Polarized primordial gravitational waves in spatial covariant gravities Tao ZhuabWen Zhaocdand Anzhong Wange aInstitute for theoretical physics and cosmology

2025-05-02 0 0 679.38KB 13 页 10玖币

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Polarized primordial gravitational waves in spatial covariant gravities

Tao Zhua,b,∗Wen Zhaoc,d,†and Anzhong Wange‡

aInstitute for theoretical physics and cosmology,

Zhejiang University of Technology, Hangzhou, 310032, China

bUnited Center for Gravitational Wave Physics (UCGWP),

Zhejiang University of Technology, Hangzhou, 310032, China

cCAS Key Laboratory for Research in Galaxies and Cosmology, Department of Astronomy,

University of Science and Technology of China, Hefei 230026, China

dSchool of Astronomy and Space Sciences, University of Science and Technology of China, Hefei, 230026, China

eGCAP-CASPER, Physics Department, Baylor University, Waco, 76798-7316, Texas, USA

(Dated: January 25, 2023)

The spatial covariant gravities provide a natural way to including odd-order spatial derivative

terms into the gravitational action, which breaks the parity symmetry at gravitational sector. A

lot of parity-violating scalar-tensor theories can be mapped to the spatial covariant framework by

imposing the unitary gauge. This provides us with a general framework for exploring the parity-

violating eﬀects in primordial gravitational waves (PGWs). The main purpose of this paper is to

investigate the polarization of PGWs in the spatial covariant gravities and their possible obser-

vational eﬀects. To this end, we ﬁrst construct the approximate analytical solution to the mode

function of the PGWs during the slow-roll inﬂation by using the uniform asymptotic approximation.

With the approximate solution, we calculate explicitly the power spectrum and the corresponding

circular polarization of the PGWs analytically. It is shown that the new contributions to power

spectrum from spatial covariant gravities contain two parts, one from the parity-preserving terms

and the other from the parity-violating terms. While the parity-preserving terms can only aﬀect

the overall amplitudes of PGWs, the parity-violating terms induce nonzero circular polarization of

PGWs, i.e., the left-hand and right-hand polarization modes of GWs have diﬀerent amplitudes. The

observational implications of this nonzero circular polarization is also brieﬂy discussed.

I. INTRODUCTION

The inﬂation which took place at the early Universe

has become a dominant paradigm in the standard cos-

mology [1–6]. In this paradigm, primordial density and

gravitational-wave ﬂuctuations are created from quan-

tum ﬂuctuations during the inﬂation process. The for-

mer provides primordial seeds for the formation of ob-

served large-scale structure and creates the temperature

anisotropy in the cosmic microwave background (CMB),

which was already detected by various CMB experiments

[7–10]. The primordial gravitational waves (PGWs), on

the other hand, also produce distinguishable signatures

in both the spectra of the CMB [11–15] and the galaxy

shaped power spectrum [16–22]. In CMB, the PGWs can

produce the TT, EE, BB, and TE spectra, but the TB

and EB spectra vanish if the parity symmetry in gravity

is respected [11–15]. These signatures are important tar-

gets of future CMB experiments [23–26]. Similarly, the

PGWs also leave distinct imprints in the B-mode of the

galaxy shaped power spectrum but with vanishing E-B

correlation due to the parity conservation of the theory

[16–22]. It is therefore expected that the future galaxy

surveys could also provide invaluable information about

the physics of PGWs [22,27,28].

