I Introduction

Warm Vector Inflation

M. R. Setare111rezakord@ipm.ir

V. Kamali222vkamali1362@gmail.com

Department of Science, University of Kurdistan,

Sanandaj, IRAN.

Abstract

In this paper we introduce the ”warm vector inflation” scenario. In warm inflation scenario radiation is produced during the inflation epoch and reheating is avoided. Slow-roll and perturbation parameters of this model are presented. We develop our model using intermediate inflation model. In this case, the model is compatible with observational data. We also study the model using another exact cosmological solution, named logamediate scenario. We present slow-roll and Hubble parameters, power spectrum and tensor-scalar ratio in terms of inflaton. The model is compatible with WMAP7 and Planck observational data.

## I Introduction

Evidence from the cosmic microwave background (CMB) radiation indicates the early universe has an accelerating phase i.e, the inflationary epoch. Moreover inflation theory is one of the most compelling solution to many long-standing problems of the standard hot big bang model, for example, the horizon, the flatness, and the monopole problems, among others 1' (); 2' (). Main successful models of inflation are presented based on weakly interacting scalar fields. Slow-roll (approximation) is an important characteristic behavior of inflation process, but higher spin fields induce an anisotropy in spatial segment of space and the effective masses of these fields are usually of the order of the Hubble scale, therefore the slow-roll inflation does not occur 3' (); 4' (); 5' (). Vector inflation model using an orthogonal triple vector set and nonminimal coupling to gravity has been presented in Ref.1 (). This model is very successful, non-minimally coupled vector fields appear to behave in precisely the same way as a massive minimally coupled scalar field in a flat universe.
One main problem of the inflation theory is how to attach the universe to the end of the inflation epoch. An interesting solution of this problem is the study of inflation in the context of warm inflation scenario n1 (). In warm model of inflation, radiation is produced during inflation period where its energy density is kept nearly constant. This is fulfilled by introducing the dissipation coefficient . Dissipation term may be derived, using quantum field theory (QFT) methods in a two-stage mechanism for field interaction 4nn (); arj (). The study of warm inflation model as a mechanism that gives an end for vector inflation theory is motivated us to consider the warm vector inflation model.
In warm inflation there has to be continuously particle (photon) production. For this to be possible, then the microscopic processes that produce these particles must occur at a timescale () much faster than Hubble expansion. Thus the decay rates (not to be confused with the dissipative coefficient) must be bigger than . Also these produced particles must thermalize. Thus the scattering processes amongst these produced particles must occur at a rate bigger than . These adiabatic conditions were outlined since the early warm inflation papers, such as Ref. 1-ne (). There has been considerable explicit calculations from QFT that explicitly compute all these relevant scattering and decay rates in warm inflation scenario 4nn (); arj ().
In all inflationary models the scale factor is exponentially or power-law. Exact solutions exist in both cases 1' (); 6' (). However exact solutions may also be presented for intermediate inflationary universes in which the scale factor is given by 7' (); 4 ().

 a(t)=exp(Atf), (1)

where and are two constants, , and . The expansion of this universe is slower than standard de Sitter inflation, but faster than other exact solutions (power-law) y (). Intermediate inflation arises as the slow-roll solution to potential which falls off asymptotically as a power-law in field, and may be modelled by an exact cosmological solution.
In one section of the present work, we will consider warm vector inflation model in the context of ”intermediate inflation”. The study of this form of scale factor is motivated by string/M theory 4-m (). If we add the higher order curvature correction, which is proportional to Gauss-Bonnent (GB) term, to Einstein-Hilbert action then a free-ghost action 5-m () may be obtained. The Gauss-Bonnent interaction is leading order of the ”” expansion to low-energy string effective action 5-m () ( is inverse string tension). This theory may be applied for black hole solutions 6-m (), acceleration of the late time universe 7-m () and initial singularity problems 8-m (). The GB interaction in with dynamical dilatonic scalar coupling leads to an intermediate form of scale factor 4-m (); 10-m (); 11-m ().
On the other hand, we would like to study our model in the context of ”logamediate inflation” with scale factor

 a(t)=a0exp(A[lnt]λ) (2)

where 12-m (). This model is converted to power-law inflation for case. This scenario is applied for a number of scalar-tensor theories 13-m (). The study of logamediate scenario is motivated by imposing weak general conditions on the cosmological models which have indefinite expansion 12-m (). The effective potential of the logamediate model has been considered in dark energy models 14-m (). These form of potentials are also used in supergravity, Kaluza-Klein theories and super-string models 13-m (); 15-m (). For logamediate models the power spectrum could be either blue or red tilted 16-m (). In Ref.12-m (), eight possible asymptotic scale factor solutions for cosmological dynamics would be presented. Three of these solutions are non-inflationary scale factor, another three one’s of solutions give de sitter, power-low and intermediate scale factors. Finally, two cases of these solutions have asymptotic expansion with logamediate scale factor. We will study warm vector inflation model in the context of intermediate and logamediate scenarios.

