The unified history of the viscous accelerating universe and phase transitions

Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 974 (2022) 115646 www.elsevier.com/locate/nuclphysb The unified history of the viscous accelerating universe and phase transitions A.V. Astashenok a,∗, S.D. Odintsov b,c,d, A.S. Tepliakov a aInstitute of Physics, Mathematics and IT, Immanuel Kant Baltic Federal University, 236041 Kaliningrad, Russia bConsejo Superior de Investigaciones Científicas, ICE/CSIC-IEEC, Campus UAB, Carrer de Can Magrans s/n, 08193 Bellaterra (Barcelona), Spain cInternational Laboratory for Theoretical Cosmology, Tomsk State University of Control Systems and Radioelectronics (TUSUR), 634050 Tomsk, Russia dICREA, Passeig Luis Companys, 23, 08010 Barcelona, Spain Received 28 August 2021; received in revised form 3 November 2021; accepted 26 December 2021 Available online 27 December 2021 Editor: Hong-Jian He Abstract We propose the unified description of the early acceleration (cosmological inflation) and the present epoch of so called “dark energy”. The inflation can be described by cosmic fluid with van der Waals equation of state and with viscosity term. Viscosity leads to slow-roll inflation with the parameters such as the spectral index, and the tensor-to-scalar ratio in concordance with observational data. Our next step is to modify this equation of state (EoS) to describe the present accelerated expansion. One can add the term into EoS so that the contribution of which is small for inflation but crucial for late-time acceleration. The key point of the model is possible phase transition which leads to decrease of the viscosity. We show that proposed model describes observational data about standard “candles” and correct dependence of Hubble parameter from redshift. Moreover, we propose the possible scenario to resolve dark matter problem. ©2021 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. *Corresponding author. E-mail addresses: [email protected] (A.V. Astashenok), [email protected].es (S.D. Odintsov), ateplyako[email protected] (A.S. Tepliakov). https://doi.org/10.1016/j.nuclphysb.2021.115646 0550-3213/©2021 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 1. Introduction The observational data from Planck [1,2] confirmed the inflation theory in its simple form [3,4]. The non-Gaussian perturbations of some type are negligible and therefore simplest singlefield models are valid. The results of Planck ruled out some alternative scenarios. The inflation is very rapid expansion of the universe in an unstable state at the top of the effective potential. Scalar field slowly rolls down to the minimum of effective potential. Due to the expansion the universe became flat and very big. During the inflation the density perturbations are generated. These perturbations are inversely proportional to the velocity of scalar field decreasing ˙ φ. In slow-roll regime perturbations generated during the inflation have a spectrum very close to flat one. Although inflation usually is described in terms of scalar field theory one can use a perfect fluid model or modified gravity [5–7]. In [8]the authors obtained conditions for preventing the occurrence of self-reproduction in inflation epoch. The case of multiple coupled viscous fluid was considered in [9–11]. The possibility of coupling between energy and dark matter was investigated in [12], and some examples of inhomogeneous viscous coupled fluids were considered in [13–15]. One can note also cosmological models with bouncing, which are caused by an inhomogeneous viscous fluid [16]. A number of papers is devoted to viscous fluid with so called van der Waals EoS [17–21]. The van der Waals fluid model can account both for the early and late-time accelerated expansion stages of the universe. That is, inflation may be described by a van der Waals fluid, with the specific properties of a cosmic fluid obeying the van der Waals equation of state having been analyzed [22]. Various versions of the van der Waals equation can be considered in seeking a better match with the observational data of the Planck satellite [23]. Another interesting cosmological phenomenon is accelerated expansion of the Universe [24, 25]. From a theoretical viewpoint the cosmological acceleration can be caused by some fluid with negative pressure and/or negative entropy (for review, see [13,26,27]). The nature of this fluid is unclear and late-time accelerated expansion is dubbed also as the dark energy era. The current observations coming from Planck [2], indicate that dark energy consist of nearly 70% of total energy density of the Universe [28]. The EoS parameter for the dark fluid, namely wd, is negative: wd=pd/ρd<0,(1) where ρdand pdare the dark energy density and pressure, correspondingly. However, it is still not clear what is the precise value of