Logamediate Inflation in f(T) Teleparallel Gravity
Kazem Rezazadeh Asrin Abdolmaleki Kayoomars Karami



















































Kazem Rezazadeh
rezazadeh86@gmail.com

0000-0001-7133-3927]Kazem Rezazadeh
Department of Physics, University of Kurdistan, Pasdaran Street, P.O. Box 66177-15175, Sanandaj, Iran

0000-0001-6371-6493]Asrin Abdolmaleki
Research Institute for Astronomy and Astrophysics of Maragha (RIAAM), P.O. Box 55134-441, Maragha, Iran



0000-0003-0008-0090]Kayoomars Karami
Department of Physics, University of Kurdistan, Pasdaran Street, P.O. Box 66177-15175, Sanandaj, Iran


















We study logamediate inflation in the context of f(T) teleparallel gravity. f(T)-gravity  is a generalization of the teleparallel gravity which is formulated on the Weitzenböck spacetime, characterized by the vanishing curvature tensor (absolute parallelism) and the non-vanishing torsion tensor.  We consider an f(T)-gravity model which is sourced by a canonical scalar field. Assuming a power-law f(T) function in the action, we investigate an inflationary universe with a logamediate scale factor. Our results show that, although logamediate inflation is completely ruled out by observational data in the standard inflationary scenario based on Einstein gravity, it can be compatible with the 68% confidence limit joint region of Planck 2015 TT,TE,EE+lowP data in the framework of f(T)-gravity.





























§ INTRODUCTION



Inflation is accepted as a paradigm to solve some problems of hot Big Bang cosmology, such as the flatness, horizon, and unwanted relics problems <cit.>. Furthermore, growth of the perturbations seeded during inflation can provide a convincing explanation for the large-scale structure (LSS) formation in the universe and also for the anisotropies of the cosmic microwave background (CMB) radiation  <cit.>.

In the standard inflationary scenario, a canonical scalar field, known as an “inflaton”,  is considered in the framework of Einstein’s general relativity (GR) to explain the accelerated expansion of the universe during the inflationary era. Various inflation models with specific potentials or scale factors have been extensively investigated in the setting of the standard inflationary scenario in the light of observational data <cit.>.

Impressive observational data have been released by the Planck 2015 collaboration <cit.> following study of the anisotropies in both the temperature and polarization of the CMB radiation. Applying these observational data, we can obtain useful information about the primordial stages of our universe. Furthermore, the observational data from CMB, LSS and other sources can be employed to probe the theory of gravity on astrophysical and cosmological scales <cit.>.

Since the early stages of our universe occurred in the regime of high-energy physics, quantum modifications to gravity may play a key role in the inflationary dynamics. Motivated by this concept, numerous inflationary models have been suggested on the basis of extended theories of gravity <cit.>. One interesting class of inflationary models is based on teleparallel gravity (TG) and its generalization, f(T)-gravity. TG was employed by Einstein in 1928 to attempt to unify gravity and electromagnetism <cit.>. Although TG and GR are equivalent, they are conceptually completely different theories. In TG, the dynamical object is not the metric but instead is a set of vierbein (or tetrad) fields which forms an orthogonal basis for the tangent space at each point of spacetime. The vierbein fields are transferred parallel in all of the manifolds, which is why TG is sometimes called teleparallelism. Also, in TG, the covariant derivative is defined using the curvature-free Weitzenböck connection rather than the torsionless Levi-Civita version in GR. Furthermore, in TG, the trajectory of motion is determined by the force equations as opposed to the geodesic equations in GR <cit.>.

By a formal analogy with f(R)-gravity, the theory of f(T)-gravity theory was established by extending the Lagrangian of TG to an f(T) function of a torsion scalar T <cit.>. Cosmological implications of f(T)-gravity have
been extensively studied in the literature <cit.>. The theory of cosmological perturbations in this scenario has been studied by <cit.>, and <cit.>. Recently, several inflationary models have been investigated in the framework of f(T)-gravity <cit.>. For a comprehensive review of f(T)-gravity and its cosmological implications, see <cit.> and references therein.

In this work, we study logamediate inflation in the framework of f(T)-gravity with a minimally coupled canonical scalar field. Logamediate inflation is specified by a scale factor of the form a(t)∝exp[A(ln t)^λ] where A>0 and λ≥ 1 <cit.>.  Logamediate inflation can be regarded as a class of possible indefinite cosmological solutions resulting from imposing weak general conditions on cosmological models. <cit.> proposed that there are eight possible asymptotic solutions for cosmological dynamics, of which three lead to non-inflationary expansions. Three others give rise to the power-law (a(t)∝ t^q where q>1), de Sitter (a(t)∝ e^Ht where H is constant) and intermediate (a(t)∝exp[At^λ] where A>0 and 0<λ<1) inflationary expansions. The remaining two inflationary solutions have asymptotic expansions in logamediate form. It is worth mentioning that logamediate inflation arises naturally in some scalar–tensor theories <cit.>.