∗zhut05@zjut.edu.cn; Corresponding author

†wzhao7@ustc.edu.cn

‡anzhong wang@baylor.edu

In most of inﬂation models that produce PGWs, the

theory of general relativity (GR) is assumed to describe

the theory of gravity. Due to the parity symmetry of this

theory, the PGWs have two polarization modes which

share exactly the same statistical properties and the

corresponding inﬂationary power spectra take the same

form. If the parity symmetry is violated, however, the in-

ﬂationary power spectra of right- and left-handed PGWs

can have diﬀerent amplitudes. The corresponding rela-

tive diﬀerence between the power spectra of right- and

left-handed PGWs measures the level of the parity vi-

olation. In CMB, such parity violating eﬀects can in-

duce nonvanishing TB and EB correlation in CMB at

large scales and thus the precise measurement of TB and

EB spectra could be an important evidence of the par-

ity violation of the gravitational interaction [29–33]. It

is also proposed that the future ground- and space-based

interferometers (such as LIGO/Virgo [34,35], the Big

Bang Observer [38], LISA and Taiji/Tianqin [36,37],

etc) are also able to detect or constrain the parity vi-

olating eﬀects in the stochastic gravitational-wave back-

ground of primordial origin. In addition, parity-violating

PGWs also leaves imprints on the large scale structure

of the Universe [39] and sources nonzero E-B correlation

in the galaxy shape power spectrum [22]. Thus the fu-

ture galaxy surveys can provide an important approach

for testing or constraining the parity violating eﬀects in

PGWs [22,39].

Theoretically, gravitational parity violation has to

somehow modify the theory of GR. This can be achieved

by adding some parity-violating terms into the gravi-

arXiv:2210.05259v2 [gr-qc] 24 Jan 2023

tational action of GR. In fact, the gravitational terms

with parity violation are ubiquitous in numerous can-

didates of quantum gravity, such as string theory, loop

quantum gravity, and Horava-Lifshitz gravity. One im-

portant example is the Chern-Simons modiﬁed gravity,

which modiﬁes the GR by adding a gravitational Chern-

Simons term, arising from string theory and loop quan-

tum gravity [40,41]. This theory has been extended to a

chiral scalar-tensor theory by including the higher deriva-

tives of the coupling scalar ﬁeld [42]. On the other hand,

by breaking the time diﬀeomorphism (or Lorentz sym-

metry) of the gravitational theory, one can naturally add

parity-violating but spatial covariant terms into the grav-

itational action. This type of parity-violating theories

includes Horava-Lifshitz gravities with parity violations

[43–46] and more generally, the spatial covariant grav-

ities [47–49]. Other parity-violating theories, to men-

tion a few, include Nieh-Yan modiﬁed teleparallel grav-

ity [52,53], parity-violating symmetric teleparallel gravi-

ties [54,55], and standard model extension [56–60], Holst

gravity [61], etc.

In all these modiﬁed theories, a basic prediction of par-

ity violation is the circular polarization of PGWs, i.e.,

the left-hand and right-hand polarization modes of GWs

propagate with diﬀerent behaviors. As we also mentioned

in the above, such asymmetry between the left- and right-

handed modes of PGWs can induce various observational

or experimental eﬀects in CMB, stochastic gravitational-

wave background, and galaxy-shaped power spectrum.

These phenomenological eﬀects have motivated a lot of

works in this directions (see Refs. [22,43–45,62–84] and

references therein for example). It is worth noting that

the gravitational-wave constraints on the parity violation

in gravity have also been extensively explored in the lit-

erature by using the gravitational-wave data realized by

LIGO/Virgo Collaboration [85–97].

Spatial covariant gravities is one of modiﬁed theory of

GR, which breaks the time diﬀeomorphism of the grav-

ity but respects spatial diﬀeomorphisms [47–50]. Such

spatial covariance provides a natural way to incorporate

the parity-violating terms into the theory [85]. With spa-

tial covariance, the parity violation can be achieved by

including the odd-order spatial derivatives into the grav-

itational action. It is shown in [47,51] that the spa-

tial covariant gravities can provide a uniﬁed description

for a lot of scalar-tensor theory by imposing the unitary

gauge, including those with parity violation, such as the

Chern-Simons modiﬁed gravity, chiral scalar-tensor the-

ory, Horava-Lifshitz gravities, etc. Therefore, the spatial

covariant gravities can provide a general framework for us

to explore the parity violating eﬀects in PGWs. For this

purpose, in this paper we study the circularly polarized

PGWs in this theory of gravity with parity violation, and

the possibility to detect the chirality of PGWs by future

potential CMB observations and galaxy surveys.

This paper is organized as follows. In the next sec-

tion, we present a brief introduction of the construction

of the spatial covariant gravities and then discuss the as-

sociated propagation of GWs in the a homogeneous and

isotropic cosmological background in Sec. III. In Sec.

