# Thermodynamics in Loop Quantum Cosmology

Authors: Li-Fang Li; Jian-Yang Zhu
Journal: Advances in High Energy Physics
 (2009)

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## Abstract

Loop quantum cosmology (LQC) is very powerful to deal with the
behavior of early universe. Moreover, the
effective loop quantum cosmology gives a successful description of
the universe in the semiclassical region. We consider the apparent
horizon of the Friedmann-Robertson-Walker universe as a
thermodynamical system and investigate the thermodynamics of LQC
in the semiclassical region. The effective density and effective
pressure in the modified Friedmann equation from LQC not only
determine the evolution of the universe in LQC scenario but
also are actually found to be the thermodynamic
quantities. This result comes from the energy definition in
cosmology (the Misner-Sharp gravitational energy) and is
consistent with thermodynamic laws. We prove that within the
framework of loop quantum cosmology, the elementary equation of
equilibrium thermodynamics is still valid.


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## Body

## 1. Introduction

Loop quantum gravity (LQG) [1–5] is a nonperturbative and
background independent quantization of gravity. One of the important and
successful applications of LQG is loop quantum cosmology (LQC). It has been
shown that LQC resolves the problem of classical singularities both in an
isotropic model [6]
and in a less symmetric homogeneous model [7]. LQC also gives a quantum suppression of classical
chaotic behavior near singularities in Bianchi-IX models [8, 9]. Furthermore, it has been shown that nonperturbative
modification of the matter Hamiltonian leads to a generic phase of inflation
[10–12]. On the other hand, we know that spacetime
thermodynamic properties result from, in a sense, quantum effects of spacetime
[13]. Therefore, it is
very interesting and important to investigate the thermodynamics of quantum gravity.
There are indeed many results on thermodynamical implications of loop quantum
gravity [14, 15], but very little discussion
on the thermodynamics of loop quantum cosmology, which will be the focus of the
present paper.In LQC, the phase space for spatially flat universe is
spanned by coordinatesc=γa˙,
being the gravitational gauge connection, and p=a2,
being the densitized triad. γ is the Barbero-Immirzi parameter, and a is the scale factor of the universe. In the
LQC scenario, the evolution of the universe can be divided into three phases.
(i) Initially, there is a truly discrete quantum phase which is described by a
difference equation [16]. In this stage, the universe may be nonequilibrium
due to the fast quantum evolution. (ii) As the volume of universe increases and
matter density decreases, the discrete quantum effect becomes less important,
the universe enters an intermediate semiclassical phase in which the evolution
equations take a continuous form but with modifications due to nonperturbative
quantum effects [10].
In this stage, the effective loop quantum cosmology is valid, and it is
reasonable to approximately treat the universe as a thermodynamic system in
equilibrium. The thermodynamic properties are subject to quantum effects, and
we are most interested in this stage. (iii) Finally, there is the classical
phase in which the quantum effects vanish and the usual continuous equations
describing cosmological behavior are established and so is the usual
thermodynamics [17, 18].In recent years, many authors endeavor to study the
thermodynamics [14, 15] of black holes in the semiclassical context and the
framework of LQG. Now people have studied the effective theory, though not
complete yet due to quantum back-reaction [19], in loop cosmology. Thus, there is considerable
interest in the thermodynamic properties of the universe in LQC scenario. With
the universe being nonstationary and evolving, the thermodynamics is different
from the black hole systems. It is conceivable that some of the mechanisms
involved in establishing thermal equilibrium may be modified, especially when
the expansion time scale becomes comparable to that of the matter processes
responsible for establishing the thermal equilibrium.To resolve this issue, we develop a procedure to study
the thermodynamic properties at the apparent horizon of the
Friedmann-Robertson-Walker (FRW) universe. Our analysis is based on the
effective theory of LQC and the homogeneous and isotropic cosmological
settings.  Fundamentally, comparing the modified
Friedmann equation with the ordinary one, we derive the effective density and
pressure of the perfect fluid. Then, we introduce the Misner-Sharp energy
[20], which is
different from the other forms of energy for its relation to the structure of
the spacetime and one can relate it to the Einstein equation. From the
expression of the Minser-Sharp energy, we get the physical meaning of the
effective density. Furthermore, from the conservation law, we get the physical
meaning of the effective pressure. To understand the intrinsic essence of the
effective density and pressure, we prove that within the framework of loop
quantum cosmology, the fundamental relation of thermodynamics is still valid.This paper is organized as follows. In Section2, we
briefly review the framework of the effective LQC. We present the dynamics in
terms of effective density and pressure, which will be defined there. Then in
Section 3, we obtain the thermodynamic origin of the effective density and
pressure. Some elementary consequences are also noted. In Section 4, we
conclude this paper with some discussions on the further implications for
phenomenology.


