# Extension of warm inflation to noncanonical scalar fields

Xiao-Min Zhang Department of Physics, Beijing Normal University, Beijing 100875, China    Jian-Yang Zhu Department of Physics, Beijing Normal University, Beijing 100875, China
July 20, 2022
###### Abstract

We extend the warm inflationary scenario to the case of the noncanonical scalar fields. The equation of motion and the other basic equations of this new scenario are obtained. The Hubble damped term is enhanced in noncanonical inflation. A linear stability analysis is performed to give the proper slow-roll conditions in warm noncanonical inflation. We study the density fluctuations in the new picture and obtain an approximate analytic expression of the power spectrum. The energy scale at the horizon crossing is depressed by both noncanonical effect and thermal effect, and so is the tensor-to-scalar ratio. Besides the synergy, the noncanonical effect and the thermal effect are competing in the case of the warm noncanonical inflation.

###### pacs:
98.80.Cq
thanks: Corresponding author

## I Introduction

Inflation is a quasiexponential expansion (strictly speaking, the inflation is an accelerated expansion and often taken to be a regime of quasiexponential expansion in the majority of models considered in the literatures) in the very early Universe Guth1981 ; Linde1982 ; Albrecht1982 , which can give successful explanation to the problems such as the horizon and the flatness. As a necessary supplement to the standard cosmological model, the inflation can also produce seeds to give rise to the large scale structure and to the observed little anisotropy of cosmological microwave background (CMB) WMAP ; PLANCK through vacuum fluctuations. Besides the standard inflation, there is also another type of inflation called warm inflation which is proposed by Berera and Fang BereraFang . Radiation is produced constantly through the interaction between the inflaton field and other subdominated boson or fermion fields during warm inflation so there is no reheating phase. The Universe can smoothly go into the big-bang phase. And the density fluctuations originate mainly from the thermal fluctuations BereraFang ; Lisa2004 ; Berera2000 rather than vacuum fluctuation. Many problems suffered in standard inflation such as eta problem etaproblem ; BereraIanRamos and the overlarge amplitude of the inflaton Berera2005 ; BereraIanRamos can be cured in warm inflation. With an additional thermal damped term added to the evolution equation of the inflaton, the slow-roll conditions are much more easily satisfied Ian2008 ; Campo2010 ; ZhangZhu .

Usually, the inflation can be realized using the canonical scalar field which has the Lagrangian density , where and is the potential of inflaton. But the noncanonical fields have many novel features as the inflaton when the Universe accelerates, such as that the equations of motion remain second order and that the slow-roll conditions become more easily satisfied compared to canonical inflationary theory refining2012 . The tensor-to-scalar ratio can drop considerably in most plausible noncanonical models refining2012 ; Cai2011 or increase in some phenomenological models Mukhanov2006 . Much work has been done about noncanonical standard inflation refining2012 ; Mukhanov2006 ; Armendariz-Picon ; Garriga1999 ; Gwyn2013 ; Tzirakis ; Franche2010 ; Eassona2013 ; Bean2008 , and noncanonical fields are the more universal case with a general Lagrangian density satisfied some conditions Franche2010 . However, warm inflation as a kind of new and realizable inflationary scenario, has always dealt with canonical fields except in Cai2011 where a warm Dirac-Born-Infeld (DBI) inflationary model was proposed. In this paper we try to extend warm inflation to a general noncanonical scalar field and thus the inflation can have a greater and broader scope. Through the new picture, we can find whether its predictions can be fitted to the observation and what attractive and new features can be obtained.

The paper is organized as follows: In Sec. II, we introduce a new noncanonical warm inflationary scenario and get the basic equations of the new picture. Then in Sec. III, we propose slow-roll inflation in the new picture and make a fully linear stability analysis to obtain the conditions that guarantee the slow-roll approximation is valid. The scalar and tensor perturbations in the new scenario are performed in Sec. IV. Finally, we draw the conclusions in Sec. V.

