# Intermediate Inflation or Late Time Acceleration?

Authors: Abhik Kumar Sanyal
Journal: Advances in High Energy Physics
 (2008)

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## Abstract

The expansion rate of “intermediate inflation” lies between the exponential and power law expansion but
corresponding accelerated expansion does not start at the onset of cosmological evolution. Present study of
“intermediate inflation” reveals that it admits scaling solution and has got a natural exit form it at a later
epoch of cosmic evolution, leading to late time acceleration. The corresponding scalar field responsible for such
feature is also found to behave as a tracker field for gravity with canonical kinetic term.


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## Body

## 1. Introduction

It is now
almost certain that the universe contains70% of dark energy, which is evolving slowly in
such a manner that at present the value of the equation of state parameter is w<−1/3,
or more precisely, w=−1,
so that the universe is  presently accelerating (see, e.g., [1] for a recent review). Λ CDM-model is the simplest one that can explain
the present observable features of the universe. However, in order to comply
the vacuum energy density, ρvac≈1074GeV4 (calculated from quantum field theory as–ρvac≈mpl4/16π2) with the critical density, ρΛ≈10−47GeV4 (related to the cosmological constant as–ρΛ=Λmpl2/8π), it requires to set up yet another energy
scale in particle physics. This problem is known as the "coincidence
problem" when stated as "why Λ took 15 billion years to dominate over other
kinds of matter present in the universe?" A scalar field with dynamical
equation of state wϕ,
dubbed as quintessence field [2], appears to get rid of this problem, which during
“slow roll” over the potential, acquires negative pressure and
finally acts as effective cosmological constant (Λeff).
Nevertheless, this quintessence field requires to be fine tuned for the energy
density of the scalar field (ρϕ) or the corresponding effective cosmological
constant (Λeff),
to be comparable with the present energy density of the universe. Tracker
fields [3–8] are introduced to overcome
the fine tuning problem. Tracker fields have attractor-like solutions in the
sense that a wide range of initial conditions (namely, a wide range of initial
values of ρϕ) rapidly converges to a common cosmic
evolutionary track with ρΛ,
and finally settles down to the present observable universe with ρϕ≈ρΛ.
Thus, tracker solutions avoid both the coincidence problem and the fine tuning
problem, of course for certain type of potentials, without any need for
defining a new energy scale.The important
parameter required to check for the existence of the tracker solutions isΓ=V′′(ϕ)V(ϕ)/V′(ϕ)2, V(ϕ) being the scalar potential. For quintessence,
the condition for the existence of the tracker solution with wϕ<wB,
(where wϕ and wB are the state parameters of the scalar field
and the background field, resp.) is Γ>1,
or equivalently, |λ|=|V′(ϕ)/κV(ϕ)|≈|H/ϕ˙| decreasing as V(ϕ) decreases. Tacker solution further requires a
nearly constant Γ,
which is satisfied if |d(Γ−1)/Hdt|≪|Γ−1|,
or equivalently, |Γ−1(d(Γ−1)/Hdt)|≈|Γ′/Γ(V′/V)|≡Γ¯≪1
[3–5]. The condition wϕ<wB is required for the present day acceleration
of the universe. So, eventually, the slope of the potential becomes
sufficiently flat ensuring accelerated expansion at late times. The same
condition for k-essence models, having noncanonical form of kinetic energy,
requires Γ>3/2 and is slowly varying [9], where Γ=g′′(ϕ)g(ϕ)/g′(ϕ)2, g(ϕ) being the coupling parameter, in the absence
of the potential. For noncanonical,
Dirac, Born, and Infeld (DBI) tachyonic action the
tracking behaviour has been investigated in [10]. Such type of scalar fields remain subdominant until
recently.General theory
of relativity with a minimally coupled scalar field admits a solution in the
forma=a0exp(Atf),
(where a is the scale factor and a0>0,A>0,
and 0<f<1 are constants) which was dubbed as
intermediate inflation in the nineties [11–13]. The expansion rate in the intermediate inflation is
faster than power law and slower than exponential ones. Some aspects of
intermediate inflation have been studied in the past years [14, 15]. Particularly, in a recent
work [16] it has been
shown that such inflationary model can encounter the observational features of
the three year Wilkinson Microwave Anisotropy Probe (WMAP) data [17] with spectral index ns=1,
considering nonzero tensor-to-scalar ratio r.
Such solutions [11–13] may also appear in
other theories of gravitation [18, 19]. In fact, the action obtained under modification of
Einstein's theory by the introduction of higher order curvature invariant terms
(which is essentially four dimensional effective action of higher dimensional
string theories) has been found to be reasonably good candidate to explain the
presently observed cosmological phenomena. In particular, in a recent work it
has been found [20]
that Gauss-Bonnet interaction in four dimensions with dynamic dilatonic scalar
coupling leads to a solution (a=a0exp(Atf)) in the above form, where the universe starts
evolving with a decelerated exponential expansion. Such solutions encompasses
the cosmological evolution, as the dilatonic scalar during evolution behaves as
stiff fluid, radiation, and pressureless dust. Solutions of these type are
known as scaling solutions [21, 22] in which the energy density of the scalar field (ρϕ) mimics the background matter energy density.
It then comes out of the scaling regime [21, 22] and eventually the Universe starts accelerating.
Asymptotically, the scalar behaves as effective cosmological constant. The
deceleration parameter corresponding to such solution is given by q=−1+(1−f)/Aftf.
Thus, unlike usual inflationary models with exponential or power law expansion,
accelerated expansion of the scale factor corresponding to intermediate
inflation [11–13] does not start at
the onset of the cosmological evolution, rather it starts after the lapse of
quite some time.Inflation
should have started at the Planck epoch so that it can solve the initial conditions,
namely, the horizon and the flatness problems of the standard model and can
lead to almost a scale invariant spectrum of density perturbation. As such, the
epoch at which the accelerated expansion of the scale factor in intermediate
inflation starts, it has also been arbitrarily taken as the Planck's era. But
it is not true. Because, as observed in the context of Gauss-Bonnet gravity
[20], such solutions
admit synchronize scaling between ρϕ and ρB,
which can happen long after the Planck's era. Thus it is required to study the
so-called intermediate inflation in some more detail.A
comprehensive study in the present work reveals that (1) under different
choices of the superpotentialH(ϕ) solutions in the form a=a0exp(Atf) are realized which lead to late time
acceleration and, therefore, should not be treated as inflationary model of early
universe. (2) It has also been observed that even for a noncanonical form of
kinetic energy, the same result is reproduced with the same form of potential.
(3) Finally, it has been shown that in the presence of background matter such
solutions are again admissible. The nature of such solution also reveals that the
equation of state of the scalar field (wϕ) follows that of the background matter (wB) closely, and finally the scalar field comes
out of the scaling regime [21, 22], leading to accelerated expansion of the universe.
For the standard form of kinetic energy the tracking behaviour of the scalar
field requires to constrain the present matter density parameter Ωmo≤0.2.In Section2,
we have started with a k-essence action
[23–26] in its simplest form, keeping only a coupling
parameter g(ϕ) in the kinetic energy term and writing down
the field equations. In Section 3, instead of choosing the form of the
potential, we have chosen different forms of the super-potential H(ϕ) [27, 28], and presented explicit solutions in the form a=a0exp(Atf),
discussed above, for standard g=1/2 and nonstandard form of kinetic energy g=g(ϕ).
Finally in Section 4, similar solutions in the presence of background matter
featuring tracking behaviour for canonical kinetic energy term have been
presented.


