Dark Energy and Cosmological Horizon Thermal Effects
Artyom V. Astashenok, Sergei D. Odintsov, Vasilis K. Oikonomou
DDark Energy and Cosmological Horizon Thermal Effects
Artyom V. Astashenok , Sergei D. Odintsov , ,V.K. Oikonomou , I. Kant Baltic Federal University, Institute of Physics,Mathematics and IT, 236041, 14, Nevsky st., Kaliningrad, Russia Instituci`o Catalana de Recerca i Estudis Avan¸cats (ICREA), Barcelona, Spain Institut de Ciencies de l’Espai (CSIC-IEEC), Campus UAB,Facultat de Ciencies, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain Department of Physics, Aristotle University of Thessaloniki, Thessaloniki 54124, Greece Laboratory for Theoretical Cosmology, Tomsk State University ofControl Systems and Radioelectronics, 634050 Tomsk, Russia (TUSUR)
We investigate various dark energy models by taking into account the thermal effects inducedfrom Hawking radiation on the apparent horizon of the Universe, for example near a finite-timefuture singularity. If the dark energy density increases as the Universe expands, the Universe’sevolution reaches a singularity of II type (or sudden future singularity). The second derivative ofscale factor diverges but the first remains finite. Quasi-de Sitter evolution can change on suddenfuture singularity in the case of having an effective cosmological constant larger than the maximumpossible value of the energy density of the Universe. Another interesting feature of cosmologicalsolution is the possibility of a transition between deceleration and acceleration for quintessence darkenergy with a simple equation of state. Finally, we investigate which fluid component can remedyBig Rip singularities and other crushing type singularities.
PACS numbers:
I. INTRODUCTION
The accelerated expansion of the Universe [1, 2] from a theoretical viewpoint can be caused by some fluid withnegative pressure and/or negative entropy (for review, see [3–5]). Due to the fact that it is still unknown how thisfluid will behave, the late-time accelerated expansion era is dubbed the dark energy era. The current observationscoming from Planck [6], indicate that dark energy controls nearly 70% of total energy density of the Universe [7]. Theequation of state (EoS) parameter for the dark fluid, namely w d , is negative, that is, w d = p d /ρ d < , (1)where ρ d and p d are the dark energy density and pressure correspondingly. However, it is still not clear what is theprecise value of w d [8, 9], although the latest Planck data constraint significantly the values that the EoS parametercan take. The standard Λ-Cold-Dark-Matter (ΛCDM) model according to which dark energy is simply a cosmologicalconstant, implies that w d = −
1. If the EoS parameter is in the range − < w d < − / w d < −
1, and in this case the Universeevolves dominated by phantom dark energy. A simple choice of the form w d = const leads to a Big-Rip singularity[10–27], where the scale factor and all the physical quantities that can be defined on the three dimensional spacelikesingularity defined of the time instance that the Rip occurs, severely diverge. There are other types of singularities forphantom Universe, for example a singularity of type III corresponds to a situation in which, the Hubble parameter H diverges at a finite time and for a finite scale factor. A sudden future singularity (or type II) is milder in comparisonwith the two previous. The scale factor and its first derivative (and therefore the energy density) are finite at themoment of the singularity, but the second derivative of scale factor diverges (and the pressure correspondingly) [28–42]. This singularity can occur for quintessential evolution too. In principle, the occurrence of finite-time singularities,during the evolution of the Universe, is not necessarily a consequence of phantom dark energy. In fact, a phantomscalar field always leads to a Big Rip singularity [10], but the opposite is not true. For some studies on alternativemodels leading to a little-rip singularity, see Refs. [43–46]. If the EoS parameter w tends to minus unity sufficientlyfast, the time instance that the singularity occurs is at infinity. Of course, by that time, the tidal forces would growinfinitely large and all the stellar structures and galactic structures would be torn apart by the tidal forces before thesingularity would ever be reached. Another possibility is the so-called pseudo-rip scenario, which can be realized whenthe energy density of dark energy approaches asymptotically a constant value (effective “cosmological constant”). Thiseffective “cosmological constant” in principle can considerably differ from the current energy density of dark energy ρ d .According to the observational data, it is marginally probable that the phantom divide line is crossed and the darkenergy era is actually a phantom dark energy era. For example for the flat wCDM model according to latest Planck a r X i v : . [ g r- q c ] J a n data, the dark energy EoS is constrained as follows, w = − . ± . , so the possibility w < − w < − p d = g ( ρ d ), where g is an arbitrary function.We shall consider various models of dark energy with the inclusion of thermal radiation effects. After discussingthe main theoretical framework, we shall study two important state-finder parameters, namely the deceleration andjerk parameters. Then some simple dark energy EoS parameters are considered, with account of thermal radiation.Also our findings show that for any equation of state which leads to infinitely increasing phantom energy density, asudden future singularity always occurs. Also the little rip regime cannot be realized in the present framework, andinterestingly enough only one scenario of non-singular evolution for phantom energy takes place, that of pseudo-ripevolution, but in that case, the value of the ”effective” cosmological constant should be less than some critical density.It is also important to note that in the present context, a phantom dark energy era can be realized without the needfor a phantom scalar field, as in ordinary Einstein-Hilbert gravity. II. FIELD EQUATIONS AND THE DESCRIPTION OF THE THEORETICAL FRAMEWORK
In this section we briefly consider the cosmological equations for the evolution of the Universe, by taking intoaccount the thermal radiation effects from the cosmological horizon. Let’s start from the spatially-flat FRW universewith metric, ds = dt − a ( t )( dx + dy + dz ) . (2)The cosmological equations are, (cid:18) ˙ aa (cid:19) = ρ , ¨ aa = −
16 ( ρ + 3 p ) , (3)where ρ and p are the total energy density and pressure, a is the scale factor,and the “dot” denotes differentiationwith respect to the cosmic time. We use the system of units in which 8 πG = c = 1, the so-called reduced Planck unitssystem.From the Friedmann and the Raychaudhuri equations it follows that,˙ ρ + 3 H ( ρ + p ) = 0 . (4)If the Universe consists of non-interacting components ρ i , this equation holds for each component ρ i .The radius of the apparent horizon of the Universe is r h ∼ /H , Therefore, the Hawking temperature of the thermal radiation appearing due to the apparent horizon is proportionalto Hubble parameter H . The energy density of radiation thermal term according to Stefan-Boltzmann law is, ρ t rad = 3 αH , where α is some constant. The multiplicative factor 3 is introduced for convenience. One can expect that the constant α could be calculated from a self-consistent quantum model describing the thermal radiation. Such a model thoughis not available, and therefore the direct usage of the thermal radiation effects related to the de Sitter horizon is notjustified. However, in order to have a concrete idea of these effects, we assumed only by analogy that the energydensity of thermal radiation is proportional to H , because the temperature of Hawking radiation of the apparenthorizon is inversely proportional to the size of the horizon.A similar physical analogy is used in the context of holographic energy, for which we usually assume that, ρ = β/L , where L is the event horizon. Regarding the values of β , there are various estimations from calculations, but its exactvalue is unknown. Therefore we use α as the unknown parameter in our calculations.The Friedmann equation for the Hubble parameter, by taking into account the density of thermal radiation, is, αH − H + ρ , (5)which can be solved with respect to the Hubble parameter as follows, H = ± (cid:32) ± (cid:112) − αρ/ α (cid:33) / . (6)Further we assume that the Universe expands, and therefore the sign “+” is chosen. One also needs to choose sign “-”in brackets for the following reason. Seing the thermal radiation as the Hawking radiation from the apparent horizonhas a quantum origin. It is known that the cosmological evolution of Universe from beginning to present date, isdescribed well by the Friedmann equations and one can expect that for α → αρ/ <<
1, we obtain the ordinary cosmological model, H = 1 − (cid:112) − αρ/ α ≈ − (1 − αρ/ α = ρ . But by choosing “+”, we obtained very strange cosmological model with a Hubble parameter which tends to infinityfor α → ∞ .For these choices of signs in Eq. (6), one can obtain the following relation for the derivative of Hubble parameter,˙ H = − ρ + p (cid:112) − αρ/ . (7)The second derivative of the scale factor is,¨ aa = ˙ H + H = − ρ + p (cid:112) − αρ/ − (cid:112) − αρ/ α . (8)We consider the case when equation of state for dark energy is given in form, p d = − ρ d − f ( ρ d ) , (9)where f ( ρ d ) is some arbitrary function. Condition f ( ρ ) > f ( ρ ) < t = √ α (cid:90) ρρ dρf ( ρ ) (cid:16) − (cid:112) − αρ/ (cid:17) / . (10)We omit the subscript d in this relation assuming ρ = ρ d . The quintessence energy density decreases with time( ρ < ρ ), while the phantom energy density increases ( ρ > ρ ).The effective pressure and density can be defined as, ρ eff = 3 H , p eff = − H − H (11)and therefore for our cosmological model we have, ρ eff = 32 α (cid:16) − (cid:112) − αρ/ (cid:17) , (12) p eff = ρ + p (cid:112) − αρ/ − ρ eff . From this relation one can conclude that the effective pressure reaches infinity