Integrated perturbation theory for cosmological tensor fields. II. Loop corrections Takahiko Matsubara1 2 1Institute of Particle and Nuclear Studies High Energy Accelerator Research Organization KEK Oho 1-1 Tsukuba 305-0801 Japan

2025-05-05 0 0 420.75KB 21 页 10玖币

Integrated perturbation theory for cosmological tensor ﬁelds. II. Loop corrections

Takahiko Matsubara1, 2, ∗

1Institute of Particle and Nuclear Studies, High Energy Accelerator Research Organization (KEK), Oho 1-1, Tsukuba 305-0801, Japan

2The Graduate Institute for Advanced Studies, SOKENDAI, Tsukuba 305-0801, Japan

(Dated: September 23, 2024)

In the previous paper [1], the nonlinear perturbation theory of the cosmological density ﬁeld is generalized

to include the tensor-valued bias of astronomical objects, such as spins and shapes of galaxies and any other

tensors of arbitrary ranks which are associated with objects that we can observe. We apply this newly devel-

oped method to explicitly calculate nonlinear power spectra and correlation functions both in real space and

in redshift space. Multidimensional integrals that appear in loop corrections are reduced to combinations of

one-dimensional Hankel transforms, thanks to the spherical basis of the formalism, and the ﬁnal expressions are

numerically evaluated in a very short time using an algorithm of the fast Fourier transforms such as FFTLog.

As an illustrative example, numerical evaluations of loop corrections of the power spectrum and correlation

function of the rank-2 tensor ﬁeld are demonstrated with a simple model of tensor bias.

I. INTRODUCTION

The large-scale structure (LSS) of the Universe, probed

by galaxies and other astronomical observables such as weak

lensing, 21 cm emission and absorption lines and so forth,

plays an essential role in cosmology. The LSS is comple-

mentary to the cosmic microwave background (CMB) radi-

ation, which mainly probes the early stages of the Universe

around the time of decoupling. The information contained in

the temperature ﬂuctuations in the CMB have been extracted

in exquisite details, and the temperature and polarization maps

obtained by the Planck satellite determined the precise values

of the cosmological parameters [2]. Beyond the cosmologi-

cal information extracted from the CMB, the LSS oﬀers a lot

of opportunities to obtain further information of the Universe

which is contained mainly in the lower-redshift Universe. In

addition, the representative values of the cosmological param-

eters determined by the Planck are obtained by combining the

observational data of LSS, such as the scale of baryon acoustic

oscillations (BAO) and weak lensing, as the CMB data alone

has degeneracies among cosmological parameters. The accel-

eration of the Universe due to dark energy is also an eﬀect that

can only be probed in the lower-redshift Universe.

While most of the physics in CMB is captured by the lin-

ear perturbation theory of ﬂuctuations, the properties of LSS

are more aﬀected by nonlinear evolutions, as the scales of

interest become smaller. On one hand, the physics of long-

wavelength modes in the density ﬂuctuations in the LSS can

still be captured by the linear perturbation theory, and the am-

plitude of density ﬂuctuations simply grows according to the

linear growth factor. However, the number of independent

modes of density ﬂuctuations included in a survey is limited

by the ﬁniteness of the survey volume V, i.e., the number of

independent Fourier modes with a wave number magnitude k

roughly scales as ∼k3Vin three-dimensional surveys. On the

other hand, short-wavelength modes are aﬀected by nonlinear

evolutions of the density ﬁeld, which mix up diﬀerent scales

of Fourier modes, and thus their analysis becomes much more

∗tmats@post.kek.jp

diﬃcult. The fully nonlinear evolutions cannot be analytically

solved because of the extreme mixture of modes, and extrac-

tion of the cosmological information from the fully nonlinear

density ﬁeld is diﬃcult. While one can resort to the numerical

simulations to solve the nonlinear evolutions, the information

contents of initial condition and cosmology are largely lost

in the nonlinearly evolved ﬁeld [3], compared to the linearly

evolved ﬁeld.

