(1) n, with m_(1)^μ P_1^μν m_ν, with the projector P_1 defined in (<ref>).]
            ^(1) = -3/8 π[ 
            2 (X  v_2)^2/|b̃_(1)|^5  
            + 5 (X b̃)/|b̃_(1)|^7(v_2(1)^2 (X b̃) 
            + 4 
            (b̃_(1) v_2) (X  v_2) 
            +7 (b̃_(1) v_2)^2/|b̃_(1)|^2 (X b̃))
            ]  .
        
        Alternatively, we can evaluate the third line of (<ref>) by twice differentiating (<ref>), which gives  
        
            ^μναβ(b̃)     = - 3/8 π|b̃_(1)|^5[ 
                ( P_1^μν P_1^αβ + P_1^μα P_1^νβ + P_1^να P_1^μβ) 
                
               + 5/|b̃_(1)|^2  (
                P_1^μνb̃_(1)^αb̃_(1)^β + P_1^μαb̃_(1)^νb̃_(1)^β + P_1^ναb̃_(1)^μb̃_(1)^β
                +
                P_1^αβb̃_(1)^μb̃_(1)^ν + P_1^μβb̃_(1)^αb̃_(1)^ν + P_1^νβb̃_(1)^αb̃_(1)^μ)
                
                + 35/|b̃_(1)|^4   b̃_(1)^μb̃_(1)^νb̃_(1)^αb̃_(1)^β]   ,
        
        where the projector P_1^μν is defined in (<ref>), 
        
        b̃_(1)^μ in (<ref>) and 
        |b̃_(1)| 
        in (<ref>). 
        
        Note that all the terms proportional to P_1^μν in 
        (<ref>) do not contribute to  the contractions in (<ref>).
        Pleasingly, we find complete agreement of the contractions of this tensor integral with the result obtained from the generating function,  
        
            X_μ X_ν v_2 α v_2β ^μναβ(b̃) = ^(1) .
        
        We then have 
        
            h^∞, (1)_G_3 (u)  
                    = - 3/2(κ/2)^6
                m_1 m_2
                /(k̂ v_2)^3[ 
                .^(1)|_b̃→b̃ + i(ã_1 - ã_2) + .^(1)|_b̃→b̃+ i(ã_1 + ã_2)] 
                  .
        
        We  note that effectively we have rewritten the spinning waveform in terms of the scalar waveform, but with shifted impact parameters: 
        
            h_G_3^∞, (1) (u) =1/2[ . h_G_3^∞, (1) (u) |_a_i=0,  b̃→b̃+ i (ã_1 + ã_2)  +  . h_G_3^∞, (1) (u) |_a_i=0,  b̃→b̃+ i (ã_1 - ã_2)]   .
         
        We call the two contributions of these two terms as arising from two “sectors”, with shifted impact parameter
        
            (1)               (2)        
            b̂ = b̃ + i (ã_1+ ã_2)        b̂ = b̃ + i(ã_1- ã_2)   ,
        
        where we recall that ã_i was defined in (<ref>). 
        The result for I_1 will not have such a simple form, but it can be expressed in terms of the same two sectors. 
        
        Finally, the  complete waveform, including contributions from both factorisation channels, is 
        
            h^∞_G_3 (u) = h^∞, (1)_G_3 (u) + h^∞, (2)_G_3 (u)  ,
        
        where
        
            h^∞, (2)_G_3 (u) = .h^∞, (1)_G_3 (u)|_(m_1, a_1,  v_1, b_1)↔ (m_2, a_2,  v_2, b_2)  .
        
        We also observe  that under this exchange the variable b̃ defined in (<ref>) changes sign, b̃↔  - b̃. 
        
        
        
        
        
        
        
        
        
        
        
         §.§ The I_1 waveform
        
        
        In the presence of the I_1 deformation, the time-domain waveform  is 
        
            h^∞, (1)_I_1 (u) = - i κ/2 m_1 m_2 (k̂ v_2)∫d^4 q_1/(2π)^3δ(2 q_1  v_1) e^i q_1b̃._I_1, q_1^2|_k = ω^∗k̂  ,
        
        where we have, from (<ref>),
        
            ._I_1, q_1^2|_k = ω^∗k̂   =   -  i (κ/2)^512    m_1^2 m_2^2  (ω^∗)^3/q_1^2  (q_1k̂)  
              (X  q_1)^2(v_2 k̂ )  e^- a_1  q_1×
               [   cosh [ a_2 (ω^∗k̂-q_1)]  (v_2 k̂) -  i sinh a_2 (ω^∗k̂-q_1)/a_2 (ω^∗k̂-q_1)ϵ ( k̂ v_2 q_1 a_2)]  ,
        
        with ω^∗  given in (<ref>). 
        Therefore we obtain  
        
            h^∞, (1)_I_1 (u)
                = - (κ/2)^6 
            6m_1 m_2/(k̂ v_2)^3∫d^4q_1/(2π)^3δ(2 q_1 v_1) (X  q_1)^2  (q_1 v_2)^3/q_1^2  (q_1k̂)×
               [ ( v_2 k̂ + i ϵ(k̂ v_2 q_1 a_2)/q_1 ã_2)  e^i q_1  [  b̃ +  i (ã_1 + ã_2) ] + ( v_2 k̂  - i ϵ(k̂ v_2 q_1 a_2)/q_1 ã_2)  e^i q_1  [  b̃ +  i (ã_1 -  ã_2) ]]
                  ,
        
        where the shifted spin vectors  ã_i were defined in (<ref>). 
        
        We now  derive using the  method of generating functions presented in Appendix <ref>
        the two master integrals  needed to evaluate the waveform integral  in  (<ref>), corresponding to the first factorisation diagram. 
        Before doing so, we  comment that 
        its integrand   contains the spurious pole 1/ (q_1k̂), which originates from the propagator i/(q^2 + i ε) =  i / ( -2 q_1k̂ + i ε) in the Compton amplitude (<ref>) onto which the five-point amplitude factorises. We have checked explicitly that this pole is cancelled by a corresponding spurious pole in the second factorisation diagram, and have double-checked this conclusion by a detailed comparison with the full five-point amplitude obtained from a diagrammatic computation. In practice it is convenient to perform the integrations of the two factorisation diagrams separately, which requires a choice of regularisation of the pole. The sum of the two diagrams should be independent of the  regularisation, since the pole is spurious; 
        we have confirmed this expectation by performing the integration with either the Feynman i ε prescription or the Principal Value (PV)  prescription,  finding complete agreement.
        [In fact, using the PV prescription the spurious contributions vanish separately in each diagram; this is related to the tracelessness of the integral 𝒦^μν
        introduced in (<ref>), see end of Appendix <ref> for a thorough discussion of this point.
        ]
        In the following we will choose the PV prescription, and for completeness    we  give in Appendix <ref> the master integral 𝒥^μ in (<ref>) with both prescriptions (while for 𝒦^μν we only present the result in the PV prescription).
        