wd[29,30], although the latest Planck data constrains significantly the values that the EoS parameter can take. The interesting question is whether it is possible the description not only of the early-time inflation but also of the late cosmological acceleration in unified way with using the viscous fluid. Note that viscous dark energy models as well as viscous inflationary models are considered in many papers [31–47]. We studied the possible unified description of early and late-time accelerated expansion of universe in terms of the van der Waals equation of state for cosmic fluid including viscous term. Slow-roll inflation is caused by viscous term in the equation of state being proportional to square of Hubble parameter i.e. the energy density if we neglect the contribution of matter and radiation. The exit from the rapid expansion is provided by decrease of viscosity in the form of a phase transition. One can add other term to the EoS of van der Waals fluid which contribution is negligible for inflation epoch but crucial for late-time acceleration. 2 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 We calculate parameters of inflation for considered model and especially consider such values of parameters which are compatible with Planck data. One should stress the role of viscosity in considered model. The agreement with Planck data can be achieved only if viscosity term is present. We start in the Section 2from the basic cosmological equations and van der Waals equation for viscous fluid which plays the role of dark energy in our model. One assumes that viscosity depends from the energy density. Then we consider the possible unification of inflation with the era of matter domination. Section 4is devoted to the comparison of results of considered model with Planck data. We calculated spectral index and slow-roll parameters for the model. Finally the adopted EoS for the description of late time acceleration is considered. One can include in the equation of state such terms which don’t affect the inflation parameters but lead to cosmological acceleration. The analysis shows that proposed model is compatible with observational data such as data about dependence “magnitude-redshift” for SN Ia and dependence of Hubble parameter from redshift. Some outlook is given in the conclusion. 2. Basic viscous universe We consider spatially flat Friedmann-Lemaître-Robertson-Walker spacetime with the metric: ds2=dt2−a2(t)(dx2+dy2+dz2)(2) In the natural system of units (c=8πG =1) the first Friedmann equation is H2=ρ 3,(3) where ρis the energy density, H(t) =˙a(t)/a(t) is the Hubble parameter, a(t) the scale factor. The energy density is ρ=ρd+ρm+ρr,(4) where ρmand ρdare the density of matter and radiation correspondingly and ρdis energy density of cosmic dark fluid. We take the following nonlinear inhomogeneous equation of state for this fluid: pd=w(ρd,t)ρ d−Hζ(H,t)+f(ρ d). (5) Here w(ρ, t) is thermodynamical parameter which depends from the energy density and time. The second term describes bulk viscosity. The viscosity ζ(H, t) in general case depends from Hubble parameter and time (see [48]). We add also arbitrary function of density into equation of state f(ρ). In the simplest case f(ρ) =0. Similar EoS is typical in theories of modified gravity [49]. One can note that equation of continuity for cosmic fluid with equation of state (5)is the same for non-viscous fluid: ˙ρd+3H(ρd+pd)=0.(6) Usually, in viscous cosmology, this equation would include a term on the right hand side containing the bulk viscosity. But, we should emphasize that the properties coming from viscosity are here included through the inhomogeneous equation of state, instead of via a more standard bulk viscosity term. 3 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 We consider the van der Waals fluid model [22]. As well known the van der Waals theory in hydrodynamics describes the gas when density is sufficiently large and due to this gas cannot be considered as the ideal one. In this case one should take into account the finite volume of molecules of the gas and viscosity. At first glance this situation is very far from the cosmological fluids. However one can assume that at the very early universe the phase transition takes place and viscosity term plays significant role in the inflation. This gives the ground to think of van der Waals fluid models as a serious and reasonable alternative in cosmological applications. Indeed, the corresponding equation fits very naturally in, as a possibility for an inhomogeneous equation of state. Following this approach we consider thermodynamic parameter to be of the form w(ρd,t)=w0 1−βρd/ρc .