IV, we ﬁrst derive the master equation that describes the

propagation of GWs during inﬂation and construct the

approximate analytical solution to the PGWs by using

the uniform asymptotic approximation. With such ap-

proximate solution we then calculate explicitly the power

spectrum and the polarization of PGWs during the slow-

roll inﬂation. The eﬀects of the parity violation in the

CMB spectra and galaxy shaped spectrum, and their de-

tectability have also been brieﬂy discussed. The paper

is ended with Sec. V, in which we summarize our main

conclusions and provide some outlooks.

Throughout this paper, the metric convention is cho-

sen as (−,+,+,+), and greek indices (µ, ν, · ··) run over

0,1,2,3 and latin indices (i, j, k) run over 1,2,3.

II. SPATIAL COVARIANT GRAVITIES

In this section, we present a brief introduction of the

framework of the spatial covariant gravity, for details

about this theory, see [47,48] and references therein.

We ﬁrst start with the general action of the spatial

covariant gravity,

S=Zdtd3xN√gL(N, gij , Kij , Rij ,∇i, εijk),(2.1)

where Nis the lapse function, gij is the 3-dimensional

spatial metric, Kij is the extrinsic curvature of

t=constant hypersurfaces,

Kij =1

2N(∂tgij − ∇iNj− ∇jNi),(2.2)

with Nibeing the shift vector, Rij the intrinsic curvature

tensor, ∇ithe spatial covariant derivative with respect

to gij , and εijk =√gijk the spatial Levi-Civita tensor

with ijk being the total antisymmetric tensor. The most

important feature of the spatial covariant gravity is that

it is only invariant under the three-dimensional spatial

diﬀeomorphism, which breaks the time diﬀeomorphism.

Normally, the violation of the time diﬀeomorphism can

lead to an extra degree of freedom, in addition to the

two tensorial degree of freedom in GR. Indeed, it has

been veriﬁed that the spatial covariant gravity described

by the action (2.1) can propagate up to three dynamical

degrees of freedom [48]. In [49,50], the above action has

also been extended by introducing ˙

Nin the Lagrangian

through 1

N(˙

N−Ni∇iN). Since such terms does not

contribute to the gravitational waves at quadratic order,

we will not consider them in this paper.

There are a lot approaches to construct the gravita-

tional theories with spatial covariance. In this paper,

we adopt the approach used in [85] which constructs the

Lagrangians of the theory by using the linear combina-

tions of the extrinsic curvature Kij , intrinsic curvature

Rij , as well as their spatial derivatives and derivatives of

the spatial metric itself. Then, up to the fourth order in

TABLE I. The building blocks of spatial covariant gravities up to the fourth order in derivatives of hij , where dt,dsare the

number of time and spatial derivative respectively, and d=dt+dsdenotes the total numbers of time and spatial derivatives.

Here ω3(Γ) denotes the spatial gravitational Chern-Simons term, and ω3(Γ) = εijk(Γm

jl ∂jΓl

km +2

3Γn

ilΓl

jmΓm

kn) with Γk

ij =

2gkm(∂jgmj +∂jgij −∂mgij ) being the spatial Christoﬀel symbols. The terms in this Table is the same as those of Table. I in

[85] except the two new terms ω3(Γ) and ω3(Γ)K.

d(dt, ds) operators

0 (0,0) 1

1(1,0) K

(0, 1) -

(2, 0) Kij ,K2

(1, 1) -

(0, 2) R

(3, 0) Kij KjkKi

k,Kij Kij K,K3

(2, 1) εijkKi

l∇jKkl

(1, 2) ∇i∇jKij ,∇2K,Rij Kij ,RK

(0, 3) ω3(Γ)

(4, 0) Kij KjkKi

kK,Kij Kij 2,Kij Kij K2,K4

(3, 1) εijk∇mKi

nKjmKkn, εijk∇iKj

mKk

nKmn, εijk∇iKj

lKklK

(2, 2) ∇kKij ∇kKij ,∇iKjk∇kKij ,∇iKij ∇kKk

j,∇iKij ∇jK,∇iK∇iK,Rij Ki

kKjk,RKij Kij ,Rij Kij K,RK2

(1, 3) εijkRil∇jKk

l, εijk∇iRj

lKkl,ω3(Γ)K

(0, 4) ∇i∇jRij ,∇2R, Rij Rij , R2

derivatives of hij , we have the building blocks as shown

in Table. Ithat are all scalars under transformation of

spatial diﬀeomorphisms. Then the general action of the

gravitational part will be given by [85]