        The first type of integral we need is
        
            ^(1)   ∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^3/q_1^2 (q_1 k̂) e^i q_1 b̃ , 
               = [  ∂^4/∂^2 t_1 ∂^2 t_2(
             𝒥 v_2 )_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0
                = 
            [  ∂^4/∂^2 t_1 ∂^2 t_2(v_2  K_1 k̂/8 π |b̃_(1)|  [(v_1 k̂)^2 |b̃_(1)|^2-(k̂b̃_(1))^2 ])_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0  ,
        
        where we have used (<ref>) and 
        K_1^μν = |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν from (<ref>).
        
        The second type is
        
            ℰ^(1)   ∫d^4q_1/(2π)^3δ(2 q_1 v_1) (q_1  X)^2 (q_1 v_2)^3 ϵ(k̂ v_2 q_1 a_2)/q_1^2 (q_1 k̂)(q_1 ã_2) 
            e^i q_1 b̃
               = -[  ∂^4/∂^2 t_1 ∂^2 t_2(Y 𝒦 v_2 )_b̃→b̃+t_1 v_2 + t_2 X]_t_1=t_2=0 ,
        
        with 
        
            Y 𝒦 v_2 = 
              - ( ã_2  K_1k̂) (Y  K_1 v_2) - (ã_2  K_1 Y) (k̂ K_1 v_2)-
            (ã_2  K_1 v_2) (k̂ K_1 Y)/8 π  |b̃_(1)| ( ã_2  K_1ã_2 ) ( k̂ K_1k̂) ,
        
        where we have used
        Y^μϵ(k̂ v_2μ a_2), as well as (<ref>) and (<ref>).
        
        Putting everything together, we find that (<ref>) evaluates to
        
            h^∞, (1)_I_1 (u)
                = - (κ/2)^6 
            6m_1 m_2/(k̂ v_2)^3×
               [ ( (k̂ v_2) ^(1)+ i ℰ^(1))_b̃→b̃ +  i (ã_1 + ã_2) + 
            ( (k̂ v_2) ^(1)- i ℰ^(1))_b̃→b̃ +  i (ã_1 - ã_2)]
                  .
        
        As for the G_3 case, the complete I_1 waveform is 
        
            h^∞_I_1 (u) = h^∞, (1)_I_1 (u) + h^∞, (2)_I_1 (u)  ,
        
        where
        
            h^∞, (2)_I_1 (u) = .h^∞, (1)_I_1 (u)|_(m_1, a_1,  v_1, b_1)↔ (m_2, a_2,  v_2, b_2)  .
        
        Finally, we  note the appearance in (<ref>) of shifted impact parameters b̃ +  i (ã_1 ±ã_2),  similarly to (<ref>) in the G_3 case. However there is an important difference with that case: while  the G_3 waveform can be obtained from the scalar  integrated waveform by performing shifts in the impact parameter as dictated by the sectors, in the I_1 case these shifts occur at the level of the  integrand. For the cosh part of the I_1 integrand they translate to shifts that can be performed on the integrated result; however the part of the I_1 proportional to the sinh function vanishes in the scalar case, and thus provides a new contribution that cannot be obtained from the spinless waveform.  
        
        
        
        
        
        
         §.§ Waveforms for parity-odd deformations
        
        
        To derive the waveforms for parity-odd deformations we only need to recall from Section <ref> that the Compton amplitudes for parity-odd deformations are obtained from the parity-even ones by multiplying them by a factor of ± i depending on whether the two gravitons have positive or negative helicity. The same property is inherited by the cut five-point amplitudes which we use to construct the waveforms, that is 
        
            -i _5^ P.O. (k^+) = i (-i _5^ P.E.(k^+))  ,     (-i _5^ P.O. (k^-))^∗ = i (-i _5^ P.E.(k^-))^∗  ,
        
        from which it follows that 
        
            W^ P.O. (k^+) = i W^ P.E.(k^+)  ,     (W^ P.O. (k^-))^∗ = i (W^ P.E.(k^-))^∗  ,
        
        where P.O. and P.E. stand for parity odd and even, respectively.
        Combining (<ref>) with (<ref>) 
        we see that 
        
            .    (W^ P.O. (k^-))^∗|_k = - ω (1,𝐱̂)  = .W^ P.O. (k^+)|_k =  ω (1,𝐱̂)  = 
            .i  W^ P.E. (k^+)|_k = ω (1,𝐱̂)   .
        
        The two terms in  
        (<ref>), can then be combined as in the parity-even case, 
        and hence we conclude that 
        
            (h^∞,  P.O._+ + i    h^∞,  P.O._×)    = 
                κ∫_-∞^+∞dω/2πe^- i ω u. 
            W^ P.O.(b, k)|_k=ω(1, 𝐱̂)
               =
                i   (h^∞, P.E._+ + i    h^∞,  P.E._×)  .
        
        This implies that  
        
            h^∞,  P.O._+ = -    h^∞,  P.E._×  ,       
                h^∞,  P.O._× = h^∞,  P.E._+  ,
        
        that is,  the “plus” and “cross” polarisations are then swapped  in the way prescribed by (<ref>).
        
