Running of effective dimension and cosmological entropy in early universe

In this paper, we suggest that the early universe starts from a high-energetic state with a two dimensional description and the state recovers to be four dimensional when the universe evolves into the radiation dominated phase. This scenario is consistent with the recent viewpoint that quantum gravity should be effectively two dimensional in the ultraviolet and recovers to be four dimensional in the infrared. A relationship has been established between the running of effective dimension and that of the entropy inside particle horizon of the universe, i.e., as the effective dimension runs from two to four, the corresponding entropy runs from the holographic entropy to the normal entropy appropriate to radiation. These results can be generalized to higher dimensional cases.


I. INTRODUCTION
It has been established that black holes have thermodynamic properties such as temperature and entropy. In particular, the thermodynamics of Schwarzschild black hole of radius R has the form where we have set = c = K B = 1. The celebrated Bekenstein-Hawking entropy A 4G is often called holographic entropy because of its proportionality with the boundary area of the system. In [1] Bekenstein and Mayo revealed a secret behind this kind of thermodynamics, that is, black holes are effectively 1 + 1 dimensional as far as entropy flow is concerned. Recently, in [2] Xiao showed more directly that the thermodynamics (1) is 1+1 dimensional in essence. Quantum gravitational (QG) particles with non-trivial phase space were introduced there in order to provide a microscopic explanation to (1), while the equation of state (EoS) w = P/ρ = 1 was derived as a byproduct.
In the context of cosmology, we mainly concern about the dominate stage of these QG particles in the evolvement history of the universe. First, we expect the veryearly high-energetic stage of the universe be controlled by QG theory. Second, by Friedmann equations the evolvement of the universe declines to lower the value of w as time increases. So it is natural to expect an early stage of the universe with w = 1 exists before the radiation dominated universe with w = 1/3. It immediately follows an interesting scenario of the evolution of the very early universe: When the dominate EoS evolves from w = 1 to w = 1/3, the effective description of the universe runs from 1 + 1 dimensional to 3 + 1 dimensional, along with a remarkable evolvement of the entropy from holographic entropy to normal entropy for radiation.
Interestingly, in recent years, there have been cumulative evidences [3][4][5] indicating that quantum gravity * Electronic address: xiaoyong@hbu.edu.cn should be effectively 1+1 dimensional at small sizes (commonly near the Planck length) and recovers to be 3 + 1 dimensional at large scales. Actually the phenomenon of short-distance dimensional reduction is obtained from various approaches to quantum gravity and various definition of effective dimensions. The universality has even been viewed as a question to be addressed [6,7]. The subtlety here is the concept of "effective dimension" which is defined and obtained by examining some specific physical behaviors sensitive to space-time dimensions. For example, the diffusive behavior of particles defines the spectral dimension, and the temperature dependence of the thermodynamic quantities determines the thermodynamic dimension. In particular, Hořava-Lifshitz gravity flows to be 2 dimensional in the ultraviolet, measured by both the generalized spectral dimension and thermodynamic dimension [5]. Hořava stressed that the behavior doesn't necessarily imply a topological change of the space-time manifold and it may mainly imply the existence of some special properties of quantum gravity such as the anisotropic scaling of space and time at short distances. Thus to avoid confusion, it would be better to say that the corresponding effective dimensions coincide with the macroscopic space-time dimension D at large scales, while deviate from it and reduce to 2 at small scales. We refer the interested reader to [6,7] for a clear discussion of the concept of effective dimensions and the phenomenon of short-distance dimensional reduction in quantum gravitational theories.
The paper is organized as follows. First we review the thermodynamics (1) can be microscopically explained by introducing the so-called QG particles and it is effectively 1 + 1 dimensional [2]. Next we study the universe filled with a single kind of constituents for simplicity and derive the concerned properties including the entropy inside particle horizon and effective dimension. Then we provide a self-consistent evolvement scenario for the early universe, with the running of effective dimension and entropy. We exhibit the general results in D = d + 1 dimensions in the Appendix.

II. QG PARTICLES AND BEKENSTEIN-HAWKING ENTROPY
The fact that black holes have temperature and entropy implies that there must be some kinds of microscopic degrees of freedom behind it. In order to provide a statistical interpretation to the thermodynamics (1), the Schwarzschild black hole is considered to be composed of microscopic particles that we called QG particles for convenience [2]. In fact such ideas are not new, for example, the charged AdS black holes have been suggested to consist of "molecules" with attractive or repulsive interactions [8,9].