for a finite value of the energy density ρ max = α (the effective energy density is finite also). Therefore, we have a type II singularity (or sudden futuresingularity in another words). We can also compute the scale factor as, a = a exp (cid:18) (cid:90) ρρ dρf ( ρ ) (cid:19) . (13)From Eq. (10) one can see that there are two main possibilities for the cosmological evolution,1. The integral (10) converges at ρ → ρ m = α and singularity of type II takes place [28, 51].The scale factor remains finite with its first derivative with respect to the cosmic time. This variant of cosmo-logical evolution is realized for phantom dark energy only. Also we have singularity of this type if f ( ρ ) has azero at some ρ f < ρ m and the integral with respect to the cosmic time converges.2. Although we do not know the equation of state for dark energy, we can forecast its qualitative behavior byassuming that the function f ( ρ ) has a zero at some value ρ f . If the integral (10) diverges at some value ofenergy density ρ → ρ f < α . In this case scale factor tends to infinity at t → ∞ . Therefore we have quasi-deSitter expansion with effective cosmological constant Λ = ρ f .Of course one can ask how it is possible to discriminate the effects of the thermal radiation from modified gravityor specific fluid effects. This question arises for many cosmological models. Theories of modified gravity can leadto cosmological dynamics similar to models with some fluids or models on brane for example. Maybe one of thearguments is in favor for the model of thermal radiation which is inherently relatively simpler (on our opinion ofcourse) in comparison with complicated form of actions of Galileon gravity for example.One of the main indicators for any cosmological model is data about distance modulus as a function of the redshiftfrom the Supernova Cosmology Project [52]. For standard cosmology the distance modulus for a supernova withredshift z = a /a − µ ( z ) = const + 5 log D L ( z ) . (14)Here D L ( z ) is the luminosity distance. For Friedmann cosmology, the luminosity distance is, D L ( z ) = cH (1 + z ) (cid:90) z dzE ( z ) , (15)where E ( z ) is the dimensionless Hubble parameter i.e. E ( z ) = H ( z ) H = (cid:18) ρρ (cid:19) / . In particular, for the well known ΛCDM model, we have, E ( z ) = (cid:0) Ω m (1 + z ) + Ω Λ (cid:1) / . (16)Here, Ω m is the fraction of the total density contributed by matter, and Ω Λ is the fraction contributed by the vacuumenergy density.For our purposes, it is convenient to use dimensionless units for the Hubble parameter, α and energy density: α → ˜ αH − , H → H E, ρ → H ˜ ρ. In these units for dimensionless Hubble parameter, we obtain, E = (cid:18) − √ − α ˜ ρ α (cid:19) / . (17)One needs to take into account that, at present time, the dimensionless density satisfies ˜ ρ (cid:54) = 1, in contrast with theordinary Friedmann cosmological model, because ρ (cid:54) = 3 H although for ˜ α << ρ ≈ q is defined according to relation, q = − aH d adt = (18)= − α (1 − αρ/ − / (cid:16) − (1 − αρ/ / (cid:17) − ( ρ + p )In Friedmann cosmology one obtains the well-known expression for q assuming simply α → q (0) = − (cid:18) pρ (cid:19) . (19)For the ΛCDM model one obtains, q (0)Λ CDM = − m . One can get also jerk parameter j [53] for our model in comparison with ordinary cosmology, j = 1 aH d adt = (20)= 2 α (cid:32) α ( ρ + p ) (1 − αρ/ / + 1 − (cid:112) − αρ/ α + 3( ρ + p )2(1 − αρ/ / dpdρ (cid:33) (1 − (1 − αρ/ / ) − For α → j (0) = 1 + 92 (cid:18) pρ (cid:19) dpdρ (21)and particularly for the ΛCDM model we have p = − Λ = const, therefore j (0)Λ CDM = 1.Data about the deceleration and jerk parameters can be obtained from astronomical observations of distant objectsat redshifts z (cid:38)
1. Estimations for deceleration and jerk parameter from current observations do not discriminatedirectly the correct model of dark energy, because statistical errors are sufficiently large, and therefore models alter-native to the ΛCDM model can be perfectly compatible to the current observational data. The ΛCDM values of thesestatefinder quantities can act as reference points for alternative to ΛCDM models.In the next sections, we shall present several dark energy models by taking into account thermal radiation effects.There are various scenarios for a Universe with dark energy in the future. From observations it is apparent thatequation of state for dark energy is very close to simple equation of state of the cosmological constant, p d = − ρ d . It is convenient to use EoS formalism for other models of dark energy. Such formalism can be applied not only tomodels in which dark energy is a fluid but for scalar fields too. Our main purpose was to investigate the role whichthermal radiation in these scenarios. Therefore the models that we shall study in the following sections, are illustrativeand simple examples of dark energy models with a variety of possible future evolutions, with a common characteristicthough, that they mimic the standard cosmological model in the cosmological past.
