# General modified Friedmann equations in Rainbow flat universe, by thermodynamics

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## Abstract

We investigate the derivation of Friedmann equations in Rainbow gravity following Jacobson thermodynamic approach. We do not restrict the rainbow functions to be constant as is customarily used, and show that the first law of thermodynamics with a corresponding ‘classical’ proportionality between entropy and surface area, supplemented eventually by a ‘quantum’ logarithmic correction, are not in general sufficient to obtain the equations in flat FRW metrics.

## 1 Introduction

Jacobson [1] derived Einstein equations using a thermodynamic approach. As general relativity (GR) equations lead to Friedmann equations (FEs), the approach was adopted [2] in a cosmological setup to reach the FEs in general Friedman–Robertson–Walker (FRW) metrics.

This approach proved successful in investigating other modified gravities, such as *f*(*R*) and scalar–tensor gravities [3] and Horava–Lifshtiz [4, 5], where the FEs were reached albeit at the expense of adopting a proper definition of energy density and pressure. Recently, the authors of [6], applied the Jacobson approach within Rainbow gravity, putting aside any ‘quantum’ correction to the entropy-surface area relation, and reached the usual simple modified FEs (MFEs) mentioned in [7]. However, the simple MFEs were obtained assuming the rainbow functions as constants. The general MFEs without this restriction were obtained first within rainbow gravity in [8], and our aim in this letter is to study these general MFEs using Jacobson approach.

Moreover, and rather than adopting an involved “energy–supply vector” technique, as is usually done, we use Kodama vector [9] concept, which is a technique allowing a direct way to compute the energy flux through horizon.

## 2 Jacobson approach and Friedamnn equations

^{1}\(k=+ \ 1,0\) and \(- \ 1\) for a closed, flat and open universe respectively, \(\tilde{r}=a(t)r\) and the indices

*a*,

*b*span 0(

*t*), 1(

*r*).

*H*is the Hubble parameter, \(H\equiv \dot{a}/a\).

^{2}The apparent horizon is a causal horizon associated with the gravitational entropy and surface gravity [10] onto which one can apply the first law of thermodynamics. If we define the total energy inside the space with radius \(\tilde{r}\) by

*S*and temperature

*T*given by [11]

## 3 Friedmann equation in rainbow gravity from the first law of thermodynamics

^{3}In the context of rainbow gravity, studies were performed [17, 18] and give the result

*S*is in its turn dependent on the Probe particle’s energy via the dependence of the surface

*A*on \(\tilde{r}_A\) (Eq. 17). Substituting in the Clausius relation (Eq. 6) and using Eq. (17) we get

^{4}

To get the second MFE (Eq. 15), it suffices to add the constraint Eq. (27) to Eq. (24). We conclude thus that the Jacobson approach cannot lead to the MFEs in Rainbow gravity (Eqs. 14, 15) unless the constraint (Eq. 27) is met.

## 4 Discussion

*g*is a constant, the ‘classical’ part of the constraint is satisfied whereas the ‘quantum’ part is not met unless

*f*is inversely proportional to

*H*, but the question arises as to whether other solutions are of physical significance. In order to simplify the discussion and get exact solutions, we shall assume that the universe apparent horizon \({\tilde{r_A}}\) is far larger than the Planck length \(L_P \propto \sqrt{G}\), which is equivalent to looking at times far beyond the Planck time. In this regime, one can neglect the quantum corrections in the entropy-area relation, proportional to \(\alpha \), and so we assume the constraint of the form:

*f*,

*g*) are not independent but are related.

*f*(

*t*),

*g*(

*t*), then using Eqs. (14, 25), we reach an ODE:

*H*(

*t*), then the temporal evolution of the scale factor

*a*(

*t*). However, we are generally given \(f(\varepsilon ), g(\varepsilon )\), and unless one knows the dependence of the probe energy on time (\(\varepsilon (t)\)), we can not solve the equations of motion. In ordinary rainbow gravity, the probes are carried during cosmologically short periods of time when one can assume the probe energy \(\varepsilon (t)\) approximately constant. One exception, where one can do the computations when the probe energy varies with time, lies in the arena of radiation-dominated early universe, where the probe particles can be taken as photons, and on dimensional grounds we have [20]

*f*now as a function of \(\rho \), and by solving it we get consecutively (\(\rho (t), f(t), g(t), H(t)\)), then (

*a*(

*t*)) can be found.

*g*(

*t*). One then solves for \(\rho (t), H(t)\), using Eqs. (14, 25), but the obtained solution may not satisfy thermodynamics first law even though it might be physically meaningful regarding other considerations. If we return to the previous example (\(f=1\)), and if we take, say, \(g(t)=t\), then we find that we have \(H=\frac{\gamma }{t}\) with \(\gamma =1+ \frac{2/3}{1+\omega }>0\), giving a power law for the expansion of the universe (\(a(t)\propto t^\gamma \)). However, clearly this choice of functions does not satisfy the thermodynamic constraint, as Eq. (29) would give instead \(H=-\frac{1}{t}\). Thus, we adopt the second strategy which consists of imposing Eq. (29) and solving for

*g*by replacing

*H*in the constraint by \(H=\frac{\dot{g}}{g}+\frac{c_1 \rho ^{1/2}}{f}\) from Eq. (14). Going back to our example where \(f=1\), we find an ODE for

*g*(

*t*) to be solved and then using Eq. (32) we get \(g(\varepsilon )\):

*f*(

*t*),

*g*(

*t*), one can advance now to compute consecutively \(\rho (t), H(t), a(t)\) while being assured the found solution will satisfy thermodynamics requirements.

*f*,

*g*which meet the thermodynamic constraint Eq. (29). Let’s discuss the important case of a constant energy density (\(\dot{\rho }=0\)), which in ordinary GR corresponds to inflationary scenario. Imposing Eq. (25), we find either (\(H-\frac{\dot{g}}{g}=0\)) leading via eq. (14) to the uninteresting result of zero energy density, or \(\rho +P=0\), so we find here (\(\omega =-1\)) exactly like in GR situation. Now the thermodynamics first law via Eq. (24) and the constraint (29) would give

*f*,

*g*). If we take the first ansatz \(f=1\) then the solution of the ODE representing the constraint is of the form

*g*increases till it reaches the value 1 approximately at the end of inflation.

We summarize our findings in that imposing the validity of deducing the equations of motion by applying Thermodynamics first law in the context of generalized rainbow gravity makes the generalized rainbow functions dependent on each other.

## Footnotes

- 1.
The volume of an

*n*-dimensional unit ball is \(\Omega _{n}=\pi ^{n/2}/\Gamma (n/2+1)\). - 2.
This radius is identical to the Hubble horizon radius \(\tilde{r}_H=1/H\) for flat universe (\(k=0)\).

- 3.
In [16], a new form of ‘non-additive’ entropy, suitable for gravitational systems, was used. However, we shall not follow this approach here, and will be content with the Bekenstein–Hawking entropy.

- 4.
We are assuming that no cosmological constant \(\Lambda \), and that Newton’s constant

*G*does not vary with time. Moreover, the parameter \(\omega \) in the equation of state \(P=\omega \rho \) is also assumed constant.

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