Crossing the cosmological constant barrier with kinetically interacting double quintessence

# Crossing the cosmological constant barrier with kinetically interacting double quintessence

## Abstract

We examine the plausibility of crossing the cosmological constant () barrier in a two-field quintessence model of dark energy, involving a kinetic interaction between the individual fields. Such a kinetic interaction may have its origin in the four dimensional effective two-field version of the Dirac-Born-Infeld action, that describes the motion of a D3-brane in a higher dimensional space-time. We show that this interaction term could indeed enable the dark energy equation of state parameter to cross the -barrier (i.e., ), keeping the Hamiltonian well behaved (bounded from below), as well as satisfying the condition of stability of cosmological density perturbations, i.e., the positivity of the squares of the sound speeds corresponding to the adiabatic and entropy modes. The model is found to fit well with the latest Supernova Union data and the WMAP results. The best fit curve for crosses at red-shift in the range , whereas the transition from deceleration to acceleration takes place in the range of . The scalar potential reconstructed using the best fit model parameters is found to vary smoothly with time, while the dark energy density nearly follows the matter density at early epochs, becomes dominant in recent past, and slowly increases thereafter without giving rise to singularities in finite future.

###### pacs:
98.80.-k, 95.36.+x, 98.80.JK

sourav.sur@uleth.ca

## 1 Introduction

A variety of recent observational probes, including in particular the type Ia Supernovae (SN Ia) [1, 2, 3, 4, 5, 6, 7, 8, 9], indicate that our universe has entered in a phase of accelerated expansion in recent past, following an early decelerating regime. Despite several alternative proposals, such as modified gravity [10] and the averaging of cosmological inhomogeneities [11], the origin of this acceleration has widely been attributed to a ‘mysterious’ energy component, namely the dark energy (DE), which constitutes about 72% of the present universe. Moreover, the cosmic microwave background (CMB) temperature fluctuation measurements by the Wilkinson Microwave Anisotropy Probe (WMAP) [12, 13, 14] as well as the large scale red-shift data from the Sloan Digital Sky Survey (SDSS) [15] indicate that our universe is very nearly spatially flat, so that spatial inhomogeneities may be neglected at large scales. Although the DE closely resembles a positive cosmological constant , for which the DE equation of state (EoS) parameter , there are some serious theoretical problems, such as fine tuning and coincidence, associated with [16]. Specifically, if the DE is supposed to be due to and the acceleration began only in recent past, then (i) what makes the DE density scale very small compared to the Planck scale? and (ii) why is the DE density is of the order of the present critical density right now? Hence, there have been suggestions that the DE may be (more appropriately) dynamic and can be modeled by one or more scalar field(s) originating from a fundamental theory. Of major interest are the DE models developed in the framework of quintessence and tracker fields [17, 18], k-essence [19, 20], Chaplygin gas [21] etc., (see [22] for extensive reviews). However, in many of these models the value of is always restricted to be , which is not desirable for a consistent statistical fit with the observational data. In fact, even with the presumption that is a constant, the recent WMAP five year data, combined with with those for SN Ia and baryon acoustic oscillation (BAO) peaks, constrain the value of , to be between and , at 95% confidence level (CL) [14]. For a time-varying DE, the same data constrain the value of the DE EoS parameter at the present epoch (i.e., at red-shift ) to be between and (at 95% CL) [14]. Since in the distant past, the value of a variable must have to be (so that the universe had a decelerated expansion and structures were formed), there is a fair plausibility that one (or possibly more) transition(s) from to (or vice versa) could have been taken place in the recent course of evolution of the DE, and at present .

