Smooth double crossing of the phantom divide line wΛ=−1 has been found possible with a single minimally
coupled scalar field for the most simple form of generalized k-essence cosmological model, in the presence of
background cold dark matter. Such crossing is a sufficiently late time transient phenomenon and does not have
any pathological behaviour.

1. Introduction

The analysis of the three year WMAP data [1–3] and the very recent one of the five-year WMAP data [4] provide no indication of any significant deviations from Gaussianity and adiabaticity of the CMBR power spectrum and therefore suggest that the Universe is spatially flat to within the limits of observational accuracy. Further, the combined analysis of the three-year WMAP data with the supernova Legacy survey (SNLS), in [1], constrains the equation of state wde, corresponding to almost 74% of dark energy present in the currently accelerating Universe, to be very close to that of the cosmological constant value. Moreover, observations appear to favour a dark energy equation of state, wde<-1 [5, 6]. The marginalized best fit values of the equation of state parameter are given by -1.14≤wde≤-0.93 at 68% confidence level. In case, one considers a flat universe apriori, then the combined data leads to -1.06=wde=-0.90. The five-year WMAP data [4], on the other hand, practically indicates no deviation from ΛCDM model and the combined analysis of WMAP; distance measurement from SN1a and Baryon Acoustic Oscillation (BAO) confirms the presence of 73% of dark energy together with the range of the equation of state parameter -1.33≤wde≤-0.79. Thus, it is realized that a viable cosmological model should admit a dynamical equation of state that might have crossed the value wΛ=-1, in the recent epoch of cosmological evolution.

So far, it has been administered by Vikman [7] and accepted almost by all [8–22], except perhaps by Andrianov et al. [23], and more recently by Cannata and Kamenshchik [24] that smooth crossing of wΛ=-1 line is not possible in minimally coupled theories, even through a generalized k-essence Lagrangian [25–28] in the form L=(1/2)g(ϕ)ϕ̇2-V(ϕ). It is not difficult to understand that the standard minimally coupled theory cannot go smoothly over to the phantom [29] domain without violating the stability both at the classical [30] and the quantum mechanical levels [31, 32] (although it has recently been inferred [33] that quantum effects which induce the wde<-1 phase are stable in the ϕ4 model). However, Vikman [7], in particular, argued that transitions from wde≥-1 to wde<-1 (or vice versa) of the dark energy described by a general scalar-field Lagrangian (ρ(ϕ),∇(ϕ)) are either unstable with respect to the cosmological perturbations or realized on the trajectories of measure zero, even in the presence of k-essence Lagrangian. As a consequence, it has given birth to further complicated models to establish a smooth crossing. Particularly, it requires hybrid model composed of at least two scalar fields [34], onethe quintessence and the other a phantom and is usually dubbed as quintom model [35, 36]. Others, even complicated models like hessence [37, 38], holographic dark energy models [39], nonminimal scalar tensor theories of gravity [40, 41], Gauss-Bonnet gravity [42–46], models with higher-order curvature invariant terms [47, 48] have also been invoked for the purpose. There is yet another mechanism [49, 50], where the crossing is achieved through a dark matter-dark energy interaction in view of exchange of energy between CDM and quintessence field. As a result the Universe appears to be dominated by CDM and a crossing dark energy. However, the same phenomena may be engineered even through the gravitational interaction between dark matter and dark energy.

In the present paper we have been able to show that the so-called phantom divide line corresponding to the state parameter, wΛ=-1, can indeed be crossed in a single minimally coupled scalar field model, only by invoking the most simple form of a generalized k-essence Lagrangian [25–28] in the background of baryonic and nonbaryonic cold dark matter. In the model under consideration the state parameter of the dark energy wde>-1, in the absence of the background matter, implies that the field is quintessencial in origin. It is only through the gravitational interaction the background matter density pushes the dark energy density down to a minima and then pulls it up to a local maxima along which the phantom divide line is crossed transiently.