        
        
        § TIME-DOMAIN WAVEFORMS FROM  TENSOR INTEGRAL GENERATING FUNCTIONS, RELOADED
        
        Alternatively, in the spirit of modern multiloop amplitude calculations, we can compute the integrals by a systematic method proposed in <cit.>.
        For the tree-level waveform, we have two D-dimensional master integrals,    
        
            _1[𝐲]∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1) q_1^2  e^i q_1 b̂ ,      _2[𝐲]∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1) q_1^2 q_1k̂  e^i q_1 b̂ ,
        
        where 𝐲 (y_1, y_2, y_3,ŷ_4,ŷ_5,ŷ_6,ŷ_7), with 
        
            y_1   σ=v_1 v_2,     y_2    v_1k̂ ,    y_3    v_2k̂ , 
            ŷ_4   b̂k̂√(-b̂b̂)=0,    ŷ_5   b̂ v_1 √(-b̂b̂),    ŷ_6   b̂ v_2√(-b̂b̂),    ŷ_7b̂b̂ .
        
        One can rewrite the two master integrals as 
        
            _j[𝐲]∫_-∞^∞dt   e^it _j[𝐲, t]   :=∫_-∞^∞dt   e^it∫d^D-4q_1/(2π)^3δ(2 q_1 v_1) q_1^2 (q_1 k)^j-1δ(q_1b̂-t)
               .
        
        The differential equation in t for _j[𝐲, t] is very simple:
        
            ∂_t  _1[𝐲, t]= D-4/t_1[𝐲, t],       ∂_t  _2[𝐲, t]= D-5/t_2[𝐲, t] .
        
        It is easy to see that  the t dependence is factorised from the other variables, and we can solve for it directly as 
        
            _1[𝐲, t]= t^D-4 D-4_1[𝐲] ,       _2[𝐲, t]= t^D-5_2[𝐲]  .
        
        Then the two integrals in (<ref>) are rewritten as  
        
            _1[𝐲]   =(∫_-∞^∞dt   e^it t^-4 + D D-4)_1[𝐲]=-π _1[𝐲]  ,
            _2[𝐲]   =(∫_-∞^∞dt   e^it t^-5 + D)_2[𝐲]=iπ _2[𝐲]  ,
        
        where we have already used D = 4 - 2ϵ and taken the → 0 limit.
        The differential equations in four  spacetime dimensions  of the  𝐲-dependent part _1,2[𝐲] are 
        
            ∂_ŷ_5_1[𝐲]= -ŷ_5/ŷ_5^2+1_1[𝐲],       ∂_y_2_2[𝐲]= -1  y_2_2[𝐲].
        
        The solutions  are 
        
            _1[𝐲]=-π/(2π)^3√(-b̂b̂)√(ŷ_5^2+1),      _2[𝐲]=c_2  y_2 ,
        
        where c_2=0 from the boundary conditions. 
        Hence we can omit _2[𝐲] in the calculation and we are thus left with a single master integral. 
        
        All other integrals can be reduced to this master integral. For example,  consider the case of the integral
        
            𝒱[𝐲]   ∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1)q_1 v_2 q_1^2 q_1k̂  e^i q_1 b̂  .
        
        We can rewrite it as 
        
            𝒱[𝐲]=∫_-∞^∞dt  e^it 𝒱[𝐲, t]   =∫_-∞^∞dt   e^it∫d^Dq_1/(2π)^D-1δ(2 q_1 v_1)q_1 v_2 q_1^2 (q_1 k)δ(q_1b̂-t)  .
        
        By  IBP reduction, the q_1-integral is reduced to the single master integral  _1[𝐲], and finally we get 
        
        
        
        
        
            𝒱[𝐲]
            	   =(-π)y_1 (y_2+ŷ_4 ŷ_5)-y_3 (ŷ_5^2+1)-ŷ_4 ŷ_6+y_2 ŷ_5 ŷ_6/y_2^2+2 ŷ_4 ŷ_5 y_2-ŷ_4^2_1[𝐲]  .
        
        We have checked that this formula is  consistent with (<ref>).   
        
        
        
        
        
        
        
        
        
         The integral in (<ref>) contains a spurious pole in the entire function (sinh x)/x (see <cit.> for general discussions). However we can eliminate it  by introducing the integral representation   <cit.>
            sinh x  x= ∫_0^1dz  cosh(z  x)  .
        
        Then there are four sectors in the waveform, with exponential factors e^i q b̂. The particular  form of the new variable b̂ depends on the  sector as follows:
        [We note that in the previous approach we employed to compute the I_1 waveform there were  two sectors, corresponding to the cases   (1) and  (2) below. We now have two additional sectors because 
         the integral representation    (<ref>) introduces a   dependence on the integration variable z.]
            (1)              (2)              (3)              (4)       
            b̂= b̃+i(ã_1+ ã_2)        b̂=b̃+i(ã_1- ã_2)       b̂= b̃+i(ã_1+ z ã_2)       b̂= b̃+i (ã_1- z ã_2)  .
        
        Then the waveform can be represented as a combination of tensor integrals:
            
            h^∞, (1)_G_3 (u)
                = - (κ/2)^6
            6m_1 m_2/(k̂ v_2)^3(1 4X^μ_1X^μ_2v_2^μ_3v_2^μ_4∑_i=1^2 ^(i)_μ_1μ_2μ_3 μ_4(y))   ,
               h^∞, (1)_I_1 (u)
                = - (κ/2)^6
            6m_1 m_2/(k̂ v_2)^3(v_2k̂ X^μ_1X^μ_2v_2^μ_3v_2^μ_4∑_i=1^2 '^(i)_μ_1μ_2μ_3 μ_4(y)
                                                  +X^μ_1X^μ_2v_2^μ_3v_2^μ_4Y^μ_5∑_i=3^4 '^(i)_μ_1μ_2μ_3 μ_4μ_5(y)) 
              ,
        
        where Y^μϵ(k̂ v_2μ a_2),  and  y (y_1, y_2, y_3, y_4, …, y_18)  denotes all the independent scalar products among external kinematic vectors v_1, v_2, k̂, ã_1, ã_2, b̃, with y_1, y_2, y_3 defined in (<ref>) and 
        
            y_4   b̃k̂√(-b̃b̃)=0,      y_5   b̃ v_1 √(-b̃b̃),     y_6   b̃ v_2√(-b̃b̃),      y_7   b̃b̃ ,  
            
                  y_8   ã_1k̂√(-b̃b̃)=0,     y_9   ã_1 v_1√(-b̃b̃),     y_10   ã_1 v_2√(-b̃b̃),      y_11   ã_1 ã_1 -b̃b̃    
             
                   y_12   ã_2k̂√(-b̃b̃)=0,      y_13   ã_2 v_1√(-b̃b̃),      y_14   ã_2 v_2√(-b̃b̃),     y_15   ã_2 ã_2 -b̃b̃    
             
                   y_16   ã_1 b̃ -b̃b̃,    y_17   ã_2 b̃ -b̃b̃,    y_18   ã_1 ã_2 -b̃b̃  .
        