To date, the power-law and intermediate inflationary models have been investigated in the f(T)-gravity  scenario, and it has been shown that, using this setting, we can resurrect these models in light of observational data  <cit.>. This motivates us to consider logamediate inflation in the framework of f(T)-gravity. Logamediate inflation has already been studied within the standard inflationary scenario <cit.>, and it seems that its predictions are not compatible with the current constraints from the Planck 2015 data <cit.>.

The structure of this paper is as follows. In Section <ref> we review the dynamics of the background cosmology in f(T)-gravity. We also explore the relations governing the power spectra of the scalar and tensor perturbations in this model. In Section <ref>, we consider a power-law form for the f(T) function and study logamediate inflation in this setting. We estimate the inflationary observables in our model and check their viability in light of the Planck 2015 data <cit.>. Finally, in Section <ref>, we present our concluding remarks.





§ THE F(T) THEORY OF GRAVITY



In the context of f(T)-gravity, the action of modified TG can be written as <cit.>

    I=1/2∫ d^4x e[f(T)+L_ϕ],

where e≡ det(e^i_μ)=√(-g). Also, T and L_ϕ are the torsion scalar and the Lagrangian of the scalar field ϕ, respectively. It should be noted that throughout of this paper we set the reduced Planck mass to be unity, M_P≡1/√(8π G)=1, for the sake of convenience.

In our notation, e^i_μ is the vierbein or tetrad field which is used as a dynamical object in TG, and satisfies the orthonormality relations

    e^μ_ie^i_ν=δ^μ_ν,    e^μ_ie^j_μ=δ^j_i.

Here, Latin and Greek indices label tangent space and spacetime coordinates, respectively. All indices take values from 0 to 3. With the help of a dual vierbein, one can obtain the metric tensor as

    g_μν(x)=η_ije^i_μ(x)e^j_ν(x),

where η_ij= diag(-1,1,1,1) is the Minkowski metric induced on the tangent space.

In GR, the Levi-Civita connection is

    Γ^λ_μν=Γ^λ_νμ≡1/2g^λρ(g_ρμ,ν+g_ρν,μ-g_μν,ρ),

where the comma denotes the partial derivative. The Levi-Civita connection (<ref>)  leads to nonzero spacetime curvature but zero torsion. In contrast, in TG, we have the Weitzenböck connection

    Γ^λ_  μν≡ e^λ_i∂_νe^i_μ=-e^i_μ∂_νe^λ_i,

which yields zero curvature but nonzero torsion.

In GR, the curvature plays the role of gravitational force, and the trajectory of motion is determined by the geodesic equations as

    du^λ/ds+Γ^λ_μνu^μu^ν=0,

where u^λ is the four-velocity of the particle. In contrast, in TG, the torsion acts as a force and gravitational interaction is given by the force equations <cit.>

    du^λ/ds+Γ^λ_  μνu^μu^ν=T^λ_  μνu^μu^ν,

where T^λ_  μν is the torsion tensor, expressed as

    T^λ_  μν≡Γ^λ_  νμ-Γ^λ_  μν=e^λ_i(∂_μe^i_ν-∂_νe^i_μ).

The difference between the Levi-Civita and Weitzenböck connections gives the contorsion tensor

    K^λ_  μν≡Γ^λ_  μν-Γ^λ_μν=1/2(T^λ_μν+T^λ_νμ-T^λ_  μν).

The torsion scalar T is defined as

    T≡ S^  μν_λT^λ_  μν,

where S_λ^  μν is the superpotential tensor given by

    S_λ^  μν≡1/2(K^μν_   λ+δ^μ_λT^ρν_   ρ-δ^ν_λ T^ρμ_   ρ).

Using the Weitzenböck connection (<ref>), the teleparallel covariant derivative, ∇_μ, of the vierbein fields vanishes, i.e.,

    ∇_μe^i_ν≡∂_μe^i_ν-Γ^λ_  νμ e^i_λ=0.

This reflects the concept of absolute parallelism or teleparallelism in TG. In GR, the metric covariant derivative, ∇_λ, of the metric is zero

    ∇_λg_μν≡∂_λg_μν-Γ^ρ_λμg_ρν-Γ^ρ_λνg_ρμ=0.