## Ii The model

One inflationary model of non-minimally coupled vector fields in an isotropic and homogeneous universe (FRW) was presented in 1 (). Vector fields in this scenario behave in the same way as a minimally coupled scalar field. The action of this model is given by

 S=−12∫d4x√−g(R+12FμνFμν−R6AμAμ−V(AμAμ)), (3)

where and …, here we work in natural unit where . By variation of this action with respect to the following field equations for the vector fields is obtained

 1√−g∂∂xμ(√−gFμν)+R6Aν+∂V∂Aν=0. (4)

In the FRW universe with the metric

 ds2=dt2−a2(t)dxidxi, (5)

the equation (4), for homogeneous vector fields () converts to

 ¨Bi+3H˙Bi+V′(BjBj)Bi=0,                  A0=0, (6)

where and prime is derivative with respect to . Where the above equation is simplified

 ¨Bi+3H˙Bi+m2Bi=0, (7)

which is similar to the equation for the massive minimally coupled scalar field. Energy-momentum tensor of the vector field model may be presented by variation of action (3) with respect to the metric. The components of energy-momentum tensor in term of have the following forms 1 ()

 T00=12(˙B2k+V(BiBi)), (8)
 Tij=[−56˙B2k−12V(B2)−23H˙BkBk−13(˙H+3H2−V′(B2))B2k]δij (9) +˙Bi˙Bj+H(˙BiBj+˙BjBi)+(˙H+3H2−V′(B2))BiBj,

where the summation over index has been used. The symmetries of the FRW universe with metric (5) do not allow the energy-momentum tensor contains off-diagonal components. Therefore to remove the off-diagonal sector of energy-momentum tensor, we have to present several fields simultaneously. A triplet of mutually orthogonal vector fields are introduced in 2 (). This vector fields are constrained by orthogonality

 ∑iBaiBbi=|B|2δba, (10)

and

 ∑aBaiBaj=|B|2δji. (11)

Using above relations and Eqs. (8),(9), the total energy-momentum tensors of the triplet of mutually orthogonal vector fields are given by

 T00=ρ=32(˙B2k+V(|B|2)), (12) Tij=−Pδij=−32(˙B2k−V(|B|2))δij,

where satisfy the equation of motion (6). In the above equation we have used an ansatz

 B(a)i=|B|δai, (13)

In slow-roll regime when we have therefore the universe undergoes the stage of inflation. So, inflation may be driven by non-minimally coupled vector fields in an isotropic and homogeneous universe which behaves similar to massive minimally coupled scalar field in FRW space-time 3 (). The dynamic of warm vector inflation in spatially flat FRW universe is presented by these equations

 ˙ρ+3H(P+ρ)=−Γ˙B2 (14) ˙ργ+4Hργ=Γ˙B2 H2=13(32(˙B2+V(B))+ργ)=12(˙B2+V(B))+13ργ

where is energy density of the radiation and is the dissipative coefficient. In the above equations dot ”.” means derivative with respect to cosmic time. Warm inflation is an example of a phase transition with dissipative effect n1 (); 4nn (). In supercool inflation model 1' (), this effect becomes important after the end of inflation (in reheating epoch,) but in warm inflation model the interactions of inflaton with other fields are important during the inflationary period. Because of these interactions the dynamic equations of inflation (14) are completely changed. Dissipation coefficient in the above equations could be calculated, by using QFT methods in a two-stage mechanism for field interaction 4nn (); arj () (for full consideration see appendix). In analogy to the QFT calculations in Refs.4nn (); arj (), one might expect a form such as

 Γ=Γ0T3B2 (15)

if the interaction structure is as given in the appendix. The above form is presented as a possible type of dissipative coefficient but there might be other forms. In the above equation is the temperature of thermal bath. In Sections (III) and (IV) we will use this form of dissipation coefficient. During inflation epoch the energy density of inflaton is the order of potential energy density () and dominates over the radiation energy density . Using slow-roll approximation where n1 () and when inflation radiation production is quasi-stable, (, ) the dynamic equations are reduced to

 3H(1+r3)˙B=−12V′ (16) ργ=34r˙B2=r(1+r3)2V′2V=CT4 H2=12V

where and ( is the number of relativistic degree of freedom.). In the above equations prime () denotes derivative with respect to filed . From above equations the temperature of thermal bath is presented

 T=[−r˙H2C(1+r3)]14 (17)

Slow-roll parameters of the warm vector inflation are

 ϵ=−1HddtlnH=112(1+r3)V′2V2 (18) η=−¨HH˙H=2ϵ−˙ϵHϵ =2ϵ+13+rV′V(2V′′V−3V′V−r′3+r)

From Eqs.(16) and (18), we could find a relation between and .