## 2. Action and the Field Equations

The generalizedk-essence [23–26]
noncanonical Lagrangian, (2.1)L=g(ϕ)F(X)−V(ϕ), where X=(1/2)∂μϕ∂μϕ,
when coupled to gravity may be expressed in the following simplest form: (2.2)S=∫d4x−g[R2κ2−g(ϕ)ϕ,μϕ,μ−V(ϕ)], where a coupling parameter, g(ϕ),
is coupled with the kinetic energy term. g(ϕ) has got a Brans-Dicke origin, g=ω/ϕ too, ω(ϕ) being the Brans-Dicke parameter. This is the
simplest form of an action in which both canonical and noncanonical forms of
kinetic energies can be treated. For the spatially flat Robertson-Walker
space-time (2.3)ds2=−dt2+a2(t)[dr2+r2{dθ2+sin2(θ)dϕ2}], the field equations are (2.4)2a¨a+a˙2a2=2H˙+3H2=−κ2[gϕ˙2−V(ϕ)]=−8πGp,(2.5)3a˙2a2=3H2=κ2[gϕ˙2+V(ϕ)]=8πGρ, where H=a˙/a is the Hubble parameter. In addition, we have
got the ϕ variation equation (2.6)ϕ¨+3a˙aϕ˙+12g′gϕ˙2+V′2g=0, which is not an independent
equation, rather it is derivable from  (2.4) and (2.5). In
the above, over-dot and dash (′) stand for differentiations with respect to
time and ϕ,
respectively. Instead of using  (2.4) and (2.5), it is always useful to
parametrize the motion in terms of the field variable ϕ [13, 29–31]. Thus, with κ2=8πG,
the above set of equations can be expressed as (2.7)ϕ˙=−(H′(ϕ)κ2g(ϕ)), and in the Hamilton-Jacobi form (2.8)[H′(ϕ)]2−3κ2H2(ϕ)g(ϕ)+κ4V(ϕ)g(ϕ)=0. The two important parameters of
the theory, namely, the equation of state wϕ and the deceleration q parameters, are expressed as (2.9)wϕ=−1−2H˙3H2;q=−1−H˙H2. Now, we are to solve for a (in view of H), ϕ, g(ϕ),
and V(ϕ) from (2.7) and
(2.8), and so, two additional assumptions are required. It is found that some
specific forms of the super-potential H(ϕ) [27, 28] lead to the so-called intermediate inflationary
solutions.


## 3. Solution in the Forma=a0e(Atf) and Its Dynamics

This section is devoted in presenting the solution of
the scale factor in the forma=a0e(Atf),
which was dubbed as intermediate inflation earlier.


### 3.1. Case-I

Let us choose
the following form of the super-potentialH(ϕ(t)): (3.1)H=hϕn, where h>0 and n>0 are constants. In view of this assumption,
(2.7) becomes (3.2)gϕ˙=nhκ2ϕ−(n+1). We can now solve the set of
(2.8),
(3.1), and (3.2), provided that we choose a particular form of g(ϕ) or V(ϕ).
In the following, we choose the standard form of kinetic energy, that is, g=1/2.
For the most natural choice g=1/2,
the field variables are found from (3.2) and (3.1) as (3.3)ϕ=[2hn(n+2)κ2t]1/(n+2),H=mt−(n/(n+2)),a=a0exp[(m(n+2)2)t(2/(n+2))], where m=h(2/(n+2))[κ2/2n(n+2)](n/(n+2)).
The above form of the scale factor, the Hubble parameter, and the scalar field ϕ can be expressed, respectively, as(3.4)a=a0exp[Atf],H=Aft(1−f),ϕ=[8h(1−f)κ2f2t]f/2, where A=m(n+2)/2>0 and 0<f=2/(n+2)<1 are related to the constants h and n.
In view of  (2.9), the state parameter and the deceleration parameter
evolve as (3.5)wϕ=−1+n3At−f;q=−1+n2At−f. The form of the potential, in
view of  (2.8), for such a solution is restricted to (3.6)V(ϕ)=h2κ4(3κ2ϕ2n−2n2ϕ2(n+1)) which has the form of double
inverse power. Thus, we have just reproduced the solutions obtained by Barrow
[11, 12] and Muslimov
[13], starting from a
particular choice of the superpotential. As previously mentioned, it was found
in the nineties and was dubbed as intermediate inflation, since the expansion
rate of the scale factor is greater than the power law but less than standard
exponential law. For such an expansion rate, (3.7)ρ+p=4n2h2κ4ϕ2(n+1)>0, that is, the weak energy
condition is always satisfied and so wϕ≥−1.
However, (3.8)ρ+3p=6κ2(−H˙−H2)=6Afκ2[(1−f)−Aftf]t−(2−f) implies that the strong energy
condition is violated at (3.9)ta>(1−fAf)1/f, when wϕ<−1/3,
and ta corresponds to the time at which the
acceleration starts.It is to be
noted that the necessary condition for inflation, namely,a¨>0 or more precisely, (d/dt)(H−1/a)<0,
is satisfied under the same above condition (3.9). Eventually, for large ϕ,
namely, ϕ≫2n/κ the potential energy starts dominating over
the kinetic energy, that is, ϕ˙2≪V(ϕ),
and thus the slow roll condition, ϵ=−H˙/H2<1,
is also satisfied under condition  (3.9). However, inflation is supposed to have
started at Planck's era, so that it can solve the initial problems of the
standard model, namely, horizon and flatness problems and the structure
formation problem and lead nearly to a scale invariant spectrum. Here we
observe that accelerated expansion of the scale factor in the so-called
intermediate inflation does not start at the onset of cosmic evolution. So, now
we can ask the following question: “what happened prior to the onset of
the accelerated expansion in the so-called intermediate inflationary era?”The solution
dictates that the universe starts evolving from an infinitely decelerated
exponential expansion withwϕ>1.
It might appear that we are considering a highly unorthodox cosmological model
involving an ultrahard equation of state and superluminal speed of sound. This
is true in some sense, since as mentioned earlier, in order to study the
situations under which such solutions emerge and its dynamics, we have not
considered the presence of any form of background matter explicitly.
Nevertheless, this situation is quite similar to the phantom models where super
negative pressure gives rise to ultranegative equation of state, indicating
that the effective velocity of sound in the medium, v=|dp/dρ|,
might become larger than the velocity of light. Likewise, here the universe
starts evolving with such a situation which actually demonstrates that the
corresponding era is classically forbidden and it is required to invoke quantum cosmology at that era.Now, during
the evolution, the universe passes transiently through the stiff fluid erawϕ=1,
the radiation dominated era wϕ=1/3,
the pressureless dust era wϕ=0,
the transition (from deceleration to acceleration) era wϕ=−1/3,
and asymptotically tends to the magic line, that is, the vacuum energy
dominated inflationary era wϕ≈−1,
ensuring asymptotic de-Sitter expansion.


