Tachyon logamediate inflation on the brane
Eur. Phys. J. C
Tachyon logamediate inflation on the brane
Vahid Kamali 0
Elahe Navaee Nik 0
0 Department of Physics, BuAli Sina University , Hamedan 65178, 016016 , Iran
According to a Barrow solution for the scale factor of the universe, the main properties of the tachyon inflation model in the framework of the RSII braneworld are studied. Within this framework the basic slowroll parameters are calculated analytically. We compare this inflationary scenario to the latest observational data. The predicted spectral index and the tensortoscalar fluctuation ratio are in excellent agreement with those of Planck 2015. The current predictions are consistent with those of viable inflationary models.

The standard model of inflation is driven by a scalar inflaton
(quanta of the inflationary field) field can be traced back to
early efforts to solve the basic problems of the BigBang
cosmology, namely horizon, flatness and monopole [
1,2
]
problems. The nominal inflationary paradigm contains two
mainly different segments: the slowroll and the (p)reheating
regimes. In the slowroll phase the kinetic part of energy
(which has the canonical form here) of the scalar field is
negligible with respect to the potential part of energy V (φ), which
implies a nearly de Sitter expansion of the universe.
However, after the slowroll epoch the kinetic energy becomes
comparable to the potential energy and thus the inflaton field
oscillates around the minimum at the (p)reheating phase and
progressively the universe is filled by radiation [
3,4
]. In order
to achieve inflation one can use tachyon scalar fields for
which the kinetic term does not follow the canonical form
(kinflation [
5
]). It has been found that tachyon fields which are
associated with unstable Dbranes [
6
] may be responsible for
the cosmic acceleration phase in early times [
5,7,8
]. Notice
that the tachyon potential has the following two properties:
the maximum of the potential occurs when φ → 0, while
the corresponding minimum takes place when φ → ∞. For
tachyonic models of inflation with ground state at φ → ∞,
the inflaton rolls toward its ground state without oscillating
about it and the reheating mechanism does not work [
9
]. For a
quasipowerlaw time dependence, which will be considered
in the present work, there is a weak scale factor dependence of
the tachyon energy density. Therefore in the postinflation era
the tachyon density would always dominate radiation unless
there is a mechanism by which tachyons decay into
radiation. Our tachyon model in the present work is an unphysical
toy model but there is a solution for the reheating problem
in the context of warm inflation [
10–17
], which, however, is
beyond the scope of the present work. From the dynamical
viewpoint one may present the equation of motion of tachyon
field using a special Lagrangian [18], which is nonminimally
coupled to gravity:
L = √−g
R
16π G − V (φ) 1 − gμν ∂μφ∂ν φ .
(1)
Considering a spatially flat Friedmann–Lemaitre–Robertson–
Walker (FLRW) (hereafter FLRW) universe the stressenergy
tensor components are represented by
T μ ∂ L
ν = ∂(∂μφ) ∂ν φ − gνμ L = diag(−ρφ , pφ , pφ , pφ )
equation where ρφ and pφ are the energy density and pressure
of the tachyon field. Combining the above set of equations
one can find
ρφ =
and
V (φ)
1 − φ˙ 2
Pφ = −V (φ) 1 − φ˙ 2
(2)
(3)
(4)
where φ is the tachyon scalar field in units of the inverse
Planck mass Mp−l1, and V (φ) is the potential associated with
the tachyon scalar field. In the past few years, there was a
debate among particle physicists and cosmologists
regarding those phenomenological models which can be produced
in extra dimensions. For example, the reduction of a
higherdimensional gravitational scale, down to the TeVscale, could
be presented by an extra dimensional scenario [
19–21
]. In
these scenarios, the gravity field propagates in the bulk,
while standard models of particles are confined to the
lowerdimensional brane. In this framework, the extra dimension
induces additional terms in the first Friedmann equation [
22–
24
]. Especially, if we consider a quadratic term in the energy
density then we can extract an accelerated expansion of the
early universe [
25–29
]. We will study the tachyon
inflation model in the framework of the Randall–Sundrum II
braneworld [
30
], which contains a single, positive tension
brane and a noncompact extra dimension. We note that
this is not the only scenario where these characteristics are
present. For example DBI Galileon inflation [
31–34
] has
these properties in its T3 brane and a cosmological inflation
analysis of this model agrees at 1σ confidence level with
the Planck data [35]. Following the lines of Ref. [
36
], we
attempt to study the main properties of the tachyon inflation
in which the scale factor evolves as a(t ) ∝ exp( A[ln t ]λ),
where λ > 1 and A > 0 (“logamediate inflation”). For the
λ = 1 case cosmic expansion evolves as ordinary
powerlaw inflation a ∝ t p where p = A [
37,38
]. More details
regarding the cosmic expansion in various inflationary
solutions can be found in the papers by Barrow [
37,38
]. In
these papers there is no comment about the behavior around
the λ = 0 case of the logamediate solution. In the
current work, we investigate the possibility of using the
logamediate solution in the case of tachyon inflation on the
brane.