The essential feature of the model is a solution of the scale factor in the form a=a0exp(tf/n), with 0<f<1 and n>0. Such a solution was dubbed as intermediate inflation in the nineties [51–53]. Recently, it has been observed [54] that Gauss-Bonnet interaction in four dimensions with dynamic dilatonic scalar coupling admits such solution leading to late time cosmic acceleration rather than inflation at the very early Universe. Under this consequence, a comprehensive analysis has been carried out [55] with such solution in the context of a generalized k-essence model. It has been observed that it admits scaling solution with a natural exit from it at a later epoch of cosmic evolution, leading to late time acceleration with asymptotic de-Sitter expansion. The corresponding scalar field has also been found to behave as a tracker field [56–61]. Unfortunately, we have not analyzed the behaviour of the state parameter in the intermediate region. In this work, we show that such solution in the presence of background matter leads to late time cosmic acceleration with a transient double crossing of the phantom divide line and the Universe is dominated by the background matter till the second crossing. In Section 3, the stability of the model under linear perturbation has been shown. Thus the model does not appear to develop any pathological features like big-rip or instabilities at the classical and quantum mechanical level, during the cosmological evolution.

2. The Model

As mentioned in the introduction, we start with generalized k-essence [25–28] Lagrangian in the form

L=g(ϕ)F(X)-V(ϕ),
where X=(1/2)∂μϕ∂μϕ, which, when coupled to gravity, may be expressed in the following most simplest form:

S=∫d4x-g[R2κ2-g(ϕ)2ϕ,μϕ′μ-V(ϕ)+Lm],
where F(X) appears linearly in X and Lm is the matter Lagrangian. This is the simplest form of an action in which both canonical and noncanonical forms of kinetic energies can be treated and a possible crossing of the phantom divide line may be expatiated. For the spatially flat Robertson-Walker space-time
ds2=-dt2+a2(t)[dr2+r2{dθ2+sin2(θ)dϕ2}],
the field equations in the units κ2=8πG=c=1 can be expressed as
Ḣ=-[12gϕ̇2+ρm+pm2],Ḣ+3H2=V(ϕ)+ρm-pm2,g(ϕ̈+3Hϕ̇)+12g′ϕ̇2+V′=0,
where the dot and the dash represent derivatives with respect to the time and ϕ, respectively. In the above equations, H=ȧ/a is the Hubble parameter, while pm and ρm stand for pressure and the energy density of the background matter. So, altogether, we have got three independent equations, namely, (2.4) through (2.6), corresponding to six variables of the theory, namely, H,ϕ,g(ϕ),V(ϕ),ρm and pm. Therefore, we need three physically reasonable assumptions to obtain complete set of solutions. Our first assumption is to neglect the amount of radiation in the present day Universe, and to consider the background matter to be filled with luminous along with baryonic and nonbaryonic cold dark matter with equation of state wm=pm/ρm=0. Further, to find a solution viable for crossing the phantom divide line, we present our second assumption, in the form of the following ansatz for the Hubble parameter:
H=fnt1-f,
with n>0 and f>0. It is clear that f=1 leads to exponential expansion. However, we choose f in between, that is, 0<f<1. Thus the complete set of solutions are

