ЖЭТФ, 2019, том 156, вып. 4 (10), стр. 581-584

QUASI-ISOTROPIC EXPANSION FOR A TWO-FLUID

COSMOLOGICAL MODEL CONTAINING RADIATION

AND STRING GAS

I. M. Khalatnikova*, A. Yu. Kamenshchika,b**, A. A. Starobinskya,c***

^a Landau Institute for Theoretical Physics, Russian Academy of Science

119334, Moscow, Russia

^b Dipartimento di Fisica e Astronomia, Università di Bologna and INFN

40126, Bologna, Italy

^c Bogolyubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research

141980, Dubna, Moscow Region, Russia

Received April 15, 2019,

revised version April 15, 2019

Accepted for publication April 16, 2019

Contribution for the JETP special issue in honor of I. M. Khalatnikov’s 100th anniversary

DOI: 10.1134/S0044451019100018

of cosmological evolution is multi-component generi-

cally, and these components may obey very different

The quasi-isotropic solution of the Einstein equa-

equations of state. Moreover, the very notion of the

tions near a cosmological singularity was first found by

equation of state appears to be not fundamental; it has

Lifshitz and Khalatnikov [1] for the Universe filled by

only a limited range of validity as compared to a more

radiation with the equation of state p = ε/3 in the early

fundamental field-theoretical description. From this

1960th. In the paper [2], we presented its generaliza-

general point of view, the generalization of the quasi-

tion to the case of an arbitrary one-fluid cosmological

isotropic solution to the case of two ideal barotropic

model. Then this solution was further generalized to

fluids with constant but different p/ε ratios seems to

the case of the Universe filled by two ideal barotropic

be a natural and important next logical step.

fluids [3].

To explain the physical sense of the quasi-isotropic

As is well known, modern cosmology deals with

solution, let us remind that it represents the most

many very different types of matter. In comparison

generic spatially inhomogeneous generalization of

with the old standard model of the hot radiation dom-

the Friedmann-Lemaître-Robertson-Walker (FLRW)

inated Universe (dubbed the Big Bang), the situation

model in which the space-time is locally FLRW-like

has been dramatically changed, first, with the devel-

near the cosmological singularity t = 0 (in particular,

opment of inflationary cosmological models which con-

its Weyl tensor is much less than its Riemann tensor).

tain an inflaton effective scalar field or/and other ex-

On the other hand, generically it is very inhomo-

otic types of matter as an important ingredient, and

geneous globally and may have a very complicated

second, with the understanding that the main part of

spatial topology. As was shown in [2, 4], such a

the non-relativistic matter in the present Universe is

solution contains three arbitrary functions of spatial

non-baryonic — cold dark matter. Furthermore, the

coordinates. From the FLRW point of view, these

appearance of brane and M-theory cosmological models

degrees of freedom represent the growing (non-de-

and the discovery of the cosmic acceleration suggests

creasing in terms of metric perturbations) mode of

that matter playing an essential role at different stages

adiabatic perturbations and the non-decreasing mode

of gravitational waves (with two polarizations) in the

^* E-mail: khalat@itp.ac.ru

^** E-mail: kamenshchik@bo.infn.it

case when deviations of a space-time metric from

^*** E-mail: alstar@landau.ac.ru

the FLRW one are not small. So, the quasi-isotropic

581

I. M. Khalatnikov, A. Yu. Kamenshchik, A. A. Starobinsky

ЖЭТФ, том 156, вып. 4 (10), 2019

solution is not a generic solution of the Einstein

A slightly different versions of the quasi-isotropic

equations with a barotropic fluid. Therefore, one

expansion were independently developed during last

should not expect this solution to arise in the course

decades which are known under the names of long-wave

of generic gravitational collapse (in particular, inside

expansion, gradient expansion, or the separate universe

a black hole event horizon). The generic solution near

approach. However, the specific of our approach is that

a space-like curvature singularity (for p < ε) has a

we consider only solutions having the local FLRW be-

completely different structure consisting of the infinite

havior near singularity at t = 0.

sequence of anisotropic vacuum Kasner-like eras with

Originally the quasi-isotropic expansion was devel-

space-dependent Kasner exponents [5-7].

