The quantum effect on Friedmann equation in FRW universe

We study the modified Friedmann equation in the Friedmann-Robertson-Walker universe with quantum effect. Our modified results mainly stem from the new entropy-area relation and the novel idea of T. Padmanabhan, who considers the cosmic space to be emerged from the cosmic time progresses, so that the expansion rate of the universe is determined by the difference of degrees of freedom between the holographic surface and the bulk inside. We also discuss the possibility of having bounce cosmological solution from the modified Friedmann equation in spatially flat geometry.


I. INTRODUCTION
In the 1970s, the thermodynamic property of black holes has been proposed [1][2][3], and it reveals that the gravitational dynamics is entwined with thermodynamics. Inspired by Bekenstein's entropy-area theorem [1], Bardeen, Carter and Hawking put forward the four thermodynamical laws of black hole systems [2]. In 1995, Jacobson considered the Einstein's field equation as an equation of state. Afterwards, he reproduced the Einstein's field equation by demanding that the fundamental relation δQ = T dS holds for all local Rindler causal horizons through each spacetime point, and δQ and T are treated as the energy flux and Unruh temperature, respectively, felt by an accelerated observer inside the horizon [4]. In 2010, Verlinde defined gravity as an entropic force due to the changes of the information related to the positions of the materials, and the space is emergent based on the holographic principle in his discussions [5]. Moreover, Verlinde's proposal has been applied to reproduce the Friedmann equation into brane cosmology [6] and Friedmann-Robertson-Walker(FRW) universe [7], respectively.
On the other hand, it was addressed in [8][9][10][11][12] that the Friedmann equation can be modified by a bounce solution of the universe as in loop quantum cosmology(LQC). Then the authors of [13] attempted to derive the Friedmann equation by borrowing the clausius relation, i.e., δQ = T dS and an entropy-area relation with quantum correction, however, they failed to reproduce the same modified Friedmann equation as that in LQC with bounce solution. Luckily, the difficulty was overcome by the authors of [14], where they proposed a modified dispersion relation at quantum phenomenological level and then obtained the modified Friedmann equation for a bounce solution to the flat FRW universe in LQC. It was found that the role of their modified dispersion relation is to explicitly modify the clausius relation. This proposal has also been extended into the spatially curved cases and the corresponding modified entropy-area relations have been derived [15].
However, the above studies only involved in the gravity as an emergent phenomenon rather than the spacetime itself as an emergent structure. This situation has been improved by Padmanabhan. In details, he proposed in [16] that the accelerated expansion of the universe is related with the difference between the surface degrees of freedom (N sur ) and the bulk degrees of freedom (N bulk ) in a region of space, i.e., ∆V = ∆t(N sur − N bulk ), where V is the Hubble volume and t is the cosmic time in Planck units. Moreover, the standard evolution of the universe was also reproduced directly from the proposed relation. This proposal inspired plenty of related studies and remarkable progress [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33]. However, in the framework of Padmanabhan's conjecture, the study of quantum effect is missing. So, it is interesting to introduce the quantum effect to Padmanabhan's conjecture and study the related cosmology.
Thus, in this paper, we will introduce the modified dispersion relation in the framework of Padmanabhan's conjecture and derive the modified Friedmann equation. Then we will analyze whether the quantum effect in Padmanabhan's conjecture will bring in the modified Friedmann equation (1) with bounce solution. It is notable that starting from the clausius relation to the apparent horizon along with the modified dispersion relation, one can easily get the modified Friedmann equation with bounce solution to the FRW universe [14,15], but the answer is not direct in Padmanabhan's conjecture in the emergent universe. Our study will give an insight to the answer.
Our paper is organized as follows. In next section we briefly review Padmanabhan's idea that the cosmic space is emergent as cosmic time progresses, and give the standard Friedmann equation governing the dynamical evolution of the FRW universe. Then in section III, we will analyze the modified Friedmann equation from Padmanabhan's conjecture based on modified entropy-area relation. Finally, we will give our summary and discussions in section IV. In this paper, we use the natural units with c = = k B = 1.