## 2. A Short Review of Effective Theory of
LQC

In this section, we give a short review of the
effective framework of LQC before we study the thermodynamics. The classical
form of Hamiltonian for spatially flat universe is(2.1)Hcl=−38πγ2pc2+HM(p,ϕ). There are two
kinds of important modifications in the LQC. The first one is based on the
modification to the behavior of inverse scale factor below a critical scale
factor (the inverse volume modification). The second one essentially comes from
the discrete quantum geometric nature of spacetime (quadratic modification), as
predicted by the LQG. Besides these two kinds of
corrections, there is also the more generic quantum back-reaction which gives
rise to effective potentials. In this paper, we only consider the corrections
coming from the quadratic modification. But it is worthy to note that our result
is valid for general effective potential. With the quadratic modification, the
effective Hamiltonian becomes [21–23](2.2)Heff=−38πγ2μ¯2psin2(μ¯c)+HM(p,ϕ).The variable μ¯ corresponds to the dimensionless length of the
edge of the elementary loop and is given by(2.3)μ¯=ξpλ,where ξ>0 and λ depend on the particular scheme in the
holonomy corrections. In this paper, we take μ¯-scheme, which gives(2.4)ξ2=23πγlp2and λ=−1/2,
where lp is Planck length. With this effective
Hamiltonian, we have the canonical equation(2.5)p˙={p,Heff}=−8πγ3∂Heff∂cor(2.6)a˙=sin(μ¯c)cos(μ¯c)γμ¯.We define energy density and
pressure of matter [24] as(2.7)ρ=a−3HM,P=−13a−2∂HM∂a.Combining with the constraint on
Hamiltonian, Heff=0,
we obtain the modified Friedmann equation(2.8)H2=8π3ρ(1−ρρc),where H≡a˙/a  denotes the Hubble rate, and ρc≡3/8πγ2μ¯2p is the quantum critical density. Compared with
the standard Friedmann equation, we can define the effective
density(2.9)ρeff=ρ(1−ρρc).Taking derivative of (2.8) and
also using the conservation equation of matter ρ˙+3H(ρ+P)=0,
we obtain the modified Raychaudhuri equation(2.10)a¨a=H˙+H2=−4π3{ρ(1−ρρc)+3[P(1−2ρρc)−ρ2ρc]}.Compared with the standard
Raychaudhuri equation, we can define the effective pressure(2.11)Peff=P(1−2ρρc)−ρ2ρc.For different quantum
corrections, the ρeff and Peff may have different
forms. But our following statement is still valid.
In terms of the effective density and the effective pressure, the modified
Friedmann, Raychaudhuri, and conservation equations
take the following forms:(2.12)H2=8π3ρeff,(2.13)a¨a=H˙+H2=−4π3(ρeff+3Peff),(2.14)ρ˙eff+3H(ρeff+Peff)=0.Till now, ρeff and Peff  are nothing but mathematical symbols to denote
the coupling of matter and gravity. They still lack a thermodynamic origin, as
noted by the authors of [25]. In the following, we will explore their intrinsic
meaning in the thermodynamic sense and discuss some elementary implications
based on above effective framework of LQC. But our result is more general and
independent on the form of ρeff and Peff which may be different when considering
different quantumcorrections and quantum back
reactions.


