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Physics Letters B

www.elsevier.com/locate/physletb

Confronting dark energy models mimicking ACDM epoch with observational constraints: Future cosmological perturbations decay or future Rip?

Artyom V. Astashenoka*, Sergei D. Odintsovb c'd e

a Baltic Federal University of I. Kant, Department of Theoretical Physics, 236041, 14, Nevsky st., Kaliningrad, Russia b Institucio Catalana de Recerca i Estudis Avanfats (ICREA), Barcelona, Spain

c Institut de Ciencies de l'Espai (CSIC-IEEC), Campus UAB, Torre C5-Par-2a pl, E-08193 Bellaterra (Barcelona), Spain d Eurasian International Center for Theor. Physics, Eurasian National University, Astana 010008, Kazakhstan e Tomsk State Pedagogical University, Tomsk, Russia

ARTICLE INFO

Article history:

Received 2 October 2012

Received in revised form 19 December 2012

Accepted 22 December 2012

Available online 28 December 2012

Editor: T. Yanagida

ABSTRACT

We confront dark energy models which are currently similar to ^CDM theory with observational data which include the SNe data, matter density perturbations and baryon acoustic oscillations data. DE cosmology under consideration may evolve to Big Rip, type II or type III future singularity, or to Little Rip or Pseudo-Rip universe. It is shown that matter perturbations data define more precisely the possible deviation from ^CDM model than consideration of SNe data only. The combined data analysis proves that DE models under consideration are as consistent as ^CDM model. We demonstrate that growth of matter density perturbations may occur at sufficiently small background density but still before the possible disintegration of bound objects (like clusters of galaxies, galaxies, etc.) in Big Rip, type III singularity, Little Rip or Pseudo-Rip universe. This new effect may bring the future universe to chaotic state well before disintegration or Rip.

© 2012 Published by Elsevier B.V.

1. Introduction

The late-time cosmic acceleration which seems to be proved by astrophysical data (see Refs. [1,2]) opened the door for a number of (often exotic) theoretical models of so-called dark energy (DE) (for recent review, see [3,4]). The dark energy contributing of nearly 72% of the total mass energy of the universe [5] has quite unusual properties like negative pressure and/or negative entropy, invisibility in the early universe, non-coupling with baryonic matter and effective non-observability, etc. The DE properties may vary in wide limits, what depends from the specific DE model under discussion.

The negative pressure leads to that the equation of state (EoS) parameter wD0 is negative:

w0 = PD0/PD0 < 0, (1)

where pD is dark fluid energy-density and pD is the pressure. The subscript 0 denotes that related quantities are considered at present time.

* Corresponding author.

E-mail address: artyom.art@gmail.com (A.V. Astashenok).

0370-2693/$ - see front matter © 2012 Published by Elsevier B.V. http://dx.doi.org/10.1016Zj.physletb.2012.12.058

The data of observations (supernova data, baryon acoustic oscillations, etc.) indicate in favour of vacuum energy/^CDM cosmology (w = — 1). However the ultimate resolution of question about nature of dark energy seems to be impossible due to sufficiently large uncertainties in the determination of the DE EoS parameter w = -1.04—0;?9 [6,7].

The phantom model (w < —1) firstly proposed in Ref. [8] probably is the most exotic model of dark energy. This model does not contradict the cosmological tests based on present data although from theoretical point of view the phantom field is unstable [9] because the violation of all energy conditions occurs.

The simplest phantom energy model with w = const leads to so-called Big Rip future singularity [8,10-13]. The scale factor becomes infinite at a finite time. Another types of finite-time future singularities are type II singularity [14] where the second derivative of scale factor becomes infinite at finite time while the first derivative is finite and type III singularity where the first derivative of scale factor diverges.

If w decreases sufficiently rapidly with increase of phantom energy-density the so-called Little Rip or Pseudo-Rip may be realized [15-17]. In the first case the acceleration of universe increases but the scale factor remains finite always. Of course, the disintegration of bound structures occurs as in a case of Big Rip cosmology. For Pseudo-Rip universe, dark fluid energy-density asymptotically

tends to constant value, i.e. this phantom energy model mimics vacuum energy from some moment of time.