III. A DECELERATING QUINTESSENCE MODEL
It is well-known that for a quintessential Universe with simple equation of state p = wρ ( − < w < − / H i = 1 √ α , (22) FIG. 1: Dependence of acceleration ¨ a/a (in units of H from scale factor for various values of w in a case of simple model ofquintessence with constant parameter of state. For ˜ α we take value 0.01. Initial value of scalar factor is a i = 1. Duration ofphase with ¨ a < | w | . Maximal value of acceleration otherwise increases. and the energy density of quintessence at the moment of the singularity is, ρ i = 34 α (23)and then it decreases with scale factor as, ρ d = ρ i a w +1) i a w +1) . Therefore we have, ¨ aa = − ρ i (1 + w ) (cid:113) − a w +1) i /a w +1) a w +1) i a w +1) + 1 − (cid:113) − a w +1) i /a w +1) α . (24)For t >> t i we have, ¨ aa = −
16 (1 + 3 w ) ρ i a w +1) i /a w +1) > . Near the initial singularity, the first term in Eq. (24) is very large and negative for quintessence, and therefore ¨ a < w and α . Duration of deceleratingphase depends from w strongly. In a phantom case first term is always positive and therefore universe filled phantomenergy expands with acceleration. IV. TRANSITION FROM LITTLE-RIP AND TYPE I AND III SINGULARITIES TO TYPE IISINGULARITY
Let’s consider the case of the following EoS, f ( ρ d ) = β ρ d ( ρ d /ρ d ) γ , β, γ ≡ const > . (25) w = − . w = − . w = − . α t f − t t − t a t f − t t − t a t f − t t − t a t f − t and time after moment when acceleration began t − t a (in units of H − ) for variousvalues of ˜ α ). For initial Ω m the value 0.28 is taken.FIG. 2: Deceleration (left panel) and jerk (right panel) parameters in model (26) for some β and α . Hereinafter for Ω d istaken 0.72 in current moment. In Friedmann cosmology without radiation term, this EoS leads to various scenarios of cosmological evolution: 1)little rip takes place for γ ≤ / / < γ ≤ γ > w = p d /ρ d for the above EoS at present time is simply, w = − − β . In our case we have only Type II singularity for this EoS. We start our consideration using a simple model: f ( ρ ) = β ρ d = const . (26)From the continuity equation for the dark energy fluid (3) we derive ρ d = ρ d (1 + 3 β ln a ) . (27)Without loss of generality we assume that the current value of the scale factor is a = 1. We consider the Universefilled matter and dark energy only and calculate the time for the final singularity for several values of w and α . Alsowe considered the behavior of cosmological acceleration in past and we found the time instance at which ¨ a changessign, and our results are presented in IV.The behavior of the deceleration q and jerk j parameters are presented in Fig. 2 in comparison with dark energymodel (25) without thermal radiation from the moment of beginning of cosmological acceleration. Roughly, one canexpect that thermal radiation effects have a measurable effect, only during the cosmological acceleration.One can describe this model in terms of scalar field theory, and in the case of a phantom scalar field, the energydensity and the pressure of the scalar field are, ρ d = − ˙ φ V ( φ ) , p d = − ˙ φ − V ( φ ) . (28), From these equations one can obtain that φ = (cid:90) √− ρ d − p d dt = (cid:90) (cid:112) f ( ρ d ) dt. (29)For f ( ρ ) from Eq. (26) we have simple linear dependence of scalar field with respect to the cosmic time, φ = φ + β √ ρ d t. (30)For a Universe dominated of dark energy without thermal radiation, t = √ (cid:90) daa √ ρ d (31)and thus we have for the integral (27), t = 1 µ (cid:0) β ln a − (cid:1) , µ ≡ β √ ρ d . For the scalar field potential we have,2 V = ρ d − p d = 2 ρ d + f ( ρ d ) = 2 ρ d (cid:112) β ln a + β ρ d . Using the expression for the cosmic time, one obtains the potential as function of scalar field has the following form, V ( φ ) = 3 β φ − φ ∗ ) + β ρ d , φ ∗ ≡ φ − √ β (32)With thermal radiation we have for the cosmic time a more complicated expression, t = 1 ν (cid:110) (2 + g ( x )) (cid:112) − g ( x ) − (2 + g (0)) (cid:112) − g (0) (cid:111) , g ( x ) ≡ (cid:112) − αρ d / − αβ x, (33) x ≡ ln a, ν ≡ √ αβ ρ d √ . Therefore, the potential of scalar field could be obtained only in parametric form. For comparison we present thedependence of the potential with respect to various parameters in Fig. 3.For γ = 2 we have the following dependence density of dark energy from scale factor ρ d = ρ d − β ln a . The value of a f = e β − / corresponds to the moment of final singularity. The energy density tends to infinity fora finite value of the scale factor. With account of thermal radiation, the Universe ends its existence earlier notapproaching a f . We compared the time for singularity occurrence in this model without thermal radiation and withaccount of it, in Table IV. For the considered values of α , the cosmological evolution in the past does not differsignificantly from the model without thermal radiation. The behavior of the parameters q and j for γ = 2 can beseen in Fig. 4. V. PSEUDO-RIP WITH ACCOUNT OF THERMAL RADIATION
