A Toy Model of Complete Cosmic History

In the present paper, we study a toy cosmological model derived from the specific behavior of the Hubble parameter and the scale factor in a spatially-flat Friedmann-Robertson-Walker (FRW) space-time. We demonstrate that our model could match in some approximation the complete history of cosmic expansion. To establish the appropriate values of the the model parameters, that is to fit the real universe, we apply some theoretical and observational tests.


Introduction
Present accelerated expansion of the universe is well proved in many papers [1][2][3][4][5][6][7][8][9]. In order to explain so unexpected behavior of our universe, one can modify the gravitational theory, or construct various field models of the so-called dark energy (DE) which equation of state satisfies w = p/ρ < −1/3. The simplest candidate of DE is the cosmological constant with w = −1. If it is quintessence then −1 < w < −1/3 and if it is phantom then w < −1. The constant equation of state w = −1 is called phantom divide.
So far, a large class of scalar-field dark energy models have been studied, including tachyon, ghost condensate and quintom, and so forth. In addition, other proposals on DE include interacting DE models, braneworld models, and holographic DE models, etc. There are some dark energies which can cross the phantom divide from both sides. Nevertheless, the puzzle of dark energy is still an unsolved problem in modern cosmology (see, e.g. [10] and references therein).
At the same time, we firmly believe in the paradigm of cosmological inflation regarding the very early universe. If this is so, any cosmological models should have at least two periods of accelerated expansion: early and late. It is hard to imagine a model that would be able to describe the extremely difficult scenario for the evolution of the universe from the origin to the present day, with all its processes and content. A variety of models of the processes in the early and late universe calls into question even perspective obtaining of such models in the near future. In this respect, it is interesting to consider any analytic model, even a toy model, capable of, even superficially, but realistically describe the whole evolution from (almost) the beginning to the present day (see, e.g. [11][12][13][14][15][16]).
We briefly study a model in which the Hubble parameter H takes a specific form, providing two accelerated epoch in the evolution of the universe. In this regard, we consider a spatially flat Friedmann-Robertson-Walker background equipped with such a time-varying H(t). To testify this model, we consider some theoretical and observational tests. In order to simplify our study, we consider only a limited number of those tests. It should be noted all that tests could essentially improve any models (see, for example, [17]) and bring them closer to the real scenario of cosmic evolution.

The Model
The Einstein's field equations without a cosmological Λ -term can be written as where we assume that the gravitational constant 8πG = 1, and all symbols have their usual meanings in the Riemannian geometry. The energy-momentum tensor of matter T ik can be derived in a usual manner from the Lagrangian of matter. Considering the matter to be a perfect fluid with the energy density ρ and pressure p , we have where u i = (1, 0, 0, 0) is 4-velocity of the co-moving observer, satisfying u i u i = 1. The line element of a Friedmann-Robertson-Walker (FRW) is represented by where a(t) is a scale factor of the Universe, ξ(r) = sin r, r, sinh r in accordance with the sign of spatial curvature k = +1, 0, −1. Given this metric and (2), we can reduce the field equation (1) to the following set of equations: where H =ȧ/a is the Hubble parameter, and an overdot stands for differentiation with respect to cosmic time t. The continuity equation follows (3) and (4) as:

A spatially flat FRW cosmology
From now on, we consider a spatially flat FRW cosmology with k = 0. In this case, one can rearrange the basic equations of the model, (3) and (4), as follows: As can be easily verified, the continuity equation (5) follows from the set of equations (6), (7). So, we have only two independent equations in the set (5)- (7). The generic procedure for solving this system lies in specification of the material or sources of gravitational field, possessing the energy density ρ and pressure p = wρ.
Here, the equation of state parameter w (the barotropic index) can be a constant or a function of time.
On the other hand, two equations, for example (6) and (7), are quite enough to find two unknown parameter such as ρ and p, if we specify the Hubble parameter H as a function of time. Let us consider one such a law for H(t), which is appeared in other context in [18,19]. This law is similar, but not coincide with the results suggested in [20][21][22], and can be written as follows where m, n are some positive dimensionless constants, and H 0 is the asymptotic (at large cosmic time) Hubble constant. Making use of this equation, we can obtain the time evolution of the scale factor represented by where a 0 is the scale factor at instant t 0 = m 1/(n + 1) /H 0 . In order of magnitude, it coincides with the age of the universe t 0 ≈ 13.7×10 9 yrs = 4.32×10 17 s. Therefore, the constant a 0 is approximately the present value of scale factor. In Fig. 1, we show the time evolution of the scale factor and the Hubble parameter, for three deferent values of n. Corresponding values of m are chosen from the condition w max = 1/3 as shown below. It is interesting to note that the so-called hybrid expansion law is a kind of the limiting case in (8): n → 0 simultaneously with mn → M = constant = 0 (see, e.g. [19,22]).
Substituting (8) into (6) and (7), we can find both ρ and p as the functions of time: the barotropic index w = p/ρ is given by Evolution of the equation of state in our model according to the last equation for the certain values of parameters is shown in Fig. 2. In order to construct a simple (and yet not completely trivial) model for the complete cosmic history, let us outline the generic features we want to reproduce by means of our model. It is easy to find that w starts at w(τ = 0) = −1 at the initial time and asymptotically tend to the same value: w(τ → ∞) = −1. However, during a certain time interval, the equation of state may become positive: 0 < w < 1. Furthermore, the expansion slows down even for a longer period. After that, the expansion again accelerates. It is interesting that this equation of state gains w = w max > 0 in its maximum. One can find out from (12) that The corresponding instant of reaching this maximum is given by From (13) and (14), we can derive the magnitude of parameter m which provides this maximum with the given value of w max in it: m = n n−1 (n + 2) n+2 6(n + 1)(w max + 1) In our view, all these features of the model bring it closer to the realistic scenarios, widely discussed at present. As one can see, the tuning of this model is possible by means Here, we use the same convention on values of n and m as in Fig. 1.
of several parameters, such as n, m, H 0 , a 0 and w max , and retains the basic features of this model that makes it so attractive.
To investigate the possibility of accelerated expansion for the universe in the framework of our model, we take into account the so-called "deceleration parameter" q = −aä/ȧ 2 = −1 −Ḣ/H 2 . From (9), we can obtain It should be noted that the time evolution of this deceleration parameter is also clear from Fig. 2 since its graph crosses the line q = 0 at the same time as w(τ ) crosses the line w = −1/3. Such a behavior of the deceleration parameter indicates the existence of two periods of accelerated expansion in our model, the initial and the final, separated by the period of decelerated expansion. Namely this feature of the model makes it so attractive.

Testing the Model
Here, we want to give several some diagnostics of our model based on theoretical background and on the observational data.

Square adiabatic sound speed
As well known [10], there are several approaches to investigate stability of a model. For instance, one can use the fact that the sound-speed squared should be nonnegative. This parameter is useful to investigate the classical stability of the models, therefore if sound speed be equal to a positive quantity the proposal can be considered as a viable model. The squared adiabatic sound speed is defined as Due to (6) and (7), the stability condition is given by Let us apply this relation to our model. Substituting (8) into (14), we conclude that this stability condition can be expressed as which the model is stable in respect to the sound-speed criteria for any values of the model parameters. It is noteworthy that increasing value of n allows the soundspeed condition of stability to be satisfied during the whole late acceleration period up to the present.

Statefinder diagnostic
In order to distinguish between various DE models, V. Sahni et al. [23,24] proposed a cosmological diagnostic pair {r, s} called statefinder. The statefinder test is a geometrical one based on the expansion of the scale factor a(t) near the present time t 0 : where a(t 0 ) = 1 and H 0 , q 0 , r 0 are the present values of the Hubble parameters, deceleration parameter and the statefinder index r = ... a /aH 3 respectively. The statefinder parameter s is the combination of r and q: s = (r − 1)/3(q − 1/2). The important feature of statefinder is that the spatially flat ΛCDM has a fixed point {r, s} = {1, 0}. Departure of a DE model from this fixed point is a good way of establishing the distance of this model from flat ΛCDM. In terms of the Hubble parameter and its derivatives with respect to cosmic time, the statefinder parameters of a flat FRW model are given by With the help of (8) and (16), one can find that r = 1 + mn(n + 1) τ n mn + τ n + 1 3 (n + 2)τ n − 3 mn + τ n + 1 , s = 2mn(n + 1) τ n 3 2mn(n + 1) τ n − 3 mn + τ n + 1 2 × (n + 2)τ n − 3 mn + τ n + 1 mn + τ n + 1 .
As can be easily seen from equations (19) and (20), our model is able to appear in the ΛCDM point {r, s} = {1, 0} at the beginning moment τ = 0, at a final time τ → ∞, and also many times at the certain intermediate instants, determined by the roots of the following equation 3 mn + τ n + 1 2 = 2mn(n + 1) τ n .
The final stage of cosmic evolution on {r, s} plane is shown in Fig. 4, for n = 1/2 and 1.