Variation of the action (<ref>) with respect to the vierbein (tetrad) e^i_λ leads to the field equations in f(T)-gravity as <cit.>

    [e^-1∂_μ(e e_i^ρS_ρ^  λμ)-e_i^σS_μ^  νλT_  νσ^μ]f_,T+1/2e_i^λf(T)+ 
           e_i^ρS_ρ^  λσ(∂_σT)f_,TT=8π G Θ_i^λ,

where we define f_,T≡ df/dT and f_,TT≡ d^2f/dT^2. Also Θ_i^λ≡ e^-1δ L_ϕ/δ e_λ^i and the usual energy-momentum tensor is given in terms of Θ^λ_i as

    Θ^μν=η^ijΘ^ν_ie^μ_j.

Note that the set of field equations (<ref>)  are second order, and are considerably simpler than the fourth-order equations of  f(R) theory <cit.>.

Contracting with e^i_ν, Equation (<ref>) can be rewritten into the form <cit.>

    R_μνf_,T-1/2g_μν[(1+T)f_,T-f(T)]+S_νμ^   λ(∇_λT)f_,TT
           =8π G Θ_μν,

which shows that for f(T)=T,  the field equations coincide completely with those of GR. This is why in the literature, TG is called the teleparallel equivalent of GR (TEGR). This can also be understood in another way. <cit.> have shown that T and R differ only by a total divergence, i.e., R=-T-2∇^μ(T^ν_  μν). Since the total divergence can be neglected inside an integral, the TG Lagrangian density is completely equivalent to the Einstein–Hilbert density.

Now, we consider a spatially flat universe described by the Friedmann-Robertson-Walker metric

    g_μν= diag(-1,a^2(t),a^2(t),a^2(t)),

where a  is the scale factor of the universe. Using this together with Equation (<ref>), we get

    e_μ^i= diag(1,a(t),a(t),a(t)).

Substituting the vierbein (<ref>) into (<ref>) yields

    T=-6H^2,

where H ≡ȧ/a is the Hubble parameter.

Taking Θ^μ_ ν= diag(-ρ_ϕ,p_ϕ,p_ϕ,p_ϕ)  for the energy–momentum tensor of the scalar field in the perfect fluid form and using the vierbein (<ref>), the field equations (<ref>) yields the Friedmann equations in f(T)-gravity as <cit.>

    2ρ_ϕ   =    12H^2f_,T+f(T), 
    
    2p_ϕ   =    48H^2Ḣf_,TT-(12H^2+4Ḣ)f_,T-f(T).

Here, ρ_ϕ and p_ϕ are the energy density and pressure of the scalar field, respectively, and satisfy the conservation equation

    ρ̇_ϕ+3H(ρ_ϕ+p_ϕ)=0.

One can rewrite Equations (<ref>) and (<ref>) in the standard form of the Friedmann equations as

    H^2   =   1/3(ρ_T+ρ_ϕ),
    Ḣ+3/2H^2   =   -1/2(p_T+p_ϕ),

where

    ρ_T   ≡   1/2(2Tf_,T-f-T),
    
    p_T   ≡    -1/2[-8ḢTf_,TT+(2T-4Ḣ)f_,T-f+4Ḣ-T],

are the torsion contribution to the energy density and pressure which satisfy the energy conservation law

    ρ̇_T+3H(ρ_T+p_T)=0.

In the case of f(T)=T, from Equations (<ref>) and (<ref>) we have ρ_T=0 and p_T=0. Therefore, Equations (<ref>) and (<ref>) are transformed to the usual Friedmann equations in GR. In the following, we assume the universe to be filled with a canonical scalar field which has energy density and pressure as follows:

    ρ_ϕ   =   1/2ϕ̇^2+V(ϕ),
    
    p_ϕ   =   1/2ϕ̇^2-V(ϕ).

Substitution of Equations (<ref>) and (<ref>) into the conservation equation (<ref>) yields the evolution equation governing the scalar field as

    ϕ̈+3Hϕ̇+V_,ϕ=0,

where V_,ϕ≡ dV/dϕ.

In order to study inflation in f(T)-gravity, it is useful to define the Hubble slow-roll parameters as follows:

    ε_1   ≡    -Ḣ/H^2,
    ε_i+1   ≡   ε̇_i/Hε_i.