### 3.2. Case-II

In this
subsection, we show that the same set of solutions obtained in case-I can be
reproduced even for a noncanonical kinetic term. To show this, we observe that
the above set of solutions does not necessarily require to fix upg=1/2,
rather they are found even for a functional form of g=g(ϕ),
that is, with a noncanonical kinetic term. This can be checked easily by
assuming solution (3.4) of the scale factor along with assumption (3.1) for the
Hubble parameter. The field ϕ,
the potential V(ϕ),
and the coupling parameter g(ϕ) are then found as (3.10)ϕ=(hAf)1/nt((1−f)/n);V=h2κ2[3ϕ2n−(hfAf)1/(1−f)(1−f)ϕn((2−f)/(1−f))];g=hn2κ2(1−f)(Afh)1/(1−f)ϕ[nf/(1−f)−2]. Since f>0 requires ((2−f)/(1−f))>2,
the potential is also found in the same above form (3.6).Thus we find
that even a noncanonical kinetic term reproduces the same set of solutions
obtained with a canonical kinetic term. So, in principle it is possible to find
a field theory with a noncanonical kinetic term, such that the cosmological
solution is exactly the same as the solution of the said theory with canonical
kinetic term. This proves our second claim which is of course a new result.It is to be mentioned
that forf=2/(n+2),
the noncanonical kinetic term turns out to be canonical and the results arrived
at in case-I are recovered.


### 3.3. Case-III

In this
subsection we show that even a different form of the superpotential in the
presence of a nonstandard form of kinetic energy leads to similar form of the
scale factor obtained in the previous subsections. Let us choose the form of
the superpotentialH(ϕ(t)) as (3.11)H=κ2e−lϕ with l>0.
So, (3.12)gϕ˙=le−lϕ;H˙H2=−l2κ2g;V=−le−lϕϕ˙+3κ2e−2lϕ. In view of (2.9) and (3.11), it is
clear that for a constant g, wϕ becomes nondynamical. So, to find the
solutions explicitly, let us further make the following choice: (3.13)ϕ˙=e−mlϕ, where m is a constant. Under this choice, (3.14)g=le(m−1)lϕ, and the potential can be
expressed as the algebraic sum of two inverse exponents as (3.15)V(ϕ)=3κ2e−2lϕ−le−(m+1)lϕ. This form of the potential
[22, 32–35] is found as a result of compactifications in superstring models and is usually
considered to exit from the scaling regime in the presence of background
matter. Solutions for the scalar field and the scale factor are obtained as (3.16)ϕ=1mlln(mlt);a=a0exp[mκ2(m−1)(ml)1/mt(m−1)/m]. Thus the scale factor can be
expressed in the same form (3.4) of the so-called intermediate inflation for m>1.
Further, the equation of state and the deceleration parameters (2.9)
are obtained as (3.17)wϕ=−1+2l3κ2(mlt)(1−m)/m;q=−1+lκ2(mlt)(1−m)/m. Now, for m>1, (3.18)ρ+p=2lexp[−(m−1)lϕ]>0, that is, the weak energy condition
is always satisfied while (3.19)ρ+3p=6(lexp[−(m−1)lϕ]−κ2exp[−2lϕ]), that is, the strong energy
condition is violated at t≥(1/ml)(1/κ2)m/(m−1),
which corresponds to exp(m−1)lϕ≥l/κ2,
the epoch of transition from decelerating to the accelerating phase. Further,
the necessary condition for inflation and the slow roll condition are satisfied
at the same epoch. The universe expands exponentially but decelerates from a
infinitely large value. The equation of state wϕ starts from indefinitely large value at the
beginning. wϕ→∞ implies a greater effective velocity of sound
than that of light in the corresponding medium, which also appears in phantom
models having super negative pressure. So classically it has got no meaning at
all. Such result only dictates the importance of invoking quantum cosmology
before the equation of state reaches stiff fluid era.Here again,
during evolution the universe passes transiently through the stiff fluid erawϕ=1,
the radiation dominated era wϕ=1/3,
the pressureless dust era wϕ=0,
the transition (from deceleration to acceleration) era wϕ=−1/3,
and asymptotically tends to de-Sitter expansion. Thus, this solution also has
got the same features as the previous one.