The transition scales of the linear and nonlinear ﬁelds are

roughly around 10–20 h−1Mpc at the present time of z=0,

and the transition scales become smaller at an earlier time of

higher redshift. On the transition scales, although the linear

theory does not quantitatively apply, the nonlinearity is still

weak and the mixing of Fourier modes is not complicated

enough. Only a countable number of modes are eﬀectively

mixed, and the nonlinear perturbation theory is applicable in

such a situation. Therefore, the theory of nonlinear perturba-

tion theory of density ﬁeld [4] is expected to play an important

role in the analysis of the LSS, in the era of large surveys in

the near future when the suﬃciently large number of Fourier

modes on the transition scales are expected to be available. In

addition, the density ﬂuctuations even on large scales, which

have been traditionally considered as the linear regime, are

more or less aﬀected by weak nonlinearity, and it is critically

important to estimate such subtle eﬀects in the era of preci-

sion cosmology. A representative example of the last case is

the nonlinear smearing eﬀects of the BAO in correlation func-

tions of galaxies around ∼100 h−1Mpc [5], which is used as a

powerful standard ruler to probe the expansion history of the

Universe and the nature of dark energy.

The higher-order perturbation theory beyond the linear the-

ory has been extensively developed for matter distributions in

the past several decades [6–13]. However, the distribution of

matter is not the same as that of objects that we can observe,

and the mass density ﬁeld is dominated by the dark matter in

the Universe. The bias between distributions of matter and ob-

servable objects is one of the most important concepts in un-

derstanding the large-scale structure of the Universe. In order

that the nonlinear perturbation theory can be compared with

observations, the eﬀect of bias is an indispensable element

that should be included in the theories with the predictabil-

ity of the observable Universe. There are many attempts to

arXiv:2210.11085v4 [astro-ph.CO] 20 Sep 2024

include the eﬀect of bias in the nonlinear perturbation theory

(for a recent review, see Ref. [14]). Understanding the bias

from the ﬁrst principle is extremely diﬃcult because of the

full nonlinearity of the problem and extremely complicated

astrophysical processes in the galaxy formation, and so forth.

The concept of bias has usually been discussed in the con-

text of number density ﬁelds of astronomical objects such as

galaxies, as probes of the underlying matter density ﬁeld. In

this case, the bias corresponds to a function, or more prop-

erly a functional, of the underlying mass density ﬁeld to give

a number density ﬁeld of the biased objects. Thus the func-

tion(al) of the bias has a scalar value in accordance with that

the density of biased objects is a scalar ﬁeld. In the previ-

ous work of Paper I [1], the concept of bias in the nonlinear

perturbation theory is generalized to the case that the bias is

given to a tensor ﬁeld. The number densities of objects are

not the only probes of the density ﬁelds in the LSS. For ex-

ample, galaxy spins and shapes are in principle determined by

the mass density ﬁelds through, e.g., tidal gravitational forces,

and other physical quantities. Recently, interests in statistics

of the galaxy sizes and shapes, or intrinsic alignments, are

growing as probes of the LSS of the Universe [15–19], and

analytical modelings of galaxy shape statistics by the nonlin-

ear perturbation theory have also been introduced [20–24].

Motivated by these recent developments, we generalize the

nonlinear perturbation theory to predict statistics of biased

ﬁelds with an arbitrary rank of tensor in Paper I [1]. We adopt

the spherical decomposition of the tensor ﬁeld, which plays an

important role in the formalism. This method of decomposi-

tion has been already adopted in the perturbation theory in lit-

erature to investigate the clustering of density peaks [25] and

galaxy shapes [23]. In the last two references, the coordinates

system of the spherical basis is chosen so that the third axis is

aligned with a radial direction of the correlation function, or a

direction of wave vector of perturbations in Fourier space. In

contrast, we do not ﬁx the coordinates system in the spherical

basis, and explicitly keep the rotational covariance apparent

throughout the formulation. The basic formalism to calculate

the power spectrum and higher-order spectra of tensor ﬁelds

of arbitrary ranks by the nonlinear perturbation theory to arbi-

trary orders is described in Paper I.