Due to having an inflationary epoch (ä > 0), according to Equation (<ref>) we must have ε_1 < 1. It should be noted that the condition ε_1 = 1 can determine the initial (or final) time of inflation if the first Hubble slow-roll parameter ε_1 is a decreasing (or increasing) function of time <cit.>.

During inflation, the scalar field ϕ and the Hubble parameter H change very slowly. This enables us to use the slow-roll conditions  given by ϕ̇^2≪ V(ϕ) and |ϕ̈|≪|3Hϕ̇|, |V_,ϕ|. Applying the slow-roll approximation to Equations (<ref>) and (<ref>), one can find

    V    =   1/2(f-2Tf_,T),
    ϕ̇^2   =    -2Ḣ(f_,T+2Tf_,TT).

With the help of the above equations, one can obtain the evolutionary behaviors of the potential V(t) and scalar field ϕ(t), if the functional form of f(T)-gravity and the scale factor a(t) are known. Combining the results of V(t) and ϕ(t) to eliminate t between them, one may get V(ϕ) determining the shape of the inflationary potential with respect to the inflaton.

In the study of inflation, we usually express the extent of the universe expansion in terms of the e-fold number, defined as

    N≡lna_e/a,

where a_e denotes the scale factor of the universe at the end of inflation. The above definition is equivalent to

    dN =  - H dt.

It is believed that the anisotropies observed in the CMB radiation and in the LSS of the universe are related to the perturbations which exit the Hubble radius around the e-fold number  N_* ≈ 50-60 before the end of inflation <cit.>. Those perturbations remain outside the horizon until a time close to the present time and this enables us to relate the late-time observations to the primordial power spectra of the perturbations produced during inflation.

In the following, we review briefly the basic results of the theory of cosmological perturbations in the f(T)-gravity scenario when a canonical scalar field is assumed to be the matter-energy content of the universe  (for more details, see <cit.>). The primordial power spectrum of the scalar perturbations in the f(T)-gravity is given by <cit.>

    P_s=.H^2/8π^2c_s^3ε_1|_c_sk=aH,

which should be evaluated at the sound horizon exit for which c_s k=a H. Here, c_s is the sound speed defined as

    c_s^2=f_,T/f_,T-12H^2f_,TT.

It is evident that in the case of TEGR (i.e., f(T)=T), we have c_s=1 from Equation (<ref>), and then Equation (<ref>) reduces to the expected relation for the standard inflationary scenario <cit.>.

The scale-dependence of the scalar power spectrum is measured by the scalar spectral index

    n_s-1≡dln P_s/dln k.

In the slow-roll approximation, it is assumed that the Hubble parameter H and the sound speed c_s are slowly varying <cit.>. Therefore, the relation c_s k=a H leads to

    dln k ≈ Hdt=-dN,

which is valid around the sound horizon exit. Now, using Equations (<ref>), (<ref>), (<ref>), (<ref>) and (<ref>), we can obtain the scalar spectral index in f(T)-gravity scenario as

    n_s=1-2ε_1-ε_2-3ε_s1,

where we have defined the sound speed slow-roll parameters as follows:

    ε_s1≡ċ_s/Hc_s,
    ε_s(i+1)≡ε̇_si/Hε_si.

We further can use Equations (<ref>), (<ref>), (<ref>), (<ref>), and (<ref>) to obtain the running of the scalar spectral index as

    dn_s/dln k=-2ε_1ε_2-ε_2ε_3-3ε_s1ε_s2.


We now focus on the tensor perturbations in the framework of f(T)-gravity. Following <cit.>, we define the parameters γ and δ as follows:

    γ   ≡   (f_,TT/f_,T)Ṫ,
    δ   ≡   | γ|/2H.

<cit.> proposed that if the δ parameter is much less than unity (δ≪ 1), then the tensor power spectrum of the tensor perturbations in f(T)-gravity reduces to the one for the standard inflationary model, which is given by

    P_t=.2H^2/π^2|_k=aH.

It should be noted that the tensor power spectrum must be calculated at the time of horizon crossing specified by k=a H. This time is not exactly the same as the time of sound horizon crossing for which c_s k=a H, but to lowest order in the slow-roll parameters the difference is negligible <cit.>.

The scale-dependence of the tensor power spectrum is specified by the tensor spectral index

    n_t≡dln𝒫_t/dln k.

Using Equations (<ref>), (<ref>), (<ref>), and (<ref>), we obtain this observable for the inflationary model based on the f(T)-gravity scenario as

    n_t=-2ε_1.

Current experimental devices are not accurate enough to measure this observable, and we may be able to determine it with more sensitive measurements in the future <cit.>.