II. THE EMERGENCE OF COSMIC SPACE OF THE FRW UNIVERSE
In this section, we will give a brief review on the process of obtaining standard Friedmann equation in the emergent universe, which was addressed by Padmanabhan in [16]. The main idea is that the expansion of the universe, (or the emergence of space) tends to fulfill the holographic equipartition condition, which is stated that the number of degrees of freedom (N bulk ) inside the Hubble volume is equal to the number of degrees of freedom (N sur ) on the spherical surface of Hubble radius, i.e., N bulk = N sur . So in our asymptotically de Sitter universe, the natural law governing the emergence of space in an infinitesimal interval dt is where V = 4π 3H 3 is the Hubble volume and t is the cosmic time. For a spatially flat FRW universe with Hubble constant H and apparent horizon r A = 1/H, we have where S = A 4G = π/GH 2 is the entropy of the apparent horizon, and where in the second equality, we recall the horizon temperature T = H 2π and Komar energy |E| = −(ρ + 3p)V for accelerating part with dark energy 1 . Subsequently, one can reduce (2) intoä which is the standard dynamical Friedmann equation of flat FRW universe in general relativity. Furthermore, recalling the continuity equatioṅ and integrating (5) gives us the standard Friedmann equation Note that in [13] the integration result is H 2 + k a 2 = 8πGρ 3 with general geometry, where the authors interpreted the integration constant k as the spatial curvature of the FRW universe.

SPACE OF THE FRW UNIVERSE
In this section, we will apply the proposal described in last section to study the quantum effect on the Friedmann equation. We only consider the quantum effect at the phenomenological level and borrow the modified dispersion relation(MDR) [14] sin(ηl p E) ηl p = p 2 + m 2 .
Here p and E are the momentum and energy of a particle with mass m, respectively. The Planck length is l p = √ 8πG = 1/M p where M p is the Planck mass. η is a dimensionless parameter and η → 0 goes to the standard dispersion relation E 2 = p 2 + m 2 .
With the use of thermodynamical description on the apparent horizon, the authors of [14] derived the modified Friedmann equation of a spatially flat universe from the MDR (8).
Later, the extended study in general FRW universe with k = 0, ±1 was presented in [15].
Here, we will derive the modified Friedmann equation by following the steps of emergent cosmic space shown in last section. According to the study in [15], the MDR (8) modified the entropy for the first energy branch as where A = 4πr 2 A = 4π/(H 2 + k/a 2 ) is the area of the apparent horizon at the classical level. To proceed, we define an effective apparent horizon area with the quantum effect Note that when η → 0, A is equal to A and recovers the usual result.
Moreover, the volume (V ) and the area (A) of the apparent horizon of an n-sphere with radius r A satisfies [34] Then one can think that the change of the effective volume mainly stems from the change of the effective area, so that we have the time evolution of the effective volume of the FRW universe [34] from which we can obtain the effective volume Also, when η → 0, V = is the usual Hubble volume.
We move on to calculate N bulk in the bulk and N sur in the boundary. Considering the Hawking temperature T = 1 2πr A and E = −(ρ + 3p) V with dark energy in the bulk, we obtain The statistical physics has shown that N sur can be calculated from the entropy [18] N sur = 4S M = 4πr 2 Substituting (12), (14) and (15) into (2), we get The expression above looks very complicated, however, with k = 0, we haveṙ A = 1−ä a r 2 A , so that equation (16) can be reduced into 2 where in the second line, we keep until the second order of because it is a small quantity.
The form (17) is the modified dynamical Friedmann equation for the flat FRW universe, which reduces to the standard dynamical Friedmann equation when η → 0. Further combining the continuity equation (6) with (17), we obtain the other modified Friedmann equatioṅ Here we also set the integration constant to be vanished. Again, η → 0 in (18) reproduces the standard result of flat FRW universe.
r 2 A which makes it difficult to simplify equation (16). We hope to solve this problem in near future.
In order to analyze whether (18) admits a bounce solution, we define so that equation (18) can be rewritten as equation (1) for bounce solution. The unsolved integral in ρ c makes it difficult to give a reliable conclusion, however, we can at least give some discussions. On one hand, when ρ c in (19) is positive, Eqs. (17) and (18)  We note that for the second energy branch whose entropy is −S M [15], the procedures above are straightforward and the modified Friedmann equation are the same as equations (17) and (18). However, for this energy branch, the effective volume ( V ), and the area ( A) of the apparent horizon are negative, which are not physical.

IV. SUMMARY AND DISCUSSIONS
In this paper, we studied the quantum effect on the Friedmann equation for the flat FRW universe with the use of Padmanabhan's conjecture in the emergent universe. We obtained the modified Friedmann equations (17) and (18) with the quantum correction on the dispersion relation (8). For the closed (k = 1) and open (k = −1) universes, we found it is difficult to simplify the dynamical equation in our process, but we still see the quantum effect on equation (16) which supposes to be the modified Friedmann equation. We also argued the condition under which the modified Friedmann equation (18) admits a bounce solution in the flat universe.
It is worthy to point out that in our paper, the modified Friedmann equation was only obtained in the flat FRW universe with k = 0, which may imply that the key equation (2) is not the basic equation of the emergent universe and it may have to be corrected at the quantum level. This is an interesting point we will study in the near future.