a=a0exp(tfn),12gϕ̇2=f(1-f)nt(2-f)-ρm0[2a03exp((3/n)tf)],V=3f2n2t2(1-f)-f(1-f)nt(2-f)-ρm0[2a03exp((3/n)tf)],ρϕ=3f2n2t2(1-f)-ρm0[a03exp((3/n)tf)],pϕ=2f(1-f)nt(2-f)-3f2n2t2(1-f),ρm=ρm0[a03exp((3/n)tf)],
where a0 and ρm0 are integration constants. The third assumption, which we do not require for the present purpose, expresses the coupling parameter g(ϕ) and the potential V(ϕ) as functions of ϕ. One can, for example, choose ϕ arbitrarily to find different forms of g and V, which does not affect crossing the phantom divide. We will give one particular form of g(ϕ) and V(ϕ) at the end. The effective equation of state wϕ=pϕ/ρϕ corresponding to the scalar field is now expressed as
wϕ=a03(2nf(1-f)-3f2tf3a03f2tf-ρm0n2t(2-f)exp(-(3/n)tf)).
The above form of the state parameter wϕ appearing in (2.9) has been found in an earlier work [55], where we just mentioned that it goes over to -1 value asymptotically. Here, our attempt is to analyze its behaviour in the interim region. For this purpose let us express the state parameter wϕ as a function of the red-shift parameter. For simplification, we choose a0=1, without loss of generality. As a result, the constant ρm0, appearing in (2.8) stands for the amount of matter density present in the Universe at t=0. The red-shift parameter z is defined as
1+z=a(t0)a(t)=exp[1n(t0f-tf)],
where a(t0) is the present value of the scale factor, while a(t) is that value at some arbitrary time t, when the light was emitted from a cosmological source. Thus, wϕ can now be expressed as
wϕ=(2nf(1-f)-3f2[tof-nln(1+z)]3f2[tof-nln(1+z)]-ρm0n2[tof-nln(1+z)](2-f)/fexp(-(3/n)[tof-nln(1+z)])).
For a graphical representation of the state parameter versus the red-shift parameter, we need to select a few parameters of the theory. Firstly, let us choose f=0.5 to find n. The motivation of choosing the value of f in the middle is simply to set a comfortable dimension of time for n2 and to obtain a reasonably better form of the potential V(ϕ). Taking the present value of the Hubble parameter Ho-1=9.78/hGyr, the age of the Universe t0=13Gyr and with, h=0.65, n can be found from the ansatz (2.7) as n=0.5(Ho-1/to)=2.08. To estimate the amount of matter density ρm0 present at the time t=0, we take the present value of the matter density parameter Ωmo=0.26, and so in view of solution (2.8), we have

Ωmo=ρmoρco=ρm0(Ho-2exp(-(3/n)tof)3)=0.26,
where ρmo and ρco are the present values of the matter density and the critical density, respectively. Thus, ρm0≈0.63 and hence, n2ρm0=2.72. With these numbers we have plotted the state parameter versus red-shift parameter in Figure 1, which clearly exhibits a smooth double crossing, one from above at z≈1.8,t≈2.2 Gyr and other from below z≈0.44,t≈8.2 Gyr (note that in the present model we have started from the value of the scale factor a=1, at t=0, corresponding to which the red-shift parameter is approximately z=4.66) which is slightly different from that predicted in view of ΛCDM model [62–67]. In Figure 2, the matter density ρm (thin line) and the dark energy density ρϕ (thick line) have been plotted against time in Gyr. It demonstrates that initially the Universe was matter dominated. The crossing from above is experienced at the minima and that from below at the local maxima of ρϕ. At the local maxima, ρϕ overtakes ρm and the Universe is dominated by dark energy. It should be mentioned at this stage that it is the background CDM that is responsible for the observed transient crossing. The scalar field alone in the absence of the background CDM is not viable of crossing the phantom divide line.

State parameters wϕ(z) have been plotted against the red-shift parameter z (with f=0.5,h=0.65,t0=13Gyr,ρm0≈0.63). Smooth double crossing of the Cosmological constant barrier is observed at sufficiently later epoch, z≈1.8 from above and z≈0.44 from below.

The tracking behaviour of the scalar field is demonstrated. Thick and the thin lines correspond to ρϕ and ρm, respectively. The crossing of the phantom divide line occurs at the minima and local maxima (where ρϕ overtakes ρm) of the dark energy density. Dashed line corresponds to ρϕ in the absence of background matter. The dark energy is found to remain subdominant till the second crossing.

Now we check how far our model fits with the standard ΛCDM model. In connection with ΛCDM model, the luminosity-redshift relation is
HodL=(1+z)∫0zdz0.74+0.26(1+z)3,
while in the present model it is
HodL=(1+z)t0∫0z[t0-nln(1+z)]dz.
Further, the expression for distance modulus is given by
m-M=5log10(dLMpc)+25=5log10(DL)+43,
where m and M are the apparent and absolute bolometric magnitudes, respectively, and DL=HodL. In view of (2.13) through (2.15), the distance modulus-redshift graph has been plotted in Figure 3, which shows a perfect fit between the ΛCDM and the present models in agreement with the very recent five-year WMAP data analysis [4].