An important inflationary observable which can be applied to discriminate between inflationary models is the tensor-to-scalar ratio

    r≡𝒫_t/𝒫_s.

In f(T)-gravity setting, using Equations (<ref>) and (<ref>) in Equation (<ref>), it is easy to see that this observable is given by

    r=16c_s^3ε_1.


From Equations (<ref>) and (<ref>), the consistency relation in f(T)-gravity takes the form

    r=-8c_s^3n_t.

It is obvious that for c_s=1, this equation reduces the well-known consistency relation r=-8n_t in the standard inflationary scenario <cit.>.





§ LOGAMEDIATE INFLATION IN F(T) TELEPARALLEL GRAVITY



<cit.> investigated logamediate inflation in the framework of the standard inflationary scenario based on Einstein gravity. From their results, it seems that logamediate inflation within the standard inflationary setting is ruled out by current observational data from the Planck 2015 collaboration <cit.>. This motivates us to examine the observational viability of logamediate inflation in f(T) teleparallel gravity.

We consider an f(T)-gravity setting in which the f(T) function in action (<ref>) has the power-law form <cit.>

    f( T ) = T_0(T/T_0)^n,

where T_0 and n are constant parameters of the model. For the case n=1, Equation (<ref>) transforms to TEGR, i.e., f(T)= T. From the definition of sound speed in Equation (<ref>), we see that the f(T) model (<ref>) gives rise to a constant sound speed as

    c_s^2 = 1/2n - 1.

The above equation leads to the requirement n ≥ 1  required to have a physical speed for propagation of the scalar perturbations, i.e., 0<c_s^2≤ 1 <cit.>. Furthermore, in our f(T)-gravity model (<ref>), the sound speed slow-roll parameters (<ref>) and (<ref>) vanish because of the constant sound speed (<ref>).

Now, we consider the logamediate scale factor which has the following form <cit.>:

    a(t)=a_0exp[A(ln t)^λ],

where a_0>0, A>0 and λ≥ 1 are constant parameters. For λ=1, the logamediate scale factor (<ref>), reduces to the power-law scale factor a(t)=a_0t^q, where q=A. With the above scale factor, the Hubble parameter reads

    H=Aλ(ln t)^λ-1/t.

Furthermore, we see from Equation (<ref>) that the first slow-roll parameter takes the form

    ε_1=ln t-λ+1/Aλ(ln t)^λ.

The above equation shows that at late times, t ≫ 1,  the first slow-roll parameter becomes a decreasing function during inflation, and hence it cannot reach unity at the end of inflation. This demonstrates that, in our model, inflation cannot end with slow-roll violation <cit.>.

To obtain the evolution of the inflationary potential, we use Equations (<ref>), (<ref>), (<ref>), and (<ref>), and obtain

    V(t)=3^n(2n-1)[2/(-T_0)]^n-1[Aλ(ln t)^λ-1/t]^2n.

We further can use Equation (<ref>) to find

    ϕ̇=       [2^nn(2n-1)(Aλ)^2n-1(3/(-T_0))^n-1]^1/2
          ×[(ln t)^2n(λ-1)-λ(ln t-(λ-1))]^1/2t^-n .

In general, it is too difficult to solve the above equation for a given value of n. Therefore, we cannot combine Equations (<ref>) and (<ref>) and find the shape of the inflationary potential V(ϕ) for a general n. However, we can check the validity of our results for the simplest case n=1 corresponding to TEGR, and we expect that it leads to the same results for logamediate inflation in the standard inflation scenario. For the case of n=1, Equations (<ref>) and (<ref>) yield

    V(t)   =   3[Aλ(ln t)^λ-1/t]^2,
    ϕ(t)   =   2√(2Aλ)/λ+1(ln t)^(λ+1)/2.

In the derivation of Equation (<ref>) we have followed the logic of <cit.>, and considered the late time limit which allows us to ignore (λ- 1) versus ln  t. If we combine the above two equations to eliminate t, we find the inflationary potential as

    V(ϕ)=V_0ϕ^αexp(-2Bϕ^β),

where

    V_0   ≡   3(Aλ B^λ-1)^2,
    
     B   ≡   (λ+1/2√(2Aλ))^2/(λ+1),
    α   ≡   4(λ-1)/λ+1,
    β   ≡   2/λ+1.

The result in Equation (<ref>)  is the potential responsible for logamediate inflation in the standard inflationary scenario, and this result is in agreement with that found by <cit.>.