Sg=Zdtd3x√gNL(0) +L(1) +L(2) +L(3) +L(4)

+˜

L(3) +˜

L(4),(2.3)

where L(0),L(1),L(2),L(3), and L(4) are the parity-

preserving terms, which are given by

L(0) =c(0,0)

1,(2.4)

L(1) =c(1,0)

1K, (2.5)

L(2) =c(2,0)

1Kij Kij +c(2,0)

2K2+c(0,2)

1R, (2.6)

L(3) =c(3,0)

1Kij KjkKi

k+c(3,0)

2Kij Kij K+c(3,0)

3K3

+c(1,2)

1∇i∇jKij +c(1,2)

2∇2K+c(1,2)

3Rij Kij

+c(1,2)

4RK, (2.7)

L(4) =c(4,0)

1Kij KjkKi

kK+c(4,0)

2Kij Kij 2

+c(4,0)

3Kij Kij K2+c(4,0)

4K4

+c(2,2)

1∇kKij ∇kKij +c(2,2)

2∇iKjk∇kKij

+c(2,2)

3∇iKij ∇kKk

j+c(2,2)

4∇iKij ∇jK

+c(2,2)

5∇iK∇iK+c(2,2)

6Rij Ki

kKjk

+c(2,2)

7RKij Kij +c(2,2)

8Rij Kij K+c(2,2)

9RK2

+c(0,4)

1∇i∇jRij +c(0,4)

2∇2R+c(0,4)

3Rij Rij

+c(0,4)

4R2,(2.8)

and ˜

L(3) and ˜

L(4) are parity-violating terms which are

given by

L(3) =c(2,1)

1εijkKi

l∇jKkl +c(0,3)

1ω3(Γ),(2.9)

L(4) =c(3,1)

1εijk∇mKi

nKjmKkn +c(3,1)

2εijk∇iKj

mKk

nKmn

+c(3,1)

3εijk∇iKj

lKklK+c(1,3)

1εijkRil∇jKk

+c(1,3)

2εijk∇iRj

lKkl +c(1,3)

3ω3(Γ)K. (2.10)

All the coeﬃcients like c(dt,ds)

iare functions of tand N.

Note that in Table. Iand Eqs. (2.9) and (2.10), we add

the spatial Chern-Simons term ω3(Γ) and its coupling to

K, which are absent in the original action in [85]. It is

interesting to note that the above action reduces to GR

if one imposes

c(2,0)

1=c(0,2)

1=−c(2,0)

2=M2

2,(2.11)

with all other coeﬃcients c(dt,ds)

ibeing setting to zero.

The spatial covariant gravity described in the above

action can represent a very general framework for de-

scribing the propagations of GWs in the low-energy ef-

fective gravities with Lorentz or parity violation. To our

knowledge, a lot of modiﬁed gravities can be casted in the

framework of the spatial covariant gravity. In addition, it

is shown that one in general can relate the spatial covari-

ant gravity to the scalar-tensor theories in the unitary

gauge [47,51].

III. GWS IN SPATIAL COVARIANT

GRAVITIES

Let us investigate the propagation of GWs in the spa-

tial covariant gravities with the action given by (2.3).

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PolarizedprimordialgravitationalwavesinspatialcovariantgravitiesTaoZhua;b,WenZhaoc;d,andAnzhongWangeaInstitutefortheoreticalphysicsandcosmology,ZhejiangUniversityofTechnology,Hangzhou,310032,ChinabUnitedCenterforGravitationalWavePhysics(UCGWP),ZhejiangUniversityofTechnology,Hangzhou,310032,Chinac...

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Polarized primordial gravitational waves in spatial covariant gravities Tao ZhuabWen Zhaocdand Anzhong Wange aInstitute for theoretical physics and cosmology

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