Om diagnostic
Recently, a new cosmological parameter named Om was proposed [25,26]. It is a combination of the Hubble parameter and the cosmological redshift and provides a null test of dark energy. As a complementary to {r, s}, this diagnostic helps to distinguish the present matter density in different models more effectively. Om diagnostic has been discussed together with statefinder for many cosmological models of dark energy. It was introduced to differentiate ΛCDM from other dark energy models. For ΛCDM model, Om = Ω m0 is a constant, independent of redshift z. The main utility for Om diagnostic is that the quantity of Om can distinguish dark energy models with less dependence on matter density Ω m0 relative to the equation of state of dark energy. For this end, we express the scale factor a in terms of redshift z by the relation a = a 0 1 + z , where a 0 is the present value of scale factor. The starting point is the Hubble parameter which is used to determine the Om diagnostic as follows: where x = 1 + z, h(x) = H(x)/H 0 . From the equations (8), (9) and the definition (22), we can conclude that the explicit dependence h(z) can be established for n = 1 only. In this case, we have and h(x) = 1 + 4m x(τ ) = exp m τ n − τ . Nevertheless, one can plot the Om diagnostic parameter on {Om, x} plane using the parametric representation as follows We find from Fig. 5 that the Om(z) with m = 27/256 is of negative slope almost everywhere, except for a certain interval of z. With m increasing, the Om(z) becomes of negative slope for all z. According to [25], it should suggest quintessence like behavior of our model (w > −1). This allows us to reconstruct this model by means of a standard scalar field in Section 4.

Supernovae type Ia constrains
As known, the SN Ia Union2 database includes 557 SNIa [27] and provides one of the possible observational restrictions on the cosmological models [28][29][30]. One should follow the maximum-likelihood approach under which one minimizes χ 2 and hence measures the deviations of the theoretical predictions from the observations. Since SN Ia behave as excellent standard candles, they can be used to directly measure the expansion rate of the Universe upto high redshift, comparing with the present rate. The SN Ia data gives us the distance modulus µ to each supernova. In the flat universe,the theoretical distance modulus is given by µ th (p s ; z) = 5 log 10 (d L /M pc) + 25 (24) where d L is the luminosity distance, and p s denotes model parameters. For theoretical calculations, the luminosity distance d L of SNe Ia is defined as follows where E(z) = H(z)/H 0 or E(z) ≡ h(z). We choose the marginalized nuisance parameter [31] for χ 2 and σ i is the 1σ uncertainty of the observed data µ obs (z i ), and p s = {H 0 , m, n} is the set of free parameters of the model to be defined through the best fit analysis.
Keeping in mind that we are dealing with the toy model, let us restrict ourself with the case n = 1, when we are able to follow an analytic calculation up to a certain point. In this case, there are only two parameters to be assessed: m and H 0 .
By substituting (22), we can represent the luminosity distance (25) as Let us rewrite this expression approximately for a small value z compared with unity, that is for the epoch close to the present time. Retaining in the series expansion of the power and logarithmic functions only linear terms, one can obtain Substituting this expression into (24), we obtain the theoretical distance modulus in terms of the redshift parameter z.
Following the usual procedure (see, e.g. [27] and bibliography therein), we can find the best-fit values of m and H 0 . As we use the approximation of z ≪ 1, it is reasonable to take into account the restricted data on SNe Ia, presented in [32] and shown in Fig. 6. There, we ignore the confidence intervals of the data for the sake of simplicity.
With the aim of a preliminary study of our model for only the case n = 1 , we have to roughly estimate its parameters. In the best-fit procedure mentioned above, we use the observational errors at the 1σ level and the data for SNeIa with z < 0.1. One could find that the best fit parameters are m = 0.2025 and H 0 = 0.00011. As can be seen, the curve µ th (z) essentially deviates from the observational data at z i close to unity. Obviously, this is the consequence of approximation which we made.

Scalar-Field Reconstruction
The question, that naturally arises, is as follows. How could this model be realized with some well-known forms of matter or field? The matter content of our Universe is complex and quite diverse. Nevertheless, let us keep in mind that we are dealing here with the toy model. It allows us to attempt to build this model with a single source of gravity.
We assume now that the origin of the gravity in the universe is a single self-interacting scalar field φ which couples minimally to gravity. The energy density ρ and pressure p for a scalar field with potential V (φ) are given by respectively. Scalar field evolution is governed by the equation of motion which can be easily obtained from (5) by means of substitution (21). Combining two equations in (27) and taking into account (10) and (11), we can obtain the following equations V (τ ) = 6H 2 0 1 + mn τ n + 1 where a prime stands for differentiation with respect to the dimensionless cosmic time τ . Integrating equation (28), we can readily get φ(τ ) = ± 8m(n + 1) n τ −n/2 + φ 0 , where φ 0 is a constant of integration. Substituting τ from (30) where A mn = [n/8m(n + 1)] (n + 1)/n . As an example, we plot the graphs of potential (31) in Fig. 7 for the cases: a) n = 1/2, m = (25/96) 5/3, b) n = 1, m = 27/256, and c) n = 2, m = 1/27.

Conclusions
Thus, we have proposed and briefly investigated the toy model of complete cosmic history. It is hard to expect a full description of all complex processes occurring in the universe in the framework of such a simple model. Our intention is to demonstrate that there are models that could cover, at some approximation, a wide period of cosmic evolution (cf., e.g. [16]). At the same time, this model itself is the analytically accurate one and therefore very suitable for the deeper studying of the main properties of cosmic expansion. One could express the hope for further development of this model in terms of some modification of the sources and the evolutionary law of the Hubble parameter. At least, we believe this model merits a deeper study.