## Ii noncanonical warm inflationary scenario

In warm inflationary case, the Universe is a multicomponent system; thus, the total matter action can be given as:

 S=∫d4x√−g[L(X,ϕ)+LR+Lint], (1)

where the Lagrangian density of the noncanonical field is , which can be an arbitrary function of the inflaton field and the kinetic term , and for brevity we use to stand for , denotes the Lagrangian of the radiation fields and denotes the interaction term between inflaton and other fields. In order to have a uniform normalization of the field, we will make the Lagrangian density in a form that can reduce to canonical case (i.e., ) in small limit. The noncanonical Lagrangian density should satisfy the conditions: and (where a subscript here denotes a derivative while the subscripts in and are just labels) to obey the null energy condition and the physical propagation of perturbations Franche2010 ; Bean2008 . Through these two conditions and normalization of the field, we can obtain . The equation of motion can be obtained by taking the variation of the action:

 ∂(L(X,ϕ)+Lint)∂ϕ−1√−g∂μ[√−g∂L(X,ϕ)∂(∂μϕ)]=0. (2)

In the spatially flat Friedmann-Robertson-Walker Universe, the mean inflaton field is homogeneous, i.e. ; hence the equation of motion reduces to

 [∂L(X,ϕ)∂X+2X∂2L(X,ϕ)∂X2]¨ϕ +[3H∂L(X,ϕ)∂X+˙ϕ∂2L(X,ϕ)∂X∂ϕ]˙ϕ −∂(L(X,ϕ)+Lint)∂ϕ = 0, (3)

where . Through the energy-momentum tensor of , we can get the energy density and pressure of the field: , . An important parameter of the noncanonical field is the sound speed which can describe the traveling speed of scalar perturbations: , where the subscript denotes a derivative.

Now we consider a special case that the Lagrangian density can be written in a separable form for the kinetic term and the potential term, i.e. , where is the noncanonical kinetic term that is weakly dependent or independent on Franche2010 , so we assume is only the function of . In this case we have and . The general Lagrangian mainly contains two kinds: a series-form Lagrangian and a closed-form Lagrangian Franche2010 .The second form can be reduced to a canonical or DBI inflation in the specific gauge Bean2008 . The interaction term in Eq. (1) is only the function of zero order of the inflaton and other fields but not of the derivative of the fields. The most successful explanation of the interaction between the inflaton and other fields is the supersymmetric two-stage mechanism BereraKephart ; MossXiong . We use to describe the dissipation effect of to all other fields BereraFang ; Berera2005 ; Berera2000 ; BereraIanRamos , which is a thermal damping term. The other terms that do not contain in the of Eq. (II) and the term are resumed as the effective potential , which is the thermal correction potential and is the function of inflaton and temperature. The detailed introduction of the temperature in warm inflation can be found in BereraFang ; Berera1999 etc. The temperature appearing in the effective potential is that of the radiation bath and does not fall to zero thanks to the dissipations of the inflaton to the bath provided that the dependence of temperature in the dissipative coefficient satisfies the condition Eq. (33) obtained in Sec. III. Under these assumptions the equation of motion can be finally gotten:

 LXc−2s¨ϕ+(3HLX+Γ)˙ϕ+Veff,ϕ(ϕ,T)=0. (4)

For simplicity, we write as hereinafter, and the subscript denotes a derivative. We can see that the Hubble damping term is times larger than that in canonical inflation.

The total energy density of the multicomponent Universe is

 ρ=2XKX−K(X)+V(ϕ,T)+Ts, (5)

where is entropy density. Through the thermodynamics relation , we can get the free energy density of the warm inflationary Universe:

 f=2XKX−K(X)+V(ϕ,T). (6)

Through the definition of entropy in thermodynamics, we can get the expression for :

 s=−∂f/∂T=−VT(ϕ,T). (7)

So, the total pressure of the Universe is

 p=K(X)−V(ϕ,T). (8)

Combining the total energy conservation equation and Eq. (4) we can get the entropy production equation:

 T˙s+3HTs=Γ˙ϕ2. (9)

If the thermal corrections to the potential are little enough (i.e. , which can be obtained in slow-roll valid regime; see next section), the radiation energy can be written as and Eq. (9) is equivalent to the radiation energy density producing equation:

 ˙ρr+4Hρr=Γ˙ϕ2. (10)

To get a successful inflation that has enough number of -folds, we should make

 ϵH=−˙HH2=322XKX+Ts2XKX−K+V+Ts≪1, (11)

which means

 Ts≪V,XKX∼K≪V (12)

i.e. the noncanonical warm inflation should be potential dominated. The number of -folds is

 N=∫Hdt=∫H˙ϕdϕ≃−1M2p∫ϕendϕ∗V(LX+r)Vϕdϕ, (13)

where is the parameter that describes the damping strength of warm inflation.