        The tensor integrals are generated as  
        
            ^(i)_μ_1…μ_r(y)   = ∂_b̃^μ_1⋯∂_b̃^μ_r _1^(i)[𝐲]= ∂_b̃^μ_1⋯∂_b̃^μ_r∫_0^1dz  _1^(i)[𝐲]  ,  
            '^(i)_μ_1…μ_r(y)   = ∂_b̃^μ_1⋯∂_b̃^μ_r 𝒱^(i)[𝐲]=∂_b̃^μ_1⋯∂_b̃^μ_r∫_0^1dz 𝒱^(i)[𝐲]  .
        
        We note that one cannot use _2 to generate the tensor integral as its value is trivial. Furthermore,  it is not possible to   set y_4=0 in 𝒱^(i) and ^(i)_1 directly,  as ∂_b̃^μy_4 is nonvanishing.  The superscripts in 𝒱^(i) and ^(i)_1 denote the tensor generating function with b̂ in different sectors. Then we only need to perform the z integration for the simple cases of _1^(3,4)[𝐲] and 𝒱^(3,4)[𝐲]. By using IBP and partial fractioning, the z integral can  then  be reduced  to a basis of  three integrals:
        
            ∫_0^1 dz/𝒴,       ∫_0^1 dz/b̂b̂ 𝒴,       ∫_0^1  dz  z/b̂b̂ 𝒴  ,
        
        where  𝒴^2  -b̂b̂ ŷ_5^2-b̂b̂, which 
         are straightforward to evaluate directly.
        
        
        
        
        
        
        
        
        
        
        
        
        
        
            
            
            
            
            
        
        
        
        
        
        § TIME-DOMAIN WAVEFORMS FROM RESIDUES 
        
        A final approach, followed in <cit.>, 
        makes use of Cauchy's residue theorem. We start by rewriting the original expression for the waveform in (<ref>) as follows,
        
            h_I^∞(u) 
               = -iκ∫_-∞^+∞dω/2πe^- i ω (u-k̂_2)∫d^4q_1/(2π)^2δ(2  p_1 q_1 ) δ(2  p_2 (k-q_1) )  e^iq_1 ℳ_I   ,
        
        where k=ωk̂. Further, we  split up the waveform into contributions coming from the two cuts (<ref>), and rescale the momentum transfers as q_i=ωq̂_i,  
        
            h_I^∞, (i) (u) 
            = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2π    ω^2 e^- i ω (u-k̂_2 -q̂_1)δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
               ℳ_I,q_i^2(ωk̂,ωq̂_1,ωq̂_2) .
        
        The basic idea now is to rewrite the integral over q_1 as a contour integral encircling the physical poles. To do this, it is worth introducing a parameterisation which is best suited to the various sectors of the problem. 
        Explicitly, we can factor out the exponential dependence on the spin for each cut, in a similar manner to (<ref>),
        
            _I,q_1^2=e^-_1  q_1∑_l=±e^l _2 q_2_I,q_1^(l) ,    _I,q_2^2=e^-_2  q_2∑_l=±e^l _1 q_1_I,q_2^(l) .
        
        Focusing on the q_1 cut, this allows us to rewrite the waveform as 
        
            h_I^∞, (1) (u) 
               = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2πω^2  
                δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
                 ∑_l=±e^- i ω (u-k̂_2 -q̂_1-i_1q̂_1+ i l_2(k̂-q̂_1))ℳ_I,q_1^2^(l)(ωk̂,ωq̂_1,ωq̂_2)
               = -iκ∫d^4q̂_1/(2π)^2∫_-∞^+∞dω/2πω^4  
                δ(2  p_1q̂_1 ) δ(2  p_2 (k̂-q̂_1) ) 
                 ∑_l=±e^- i ω (_l-_l q_1)ℳ_I,q_1^2^(l)(k̂,q̂_1,q̂_2)  ,
        
        where we have extracted the ω dependence of ℳ_I,q_i^2^(l), which is simply a factor of ω^2, and we have defined
        
            _l u-k̂_2+ i   l _2k̂  ,   _l^μ^μ + i _1^μ +i   l _2^μ .
        
        To evaluate (<ref>), we parameterise the q_1 integral in each sector using <cit.>
            z_1 v_1 q_1,    z_2    v_2 q_1,     z__l_l q_1,    z_o_l o_l q_1 ,
        
            with 
            
            o_l^μϵ(v_1 v_2 _lμ ) .
        
        The Jacobian for this transformation is simply 
        
            | ∂ q_1^μ/∂ z_j |= 1/|o_l o_l| .
        
        With this, the q_1 cut contribution to the waveform becomes
        
            h_I^∞, (1) (u)   = -iκ/(4π)^2 m_1 m_2(∂/∂ u)^4 ∑_l=±∫d^4 z_j/|o_l o_l|δ(z_1) δ(z_2-v_2k̂)δ(z__l-_l) _I,q_1^2^(l)
               =-iκ/(4π)^2 m_1 m_2(∂/∂ u)^4 ∑_l=±∫_-∞^∞d z_o_l/|o_l o_l|_I,q_1^2^(l)|_z_1→0, z_2→ v_2k̂, z__l→_l .
        
        The final integral in z_o_l can be computed using Cauchy's residue theorem. There are (at most) three poles present in _I,q_1^2^(l):
        
            Physical Pole:    1/q_1^2∼1/(z_o_l-A)(z_o_l-A^⋆) ,  
               Spurious Poles:  1/q_1 k∼1/(z_o_l-B) ,  1/(k-q_1) a_2∼1/(z_o_l-C) ,
        
        where A,B,C are functions of the external kinematics. The residues in 1/q_1 k and 1/(k-q_1) a_2 must cancel upon summing over the sectors l and cuts since the poles are spurious.[ More precisely the residue in  1/(k-q_1) a_2 must cancel when summing over the sectors l in each cut. The residue in 1/q_1 k will only cancel after summing both cut contributions.] Thus, in practice, we only need to compute the residue on the physical pole 1/q_1^2. This way of computing the integrals also makes it clear why terms 
        with only spurious poles do not contribute to the waveform: they have vanishing residues. 
        