Specifically, the structure of the article is as follows: In
Sect. 2 we briefly discuss the main properties of tachyon
inflation, while in Sect. 3 we provide the perturbation
parameters. In Sect. 4 we study the performance of our predictions
against the Planck 2015 data. Finally, the main conclusions
are presented in Sect. 6.
2 Tachyon inflation
In this section we consider a FLRW universe with tachyon
component in the inflation era; the basic cosmological
equations in the context of the Randall–Sundrum II (RSII)
brane [
30
] are
ρ˙φ + 3H (ρφ + pφ ) = 0,
φ˙ =
H
˙
− 3H 2 ,
and this will be used. Using Eqs. (7) and (8) we can find a real
velocity field φ˙ by Eq. (9). Now we consider the logamediate
inflation model in which its scale factor behaves as [
37,38
]
a(t ) = a0 exp( A[ln t ]λ) ,
and one can present the compact solution of Eq. (9) thus:
dφ
dt
1 1
Aλ(ln t )λ−1 ⇒ φ − φ0 = √ Aλ
(ln t ) 1−2λ dt,
(5)
(6)
which leads to
φ = φ0 +
g(t ) = γ
g(t )
K
3 − λ
2
, − ln t
where H = aa˙ and a are the Hubble parameter and the scale
factor, respectively, a dot means the derivative with respect
to the cosmic time. The parameter τ in Eq. (5) represents
the brane tension [
22–24
]. The value of this term is
constrained to be τ > (1M eV )4 by considering the
nucleosynthesis epoch [
26
]. Another, stronger limitation for the value
of τ is found by the usual tests of the deviation from
Newton’s law τ ≥ (10T ev)4 [
39
]. The model is described in
natural units, 8π G = 2hπ = c = 1. Using Eqs. (3)–(7),
we can derive the background evolution motion of a tachyon
scalar field coupled by the scale factor in the high energy
limit ρφ τ . We have
3H 2 =
V 2(φ)
2τ 1 − φ˙ 2
2 2
ρ φ˙
H˙ = − 2M 2pτ .
φ¨ 1 dV
1 − φ˙ 2 + 3H φ˙ + V dφ = 0,
In Ref. [
38
], a complete analysis around the slowroll
parameters was made for canonical scalar fields which leads to
slowroll condition 3H φ˙ − dVdφ(φ) . We will consider our
model in the slowroll limit of tachyonic scalar fields, φ˙ 1,
φ¨ 3H φ˙ , which leads to 3H V φ˙ − dVdφ(φ) [
40
]. In the
slowroll regime of tachyon fields, from Eqs. (8) φ˙ can be
presented in terms of the Hubble parameter and its
derivative:
and
1 ρφ
H˙ = − 2M 2p (ρφ + Pφ ) 1 + τ
(7)
(8)
(9)
(10)
(11)
(12)
1
= ε −1+ g−1[K (φ − φ0)] ,
(13)
PR =
.
1
2
where K = √3λ A and γ (a, x ) is the incomplete gamma
function [
41,42
], where a is an integer constant and x is
a variable, for example in our case (a, x ) = ( 3−2 λ , − ln t ).