## Iii Stability analysis

In order to make a systematic stability analysis, we define some potential slow-roll parameters which are different from but have relations with the Hubble slow-roll parameters Liddle1994 ,

 ϵ=M2p2(VϕV)2,  η=M2pVϕϕV,  β=M2pVϕΓϕVΓ, (14)

and two parameters about the temperature dependence:

 b=TVϕTVϕ,   c=TΓTΓ. (15)

We define , and Eqs. (4) and (9) can be rewritten as

 ˙u=−L−1Xc2s[(3HLX+Γ)u+Vϕ(ϕ,T)], (16)

and

 ˙s=−3Hs+Γu2T. (17)

The Friedmann equation is .

Inflation is often associated with slow-roll approximation, which consists of neglecting the highest order terms in Eqs. (4) and (9). The slow-roll approximation implies that the energy is potential dominated, the evolution of inflaton is slow and the production of radiation is quasistatic.

We use , and to denote the slow-roll solutions that satisfy slow-roll equations below:

 (3HLX+Γ)u0+Vϕ(ϕ,T)=0, (18)
 3H0T0s0−Γu20=0. (19)

The variables , and can be expanded around the slow-roll solutions: , . The perturbation terms , and are much smaller than the background ones , and . The stability is done around the slow-roll solutions, for we should obtain the conditions to guarantee they can really act as formal attractor solutions for the dynamical system.

Using the new variable, , then , and . Varying the Friedmann equation we obtain . Through the thermal relation , we have . Then we can get the variations of , etc. by using the definition of the slow roll parameters.

Taking the variation of Eqs. (16) (17), we can get

 ⎛⎜⎝δ˙ϕδ˙uδ˙s⎞⎟⎠=E⋅⎛⎜⎝δϕδuδs⎞⎟⎠−F. (20)

The matrices and can be expressed as

 E=⎛⎜⎝010Aλ1BCDλ2⎞⎟⎠,   F=⎛⎜⎝0˙u0˙s0⎞⎟⎠. (21)

The matrix elements of can be calculated out:

 A = 3H20LXc−2s[LXLX+rϵ−η+rLX+rβ (22) −(LX+r)2rb2+(LX+r)bc],
 B=H0T0u0c2s[−ϵ(LX+r)2−cLX+LX+rLXrb], (23)
 C=3H20u0T0[rLX+rϵ−rLX+rβ+(LX+r)(1−c)b], (24)
 D=H0u0T0[6r−rLXc−2s(LX+r)2ϵ], (25)
 λ1=−3H0(1+rc2sLX)−H0ϵLX(LX+r)2, (26)
 λ2=−H0(4−c)−H0rϵ(LX+r)2. (27)

The slow-roll solution can be an attractor for warm inflationary dynamic system only when the eigenvalues of the matrix are negative or possibly positive but of order (i.e. we would have slow growth) and the ”forcing term” is small enough, i.e. , . Now we study the forcing term first. Taking the time derivative of the slow roll equations (18) (19), we get

 ˙u0H0u0 = c2sΔ[1LX+r(4−c−crLX)ϵ+1LX+r4rLXβ + c−4LXη+3cLX+rLXb+(c−4)(LX+r)2rLXb2],
 ˙s0H0s0 = c2sΔ{1LX+r(6+3c2s+3rLX)ϵ−6LXη + 1LX+r(9rLX+3c2s)β+6(LX+r)2LXrb2 + (LX+r)r[3r(3c−1)LX−3(c−1)c2s]b},

where . The Hubble parameter should also be slowly varying, i.e. . Then we can get the sufficient conditions to satisfy the above requirements:

 ϵ≪LX+rc2s,  β≪LX+rc2s,  η≪LXc2s,  b≪min{LX,r}(LX+r)c2s; (30)

where is not far less than unity, and when ,

 ϵ≪LX+r9,  β≪LX+r9,  η≪LXc2s,  b≪r9(LX+r). (31)

We can reach the conclusion that the slow-roll conditions in our new case are much broader than the canonical warm inflation, let alone standard inflation. The good features are guaranteed by the two large overdamped terms: the larger Hubble damped term and the thermal damped term in Eq. (4). Thus the potential can have a much broader choice and many new models can be embedded into the cosmological inflation. This is the synergy of the two kind effect. And noncanonical effect and thermal effect also has competitive effect. If thermal dissipation dominates over Hubble damping effect, i.e. , the case approximates to the canonical warm inflationary one. In the opposite case , thermal effect is weak but still different from cold noncanonical inflation, and the reason we will see later in this paper. The slow-roll condition for implies thermal correction to the inflaton potential should be small as in canonical warm inflation Ian2008 ; Campo2010 . Thus the total energy density can have a nearly separable form .