To describe dark energy fluid we use the equation of state (EoS):

Pd = F (Pd ),

where F is a function of the energy-density. The evolution of the universe then depends on the choice of the EoS.

In this Letter we confront DE models mimicking currently ^CDM cosmology with the combined observational data including the luminosity distance modulus vs redshift for SNe la, the baryon acoustic oscillations (BAO) and matter and DE density perturbations. We demonstrate that the models under discussion are totally viable and not-distinguishable from ^CDM, while their future evolution may vary in a number of ways. The Letter is organized as follows. In Section 2 we present brief overview of the EoS fluid formalism. In Section 3 the main constraints from observational data are discussed. In Section 4 we confront DE models with Big Rip or type Ill future singularity as well as Little Rip model with SNe data, BAO data and matter perturbations. It is established the excellent coincidence with ^CDM model predictions at current universe. The consideration of DE and matter perturbations shows that growth of the energy-density may occur at sufficiently small background density but still before the possible disintegration of bound objects in the Rip universe. This new effect may bring the future universe to the separation in the several domains with different values of energy-density, i.e. kind of chaotic universe may emerge. Section 5 is devoted to comparison of DE models of Pseudo-Rip type or with type ll future singularity with combined data. The region of parameters where such models effectively coincide with current ^CDM cosmology are established. ln Section 6 some outlook is given. ln Appendix A we briefly describe the properties of Quasi-Rip universe.

2. Scalar dark energy models

For the spatially-flat Friedmann-Robertson-Walker (FRW) universe with metric

ds2 = dt2 - a2(t)(dx2 + dy2 + dz2), the FRW equations are given by

3, p = -3> + ^

where p and p are the total energy-density and pressure, a is the scale factor, ' = d/dt, and we use the natural system of units in which G = c = 1. For simplicity we assume that universe is filled only by cold dark matter, baryon matter and dark energy, i.e. p = pa + Pm and p = pD.

One can rewrite the dark energy EoS (2) in the following form:

Pd = -Pd - g(PD),

where g(pD ) is some function. The case g(pD) > 0 corresponds to the EoS parameter w < —1 (phantom) while the case g(pD) < 0 corresponds to the EoS parameter w > —1. Assuming that dark energy dominates, one can neglect the contribution of other components (matter, dark matter). Then from Eq. (4), one can get the following expression for time variable:

t — to-

Pd2 g(PD )

For current time we choose t0 = 0. The quintessence energy-density decreases with time (pD < pD0), while the phantom energy-density increases (pD > pD0). For scale factor we have the following expression:

a = ao exp | 3

1 i dpD \ 3 J g(PD))'

For simplicity a0 = 1 is chosen. A finite-time singularity occurs if the integral (6) converges at p The scale factor in this case

may become infinite (Big Rip) or may remain finite but a singularity (p ^ ro) occurs. This is a type lll singularity [18]. The type ll singularity realizes if g(pD) ^ at pD = pf. The pressure of the dark energy becomes infinite at a finite energy-density. The second derivative of the scale factor diverges while the first derivative remains finite.

Non-singular evolution corresponds to the case when integral (6) diverges at pD (Little Rip [15,17,22]) or (6) diverges at

pd ^ pf [15]. ln the last case the dark energy-density asymptotically tends to a constant value ("effective cosmological constant" or Pseudo-Rip).

The so-called Quasi-Rip [19,20] corresponds to (inverse) crossing of phantom divide line ("(de)phantomization" [21]). One can show that in terms of EoS this case corresponds to multiply-valued function g(pD) and phase transition at some value of scale factor may occur.

3. Observational data

Confrontation of the theoretical models with observational data includes the comparison with several observational constraints:

(i) the luminosity distance moduli to type la supernovae from the Supernova Cosmology Project [23],

(ii) BAO (see for example [24]),

(iii) the data for the growth factor for density perturbations (see, for instance, Refs. [25,26]) from Lyman-a forest in the Sloan Digital Sky Survey and galaxy redshift distortions (for instance, Refs. [27,28]).

Let us consider these observational bounds in detail.

(i) The distance modulus for a supernova with redshift z = a0/a — 1 is

¡i{z) = const + 5log D (z), (8)

where D(z) is the luminosity distance. As is well known

Dl (z) = — (1 + z) H 0

h-\z) dz, h2(z) = P(z)/I0.