oped as a technique of generation of some kind of per-

For this reason, the quasi-isotropic solution had not

turbative expansion in the vicinity of cosmological sin-

attracted much interest for about twenty years. Its new

gularity at t = 0, where the synchronous time t serves

life began after the development of successful inflation-

as a small parameter. However, the more general treat-

ary models (i.e., with “graceful exit” from inflation) and

ment of the quasi-isotropic expansion is possible if one

the theory of generation of perturbations during in-

notices that the next order of the quasi-isotropic ex-

flation, because it had immediately become clear that

pansion contains higher orders of spatial derivatives of

generically (without fine tuning of initial conditions)

metric coefficients. Thus, it is possible to construct a

scalar metric perturbations after the end of inflation re-

natural generalization of the quasi-isotropic solution of

main small in a finite (though still very large compared

the Einstein equations which would be valid not only

to the presently observable part of the Universe) region

in the vicinity of cosmological singularity, but in the

of space which is much less than the whole causally

full time range. In this case simple algebraic equa-

connected space volume produced by inflation. It ap-

tions, which one resolves to find higher orders of the

pears that the quasi-isotropic solution can be used for

quasi-isotropic approximation in the vicinity of singu-

a global description of a part of space-time after infla-

larity are substituted by differential equations, where

tion which belongs to “one post-inflationary universe”.

the time dependence of the space-time metric can be

The latter is defined as a connected part of space-time

rather complicated in contrast to the power-law be-

where the hyper-surface t = t_f (r) describing the mo-

havior of its coefficients of the original quasi-isotropic

ment when inflation ends is space-like and, therefore,

expansion.

can be made the surface of constant (zero) synchronous

In the present paper we construct this expansion for

time t by a coordinate transformation. This directly

a relatively simple two-fluid cosmological model con-

follows from the derivation of perturbations generated

taining radiation and the cosmic string gas (see e.g.

during inflation given in [8] which is valid in case of

[11]). Such a model has a technical advantage: the

large perturbations, too. Thus, when used in this con-

corresponding Friedmann equation is exactly solvable

text, the quasi-isotropic solution represents an interme-

in terms of the synchronous time t and, hence, t is a

diate asymptotic regime during expansion of the Uni-

natural parameter for constructing the quasi-isotropic

verse after inflation. The synchronous time t appearing

solution. In the second section of the paper (full text)

in it is the proper time since the end of inflation, and

we explicitly construct next order terms of the quasi-

the region of validity of the solution is from t = 0 up to

isotropic expansion for the metric tensor in the syn-

a moment in future when spatial gradients become im-

chronous reference frame, energy densities and veloci-

portant. For sufficiently large scales, the latter moment

ties of two fluids, and determine their asymptotic be-

may be rather late, even of the order or larger than the

havior at early and late times. The last section contains

present age of the Universe. Note also analogues of the

some concluding remarks. In the Appendix we apply

quasi-isotropic solution related to the Universe behav-

the developed formalism to the case of a one-fluid cos-

ior before the end of inflation which are produced by

mological model. In this case the solutions valid in the

generic globally inhomogeneous late-time asymptotic

full time range coincide with those valid in the vicinity

solutions of the Einstein equations either with a cos-

of singularity [2].

mological constant [9] or with a scalar field having the

Summing up, we can say that we have calculated

exponential potential (power-law inflation) [10]. These

explicitly the next order terms in the quasi-isotropic

solutions can be smoothly matched across the hyper-

solution for the metric tensor,

surface of the end of inflation to a post-inflationary

ds² = dt² - γ_αβ dx^αdx^β ,

(1)

quasi-isotropic solution of the type we are studying (of

course, the matter content has to be changed beyond

γ_αβ = a_αβ(x)(t + b(x)t²) + c_αβ(x, t),

(2)

this hypersurface, too).

582

ЖЭТФ, том 156, вып. 4 (10), 2019

Quasi-isotropic expansion for a two-fluid cosmological model. ..

