# Modified cosmology through nonextensive horizon thermodynamics

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## Abstract

We construct modified cosmological scenarios through the application of the first law of thermodynamics on the universe horizon, but using the generalized, nonextensive Tsallis entropy instead of the usual Bekenstein–Hawking one. We result to modified cosmological equations that possess the usual ones as a particular limit, but which in the general case contain extra terms that appear for the first time, that constitute an effective dark energy sector quantified by the nonextensive parameter \(\delta \). When the matter sector is dust, we extract analytical expressions for the dark energy density and equation-of-state parameters, and we extend these solutions to the case where radiation is present too. We show that the universe exhibits the usual thermal history, with the sequence of matter and dark-energy eras, and according to the value of \(\delta \) the dark-energy equation-of-state parameter can be quintessence-like, phantom-like, or experience the phantom-divide crossing during the evolution. Even in the case where the explicit cosmological constant is absent, the scenario at hand can very efficiently mimic \(\Lambda \hbox {CDM}\) cosmology, and is in excellent agreement with Supernovae type Ia observational data.

## 1 Introduction

Recent cosmological observations from various and different fields reveal that the universe has experienced two accelerated expansion phases, one at early and one at late times. Since the established knowledge of general relativity and Standard Model of particles is not sufficient to explain this behavior, there has been a lot of effort in constructing theories beyond the above, in order to acquire the necessary extra degrees of freedom. On one hand, one can introduce new forms of matter, such as the inflaton field [1, 2] or the concept of dark energy [3, 4], which in the framework of general relativity can lead to the aforementioned accelerated behaviors. On the other hand, one can construct gravitational modifications, which possess general relativity as a particular limit, but at large scales can provide extra degrees of freedom capable of driving the acceleration (for reviews see [5, 6, 7, 8]). Note that this last approach has the additional theoretical advantage that may improve renormalizability, which seems to be necessary towards quantization [9, 10].

The usual approach of constructing modified gravitational theories is to start from the Einstein–Hilbert action and add correction terms. The simplest extension is to replace the Ricci scalar *R* by a function *f*(*R*) [11, 12, 13, 14]. Similarly, one can proceed in constructing many other classes of modification, such as *f*(*G*) gravity [15, 16], Lovelock gravity [17, 18], Weyl gravity [19, 20] and Galileon theory [21, 22, 23]. Alternatively, one can start from the torsional formulation of gravity and build various extensions, such as *f*(*T*) gravity [24, 25, 26], \(f(T,T_G)\) gravity [27, 28], etc.

On the other hand, there is a well-known conjecture that one can express the Einstein equations as the first law of thermodynamics [29, 30, 31]. In the particular case of cosmology in a universe filled with the matter and dark-energy fluids, one can express the Friedmann equations as the first law of thermodynamics applied in the universe apparent horizon considered as a thermodynamical system [32, 33, 34, 35]. Reversely, one can apply the first law of thermodynamics in the universe horizon, and extract the Friedmann equations. Although this procedure is a conjecture and not a proven theorem, it seems to work perfectly in a variety of modified gravities, as long as one uses the modified entropy relation that corresponds to each specific theory [35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Nevertheless, note that in order to know the modified entropy relation of a modified gravity, ones needs to know this modified gravity a priori and investigate it in spherically symmetric backgrounds. In this sense the above procedure cannot provide new gravitational modifications, offering only a way to study their features.

In the present work we are interested in following the above procedure in a reverse way, in order to construct new cosmological modifications. In particular, we will apply the first law of thermodynamics, but instead of the usual entropy relation we will use the nonextensive, Tsallis entropy [45, 46, 47], which is the consistent generalization of the Boltzmann–Gibbs additive entropy in non-additive systems, such as gravitational ones. In this way we will obtain new modified Friedmann equations that possess the usual ones as a particular limit, namely when the Tsallis generalized entropy becomes the usual one, but which in the general case contain extra terms that appear for the first time. Hence, we will investigate in detail the cosmological implications of these new extra terms.

The plan of the work in the following: In Sect. 2 we present the construction of the scenario, applying the first law of thermodynamics in the universe horizon, but using the generalized, nonextensive Tsallis entropy instead of the usual Bekenstein–Hawking one. In Sect. 3 we investigate the cosmological evolution, focusing on the behavior of the dark energy density and equation-of-state parameters, studying separately the cases where an explicit cosmological constant is present or absent. Finally, in Sect. 4 we summarize our results.

## 2 The model

*a*(

*t*) is the scale factor, and with \(k=0,+\,1,-\,1\) corresponding to flat, close and open spatial geometry respectively.