(7) Here w0is the constant value. For ρd<< ρc/β we have simply w≈w0. Parameter βplays the role a critical thermodynamic parameter. The critical value ρcindicates that cosmic fluids change phases under certain thermodynamic conditions. Whereas perfect fluids do not permit phase transitions to occur, here the phase transition phenomenon can be occur by means of the twophase fluid described by the van der Waals equation. For bulk viscosity the following dependence can be chosen: ζ(H,t)=ξ(H)(H)n. We assume that function ξ(H) weakly depends from the Hubble parameter in the sense that dξ dH << n(H)n−1. We consider the simplest case n =1for the bulk viscosity. In this case equation of state (5) takes the following form pd=w0ρd 1−βρd/ρc +f(ρ d)−ξρ/3(8) For inflationary acceleration epoch one can neglect the densities of radiation and matter and assume that ρ≈ρd. Next, using equations (6) and (8) one obtains the following differential equation relating the scale factor to the energy density: a1−βρd ρcdρd da +3ρd(1−ξ/3)(1−βρd/ρc)+w0+f(ρ d) ρd (1−βρd/ρc)=0.(9) Let’s assume that viscosity depends from the fluid density in the following way: ξ(ρd)=ξ0−ξ exp(−(ρd−ρ0)2/ρ2), ξ > 0.(10) If ρd>> ρ0, ρ we have ξ≈ξ0. In vicinity of ρ0viscosity drops down. The main motivation for Eq. (10)is describing of the possible phase transition for viscous fluid when the value of viscosity sharply decreases. For f(ρ d)we can choose for simplicity: f(ρ d)=−αρ2 d ρc (11) The equation (9) can be rewritten in the following form: dlna dx =− (1−βx) x(a1x2+a2x+a3+d(x)).(12) 4 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 Fig. 1. Dependence of e-foldings ln(a/ainf )from time (hereafter in units of H−1 inf , Hinf means value of Hubble parameter in the beginning of inflation) for various parameters of the model with equation of state (8). Parameters α=0.005, β=0.05 are fixed. (For interpretation of the colors in the figure(s), the reader is referred to the web version of this article.) Here x=ρd/ρcis dimensionless density. The coefficients aiare a1=αβ, a2=βξ0/3 −α− β, a3=w0−ξ0/3 +1. The function d(x) is d(x)=1 3ξ exp(−(x −xf)2/x2)(1−βx), x =ρ/ρc. For x>>x fthe function d(x) ≈0 with good accuracy. For the epoch of cosmological expansion xdecreases and scale factor aincreases. Therefore r.h.s. of Eq. (12) should be positive. Initially consider the case when ξ =0. If equation a1x2+a2x+a3=0 has at least positive root x1and initial value of x(0) >x 1the scale factor a→∞for x→x1. Therefore we have the expansion according to the exponential law at large times with some constant energy density and pd/ρd→−1for t→∞. Possible sharp decrease of viscosity (one can consider this process as phase transition) leads to decrease of the cosmological expansion rate and for some parameters to exit from the inflation. There are two possible scenarios of this phase transition: 1) ρf= 0i.e. viscosity decreases in the range ρf<ρ d<ρ cand then viscosity increases. 2) ρf=0, viscosity decreases with density. For the illustration we consider the model with fixed parameters α=0.005, β=0.05 and assume that ρ(0) =ρc, ρ0=0. On Fig. 1we give the dependence of e-foldings number ln(a/ainf ) where ainf is initial scale factor. Due to the decrease of the viscosity inflation ends. The time of exit from the phase of fast expansion (exit on plateau) depends from ξ and x2. Hubble parameter sharply decreases (see Fig. 2). For illustration we choose such values of ξ and x2 that numbers of e-foldings lie withing realistic limits. 3. The unification of the inflation with matter domination epoch One can construct models in which dark fluid mimics dark matter with the effective value of state parameter close to w≈0. The asymptotic value of wafter inflation depends from parameter 5 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 Fig. 2. Dependence of Hubble parameter (in units of Hinf ) from time for various parameters of the model with equation of state (8). ξ (for given w0and αand β. On Fig. 3we depicted the dependence of effective equation of state parameter w=pd/ρd from time for various values of x2. For α=0.005, β=0.05, ξ0=3.42 we have asymptotic value w≈0on exit from the inflation if ξ =0.87ξ0. Phase transition leads to sharp increase of wfrom the value nearly −1to 0. The change of ξ leads to various asymptotical value of EOS parameter. Therefore our example can describe transition to matter domination epoch at least qualitatively. The possible interaction between viscous fluid and baryon matter also leads to interesting consequences. Let’s consider the Universe which is filled with dark energy and baryon matter ρ=ρd+ρm, and assume the interaction between matter and dark energy. Then the following equations for dark energy and matter density are satisfied ˙ρd+3H(ρd+pd)+Q(ρd,ρ m)=0,(13) ˙ρm+3Hρm−Q(ρd,ρ m)=0.