## 3. Thermodynamics in LQC

Let us begin with the effective LQC description of the
universe evolution. For a spatially homogenous and isotropic universe described
by the FRW metric, the line element is represented by(3.1)ds2=−dt2+a(t)2(dr2+r2dΩ2),where a(t) is the scale factor of the universe, t
is the cosmic
time, and dΩ2 is the metric of sphere with unit radius.
Thus, it is clear that all dynamical behaviors of the universe are determined
by the scale factor a(t).
The metric (3.1) can be rewritten as(3.2)ds2=habdxadxb+r˜2dΩ,where r˜=a(t)r and x0=t, x1=r and the two-dimensional metric hab=diag(−1,a2).For the FRW universe, the dynamical apparent
horizon,  (without the whole evolution history of the universe, one
cannot know whether there is a cosmological event horizon.
However, apparent horizon always exists in the FRW
universe since it is a local quantity of spacetime.) defined as the sphere with vanishing
expansion [18], can be
determined by the relation hab∂ar˜∂br˜=0 as(3.3)RA=1H,which coincides with the Hubble
horizon in this case. According to the definition of the surface
gravity(3.4)κ=12−h∂a(−hhab∂br˜),its explicit evaluation at the
dynamical apparent horizon RA of the FRW universe reads(3.5)κ=−1RA(1−R˙A2HRA).We now introduce the Misner-Sharp spherically
symmetric gravitational energyE, or simply the MS energy, defined in natural
units by [20](3.6)E=r2(1−hab∂ar∂br),which is the total energy (not
only the passive energy) inside the sphere with radius r.
The MS energy is a pure geometric quantity and is extensively used in
literatures about thermodynamics of spacetime [26–28]. Its physical meaning and the comparison to the ADM
mass and Bondi-Sachs energy have been given in [29]. For spherical space-time,
Brown-York energy [30]
agrees with the Liu-Yau energy [31], but they both differ from the MS energy. For
example, for the four-dimensional Reissner-Nordström black hole, the MS energy
differs from the Brown-York or Liu-Yau mass by a term which is the energy of
the electromagnetic field inside the sphere, as discussed in [29].In terms of the apparent horizon radius (3.3), the
Friedmann equation (2.12) can be rewritten as(3.7)1RA2=8π3ρeff.Now we consider the MS energy
(3.6) within the apparent horizon r=RA of the FRW universe, given by(3.8)E=RA2.Using (3.7), we
get(3.9)E=4πRA33ρeff=ρeffV.It shows that it is reasonable
to say that ρeff is indeed the energy density, not just a
mathematical symbol. Then from the conservation equation (2.14), which implies
energy and momentum conservation, it is also reasonable to take Peff as pressure. That is to say that the
gravitational effects contribute to energy density and pressure in the
thermodynamical sense. In the following, we will find that this physical
meaning is consistent with the fundamental relation of thermodynamics, which in
turn supports this physical interpretation.To examine the fundamental relation of thermodynamics
in the setup of LQC, we consider the apparent horizon of the FRW universe as a
thermodynamical system. An ansatz is made. Assume that the apparent horizon has
an associated Hawking temperatureT and entropy S expressed, respectively, as(3.10)T=κ2π,S=A4,where A=4πRA2 is the area of the apparent horizon.Taking derivative of the energy equation (3.9) and
using conservation equation (2.14), we get(3.11)dE=4πRA2ρeffR˙Adt−4πRA3H(ρeff+Peff)dt.Beside this, by taking
derivative of the Friedmann equation (3.7) and using the conservation equation
(2.14), we get the differential form of the Friedmann equation(3.12)1RA3dRA=4π(ρeff+Peff)Hdt.Considering the surface gravity
on the apparent horizon (3.5), we can multiply both sides of the above equation by
a factor RA(1−R˙A/2HRA) and get(3.13)κ2πd(πRA2)=−4πRA3(ρeff+Peff)H(1−R˙A2HRA)dt.Therefore, in virtue of the
ansatz (3.10) and combining the (3.11) and (3.13), one gets(3.14)dE=TdS+WdV,where W=(ρeff−Peff)/2 is the work density if we take ρeff and Peff as the energy density and pressure physically
[26]. Again, we see
that taking ρeff and Peff  as the energy density and pressure in the
thermodynamical sense is consistent with the fundamental relation of
thermodynamics. However, if we take ρ and P as the thermodynamical quantities, we will
find that(3.15)dE=TdS+W′dV+ρρcPdVwith work density W′=(ρ−P)/2.
This equation means that the fundamental relation of thermodynamics breaks down
unless we consider that the work term now does not
take the form suggested by [26]. But this complicated expression for work term seems
not reasonable. In contrast, the physical interpretation that ρeff and Peff  are thermodynamical quantities in LQC is
consistent with the fundamental relation of thermodynamics. Or to say in terms
of thermodynamical quantities ρeff and Peff,
the fundamental relation of thermodynamics is valid in LQC too.


## 4. Conclusion

In Conclusion, we have investigated the thermodynamic
properties of the universe in LQC scenario and found that the fundamental
relation of thermodynamics is valid in the effective LQC scenario. We found
that the effective densityρeff and the effective pressure Peff are not only a symbol to denote the coupling
between the gravity and matter, but also actually
the energy density and pressure in thermodynamical sense. This result comes
from the energy definition in cosmology (the Misner-Sharp spherically symmetric
gravitational energy) and is consistent with fundamental relation of
thermodynamics.In the following, we briefly comment on the physical
meanings from the expressions of the effective energy density and pressure.
When the energy density is much smaller than the quantum critical density (ρ≪ρc), the effective density ρeff and the effective pressure Peff come back to the traditional ones, that is, ρ and P,
and the classical picture is recovered. Apart from the contribution of the
matter sector, the effective density and pressure also receive the contribution
from the spatial curvature. Also note that while for large volumes the spatial curvature
is negligible to the density and pressure, for small volumes it is important.
Since the ρeff and Peff have thermodynamic meanings, and
nonperturbative modification to the matter field at short scales implies
inflation which also means a violation of the strong energy condition [32], we can expect that the
wormhole solution maybe a normal object in effective LQC. Similarly, the
spectrum of fluctuation of ρeff may be more important than ρ itself which contributes to the large-scale
structure of the universe. All these are interesting topics for further study.



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