It is known from the literature that several phantom and quintessence models can lead to the so-called pseudo-Rip singularities [43, 46]. The Universe expands asymptotically according to exponential law with some effective“cosmological constant”. It is interesting to investigate influence of thermal radiation on the pseudo-rip expansion ofthis sort. As an example of such model we shall consider the following, f ( ρ d ) = ± β ρ d ρ f − ρ d ρ f − ρ d , ρ f = const . (34) FIG. 3: Potential of scalar field for model (26) for the same parameters as on Fig. 2. w = − . w = − . w = − . α t f − t t − t a t f − t t − t a t f − t t − t a t f − t and time after moment when acceleration began t − t a in model with thermal radiation(in units of H − ) for γ = 2 in EoS (25). For initial Ω m the value 0.28 is taken. We again choose the EoS so that the present time EoS parameter is simply, w = p d ρ d = − ∓ β . The choice of “+” ( ρ f > ρ d ) corresponds to phantom model, while the sign “-” ( ρ f < ρ d ) describes quintessence.For dark energy density as function of scale factor we obtain, ρ d = ρ f − ( ρ f − ρ d ) a − δ , (35) δ ≡ β ρ d | ρ f − ρ d | . If ρ f < / α then for large a , the energy density tends to ρ f , for phantom energy we have that ρ d → ρ f − (cid:15) while asfor quintessence ρ d → ρ f + (cid:15) .The effective value of ”vacuum energy” due to thermal radiation is larger in comparison to ρ f . For ρ f → α we have ρ eff /ρ f →
2. Therefore, the Universe expands faster due to the thermal radiation. The dimensionless parameters q and h behave very similar in both models (see Fig. 5). For ρ f > / α , a sudden future singularity occurs before exiton quasi-de Sitter expansion.0 FIG. 4: Deceleration (left panel) and jerk (right panel) parameters in model (25) for γ = 2 and some β and α . Nearthe singularity state-finder parameters q → −∞ , j → ∞ and its behaviour significantly differs from model without thermalradiation.FIG. 5: Deceleration (left panel) and jerk (right panel) parameters in model (34) for ρ f = 20 and some β and α . VI. TYPE II FUTURE SINGULARITY DARK ENERGY
Some models of dark energy can lead to sudden singularity in future without the effect of thermal radiation forexample, f ( ρ d ) = ± β ρ d − ρ d /ρ f − ρ/ρ f , (36)If we choose the sign “+”, the energy density increases as the Universe expands (phantom energy). The current EoSparameter w is simply w = − − β . (37)1 w = − . w = − . w = − . α t f − t t − t a t f − t t − t a t f − t t − t a t f − t and time after moment when acceleration began t − t a in model with sudden futuresingularity (36) (in units of H − ). For initial Ω m the value 0.28 is taken. Parameter ρ f is 20 (in units of 3 H ).FIG. 6: Deceleration (left panel) and jerk (right panel) parameters in model (36) for some β and α . Parameter ρ f is 20 (inunits of 3 H ). The pressure approaches infinity and a sudden future singularity takes place. For quintessence dark energy, f ( ρ d ) isnegative and the final singularity corresponds to ¨ a < ρ = ρ f (cid:16) − (cid:0) (1 − ∆) − β ∆(1 − ∆) ln a (cid:1) / (cid:17) , ∆ = ρ /ρ f . (38)The scale factor in past can be expressed as a function of the redshift, which is defined as, a = 11 + z assuming that the present time scale factor is equal to unity, and the dependence of the luminosity distance D L fromthe redshift z is D L = cH (1 + z ) (cid:90) z (cid:0) Ω m (1 + z ) + Ω D h ( z ) (cid:1) − / dz, (39) h ( z ) = ∆ − (cid:18) − (cid:16) (1 − ∆) + 6 β ln(1 + z ) (cid:17) / (cid:19) . The model under discussion is indistinguishable from theΛCDM cosmology for ∆ <<
1. The deceleration and jerkparameters are given on Fig. 6 for some values of α and β . The example considered above is a good theoreticalillustration of dark energy models mimicking vacuum energy, but these models lead to Type II singularities.2 VII. A PROPOSAL FOR EVADING THE BIG RIP CHAOS: A DARK FLUID MIMICKING R GRAVITY