Many diﬀerent methods have been considered in the liter-

ature to include the bias in the nonlinear perturbation theory

[14]. Most methods rely on a local or semilocal ansatz of the

bias function which relates the mass density ﬁeld and the bi-

ased density ﬁeld. The locality or semilocality of the relation

is given in either Eulerian or Lagrangian space of the density

ﬁeld. However, (semi)local biases in Eulerian and Lagrangian

spaces are not compatible with each other in general, because

the dynamically nonlinear evolution by gravity is essentially

nonlocal. Therefore, the bias relation should be given by a

nonlocal functional, in either Eulerian or Lagrangian space,

and the (semi)local Ans¨atze are approximations to the reality.

A general formulation to systematically incorporate the non-

local bias into the nonlinear perturbation theory is provided

by the integrated perturbation theory (iPT) [26, 27]. The local

and semilocal Ansatz of the bias can also be derived from this

formulation by restricting the form of bias functional in the

class of local or semilocal function. Moreover, the iPT also

provides a natural way of including the eﬀect of redshift space

distortions, which should be taken into account for predicting

observable statistics in redshift surveys. Our formulation of

Paper I is built upon and generalizes the method of iPT and

establishes a nonlinear perturbation theory of tensor ﬁelds in

general. Paper I describes the basic formulation of the theory

and gives some results of lowest-order approximations of the

perturbation theory.

In this second paper of the series, we apply the formula-

tion of Paper I to concretely calculate the one-loop correc-

tions of the perturbation theory. The strategy of the calcula-

tion is fairly straightforward according to Paper I. Some tech-

niques are introduced to reduce the higher-dimensional inte-

grals to the lower ones, which are generalizations of an ex-

isting method using a fast Fourier transform applied to the

nonlinear perturbation theory [28]. In particular, all the nec-

essary integrations to evaluate the one-loop corrections in the

perturbation theory with the (semi)local models of tensor bias

reduce to essentially one-dimensional Hankel transforms. As

an illustrative example, we calculate the power spectrum and

correlation function with one-loop corrections for a simple

model of a rank-2 tensor which is biased from spatially second

derivatives of the gravitational potential in Lagrangian space.

This paper is organized as follows. In Sec. II, the propa-

gators, elements of the nonlinear perturbation theory, in the

spherical basis of our formalism are calculated, up to neces-

sary orders for evaluating one-loop corrections of the power

spectrum and correlation function. In Sec. III, our main result,

the one-loop approximations of the power spectra of the ten-

sor ﬁeld are explicitly derived in analytic forms, both in real

space and in redshift space. In Sec. IV, a simple example of

the tensor bias with a semilocal model is explicitly calculated

and numerically evaluated. Conclusions are given in Sec. V.

In the Appendix, a formal expression of the all-order power

spectrum of the tensor ﬁeld is derived beyond the one-loop

approximation.

II. PROPAGATORS OF TENSOR FIELDS AND LOOP

CORRECTIONS

The fundamental formulation of the iPT of tensor ﬁelds is

described in Paper I [1]. One of the essential ingredients of the

theory is the evaluation of propagators, with which statistics of

tensor ﬁelds, such as the power spectrum, bispectrum, corre-

lation functions, etc. are represented. Several examples in rel-

atively simple cases with lowest-order approximation are ex-

plicitly derived in Sec. V of Paper I. In this section, we further

derive the propagators that are required to evaluate next-order

approximation with loop corrections. We cite many equations

from Paper I, which readers are assumed to have in hand.

A. Invariant propagators

The propagators of tensor ﬁelds can be represented by ro-

tationally invariant functions as well as the renormalized bias

functions as extensively explained in Paper I. First, we sum-

marize the essential equations and introduce various quantities

and functions which are used in later sections. The concepts

of propagators and renormalized bias functions are explained

in detail in Sec. III of Paper I. The details of the deﬁnitions are

not explained here. In short, they are response functions of the

nonlinear evolutions from the initial density ﬁeld. Below we

summarize their properties which are essential to derive the

main equations in later sections of this paper.