The crossing of the cosmological constant barrier () can, most simply, be achieved in the so-called quintom scenario [23], where there are two (or more) scalar fields, (at least) one of which is of ‘phantom’ nature, i.e., carries a wrong sign in front of the kinetic term in the Lagrangian [24]. Such a phantom field is quantum mechanically unstable [25] and also gives rise to singularities in finite future [26, 27]. Moreover, classical instabilities could arise as the dominant energy condition gets violated in the models involving the phantom fields [28]. Attempts have therefore been made to circumvent the problem of crossing in various alternative ways. Notable among these are the scalar-tensor models [29], brane-world models [30], multi-field k-essence models [31, 32], modified gravity models [33], string-inspired dilatonic ghost condensate models [34], quantum-corrected Klein-Gordon models with quartic potential [35], coupled DE models [36], H-essence (complex scalar) models [37], etc. However, apart from a very few exceptions (such as the scalar-tensor models [29], or models where the kinetic term abruptly flips sign due to some extraordinary nature of the potential [38]) the crossing is hard to be realized with a single-field DE. Even in the case of a single-field k-essence DE, with a generic non-linear dependence of the Lagrangian on the kinetic term, such a crossing either leads to instabilities against cosmological perturbations or is realized by a discrete set of phase space trajectories1 [40]. In multi-field DE models, however, the crossing could be made possible, as is shown for example in refs. [31, 32], although the field configuration may be severely constrained by the criterion of stability, i.e., the square of the effective speed of propagation of cosmological perturbations should be positive definite [32].

In this paper we explore the plausibility of the -barrier crossing in the framework of a two-field quintessence model with a kinetic interaction between the individual fields. Such a model may be looked upon as a specialization of a more general (interacting) multi-field k-essence scenario, which involves non-canonical (higher order) kinetic terms for the scalar fields [19, 32, 41, 42, 43]. Moreover, the kinetically interacting double quintessence (KIDQ) Lagrangian may, under certain approximations, be derived from the four dimensional effective two-field version of the Dirac-Born-Infeld (DBI) Lagrangian describing the evolution of D3-branes in higher dimensional string theoretic manifolds [44]. The biggest advantage with such a Lagrangian, compared to those in other -barrier crossing multiple k-essence models [31, 32], is that the total DE Hamiltonian consists of a positive definite kinetic part, which ensures that it is bounded from below and the model is quantum mechanically consistent. Stability against cosmological density perturbations further requires the squares of the effective (sound) speeds of propagation of the adiabatic and entropy modes to be positive definite as well. For the DBI multiple scalar fields in homogeneous cosmological backgrounds, both these sound speeds turn out to be the same, implying isotropic propagation of the adiabatic and entropy modes [41, 42]. Assuming this result to hold approximately for KIDQ (which is an approximation to the DBI two-scalar scenario), we find the square of the effective (isotropic) sound speed to be positive definite, ensuring the stability of the KIDQ model2.

We consider certain specific ansatze to solve for the KIDQ field equations, and obtain the condition under which the line could be crossed in some regime. In choosing the ansatze, we particularly emphasize on the following:

(i) the kinetic energy densities of the interacting scalar fields should always be positive definite,

(ii) the DE density should be less but not very smaller than the matter density at early epochs, and should dominate the latter at late times, and

(iii) the DE density should not grow rapidly with increasing scale factor (i.e., decreasing red-shift ) and reach to abnormally high values in finite future.

These are important in order to avoid (i) ghosts or phantoms, (ii) coincidence or fine-tuning related problems, and (iii) occurance of future singularities, respectively.

We then constrain the parameters of the model with the latest Supernova Ia data compiled in ref. [8], viz., the 307 Union data-set, as well as with the WMAP 5-year [14] update of the CMB-shift parameter and the scalar spectral index , which determines the BAO peak distance parameter from the SDSS luminous red galactic distribution [15]. After uniformly marginalizing over the Hubble constant , we obtain good fits of the model with the data. The minimized value of the total (SN+CMB+BAO) is found to be , which is better than the minimized found with the Union data-set in ref. [9] for the cosmological constant DE coupled with cold dark matter – the so-called CDM model. The best fit values of the parameters of our KIDQ model indicate that the crossing from to takes place at a red-shift range , whereas the transition from the decelerated regime to the accelerated regime occurs in the range . At the present epoch (), the best fit values of the matter density parameter and the DE EoS parameter, are respectively found to lie within the ranges and . All these results are fairly in agreement with those found with other model-independent or model-dependent parameterizations of the DE in the literature [46, 47, 48, 49, 50].