The fit is almost perfect and the models are nearly indistinguishable. Only after z=3.5, the present model deviates slightly by moving up a little bit from the ΛCDM model.

Surprisingly, as already mentioned, the crossing depicted here does not depend on a particular form of the potential V(ϕ) or the coupling parameter g(ϕ). One can choose ϕ arbitrarily to find different forms of potential and the coupling parameter without affecting the results. The importance of this fact will be described in the following paragraph. But before that, as an example, let us consider a very trivial choice ϕ=t. Thus, we have in view of (2.8),
V=3f2n2ϕ2(1-f)-f(1-f)nϕ(2-f)-ρm(0)2[a03e(3/n)ϕf],
which has already been demonstrated to be a tracker potential [55], and
12g(ϕ)=f(1-f)nϕ(2-f)-ρm02a03e(3/n)ϕf.
With the above numerics we also present the plots of g(ϕ) and V(ϕ), in Figures 4 and 5, respectively.

The figure depicts how g(ϕ) smoothly phantomizes and dephantomizes the model.

The potential has a maxima at the first crossing and thereafter it is almost flat without any peculiarity during the second crossing.

Vikman's argument [7] that the transition across wΛ=-1 occurs on trajectories of measure zero has been mentioned in the introduction. On the other hand, Caldwell and Doran [68] argued that there exists a fixed point (ϕ*,X*) through which the field must pass to achieve crossing and that for crossing through such a single point, the initial condition must be exceptionally tuned to prevent the slightest deviation from the special trajectory. This means, one has to start from V=V(ϕ) and g=g(ϕ) and solve (2.6) for several choices of ϕ=ϕ(t) and X*=X*(t) with a pair of initial conditions and show that not only a single precise choice of ϕ and X* leads to crossing. Following this procedure, it has not been possible to solve (2.6), for the form of the potential V(ϕ) and the coupling parameter g(ϕ) given by (2.16) and (2.17). Thus the arguments against such crossing could not be tested directly. However, it has been pointed out in the previous paragraph that the transient crossing illustrated here is independent of the choice of ϕ=ϕ(t), that is, crossing is admissible if one starts from different initial values of ϕ. Thus we can assert that the transient crossing occurs generically for the majority of possible initial conditions of the field and thus is free from the disease of fine tuning. Additionally, we can show that different model parameters admit such transient crossing even for different initial values of ρm0. For this, let us make another plot of the state parameter versus red-shift parameter in Figure 6, taking f=0.5, and t0=13 Gyr, as before, but with n=2, for which Ho-1=14.42 Gyr, which corresponds to h≈0.68. With these values one can find
n2ρm0=3.09,
for Ωmo=0.24 and thus ρm0≈0.77 which is quite different from the value ρm0≈0.63 obtained previously. The smooth double crossing clearly depicts that it is viable for a wide range of initial data ρm0.

State parameters wϕ(z) have been plotted against the red-shift parameter z (with f=0.5,h=0.68,t0=13Gyr). Smooth double crossing of the Cosmological constant barrier is observed once again for a different initial data ρm0≈0.77.

This fact can be illustrated analytically also. It is clear from solutions (2.8) that the kinetic energy term (1/2)gϕ̇2, rather, g(ϕ) in particular, vanishes at the crossing. This has also been depicted in Figure 4 under a trivial choice of ϕ=t. Now, in view of the scalar field (2.6), it is observed that at the crossing, that is, when g(ϕ)=0, the term (1/2)g′ϕ̇2+V′ must also vanish. However, this condition is trivially satisfied, as one can see in view of the solutions presented in (2.16) and (2.17) that
12g′ϕ̇2+V′=-6fnϕ1-fg(ϕ)=0.
In addition, the above condition yields
ρm0=2f(1-f)ne(3/n)ϕ*fϕ*2-f,
which makes it apparent that ϕ*, through which crossing occurs, is different for different initial condition ρm0, and as a result, the crossing does not take place through a particular fixed point (ϕ*,X*), rather it occurs through a line. Other way round, it can be said that such crossing is viable for a wide range of initial conditions (ρm0), and so the model does not suffer from the disease of fine tuning.

