Growth Rate and Configurational Entropy in Tsallis Holographic Dark Energy

In this work, we analyzed the effect of different prescriptions of the IR cutoffs, namely the Hubble horizon cutoff, particle horizon cutoff, Granda and Oliveros horizon cutoff, and the Ricci horizon cutoff on the growth rate of clustering for the Tsallis holographic dark energy (THDE) model in an FRW universe devoid of any interactions between the dark Universe. Furthermore, we used the concept of configurational entropy to derive constraints (qualitatively) on the model parameters for the THDE model in each IR cutoff prescription from the fact that the rate of change of configurational entropy hits a minima at a particular scale factor $a_{DE}$ which indicate precisely the epoch of dark energy domination predicted by the relevant cosmological model as a function of model parameter(s). By using the current observational constraints on the redshift of transition from a decelerated to an accelerated Universe, we derived constraints on the model parameters appearing in each IR cutoff definition and on the non-additivity parameter $\delta$ characterizing the THDE model and report the existence of simple linear dependency between $\delta$ and $a_{DE}$ in each IR cutoff setup.


November 30, 2020
In this work, we analyzed the effect of different prescriptions of the IR cutoffs, namely the Hubble horizon cutoff, particle horizon cutoff, Granda and Oliveros horizon cutoff, and the Ricci horizon cutoff on the growth rate of clustering for the Tsallis holographic dark energy (THDE) model in an FRW universe devoid of any interactions between the dark Universe. Furthermore, we used the concept of configurational entropy to derive constraints (qualitatively) on the model parameters for the THDE model in each IR cutoff prescription from the fact that the rate of change of configurational entropy hits a minima at a particular scale factor a DE which indicate precisely the epoch of dark energy domination predicted by the relevant cosmological model as a function of model parameter(s). By using the current observational constraints on the redshift of transition from a decelerated to an accelerated Universe, we derived constraints on the model parameters appearing in each IR cutoff definition 1 Introduction The dynamics of the Universe is well explained by the ΛCDM cosmological model, albeit with the presumption that the Universe contain dark matter and dark energy in overwhelming quantities (1) where the former dictate how the objects in the Universe cluster, and the latter on how should the Universe expand. Much of modern cosmology has therefore been focused in pinpointing the properties of these quantities to understand the dynamical evolution of the Universe in the past and in the future. In this spirit, many dedicated experiments have been proposed or are in operation but have so far not yielded any conclusive evidence for the existence of the dark Universe and therefore critically affect the viability of the standard cosmological model. Additionally, the existence of the so-called Hubble tension (2) makes things worse and hints at the possible existence of unknown physics and on the possibility of the Universe being governed by an entirely different cosmological model. It therefore compels us to explore other avenues to explain these profound cosmological enigmas.
Holographic dark energy (HDE) models are proving to be viable alternatives to the static cosmological constant in addressing the cosmic acceleration. Constructed upon the holographic principle (3,4,5), HDE models are also in harmony with latest observational results (6,7,8,9,10). In HDE, the horizon entropy plays the most crucial role which if changed, changes the dynamics of the model significantly (11).
Exercising the modified entropy-area relation reported in (12) and in conjunction with the holographic principle, the authors in (13) proposed the Tsallis HDE which could explain the observed accelerated expansion of the universe and has been widely studied in literature. For example, in (11), the authors studied the dyanmics of THDE for various IR cutoff scales and also for non-interacting and interacting scenarios. In (14), the dyanmics of THDE has been studied for the hybrid expanion law. Interested readers can further look into (15) for a recent review on various HDE models.
As first proposed in (16), the cosmic evolution from a near perfect gaussian matter distribution to the highly non-linear state owing to the clustering of dark matter haloes could be due to the dissipation of configurational entropy. Additionally, the depletion of configurational entropy could also be responsible for the presently observed cosmic expansion (17). The depletion of configurational entropy is inevitable for a static Universe since it is unstable due to the presence of large scale structures. Therefore, if we assume the Universe as a whole to comprise a thermodynamic system, it therefore must obey the second law of thermodynamics which warrants cosmic expansion to suppress the growth of structure formation due to the absence of entropy generation processes to check the dissipation of configurational entropy (17). It must also be noted that the configurational entropy dissipates for a dust Universe and only damps out in an accelerating Universe and that the first derivative of configurational entropy attains a minima at a scale factor corresponding to the epoch of the dark energy domination for that model (17).
The concept of configurational entropy has been used to study cosmology in power law f (T ) gravity (25). In this work, we shall investigate the growth rate of clustering in the THDE model for various IR cutoffs to understand how does the different IR cutoffs influence the growth rate with respect to the standard ΛCDM model and also to put qualitative constraints on the model parameters from the first derivative of the configurational entropy.
The manuscript is organized as follows: In Section 2, we briefly summarize the concept of configurational entropy and growth rate. In Section 3, we discuss the Tsallis holographic dark energy model in different IR cutoffs and discuss the results and in Section 4 we conclude.