Finally, we integrate the scalar field equations of motion and reconstruct the interacting double quintessence potential using the best fit model parameters. We show that the reconstructed potential has a smooth dependence (i.e., without any discontinuity or multi-valuedness) on the scale factor . We work out the approximate analytic expressions for the potential as function of the scalar fields, and find that they also exhibit the same smooth nature at early and late stages of the evolution of the universe.

This paper is organized as follows: in sec. 2 we describe the general framework of the multi-scalar (k-essence) DE scenario, following the formalism shown in refs. [42, 43] in the context of multi-field k-inflation. In sec. 3 we emphasize on a special case which involves two quintessence type of scalar fields with canonical kinetic terms in the Lagrangian, and with a specific kinetic interaction between the individual fields. Assuming suitable ansatze for the solutions of the field equations we work out the condition under which the cosmological constant barrier could be crossed, and find the expression for the Hubble parameter maintaining this condition. In sec. 4 we fit our KIDQ model with the Union SN Ia data [8], combined with the CMB+BAO results from WMAP and SDSS, to obtain the DE density and EoS profiles. In sec. 5 we use the best fit values of the model parameters to reconstruct phenomenologically the interacting double quintessence potential and determine the temporal variations of the scalar fields. We also work out the approximate analytic functional forms of the potential in terms of the scalar fields, at early and late stages of the evolution of the universe. In sec. 6, we conclude with a summary and some open questions. In the Appendix, we show how the KIDQ action, that we consider, could be derived from the two-field DBI action under certain approximations.

## 2 General Formalism

Let us consider the following action, in dimensions, for gravity minimally coupled with matter fields and number of kinetically interacting (k-essence) scalar fields ():

 S=∫d4x√−g[R2κ2 + Lm + P(XIJ,ϕK)], (1)

where is the gravitational coupling constant, is the Lagrangian density for matter, that is considered to be pressureless dust. is the multi-scalar Lagrangian density, with

 XIJ = − 12 gμν ∂μϕI ∂νϕJ,(I,J=1,…,N), (2)

describing the kinetics of the scalar fields [42].

In a spatially flat Friedmann-Robertson-Walker (FRW) background, with line element

 ds2=−dt2+a2(t)[dr2+r2(dϑ2+sin2ϑdφ2)] (3)

the above expression for reduces to

 XIJ = XJI = 12 ˙ϕI ˙ϕJ = a2H22 ϕ′I ϕ′J, (4)

where the dot denotes time derivative () and the prime denotes derivative () with respect to the scale factor , which has been normalized to unity at the present epoch . is the Hubble parameter.

The Friedmann equations and the scalar field equations of motion are given by

 (5)
 ddt(a3 ∂P∂XIJ ˙ϕJ)= a3 ∂P∂ϕI, (6)

where is the energy density of matter in the form pressureless dust, and are the multi-field dark energy density and pressure, given respectively as

 ρX = 2XIJ∂P∂XIJ − P,pX = P. (7)

Assuming that there is no mutual interaction between matter and dark energy, the Friedmann equations (5) integrate to give , where is the matter density at the present epoch (). One also has the continuity equation for the dark energy

 ˙ρX=−3H(ρX+pX)⇒ρ′X=−3a(ρX+pX). (8)

From the Friedmann equations (5) one obtains the expressions for the DE EoS parameter , the total EoS parameter , and the deceleration parameter :

 wX = pXρX = −1 + 2XIJρX ∂P∂XIJ, (9) w = pXρm+ρX = wX(1−Ω0m~H2a3), (10) q ≡ −¨aaH2 = 1+3w2, (11)

where is the present critical density; being the value of at the present epoch ().

 ~H ≡ HH0 = √ρXρ0c + Ω0ma3, (12)

is the normalized Hubble parameter and is the present matter density parameter.