Here c is speed of light and H0 is Hubble parameter. We used H0 = 72 km/s/Mpc from the Hubble Space Telescope key project [29]. The best-fit for SNe is given in the framework of ^CDM cosmology. For such a model (which we call the "standard cosmology" (SC)), one obtains

h(z) = (Qm0(1 + z)3 + ^0)1/2. (10)

Here, Qm0 is the fraction of the total density contributed by matter at present time, and Qa is the fraction contributed by the vacuum energy-density. Following the approach of Ref. [30] one can exclude the SNe samples satisfying the condition \^obs — I^sc\/aobs > 1.9. Therefore in the data set of 580 SNe from [31] we exclude the following 50 SNe samples:

1992bs, 1992bp, 1995ac, 1999bm, 1997o, 2001hu, 1998ba, 04Pat, 05Red, 03D4au, 04D3gt, 04D3cp, 03D4at, 03D1fc, 04D3co, 03D4dy, 04D3oe, 04D1ak, 03D1co, b010, 1995aq, f076, g050, k430, m138, 2006br, 2006cm, 2006cf, 2007ca, 2004gc, 10106, 2005ll, 2005lp, 2005fp, 2005gr, 2005ia, 1997aj, f308, e140, d084, 2002hu, 2002ju, 2003ch, 2005gs, 2005hv, 2005ig, 2005jj, 1997k, g120, 05Str.

(ii) In Ref. [24] it was suggested that instead of taking the position of an acoustic peak one should measure the large-scale correlation function at 100h—1 Mpc separation using the sample of 46748 luminous red galaxies (LRG) selected from the SDSS (Sloan Digital Sky Survey) sample. The appropriate quantity to be measured is known as A parameter and reads as

A = Dv (Z0)

^o2 H0.

In Eq. (11) the dilaton scale Dv (Z) is defined as

Dv (z) =

Dm (z)2cz' H (z)

where DM (z) is the comoving angular diameter distance. The red-shift z0 = 0.35. We have for the parameter A the following relation

a=y^m0h(z0)—v3

1 f> z0

h—1(z) dz

The observational value of BAO parameter is A = 0.469 ± 0.017. (iii) As it was shown in Ref. [26] one can neglect density perturbations of dark energy. In this case the dark matter perturbations effectively decouple from DE perturbations. The equation that determines the evolution of the density contrast 8 in a flat background filled by the matter with density pm is

8m + 2H 8m = 2 Pm 8m. (13)

It is convenient to introduce the function of growth rate of perturbations f = d ln 8m/d ln a. Using FRW equations one can get the following equation for f:

df 2 / H \ 3

dL + f 2 + H2 + 2 f — 2 ^m = 0,

where Qm is matter fraction of the total energy-density: Qm = Qm0(1 + z)3/h2(z). Finally, using relation

--" + z)T

d ln a dz

and taking into account that

a a 2- + = —Pd

we get

— (1 + z) f + f2 + dz

1 + 3 Qd + 3 gp)

2 2 D 2 p

f — - Qm = 0,

where QD = pD/p. For dark fluid with given EoS one can find DE density as function of redshift z. Then, Eq. (15) can be solved numerically. The observational data for growth function f at various redshifts are given in Table 1.

Table 1

The available data growth function f at various redshifts from the change of power spectrum Ly-a forest data in SDSS.

z fobs Ref.

0.15 0.51 ± 0.11 [32,33]

0.35 0.70 ± 0.18 [34]

0.55 0.75 ± 0.18 [35]

1.4 0.90 ± 0.24 [36]

3.0 1.46 ± 0.29 [37]

In the data analysis we use x2 statistics. The x2 value for some physical quantity x is given by equation

(xth — xobs)2

where xth is theoretically predicted value of x, xth is experimentally measured value and ax is standard deviation. For the data set the total x2 is the sum of all xx2.

4. Dark energy model I: Little Rip, Big Rip and type III future singularity

Let us start from the simple model with following EoS:

g(pD) = a2 pdJ —J

where a and P are dimensionless constants. If P = 1 we have ordinary phantom energy model with constant EoS parameter w = —1 — a2. From Eqs. (6), (7) one can see that for various P the model (17) describes three types of future universe evolution:

(a) Little Rip if P < 1/2,

(b) Big Rip if 1/2 < P < 1, and

(c) type III singularity if P > 1.

