ЖЭТФ, 2022, том 162, вып. 3 (9), стр. 373-376

HOLOGRAPHIC DESCRIPTION OF THE DISSIPATIVE MODEL OF

UNIVERSE WITH CURVATURE

I. Brevika*, A. V. Timoshkinb,c**

^a Department of Energy and Process Engineering, Norwegian University of Science and Technology

N-7491 Trondheim, Norway

^b Institute of Scientific Research and Development, Tomsk State Pedagogical University (TSPU )

634061, Tomsk, Russia

^c International Centre of Gravity and Cosmos,

Tomsk State University of Control Systems and Radioelectronics (TUSUR)

634050, Tomsk, Russia

Received April 21, 2022,

revised version April 29, 2022

Accepted May 4, 2022

DOI: 10.31857/S0044451022090097

Furthermore all known models of HDE are a conse-

EDN: EKQOFH

quence of the Nojiri-Odintsov model [14-16]. The de-

velopment of various approaches and generalizations of

A description of the accelerated expansion of the

HDE is given in various reviews [17-19]. Holographic

Universe [1, 2] is possible with the help of dark energy

theory is shown to be in agreement with astronomical

or through modified gravity [3,4]. Dark energy can be

observations [17, 20-24].

represented within the framework of the cosmological

The motivation of this study is that, despite the

model using a dark fluid with an unusual equation of a

observations tell us that the universe is essentially

state [5, 6].

flat [25], it is nevertheless impossible to exclude by

The most realistic cosmological picture of the evo-

100% certainty that the universe has a finite curvature;

lution of the Universe is obtained taking into account

cf., for instance, the discussion in Refs. [26, 27].

the properties of the viscosity of the dark fluid. This is

Further, we will consider the Friedmann-

due to the fact that in the era of dark energy the occur-

rence of cosmological singularities with the final time

Robertson-Walker (FRW) metric with a finite

curvature of our Universe, and will investigate the

of formation is possible. The influence from (bulk) vis-

cosity affects the behavior of the universe near cosmo-

consequences of this assumption for the Friedmann

equation. We will obtain a holographic description

logical singularities [7, 8]; cf. the Big Rip phenomena

in cosmologies of type II, III, and IV (what is called

of cosmological models associated with an inhomoge-

a N ojiri-Odintsov-Tsujikawa classification of singulari-

neous viscous dark fluid, and will discuss the singular

ties is given in [9]) and is important in connection with

behavior of the Universe determined by this model.

turbulence effects [10]. Thus, the assumption about no

According to the holographic principle, all physi-

viscous (ideal) fluid is inaccurate.

cal quantities in the universe including the dark energy

We will consider the holographic description of the

can be described by specified values on the space-time

universe, which implies that all information about the

boundary [28,29]. The typical HDE density can be de-

system parameters can be described in the form of a

scribed via the Planck mass M_p and a characteristic

hologram, associated with the surface area of cosmic

length L_IR (infrared radius) [11]

space [11].

The generalized holographic dark energy (HDE)

ρ_hol = 3c²M2pL-2IR,

(1)

model was proposed by Nojiri and Odintsov [12, 13].

^* E-mail: iver.h.brevik@ntnu.no

where c is an arbitrary dimensionless positive parame-

^** E-mail: alex.timosh@rambler.ru

ter while the universe is expanding.

373

I. Brevik, A. V. Timoshkin

ЖЭТФ, том 162, вып. 3 (9), 2022

Let us consider the homogeneous and isotropic

We apply the holographic principle for cosmological

FRW universe in which the metric has the form

models with a constant value of the thermodynamic

parameter ω(ρ, t)) = ω₀ and various values of the bulk

)

( dr

ds² = -dt² + a²(t)

+ r²dΩ² ,

(2)

viscosity ζ(H, t), and distinguish between two cases.

1 - λr²

Case 1. Fluid model with constant ω(ρ, t) = ω₀ and

where a(t) is the scale factor and dΩ² = dθ² +sin² θdϕ²

constant viscosity ζ(H, t) = ζ₀.

is the metric of the space. The quantity λ characterizes

In this simple case the EoS will take the form

the curvature of the three-dimensional space.

As is known, the metric (2) describes a homoge-

p = ω₀ρ - 3ζ₀H.