(14) For the pressure of dark fluid we again use the equation of state in the form (8). The function Q describes the interaction between the components. For example, let Qbe Q=δ(ρc−ρd)ρd,δ=const (15) For some δand ξ one can construct the models with the exit from the inflation and transition between rapid acceleration and deceleration. On Fig. 4we give some examples of such models for which w≈0after exit from the inflation. Some amount of matter appears due to the interaction between dark energy and matter if we start from the moment when ρm=0. Then parameter wd for dark energy asymptotically tends to zero. In some sense we have usual baryon matter and dark matter. One can in principle obtain required relation between density of baryon matter and dark energy (see left upper panel of Fig. 4). 6 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 Fig. 3. The dependence of the equation-of-state parameter w=pd/ρdfrom time (upper panel) and number of e-foldings as the function of w(down panel). Other parameters of the model are α=0.005, β=0.05, w0=0.145, ξ0=3.42, ξ =0.87ξ0. One note that we consider unification in sense that one equation of state for dark energy can describe early inflation and transition to matter domination epoch. Effective EOS parameter tends to 0after inflation for some parameters of our model and therefore viscosity fluid can mimic dark matter. Interaction with baryon matter leads to creation of some amount of baryon matter. In result after inflation we have baryon matter and viscous fluid with w=0 which can be considered as dark matter. Of course these examples are only illustration but probably in this way one can solve dark matter problem (for review of dark matter problem see [51]). 4. Late-time acceleration in the model of fluid with viscosity Due to the viscosity the cosmological acceleration decreases. For the equation of state considered above energy density of van der Waals fluid asymptotically approaches 0at t→∞. Hubble parameter also tends to zero. However, it is possible to construct solution with second phase of slow acceleration. For this one needs, for example, to change the function f(ρ d)in the equation of state as 7 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 Fig. 4. The relation between the matter density and dark energy density (upper left panel) for late times, number of efoldings as function of time (upper right panel), parameter p ρd+ρm(down panel) in the model with interaction between matter and viscous fluid. Other parameters of the model are α=0.005, β=0.05, w0=0.145, ξ0=3.42 and ξ = 0.88ξ0for δ=0.001 (ξ =0.89ξ0for δ=0.002). We choose ξ0and ξ so that asymptotic value of wis 0. f(ρ d)=−αρ2 d ρc −δ1ρcexp(−δ2ρd/ρc), (16) where δ1is small dimensionless positive value and δ2is some constant. The value ρcis used only for dimensional reasons in (16). At the beginning of the inflation this term is very small but it plays role at small energy densities. The possible physical motivation for this choice of course is not clear. But this is simplest way to modify EOS of viscous fluid for obtaining of late-time acceleration. One need to stress that the EOS (16) doesn’t coincide with standard EOS for late time acceleration. For ρd<< ρcwe have that pressure of dark energy is pd≈w0ρd−ξρd/3−C, C =δ1ρc. The Eq. (12)is rewritten in this case as dlna dx =− (1−βx) x(a1x2+a2x+a3+d(x)+g(x)),(17) g(x) =−δ1 x(1−βx)exp(−δ2x). 8 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 Fig. 5. The dependence of e-foldings (upper left panel), parameter of state (upper right panel) from time after initial fast acceleration and dependence of relation ρm/ρdfrom time for various values of δ1and interaction between matter and viscous fluid. Other parameters of the model are α=0.005, β=0.05, w0=0.145, ξ0=3.42 and ξ =0.88ξ0, δ=0.001, δ2=ρ−1 c. The main point is that for some small xthe expression in denominator has a zero at some x=xf. Therefore if inflation begins from some density xinf for which a1x2 inf +a2xinf +a3+d(xinf ) + g(xinf ) >0, the energy density asymptotically approaches xfat t→∞. The Universe expands according to de Sitter law. a∼exp(ρf/3t), ρf≈δ1 ρc c+d(0). We have two epochs of acceleration, one is the inflation (when parameter of equation-of-state is close to −1) and second one is de Sitter expansion (again the equation-of-state parameter is ≈−1). As on only illustrative examples we consider the cases when δ1=10−6, 10−7and δ1=10−8 and include in our model the interaction between viscous fluid and matter in form (15) (Fig. 5). At late times (for δ1=10−8after ≈1000H−1 inf ) the EoS parameter approaches −1. The rate of the expansion is relatively slow (in comparison with the early inflation). Of course the considered values of δ1are non realistic. The current Hubble parameter H0 is many orders of magnitude smaller than Hinf . However, simple model (8) with function f 9 A.V. Astashenok, S.D. Odintsov and A.S. Tepliakov Nuclear Physics B 974 (2022) 115646 References [1] N. Aghanim, et al., Planck collaboration, Astron. Astrophys. 641 (2020) A1. [2] Y. 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