## 3.1. Case-I

Let us choose
the following form of the super-potentialH(ϕ(t)): (3.1)H=hϕn, where h>0 and n>0 are constants. In view of this assumption,
(2.7) becomes (3.2)gϕ˙=nhκ2ϕ−(n+1). We can now solve the set of
(2.8),
(3.1), and (3.2), provided that we choose a particular form of g(ϕ) or V(ϕ).
In the following, we choose the standard form of kinetic energy, that is, g=1/2.
For the most natural choice g=1/2,
the field variables are found from (3.2) and (3.1) as (3.3)ϕ=[2hn(n+2)κ2t]1/(n+2),H=mt−(n/(n+2)),a=a0exp[(m(n+2)2)t(2/(n+2))], where m=h(2/(n+2))[κ2/2n(n+2)](n/(n+2)).
The above form of the scale factor, the Hubble parameter, and the scalar field ϕ can be expressed, respectively, as(3.4)a=a0exp[Atf],H=Aft(1−f),ϕ=[8h(1−f)κ2f2t]f/2, where A=m(n+2)/2>0 and 0<f=2/(n+2)<1 are related to the constants h and n.
In view of  (2.9), the state parameter and the deceleration parameter
evolve as (3.5)wϕ=−1+n3At−f;q=−1+n2At−f. The form of the potential, in
view of  (2.8), for such a solution is restricted to (3.6)V(ϕ)=h2κ4(3κ2ϕ2n−2n2ϕ2(n+1)) which has the form of double
inverse power. Thus, we have just reproduced the solutions obtained by Barrow
[11, 12] and Muslimov
[13], starting from a
particular choice of the superpotential. As previously mentioned, it was found
in the nineties and was dubbed as intermediate inflation, since the expansion
rate of the scale factor is greater than the power law but less than standard
exponential law. For such an expansion rate, (3.7)ρ+p=4n2h2κ4ϕ2(n+1)>0, that is, the weak energy
condition is always satisfied and so wϕ≥−1.
However, (3.8)ρ+3p=6κ2(−H˙−H2)=6Afκ2[(1−f)−Aftf]t−(2−f) implies that the strong energy
condition is violated at (3.9)ta>(1−fAf)1/f, when wϕ<−1/3,
and ta corresponds to the time at which the
acceleration starts.It is to be
noted that the necessary condition for inflation, namely,a¨>0 or more precisely, (d/dt)(H−1/a)<0,
is satisfied under the same above condition (3.9). Eventually, for large ϕ,
namely, ϕ≫2n/κ the potential energy starts dominating over
the kinetic energy, that is, ϕ˙2≪V(ϕ),
and thus the slow roll condition, ϵ=−H˙/H2<1,
is also satisfied under condition  (3.9). However, inflation is supposed to have
started at Planck's era, so that it can solve the initial problems of the
standard model, namely, horizon and flatness problems and the structure
formation problem and lead nearly to a scale invariant spectrum. Here we
observe that accelerated expansion of the scale factor in the so-called
intermediate inflation does not start at the onset of cosmic evolution. So, now
we can ask the following question: “what happened prior to the onset of
the accelerated expansion in the so-called intermediate inflationary era?”The solution
dictates that the universe starts evolving from an infinitely decelerated
exponential expansion withwϕ>1.
It might appear that we are considering a highly unorthodox cosmological model
involving an ultrahard equation of state and superluminal speed of sound. This
is true in some sense, since as mentioned earlier, in order to study the
situations under which such solutions emerge and its dynamics, we have not
considered the presence of any form of background matter explicitly.
Nevertheless, this situation is quite similar to the phantom models where super
negative pressure gives rise to ultranegative equation of state, indicating
that the effective velocity of sound in the medium, v=|dp/dρ|,
might become larger than the velocity of light. Likewise, here the universe
starts evolving with such a situation which actually demonstrates that the
corresponding era is classically forbidden and it is required to invoke quantum cosmology at that era.Now, during
the evolution, the universe passes transiently through the stiff fluid erawϕ=1,
the radiation dominated era wϕ=1/3,
the pressureless dust era wϕ=0,
the transition (from deceleration to acceleration) era wϕ=−1/3,
and asymptotically tends to the magic line, that is, the vacuum energy
dominated inflationary era wϕ≈−1,
ensuring asymptotic de-Sitter expansion.


## 3.2. Case-II

In this
subsection, we show that the same set of solutions obtained in case-I can be
reproduced even for a noncanonical kinetic term. To show this, we observe that
the above set of solutions does not necessarily require to fix upg=1/2,
rather they are found even for a functional form of g=g(ϕ),
that is, with a noncanonical kinetic term. This can be checked easily by
assuming solution (3.4) of the scale factor along with assumption (3.1) for the
Hubble parameter. The field ϕ,
the potential V(ϕ),
and the coupling parameter g(ϕ) are then found as (3.10)ϕ=(hAf)1/nt((1−f)/n);V=h2κ2[3ϕ2n−(hfAf)1/(1−f)(1−f)ϕn((2−f)/(1−f))];g=hn2κ2(1−f)(Afh)1/(1−f)ϕ[nf/(1−f)−2]. Since f>0 requires ((2−f)/(1−f))>2,
the potential is also found in the same above form (3.6).Thus we find
that even a noncanonical kinetic term reproduces the same set of solutions
obtained with a canonical kinetic term. So, in principle it is possible to find
a field theory with a noncanonical kinetic term, such that the cosmological
solution is exactly the same as the solution of the said theory with canonical
kinetic term. This proves our second claim which is of course a new result.It is to be mentioned
that forf=2/(n+2),
the noncanonical kinetic term turns out to be canonical and the results arrived
at in case-I are recovered.