### 2.1 Friedmann equations as the first law of thermodynamics

*t*. The apparent horizon is a marginally trapped surface with vanishing expansion, defined in general by the expression \(h^{ij}\partial _i\tilde{r}\partial _j \tilde{r}=0 \) (which implies that the vector \(\nabla \tilde{r}\) is null or degenerate on the apparent horizon surface) [49]. For a dynamical spacetime, the apparent horizon is a causal horizon associated with the gravitational entropy and the surface gravity [49, 50, 51]. Finally, note that in flat spatial geometry the apparent horizon becomes the Hubble one.

*G*the gravitational constant. Thus, in the case of a universe governed by general relativity, the horizon entropy will be just

^{1}

*dt*, the heat flow that crosses the horizon can be straightforwardly found to be [33]

Interestingly enough, we saw that applying the first law of thermodynamics to the whole universe resulted to the extraction of the two Friedmann equations, namely Eqs. (6) and (8). The above procedure can be extended to modified gravity theories too, where as we discussed the only change will be that the entropy relation will not be the general relativity one, namely (4), but the one corresponding to the specific modified gravity at hand [35, 36, 37, 38, 39, 40, 41, 42, 43, 44]. Nevertheless, we have to mention here that although the above procedure offers a significant tool to study the features and properties of various modified gravities, it does not lead to new gravitational modifications, since one needs to know the entropy relation, which in turn can be known only if a specific modified gravity is given a priori.

### 2.2 Tsallis entropy

*L*,

*G*is the gravitational constant, \(\tilde{\alpha }\) is a positive constant with dimensions \([L^{2(1-\delta )}]\) and \(\delta \) denotes the non-additivity parameter.

^{2}Under the hypothesis of equal probabilities the parameters \(\delta \) and \(\tilde{\alpha }\) are related to the dimensionality of the system [56] (in particular the important parameter \(\delta =d/(d-1)\) for \(d>1\)), however in the general case they remain independent and free parameters. Obviously, in the case \(\delta =1\) and \(\tilde{\alpha }=1\), Tsallis entropy becomes the usual Bekenstein–Hawking additive entropy.

### 2.3 Modified Friedmann equations through nonextensive first law of thermodynamics

In Sect. 2.1 we presented the procedure to extract the Friedmann equations from the first law of thermodynamics. This procedure can be applied in any modified gravity, as long as one knows the black hole entropy relation for this specific modified gravity. Hence, as we mentioned above, although it can be enlightening for the properties of various modified gravities, the thermodynamical approach does not lead to new gravitational modifications since one needs to consider a specific modified gravity a priori.

In the present subsection however, we desire to follow the steps of Sect. 2.1, but instead of the standard additive entropy relation to use the generalized, nonextensive, Tsallis entropy presented in Sect. 2.2 above. Doing so we do obtain modified Friedmann equations, with modification terms that appear for the first time, and which provide the standard Friedmann equations in the case where Tsallis entropy becomes the standard Bekenstein–Hawking one.

*T*by (3), but we will consider that the entropy is given by Tsallis entropy (9). In this case, and recalling that \(A=4\pi \tilde{r}_a^2\) we acquire

*H*(

*t*) that can be solved similarly to all modified-gravity and dark-energy models.

## 3 Cosmological evolution

In this section we proceed to a detailed investigation of the modified cosmological scenarios constructed above. The cosmological equations are the two modified Friedmann equations (13) and (14), along with the conservation equation (7). In the general case of a general matter equation-of-state parameter, \(w_m\equiv p_m/\rho _m\), analytical solutions cannot be extracted, and thus one has to solve the above equations numerically. However, we are interested in providing analytical expressions too, and thus in the following we focus to the case of dust matter, namely \(w_m=0\).

*z*as the independent variable, defined as \( 1+z=1/a\) for \(a_0=1\). Thus, differentiating (27) we can obtain the useful expression

*z*.

### 3.1 Cosmological evolution with \(\Lambda \ne 0\)

We first examine the case where the explicit cosmological constant \(\Lambda \) is present. In this case when \(\delta =1\) and \(\alpha =1\) we obtain \(\Lambda \hbox {CDM}\) cosmology, and thus we are interested in studying the role of the nonextensive parameter \(\delta \) on the cosmological evolution.

*q*(

*z*) from (33). We mention that for transparency we have extended the evolution up to the far future, namely up to \(z \rightarrow -1\), which corresponds to \(t \rightarrow \infty \).

As we observe, we acquire the usual thermal history of the universe, with the sequence of matter and dark energy epochs, with the transition from deceleration to acceleration taking place at \(z\approx 0.45\) in agreement with observations. Additionally, in the future the universe tends asymptotically to a complete dark-energy dominated, de-Sitter state. We mention the interesting bahavior that although at intermediate times the dark-energy equation-of-state parameter may experience the phantom-divide crossing and lie in the phantom regime, at asymptotically large times it will always stabilize at the cosmological constant value \(-\,1\). Namely, the de-Sitter solution is a stable late-time attractor, which is a significant advantage (this can be easily showed taking the limit \(z \rightarrow -1\) in (29),(31) and (32), which gives \(\Omega _{DE}\rightarrow 1\), \(\Omega '_{DE}\rightarrow 0\), and \(w_{DE}\rightarrow -1\), respectively).