        One may check that the integral (<ref>) has no pole at infinity (after applying the ∂/∂ u derivatives) and that the integrand falls off sufficiently fast at infinity to close the contour above or below the axis. This contrasts with the case of  Einstein-Hilbert gravity <cit.> where the pole at infinity exists and must be evaluated using a principal value prescription. In fact, we could have predicted this from the start. In  <cit.>, it was shown that the pole at infinity is given by the leading soft theorem of the five-point amplitude. Our five-point amplitude is built from higher-derivative corrections to general relativity and cannot change this universal leading soft theorem. Indeed, we can check that our amplitude vanishes in the soft limit ω→ 0, which manifests in the time-domain waveform as vanishing linear memory, as discussed in Section <ref>. 
        
        To compute the contribution of the q_2 cut, we follow an identical procedure to the above but using the sectors for _q_2^2 in (<ref>). Upon summing both contributions, we obtain the full waveform for I_1 and G_3:
        
            h_I^∞ (u)= h_I^∞, (1) (u)+h_I^∞, (2) (u) .
        
        The expressions obtained using this method are lengthier than those found in Section <ref>. Reassuringly, they are in complete agreement.
        
        
        
        
        § SHOWCASE OF WAVEFORMS
        
        
         
        In this section we show several  plots of the waveforms for the  G_3 and I_1 deformations, with and without spin. We recall that the waveforms for the parity-odd interactions G̃_3 and Ĩ_1 are related to the parity-even ones by (<ref>), that is a simple swap of their real and imaginary parts; thus  separate plots for the case of parity-odd deformations are not needed. 
        We will   parameterise our external kinematics as follows:
        [Our waveforms are valid for arbitrary spin configurations; however in the figures shown in this section we will focus on the case where the spins of the two bodies are aligned.]
            v_1^μ=(σ,0,0,√(σ^2-1)) ,      v_2^μ   =(1,0,0,0) ,
               k̂^μ=(1,sinθcosϕ ,sinθsinϕ ,cosθ ) ,     b^μ   =b(0, 1, 0, 0) , 
               a_1^μ=a_1(0,0,1,0) ,     a_2^μ   =a_2(0,0,1,0) , 
               
               [k|_α̇=√(2)[         sinθ2; -cosθ2  e^-iϕ                             ]  .
        
        For the graphs below, we will choose θ=ϕ=π/3 and fix the spins to be either aligned or anti-aligned with the orbital angular momentum. We will also set b = 1 which means in practice 
        that the spins a_i and the retarded time u are measured in units of b. Finally we will ignore a prefactor so that the waveforms are plotted in units of κ^6 m_1 m_2 β_i.
        
        
        
        
        
         §.§ Spinless case
        
        
        We begin by showing the G_3 and I_1 waveforms for the scalar case, for various values of σ=v_1 v_2. 
        
        
        
        
         §.§ Spinning case
        
        We now show a few  G_3 and I_1 waveforms for  spinning objects. In Figures <ref> and <ref> we consider the cases where only the spin of one of the two bodies is nonvanishing, say a_1≠ 0 and a_2=0, and  do so for increasing values of a_1.
        In Figures <ref> and <ref> we show the G_3 and I_1  waveforms when both objects are spinning,  with their spins aligned, for various choices of the ratio a_2/a_1.
        Note that the  gravitational memory is absent  also in the spinning case. 
        We also recall that in setting b=1 we are measuring the spins a_i in units of b. Thus, the values we are plotting here, e.g. a_i=± 0.5, are rather large. Finally,  the waveforms also have poles at b=± a_1± a_2,
        [These poles arise  from treating  the  spin dependence   exactly and are  also present  in the scattering  angle <cit.>.] although  these points are not problematic since we assume |a_i|≤  G m_i< b.
        
        
        
        
        § (NO) GRAVITATIONAL MEMORY
        
        
        
        As it is well known, gravitational memory is related to soft limits of the five-point amplitude <cit.>, and a discussion of the memory in the General Relativity waveforms to all orders in spin was presented in <cit.>. We now wish to study if   cubic deformations alter in any way the gravitational memory. 
        Defining the memory  as 
        
            Δ (h^∞_+± i  h^∞_×) . (h^∞_+± i  h^∞_×)|_u→ + ∞ - . (h^∞_+± i  h^∞_×)|_u→ -  ∞  ,
        
        one finds that (see e.g. <cit.>)
        
            Δ (h^∞_+± i  h^∞_×)    = 
              -i  κ S_ W^ HEFT(k̂, -i ∂/∂ b; ±) δ_ HEFT  ,
        
        where
        
            δ_ HEFT∫dμ^(D) e^iq_ b ( -i _4^ HEFT)(q)  ,
        
        is the Fourier transform to impact parameter space of the four-point classical amplitude _4^ HEFT<cit.>, which now has to be evaluated in the presence of cubic deformations; while 
        and S_ W^ HEFT is the classical (or HEFT) Weinberg soft factor   <cit.>
            S_ W^ HEFT   = 
               - κ/2 1/ωε_μν^(h)(k) [
            		p_1^μ q^ν +p_1^ν q^μ/p_1k̂ - p_1^μ p_1^νqk̂/(p_1k̂)^2  -   1↔ 2]
                 .
        
         However, at leading order (tree level) in the presence of cubic deformations the classical four-point amplitude vanishes,  _4^ HEFT, (0)=0; indeed,  this quantity  starts receiving corrections at one loop, or 2PM, which were computed  in <cit.>. We conclude that the leading-order gravitational memory in the presence of cubic deformations vanishes. This is also confirmed by the explicit plots shown in 
         the 
         
         figures  in Section <ref>. 
        
        
        § DISCUSSION
        
        
        
        
         
        
        In this paper we have considered  parity-even and -odd cubic deformations to general relativity, and computed analytically the corresponding corrections to the waveforms to leading order in the deformations and in Newton's constant. We have done so following three independent approaches, yielding  results in complete agreement. The direct integration method provides the most compact expressions, though the IBP reduction and residue methods are possibly more easily extendable beyond tree level. The residue method is especially   rewarding in that it easily shows  the absence of contributions from spurious poles. 
        