Here, we are interested in showing that our result is also in agreement with the analysis performed by <cit.>, who presented asymptotic solutions of the potential

    V(ϕ)=V_0ϕ^lexp(-κϕ^m),

in the slow-roll approximation. In the above equation, V_0, l, κ and m are positive constant parameters. <cit.> found that for the case m=1 and l=0, and provided that κ^2<2, the potential (<ref>) leads to an inflationary expansion in the power-law form

    a(t)∝ t^2/κ^2.

This is just the well-known fact that in the standard inflationary scenario, the exponential potential gives rise to power-law inflation <cit.>. In addition, <cit.> obtained that for 0<m<1 and in the limit of t →∞, the potential (<ref>) provides an inflationary scale factor in the form of

    a(t)∝exp[1/κ m(2-m)(2/κ)^2-m/m(ln t)^2-m/m].


By comparing the potentials (<ref>) and (<ref>), we find

    α   =    l,
    β   =    m,
    
     2B    =   κ.

First, we focus on the case m=1 and l=0. In this case, Equations (<ref>) and (<ref>) give α=0 and β=1. As a result, from Equation (<ref>) we get λ=1. Using this, we see that in Equation (<ref>) the logamediate scale factor reduces to the power-law one 
    a(t)∝ t^A.

In addition, Equation (<ref>) gives B=1/√(2A). This together with Equation (<ref>) gives A=2/κ^2. Finally, using this result in Equation (<ref>), we reach Equation (<ref>) obtained by <cit.>.

Second, we proceed to examine the case 0<m<1. By use of Equations (<ref>) and (<ref>), we obtain

    λ=2-m/m.

Applying this together with Equations (<ref>) and (<ref>), we find

    A=1/κ m(2-m)(2/κ)^2-m/m.

Now, it is obvious that substitution of Equations (<ref>) and (<ref>) into (<ref>) yields Equation (<ref>) obtained by <cit.>. Therefore, we showed that the results (<ref>) and (<ref>) given by <cit.> are recovered in our model.

For the f(T) function given in Equation (<ref>) with a general n, and considering the logamediate scale factor (<ref>), the scalar power spectrum (<ref>)  becomes

    𝒫_s=(Aλ)^3(2n-1)^3/2(ln t)^3λ-2/8π^2t^2(ln t-λ+1).

We can also obtain the scalar spectral index from Equation (<ref>) as

    n_s=       [Aλ(ln t)^λ(ln t-λ+1)]^-1
          ×[Aλ(ln t)^λ+1-Aλ(λ-1)(ln t)^λ-2(ln t)^2. 
          .+5(λ-1)ln t-3λ^2+5λ-2].

In addition, the running of the scalar spectral index follows from Equation (<ref>) as

    dn_s/dln k=        (λ-1)[Aλ(ln t)^λ(ln t-λ+1)]^-2
          ×[2(ln t)^3-(7λ-4)(ln t)^2+(8λ^2-9λ+3)ln t. 
          . -λ(3λ^2-5λ+2)].


In order to find the expression of the tensor power spectrum for the model under consideration, we note that for the f(T) function (<ref>), the δ parameter can be simplified as

    δ=(n-1)ε_1.

In this paper, we are only dealing with values of n of order unity. Hence, the δ parameter takes the order of the first slow-roll parameter and therefore it becomes much less than unity in the slow-roll regime. This allows us to use Equation (<ref>)  for the tensor power spectrum to obtain

    𝒫_t=2[Aλ(ln t)^λ-1/π t]^2.

Then, using this together with Equation (<ref>), the tensor spectral index is obtained as

    n_t=-2(ln t-λ+1)/Aλ(ln t)^λ.

If we use Equation (<ref>), we can easily show that the tensor-to-scalar ratio becomes

    r=16(ln t-λ+1)/Aλ(2n-1)^3/2(ln t)^λ.


It is interesting to find simplified forms of the equations for the inflationary observables in the case of λ=1 for which logamediate inflation reduces to power-law inflation a(t)∝ t^q, where q=A. For λ=1, Equations (<ref>), (<ref>), (<ref>), and (<ref>) reduce to

    n_s = 1-2/A,
          dn_s/dln k = 0,
           n_t = -2/A,
           r = 16/(2n-1)^3/2A.