3. Stability Criteria

Primarily, we note that since there is a dynamical transition of the equation of state from below, so the model avoids big-rip singularity [30] and also prevents undesirable quantum mechanical negative energy graviton and phantom particle production [31, 32]. As a result classically the model is free from future singularity and quantum mechanical stability is guaranteed. Further, the velocity of sound [69–71], ĉs2=∂p/∂ρ=pϕ,X/ρϕ,X, is always 1, for the model under consideration. Now the question is, if the model is stable under appropriate perturbation [72, 73].

First, we recall that in the present model, we have two fluids, one is barotropic (wm=0) and the other, a nonadiabatic scalar field. Now, in the absence of the background matter, ρm=0 and pm=0, the field equations are
ρϕ0=3H2,pϕ0=2Ḣ+3H2,
together with (2.6), where, ρϕ0 and pϕ0 are the energy density and the pressure in the absence of the background, respectively. In view of solutions (2.8) and (2.9) it is clear that in the absence of the background matter, gϕ̇2>0 and wϕ=-1+2n(1-f)/3ftf>-1. Thus the scalar field is of quintessence origin and is not viable of crossing the phantom divide line of its own. Therefore, the corresponding density perturbation equation given by
δρϕ0=(-g′F+V′)δϕ-gF,XδX=(12g′ϕ̇2+V′)δϕ-gδX
remains finite. Now, although there is no nongravitational interaction between the two fluids, however they interact strongly under gravitational influence. In Figure 2, the dashed line exhibits the scalar field energy density in the absence of background. It is the background matter that not only reduces the energy density of the scalar field, but also generates a minima through which phantomization and a local maxima through which dephantomization take place. The energy density of the scalar field that has undergone crossing is now given by
ρϕ=3H2-ρm=ρϕ0-ρm.
We can now easily see how the problem encountered with energy-density perturbation raised by Caldwell and Doran [68] is cured. The equation for density perturbation is
δρϕ=δ(ρϕ0-ρm)=(-g′F+V′)δϕ-gF,XδX-δρm=-(12g′ϕ̇2+V′)δϕ-gδX-δρm.
We have already demonstrated that if we restrict to ϕ̇2>0, then g(ϕ) vanishes at the crossing, and the coefficient of δϕ also vanishes due to the scalar field equation (2.6). However, δρϕ still remains finite at the crossing due to the presence of finite matter density perturbation δρm. Next, let us look for the pressure perturbation equation [74, 75]
δp=ĉs2δρ+3ℋ(1+w)(ĉs2-ca2)ρθk2,
where the symbols have their usual meaning. In expression (3.5) for pressure perturbation, the adiabatic sound velocity ca2 diverges at the crossing for one component nonadiabatic fluid and so pressure perturbation also diverges. However, in the present model, the expression for the adiabatic sound velocity is
ca2=ṗϕρ̇ϕ=ṗϕ0ρ̇ϕ0-ρ̇m=[3ℋwϕ(1+wϕ)-ẇϕ]ρϕ03ℋ[(1+wϕ)ρϕ0-ρm].
Here, dot corresponds to derivative with respect to conformal time and ℋ is the Hubble parameter in conformal time. It is observed that at the crossing ca2 remains finite yielding finite pressure perturbation. So the model is stable under linear perturbations and it appears that everything goes right with it.

4. Summary

First of all it should be admitted that there is nothing crazy in the present model. That the dark energy alone is not viable of crossing the phantom divide line without classical and quantum mechanical pathological behaviour has been proved by several authors and that has not been challenged here. One should also admit that a real cosmic fluid carries baryonic and non-baryonic CDM along with the dark energy and they interact gravitationally. Both Vikman [7] and Caldwell and Doran [68] have studied the behaviour of the k-essence Lagrangian which alone admits crossing in the background of dark matter, while the dark matter in the present model is a quintessence field with wϕ>-1 in the absence of the background. The fact that quintessence field is viable of crossing the barrier in the presence of background matter has already been accounted for [49, 50], where there exists nongravitational interaction between the two. The most interesting feature of the present model is the observation that though there is no non-gravitational interaction between the background matter and the dark energy, yet the background matter pushes the energy density of the dark energy (scalar field) to a minima and a local maxima, around which the transient crossing of the phantom divide line has been experienced. The other interesting feature is that the phenomena of such transient crossing is independent of any particular form of the potential V(ϕ) and the coupling parameter g(ϕ). Yet another important consequence of the present model, as revealed in Figure 2, is that the matter density had dominant contribution until the second crossing. Finally, the tremendous agreement of the present model with ΛCDM model as shown in Figure 3 supports the cosmological interpretation of the recent five-year WMAP observation [4], that there is practically no deviation from ΛCDM model.