 ργ=32r3+rϵρ (19)

In high dissipative regime () we have

 ργ=32ϵρ (20)

Condition of inflation epoch () may be presented by inequality .

 ρ>23ργ (21)

This is the condition for warm vector inflation era. Warm inflation epoch ends when . The number of e-folds has the following form:

 N=∫Hdt=−∫ϕfϕ∗(3+r)VV′dt (22)

where denotes the end of inflation and the epoch when the cosmological scale exits the horizon is denoted by subscript .
Perturbations of our model at the smallest level in spatially flat FRW background, will be studied. We will present the perturbation theory in isotropic universe using variation of inflaton . In warm inflation scenario the variation of inflaton is presented by thermal fluctuation. In non-warm inflation scenarios, fluctuation of inflaton may be deriven by quantum fluctuation n2 (); n3 ()

 ⟨δB⟩quantum=H22π (23)

In warm inflation model the thermal fluctuation provides

 ⟨δB⟩thermal=(ΓHT2(4π)3)14 (24)

where is the temperature of thermal bath. We will study Tensor and scalar perturbations emerge during inflation epoch for warm vector inflation model. These perturbations may leave an imprint in the CMB anisotropy and on the LSS n2 (); n3 (). Power spectrum and a spectral index, are characteristics of each fluctuation: and for scalar perturbation, and for tensor perturbation. In warm and cool inflation models, scalar power spectrum is given by

 Δ2R(k)=(H˙B⟨δB⟩)2 (25)

where is co-moving wavenumber. At the reference wavenumber the combined measurement from WMAP+BAO+SN of is reported by WMAP7 data 2-i ()

 Δ2R(k0)=(2.445±0.096)×10−9 (26)

In our model the power-spectrum of scalar perturbation is presented from Eqs.(24) and (25)

 Δ2R(k)=−(Γ3T236(4π)3)12H32˙H=(16Γ5T29√2(4π)3)12V54V′2 (27)

The largest value of density perturbation is produced when n4 (). Scalar spectral index of our model is presented by

 ns−1=−dlnΔ2R(k)dlnk (28)

In warm inflation scenario, thermal fluctuations are considered instead of quantum fluctuations that generate scalar perturbations, therefor density fluctuation of scalar perturbation is modified while the tensor perturbation shows the same spectrum as in the usual non-warm inflation n5 ()

 Δ2T=2H2π2=Vπ2 (29)

Spectral index may be found as

 nT=−2ϵ (30)

From WMAP data, we could not constraint directly, but the tensor-scalar ratio

 R=−(144(4π)3Γ3π4T2)12˙HH12 (31)

may be constrained using the WMAP+BAO+SN measurement 2-i ():

 R<0.22 (32)

## Iii Intermediate inflation

Intermediate inflation will be studied in this section, where the scale factor of this model has the following form:

 a=a0exp(Atf)      0

where is positive constant. The number of e-folds in this case is given by using the above equation

 N=∫tt1Hdt=A(tf−tf1) (34)

where is the begining time of inflation.

### iii.1 Γ=Γ0T3B2

From Eqs. (15,) (16,) (17) and (33) we could find the Hubble parameter and scalar field

 H=fA(lnB−lnB0ω)8(f−1)5f+2 (35) B=B0exp(ωt5f+28)

where and . Important slow-roll parameters and are presented

 ϵ=1−ffA(lnB−lnB0ω)−8f5f+2 (36) η=2−ffA(lnB−lnB0ω)−8f5f+2

respectively. Energy density of radiation in this case has the following form:

 ργ=3(1−f)fA(lnB−lnB0ω)8f−25f+2 (37)

Number of e-fold between two fields and is presented by using Eqs.(34) and (35)

 N=A[(lnB−lnB0ω)8f5f+2−(lnB1−lnB0ω)8f5f+2] (38)

At the begining of the inflation epoch where the scalar field in term of constant parameters of the model is presented