In the process of approaching the Big Rip singularity, no matter how this singularity occurs, the Hubble rate growsinevitably, and a cosmological horizon of some sort is expected, as we already discussed in previous sections. Oneintriguing theoretical question is whether the singularity can be avoided in the first place, if during the dark energyera some term appearing already in the Lagrangian of the theory, makes the singularity milder, or even disappear.In standard modified gravity contexts, such a possibility is realized by adding R terms in the Lagrangian, see forexample Ref. [54].Thus it is tempting to add a geometric fluid with EoS parameter of the form w = w ( H, ˙ H, ¨ H ) that mimics the R gravity energy density and pressure, and adding this near a Big Rip singularity. Assuming a flat FRW spacetime, theFriedmann equation near the Big Rip would be at leading order,3 H κ (cid:39) αH + ρ G , (40)where ρ G is the R fluid energy density, the analytic form of which as a function of the Hubble rate and its derivativeswith respect to the cosmic time is, ρ G = − βH ¨ Hκ − βH ˙ Hκ + 18 β ˙ H κ + 3 H κ , (41)where β is a parameter with mass dimensions [ m ] − . Basically, the energy density ρ G is essentially the energy densitycorresponding to an R geometric fluid of the form βR , with the Ricci scalar being as usual R = 12 H + 6 ˙ H for theflat FRW background. In order to see whether the Big Rip singularity is avoided in the presence of the R geometricfluid, one must solve the Friedmann equation (40) analytically, which can be cast as follows,36 βH ¨ Hκ + 108 βH ˙ Hκ − β ˙ H κ − αH (cid:39) . (42)However, this is not an easy task, therefore we shall investigate the behavior of the solutions numerically. We shalladopt the reduced Planck units system, for convenience, in which κ = 1. The initial time instance of the time loopfor our numerical integration will be some initial point when the thermal effects start to occur, so suppose that it isnear, t ∼ O (1 /H B ) where H B is the value of the Hubble rate when the initial Big Rip singularity is approached, soit is expected to grow significantly near unity in reduced Planck units, in contrast to the tiny present day value inreduced Planck units. So assuming that we run the numerical integration from t ∼ . in reduced Planck units, whichis basically at the beginning of the thermal effects, and by running the numerical integration for O (10) time units,we obtain the results presented in Fig. 7. Specifically, in Fig. 7 the behavior of the Hubble rate as a function of thecosmic time in reduced Planck units is presented. The time interval is significantly large in reduced Planck units, sothis numerical integration actually covers the a large time interval in the far future after the thermal effects startedto have a measurable effect in the Friedmann equations. We used various initial conditions for the derivative of theHubble rate in reduced Planck units, and and for all the plots, the Hubble rate at the beginning of the time loopwas assumed to be H B (cid:39) O (10 − ) and the values of α and β where assumed to be of the order of α, β ∼ O (1). Thered dashed curve corresponds to ˙ H (0 . ) ∼ O (10 − ), the red curve to ˙ H (0 . ) ∼ O (10 − ), while the blue dotted curveto ˙ H (0 . ) ∼ O (10 − ). As it is obvious in all these cases, the Hubble rate grows significantly, but instead of blowingup at finite-time, it reaches a plateau value, which is different for the three different initial conditions, and obviouslythis is a pure de Sitter state. This final de Sitter approach of the Hubble rate is also common to pure f ( R ) gravityworks, where the R term eliminates completely the Big Rip singularity from the cosmological evolution [54–56].Clearly this result indicates that the R fluid stabilizes the cosmological evolution, and definitely eliminates the BigRip singularity. Furthermore, as it was clearly shown in Refs. [54–56], in the presence of the R term or other f ( R )gravity models, which yield an exact de Sitter solution at future, apart from the Big Rip singularity, it is also possibleto eliminate Type II and III singularities. Hence, the fluid mimicking the R term in fact eliminates not only BigRip singularities, but also Type II and Type III singularities, although in principle smooth types of singularities, likethe Type IV, can still occur. Furthermore, eventually other fluids which lead to an accelerating Universe, but yieldasymptotically de Sitter solutions in the future, also effectively cancel future singularities of Type I,II and III even inpresence of thermal effects.3 FIG. 7: The Hubble rate as a function of the cosmic time in reduced Planck units, in the presence of an R fluid. The final deSitter point is reached and the Big Rip singularity is avoided, for various distinct initial conditions of ˙ H , at the moment thatthe thermal effects affect the evolution. VIII. CONCLUSION