Now we study the matrix to give an additional slow-roll condition. Through the slow-roll conditions we have got, we obtain that

 det(λI−E) = ∣∣ ∣∣λ−10−Aλ−λ1−B−C−Dλ−λ2∣∣ ∣∣ (32) = (λ−λ2)[λ(λ−λ1)−A]−BDλ−BC = 0

has a very small eigenvalue The other two eigenvalues satisfy . The two eigenvalues are both negative when and . Finally we get

 |c|<4. (33)

The radiation energy density is subdominated during the slow-roll inflationary epoch: which is consistent with the requirement that the inflation is potential dominated.

## Iv Cosmological perturbations

Now we develop the theory of cosmological perturbations in the warm noncanonical inflationary theory. The origin of density fluctuations is thermal fluctuations and both entropy and curvature perturbations must be present in warm inflationary scenarios. Since the energy density of radiation is subdominant, and its fluctuation only contributes to entropy perturbations and entropy perturbations decay on large scales Ian2008 ; Cai2011 ; Lisa2004 , we only focus on the pure curvature perturbation that can survive on large scales. Considering the small perturbations, we expand the inflaton field as , where is the linear response due to the thermal stochastic noise in thermal system. In the high temperature limit , the noise source is Markovian: Lisa2004 ; Gleiser1994 . Introducing the noise term and substituting the expansion of inflaton we can get a second order Langevin equation:

 LXc−2s[¨ϕ(t)+δ¨ϕ(x,t)]+(3HLX+Γ)[˙ϕ(t)+δ˙ϕ(x,t)] +Vϕ+(Vϕϕ−LX∇2a2)δϕ(x,t)=ξ(x,t).

Then we take the Fourier transform and obtain the evolution equation for the fluctuation:

 LXc−2s¨δϕk+(3HLX+Γ)δ˙ϕk+(LXk2a2+m2)δϕk=ξk. (35)

The second order Langevin equation is hard to solve and we only want to get the power spectrum when horizon crossing. Horizon crossing is well inside the slow-roll inflationary regime LiddleLyth and the slow-roll regime is overdamped so the inertia term can be neglected. Then the Langevin equation (35) can be reduced to first order as in Berera2000 ; TalyorBerera :

 (3HLX+Γ)δ˙ϕk+(LXk2a2+m2)δϕk=ξk. (36)

The approximate analytic solution is

 δϕk(t) ≈ {13HLX+Γ∫tt0exp[t′−t0τ(ϕ0)]ξ(k,t′)dt′ (37) +δϕ(k,t0)}exp[−t−t0τ(ϕ0)]

where , which describes the efficiency of the thermalizing process. The relation between physical wave number and comoving wave number is . In the expanding Universe, we can see from Eq. (37) that the larger is, the faster the relaxation rate is. If is sufficiently large for the mode to relax within a Hubble time, then that mode thermalizes. As soon as the physical wave number of a field mode becomes less than , it essentially feels no effect of the thermal noise during a Hubble time Berera2000 . Based on the criterion, the freeze-out physical momentum is defined as . The mass term is negligible compared to other terms in slow-roll inflation. Then we can work out

 kF=√3H2(LX+r)LX. (38)

Based on the field perturbation relation in warm inflation Berera2000 ; TalyorBerera , and using , we can finally get the scalar power spectrum in warm noncanonical inflationary model:

 PR=H3T2π2u2√3(LX+r)LX=9H5T(LX+r)522π2V2ϕ√3LX. (39)

CMB observations provide a good normalization of the scalar power spectrum on large scales, so we can see from the in the numerator that the energy scale when horizon crossing can be depressed by both the noncanonical effect and thermal effect, which is good news to the assumption that the Universe inflation can be described well by effective field theory. The spectral index

 ns−1=dlnPRdlnk (40)

is given by

 ns−1=α1c2sLX+rϵ+α2c2sLXη+α3c2sLX+rβ+α4c2s(LX+r)min{LX,r}b+α5c2s(LX+r)2LXrb2, (41)

where the expressions for , and are used. The parameters , , , and are given by:

 α1=−3c2s+rc2s2(LX+r)+1Δ{(6+3c2s+3rLX)[1+cr2(LX+r)]−3(4−c−crLX)[2+r(c−2s−1)2(LX+r)]}, (42)
 α2=1Δ{−6−3crLX+r+(12−3c)[2+r2(LX+r)(c−2s−1)]}, (43)
 (44)
 α4 = 1Δmin{LX,r}r{3(2rc−6r−3)LX−cr2(LX+r)[3r(3c−1)LX−3(c−1)c2s]} (45) −1Δmin{LX,r}LX[2+r2(LX+r)(c−2s−1)]−12cmin{LX,r}c2s(LX+r),
 α5=3Δ{2[1+cr2(LX+r)]−(c−4)[2+r(c−2s−1)2(LX+r)]}. (46)

The five parameters above are all of order unity, so we can find that is of order , where refer to the slow-roll parameters in general. We obtained a nearly scale-invariant power spectrum that is consistent with observations. The running of the spectral is calculated to find that it is of order , which coincides with observations qualitatively. And we can study some concrete models in the new theory numerically and fix the physical quantities by comparing with new observations given by PLANCK satellite in the future.

The tensor perturbations do not couple to the thermal background, and so gravitational waves are only generated by the quantum fluctuations as in standard inflation TalyorBerera

 PT=2M2p(H2π)2. (47)

The spectral index of tensor perturbation is , and the tensor-to-scalar ratio is

 R=PTPR=HT2ϵ√LX√3(LX+r)5/2. (48)

We can see that the tensor perturbation can be much weaker thanks to both the noncanonical effect and thermal effect if both the effects are strong, which is another synergy of both effects. Considering the slow-roll condition , we can get the upper bound of the tensor-to-scalar ratio . As BICEP2 suggests recently, the tensor-to-scalar ratio is significant at a level BICEP2 , the upper bound of R in our case should be large enough. We can obtain that a significant R prefers a weaker () noncanonical warm inflationary scenario with a big sound speed ( is order of unity). The insignificant non-Gaussianity suggested by PLANCK PLANCK2 also prefers a big sound speed of the noncanonical inflaton Cai2011 ; Franche2010 . The amount of expansion is while the scales corresponding to are leaving the horizon, the corresponding variation of field is

The radiation energy density and the universal temperature has the Stefan-Boltzmann relationship , and using the slow-roll equations we obtained

 TH=(rg∗PR)1/3(454π2)1/3[3(1+rLX)]1/6 (49)

from the scalar power spectrum. The ratio is smaller than that of warm canonical inflation Ian2008 ; ZhangZhu for the variable in the denominator in the last factor. We can see that a larger can enhance the ratio ; thus, the thermal effect is more obvious and the case is opposite when we have a smaller , which is the competitive effect of the noncanonical effect and thermal effect. The criterion for the happening of warm inflation can be easily and sufficiently satisfied by by analyzing Eq. (49). Considering that is of order and is of order , we can find very small amounts of dissipation that can result in warm inflation. So the warm inflation can describe the very early Universe more realizably and even in weak dissipative regime , and the thermal fluctuation amplitude dominates over its quantum counterpart, which is the consequence of Eqs. (39) and (49).

## V Conclusions

We summarize with a few remarks. We develop a theory of warm noncanonical inflationary scenario and generalize the scope of the inflation. Through the action of the warm Universe system, we get the equation of motion for the inflaton and other basic equations of the new scenario. The Hubble damping term is enhanced by an important physical quantity in noncanonical field. The stability analysis is made to give out broader slow-roll conditions thanks to the thermal and noncanonical effect. We obtain a new form but still nearly scale-invariant scalar power spectrum and we find the energy scale during horizon crossing can be depressed by the synergy of the two effects. The tensor-to-scalar ratio can be significant in weak noncanonical warm inflation with a big sound speed and insignificant in the opposite case. Warm noncanonical inflation in strong regime is also a kind of scenario to cure the eta problem and overlarge amplitude of the inflaton. We will focus on some concrete models of the new theory to give more precise comparison with the observations in the future. And the detailed issue of non-Gaussianity in the new scenario also deserves more cognition and research.

###### Acknowledgements.
This work was supported by the National Natural Science Foundation of China (Grants No. 11175019 and No. 11235003).
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