We take the QG particles as massless bosonic particles with the logarithm of partition function written as [31] where β = 1/T , D(ε)dε is the number of quantum states with energy between ε and ε + dε, and g represents other possible degrees of freedom such as polarization.
For comparison, we first review the familiar statistical mechanics of photon gas system. Quantum mechanics and special relativity respectively tells that △q i △p i ≥ 2 and ε = cp (we only restore the fundamental constants temporarily). So the quantum states of a photon can be labeled by p = 2π L (m 1 , m 2 , m 3 ) with energy spectrum ε = c| p|. Then the number of quantum states between ε and ε + dε can be evaluated by D(ε)dε = 1 2π 2 V ε 2 dε. Substituting it and g = 2 into eq.(2), there is ln Ξ = π 2 45 V β 3 . The standard photon gas thermodynamics follows When the energy is of the same order of that of a black hole of the same size, Substituting it into eq.(4), we get the entropy bound for conventional quantum field theory [10-13] Now turn back to the system consisting of QG particles. According to [2], in order to account for (1), we should take The special form of D(ε)dε implies QG particles have a distinctive energy spectrum from that of photons, which may either hint some radical modification to the quantum uncertainty and energy-momentum relation or simply reflect the peculiarity of the space-time structure. Whatever, all the possible effects have been encapsulated in the the form (7). Using it as the starting point, we get and other thermodynamic properties 2G , we deduce the exact Hawing temperature T = 1 4πR . Substituting it into eq.(10), the exact Bekenstein-Hawking entropy is obtained The derivation can be generalized to higher dimensions and the necessity of w = 1 for getting the exact Bekenstein-Hawking entropy is emphasized in [2]. In the Appendix, we further generalize the derivation above to general EoS w in D = d+1 dimensions, where an intriguing knowledge is found that the exponent 2 of S ∼ M 2 is physically identical to the coefficient of the Smarr relation M = 2T S.
Observing the logarithm of partition function (8), we soon realize that it is the same as that of a 1 + 1 dimensional quantum system with length L s = 9V /G [32] Obviously this means that the Schwarzschild black hole can be described as a 1 + 1 dimensional system at least from the thermodynamic viewpoint, despite the fact that it actually lives in a 3 + 1 or higher dimensional spacetime. With size L s , the momentum can be quantized as Acting the corresponding creation operators a † i on the vacuum state, all the quantum states of the system can be listed as Then the number of quantum states satisfying the constraint ψ s |H|ψ s ≤ E bh can be counted to be e A 4G by Hardy-Ramanujan partition formula. Thereupon, all the quantum states of a black hole can be one-to-one mapped here.
Actually the thermodynamics (9)-(11) has many interesting properties that may attach to quantum gravity.
First, it has the EoS w = P/ρ = 1. The fluid with w = 1 is usually called stiff fluid for that it is the most incompressible fluid permitted by relativistic causality. In contrast, black hole is also incompressible in some sense. If you want to accumulate more matter or entropy into it, the only way you can do is to increase its horizon size.
Second, we can observe S = √ 6π EV G from these formulas. It has been shown that the expression S ∼ EV G is invariant under the T and S dualities, and it even keeps its form when we curl up some extra dimensions because G with (L c ) D−4 the volume of the extra dimensions [14].

III. COSMOLOGICAL ENTROPY AND EFFECTIVE DIMENSION
Now we have two typical kinds of constituents at hand, i.e., QG particles with w = 1 and photons with w = 1/3. However, for simplicity we start from a universe filled with a single kind of constitutes but with a general EoS w. Consider the spatial-flat, homogenous and isotropic universe which is described by the Friedmann-Lemaître-Robertson-Walker metric and obeys the Friedmann equations The scaling behavior of the entropy inside the particle horizon can be obtained in the following way [15]. From eq.(16) there is Substituting it into eq.(15), there is which in turn gives Then the physical size of the particle horizon is The energy inside the particle horizon is The gravitational mass is thus The expansion of universe is an adiabatic process, so the entropy of the constituents in a co-moving volume a(t) 3 s must be conserved. This leads to s = C 2 a(t) −3 = C 3 t − 2 1+w . Thus the entropy inside the particle horizon is Using eq. (20), the relation between the entropy S and the particle horizon area A = 4πR 2 ph can be written as It shows the available entropy for an observer in the universe increases as the particle horizon expands, with different rates depending on w. This kind of cosmological entropy was first analyzed by Fischler and Susskind in applying holographic principle to cosmology [15]. The result is reliable since only the standard cosmological principles are used. It can also be written as S ∼ E 1+3w 1+w and generalized to the D = d + 1 dimensional cases as [16]. Obviously, different kinds of constituents have different strategies of distributing energy into space.