Configurational entropy and growth rate
Let us consider a large comoving volume V in the cosmos where the assumptions of isotropy and homogeneity hold appropriate. Now, V can be divided into a number of smaller volume elements dV with energy density ρ( − → x , t) where − → x and t represent respectively the comoving coordinates and time.
Now, from the definition of configurational entropy in (16) which was further motivated from the definition of information entropy in (19), we can write Next, the equation of continuity for an expanding Universe can be expressed as where a and − → ν represent respectively the scale factor and the peculiar velocity of cosmic fluid in dV .
Next, multiplying Eq. 2 with (1 + logρ), and then integrating over volume V , we arrive at the following differential equation (16) dΦ(a) daȧ where Now, the divergence of peculiar velocity ν can be written as (17) .
where D(a) is the growing mode of fluctuations and δ( − → x ) represents the density contrast at a given position − → x . Upon substituting Eq. 4 in Eq. 3, yields after some calculation (17) dΦ(a) da where Ψ = ρ( − → x , a)dV defines the total mass enclosed within the volume V and ρ represents the critical density of the cosmos and represents the dimensionless growth rate with Ω m (a) being the matter density parameter and γ being the growth index. The growth index in Einstein's gravity is roughly equal to 6/11 (27) while it is slightly different for modified gravity models and also for dynamical dark energy models. Ref (26) Solution of Eq. 5 provides the evolution of configurational entropy in the cosmological model under consideration. In order to numerically solve Eq. 5, we assume the time independent quantities in Eq. 5 to be equal to 1 and set the initial condition Φ(a i ) = Ψ.
The interesting feature of Eq. 5 is that the first derivative of configurational entropy (i.e, dΦ(a) da ) attains a minima at a particular scale factor a DE after which the dark energy domination takes place (17). So, by comparing the minima obtained from the ΛCDM model with that obtained from other dark energy models, we can put qualitative constraints on the model parameters of different dark energy models and this is precisely the motivation for this work.

Expression of energy density
The modified black hole's horizon entropy suggested in (12) takes the form S δ = γD δ , where δ represent the Tsallis or the non-additivity parameter, γ a constant and D the surface the surface area of the Black hole's event horizon and for δ = 1, the usual Bekenstein entropy can be recovered.
The relationship between the entropy S, IR and UV cutoffs denoted respectively by Π and Θ is given by (3, 4, 5) Now, since the energy density for the HDE models scales as ρ DE ∝ Θ 4 , the general expression for the energy density of the THDE can be written as (13) where A is an unknown parameter. Considering a flat FRW background with − , + , + , + metric signature, the first Friedmann equation assumes the form where ρ m and ρ DE are the energy density of the dark matter and the dark energy respectively and therefore the corresponding dimensionless density parameters are defined as with ρ c = 3H 2 being the critical density of the Universe.
It can be clearly seen from Eq. 9 that different choices for the IR cutoffs will generate different expressions for the THDE energy density with significantly different dark energy EoS parameter ω DE . There are currently 4 different prescriptions for the IR cutoffs, namely the Hubble horizon cutoff, particle horizon cutoff (21), Granda and Oliveros horizon cutoff (22) and the Ricci horizon cutoff (23). In the subsequent subsections, we shall numerically compute the growth rate and the rate of change of configurational entropy for the THDE model with the aforementioned IR cutoffs.