Acknowledgment

Acknowledgement is due to Professor Claudio Rubano and to the referee for some illuminating discussion.

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D.Interacting quintessence, the coincidence problem, and cosmic accelerationBarrowJ. D.Graduated inflationary universesBarrowJ. D.SaichP.The behaviour of intermediate inflationary universesMuslimovA. G.On the scalar field dynamics in a spatially flat Friedmann universeSanyalA. K.abhikkumar@gmail.comIf Gauss-Bonnet interaction plays the role of dark energySanyalA. K.Intermediate inflation or late time acceleration?ZlatevI.WangL.SteinhardtP. J.Quintessence, cosmic coincidence, and the cosmological constantSteinhardtP. J.WangL.ZlatevI.Cosmological tracking solutionsZlatevI.SteinhardtP. J.A tracker solution to the cold dark matter cosmic coincidence problemde RitisR.MarinoA. A.RubanoC.ScudellaroP.Tracker fields from nonminimally coupled theoryJohriV. B.vinodjohri@hotmail.comSearch for tracker potentials in quintessence theoryRubanoC.ScudellaroP.PiedipalumboE.CapozzielloS.CaponeM.Exponential potentials for tracker fieldsAlamU.ujjaini@iucaa.ernet.inSahniV.varun@iucaa.ernet.inSainiT. D.tarun@ast.cam.ac.ukStarobinskyA. A.alstar@landau.ac.ruIs there supernova evidence for dark energy metamorphosis?AlamU.ujjaini@iucaa.ernet.inSahniV.varun@iucaa.ernet.inStarobinskyA. A.alstar@landau.ac.ruThe case for dynamical dark energy revisitedHutererD.dhuterer@bombur.phys.cwru.eduCoorayA.asante@caltech.eduUncorrelated estimates of dark energy evolutionWangY.wang@nhn.ou.eduTegmarkM.tegmark@mit.eduUncorrelated measurements of the cosmic expansion history and dark energy from supernovaeNesserisS.me01629@cc.uoi.grPerivolaropoulosL.leandros@cc.uoi.grComparison of the legacy and gold type Ia supernovae dataset constraints on dark energy modelsSeljakU.seljak@ictp.itSlosarA.anze.slosar@fmf.uni-lj.siMcDonaldP.pmcdonal@cita.utoronto.caCosmological parameters from combining the Lyman-α forest with CMB, galaxy clustering and SN constraintsCaldwellR. R.Robert.R.Caldwell@Dartmouth.eduDoranM.M.Doran@gmx.deDark-energy evolution across the cosmological-constant boundaryGarrigaJ.MukhanovV. F.Perturbations in k-inflationCarrollS. M.carroll@theory.uchicago.eduHoffmanM.mb-hoffman@uchicago.eduTroddenM.trodden@phy.syr.eduCan the dark energy equation-of-state parameter w<−1?MelchiorriA.MersiniL.ÖdmanC. J.TroddenM.The state of the dark energy equation of stateMaC.-P.cpma@daffy.tapir.caltech.eduBertschingerE.bertschinger@mit.eduCosmological perturbation theory in the synchronous and conformal Newtonian gaugesHuW.Covariant linear perturbation formalismhttp://arxiv.org/abs/astro-ph/0402060BeanR.rbean@astro.princeton.eduDoréO.olivier@astro.princeton.eduProbing dark energy perturbations: the dark energy equation of state and speed of sound as measured by WMAPKunzM.Martin.Kunz@physics.unige.chSaponeD.Domenico.Sapone@physics.unige.chCrossing the phantom divide