        To conclude, we  briefly muse on the size  of the corrections due to the cubic deformations studied here and in <cit.>. Specifically, we   focus  on an observable quantity such as the impulse (or momentum kick) .Δℙ|_R^3, and   ask when the corrections arising from cubic deformations become comparable to those  in general relativity at a certain PM order. 
        
        Our cubic vertices have the form (schematically) ∫d^4x βκ^3 (∂^2h)^3. We  redefine the coupling constant in such a way that the three-point vertex scales as κ, thus introducing β = β̂ / κ^2; in terms of β̂ the vertex has the form κβ̂∫d^4x  (∂^2h)^3, and we  note that the dimension of β̂ is ( length)^4, so that  we will set β̂ = ℓ_ EFT^4. 
        We found in <cit.> that, 
        to leading order in the deformation and in G, the corrections to the impulse arising from cubic deformations have the form (see (6.61)–(6.64) of that paper)
        
            .Δℙ|_R^3∼βκ^6 m^3/b^6∼β̂/κ^2 G^3 m^3/b^6∼ m ×(ℓ_ EFT/b)^4 (Gm/b)^2   .
        
        On the other hand, to nPM order in general relativity the impulse  receives corrections    of the type 
        
            .Δℙ|_ EH,  n∼ m ×(Gm/b)^n
                  ,
        
        so that 
        
            .Δℙ|_R^3/.Δℙ|_ EH,  n∼(ℓ_ EFTb)^4/(Gmb)^n-2  .
        
        For hyperbolic encounters of interest we can choose for illustration  G m/b ∼  1/5, which ensures that  perturbation theory is still valid while also giving rise to a sizeable effect. By further requiring that, for observability,  .Δℙ|_R^3/ .Δℙ|_ EH,  n∼ 1, we arrive at  the estimate 
        
            ℓ_ EFT∼ b   (1/5)^n-2/4  .
        
        As an   example, we can ask when   the corrections  from cubic deformations  become comparable to those arising at 5PM (or four loops)  in general relativity.
        [Parts of  the 5PM corrections to the impulse and the scattering angle for non-spinning objects were recently computed in <cit.>.]
        From  (<ref>), and further choosing b∼ 100   km we arrive at the estimate  ℓ_ EFT∼ 30   km, which is within the regime of validity of the effective field theory,
        and also happens to be in agreement with the estimate   provided 
        in <cit.>. We also note that in <cit.>, bounds on the fundamental length scale  ℓ_R^4 in a quartic effective field theory considered earlier in <cit.> were established, resulting in ℓ_R^4≲ 150  km.  
        
        
        
        Similarly, we find that the waveform 
        h^∞ in general relativity at order n in the PM expansion scales as
        
            h^∞|_ EH,  n∼ b ×( G m/b)^n  ,
        
        where now  n=2 corresponds to tree level and n=3  to one loop. 
        For the leading cubic correction computed in this paper we found (see for example (<ref>) and (<ref>))
        
            h^∞|_R^3∼ b ×( G m/b)^2 ( ℓ_ EFT/b)^4  ,
        
        leading to similar conclusions as above for the impulse (<ref>). Furthermore, if we assume that ℓ_ EFT∼ G m we can say that the effects of cubic interaction are in practice four orders higher in the PM expansion. This may be challenging given the currently available state of the art results in the PM expansion, but results in the PN framework might already give access to higher orders in G.
        
        A final comment on the spin radii a_i. In the expressions for the impulse and waveform,  they appear through the shifted impact parameters (schematically) b̃ = b + a (omitting Lorentz indices and numerical factors). Recalling that for a physical Kerr black hole
        a/(G m) ≤ 1,  we can express this as
        
            b̃ = b ( 1 + a/G mG m/b)  .
        
        This implies that effectively every factor of a increases the  PM order by one,  since GM/b is the effective loop counting parameter. For example,  the linear in spin part of our results  should be accompanied  by the one-loop correction of the scalar waveform in the cubic theory.
        We hope to return to these questions in the  near future. 
        
        
        
        
        
        
        
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        We would like to thank  
        Joshua Gowdy and Paolo Pichini 
        for stimulating conversations.
        GT thanks the Physics Department at the University of Rome “Tor Vergata” for their warm hospitality.   
        This work was supported by the Science and Technology Facilities Council (STFC) Consolidated Grants   ST/T000686/1 and  ST/X00063X/1 “Amplitudes, Strings  & Duality”.
        The work of GRB and PVM is supported by STFC quota studentships.   GC has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Skłodowska-Curie grant agreement No. 847523 “INTERACTIONS”. 
        GT is also supported by a Leverhulme research fellowship RF-2023-279\ 9.
        No new data were generated or analysed during this study.
        
        
          
        
        
        § ALL TENSOR INTEGRALS WE NEED
        
        
        We summarise here relevant integrals needed to compute the time-domain waveforms in our paper. We focus here on integrals involving the 1/q_1^2 propagator, while integrals involving the 1/q_2^2 propagator can be  obtained by appropriate relabelling of the results quoted below. In the following we define q^μ q_1^μ and introduce the projector
        
            P_1^μνη^μν - v_1^μ v_1^ν .
        
        Henceforth, we denote projected four-vectors  and their modulus as
        
            V^μ_(1) P_1^μν V_ν  ,
        
            and 
            
            |V_(1)| √(-V_(1) V_(1)) = √(-V  P_1  V).
        
        The first and simplest integral we encounter is
        
            ℐ    = ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃/q^2
                = ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃_(1)/q^2
                = -1/8 π |b̃_(1)| ,
        
        where in the second line we exploited the fact that due to the δ-function we can insert for free the projector P_1 between q and b̃ in the exponent. We actually need tensor versions of this integral, 
        
            ℐ^μ_1 μ_2 …μ_n = 
                ℐ[q^μ_1 q^μ_2⋯ q^μ_n] = 
                ∫d^4q/(2 π)^3δ(2 q v_1 )   e^i q b̃ q^μ_1 q^μ_2⋯ q^μ_n/q^2 ,
        
        which can be simply obtained by acting  n times on ℐ  with the derivative operator
        
            -i ∂/∂b̃_(1) μ_i ,
        
        and using the identity 
        <cit.>
            ∂b̃_(1)^μ/∂b̃_(1)ν = P_1^μν .
        