The time left before finite-time future singularity can be estimated as following

tf —10

31/2(2P — 1)a2pD/2'

Note that pD0 = QD0p0 = 3QD0H^ and therefore 21

(14) tf — t0 =

3a2 (2P — 1)QD/2 H0

Let us restrict ourselves to cases (b) and (c). The results for Little Rip models are similar. From Eq. (7) one can derive the dependence of dark energy from redshift:

pd (z) =

{ pD0{1 — 3a2(1 — P)ln(1 + z)}P = 1, lpD0(1 + z)—3a2, P = 1.

SNe data analysis only. Using Eq. (19) one can calculate the theoretical dependence ¡i{z) from (9). Of course the best-fit of SNe data is ^CDM-model (a = 0). For QD0 = 0.72 the value of xSN is minimal: xSN = 347.06. For a = 0 and given P one can select the parameter QD0 to describe the experimental data with good accuracy. In Table 2 the optimal QD0 for some values of parameters P and a are given. We also calculated the xSN for comparison with ^CDM-model and the time left for future singularity tf (we restrict ourselves to case P > 1/2).

One can see that in principle the parameters of our model can vary in sufficiently wide limits. However, the consideration of data

Table 2

The optimal value of the parameter QDo, time left before future singularity and corresponding value of x2 for some a2 and p.

DE models with type III singularity: analysis of SNe data

a 2 ß = 1.5 ß = 2.0 ß = 3.0

^ D 0 tf, Gyr X SN fi D 0 tf, Gyr X SN fiD0 tf, Gyr X SN

0. 1 0.68 55.0 347.44 0.68 36.7 347.42 0.68 22.0 347.41

0.2 0.65 28.1 348.62 0.65 18.7 348.57 0.65 11.2 348.57

0.3 0.62 19.1 350.18 0.62 12.7 350.28 0.63 7.6 350.30

0.4 0.60 14.6 352.36 0.60 9.7 352.40 0.61 5.8 352.41

0.5 0.58 11.9 354.80 0.58 7.9 354.93 0.59 4.7 354.90

DE models with Big Rip singularity: analysis of SNe data

a2 ß = 0.55 ß = 0.75 ß = 0.95

fi D 0 tf, Gyr X SN fiD0 tf, Gyr X SN fiD0 tf, Gyr X SN

0.1 0.68 1.01 x 103 347.48 0.68 219.90 347.47 0.68 123.07 347.46

0.2 0.64 566.67 348.41 0.64 112.5 348.45 0.64 62.5 348.51

0.3 0.61 386.96 349.91 0.61 76.8 350.04 0.61 42.7 350.19

0.4 0.59 295.09 352.03 0.59 59.0 352.01 0.59 32.8 352.09

0.5 0.56 242.31 354.06 0.56 48.0 354.48 0.57 26.7 354.49

Table 3

The maximal value of parameter a2 (and correspondingly the minimal time before singularity) for given p at which the model (17) describes the density perturbations data with 95% C.L. The optimal value of QD0 for fitting SNe data, xSN and BAO parameter are also given.

DE models with Big Rip or type II future singularity: analysis of SNe + DP data

ß 0.55 0.75 0.95 1.5 2 3

a2 wmax 0.03

fiD0 0.71

f (0) 0.514

tf min. Gyr 3.6 x 103 717.3 398.5 179.3 119.6 71.7

X SN 347.28

A 0.483

for matter perturbations allows to restrict the range of these parameters.

Analysis of matter density perturbations data. Again the best-fit for observational data is ACDM-model. For QD0 = 0.72 we find that xDP + XSN is minimal at f (0) = 0.503. The value of xDp is 0.879 (within 95% C.L). The consideration of matter perturbations data only leads to minimal value xDP = 0.34 at QD0 = 0.78 and f (0) = 0.441. However, in this case the xSN is sufficiently large: XSN = 370.01 although this lies within 95% confidence level.