## 3.3. Case-III

In this
subsection we show that even a different form of the superpotential in the
presence of a nonstandard form of kinetic energy leads to similar form of the
scale factor obtained in the previous subsections. Let us choose the form of
the superpotentialH(ϕ(t)) as (3.11)H=κ2e−lϕ with l>0.
So, (3.12)gϕ˙=le−lϕ;H˙H2=−l2κ2g;V=−le−lϕϕ˙+3κ2e−2lϕ. In view of (2.9) and (3.11), it is
clear that for a constant g, wϕ becomes nondynamical. So, to find the
solutions explicitly, let us further make the following choice: (3.13)ϕ˙=e−mlϕ, where m is a constant. Under this choice, (3.14)g=le(m−1)lϕ, and the potential can be
expressed as the algebraic sum of two inverse exponents as (3.15)V(ϕ)=3κ2e−2lϕ−le−(m+1)lϕ. This form of the potential
[22, 32–35] is found as a result of compactifications in superstring models and is usually
considered to exit from the scaling regime in the presence of background
matter. Solutions for the scalar field and the scale factor are obtained as (3.16)ϕ=1mlln(mlt);a=a0exp[mκ2(m−1)(ml)1/mt(m−1)/m]. Thus the scale factor can be
expressed in the same form (3.4) of the so-called intermediate inflation for m>1.
Further, the equation of state and the deceleration parameters (2.9)
are obtained as (3.17)wϕ=−1+2l3κ2(mlt)(1−m)/m;q=−1+lκ2(mlt)(1−m)/m. Now, for m>1, (3.18)ρ+p=2lexp[−(m−1)lϕ]>0, that is, the weak energy condition
is always satisfied while (3.19)ρ+3p=6(lexp[−(m−1)lϕ]−κ2exp[−2lϕ]), that is, the strong energy
condition is violated at t≥(1/ml)(1/κ2)m/(m−1),
which corresponds to exp(m−1)lϕ≥l/κ2,
the epoch of transition from decelerating to the accelerating phase. Further,
the necessary condition for inflation and the slow roll condition are satisfied
at the same epoch. The universe expands exponentially but decelerates from a
infinitely large value. The equation of state wϕ starts from indefinitely large value at the
beginning. wϕ→∞ implies a greater effective velocity of sound
than that of light in the corresponding medium, which also appears in phantom
models having super negative pressure. So classically it has got no meaning at
all. Such result only dictates the importance of invoking quantum cosmology
before the equation of state reaches stiff fluid era.Here again,
during evolution the universe passes transiently through the stiff fluid erawϕ=1,
the radiation dominated era wϕ=1/3,
the pressureless dust era wϕ=0,
the transition (from deceleration to acceleration) era wϕ=−1/3,
and asymptotically tends to de-Sitter expansion. Thus, this solution also has
got the same features as the previous one.


## 4. Presence of Background Matter

Observations
suggest that our universe is presently filled with at least70% of dark energy, 26% or less amount of cold dark matter, about 4% of Baryons, and 0.005% of radiation [36]. So, to consider a
realistic model, presence of background distribution of all types of Baryonic
and non-Baryonic matter should be accounted for explicitly. In this section,
our motivation is to check if the above form of the scale factor admits viable
cosmological solution in the presence of background matter. The field equations
now can be arranged as (4.1)H˙=−κ2[gϕ˙2+∑ρBi+pBi2],(4.2)H˙+3H2=κ2[V(ϕ)+∑ρBi−pBi2], where ρBi and pBi are the energy density and pressure of the
background matter, respectively, and the sum over i stands for the sum of different types of
background matter present in the universe (radiation + cold dark matter +
baryonic matter). Further, since the scalar field is minimally coupled to the
background, so continuity equations for the background matter and the scalar
field hold independently. Hence, we can write (4.3)ρ˙Bi+3H(1+wBi)ρBi=0=ρ˙ϕ+3H(1+wϕ)ρϕ. Thus we have (4.4)ρBi=ρBiia−3(1+wBi),(4.5)ρϕ=ρϕia−3(1+wϕ), where ρBii and ρϕi are the initial values of the background and the scalar field energy
densities. If we now plug in the solution of the scale factor in the form a=a0expAtf,
with a0>0,A>0,
and 0<f<1,
then the potential is found in view of (4.2) as (4.6)V=1κ2[3A2f2t2(1−f)−Af(1−f)t(2−f)]−∑((1−wBi)ρBi2), while the kinetic term can be
obtained in view of (4.1) as (4.7)gϕ˙2=Af(1−f)κ2t(2−f)−∑((1−wBi)ρBi2). In (4.6) and (4.7), we have used
the equation of state pBi=wBiρBi,
for the background matter. An expression of wϕ is given as a function of t: (4.8)wϕ=[2a04Af(1−f)−3a04A2f2tf]e4Atf−κ2(ρri/3)t2−f3a04A2f2tfe4Atf−κ2[ρri+a0ρmieAtf]t2−f. Now, as already
discussed in Section 3, it is not difficult to see that during evolution the
universe passes transiently through the stiff fluid era wϕ=1,
the radiation dominated era wϕ=1/3,
the pressureless dust era wϕ=0,
the transition (from deceleration to acceleration) era wϕ=−1/3,
and asymptotically tends to the vacuum energy-dominated inflationary era wϕ≈−1.
Thus, the equation of state wϕ follows the matter equation of state wB closely and so it corresponds to the scaling
solution [21, 22]. It finally comes
out of the scaling regime and enters into the transition era.