 B1=B0exp(ω(1−ffA)5f+28f) (39)

From above equations we could find the inflaton in term of number of e-folds

 B=B0exp(ω(NA+1−ffA)5f+28f) (40)

Perturbation parameters versus the scalar field and constant parameters of intermediate scenario are presented for warm vector inflation model. In slow-roll limit the power-spectrum of scalar perturbation could be found using Eqs.(24), (25) and (40)

 Δ2R=(Γ3036(4π)3)12(311(fA)15(1−f)3(2C)11)18B3(lnB−lnB0ω)15f−185f+2 (41) =(Γ3036(4π)3)12(311(fA)15(1−f)3(2C)11)18B−30exp(−3ω(NA+1−ffA)5f+28f)(NA+1−ffA)15f−188f

Another important perturbation parameter is spectral index which is given by

 ns−1≃15f−188fA(lnB−lnB0ω)−8f5f+2=15f−188fA(NA+1−ffA)−1 (42)

In Fig.(1), the spectral index in term of the number of e-folds is plotted (where ). From this graph, it is observed that the model is compatible with observational data 2-i (); 2-m (), ( case leads to ). Tensor power spectrum and its spectral index are given by

 Δ2T=2(fA)2π2(lnB−lnB0ω)16(f−1)5f+2=2(fA)2π2(NA+1−ffA)2(f−1)f (43) nT=−2−2ffA(lnB−lnB0ω)−8f5f+2

Tensor-scalar ratio has the following form

 R=(144(4π)3(fA)4Γ30π4)12(311(fA)15(1−f)3(2C)11)−18B3(lnB−lnB0ω)f+25f+2 (44) =(144(4π)3(fA)4Γ30π4)12(311(fA)15(1−f)3(2C)11)−18B30exp(−3ω(NA+1−ffA)5f+28f)(NA+1−ffA)f+28f =(144(4π)3(fA)4Γ30π4)12(311(fA)15(1−f)3(2C)11)−18B30exp(−3ω(18−15f8fA(1−ns))5f+28f)(18−15f8fA(1−ns))f+28f

The graph of the tensor-scalar ratio in term of spectral index is presented in Fig.(2). Standard case , my be found in for important case arj (), which is agree with observational data 2-i (); 2-m ().

### iii.2 Γ=Γ1=const

From Eqs.(16) and (33) the Hubble parameter and the scalar field could be found

 H=fA(B−B0ω)2(f−1)2f−1 (45) B=B0+ωt2f−12

where and . Important slow-roll parameters and are given by

 ϵ=1−ffA(B−B0ω)2f1−2f (46) η=2−ffA(B−B0ω)2f1−2f

respectively. The energy density of radiation in this case has this form:

 ργ=3(1−f)fA(B−B0ω)4−2f1−2f (47)

The number of e-fold between two fields and is presented (We have used Eqs.(34) and (45))

 N=A[(B−B0ω)2f2f−1−(B1−B0ω)2f2f−1] (48)

At the beginning of the inflation epoch where the scalar field in terms of constant parameters of the model is given by

 B1=B0+ω(1−ffA)2f−12f (49)

Using above equations we could find the inflaton in terms of the number of e-folds

 B=B0+ω(NA+1−ffA)2f−12f (50)

The perturbation parameters versus the scalar field and constant parameters of intermediate scenario in this case are presented for warm vector inflation model. In slow-roll limit, by using Eqs. (27) and (50), we could find the power-spectrum of scalar perturbation

 Δ2R=(Γ3131236(2C)12(4π)3(fA)12(1−f)32)12(B−B0ω)3f2(2f−1) (51) (Γ3131236(2C)12(4π)3(fA)12(1−f)32)12(NA+f+12f)34

Another important perturbation parameter is spectral index which is presented by

 ns−1=−34A(B−B0ω)2f1−2f=−34A(NA+1−ffA)−1 (52)

In Fig.(3), we plot the spectral index in term of the number of e-folds (where ). From this graph, it is observed that the model is compatible with observational data 2-i (); 2-m (), ( case leads to ). The tensor power spectrum and its spectral index are presented by

 Δ2T=2(fA)2π2(B−B0ω)4(f−1)2f−1=2(fA)2π2(NA+1−ffA)2(f−1)f (53) nT=−2−2ffA(B−B0ω)2f1−2f

The tensor-scalar ratio has the following form:

 R=(144(2C)12(4π)3(fA)52(1−f)32Γ31π4312)12(B−B0ω)5f−82(2f−1) (54) =(144(2C)12(4π)3(fA)52(1−f)32Γ31π4312)12(NA+1−ffA)5f−84f =(144(2C)12(4π)3(fA)52(1−f)32Γ31π4312)12(4A3(1−ns))8−5f4f

The graph of the tensor-scalar ratio in terms of spectral index is given by Fig.(4). The standard case , my be found in which is agree with WMAP data 2-i (); 2-m ().