(

b_|α

b²t² - bt + 2bt(1 + bt)ln(bt + 1) + (2bt + 1)ln(2bt + 1) -

b³

)

√

(2bt+1)b_,αb^α,

⁽ -19b²t²-85bt

4(bt+1) ln(bt+1)

- (2bt+1)

b²t²+btArch(2bt+1)+

Arch²(2bt + 1)

b⁴

24(2bt+1)

2bt+1

√

9 ln(bt+1)

61 ln(2bt+1)

3 Arch²(2bt+1)

2 Arch(2bt+1)

b²t²+bt

−

Arch²(2bt+1) ln

4(2bt + 1)

2bt + 1

bt + 1

(

)

− Arch(2bt + 1) Li₂(e-Arch(2bt+1)) -

Li₂(e-2Arch(2bt+1))

(

)

− Li₃(e-Arch(2bt+1)) -

Li₃(e-2Arch(2bt+1))

ζ_R(3)

(3)

(

√

)

bt² + t

(2bt + 1)

b²t²

+ bt

cαβ =

Arch(2bt + 1)

αβ +

2b²

(

)(

)

bt² + t

3(2bt + 1) Arch(2bt + 1)

8 ln(1 + bt) + 6 -

√

b;αβ -

a_αβb^γ

;γ

b²

b²t² + bt

(

bt²+t

2bt

-16 ln(1+bt)+ζ_R(3) - Li₃(e-2Arch(2bt+1)) - 2 Arch²(2bt + 1)Li₂(e-2Arch(2bt+1)) -

b³

1+bt

(2bt + 1) Arch³(2bt + 1)

−

Arch³(2bt + 1) -

√

+ Arch²(2bt + 1) ln(bt) + 4 Arch²(2bt + 1) ln 2 +

b²t² + bt

)(

)

Arch²(2bt + 1)

29(2bt + 1)Arch(2bt + 1)

+ Arch²(2bt + 1) ln(1 + bt) +

√

b_,αb_,β -

a_αβb_,γb^γ

(4)

2(b²t² + bt)

b²t² + bt

and for energy-densities

(

b^α|α

ε(1)R =

- b²t² + bt - bt(1 + bt)ln(1 + bt) - (1 + 2bt)ln(1 + 2bt) +

4(bt² + t)³

2(b²t² + bt)³

)

√

b_,αb^α,

+ (1 + 2bt)

b²t² + bt Arch(2bt + 1) -

(2b²t² + 2bt + 1) Arch²(2bt + 1)

b⁴(bt² + t)³

)

√

⁽115b²t²+85bt

⁽5

192b²t²+318bt+61

bt+

ln(1+bt) -

ln(1 + 2bt)-

b²t² + bt Arch(2bt+1)-

2bt + 1

3(2b²t² + 2bt + 1)

2bt + 1

−

Arch²(2bt + 1) ln

Arch²(2bt + 1) +

Arch(2bt + 1) ×

1 + bt

(

)

(

)

× Li₂(e-Arch(2bt+1))-

Li₂(e-2Arch(2bt+1))

+ Li₃(e-Arch(2bt+1))-

Li₃(e-2Arch(2bt+1))

ζ_R(3)

(5)

(

)

b^α|α

Pbt

b_,αb^α,

ε(1)S =

b²t² + 3bt - (1 + 2bt)ln(1 + 2bt) -

Arch²(2bt + 1)

2(bt² + t)²

b²(bt² + t)²

b³(bt² + t)²

(

)

⁽18b²t²+37bt-6

61(1 + 2bt)ln(1+2bt)

1+2bt

+ 5bt+

ln(1+bt) -

Arch(2bt+1) ln

1+bt

1 - 4bt - 8b²t

12b²t² + 12bt - 1

√

Arch(2bt + 1) +

Arch²(2bt + 1) + (1 + 2bt) Arch(2bt + 1) ×

b²t² + bt

16b(bt² + t)

(

)

(

)

× Li₂(e-Arch(2bt+1))-

Li₂(e-2Arch(2bt+1))

+(1+2bt) Li₃(e-Arch(2bt+1))-

Li₃(e-2Arch(2bt+1))

)

7(1 + 2bt)

ζ_R(3)

(6)

583

I. M. Khalatnikov, A. Yu. Kamenshchik, A. A. Starobinsky

ЖЭТФ, том 156, вып. 4 (10), 2019

and velocities

REFERENCES

√

bt² + t

v_R = -

Arch(2bt + 1) +

(7)

E. M. Lifshitz and I. M. Khalatnikov, Zh. Eksp. Teor.

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Fiz. 39, 149 (1960).

v_S = -

√

Arch(2bt + 1)

(8)

2b²

4b5/2

bt² + t

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