### 4.1. Canonical Lagrangian

For the canonical Lagrangian with the standard form of
kinetic energy (g=1/2), it is not possible to find a solution of ϕ vide equation (4.7) in closed form and so one
cannot express t=t(ϕ).
Hence, the form of the potential V=V(ϕ) remains obscure. However, it is still possible
to check if the field is tracker. As mentioned in the introduction, the
condition to check the existence of tracking solution is (4.9)Γ=V′′VV′2>1⇒Γ=V¨VV˙2−Vϕ¨V˙ϕ˙>1, along with the fact that it is
nearly constant which is true provided (4.10)|Γ¯|=|Γ′ΓV′/V|≪1⇒|Γ¯|=|Γ˙ΓV˙/V|≪1. Using the scalar field equation
(2.6) with g=1/2, the first condition translates
to (4.11)Γ=V¨VV˙2+3HVV˙+Vϕ˙2.
Now, in the
unit κ2=8πG=1 and taking A=0.5,f=0.5,
as before, for which a=et/2 and H=1/4t,
where we have chosen a0=1 without any loss of generality, Γ is finally expressed in view of the solutions
(4.6) and (4.7) as (4.12)Γ=(3e2t(−2+3t)−24et/2t3/2ρmi−16t3/2ρri12e2t−48et/2t3/2ρmi−64t3/2ρri+9e2t(−2t+3t)−72et/2t2ρmi−48t2ρri4(−9e2t(−1+t)+18et/2t2ρmi+16t2ρri)+((3e2t(−2+3t)−24et/2t3/2ρmi)−16t3/2ρri)×(9e2t(−5+4t)−9et/2(2+3t)t2ρmi−16(t2+2t5/2)ρri))×(2((−9e2t)(−1+t)+18et/2t2ρmi+16t2ρri)2)−1.    One can express ρri,
the initial amount of radiation density, in terms of its present value ρr0,
in view of (4.4), with wBi=wr=1/3,
as (4.13)ρri=e4At0fρr0, which is related to the present
value of the density parameter Ωr0 and the critical density ρc0=3H02 through the relation (4.14)ρr0ρc0=Ωr0. Hence, for H0−1=14.4Gyr.,
that is, h=0.68,
which yields a comfortable age of the universe t0=13Gyr as before, and taking Ωr0=0.00005,
one finds ρri=9.77×10−4. Likewise, ρmi,
the initial amount of matter density (Baryonic and CDM), may be fixed in view
of (4.4) with wBi=wm=0,
knowing Ωm0 as (4.15)ρmi=e3At0fρm0=e3At0fρc0Ωm0=3H02e3At0fΩm0. However, we will not fix up Ωm0 apriori, rather we use the manipulation
programme of Mathematica to set up Ωm0 corresponding to a tracker field. Using the
relation between the redshift parameter z and the proper time t,
namely, (4.16)1+z=a(t0)a(t)=exp[A(t0f−tf)], where a(t0) is the present value of the scale factor,
while a(t) is the value of the same at any arbitrary time t,
we plot Γ against redshift, in Figure 1, for ρmi=0.5796,
which corresponds to Ωm0=0.18.
It shows that Γ>1 for z<0.7 and remains nearly constant for z<0.6,
implying that the field starts tracking from z<0.6.Figure 1
The plot ofΓ against z for Ωm0=0.18 shows that Γ>1 for z<0.7,
and is nearly constant for z<0.55.
Thus, the field starts tracking for z≤0.5.Γ¯ can be also calculated in a straightforward
manner, which appears in a very cumbersome form and therefore has not been
presented here. However, the plot of Γ¯ versus the redshift z,
in Figure 2, shows that |Γ¯|<1,
for z<0.45.
In fact, the usage of "Manipulate Plot" of “Mathematica 6”
shows that the tracking behaviour of the field starts for Ωm0<0.2.
Thus, the tracking behaviour requires somewhat lesser amount of Baryonic and
CDM, which has not been ruled out by the presently estimated amount of matter
through different observations. It is also interesting to note that to keep Γ nearly constant, it is sufficient to consider |Γ¯|<1,
rather than |Γ¯|≪1.Figure 2
The plot ofΓ¯ against z for Ωm0=0.18 shows that |Γ¯|<1,
for z<0.45 and so the field starts tracking from that
epoch.It is also possible to give an estimate of the value
of the redshift parameterz at which the universe started accelerating,
which corresponds to wϕ≤−1/3.
In view of (4.8) and the values of the parameters of the theory already
taken up, namely, f=A=0.5, a0=1, t0=13Gyr, ρri=9.77×10−4,
and ρmi=0.5796,
the acceleration is found to start at ta≥2.14Gyr.
Hence using (4.16), the redshift za at which acceleration started is given
by (4.17)za=exp[A(t0f−taf)]−1=1.9. Thus in the present model, we
observe that the acceleration has started somewhat earlier than usually
estimated in view of ΛCDM and other existing models. Finally, the
present value of the state parameter under the above parametric choices is
found from (4.8) to be wϕ0=−0.77.
It of course increases with Ωm0,
but then one has to sacrifice the tracking behaviour.


### 4.2. Noncanonical Lagrangian

In the Section4.1, we observed that for canonical
Lagrangian, explicit forms of the potential V(ϕ) remains obscure since it was not possible to
find an expression for t=t(ϕ) in view of (4.7). However, choosing
some particular form of the scalar field ϕ,
it is possible to express the potential V(ϕ) and the coupling parameter g(ϕ) as functions of ϕ.
It is also noticed that the first term of the potential is predominantly
dominating as the universe evolves. Thus the potential remains positive
asymptotically, though it starts from an indefinitely large negative value. As
an example, let us choose ϕ as a monotonically increasing function of
time, (4.18)ϕ=t. So the potential and the kinetic
energy of the scalar field take the following forms, respectively,(4.19)V=1κ2[3A2f2ϕ2(1−f)−Af(1−f)ϕ(2−f)]−∑((1−wBi)ρBii2[a0exp{Aϕf}]3(1+wBi)),gϕ˙2=g(ϕ)=Af(1−f)κ2ϕ(2−f)−∑((1+wBi)ρBii2[a0exp{Aϕf}]3(1+wBi)).    The energy density and the
pressure of the scalar field are expressed as (4.20)ρϕ=3A2f2κ2ϕ2(1−f)−∑(ρBii[a0exp{Aϕf}]3(1+wBi)),pϕ=1κ2[2Af(1−f)ϕ(2−f)−3A2f2ϕ2(1−f)]−∑(wBiρBii[a0exp{Aϕf}]3(1+wBi)).   The scale factor admits scaling
solution as already mentioned and the scaling of ρϕ becomes sloth as V(ϕ) starts dominating over the kinetic energy.
However, as mentioned in the introduction, the condition given by Chiba
[9] to check the
tracking behaviour of the scalar field in k-essence model requires absence of
the potential V(ϕ)=0.
Since there is no method to check such behaviour in the presence of the
potential so, it is not possible to see if the scalar field is a tracker field.
However, we can make certain estimate relevant with the present observations.
In the presence of both the radiation and matter in the form of dust, the
potential and the state parameters have the following expression: (4.21)V=1κ2[3A2f2ϕ2(1−f)−Af(1−f)ϕ(2−f)]−ρr(0)3a04e4Aϕf−ρm(0)2a03e3Aϕf, which contains algebraic sum of
two inverse powers and two inverse exponents and (4.22)wϕ=pϕρϕ=[2Af(1−f)−3A2f2ϕf]e4Aϕf−(ρri/3)ϕ2−f3A2f2ϕfe4Aϕf−[ρri+ρmieAϕf]ϕ2−f. As before, to
have an idea of the redshift value (za) of transition from deceleration to
acceleration of the universe and the present value of the state parameter (wϕ0),
we make the same comfortable choice as before, that is, f=0.5,A=0.5,
and H0−1=14.42Gyr,
corresponds to h=0.68,
which set the age of the universe to t0=13Gyr.
Further, taking ρri=9.77×10−4 as calculated in the previous subsection and ρmi=0.969,
corresponding to Ωm0=0.3,
the time of transition to accelerated expansion from deceleration is calculated
from (4.8) to be ta=1.6Gyr. Corresponding redshift value is found in view
of (4.16) to be za=2.2.
Finally, one can find the present value of the state parameter in view of
(4.8) to be wϕ0=−0.9. In this case the acceleration appears to start
even earlier, but the present value of the state parameter is pretty close to
the values calculated in other models. It is to be mentioned that we have
chosen A=f=0.5 just to give a somewhat good looking
appearance to the scale factor and particularly the potential. However, a
judicious choice of the parameters A and f can give much smaller value of za and wϕ0≈−1.Finally, we
have plottedV(ϕ) and g(ϕ) against ϕ in Figures 3 and 4, respectively. It is interesting to
note that g(ϕ) remains almost flat at the later epoch,
ensuring ever accelerating and asymptotic de-Sitter universe. On the other
hand, a cusp in V(ϕ) separates early deceleration and late time
acceleration.Figure 3
The form of the potentialV(ϕ) has been plotted against ϕ (with f=0.5,h=0.68,t0=13Gyr,Ωmo=0.3). Early deceleration and late time
acceleration are clearly distinguished by a cusp.Figure 4
The form of the coupling parameterg(ϕ) has been plotted against ϕ (with f=0.5,h=0.68,t0=13Gyr,Ωmo=0.3). It remains almost flat at the later epoch
of cosmic evolution, ensuring late time acceleration.