1. Real space

The reduced propagators (see Sec. III A of Paper I for their

deﬁnitions) up to the second order are represented by invariant

functions as

Γ(1)

Xlm(k)=(−1)l

√2l+1

Γ(1)

Xl (k)Clm(ˆ

k),(1)

for the ﬁrst order, and

Γ(2)

Xlm(k1,k2)=X

l1,l2

Γ(2) l

Xl1l2(k1,k2)Xl1l2

lm (ˆ

k1,ˆ

k2),(2)

for the second order. In the above equations, the index Xspec-

iﬁes the class of tensor-valued objects in general, such as the

density (scalar), angular momentum (vector) or shape (ten-

sors) of a certain type of galaxies etc. The functions ˆ

Γ(1)

Xl (k)

and ˆ

Γ(2) l

Xl1l2(k1,k2) are the invariant propagators which are in-

variant under the coordinates rotations. We use the spherical

harmonics with Racah’s normalization,

Clm(θ, ϕ)≡r4π

2l+1Ylm(θ, ϕ)=s(l−m)!

(l+m)! Pm

l(cos θ)eimϕ,

(3)

instead of standard normalization of spherical harmonics Ylm.

The arguments of the spherical harmonics are alternatively

represented by a unit vector n, instead of the corresponding

angular coordinates (θ, ϕ) of n. In the above notation, the

Condon-Shortley phase is included in the associated Legen-

dre polynomials Pm

las

l(x)=(−1)m

2ll!1−x2m/2dl+m

dxl+m1−x2l.(4)

The function of the last factor in Eq. (2) is the bipolar spherical

harmonics with an appropriate normalization,

Xl1l2

lm (n1,n2)=(l l1l2)m1m2

mCl1m1(n1)Cl2m2(n2),(5)

where azimuthal indices m1and m2are summed over from −l1

to +l1and from −l2to +l2, respectively, without summation

symbols following the Einstein convention, and

(l l1l2)m1m2

m=(−1)m1+m2 l l1l2

m−m1−m2!(6)

is a Wigner’s 3 j-symbol.

It is convenient to use the metric tensor of spherical basis,

deﬁned by

gmm′

(l)=g(l)

mm′=(−1)mδm,−m′,(7)

where δm,−m′is the Kronecker’s symbol which is unity when

m+m′=0 and is zero otherwise. With this notation, Eq. (6)

is represented by

(l l1l2)m1m2

m=gm1m′

(l)gm2m′

(l)(l l2l3)mm′

1m′

2,(8)

where

(l1l2l3)m1m2m3= l1l2l3

m1m2m3!(9)

is the usual 3 j-symbol, and Einstein’s summation convention

for the azimuthal indices m,m1, etc. are assumed throughout

this paper, unless otherwise stated. The two spherical metric

tensors satisfy gmm′′

(l)g(l)

m′′m′=δm

m′. Similarly to Eq. (8), we un-

derstand that the azimuthal indices can be raised or lowered

by the spherical metric tensor, for example,

(l1l2l3)m3

m1m2

=gm3m′

(l)(l1l2l3)m1m2m′

3,(10)

and so forth.

With the above notation, the complex conjugate of the

spherical harmonics is represented by

C∗

lm(n)=gmm′

(l)Clm′(n),(11)

and similarly, that of the bipolar spherical harmonics is repre-

sented by

Xl1l2∗

lm (ˆ

k1,ˆ

k2)=(−1)l1+l2+lgmm′

(l)Xl1l2

lm′(ˆ

k1,ˆ

k2).(12)

The orthonormality relation of spherical harmonics is given

Zd2n

4πC∗

lm(n)Cl′m′(n)=δll′

2l+1δm

m′,(13)

and those of bipolar spherical harmonics is given by

Zd2ˆ

4π

d2ˆ

4πXl1l2∗

lm (ˆ

k1,ˆ

k2)Xl′

1l′

l′m′(ˆ

k1,ˆ

k2)

δll′δl1l′

1δl2l′

2δ△

l1l2l

(2l+1)(2l1+1)(2l2+1) δm

m′,(14)

where δ△

l1l2lis unity when the set of integers (l1,l2,l) satisﬁes

triangle inequality |l1−l2| ≤ l≤l1+l2, and is zero otherwise.