For given p one can further vary parameter a and find the optimal values of QD0 for fitting SNe data. Then one can find f (0) from approximation of matter perturbations data. Hence, we find the maximal value of a at which our model fits the matter perturbations data with 95% C.L. The results of our calculation for the model with future Big Rip singularity are given in Table 3. For DE model with type III future singularity the results are similar (see

also [40]). We find that maximal values of parameters am

fi D 0

and f (0) in fact do not depend from p. It is interesting to note that SNe data description for these models coincides with ACDM cosmology up to excellent accuracy.

The current EoS parameter for the model under consideration

w 0 = — 1 — a 2.

Therefore small values of parameter a2 correspond to small deviation of w from —1.

The dependence of ln a as function of the time is depicted for some parameters (Fig. 1). For the illustration, in Fig. 2 the comparison between observational data for density perturbations and

Fig. 1. The dependence of scale factor from the time for model (17) with a2 = 0.02, p = 0.95 (Big Rip) and p = 3 (type III singularity). In the interval ~ 50 Gyr the universe expansion in these models is very close to the one expected from ACDM cosmology.

theoretical predictions is made. There is no significant difference between standard cosmology and our models.

Another DE model with type III singularity. One can consider another DE model which is more close to ACDM-cosmology. This model firstly was considered in [38,39]. From the following EoS

F (p) = -ß 2 aj p D+€/3,

where p, af, and e are positive constants one can find the dependence of the dark energy-density on the scale factor

pD = ß-6/j (aj - aj)

For dimensionless Hubble parameter as function of redshift we have therefore

Table 4

The optimal parameters QD0 and f (0) for fitting SNe and density perturbations data for model (20) for various w0 and e. The time before future singularity is given also. The coincidence with ACDM model is better for larger values of e and smaller values of w0.

e w 0 = —1.10 w 0 = —1.05 w 0 = —1.02

2 5 10 2 5 10 2 5 10

QD0 0.71

f (0) 0.514 0.514 0.514 0.514 0.513 0.514 0.513 0.513 0.513

XsN 350.97 349.04 347.88 347.68 347.29 347.08 346.98 346.97 346.98

xDP 1.26 1.20 1.15 1.15 1.13 1.11 1.10 1.08 1.07

tf, Gyr 13.5 6.1 3.2 18.1 8.0 4.1 24.7 10.7 5.4

Fig. 2. The comparison of observational data with theoretical predictions for the model (17) with a2 = 0.02, P = 0.95 (Big Rip) and P = 3 (type III singularity). One can see the coincidence with ACDM cosmology.

h2 (z) = Qm (1 + z)3 + Q D (1 + z)J

N 0 — 1

N 0 (1 + z)e — 1

N0 = (af /a0 )e. (22)

One can see that for large N0 our model mimics ACDM cosmology with excellent precision. Therefore our model can fit the Supernova Cosmological Project data. For N0 > 1 the dark energy-density is nearly constant in the interval 0 < t < 10, i.e. the model (20) mimics a cosmological constant in the past but it leads to a finite-time future singularity.

The current EoS parameter is

w 0 = —^-r.

N0 — 1

For given value of w0 and e one can find that such model describes the observational data with good accuracy. The results are given in Table 4. One can see that model (20) is more close to ACDM cosmology than DE model with type III singularity considered above. The model describes supernova data with the same precision as the ACDM model although for density perturbations data the agreement is slightly worse than in the case of standard cosmological model. It is interesting to note that agreement between this DE model and observational data is better for large

Fig. 3. The dependence of scale factor from time for DE model (20) with various parameters choice. For comparison the same dependence for ACDM cosmology (straight line) is given.

values of e. The model (20) coincides with ACDM model in past with excellent precision but its future evolution shows radically different dynamics (see Fig. 3).

The future evolution of dark energy and matter density perturbations. To conclude this section we consider the question about evolution of matter density perturbations in future. In such a case one should account the dark energy-density perturbations. The complete system of cosmological perturbations equations can be found for example in Refs. [42,43]. In the system of units in which 8n G = c = 1 and for the case when effective sound speed for dark energy is equal to 1 these equations can be written in the following form

8 D + 3(1 — wd )aH8 D — (1 + wd )8m + (1 + wd )(k + 9a2 H211—cA Vd = 0,

VfD — 2aHV d — --8 d = 0,

D 1 + w D

8rm + a 8m — 2 {pm8m + (1 + 3 wd)pD 8D ) = 0.