In summary, we can see that the nonextensive parameter \(\delta \), that lies in the core of the modified cosmology obtained in this work, plays an important role in giving to dark energy a dynamical nature and bringing about a correction to \(\Lambda \hbox {CDM}\) cosmology. We mention that in all the above examples we kept the parameter \(\alpha \) fixed, in order to maintain the one-parameter character of the scenario. Clearly, letting \(\alpha \) vary too, increases the capabilities of the model and the obtained cosmological behaviors.

### 3.2 Cosmological evolution with \(\Lambda =0\)

In the previous subsection we investigated the scenario of modified Friedmann equations through nonextensive thermodynamics, in the case where the cosmological constant is explicitly present. Thus, we studied models that possess \(\Lambda \hbox {CDM}\) cosmology as a subcase, and in which the nonextensive parameter \(\delta \) and its induced novel terms lead to corrections to \(\Lambda \hbox {CDM}\) paradigm.

In the present subsection we are interested in studying a more radical application of the scenario at hand, namely to consider that an explicit cosmological constant is not present and let the model parameters \(\delta \) and \(\alpha \) to mimic its behavior and produce a cosmology in agreement with observations.

From the analytical expression (38) we can see that we acquire the thermal history of the universe, with the sequence of matter and dark energy epochs and the onset of late-time acceleration. Furthermore, in the future (\(z \rightarrow -1\)) the universe tends asymptotically to the complete dark-energy domination. Additionally, as can be seen from expression (39), the asymptotic value of \(w_{DE}\) in the far future is not necessarily the cosmological constant value \(-1\). In particular, we deduce that for \(1\le \delta <2\) \(w_{DE}\rightarrow 0\) as \(z \rightarrow -1\), while for \(\delta <1\) \(w_{DE}\rightarrow (\delta -1)/(2-\delta )\) as \(z \rightarrow -1\). Hence, the case \(\delta <1\) is the one that exhibits more interesting behavior in agreement with observations, and we observe that for decreasing \(\delta \) the \(w_{DE}(z)\) tends to lower values.

We close this subsection mentioning that according to the above analysis the cosmological behavior is very efficient for low redshifts and up to the far future, despite the fact that an explicit cosmological constant is absent. However, as can be seen from (36), for high redshifts the behavior of \(\Omega _{DE}(z)\) is not satisfactory, since as it is this expression leads to either early-time dark energy or to the unphysical result that \(\Omega _{DE}(z)\) becomes negative. In order to eliminate this behavior and obtain a universe evolution in agreement with observations at all redshifts one needs to include the radiation sector too, which indeed can regulate the early-time behavior. This is performed in the next subsection.

### 3.3 Cosmological evolution including radiation

#### 3.3.1 Cosmological evolution with \(\Lambda \ne 0\)

#### 3.3.2 Cosmological evolution with \(\Lambda = 0\)

Let us examine this scenario in more detail, and in particular study the effect of \(\delta \) on the cosmological evolution. In Fig. 3 we present \(w_{DE}(z)\) for various choices of \(\delta \), extending the evolution up to the far future. In all cases the parameter \(\alpha \) is set according to (53) in order to obtain \(\Omega _{m}(z=0)=\Omega _{m0}=0.3\) and \(\Omega _{r}(z=0)=\Omega _{r0}=0.000092\) [57], and the expected thermal history of the universe. As we observe, for decreasing \(\delta \) the \(w_{DE}(z)\) tends to lower values. Moreover, although the asymptotic value of \(\Omega _{DE}(z)\) as \(z \rightarrow -1\) is 1, as can be seen immediately from (52), namely the universe tends to the complete dark-energy domination, the asymptotic value of \(w_{DE}\) is not the cosmological constant value \(-1\), i.e the universe does not result in a de Sitter space. In particular, from (55) we can see that for \(1\le \delta <2\), \(w_{DE}\rightarrow 0\) as \(z \rightarrow -1\), while for \(\delta <1\), \(w_{DE}\rightarrow (\delta -1)/(2-\delta )\) as \(z \rightarrow -1\). These asymptotic values are the same with the ones in the absent of radiation mentioned in Sect. 3.2, which was expected since at late times the effect of radiation is negligible.