The transition from the decelerating regime to the accelerating regime takes place when the deceleration parameter changes sign, i.e., the total EoS parameter becomes less than , by Eq. (11), and the DE EoS parameter is further less, by Eq. (10). The crossing from to , on the other hand, requires a flip of sign of the quantity , presuming that the DE density is positive definite. In the next section, we examine the plausibility of such a crossing by considering for simplicity a model involving only two fields () with usual (canonical) kinetic terms (quintessence type), but with a specific type of kinetic interaction, which could have its origin in the two-field DBI action, as we show in the Appendix.

## 3 Kinetically interacting double quintessence

Let us take into account the following special form of the Lagrangian density for the DE, consisting of only two scalar fields:

 P = δIJXIJ − γ√1 − β2(δI,J−1 + δI−1,J)XIJ − V(ϕI), (13)

where are positive constants, is the scalar potential, and the indices run for . Denoting the two fields as , we can re-write the above Lagrangian as

 P = ˙ϕ22 + ˙ξ22 − γ Q(˙ϕ,˙ξ) − V(ϕ,ξ),whereQ(˙ϕ,˙ξ) = √1 − β2 ˙ϕ˙ξ. (14)

This implies that the scalar fields and have usual (canonical) kinetic energy densities (given respectively by the first two terms on the right hand side), and therefore are similar to ordinary quintessence fields. However they have a mutual kinetic interaction of a specific form proportional to , given above, which may originate from the two-scalar DBI action, approximated for and (but ) as shown in the Appendix.

The dark energy pressure is equal to in Eq. (14), whereas the expression (7) for the dark energy density reduces to

 ρX = ˙ϕ22 + ˙ξ22 + γQ(˙ϕ,˙ξ) + V(ϕ,ξ). (15)

The DE equation of state parameter , Eq. (9), now takes the form

 wX = pXρX = −1 + 1ρX[˙ϕ2 + ˙ξ2 + β γ ˙ϕ ˙ξ2Q(˙ϕ,˙ξ)]. (16)

The presumption that the parameter , is in support of the positivity of the term under the square root in the expression for given in Eq. (14). That is, the requirement for the validity of the model, could be fulfilled when , even if and vary fairly rapidly with time and the product . Considering further, itself to be positive, the kinematical part of the DE density, given by the first three terms (kinetic energy densities of the fields plus their kinetic interaction) on the right hand side of Eq. (15), remains positive definite. As such, the total DE Hamiltonian is bounded from below and the model is quantum mechanically consistent. Moreover, since and are all positive, it follows from Eq. (16) that, (in some regime) necessarily implies the product . In other words, the condition for the crossing of the barrier at a particular epoch, is that one of the two fields () must fall off with time, whereas the other one should increase with time.

Now, using Eqs. (15), (16) and the continuity equation (8), one obtains the following expression for the potential as a function of the scale factor :

 V(a) = −˙ϕ2(a)+˙ξ2(a)2 − γQ(a) + Λ − 3∫ad~a~a[˙ϕ2(~a)+˙ξ2(~a)+βγ ˙ϕ(~a) ˙ξ(~a)2 Q(~a)], (17)

where is an integration constant. Plugging Eq. (17) back in Eq. (15) we get the DE density as a function of :

 ρX(a) = Λ − 3∫ad~a~a[˙ϕ2(~a)+˙ξ2(~a)+βγ ˙ϕ(~a) ˙ξ(~a)2 Q(~a)]. (18)

One may note that Eq. (17) could also have been obtained by using the scalar field equations of motion (6), which in the present scenario reduce to

 ddt[a3(˙ϕ + βγ ˙ξ4Q(˙ϕ,˙ξ))]= a3 ∂V∂ϕ, ddt[a3(˙ξ + βγ ˙ϕ4Q(˙ϕ,˙ξ))]= a3 ∂V∂ξ. (19)

Under a dimensional re-scaling:

 ϕ↔ϕ√ρ0c,ξ↔ξ√ρ0c,Λ↔Λρ0c,β↔βρ0c,γ↔γρ0c, (20)

the DE density, pressure, and the scalar potential change as

 ρX↔ρXρ0c,pX↔pXρ0c,V↔Vρ0c, (21)

while all the above equations (15) - (3) remain invariant. On the other hand, the expression (12) for the normalized Hubble parameter reduces to

 ~H2(a) = ρX(a) + Ω0ma3 = Λ + Ω0ma3 − 3∫ad~a~a[˙ϕ2(~a)+˙ξ2(~a)+βγ ˙ϕ(~a) ˙ξ(~a)2 Q(~a)]. (22)

Let us now consider the following ansatze for the kinetic energy densities of the scalar fields:

 ρKϕ(a) = 12 ˙ϕ2(a) = 12[f(a) + √f2(a) − k2], ρKξ(a) = 12 ˙ξ2(a) = 12[f(a) − √f2(a) − k2], (23)

where is taken to be a positive definite and well-behaved function of , is a positive constant, and at all epochs. Eqs. (3) imply that

 ˙ϕ2 + ˙ξ2 = 2f(a),and˙ϕ ˙ξ = ±k. (24)

We choose to take , so that the DE EoS parameter , Eq. (16), could be made less than in some regime. Moreover, this choice guarantees the positivity of the square of the kinetic interaction, which now reduces to a constant:

 Q2 = 1 + βk2. (25)

The expressions for the time derivatives of the scalar fields are given by

 ˙ϕ(a) = √f(a)−k+√f(a)+k√2,˙ξ(a) = √f(a)−k−√f(a)+k√2, (26)

whereas from Eqs. (16), (18) and (17), we respectively obtain the following expressions for the DE EoS parameter and density, and the scalar potential:

 wX(a) = −1 + 1ρX(a)[2f(a) − βγk2Q], (27) ρX(a) = Λ + 3βγk2Q lna − 6∫af(~a)~ad~a, (28) V(a) = Λ + 3βγk2Q lna − f(a) − γQ − 6∫af(~a)~ad~a. (29)

Let us now assume a specific form of the function , given by

 f(a) = Aa−ν + k,whereA>0,0<ν<3, (30)

so that the criterion is automatically satisfied. Furthermore, ensures that , and hence the kinetic energy densities and of the scalar fields, fall off with increasing values of the scale factor . However, these fall offs are not faster than that of the matter density (). This is essential in order that the quantities and , which compose the total DE density , come to dominate at late times, i.e., for large values of .

Eqs. (26) reduce to

 ˙ϕ(a) = √Aa−ν+√Aa−ν+k√2,˙ξ(a) = √Aa−ν−√Aa−ν+k√2, (31)

and the Eqs. (27) - (29), for and , take the form

 wX(a) = −1 + 2ρX(a)(Aa−ν − B), (32) ρX(a) = 6Aνa−ν + 6Blna + Λ, (33) V(a) = V0 + (6ν−1)A(a−ν−1)+ 6Blna, (34)

where we have defined

 B = k(βγ4Q − 1)= constant, (35)

and is the value of the scalar potential at the present epoch ():

 V0 = (6ν−1)A +(Λ − k − γQ). (36)

From Eqs. (22) and (33) one also obtains the following expression for the normalized Hubble rate :

 ~H2(a) = 6Aνaν + Ω0ma3 + 6Blna + Λ. (37)

At (present epoch), , whence

 Λ = 1 − Ω0m − 6Aν, (38)

and the above expression (37) reduces to

 ~H2(a) = 1 + 6Aνaν(1−aν)+ Ω0ma3(1−a3)+ 6Blna. (39)

In the next section, we fit this Eq. (39) with the latest Supernova Ia data [8], as well as with the CMB+BAO results from WMAP and SDSS [14, 15], and determine the DE density and EoS profiles over the red-shift range that is probed.

## 4 Observational constraints

We perform a analysis so as to constrain the model parameters and , for two specific choices of the index () in the ansatze (30). The SN Ia Union data-set [8], which we use, consists of most reliable data points that range up to red-shift , and include large samples of SN Ia from older data-sets [1, 2, 3, 4], high- Hubble Space Telescope (HST) observations and the SN Legacy Survey (SNLS) [5].