Here ' denotes derivative on conformal time t (t = f a-1 dt), 8D = 8PD/ PD, VD is the velocity perturbation of dark energy and c2 = dpD /dpD is the adiabatic speed of sound. The k is a wavenumber of the corresponding mode. One can rewrite Eqs. (23)-(25) using the relation

a^H. Thus, we have

d8d , 3 d8m

—— + - (1 — Wd )8 D — (1 + Wd ) —— da a da

+ a1h (1 + Wd)(k + 9a2h2 iy^ ) Vd = 0,

dVD — 2 — 1 k da a h 1 + wD d2 8m + / 3 + dh_ \ d8m da2 V a hda ) da

8D = 0,

2a2h2 H2

(Pm8m + (1 + 3 Wd)PD8D) ,

where h = H/H0 is dimensionless Hubble parameter and k is di-mensionless wavenumber measured in the units of H0/c. One notes that last equation of this system is equivalent to (13) if the dark energy-density perturbations are neglected. However, the future DE density increases and even its small perturbations may become important for the analysis of matter density perturbations (see r.h.s. of Eq. (28)).

For the model (17) the EoS parameter and adiabatic speed of sound are given as functions of scale factor

w(a) = -1 - a2(1 + 3a2(1 - p) lna)-1, c2 = -1 + p (1 + w (a)).

For current value of the scale factor we put simply a0 = 1. Initial conditions for integration are given for example in Ref. [42]. The main results of numerical analysis of system (26)-(28) are the following. For DE model (17) which evolves to Big Rip at acceptable values of parameters (see Table 3) at small scales (large wavenumbers) the amplitudes of matter density perturbations increase insignificantly in comparison with 8m0 and remain constant. The same picture takes place in ACDM cosmology. However, for the standard cosmological model all modes of matter density perturbations evolve in the same way. In the case under consideration for sufficiently large scales (> 2000 Mpc) the decay of matter perturbations occurs. This decay occurs faster for perturbations with larger scales (see Fig. 4). The value of dark energy perturbations grows very quickly before a ~ 10 but then increases very slowly. If p > 1 (type III singularity) the picture looks the same for matter density perturbations but character of dark energy perturbations evolution changes. We observe the approximately linear growth of 8D as function of ln a in the interval 1 < a < 10 but then 8m increases faster according to the law 8D ~ (ln(af /a))-1. Shortly before the singularity the perturbations become large. For example for p = 3, a2 = 0.03 the moment of singularity corresponds to af = 258.67 (tf = 71.7 Gyr) and 8d ~ 1 for k = 0.0005 Mpc-1 at a ~ 235 (t = 70.4 Gyr). It is interesting to note that growth of density contrast occurs at sufficiently small background density pD < 10pD0. The sharp growth of density perturbations occurs before possible disintegration of such bound structures as Solar System or Milky Way due to enormous acceleration of universe. One can assume that like the formation of large-scale structure due to matter density perturbations in the early universe this growth

Fig. 4. The evolution of matter density perturbations (top) and DE energy-density perturbations (bottom) in the model (17) for the cases with Big Rip (bottom lines) and type III singularity (dotted lines). For DE perturbations the evolution of the mode with k = 0.0005 Mpc-1 is depicted. For matter density perturbations the evolution of modes with k = 0.05 Mpc-1 (1) and k = 0.0005 Mpc-1 (2) is shown. The model parameters are chosen as a2 = 0.03 and p = 0.95 and p = 3.

can lead to the formation of separate domains with various values of dark energy-density ("dark large-scale structure"). The cosmological dynamics within these "sub-universes" is defined by the background dark energy-density. In fact, we cannot speak about the uniform evolution of such universe. It is possible that in some domains the perturbations lead to the decrease of DE density and induce the de-phantomization (one can assume for example that EoS in the form (17) is valid only for pD > pD but if p < pD the phase transition occurs such that w > -1). Yet, in another domains the perturbations of DE energy-density lead to singularity faster than it would be expected without consideration of density perturbations evolution. In a sense this picture maybe similar to chaotic inflation [41] reversed in time. The homogeneous universe ends its existence in "chaotic" state. Of course, this is rather speculative possibility.