(6)

neous and isotropic expanding space. We may have

λ = 0 (spatially flat space), λ = +1 (closed universe),

The solution of the energy conservation equation is

or λ = -1 (open universe). The open universe expands

√

forever; the flat universe also expands forever but at

a(t) = C₁ + C₂e^ζ0^t + θt,

(7)

t → +∞ the expansion occurs at constant speed; the

closed universe expands to a certain instant, after which

where θ is a parameter associated with the spatial cur-

the expansion is replaced by a compression leading to

vature,

ζ₀ is a modified viscosity parameter, and C₁, C₂

a collapse.

are arbitrary constants. The expression (7) for the scale

The Friedmann equation for a one-component fluid

factor contains the correction θt, which is associated

with nonzero curvature has the form

with the spatial curvature.

For the Hubble function we will obtain

H² =

ρ-

(3)

C₂

ζ₀eζ0t + θ

H(t) =

(8)

where ρ is the HDE density, k² = 8πG with G the New-

2 C₁ + C₂e^ζ0^t + θt

tonian gravitational constant, and H(t) = a(t)/a(t) is

Let us analyze the Hubble function with respect to the

the Hubble function.

singular behavior. If C₁ and C₂ are positive, then in a

The infrared radius L_IR can be identified with size

flat universe (θ = 0) there is no singularity. In an open

of the horizon of particles L_p or with size of the event

universe (θ < 0) it is possible to form a singularity after

horizon L_f [12]. However, not all ways of choosing in-

a finite time span (type Big Rip [6]).

frared radius will lead to an accelerated expansion of

In the asymptotic limit t → 0 (inflationary epoch)

the universe so that the choice of an infrared radius is

the Hubble function tends to the constant value

not arbitrary.

The holographic energy density is known to be ba-

C₂ ζ₀ + θ

sically the same as the energy of the infrared (IR) radi-

H(t) =

C₂

ζ₀ + C₁

ation. If we identify the energy ρ in equation (3) with

the HDE ρ_hol in equation (1), then we obtain the first

while in the other limit t → ∞ (dark energy epoch) it

Friedmann equation in another form:

goes again to a constant H(t) →

ζ₀ regardless of the

√(_c

)₂

spatial curvature. In both cases, the universe is in a

H =

(4)

state of accelerated expansion.

L_IR

a²

The particle horizon L_p for nonzero spatial curva-

Further, we will suppose that the viscous dark fluid

ture is

∫

driving the evolution of the universe, has a holographic

L_p =

√

(9)

origin.

αe^βx + x

We will investigate the cosmological models of a

viscous dark fluid, obeying the inhomogeneous equa-

where α = C₂e-C1β, β =

ζ₀/θ, and θ = 0.

tion of state (EoS) in a FRW universe [5, 30]. Dissipa-

In the particular case when C₁ = 0, C₂ = 1 (α = 1),

tive processes are described by the bulk viscosity in the

we obtain in the initial stage inflation

form [31].

√

Let us assume that the universe is filled with a one-

L_p(t → 0) =

1 + γt(

1 + γt - 1),

(10)

component fluid, obeying the energy conservation law

ρ + 3H(ρ + p) = 0.

(5)

where γ = θ +

ζ₀.

374

ЖЭТФ, том 162, вып. 3 (9), 2022

Holographic description of the dissipative model of universe with curvature

The Hubble function can be expressed in terms of

Summarizing the main results of this work, we have ob-

the particle horizon and its derivatives as [13]

tained, on the basis of the holographic principle, a de-

scription of two distinguished viscous dark fluid cosmo-

L_p

L_p - 1

logical models, and we have discussed the singular be-

H =

(11)

L_p

L2p

L²

havior of the universe when determined by these mod-

els. We have shown that the inclusion of nonzero cur-

As a result, the energy conservation law can be ex-

vature in the Friedmann equation leads to additional

pressed in a holographic form as

singularities of type Big Rip in the Universe. The ap-

L_p

)₂

plication of the holographic method has led to theoret-

( L_p -1

+ω₀

ical predictions in good agreement with astronomical

L_p

L2p

L_p

observations [32].

[

] L_p -1

2λ(ω₀ + 1)(C₂ ζ₀eζ0t

+ θ) -

ζ₀

= 0.

(12)

L_p

Funding. This work was supported by Rus-

sian Fund for Fundamental Studies, Project

Case 2. Fluid model with constant ω(ρ, t)

= ω₀

No. 20-52-05009 (A. V. T.).

and viscosity proportional to the Hubble function,

ζ(H, t) = 3τH.

In this case the EoS takes the form

The full text of this paper is published in the English

version of JETP.

p = ω₀ρ - 9τH²,

(13)

where τ is a positive dimensional constant.

The solution of the differential equation (5) becomes

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