Hubble horizon cutoff
For the standard HDE model, the Hubble horizon does not give rise to an accelerated expansion (21). However, as shown in (13), for the THDE model, an accelerated expansion is possible for such an cutoff even in the absence of interacting scenarios.
Therefore, for the first case, we shall assume the IR cutoff to be the Hubble horizon represented by Substituting Eq. 12 in Eq. 9, the expression of ρ DE takes the form The evolution of the dark energy density parameter (Ω DE ) can be obtained by numerically solving the following differential equation (13) where Now, in order to compute the growth rate (f (a)), we substitute Eq. 15 into Eq. 7 and Eq. 13 into Eq. 6 since Ω m (a) = 1 − Ω DE (a) and then substitute the result in Eq. 5 to compute the rate of change of configurational entropy.

Particle horizon cutoff
In the second case, we set the IR cutoff to be the particle horizon defined as (21) which satisfies the equationṘ The expression of the energy density (ρ DE ) in this setup reads (11) The evolution of Ω DE in this setup can be obtained by solving the following differential equation where and the expression for the EoS parameter (ω DE ) reads (11)

Granda and Oliveros (GO) horizon cutoff
The GO cutoff was proposed by Granda and Oliveros (22) to resolve the causality and coincidence problems in cosmology and is defined as Π = (mH 2 + nḢ) −1/2 for which the expression of the energy density takes the form (11) The differential equation required to solve for the evolution of Ω DE and the expression of the EoS parameter (ω DE ) reads respectively as (11) and The growth rate f (a) for the THDE model with the GO cutoff differ substantially from the ΛCDM model. The growth rate diminishes rapidly as δ is lowered. It is also evident that in all cases f (a) is lower than that of the ΛCDM model.  (24) and also very close to the same obtained from the ΛCDM model. A strict linear dependency between δ and a DE is observed with the best fit relation δ = −0.09a DE + 0.9.
In (11) the authors used the same combination of the free parameters (i.e, m, n and δ) and reported viable estimates of kinematical parameters in all cases. However, from this work it is clear that only for δ = 0.8, an accelerating universe is possible. Therefore, it is transparent that rate of change of configurational entropy provides an alternative method to constrain the free parameters and also furnish a robust consistency check to the constraints obtained from other methods and statistics.

Ricci horizon cutoff
For the fourth and final case, we shall set the Ricci horizon as the IR cutoff for which the expression of the THDE model reads (23,11) where λ is an unknown parameter as usual (21). Similar to the previous case, the first order differential equation which generate the evolution of Ω DE and the expression of the EoS parameter (ω DE ) reads respectively as (11) and , and the lower panel shows the best fit relation between the non-additivity parameter δ and the scale factor a DE at which dΦ(a) da attains a minima. The plots are drawn keeping λ unity.
The f (a) plot in this case shows the ΛCDM profile to predict significantly higher growth rate than the THDE model prior a specific scale factor dependent solely on the non-additivity parameter δ following which the growth rate in the THDE model exceeds the one for the ΛCDM.

Conclusions
Holographic dark energy (HDE) models are a promising alternative to the static cosmological constant in explaining the late-time dynamics of the Universe and have already alleviated some of the major hitches plaguing the ΛCDM cosmological model (6,7,8,9,10). These models are built upon the holographic principle (3,4,5) where the horizon entropy plays the most crucial role with different IR cutoffs predicting completely different dyanmics for the HDE model under consideration (11).
Ref (17) reported that the first derivative of configurational entropy attains a minima at a particular scale factor a DE after which the dark energy domination takes place. Thus, upon employing the idea, we tried to constrain the non-additivity parameter δ for the THDE model and other free parameters appearing in each IR cutoff case by juxtaposing the theoretical estimate of a DE in each case with that reported by current observations. We find that there exists suitable parameter range between which the THDE model predicts an accelerating universe in each IR cutoff recipe at a suitable redshift consistent with observations and report the existence of simple linear dependencies between the non-additivity parameter δ and a DE in each IR cutoff prescription.