5. DE model II: Pseudo-Rip and type II singularity

Let us consider another DE model with EoS

g (p) = a 2 pd 0

1 — PD ) P f J

0 <PD <Pf, p = 0.

Such dark fluid leads to the following variants of evolution:

(i) DE energy-density asymptotically tends to pf if p > 1/2. Therefore, the universe expands according to de Sitter law at t ^ro (Pseudo-Rip).

(ii) DE energy-density reaches pf for tf < ro if 0 < p < 1/2.

Table 5

The optimal value of parameter i2D0 and corresponding value of xSN for some a2 and A.

DE models with Pseudo-Rip: analysis of SNe data

a2 a = 0.1 a = 0.2 a = 0.5 a = 0.8 a = 0.95

&D0 XSN ®D0 xSn &D0 xSn &D0 xSn &D0 xSn

0.1 0.2 0.3 0.4 0.5

0.68 0.64 0.61 0.59 0.56

347.30 348.17 349.17 350.54 351.66

0.68 0.65 0.62 0.60 0.57

347.25 347.85 348.72 349.83 350.69

0.69 0.67 0.65 0.63 0.61

347.15 347.38 347.71 348.10 348.50

0.71 0.70 0.69 0.68 0.67

347.04 347.07 347.12 347.20 347.28

0.71 0.71 0.71 0.71 0.70

347.00 346.96 346.99 347.09 347.00

Table 6

The maximal value of a2 for various A and f (0) at which the model (29) describes the density perturbations data with 95% C.L. The optimal value of S2n0 for fitting SNe data, xSn and BAO parameter are also given.

DE models with Pseudo-Rip: analysis of SNe + DP data

a 0.1 0.2 0.5 0.8 0.95

a2 max 0.03 0.03 0.06 0.13 0.45

&D0 0.71

f (0) 0.514

xSn 347.19 347.11 347.30 347.19 346.98

A 0.482

Table 7

The optimal value of parameter ^D0, time before singularity and corresponding value of xSN for some a2 and A.

DE models with type II singularity: analysis of SNe data

a2 a = 0.2 A = 0.5

&D0 tf, Gyr xSn &D0 tf, Gyr xSn

0.1 0.67 27.6 347.67 0.64 8.7 348.64

0.2 0.62 14.2 349.02 0.59 4.6 351.75

0.3 0.59 9.8 351.04 0.54 3.2 356.14

0.4 0.55 7.6 353.18 0.51 2.5 360.67

0.5 0.52 6.2 355.65 0.48 2.0 365.51

(iii) Type II singularity occurs if f < 0. The second derivative of scale factor diverges while first derivative remains finite.

Let us consider two simplest cases: ff = 1, —1. (a) ff = 1. For EoS (29) the dark energy-density as function of redshift is

3a2 A)

Pd(z) = PD0A-1(1 - (1 - A)(1 + z)3a A), A = PD0/Pf.

As in previous section we consider only SNe data first. For various A one can find optimal values of parameters a and QD0. Results are given in Table 5.

The analysis of density perturbations data shows that only in the narrow range of a2 these data can be described with 95% C.L. (see Table 6). This fact simply means that current value of EoS parameter

w 0 = —1 — a2(1 — A)

only insignificantly deviates from —1.

Note that same analysis can be performed for quintessence DE model with asymptotic de Sitter evolution. The parameter A in this case is larger than 1. Eq. (30) remains correct. The analysis shows that for A ~ 1 and small a2 the model describes observational data with good precision.

(b) ff = —1. In this case for DE energy-density as function of redshift we can derive the relation

Pd (z) = Pd0A—1 {1 — ((1 — A)2 + 6a2 A ln(z + 1))1/2}. (31)

The time left for future singularity can be found from (6) by integrating from pD0 to pf. We have

tf - t0 =

3a2 qDo A1/2

2 - a1/2(1 - - A

Next, as in the previous cases we calculated the optimal value of QD0 for fitting SNe fata for given A and a2 (Table 7). One sees that only at sufficiently large a2 and A our model declines from observational data significantly. For A ^ 1 the model (29) fits the observational data only at a2 ^ 0.

Table S

The maximal value of a2 for various A and f (0) at which the model (29) describes the density perturbations data with 2a CL. The optimal value of ßD0 for fitting SNe data, xSn and BAO parameter are also given.