The invariant propagator of the second order satisﬁes an

interchange symmetry,

Γ(2) l

Xl2l1(k2,k1)=(−1)l1+l2+lˆ

Γ(2) l

Xl1l2(k1,k2).(15)

The above expansions of Eqs. (1) and (2) are inverted by the

above orthonormality relations, and we have

Γ(1)

Xl (k)=(−1)l√2l+1gmm′

(l)Zd2ˆ

4πˆ

Γ(1)

Xlm(k)Clm′(ˆ

k) (16)

for the ﬁrst-order propagator, and

Γ(2) l

Xl1l2(k1,k2)=(2l1+1)(2l2+1) gmm′

(l)

×Zd2ˆ

4π

d2ˆ

4πˆ

Γ(2)

Xlm(k1,k2)Xl1l2

lm′(ˆ

k1,ˆ

k2) (17)

for the second-order propagator. In practice, one can always

represent the propagators with polypolar spherical harmonics

in the form of Eqs. (1) and (2), and can readily read oﬀthe

expression of the invariant functions from the results.

2. Redshift space

In redshift space, the propagators also depend on the direc-

tion of the line of sight. We can decompose the dependence

on the line of sight in spherical harmonics, together with the

dependencies on the directions of wave vectors. For the ﬁrst-

order propagator in redshift space, we have

Γ(1)

Xlm(k;ˆ

z;k, µ)=X

lz,l1

Γ(1) l lz

Xl1(k, µ)Xlzl1

lm (ˆ

z,ˆ

k),(18)

where ˆ

zis the direction to the line of sight. We assume the

distant-observer approximation, and the direction to the line

of sight is ﬁxed in space. Unlike the common practice, we

do not ﬁx the line of sight in the third direction of the coor-

dinates, but allow to point in any direction. In the above ex-

pression, the direction cosine to the line of sight, µ≡ˆ

z·ˆ

k, is

included. This dependence is not necessarily included there,

because the angular dependence on the left-hand side (lhs) of

the equation can be completely expanded into spherical har-

monics (Sec. IV B 2 of Paper I). However, it is sometimes

convenient to leave some part of the dependence in the form

of the direction cosine µbetween the line of sight and the di-

rection of the wave vector. Which part of the dependence is

kept unexpanded is arbitrary. Even though the arguments k

and µof the propagator on the lhs of Eq. (18) is a function of

kand ˆ

z, the explicit arguments of kand µspecify which parts

of the angular dependence in these parameters are unexpanded

in the spherical harmonics on the right-hand side (rhs).

For the second-order propagator in redshift space, we have

Γ(2)

Xlm(k1,k2;ˆ

z;k, µ)

lz,l1,l2,L

Γ(2) l lz;L

Xl1l2(k1,k2;k, µ)Xlzl1l2

L;lm (ˆ

z,ˆ

k1,ˆ

k2),(19)

where

Xl1l2l3

L;lm (n1,n2,n3)=(−1)L√2L+1(l l1L)m1M

m(L l2l3)m2m3

×Cl1m1(n1)Cl2m2(n2)Cl3m3(n3) (20)

is the tripolar spherical harmonics with an appropriate normal-

ization. In the argument of propagators, the variables k=|k|

and µ=ˆ

z·ˆ

kare given by the total wave vector k=k1+k2,

which is optionally allowed to be included, because keeping

the angular dependencies in these variables signiﬁcantly sim-

pliﬁes the analytic calculations. The invariant propagator of

the second order satisﬁes an interchange symmetry,

Γ(2) l lz;L

Xl2l1(k2,k1;k, µ)=(−1)l1+l2+Lˆ

Γ(2) l lz;L

Xl1l2(k1,k2;k, µ).(21)

The complex conjugate of the tripolar spherical harmonics is

given by

Xl1l2l3∗

lm (n1,n2,n3)=(−1)l+l1+l2+l3gmm′

(l)Xl1l2l3

lm′(n1,n2,n3),(22)

and the orthonormality relation is given by

Zd2n1

4π

d2n2

4π

d2n3

4πXl1l2l3

L;lm (n1,n2,n3)Xl′

1l′

2l′

L′;l′m′(n1,n2,n3)

