Thermodynamic solution of the homogeneity isotropy and atness puzzles and a clue to the cosmological constant Latham Boyle1and Neil Turok12

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Thermodynamic solution of the homogeneity, isotropy and ﬂatness puzzles

(and a clue to the cosmological constant)

Latham Boyle1and Neil Turok1,2

1Perimeter Institute for Theoretical Physics, Waterloo, Ontario, Canada, N2L 2Y5

2Higgs Centre for Theoretical Physics, University of Edinburgh, Edinburgh, Scotland, EH8 9YL

(Dated: October 2022)

We obtain the analytic solution of the Friedmann equation for fully realistic cosmologies includ-

ing radiation, non-relativistic matter, a cosmological constant λand arbitrary spatial curvature κ.

The general solution for the scale factor a(τ), with τthe conformal time, is an elliptic function,

meromorphic and doubly periodic in the complex τ-plane, with one period along the real τ-axis,

and the other along the imaginary τ-axis. The periodicity in imaginary time allows us to compute

the thermodynamic temperature and entropy of such spacetimes, just as Gibbons and Hawking did

for black holes and the de Sitter universe. The gravitational entropy favors universes like our own

which are spatially ﬂat, homogeneous, and isotropic, with a small positive cosmological constant.

INTRODUCTION

Soon after the discovery of black hole thermodynam-

ics [1–4], Gibbons and Hawking [5] showed that one could

elegantly compute the temperature and entropy of a gen-

eral (charged, spinning) black hole, and of de Sitter (dS)

spacetime, by noticing that these spacetimes are periodic

in the imaginary time direction (see also the preceding

work of Gibbons and Perry [6, 7]). In this paper, we ﬁnd

the exact solution of the Friedmann equation for a general

FRW universe including arbitrary amounts of radiation,

non-relativistic matter (including baryons and dark mat-

ter), a cosmological constant and spatial curvature [8, 9].

Noticing that the scale factor a(τ) is periodic in the imag-

inary τdirection, we are able to compute the correspond-

ing temperature and gravitational entropy. Remarkably,

the entropy obtained in this way favors universes like

our own: homogeneous, isotropic and spatially ﬂat, with

small positive cosmological constant.

In earlier papers [10, 11], we obtained similar results

in the context of a simpliﬁed cosmology with radiation,

a cosmological constant λ, and spatial curvature κbut

without non-relativistic matter. Our new result strength-

ens the case that we have been developing [10–15] for a

simpler theory of the universe, not requiring inﬂation.

GENERAL SOLUTION FOR a(τ)

The action for Einstein gravity coupled to matter is

S=Zd4x√−gR

2−λ+Lmatter(1)

where λis the dark energy (or cosmological constant).

We shall use Planck units with ~=c=kB= 8πGN= 1

throughout. Now take the FRW line element

ds2=a2(τ)(−n2dτ2+γij dxidxj) (2)

where nis the lapse and γij is the metric on a maximally-

symmetric 3-space of constant curvature κ; and take the

matter to consist of radiation with energy density r/a4

and non-relativistic matter (baryons and dark matter)

with energy µ/a3, where rand µare positive constants.

Then the action becomes

S=V3Zdτ −3˙a2

n+nV (a)(3)

where V3is the comoving spatial volume [28], ˙a=da/dτ,

and

V(a) = −λa4+ 3κa2−µa −r(4a)

=−λ(a4−3˜κa2+ ˜µa + ˜r) (4b)

where we have deﬁned ˜κ≡κ/λ, ˜µ=µ/λ, ˜r=r/λ.

Varying with respect to n(and then choosing the gauge

n= 1) yields the Friedmann equation

3 ˙a2+V(a) = 0 (5)

while varying with respect to a(and again taking n= 1)

yields the acceleration equation

6¨a+V0(a)=0.(6)

First consider the critical “Einstein static universe”

(ESU) solutions, with constant scale factor aesu: these

require κ > 0 and λ > 0, with the parameters related by

2˜κ3

˜µ2=(1+8x)+(1+ 8

3x)q1+ 32

1+3q1 + 32

(x≡˜κ˜r

˜µ2).(7)

Eq. (7) deﬁnes an important boundary between diﬀerent

dynamical phases. If the lhs of Eq. (7) is greater than the

rhs (which can only happen when κand λare both pos-

itive), we call it a “turnaround” universe: its curvature

κis suﬃciently positive to cause a non-singular reversal

from expansion to re-contraction, or contraction to re-

expansion. Otherwise (i.e., if λ > 0 and κ<κesu, the

positive, critical value set by Eq. (7)), the universe ex-

pands monotonically from the bang (a= 0) to the dS-like

boundary (a→ ∞), or the reverse.

arXiv:2210.01142v2 [gr-qc] 6 Oct 2022

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摘要：

Thermodynamicsolutionofthehomogeneity,isotropyandatnesspuzzles(andacluetothecosmologicalconstant)LathamBoyle1andNeilTurok1;21PerimeterInstituteforTheoreticalPhysics,Waterloo,Ontario,Canada,N2L2Y52HiggsCentreforTheoreticalPhysics,UniversityofEdinburgh,Edinburgh,Scotland,EH89YL(Dated:October2022)Weobt...

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Thermodynamic solution of the homogeneity isotropy and atness puzzles and a clue to the cosmological constant Latham Boyle1and Neil Turok12

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