        Two examples are 
        
            ^μ = -i ∂ℐ/∂b̃_(1)μ = i/8 πb̃_(1)^μ/|b̃_(1)|^3  ,
        
            and 
            
            ^μν = - ∂^2 ℐ/∂b̃_(1)μ∂b̃_(1)ν = |b̃_(1)|^2 P_1^μν+3 b̃^μ_(1)b̃^ν_(1)/8 π |b̃_(1)|^5 .
        
        In a similar fashion, we can obtain the other two more complicated tensor integral families. 
        The first one is 
        
            𝒥^μ∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃ q^μ/q^2  k̂ q ,
        
        which, as already mentioned at the end of Section <ref>, requires a regulator because of the 1/k̂ q pole.
        This is a spurious pole that cancels in the sum of the two diagrams contributing to  the waveform and, of course, the final result is independent of the regulator. For  illustration we will present here  the evaluation of 𝒥^μ using both the principal value (PV) and iε prescriptions, while in the main text we  only make use the PV prescription.
        
        We begin discussing  the PV prescription.
        We  denote the PV regulated integral by 𝒥^μ,  which can  be expressed in terms of the scalar integral 
        
            𝒥∫d^4q/(2 π)^3δ(2 q  v_1) e^i q ·b̃/q^2  k̂ q ,
        
        as
        
            𝒥^μ = -i ∂𝒥/∂b̃_(1)μ .
        
        Following <cit.>,  we now notice that  𝒥 is invariant under rescaling of b̃. Indeed a rescaling b̃→α b̃ can be undone by the rescaling q → q/α which leaves the integrand unchanged. Hence, we infer that 
        
            b̃_(1)μ𝒥^μ = -i  b̃_(1)μ∂𝒥/∂b̃_(1)μ = 0  ,
        
        and we further notice that v_1 μ𝒥^μ = 0 since v_1  q = 0 due to the δ-function in the integral. Hence 𝒥^μ must live in a two-dimensional subspace orthogonal to v_1^μ and b̃_(1)^μ. In order to implement this,  it is convenient  to introduce the symmetric tensor <cit.>
            K_1^μν = |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν ,
        
        obeying K_1^μν v_1ν = K_1^μνb̃_(1)ν = 0. This allows us to write an ansatz for 𝒥^μ as
        
            𝒥^μ = c   K_1^μνk̂_ν .
        
        We can now solve for c by observing that 
        
            𝒥^μk̂_μ = ℐ = -1/8 π |b̃_(1)| = c  k̂ K_1 k̂  ,
        
            from which it follows that 
            
            c = -1/8 π |b̃_(1)| k̂ K_1 k̂ .
        
        Therefore, we find that 
        
            𝒥^μ = K_1^μνk̂_ν/8 π |b̃_(1)|  [(v_1 k̂)^2 |b̃_(1)|^2-(k̂b̃_(1))^2 ] ,
        
        and from this result we can obtain 𝒥^μμ_1 …μ_n with any number of extra q^μ_i insertions by taking derivatives with respect to b̃_(1) μ_i as explained above.
        
        In order to obtain tensor integrals,  the form above is preferred as it makes the b̃_(1)-dependence fully transparent. On the other hand, one can obtain a more compact expression  in terms of the shifted, unprojected impact parameter b̃ as follows. 
        First, it is important to note that b̃k̂=0 as assumed throughout the paper. Note that any shift of b̃ by ã_i
        preserves this property
        since ã_i k̂=0.
        Under this assumption we can then write 
        
            b̃^μ = b̃_(1)^μ - k̂b̃_(1)/k̂ v_1 v_1^μ  ,
        
        from which it follows that 
        
            |b̃  |^2 -b̃b̃ =  |b̃_(1)|^2 - 
            (k̂b̃_(1))^2/(k̂ v_1)^2 .
        
        We  can then  rewrite  𝒥^μ in the more compact form (if b̃k̂=0)
        
            𝒥^μ =  K_1^μνk̂_ν/8 π  (v_1 k̂)^2 |b̃_(1)|   |b̃  |^2 .
        
        Next, we examine  the same integral,  this time  regulated with an iε prescription:
        
            𝒥_ε^μ   ∫d^4q/(2 π)^3δ(2 q  v_1) q^μ e^i q ·b̃/q^2  (k̂ q - i ε)
                =   i ∫_-∞^0 dτ∫d^4q/(2 π)^3δ(2 q  v_1) q^μ e^i q · (b̃ + τk̂)  + τε/q^2
                = -1/8 π∫_-∞^0 dτb̃^μ_(1) + τk̂^μ_(1)/[ -(b̃+τk̂)  P_1  (b̃+τk̂)]^3/2 ,
        
        where in the second line we have used the Schwinger trick and in the last line we have used (<ref>). Performing the τ integration gives
        
            𝒥_ε^μ =
            |b̃_(1)| k̂^μ_(1) - v_1 k̂ b̃^μ_(1)/8 π (v_1 k̂) |b̃_(1)| (|b̃_(1)| v_1 k̂ + b̃_(1)k̂ ) ,
        
        and similarly, replacing i ε→ -i ε we find
        
            𝒥_-ε^μ =
            |b̃_(1)| k̂^μ_(1) + v_1 k̂ b̃^μ_(1)/8 π (v_1 k̂) |b̃_(1)| (|b̃_(1)| v_1 k̂ - b̃_(1)k̂ ) .
        
        It can then be checked that these integrals are related to the PV integral (<ref>) as
        
            𝒥^μ = 1/2( 𝒥^μ_ε + 𝒥^μ_-ε)  .
        
        Finally, we tackle (with PV regularisation) the most complicated integral,
        
            𝒦^μν = ∫d^4q/(2 π)^3δ(2 v_1  q)
            e^i q b̃ q^μ q^ν/q^2  k̂ q  a_2  q_2 
            = ∫d^4q/(2 π)^3δ(2 v_1  q)
            e^i q b̃ q^μ q^ν/q^2  k̂ q  (- ã_2  q) ,
        
        where 
        we have introduced the shifted spin radii 
        
            ã_i^μ = a_i^μ - a_i k̂/v_i k̂ v_i^μ  ,      ã_i k̂ = 0   ,
        
        for i=1,2.
        Also note that q_2^μ = k^μ - q_1^μ = ω^* k̂^μ - q^μ = v_2  q/v_2 k̂k̂^μ - q^μ, and therefore we can rewrite a_2  q_2 = - ã_2  q.
        𝒦^μν has the following important properties:
        
            𝒦^μν v_1ν = 𝒦^μνb̃_ν
                = 𝒦^μνb̃_(1)ν =𝒦^μνη_μν = 0  .
        