DE models with type II singularity: analysis of SNe + DP data

a 0.2 0.5

a2 max 0.02 0.012

&D0 0.71

tf min Gyr 134.1 69.4

f (0) 0.513

xsn 347.13 347.10

A 0.482

For current value of EoS parameter in the model with type II singularity we have

w 0 = —1 — 1 — A. Again, w0 is close to -1.

The combined analysis of SNe data and data for density perturbations gives the results similar to previous cases. The deviation from ^CDM model is sufficiently small for fitting observational data although these data in principle do not prevent the possibility of future singularity within ~ 100 Gyr (Table 8).

The analysis of the evolution of density perturbations can be performed as at the end of previous section. Let us briefly recall the main results. For Pseudo-Rip the matter density perturbations evolve in the same way as in ^CDM model. For dark energy-density perturbations we have rapid growth before a ~ 10 and then slow decay. For models with type II singularity the picture coincides with that of (17) with type III singularity. Hence, DE models mimicking ^CDM and fitting current observational bounds may show different exotic behaviour in the future: finite-time singularities, disintegration of bound structures(Little Rip or Pseudo-Rip cosmologies) or decay of cosmological perturbations.

6. Conclusion

In summary, we confronted number of DE models mimicking ^CDM epoch with current EoS parameter being very close to —1

Fig. 5. The pressure as function of energy-density for model (33). The dotted line corresponds to vacuum energy with w = -1.

with combined observational data: SNe data, baryon acoustic oscillations data and DE and matter energy-density perturbations. It is explicitly demonstrated that there exists sufficiently wide region of parameters for each of DE model under discussion where these theories are not less viable than the standard ACDM model. On the same time, DE models under consideration show qualitatively different future behaviour with Big Rip, type II and type III future singularities, Little Rip, Pseudo-Rip or Quasi-Rip evolution. Nevertheless, current observational data cannot determine whether or not the universe will end in a future singularity.

It should be noted that the observational data for density perturbations are more sensitive to the deviation from standard cos-mological model in comparison with SNe data. Moreover, the account of density perturbations in Rip cosmology indicates to sharp growth of density at sufficiently small background density still before the possible disintegration of bound objects. This growth from our viewpoint can lead to possibility that future universe may split in the number of separate regions so that it becomes chaotic and never reaches the Rip singularity. Further consequences of above effect will be considered elsewhere.

Acknowledgements

We are grateful to M. Trodden, M. Sasaki and R. Scherrer for useful discussions. This work has been partly supported by MICINN (Spain), project FIS2010-15640 and by AGAUR (Generali-tat de Catalunya), contract 2009SGR-994.

Appendix A. DE model with dephantomization: Quasi-Rip

In this Appendix A we briefly discuss the possibility of so-called "Quasi-Rip" scenario considered in Ref. [19]. Let us consider the case when DE pressure depends from its energy-density as follows:

-pD - 3 a2 PD (in ^ )1/2

a < aT

-Pd + 2a2pd (in ^)x>2, a > aj

where PD < Pm and ln aT = a—2 (ln(Pm/PD0))1/2 is a value of scale factor at which the dephantomization occurs. In this moment the value of DE energy-density reaches the maximal value Pm. For small a2 the universe acceleration is maximal for PD ^ Pm exp(—a2/2). The pressure as function of energy-density is depicted on Fig. 5.

The dependence of DE energy-density from redshift z for EoS (33) is

One can rewrite this equation as

P d = P d o exp( -y in2 (z + 1) - ß in(z + 1)) Y = a4, ß = 2a4 in aT.

P d ( z) = Pm exp(-a4 (in aj + in(z + 1)) ).

Eq. (35) coincides with the equation considered in Ref. [19]. As is demonstrated in above work this model fits modern SNe data in the large interval of free parameters. One can show that the model (33) is consistent with matter density perturbations data also. Hence, the Quasi-Rip model maybe consistent with current observational data.

The specific feature of Eq. (33) is that scale factor is double-valued function of the energy-density. The branch point corresponds to dephantomization: the pressure in this point is defined uniquely. It is obvious that the described scheme can be generalized. For equations of state with branch points the evolution of universe contains (de)phantomization and Quasi-Rip epochs.

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