## Iv Logamediate inflation

In this section we study warm vector field logamediate inflation where the scale factor has this form

 a(t)=a0exp(A[lnt]λ)      λ>1, (55)

is a constant parameter. The number of e-folds may be found, using the above equation

 N=∫tt1Hdt=A[(lnt)λ−(lnt1)λ] (56)

where denotes the begining time of inflation epoch.

### iv.1 Γ=Γ0T3B2

Using Eqs.(15,) (16,) (17,) and (55) we may find the inflaton

 lnB−lnB0=¯¯¯ωΞ(t) (57)

where and ( is incomplete gamma function gamma ()). Potential in term of scalar field is presented as:

 V=2λ2A2[ln(Ξ−1(lnB−lnB0¯¯¯ω))]2λ−2(Ξ−1(lnB−lnB0¯ω))2 (58)

Slow-roll parameters of the model in this case are given by

 ϵ=[lnΞ−1(lnB−lnB0¯¯¯ω)]1−λλA (59) η=2[lnΞ−1(lnB−lnB0¯¯¯ω)]1−λλA

The number of e-folds between two fields and may be determined, using Eqs.(56) and (57).

 N=A((lnΞ−1(lnB−lnB0¯¯¯ω))λ−(lnΞ−1(lnB1−lnB0¯¯¯ω))λ) (60) =A((lnΞ−1(lnB−lnB0¯¯¯ω))λ−(λA)λ1−λ)

where is inflaton at the begining of inflation epoch when (). Inflaton field in the inflation period could be obtained in term of number of e-folds by using above equation

 B=B0exp(¯¯¯ωΞ[exp([NA+(λA)λ1−λ]1λ)]) (61)

Power spectrum and tensor-scalar ratio in this case are presented

 Δ2R=(Γ3036(4π)3)12(311(λA)15(2C)11)18B−30exp(−158[NA+(λA)λ1−λ]1λ) (62) (NA+(λA)λ1−λ)15λ−18λexp(−3¯¯¯ωΞ(exp([NA+(λA)λ1−λ]1λ))) Δ2T=2λ2A2π2(lnΞ−1(lnB−lnB0¯¯¯ω))2−2λ(Ξ−1(lnB−lnB0¯¯¯ω))2 =2λ2A2π2exp[−2(NA+(λA)λ1−λ)1λ](NA+(λA)λ1−λ)2−2λλ

Spectral indices for our model have the following forms

 ns−1=−15(λ−1)8λA[NA+(λA)λ1−λ]−1 (63) nT=−2[lnΞ−1(lnB−lnB0¯¯¯ω)]1−λλA

In Fig.(5), the spectral index in term of number of e-folds is plotted (for , , cases). We may see the small values of number of e-folds are assured for large values of parameter. We could find the tensor-scalar ratio in term of number of e-folds and spectral index

 R=(144(4π)3Γ30π4)12((2C)11(λA)311)18B30exp[−18(NA+(λA)λ1−λ)1λ] (64) (NA+(λA)λ1−λ)−31(λ−1)8λexp(3¯¯¯ωΞ(exp([NA+(λA)λ1−λ]1λ))) =(144(4π)3Γ30π4)12((2C)11(λA)311)18B30exp(−18(15(λ−1)8λA(1−ns))1λ) (15(λ−1)8λA(1−ns))−31(λ−1)8λexp(3¯¯¯ωΞ[exp((15(λ−1)8λA(1−ns))1λ)])

In Fig.(6), the tensor-scalar ratio versus the spectral index is plotted for , . We find the model is compatible with observational data (for case)2-i (); 2-m ().

### iv.2 Γ=Γ1

Using Eqs.(16) and (55) we could find the inflaton field

 B−B0=¯¯¯ωΞ(t) (65)

where and ( is incomplete gamma function gamma ()). Potential in terms of scalar field is given by

 V=2(λA[ln(Ξ−1(B−B0¯¯¯ω))]λ−1Ξ−1(B−B0¯¯¯ω))2 (66)

The slow-roll parameters of the model in this case are presented as

 ϵ=[lnΞ−1(B−B0¯¯¯ω)]1−λλA (67) η=2[lnΞ−1(B−B0