In this work we considered dark energy models by taking into account thermal radiation effects. We assumed thatthis radiation contributes to the total energy density with a energy density term which is proportional to H , where H is the Hubble rate. Consequently, if the dark energy density values grow infinitely as the Universe expands (phantomenergy), then we can be certain that a sudden future singularity takes place, or is approached inevitably in the nearfuture. For some finite energy density ρ m and scale factor, the second derivative of scale factor diverges. Thermalradiation leads to the so-called little rip scenario, when the dark energy density increases infinitely with time and theEoS parameter approaches the value − t → ∞ . The cosmological expansion with a future Big Rip,without thermal radiation, also changes to an evolution leading to a sudden singularity. It is interesting to note thatthe phase of accelerated expansion in models with thermal radiation, begins later in comparison with cosmologicalmodels without this contribution, but the final singularity takes place earlier.The realization of a quasi-de Sitter expansion (pseudo-rip) depends on the dark energy EoS. If the EoS parameterfor dark energy develops a zero value at some density ρ f , then if ρ f > ρ m , the sudden singularity still occurs. Inthe opposite case the Universe evolves to a de Sitter regime with some effective “cosmological constant”, larger incomparison with ρ f . For ρ f close to ρ m we have ρ eff /ρ f →
2. In effect, the Universe expands faster and evolves tothe de Sitter expansion earlier. In the case of an EoS with singularity (pressure tends to ∞ at some finite ρ f ) wehave in any case a sudden future singularity. The inclusion of thermal radiation does not make worse the complianceof the cosmological models with the observational data. One can construct (as we have here) models that mimicthe standard ΛCDM model up to present time, but the contribution of the thermal radiation can switch the futurecosmological evolution to a regime with a sudden future singularity.At present time, dark energy remains a mystery and quite many questions related to it are still not answeredconcretely. The current models describing dark energy only answer some questions, but to date no definitive answeris given. The main model which remains with good compliance with the observational data is the ΛCDM model, andthus many dark energy models originating from various theoretical contexts, are mainly designed to mimic the ΛCDMmodel. But the ΛCDM model has its weak points, two of which already appear in its name, Λ and Cold Dark Matter.With regards to the latter, dark matter is a speculation, but no dark matter particle has ever been observed. Withregard to Λ, the cosmological constant, its nature is unknown, and if it is seen as the vacuum energy, its present dayvalue is extremely small and infinitely smaller from the predicted vacuum energy value from quantum field theories.Apart from these two, there exist other conceptual issues to be resolved in the future, such as if the dark energy itselfis dynamical or not, and more importantly, is the dark energy era a phantom dark era? If one sticks on the generalrelativistic approach and insist on using he general relativistic recipe to describe the dark energy era, so the usage ofΛ or scalar fields, the last two questions will probably make their presence in a predominant way. This is because inthe case of dynamical dark energy with a varying EoS, the cosmological constant would not fit at all, since it yieldsa constant de Sitter EoS. Regarding the phantom question, things are getting worse if one sticks with the general4relativistic recipe, scalar fields, since phantom scalar fields must be used and phantom scalar fields from a theoreticalpoint of view are instabilities. Nevertheless, even if one adopts the phantom scalar field description for the phantomdark energy era, the result would be a Big Rip singularity as was demonstrated in Ref. [11]. Thus the study of futurefinite-time singularities is a possible eschatological scenario of our Universe. In view of these questions, modifiedgravity in its various forms, stands on a promising solid ground, since it answers in a relatively successful way manyof these questions. In the models we presented in this work, we were able to generate phantom evolution withoutthe need for phantom scalar fields, and we also indicated that finite-time singularities can actually occur by usingvarious fluid approaches, without again the need for phantom fluids, like for example in general relativistic contexts.In addition, some of the models can mimic at present-time the ΛCDM model, and also can provide a viable presentday dark energy era, compatible with the Planck data. In addition, we discussed how the evolution would changeif we took into account the thermal effects, thus our model indicates a road map towards possible future evolutions.Of course our approach is one of the many possible scenarios, however the future observations are promising, thus astheorists we try to investigate all the possible scenarios. Acknowledgments