Amazingly, even without knowing the microscopic physics of the constituents, eq.(23) reproduces the correct scaling behaviors of the entropies. In particular, Nevertheless, our calculation in last section provides a clear statistical origin for eq.(24). And below we need the corresponding thermodynamic properties to derive the temperature-time relation and the effective dimension of the universe. We first discuss the temperature-time relation T (t) of the universe and show that the thermodynamics properties are consistent with the cosmological evolvement laws. For a universe filled with photon gas, from eqs. (3) and (4) the entropy and energy density are respectively s ∼ T 3 and ρ ∼ T 4 . Since the entropy in a co-moving volume a 3 T 3 is conserved, we get the familiar relation T ∼ a −1 ∼ 1/ √ t for radiation dominated universe. And the energy density ρ ∼ T 4 ∼ a −4 is consistent with the evolvement law (17) for w = 1/3 universe. Similarly, for a universe filled with QG particles, from eqs. (9) and (10) there are s ∼ T and ρ ∼ T 2 . Because a 3 T is conserved now, we have T ∼ a −3 ∼ t −1 which means the temperature changes more abrupt than the radiation case with time increases. And ρ ∼ T 2 ∼ a −6 is consistent with the evolvement law (17) for w = 1 universe.
Then we consider the effective dimension of the universe. We have shown in last section that the QG system with w = 1 has a 1 + 1 dimensional description. More formally, we can measure the effective dimension of a system using the concept of thermodynamic dimension [6]. The spirit of thermodynamic dimension is that partition function should depend on the dimension of phase space which certainly reveals the physically relevant dimensions at the quantum level. This can be translated to the temperature dependence of energy density. Thus, for a system consisting of relativistic massless particles, the effective dimension can be defined by ρ ∼ T De or written as D e = d ln ρ d ln T . In the cosmological situation, due to ρ ∼ t −2 , we have D e = −2 d ln t d ln T showing that the effective dimension can be coded in the temperature-time relation T (t). As expected, for the radiation dominated universe with T (t) ∼ 1/ √ t the effective dimension is 4 and for QG particle dominated universe with T (t) ∼ 1/t the effective dimension is 2, written clearly as D e = 4 for w = 1 3 ; Generally the effective dimension is D e = 1 + 1 w for a D dimensional universe filled with relativistic massless particles with EoS w.

IV. THE RUNNING OF EFFECTIVE DIMENSION AND ENTROPY
The realistic universe with various kinds of constituents mixed together is far more complicated than that described above. When the constituents do not interact with each other, the energy density evolves like ρ i /ρ j ∼ a −3(wi−wj ) , representing the overall trend of the universe to dilute the constituents with large w and lower the average w, from w = 1 to the conventional w = 1/3 and w = 0 and finally approaching w = −1. On the other hand, at high-energetic stages of the universe and above some characteristic temperatures, the constituents actually interact strongly with each other and translate between. Only below the temperatures, they decouple from each other and evolve independently. Below we provide a scenario of evolvement of the universe, with emphasis on the running of effective dimension and entropy.
The first characteristic temperature we concern is roughly k B T = mc 2 with m the typical mass of nuclei. The interactions of standard model are responsible here to create massive particles of standard model from the high-energetic photos. The conventional matter has entropy density s m ∼ ρ/m. At this temperature it is of the same order of that of radiation, ρ/m ∼ T 3 , so the entropy can vary continuously in this process. Above the temperature the universe is radiation dominated, and below the temperature the radiation and matter decouple from each other, the universe evolves towards matter dominated. As for the effective dimension, the energy density for massive particles is ρ = nmc 2 + 3 2 nk B T , with the second term commonly omitted at lower temperature k B T ≪ mc 2 . The spatial dimension for massive particles is obviously 3, since they can freely move in 3 directions. More formally the fact can be read from the energy equipartition term D−1 2 nk B T . Thus the effective dimension is fixed to be 3 + 1 in this process.