=(−1)l+l1+l2+l3δll′δl1l′

1δl2l′

2δl3l′

3δLL′δ△

l l1Lδ△

Ll2l3

(2l+1)(2l1+1)(2l2+1)(2l3+1) g(l)

mm′.(23)

Applying the above orthonormality relations for bipolar and

tripolar spherical harmonics, Eqs. (14) and (23), the ﬁrst- and

second-order propagators in redshift space of Eqs. (18) and

(19) are inverted as

Γ(1) l lz

Xl1(k, µ)=(2lz+1)(2l1+1) gmm′

(l)

×Zd2ˆz

4π

d2ˆ

4πˆ

Γ(1)

Xlm(k;ˆ

z;k, µ)Xlzl1

lm′(ˆ

z,ˆ

k) (24)

for the ﬁrst-order propagator, and

Γ(2) l lz;L

Xl1l2(k1,k2;k, µ)=(2lz+1)(2l1+1)(2l2+1) gmm′

(l)

×Zd2ˆz

4π

d2ˆ

4π

d2ˆ

4πˆ

Γ(2)

Xlm(k1,k2;ˆ

z;k, µ)Xlzl1l2

L;lm′(ˆ

z,ˆ

k1,ˆ

k2) (25)

for the second-order propagator. In the above equations, the

angular integrations on the rhs in variables kand µare for-

mally ﬁxed, as if they do not depend on ˆ

k,ˆ

k1,ˆ

k2and ˆ

z. The

expressions of Eqs. (24) and (25) should be considered as for-

mal, and should only be used in order to invert the expansions

of Eqs. (18) and (19), formally ﬁxing the variables kand µon

both sides of the equations.

In practice, one can always represent the propagators with

polypolar spherical harmonics in the form of Eqs. (18) and

(19), and can readily read the expression of the invariant

functions from the results. The invariant propagators in real

space, Eqs. (16) and (17), correspond to ˆ

Γ(1)

Xl (k)=ˆ

Γ(1) l0

Xl (k),

Γ(2) l

Xl1l2(k1,k2)=ˆ

Γ(2) l0;l

Xl1l2(k1,k2) when the propagators do not

contain redshift-space distortions.

B. First-order propagators with loop corrections

1. The propagators of integrated perturbation theory

The ﬁrst-order and second-order propagators, ˆ

Γ(1)

Xlm(k) and

Γ(2)

Xlm(k1,k2) are evaluated by the iPT. The results are given by

(Sec. III A of Paper I)

Γ(1)

Xlm(k)=c(1)

Xlm(k)+[k·L1(k)]c(0)

Xlm

+Zd3p

(2π)3PL(p)k·L1(−p)c(2)

Xlm(k,p)

+k·L1(−p)[k·L1(k)]c(1)

Xlm(p)

+k·L2(k,−p)c(1)

Xlm(p)

2k·L3(k,p,−p)c(0)

Xlm

+k·L1(p)k·L2(k,−p)c(0)

Xlm(26)

for the ﬁrst-order propagator, where PL(k) is the linear power

spectrum of the mass density ﬁeld, and

Γ(2)

Xlm(k1,k2)=c(2)

Xlm(k1,k2)

+[k12 ·L1(k1)]c(1)

Xlm(k2)+[k12 ·L1(k2)]c(1)

Xlm(k1)

+{[k12 ·L1(k1)] [k12 ·L1(k2)]+k12 ·L2(k1,k2)}c(0)

Xlm.

(27)

for the second-order propagator. It is suﬃcient to include one-

loop corrections only in the ﬁrst-order propagator ˆ

Γ(1)

Xlm, and

not in the second-order propagator ˆ

Γ(2)

Xlm, because the second-

order propagator always appears with loop integrals in evalu-

ating the nonlinear power spectrum [26, 27].

In the above equations, c(0)

X,c(1)

Xlm(k), and c(2)

Xlm(k1,k2) are

the renormalized bias functions, which are determined from

complicated physics involving nonlinear dynamics of galaxy

formation, etc., and the deﬁnitions of these functions are given

in Sec. III A of Paper I. The vector functions Ln(k1,...,kn)

are the kernel functions of the Lagrangian perturbation theory.