This work was supported by MINECO (Spain), project PID2019-104397GB-I00 and PHAROS COST Action(CA16214) (SDO). [1] A. G. Riess et al., Astron. J. , 1009 (1998).[2] S. Perlmutter et al., Ap. J. , 565 (1999).[3] K. Bamba, S. Capozziello, S. Nojiri and S. D. Odintsov, Astrophys. Space Sci. (2012), 155-228 doi:10.1007/s10509-012-1181-8 [arXiv:1205.3421 [gr-qc]].[4] M. Li, X. Li, S. Wang and Y. Wang, Commun. Theor. Phys. , 525 (2011).[5] Y. -F. Cai, E. N. Saridakis, M. R. Setare and J. -Q. Xia, Phys. Rept. (2010) 1 [arXiv:0909.2776 [hep-th]].[6] Y. Akrami et al. [Planck Collaboration], arXiv:1807.06211 [astro-ph.CO].[7] M. Kowalski, Ap. J. , 74 (2008).[8] K. Nakamura et al. [Particle Data Group Collaboration], J. Phys. G G , 075021 (2010).[9] R. Amanullah et al., Ap. J. , 712 (2010).[10] R. R. Caldwell, Phys. Lett. B 545
23 (2002).[11] R. R. Caldwell, M. Kamionkowski and N. N. Weinberg, Phys. Rev. Lett. , 071301 (2003).[12] S. M. Carroll, M. Hoffman and M. Trodden, Phys. Rev. D68 , 023509 (2003).[13] S. Nesseris and L. Perivolaropoulos, JCAP (2007) 018 [astro-ph/0610092].[14] P. H. Frampton and T. Takahashi, Phys. Lett. B , 135 (2003).[15] S. Nojiri and S. D. Odintsov, Phys. Lett. B , 147 (2003) [arXiv:hep-th/0303117].[16] V. Faraoni, Int. J. Mod. Phys. D , 471 (2002) [arXiv:astro-ph/0110067].[17] P. F. Gonzalez-Diaz, Phys. Lett. B , 1 (2004) [arXiv:astro-ph/0312579].[18] E. Elizalde, S. Nojiri and S. D. Odintsov, Phys. Rev. D , 043539 (2004) [arXiv:hep-th/0405034].[19] P. Singh, M. Sami and N. Dadhich, Phys. Rev. D , 023522 (2003) [arXiv:hep-th/0305110].[20] C. Csaki, N. Kaloper and J. Terning, Annals Phys. , 410 (2005) [arXiv:astro-ph/0409596].[21] P. X. Wu and H. W. Yu, Nucl. Phys. B , 355 (2005) [arXiv:astro-ph/0407424].[22] S. Nesseris and L. Perivolaropoulos, Phys. Rev. D , 123529 (2004) [arXiv:astro-ph/0410309].[23] H. Stefancic, Phys. Lett. B , 5 (2004) [arXiv:astro-ph/0310904].[24] L. P. Chimento and R. Lazkoz, Phys. Rev. Lett. , 211301 (2003) [arXiv:gr-qc/0307111].[25] J. G. Hao and X. Z. Li, Phys. Lett. B , 7 (2005) [arXiv:astro-ph/0404154].[26] M. P. Dabrowski and T. Stachowiak, Annals Phys. , 771 (2006) [arXiv:hep-th/0411199].[27] S. Nojiri and S. D. Odintsov, Phys. Rev. D (2004) 103522 [hep-th/0408170].[28] J. Barrow, Class. Quant. Grav. , L79 (2004).[29] Y. Shtanov and V. Sahni, Class. Quant. Grav. (2002) L101 [gr-qc/0204040].[30] S. Nojiri and S. D. Odintsov, Phys. Lett. B , 1 (2004) [arXiv:hep-th/0405078].[31] S. Cotsakis and I. Klaoudatou, J. Geom. Phys. , 306 (2005) [arXiv:gr-qc/0409022].[32] M. P. Dabrowski, Phys. Rev. D , 103505 (2005) [arXiv:gr-qc/0410033].[33] L. Fernandez-Jambrina and R. Lazkoz, Phys. Rev. D , 121503(R) (2004) [arXiv:gr-qc/0410124].[34] Phys. Lett. B , 254 (2009) [arXiv:0805.2284 [gr-qc]].[35] J. D. Barrow and C. G. Tsagas, Class. Quant. Grav. , 1563 (2005) [arXiv:gr-qc/0411045].[36] H. Stefancic, Phys. Rev. D , 084024 (2005) [arXiv:astro-ph/0411630].[37] C. Cattoen and M. Visser, Class. Quant. Grav. , 4913 (2005) [arXiv:gr-qc/0508045]. [38] P. Tretyakov, A. Toporensky, Y. Shtanov and V. Sahni, Class. Quant. Grav. , 3259 (2006) [arXiv:gr-qc/0510104].[39] A. Balcerzak and M. P. Dabrowski, Phys. Rev. D , 101301(R) (2006) [arXiv:hep-th/0604034].[40] A. V. Yurov, A. V. Astashenok and P. F. Gonzalez-Diaz, Grav. Cosmol. , 205 (2008) [arXiv:0705.4108 [astro-ph]].[41] J. D. Barrow and S. Z. W. Lip, Phys. Rev. D , 043518 (2009) [arXiv:0901.1626 [gr-qc]].[42] M. Bouhmadi-Lopez, Y. Tavakoli and P. V. Moniz, JCAP , 016 (2010) [arXiv:0911.1428 [gr-qc]].[43] P. H. Frampton, K. J. Ludwick and R. J. Scherrer, Phys. Rev. D (2011) 063003 [arXiv:1106.4996 [astro-ph.CO]].[44] P. H. Frampton, K. J. Ludwick and R. J. Scherrer, arXiv:1112.2964 [astro-ph.CO].[45] P. H. Frampton, K. J. Ludwick, S. Nojiri, S. D. Odintsov and R. J. Scherrer, Phys. Lett. B (2012) 204 [arXiv:1108.0067[hep-th]].[46] A. V. Astashenok, S. Nojiri, S. D. Odintsov and A. V. Yurov, arXiv:1201.4056 [gr-qc].[47] G. W. Gibbons and S. W. Hawking, Phys. Rev. D 15 (1977) 2738.[48] R. G. Cai, L. M. Cao and Y. P. Hu, Class. Quant. Grav. 26 (2009).[49] R. Ruggiero, [arXiv:2005.12684 [gr-qc]].[50] S. Nojiri and S. D. Odintsov, Phys. Dark Univ. (2020) 100695 [arXiv:2006.03946 [gr-qc]].[51] S. Nojiri, S. D. Odintsov and S. Tsujikawa, Phys. Rev. D (2005) 063004 [hep-th/0501025];S. Nojiri and S. D. Odintsov, Phys. Rev. D (2005) 023003 [hep-th/0505215].[52] R. Amanullah, C. Lidman, D. Rubin, G. Aldering, P. Astier, K. Barbary, M. S. Burns and A. Conley et al. , Astrophys. J. , 712 (2010) [arXiv:1004.1711 [astro-ph.CO]].[53] V. Sahni, T. D. Saini, A. A. Starobinsky and U. Alam, JETP Lett. (2003) 201 [Pisma Zh. Eksp. Teor. Fiz. (2003)249] [astro-ph/0201498].[54] K. Bamba, S. Nojiri and S. D. Odintsov, JCAP (2008), 045 doi:10.1088/1475-7516/2008/10/045 [arXiv:0807.2575[hep-th]].[55] S. Nojiri and S. D. Odintsov, Phys. Rev. D (2008) 046006 doi:10.1103/PhysRevD.78.046006 [arXiv:0804.3519 [hep-th]].[56] S. Capozziello, M. De Laurentis, S. Nojiri and S. D. Odintsov, Phys. Rev. D79