The SN Ia data provide the observed distance modulus , with the respective uncertainty , for SN Ia located at various red-shifts . The for the SN observations is, on the other hand, expressed as

 χ2SN(μ0;Ω0m,A,B) = 307∑i=1[μobs(zi)−μ(zi)]2σ2i(zi), (40)

where

 μ(zi) = 5log10[DL(zi)]+μ0, (41)

is the theoretical distance modulus.

 DL(zi) = (1+zi)∫zi0d~zi~H(~zi;Ω0m,A,B), (42)

is the Hubble free luminosity distance in terms of the parameters , and

 μ0 = 5log10[H−10Mpc]+ 25 = 42.38 − 5log10h, (43)

being the Hubble constant in units of Km s Mpc. The parameter is a nuisance parameter, independent of the data points, and has to be uniformly marginalized over (i.e., integrated out). For such a marginalization one may follow the procedure shown in refs. [32, 51, 52], where is first expanded suitably in terms of . Then one finds the value of for which such an expanded form of is minimum. Substituting this value of back in , finally enables one to perform the minimization of the resulting expression with respect to the parameters , in order to determine the values of the latter best fit with the SN Ia observations.

The CMB shift parameter , that relates the angular diameter distance to the last scattering surface (at red-shift ) with the co-moving sound horizon scale at recombination and the angular scale of the first acoustic peak in the CMB temperature fluctuations power spectrum [15, 53], is given by

 R(z⋆) = Ω1/20m ∫z⋆0d~z~H(~z;Ω0m,A,B), (44)

where is the red-shift of recombination. The WMAP five year data [14] updates and the observed shift parameter . The for the CMB observations is given by

 χ2CMB = (Robs − R)2σ2R, (45)

where is the error in the WMAP data [14].

Now, the scalar spectral index , which determines the observed value of the BAO peak distance parameter from the distribution of the SDSS luminous red galaxies [15] through the relation , is updated by the WMAP five year data as (see the first ref. of [14], see also ref. [52]). The theoretical expression for the distance parameter is, on the other hand, given by

 A = Ω1/20m⎡⎢ ⎢⎣1zb√~H(zb) ∫zb0d~z~H(~z;Ω0m,A,B)⎤⎥ ⎥⎦2/3, (46)

where . The for the BAO observations is expressed as

 χ2BAO = (Aobs − A)2σ2A, (47)

being the error in the SDSS data [15].

The total , which needs to be minimized in order to determine the likelihood of the model parameters with the entire SN+CMB+BAO data, is thus given as

 χ2total = χ2SN + χ2CMB + χ2BAO. (48)

Of course, has already been minimized with respect to the nuisance parameter , Eq. (43), by the process discussed above.

For two specific choices of the index (), that appears in the ansatz (30), the best fit values the parameters , as well as the minimized value of , are shown in the table 1. Fig. 1 shows the evolution of and (alongwith the corresponding errors) throughout the entire red-shift range of the available data, for both the choices of . The maximum likelihood of the present value of the DE EoS parameter is found to be for and for . Both these values are well within the limits, viz., , obtained in model-independent estimates with the SN+CMB+BAO data in ref. [14]. On the other hand, the red-shift at which the best fit makes a transition from a value to a value is found to be for and for . However, stays well below zero even for , implying that DE is varying slowly with red-shift. The above values of also agree fairly well with other independent studies [50]. The best fit DE density at the present epoch, , is found to be equal to for and for . Remembering the dimensional re-scaling of the DE density, viz., , that we have performed earlier in Eq. (21), one may note that the shown in Fig. 1 is identical with the present DE density parameter (by virtue of the dimensional re-scaling). In other words, since the DE density is effectively measured in units of the present critical density , one has . It may also be noted that the sum of the best fit and the best fit is exactly equal to (for both and ), as it should be in accord with our prior assumption of the spatial flatness of the metric. This therefore proves the correctness of the -fitting of the model with the observational data.