These are in agreement with the results of <cit.>, where the authors investigated power-law inflation in the f(T)-gravity setup (<ref>). The obtained results for the scalar spectral index n_s and the tensor-to-scalar ratio r in Equations (<ref>) and (<ref>) are independent of the dynamical quantities such as t, N or ϕ. This behavior is familiar for power-law inflation in other inflationary scenarios, for instance, the standard inflationary setting <cit.>, Brans-Dicke inflation <cit.>, tachyon inflation <cit.>,  and non-canonical power-law inflation <cit.>.  Consequently, we can combine Equations (<ref>) and (<ref>) to eliminate A between them, and obtain

    r=8/(2n-1)^3/2(1-n_s),

implying a linear relation between r and n_s. For n=1, i.e., f(T)=T, the above equation reduces to r=8(1-n_s) which is the well-known result obtained for power-law inflation in the standard inflationary scenario <cit.>.







We come back to our investigation with general n. So far, we have found the inflationary observables in terms of time. In order to estimate these observables, it is necessary to evaluate them at the time of horizon exit which has a specified e-fold number. Consequently, we should obtain the relation between time and e-fold number in our model. To this end, we solve the differential Equation (<ref>) for the logamediate scale factor (<ref>) and obtain

    t=exp[((ln t_e)^λ-N/A)^1/λ],

where t_e refers to the end time of inflation. To get the above result, we have applied the initial condition N_e≡ N(t=t_e)=0 which is a direct implication of definition (<ref>) for the e-fold number. Here, it is essential to note that we cannot determine t_e in our model by setting ε_1=1, because, as we have mentioned before, inflation in our model cannot end with slow-roll violation <cit.>. To overcome this problem, we follow the approach of <cit.> and <cit.>,  and retain t_e as an extra parameter. In the following, we determine it by fixing the amplitude of the scalar perturbations from the observational results

Inserting Equation (<ref>) into (<ref>), we obtain the scalar power spectrum at the horizon exit as

    𝒫_s(N_*)=       (Aλ)^3(2n-1)^3/2((ln t_e)^λ-N_*/A)^(3λ-2)/λ/8π^2[((ln t_e)^λ-N_*/A)^1/λ-λ+1]
          ×exp[-2((ln t_e)^λ-N_*/A)^1/λ].

The Planck 2015 data provided an estimation for the amplitude of the scalar perturbations as ln[10^10𝒫_s(N_*)]=3.094±0.034 (68% CL, Planck 2015 TT,TE,EE+lowP) <cit.>. We use this constraint in the above equation to fix the amplitude of the scalar power spectrum in our model and determine the parameter t_e in terms of the other parameters for a given horizon crossing  e-fold number N_*. Since we cannot determine t_e analytically, we use a numerical approach. Inserting the result of the numerical solution for t_e in Equation (<ref>), we can obtain the time of horizon exit t_* for given parameters n, A, λ, and N_*. Surprisingly, our computations show that t_* does not depend on N_* at all. To explain this unexpected result, we take the partial derivative of both sides of Equation (<ref>) with respect to N_*, and, keeping in mind that ∂𝒫_s(N_*)/∂ N_*=0, we obtain

    ∂ t_e/∂ N_*=t_e/Aλ(ln t_e)^λ-1.

On the other hand, if we evaluate Equation (<ref>) at the horizon exit with the e-fold number N_*, and calculate the partial derivative of the result with respect to N_*, then we will have

    ∂ t_*/∂ N_*=       exp[((ln t_e)^λ-N_*/A)^1/λ]/((ln t_e)^λ-N_*/A)^(λ-1)/λ
          ×((ln t_e)^λ-1/t_e∂ t_e/∂ N_*-1/Aλ).

It is obvious that substitution of ∂ t_e/∂ N_* from Equation (<ref>) into Equation (<ref>) leads to ∂ t_*/∂ N_*=0. Therefore, in our model, and after fixing the amplitude of the scalar perturbations from the observational data, the time of horizon exit t_* is independent of its e-fold number N_*. As an important result, we conclude that the inflationary observables (<ref>), (<ref>), (<ref>), and (<ref>) evaluated at t_* are independent of N_*.

Now, we can estimate the inflationary observables in our model and check their consistency versus the cosmological data. To do so, first we solve Equation (<ref>) numerically to find t_e for given parameters n, A, and λ. Then, we use the obtained value for t_e in Equation (<ref>) and find t_*. Subsequently, we evaluate the inflationary observables (<ref>), (<ref>), and (<ref>)  at the time of horizon exit t_*.