The dimensionless slowroll parameters of the model can be
introduced by the standard definition in terms of the scalar
field,
H
˙
ε = − H 2 =
[ln(g−1[K (φ − φ0)])]1−λ
λ A
1
η = 2H
V¨ H˙ V˙
− V + H + V
˙
where g−1(t ) is the inverse of the function g(t ). In the above
relations we have used the approximation ln t λ−1, which
may be used at the early time. The number of efoldings can
be represented for the model by
t
t1
N =
H dt =
φ H
φ1 φ˙
dφ
λ
N = (ln[g−1(K [φ − φ0])])λ − (λ A) 1−λ
where φ1 is introduced at the beginning of the inflation when
ε = 1. Using Eq. (14) we can find the tachyon scalar field in
terms of the variable of the number of efolds N ,
λ 1
φ = φ0 + g exp [N + (λ A) 1−λ ] λ
,
which will be used. The potential of the tachyon field may
be represented by using Eq. (8)
V (φ) = 6τ (λ A)2 (ln g−1[K (φ − φ0)])2(λ−1) .
(g−1[K (φ − φ0)])2
(14)
(15)
(16)
3 Perturbation
Although assuming a spatially flat, isotropic and
homogeneous FRW universe may be useful and reasonable, there are
observed deviations from isotropy and homogeneity in our
universe. These deviations motivate us to use perturbation
theory in cosmology. In the context of general relativity and
gravitation, the inhomogeneity grows with time, so it was
very small in the past time. Therefore first order or linear
perturbation theory can be used for scalar field models at the
inflation epoch. Considering Einstein’s equation, the inflaton
field in the FRW universe connects to the metric components
of this universe, so the perturbed inflaton field must be
studied in the perturbed FRW geometry. The most general linear
perturbation of spatially flat FRW metric is presented by
ds2 = −(1 + 2C )dt 2 + 2a(t )D,i dxi dt
+ a2(t )[(1 − 2ψ )δi j + 2E,i j + 2hi j ]dxi dx j ,
(17)
which includes scalar perturbations C, D, ψ, E and
tracelesstransverse tensor perturbations hi j . The powerspectrum of
the curvature perturbation PR that is derived from the
correlation of first order scalar field perturbation in the vacuum state
can be constrained by observational data. For tachyon fields,
this parameter at the first level is represented by [
36,43
]
This parameter is essential for our perturbed analysis, which
is presented in Ref. [43]. In the slowroll and high energy
limit, using Eq. (8), we may simplify the above relation to
1
PR = 4π 2V
3( Aλ)6
4π 2√6τ
V 2
2τ V
exp −
N
A
1
λ
N
A
4(λλ−1)
where V
= ddVφ . Two other important perturbation
parameters are the spectral index ns = 1 + ddlnlnPkR and its running
nrun = ddlnnsk . From Eq. (18), in the slowroll limit, these
parameters are represented by
1
ns = 1+2ε −2 + g−1(K [φ − φ0])
nrun =
4(λ − 1)
(λ A)2 (ln g−1[K (φ − φ0)])1−2λ.
1+
4(1 − λ) 1
λ N
These parameters also may be constrained by observational
data. Up to now, we have considered scalar perturbation
parameters. During the inflation era, there are two
independent components of gravitational waves, h+, h×, or tensor
perturbations of the metric with the same equation of motion.
The amplitude of the tensor perturbation is given by
Pg = 8
H
2π
2
3
τ 2
The r –ns , trajectory for inflation models can be compared to
the Planck observational data.
4 Comparison with observation
The analysis of Planck data sets has been done in Ref. [
45
].
The results of this analysis indicate that the single scalar
field models of inflation in the slowroll limit have a limited
spectral index, and a very low spectral running and
tensortoscalar ratio. We have
r =
Pg
< 0.11,
The upper bound set on the tensortoscalar ratio function
and running of the tensortoscalar ratio has been obtained
by using the results of the Planck team and joint analysis of
BICEP2/Keck Array/Planck [
46
]. In the present section we
will try to test the performance of tachyon inflation against
the results of observation (23). In Fig. 1 we render the
confidence contours in the (ns , r ) plane. The values of the pair
(λ, A) are fixed for each trajectory. The curves are related to
the pairs (λ, A) as (29.4×10−12), (39×10−15), (19.3×10−6)
and (49.5 × 10−4) from top to bottom. The main
difference between our braneworld model and ordinary scalar field
models [
36, 37
] is that there is no transition from ns < 1 to
ns > 1 for all values of λ. For the big values of λ with special
combinations of (λ, A) there are curves which behave as the
Harrison–Zel’dovich spectrum i.e. ns = 1.