The and contour plots of (i) versus (with fixed at its best fit value), (ii) versus (with best fit ), and (iii) versus (with best fit ), are shown in Fig. 2, for the choices (upper panels) and (lower panels). The case , which resembles a cosmological constant DE, is found to be about away from the best fit point in the versus contours (left panels), for both the choices.

The upper and lower left panels of Fig. 3 depict the variations of the best fit DE EoS parameter , as well as the total EoS parameter , Eq. (10), and the deceleration parameter , Eq. (11), obtained as functions of the scale factor (using the best fit values of the parameters ) and extrapolated to the range , for the choices and respectively. The range covers all of the past, i.e., right from the big bang () to the present (), and a considerable part in the future, up to (), i.e., when the present size of the universe gets doubled. Both and are negative in the past and tend to become constant at a value close to each other and a little less than in the future. The value of , on the other hand, changes from positive to negative, i.e., the transition from deceleration to acceleration takes place at () for and at () for . In the future, also remains negative and tends to be steady at a value close to and . Thus the accelerated regime , as well as the ‘super-acceleration’ (), do not appear to be transient in the present model.

The variations of the extrapolated best fit DE density and the matter density , with the scale factor in the range , are shown respectively for the choices and , in upper and lower right panels of Fig. 3. For a considerable period in the past the DE density nearly follows the the track of the matter density, until exceeding the latter at scale factor , and dominant thereafter. In other words, decreases with in a similar manner as does in the early regimes, until at a recent epoch , when the DE begins to dominate. This behaviour, although not distinctly similar to that due to the tracker quintessence fields [18], may perhaps stand as a possible resolution to the coincidence problem [54]. One can, in fact, trace the similarity of the early universe profiles of and to the form of the chosen ansatz (30) for the field solutions and the resulting expression (33) for . In the early epochs, i.e., for small values of , the DE density in Eq. (33) is dominated by the inverse power-law term , similar to the matter density . However, since and the best fit value of is of the order of the best fit , is smaller than , and decreases less rapidly than the latter, for sufficiently smaller values of . As increases, the value of eventually exceeds due to the presence of the positive constant term (, given by Eq. (38)) in the expression (33) for . The term in Eq. (33), which is negative for (i.e., past), is on the other hand, rather sub-dominant compared to and does not play a very significant role either in the past or in near future. This is the reason why, the DE density increases slowly and does not shoot up to very high values even at a scale factor as large as , giving rise to singularities in finite future. Admittedly, of course as due to the presence of the logarithmic term in . Thus, the extrapolations of the cosmological quantities using the best fit values of the model parameters, obtained in the red-shift range , appear to hold for very distant past and future.

In what follows, we integrate the expressions (31) numerically in the next section and use the values of and best fit with the data, so as to determine the variations of the scalar fields and with the scale factor . We also reconstruct the potential , given in Eq. (34), as a function of , using these values of the parameters , and finally, we work out the approximate analytic expressions for the functional variation of with and , in the regimes (distant past) and (recent past).

## 5 Reconstruction of the scalar potential

Let us recall Eqs. (31), from which one can derive the following equations for the derivatives of the scalar fields and with respect to the scale factor :

 H0 ϕ′(a) = √Aa−ν+√Aa−ν+k√2 a ~H(a),H0 ξ′(a) = √Aa−ν−√Aa−ν+k√2 a ~H(a), (49)

where is as given by Eq. (37) or (39), in terms of the model parameters .

Assuming the initial condition that at , one may re-write the above equations in integral form as

 H0 ϕ(a) = √A2[I+(a)−I+(0)],H0 ξ(a) = √A2[I−(a)−I−(0)], (50)

where

 I±(a) = ∫ad~a~a(1+ν/2)~H(~a)[1 ± √1 + 2k~aνA]. (51)

Again, denoting and at the present epoch (), we have

 H0 ϕ0 = √A2[I+(1)−I+(0)],H0 ξ0 = √A2[I−(1)−I−