## 4.1. Canonical Lagrangian

For the canonical Lagrangian with the standard form of
kinetic energy (g=1/2), it is not possible to find a solution of ϕ vide equation (4.7) in closed form and so one
cannot express t=t(ϕ).
Hence, the form of the potential V=V(ϕ) remains obscure. However, it is still possible
to check if the field is tracker. As mentioned in the introduction, the
condition to check the existence of tracking solution is (4.9)Γ=V′′VV′2>1⇒Γ=V¨VV˙2−Vϕ¨V˙ϕ˙>1, along with the fact that it is
nearly constant which is true provided (4.10)|Γ¯|=|Γ′ΓV′/V|≪1⇒|Γ¯|=|Γ˙ΓV˙/V|≪1. Using the scalar field equation
(2.6) with g=1/2, the first condition translates
to (4.11)Γ=V¨VV˙2+3HVV˙+Vϕ˙2.
Now, in the
unit κ2=8πG=1 and taking A=0.5,f=0.5,
as before, for which a=et/2 and H=1/4t,
where we have chosen a0=1 without any loss of generality, Γ is finally expressed in view of the solutions
(4.6) and (4.7) as (4.12)Γ=(3e2t(−2+3t)−24et/2t3/2ρmi−16t3/2ρri12e2t−48et/2t3/2ρmi−64t3/2ρri+9e2t(−2t+3t)−72et/2t2ρmi−48t2ρri4(−9e2t(−1+t)+18et/2t2ρmi+16t2ρri)+((3e2t(−2+3t)−24et/2t3/2ρmi)−16t3/2ρri)×(9e2t(−5+4t)−9et/2(2+3t)t2ρmi−16(t2+2t5/2)ρri))×(2((−9e2t)(−1+t)+18et/2t2ρmi+16t2ρri)2)−1.    One can express ρri,
the initial amount of radiation density, in terms of its present value ρr0,
in view of (4.4), with wBi=wr=1/3,
as (4.13)ρri=e4At0fρr0, which is related to the present
value of the density parameter Ωr0 and the critical density ρc0=3H02 through the relation (4.14)ρr0ρc0=Ωr0. Hence, for H0−1=14.4Gyr.,
that is, h=0.68,
which yields a comfortable age of the universe t0=13Gyr as before, and taking Ωr0=0.00005,
one finds ρri=9.77×10−4. Likewise, ρmi,
the initial amount of matter density (Baryonic and CDM), may be fixed in view
of (4.4) with wBi=wm=0,
knowing Ωm0 as (4.15)ρmi=e3At0fρm0=e3At0fρc0Ωm0=3H02e3At0fΩm0. However, we will not fix up Ωm0 apriori, rather we use the manipulation
programme of Mathematica to set up Ωm0 corresponding to a tracker field. Using the
relation between the redshift parameter z and the proper time t,
namely, (4.16)1+z=a(t0)a(t)=exp[A(t0f−tf)], where a(t0) is the present value of the scale factor,
while a(t) is the value of the same at any arbitrary time t,
we plot Γ against redshift, in Figure 1, for ρmi=0.5796,
which corresponds to Ωm0=0.18.
It shows that Γ>1 for z<0.7 and remains nearly constant for z<0.6,
implying that the field starts tracking from z<0.6.Figure 1
The plot ofΓ against z for Ωm0=0.18 shows that Γ>1 for z<0.7,
and is nearly constant for z<0.55.
Thus, the field starts tracking for z≤0.5.Γ¯ can be also calculated in a straightforward
manner, which appears in a very cumbersome form and therefore has not been
presented here. However, the plot of Γ¯ versus the redshift z,
in Figure 2, shows that |Γ¯|<1,
for z<0.45.
In fact, the usage of "Manipulate Plot" of “Mathematica 6”
shows that the tracking behaviour of the field starts for Ωm0<0.2.
Thus, the tracking behaviour requires somewhat lesser amount of Baryonic and
CDM, which has not been ruled out by the presently estimated amount of matter
through different observations. It is also interesting to note that to keep Γ nearly constant, it is sufficient to consider |Γ¯|<1,
rather than |Γ¯|≪1.Figure 2
The plot ofΓ¯ against z for Ωm0=0.18 shows that |Γ¯|<1,
for z<0.45 and so the field starts tracking from that
epoch.It is also possible to give an estimate of the value
of the redshift parameterz at which the universe started accelerating,
which corresponds to wϕ≤−1/3.
In view of (4.8) and the values of the parameters of the theory already
taken up, namely, f=A=0.5, a0=1, t0=13Gyr, ρri=9.77×10−4,
and ρmi=0.5796,
the acceleration is found to start at ta≥2.14Gyr.
Hence using (4.16), the redshift za at which acceleration started is given
by (4.17)za=exp[A(t0f−taf)]−1=1.9. Thus in the present model, we
observe that the acceleration has started somewhat earlier than usually
estimated in view of ΛCDM and other existing models. Finally, the
present value of the state parameter under the above parametric choices is
found from (4.8) to be wϕ0=−0.77.
It of course increases with Ωm0,
but then one has to sacrifice the tracking behaviour.