In order to check the viability of logamediate inflation (<ref>) in our f(T)-gravity model (<ref>), we use Equation (<ref>) and (<ref>) and plot the prediction of our model in r-n_s plane as shown in Figure <ref>. In this figure, the marginalized joint 68% CL and 95% CL regions of the Planck 2015 data <cit.> have been specified. We have represented the results of our model with n=1 and n=2  in the figure as black and orange lines, respectively. Each line is related to a specific value for the parameter λ, while the parameter A varies. In each case, as A increases, n_s approaches 1 and r approaches 0. The case n=1 corresponds to TEGR, which provides the same results of the standard inflationary scenario. It is obvious in the figure that for n=1, logamediate inflation is completely ruled out by Planck 2015 TT,TE,EE+lowP data <cit.>. But, our study indicates that if we take the parameter n greater than 1, then logamediate inflation (<ref>) in our f(T)-gravity model (<ref>) can be compatible with the Planck 2015 results. For instance, as we see in Figure <ref>, for n=2, logamediate inflation is consistent with the joint 68% CL region of Planck 2015 TT,TE,EE+lowP data <cit.>. In Figure <ref>, we have specified the parameter space of A and λ for which our model with n=2 is compatible with the 68% CL or 95% CL regions of Planck 2015 TT,TE,EE+lowP data <cit.>. From the figure, we conclude that for λ≲ 6 (λ≲ 8), our model is compatible with the joint 68% CL (95% CL) region of Planck 2015 TT,TE,EE+lowP data <cit.>. In Table <ref>, we present the ranges of the parameter A for which our model with n=2 and with some typical values of λ is consistent with the Planck 2015 observational data <cit.>. In Table <ref>, we also present the predicted values for the running of the scalar spectral index dn_s/dln k obtained using Equation (<ref>). The predicted values for dn_s/dln k in our model are compatible with the 95% CL constraint provided by Planck 2015 TT,TE,EE+lowP data <cit.>.

At the end of this section, it is useful to provide some explicit estimations for the inflationary observables in our model. We choose n=2, A=1.8, and λ=2. Using Equation (<ref>), (<ref>), and (<ref>), we obtain n_s=0.9657, dn_s/dln k=3.99×10^-5, and r=0.0546, respectively, and these are in good agreement with Planck 2015 TT,TE,EE+lowP data <cit.>. Using Equation (<ref>), our model predicts the tensor spectral index to be n_t=-0.0354, and this value may be verified by more precise measurements in the future. Within our model, we can also provide some predictions for other parameters, including the time of horizon exit t_* and the end time of inflation t_e. With the chosen values for n, A, λ, and taking the horizon exit e-fold number as N_*=60, we obtain the end time of inflation from the numerical solution of Equation (<ref>) as t_e=6.47×10^6M_P^-1=1.75×10^-36 sec. Applying this in Equation (<ref>) gives the time of horizon crossing as t_*=2.15×10^6M_P^-1=5.82×10^-37 sec. Here, we recall that in our logamediate inflationary model, although the value of t_e depends on N_*, the value of t_* is completely independent of it.





§ CONCLUSIONS



We have investigated logamediate inflation in the framework of f(T)-gravity which is sourced by a canonical and minimally coupled scalar field. For this purpose, we first briefly reviewed the basic equations governing the cosmological background evolution in f(T)-gravity and presented the relations of the scalar and tensor power spectra in this scenario. Then, we considered a setting in which the f(T) function in the action has the power-law form f(T)=T_0(T/T_0)^n, where n and T_0 are constant parameters. For n=1, this reduces to f(T)=T which provides the same results as for Einstein GR. In addition, in our work we considered the logamediate scale factor a(t)=a_0exp[A(ln t)^λ], where a_0>0, A>0 and λ≥ 1 are constant parameters. For λ=1, the logamediate scale factor turns into the power-law scale factor a(t)=a_0t^q, where q=A.

Our investigation implies that, although logamediate inflation is not consistent with the the Planck 2015 data <cit.> in the standard framework based on Einstein gravity, we can make it compatible with the observational data in our f(T)-gravity model, if we take the parameter n greater than 1. For instance, we showed that for n=2, the result of the logamediate inflation in r-n_s plane can lie inside the 68% CL region favored by Planck 2015 TT,TE,EE+lowP data <cit.>. Using the r-n_s test, we determined the parameter space for A and λ in our model with n=2, and showed that for λ≲ 6 (λ≲ 8), our model is consistent with the joint 68% CL (95% CL) region of Planck 2015 TT,TE,EE+lowP data <cit.>. We further estimated the running of the scalar spectral index dn_s/dln k in our model, and concluded that it satisfies the 95% CL bound from Planck 2015 TT,TE,EE+lowP data <cit.>.





The authors thank the referee for his/her valuable comments. The work of A.A. was supported financially by the Research Institute for Astronomy and Astrophysics of Maragha (RIAAM) under research project No. 1/4717-106.











































































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