*l*(

*z*), or equivalently the apparent magnitude

*m*(

*z*), are measured as functions of the redshift, and are related to the luminosity distance as

*M*and

*L*are the absolute magnitude and luminosity respectively. Additionally, for any theoretical model one can calculate the predicted dimensionless luminosity distance \(d_{L}(z)_\text {th}\) using the predicted evolution of the Hubble function as

*H*(

*z*) can be immediately calculated analytically from (44), knowing (46) and (47). In Fig. 4 we depict the theoretically predicted apparent minus absolute magnitude as a function of

*z*, for two \(\delta \) choices, as well as the prediction of \(\Lambda \hbox {CDM}\) cosmology, on top of the 580 SN Ia observational data points from [58]. As we can see the agreement with the SN Ia data is excellent. The detailed comparison with observations, namely the joint analysis using data from SN Ia, Baryon Acoustic Oscillation (BAO), Cosmic Microwave Background (CMB), and direct Hubble parameter observations, lies beyond the scope of the present work and it is left for a future project.

We close this subsection mentioning that the present scenario is very efficient in mimicking the cosmological constant, despite the fact that in this case the exact \(\Lambda \hbox {CDM}\) cosmology cannot be obtained for any parameter values. In particular, choosing the nonextensive parameter \(\delta \) suitably (namely \(\delta \sim 0.5-0.6\)) we acquire agreement with observations. This is a significant result that shows the capabilities of the modified cosmology through nonextensive thermodynamics.

## 4 Conclusions

In this work we constructed a modified cosmological scenario through the application of the first law of thermodynamics, but using the generalized, nonextensive Tsallis entropy instead of the usual Bekenstein–Hawking one. In particular, there is a well-studied procedure in the literature, which works for a variety of modified gravities, where one can apply the first law of thermodynamics in the universe horizon and extract the Friedmann equations. The crucial part in this procedure is the use of the modified entropy relation of the specific modified gravity, which is known only after this modified gravity is given, and thus in this sense it cannot provide new gravitational modifications. However, if we apply this approach using the nonextensive, Tsallis entropy, which is the consistent concept that should be used in non-additive gravitational systems such us the whole universe, then we result to modified cosmological equations that possess the usual ones as a particular limit, but which in the general case contain extra terms that appear for the first time.

The new terms that appear in the modified Friedmann equations are quantified by the nonextensive parameter \(\delta \) and constitute an effective dark energy sector. In the case where Tsallis entropy becomes the usual Bekenstein–Hawking entropy, namely when \(\delta =1\), the effective dark energy coincides with the cosmological constant and \(\Lambda \hbox {CDM}\) cosmology is restored. However, in the general case the scenario of modified cosmology at hand presents very interesting cosmological behavior.

When the matter sector is dust, we were able to extract analytical expressions for the dark energy density and equation-of-state parameters, and we extended these solutions in the case where radiation is present too. These solutions show that the universe exhibits the usual thermal history, with the sequence of matter and dark-energy eras and the onset of acceleration at around \(z\approx 0.5\) in agreement with observations. In the case where an explicit cosmological constant is present, according to the value of \(\delta \) the dark-energy equation-of-state parameter exhibits a very interesting behavior and it can be quintessence-like, phantom-like, or experience the phantom-divide crossing during the evolution, before it asymptotically stabilizes in the cosmological constant value \(-\,1\) in the far future.

An interesting sub-case of the scenario of modified cosmology through nonextensive thermodynamics is when we set the explicit cosmological constant to zero, since in this case the universe evolution is driven solely by the news terms. Extracting analytical solutions for the dark energy density and equation-of-state parameters we showed that indeed the new terms can very efficiently mimic \(\Lambda \hbox {CDM}\) cosmology, although \(\Lambda \) is absent, with the successive sequence of matter and dark energy epochs, before the universe results in complete dark-energy domination in the far future. Moreover, confronting the model with SN Ia data we saw that the agreement is excellent.

In summary, modified cosmology through nonextensive thermodynamics is very efficient in describing the universe evolution, and thus it can be a candidate for the description of nature. In the present work we derived the cosmological equations by applying the well-known thermodynamics procedure to the universe horizon. It would be interesting to investigate whether these equations can arise from a nonextensive action too. Such a study is left for a future project.

## Footnotes

- 1.
Note that although this will certainly be the situation at late times, when the universe fluid and the horizon will have interacted for a long time, it is not assured that it will be the case at early or intermediate times. However, in order to avoid applying non-equilibrium thermodynamics, which leads to mathematical complexity, the assumption of equilibrium is widely used [31, 32, 33, 34, 42, 53]. Thus, we will follow this assumption and we will have in mind that our results hold only at late times of the universe evolution.

- 2.

## Notes

### Acknowledgements

This article is based upon work from CANTATA COST (European Cooperation in Science and Technology) action CA15117, EU Framework Programme Horizon 2020.

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