Another characteristic temperature is around the Planck scale, where the entropy density s r = T 4 for radiation and s QG = 1 G T 2 for QG particles are of the same order. As we have shown in last section, the effective dimension changes from 2 to 4 when the universe evolves from w = 1 to w = 1/3. The change may happen abruptly or continuously, since we don't know the exact interaction taking charge here. The universe with EoS w = 1 should be some analogue of black hole, and black hole can translate into radiation by Hawking evaporation, so maybe the same QG mechanism also plays its role here [33]. Hawking evaporation has the ability to create all kinds of particles. The standard model particles would be broken by the ultra-high-energetic photons, while those particles that do not interact with photons would retain and be explained as dark matter.
When trace back to the time even earlier, it is basically not permitted to imagine a universe with w > 1 which violets the relativistic causality. More likely, at this stage quantum effects are so strong that the classical geometric description of space-time is not applicable any more. The stage may be controlled by highly-excited strings or the so-called string-holes as suggested in [16,17]. Besides, requiring D = 2 or w = 1 naturally leads to a scaleinvariant spectrum for cosmological perturbations even without inflation [18][19][20][21][22]. Though this, our scenario puts no specific constraints on inflationary models and there are attempts to combine inflation with w = 1 universe [14,23].
It is interesting to note that various approaches to quantum gravity have suggested the dimensional reduction from 4 to 2 near Planck scale [3][4][5][6][7]. Even more, our result shares a similar pattern with those of Hořava-Lifshitz gravity. Hořava suggested gravitational theories have the space-time anisotropy x → b x, t → b z t [24]. With z flows from z = 3 in the ultraviolet to z = 1 in the infrared, the corresponding effective dimension changes from 4 to 2. In D = d + 1 dimensions, the effective dimension for general z is given by [5] What happens in our context is that the number of quantum state D(ε)dε is unchanged under the scaling trans- . We suggest z = 3 for QG particles and z = 1 for photons with effective dimensions respectively 2 and 4. And in the Appendix, we also provide the effective dimension for general cases as Though the expressions (26) and (27) exactly match with each other, we may not naively take the whole frameworks to be conceptually equivalent. For example, in Hořava-Lifshitz gravity the anisotropic scaling of spacetime is proposed to insure power counting renormalizability and it modifies the Einstein-Hilbert action and the gravitational field equation. In contrast, we are searching for a non-trivial quantum matter satisfying the thermodynamics (1). And our QG particles determine the space-time geometry through the standard Friedmann equations with no modification (or else we can't get the expected cosmological entropy with holographic form).
Thus for now we regard this match as mainly reflecting the universality of the fundamental QG theory.

V. CONCLUSION AND DISCUSSIONS
In conclusion, we have suggested that, when the universe evolves from a QG particle dominated universe with w = 1 to a radiation dominated universe, the effective dimension runs from 2 to 4 and the cosmological entropy runs from A to A 3/4 . This may correspond to the phenomenon of dimensional reduction in the ultraviolet that has been found in various approaches to quantum gravity. The effective dimension afterwards is fixed to be 3 + 1 even when the universe evolves to be matter dominated. The available entropy for an observe in the universe increases as the particle horizon expands, but with deceasing w, the configuration of distributing energy is becoming simpler and simpler.
For general EoS w in D = d + 1 dimensions, we have shown in the Appendix the effective dimension is . Interestingly, the same entropy expression was obtained from standard cosmological analysis in [15,16], and the same expression for effective dimension was obtained from Hořava-Lifshitz gravity in [5]. Since both of them have been naturally deduced from our microscopic setting, we consider it reflect something deep and fundamental of quantum gravity.
We make some discussions here to justify D(ε)dε ∼ V G 2 dε for QG particles. First, it can successfully provide a microscopic explanation to the thermodynamics (1) and is consistent with cosmological analysis. The thermodynamics (1) is distinctive from those of the conventional radiation and matter, so we have the right to introduce something unfamiliar to avoid going around in the circle of ordinary quantum field theory. Second, its form is invariant under the transformation x → b x, ε → b −3 ε, which may reflect the space-time anisotropy of quantum gravity as suggested in Hořava-Lifshitz gravity. Third, it is reasonable that physical quantities of quantum gravity should depend on both G and . For example, a simple QG analysis has shown that the number density of gravitons in a gravitational plane wave is proportional to 1 G 2 [25]. Finally, our work suggests to take seriously the w = 1 stage of the early unverse. Fortunately the w = 1 stage has already been conjectured and studied in cosmology for many years from a number of different physical motivations, and it has the properties like enhancing stochas-tic gravitational waves, dark matter and baryon asymmetry [26][27][28][29][30]. The future observational evidence of the existence of such a stage would have profound implications for the understanding of quantum gravity.