In real space, they are explicitly given by [29, 30]

L1(k)=k

k2,(28)

L2(k1,k2)=3

k12

k1221− k1·k2

k1k2!2,(29)

L3(k1,k2,k3)=1

3h˜

L3(k1,k2,k3)+cyc.i,(30)

L3(k1,k2,k3)

=k123

k1232









71− k1·k2

k1k2!21− k12 ·k3

k12k3!2

−1

31−3 k1·k2

k1k2!2

+2(k1·k2)(k2·k3)(k3·k1)

k12k22k32









k123

k1232×(k1×k23)(k1·k23)

k12k2321− k2·k3

k2k3!2,

(31)

where k12 =k1+k2,k123 =k1+k2+k3, and +cyc.corre-

sponds to the two terms which are added with cyclic permuta-

tions of each previous term. Weak dependencies on the time

in the kernels are neglected [31, 32]. In Ref. [31], complete

expressions of the displacement kernels of Lagrangian per-

turbation theory up to the seventh order including the trans-

verse parts are explicitly given, together with a general way

of recursively deriving the kernels including weak dependen-

cies on the time in general cosmology and subleading grow-

ing modes. The redshift-space distortions can be simply taken

into account as well in the Lagrangian perturbation theory, just

replacing the displacement kernel in real space given above

with the linearly mapped kernels [33]

Ln→Ls

n=Ln+n f (ˆ

z·Ln)ˆ

z,(32)

where f=dln D/dln a=˙

D/HD is the linear growth rate,

D(t) is the linear growth factor, a(t) is the scale factor, and

H(t)=˙a/ais the time-dependent Hubble parameter, and the

unit vector ˆ

zdenotes the line-of-sight direction, as already

mentioned above.

The ﬁrst-order propagators Γ(1)

Xlm in the lowest order approx-

imation, i.e., without loop corrections, are explicitly derived in

Sec. IV C 3 of Paper I. The results are given by

Γ(1)

Xl (k)=c(1)

Xl (k)+δl0c(0)

X(33)

in real space, and

Γ(1)l lz

Xl1(k, µ)=δlz0δl1lhc(1)

Xl (k)+δl0(1 +fµ2)c(0)

Xi.(34)

in redshift space. In the last expression in redshift space, the

dependence on the direction cosine µis kept unexpanded into

spherical harmonics. Another expression with the complete

expansion is given in Sec. IV C 1 and 2 of Paper I. Below we

consider the generalization of the lowest-order results, and de-

rive necessary propagators for evaluating one-loop corrections

in the nonlinear power spectrum.

2. Real space

In the expression of propagators, Eqs. (26)–(31), many

scalar products between pairs of wave vectors appear, and they

can always be represented by the spherical harmonics by ap-

plying the addition theorem of the spherical harmonics,

Pl(n·n′)=C∗

lm(n)Clm(n′),(35)

where n,n′are normal vectors, and Pl(x) is the Legendre

polynomial. Einstein’s summation convention for the index

mis applied as before. For example, we have

k·L1(±p)=±k·p

p2=±k

pC∗

1m(ˆ

k)C1m(ˆ

p).(36)

Similarly, all the directional dependencies on the wave vec-

tors are expanded into the spherical harmonics. In such re-

ductions, representing simple polynomials by Legendre poly-

nomials, such as

1=P0(x),x=P1(x),x2=1

3P0(x)+2

3P2(x),

x3=3

5P1(x)+2

5P3(x),x4=1

5P0(x)+4

7P2(x)+8

35 P4(x),

(37)

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Integratedperturbationtheoryforcosmologicaltensorfields.II.LoopcorrectionsTakahikoMatsubara1,2,∗1InstituteofParticleandNuclearStudies,HighEnergyAcceleratorResearchOrganization(KEK),Oho1-1,Tsukuba305-0801,Japan2TheGraduateInstituteforAdvancedStudies,SOKENDAI,Tsukuba305-0801,Japan(Dated:September23,20...

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Integrated perturbation theory for cosmological tensor fields. II. Loop corrections Takahiko Matsubara1 2 1Institute of Particle and Nuclear Studies High Energy Accelerator Research Organization KEK Oho 1-1 Tsukuba 305-0801 Japan

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