        The first is due to the δ-function, the second and third property are consequence of the rescaling invariance of 𝒦^μ, and the fourth property (tracelessness) can be seen as follows: taking the trace and
        choosing to work in the rest frame v_1=(1,0,0,0), we find
        
            𝒦^μνη_μν    = ∫d^4q/(2 π)^3 δ(2 q v_1 )
            e^i q b̃/k̂ q   (- ã_2  q)
                = -∫_-∞^0dt_1 dt_2 ∫d^3q⃗/2 (2 π)^3  e^-i q⃗ (b⃗̃⃗+t_1 k⃗̂⃗ + t_2 ã⃗_2)
                = -1/2∫_-∞^0 dt_1 dt_2  δ^(3)(b⃗̃⃗+t_1 k⃗̂⃗ + t_2 ã⃗_2)  ,
        
        which vanishes for generic choices of the vectors.
        
        Due to the properties mentioned above, the integral (<ref>) must be a symmetric tensor living in the two-dimensional subspace orthogonal to v_1 and b̃, 
        
            𝒦^μν = c_1 K_1^μν + 2 c_2 (ã_2  K_1)^(μ (k̂ K_1)^ν) ,
        
        where 
            K_1^μν |b̃_(1)|^2 P_1^μν + b̃_(1)^μb̃_(1)^ν  .
        
            Notice that no other structures are allowed in this ansatz, since the result must be symmetric under swapping k̂↔ -ã_2 and be rescaled by 1/(α_1 α_2) if we perform the replacement  {k̂, ã_2}→{α_1 k̂, α_2 ã_2}. 
        Requiring also tracelessness, we instantaneously find
        c_1 = -c_2 ( ã_2  K_1k̂).
        On the other hand
        
            𝒦^μνk̂_μã_2ν = - ℐ = 1/8 π |b̃_(1)| = c_2 ( ã_2  K_1ã_2 )( k̂ K_1k̂) = - c_2 ( ã_2  K_1ã_2 ) (v_1 k̂)^2 |b̃|^2  ,
        
        and hence we find
        
            𝒦^μν = -( ã_2  K_1k̂) K_1^μν -2 (ã_2  K_1)^(μ (k̂ K_1)^ν)/8 π  |b̃_(1)| ( ã_2  K_1ã_2 ) ( k̂ K_1k̂) .
        
        If we further assume b̃k̂=0, which is
        true for all possible shifted versions of the impact parameter b̃± i ã_1 ± i ã_2,
        we find
        
            𝒦^μν = ( ã_2  K_1k̂) K_1^μν -2 (ã_2  K_1)^(μ (k̂ K_1)^ν)/8 π (v_1 k̂)^2 |b̃|^2 |b̃_(1)| ( ã_2  K_1ã_2 )  .
        
        Finally, we comment on potential terms in the waveform integrand that do not have physical poles 1/q_i^2 with i=1,2, but instead have a pole of the form 1/(w  q) (or a product of such factors). In such cases the same manipulation as in (<ref>) guarantees that such terms make vanishing contributions to the waveform. In  other words,  the tracelessness of integrals like 𝒦^μν in (<ref>) guarantees that terms with only spurious poles do not contribute to the waveform. This fact was also noticed in the Einstein-Hilbert case in <cit.>. For this reason, it is not necessary to use the full five-point amplitude,  but the sum of the two factorisation channels (<ref>) suffices. Similarly, all polynomial terms that only give rise to (derivatives of) δ-functions can be ignored.
        
        
        
        
        
        § TENSOR INTEGRALS FROM GENERATING FUNCTIONS
        
        We now describe an alternative (and faster) way to compute tensor integrals. All of the integrals we are interested in  have the form
        
            _m_1… m_n^ [] (b̃)     ∫d^4q/(2 π)^3 δ(2 q v_1 )  
            e^i q b̃ (q_1 c_1)^m_1⋯  (q_1 c_n)^m_n/ ,
        
        for some denominator , and where c_i, i=1, … , n are four-vectors. These can be obtained very easily from the integral 
        
            ^[] (b̃; t_1, …, t_n)    ∫d^4q/(2 π)^3 δ(2 q v_1 ) e^i q ( b̃ + ∑_i=1^n t_i c_i)/
                = ^[] (b̃ + ∑_i=1^n t_i c_i; 0, …, 0)
               ,
        
        as
        
            _m_1… m_n^[]   = .(-i)^∑_i=1^n m_i∂^∑_i=1^n m_i/∂^m_1t_1 ⋯∂^m_nt_n^[] (b̃; t_1, …, t_n)|_t_1 = t_2 = ⋯ = t_n = 0  .
        
        For instance, for the case of the G_3 waveform we need an integral of the form 
        
            ∫d^4q/(2 π)^3 δ(2 q v_1 )  
            e^i q b̃ (q  v_2)^2(q  X)^2/q^2  ,
        
        where X^μ[k̂| v_1 
        σ^μ | k̂]. This can be evaluated as 
        
            . ∂^4/∂^2t_1  ∂^2t_2∫d^4q/(2 π)^3 δ(2 q v_1 ) e^i q ( b̃ + t_1v_2 + t_2 X)/q^2|_t_1 = t_2  
            = 0
                
            = -1/8 π(∂^4/∂^2t_1  ∂^2t_21/
            |b̃_(1) + t_1 v_2(1) + t_2 X_(1)|)_t_1 = t_2 = 0 ,
        
        where we used (<ref>), and as usual we adopt the notation defined in (<ref>) for projected vectors (but note that X_(1) = X). 
        
          
        Summarising, in this approach we can easily relate tensor integrals to simpler ones (not necessarily scalar integrals, as seen in the previous section), without the need to perform Lorentz contractions and with the advantage of having to perform differentiations with respect to scalar parameters rather than vectors. 
        
        JHEP