## 4.2. Noncanonical Lagrangian

In the Section4.1, we observed that for canonical
Lagrangian, explicit forms of the potential V(ϕ) remains obscure since it was not possible to
find an expression for t=t(ϕ) in view of (4.7). However, choosing
some particular form of the scalar field ϕ,
it is possible to express the potential V(ϕ) and the coupling parameter g(ϕ) as functions of ϕ.
It is also noticed that the first term of the potential is predominantly
dominating as the universe evolves. Thus the potential remains positive
asymptotically, though it starts from an indefinitely large negative value. As
an example, let us choose ϕ as a monotonically increasing function of
time, (4.18)ϕ=t. So the potential and the kinetic
energy of the scalar field take the following forms, respectively,(4.19)V=1κ2[3A2f2ϕ2(1−f)−Af(1−f)ϕ(2−f)]−∑((1−wBi)ρBii2[a0exp{Aϕf}]3(1+wBi)),gϕ˙2=g(ϕ)=Af(1−f)κ2ϕ(2−f)−∑((1+wBi)ρBii2[a0exp{Aϕf}]3(1+wBi)).    The energy density and the
pressure of the scalar field are expressed as (4.20)ρϕ=3A2f2κ2ϕ2(1−f)−∑(ρBii[a0exp{Aϕf}]3(1+wBi)),pϕ=1κ2[2Af(1−f)ϕ(2−f)−3A2f2ϕ2(1−f)]−∑(wBiρBii[a0exp{Aϕf}]3(1+wBi)).   The scale factor admits scaling
solution as already mentioned and the scaling of ρϕ becomes sloth as V(ϕ) starts dominating over the kinetic energy.
However, as mentioned in the introduction, the condition given by Chiba
[9] to check the
tracking behaviour of the scalar field in k-essence model requires absence of
the potential V(ϕ)=0.
Since there is no method to check such behaviour in the presence of the
potential so, it is not possible to see if the scalar field is a tracker field.
However, we can make certain estimate relevant with the present observations.
In the presence of both the radiation and matter in the form of dust, the
potential and the state parameters have the following expression: (4.21)V=1κ2[3A2f2ϕ2(1−f)−Af(1−f)ϕ(2−f)]−ρr(0)3a04e4Aϕf−ρm(0)2a03e3Aϕf, which contains algebraic sum of
two inverse powers and two inverse exponents and (4.22)wϕ=pϕρϕ=[2Af(1−f)−3A2f2ϕf]e4Aϕf−(ρri/3)ϕ2−f3A2f2ϕfe4Aϕf−[ρri+ρmieAϕf]ϕ2−f. As before, to
have an idea of the redshift value (za) of transition from deceleration to
acceleration of the universe and the present value of the state parameter (wϕ0),
we make the same comfortable choice as before, that is, f=0.5,A=0.5,
and H0−1=14.42Gyr,
corresponds to h=0.68,
which set the age of the universe to t0=13Gyr.
Further, taking ρri=9.77×10−4 as calculated in the previous subsection and ρmi=0.969,
corresponding to Ωm0=0.3,
the time of transition to accelerated expansion from deceleration is calculated
from (4.8) to be ta=1.6Gyr. Corresponding redshift value is found in view
of (4.16) to be za=2.2.
Finally, one can find the present value of the state parameter in view of
(4.8) to be wϕ0=−0.9. In this case the acceleration appears to start
even earlier, but the present value of the state parameter is pretty close to
the values calculated in other models. It is to be mentioned that we have
chosen A=f=0.5 just to give a somewhat good looking
appearance to the scale factor and particularly the potential. However, a
judicious choice of the parameters A and f can give much smaller value of za and wϕ0≈−1.Finally, we
have plottedV(ϕ) and g(ϕ) against ϕ in Figures 3 and 4, respectively. It is interesting to
note that g(ϕ) remains almost flat at the later epoch,
ensuring ever accelerating and asymptotic de-Sitter universe. On the other
hand, a cusp in V(ϕ) separates early deceleration and late time
acceleration.Figure 3
The form of the potentialV(ϕ) has been plotted against ϕ (with f=0.5,h=0.68,t0=13Gyr,Ωmo=0.3). Early deceleration and late time
acceleration are clearly distinguished by a cusp.Figure 4
The form of the coupling parameterg(ϕ) has been plotted against ϕ (with f=0.5,h=0.68,t0=13Gyr,Ωmo=0.3). It remains almost flat at the later epoch
of cosmic evolution, ensuring late time acceleration.


## 5. Concluding Remarks

Several interesting
features have been revealed in the present model. Firstly, it has been found
that the cosmological solution in the forma=a0e(Atf) with a0>0,A>0,
and 0<f<1 may be treated as dark energy model leading to
late time cosmic acceleration, rather than intermediate inflation at early
universe. The second interesting result is that the gravitational field
equations with both canonical g=1/2 and noncanonical kinetic term g=g(ϕ) produce the same set of solutions with the
same form (sum of double inverse power) of potential. Thirdly, the sum of
double inverse exponential potential has also been found to produce the same
form of solution for noncanonical kinetic term g=g(ϕ).
Although, in view of such solution a=a0e(Atf),
in the presence of background matter, it is not possible to express the
potential V as a function of the scalar field V(ϕ),
for a canonical kinetic term with g=1/2,
however, tracking behaviour, that is, Γ>1 and its slowly varying nature have been
expatiated in Figures 1 and 2. Such behaviour constraints the present value of
density parameter to Ωm0<0.2.
For the noncanonical form of kinetic energy, the potential is in the form of
the sum of double inverse powers and double inverse exponents. The present
value of the equation of state wϕ parameter has been found to be wϕ0=−0.9,
which is close to the present observational result obtained from other models.
Thus, the particular choice of the scale factor and the form of the potential
lead to a viable dark energy model with late time cosmic acceleration which has
been depicted in both the plots of the potential V(ϕ) and the coupling parameter g(ϕ).
A different choice of parameters A and f might lead to acceleration even at a lower
redshift value z with wϕ0≈−1.
Thus, we conclude that a minimally coupled scalar field admitting the above
form of solution of the scale factor a=a0e(Atf) has all the nice features to account for the
dark energy of the present universe.



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*Source: 630414-2008-10-29.xml*