In Fig. 2, the dotdashed green line and dashed blue line are
related to the pairs (29.12 × 10−3), (3 × 10−12) respectively.
In this figure for the large value of λ = 69 and small value of
(23)
A = 10−50, the trajectory located out of the 95% confidence
means that large values of λ are not compatible with the
Planck data.
In Figs. 1 and 2 the curves of our model compared with
68 and 95% confidence regions from Planck 2015 result [
45
]
at k∗ = 0.05 Mpc−1.
In Fig. 3, we plotted nrun–ns trajectories for some pairs
(λ, A) which have been used in the previous figures. There
is no running in the scalar spectral index for the combination
(19.6 × 10−9).
5 Comparison with other models
Below we will compare the current predictions with those
of viable potentials from the literature. This can help
us to understand the variants of the tachyon–brane
inflationary model from the observationally viable inflationary
scenarios.
– The Starobinsky or R2 inflation model [
48
]: in the
Starobinsky inflation model the asymptotic behavior of
the effective potential is presented as V (φ) ∝ [1 −
2e−Bφ/Mpl + O(e−2Bφ/Mpl )], which provides us with
the following predictions in the slowroll limit [
49,50
]:
r ≈ 8/B2 N 2 and, ns ≈ 1 − 2/N where B2 =
2/3. Therefore, if we select N = 50 then we obtain
(ns , r ) ≈ (0.96, 0.0048). For N = 60 we find (ns , r ) ≈
(0.967, 0.0033). It has been found that the Planck
data [45] favors the Starobinsky inflation. Obviously, our
results (see Figs. 1 and 2) are consistent with those of R2
inflation.
– The chaotic model of inflation [
51
]: in this
inflationary model the potential is given by V (φ) ∝ φk . The
basic slowroll parameters for this potential are
represented as = k/4N , η = (k − 1)/2N , which leads to
ns = 1 − (k + 2)/2N and r = 4k/N . It has been found
that monomial potentials with k ≥ 2 are not in agreement
with the Planck data [
45
]. Using k = 2 and N = 50 we
present ns 0.96 and r 0.16. For N = 60 we find
ns 0.967 and r 0.133. It is interesting to note that
this model also corresponds to the results of
intermediate inflation [
52–55
] with a Hubble rate during inflation
which is given by H ∝ t k/(4−k) with ns = 1−(k +2)r/8k
and the k = −2 case gives ns = 1 exactly to the first
order.
– Hyperbolic model of inflation [56]: in hyperbolic
inflation the potential is represented by V (φ) ∝ sinhb(φ/ f1).
Initially, this potential was proposed in the context of
the late time acceleration phase or dark energy [
57
].
Recently, the properties of this scalar field potential have
been investigated back in the inflationary epoch [
56
]. The
slowroll parameters are written as
ε = b22 Mf12p2l coth2(φ/ f1),
η = b Mf2p2l (b − 1)coth2(φ/ f1) + 1 ,
1
and
φ = f1 cosh−1 eN bMp2l/ f 2 cosh(φend/ f1) ,
where φend 2f ln θθ+−11 . Comparing this model to
the observational data, it is found that ns 0.968,
r 0.075, 1 < b ≤ 1.5 and f1 ≥ 11.7Mpl [
56
].
– Other models of inflation: The origin of brane [
58,59
]
which is motivated by the physics of extra dimensions
and, on the other hand, the exponential [
60,61
]
inflationary models are motivated by the physics of extra
dimensions. It has been found in our study that these models are
in agreement with the Planck data, although the
Starobinsky inflation is the winner from the comparison [
45
].
6 Conclusions
In this work we investigated the tachyon inflation on the
brane in the context of a spatially flat Friedmann–Robertson–
Walker universe. We adopted a specific form of scale factor
from Barrow [
37
] solutions, namely the logamediate scale
factor. Within this context, we estimated analytically the
slowroll parameters potential of the model and compared
the predictions with those of other famous inflationary
models in the literature. Confronting the model with the latest
observational data, we found that the tachyon inflationary
model on the brane is consistent with the results presented
in Planck 2015 within 1σ uncertainties for a special class of
parameters (λ, A).
Acknowledgements We would like to thank Ahmad Mehrabi for
useful discussions and Mohammad Malekjani for comments on the
manuscript and useful discussions.
Open Access This article is distributed under the terms of the Creative
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