Two-field Warm Inflation and Its Scalar Perturbations on Large Scales
Yang-Yang Wang Jian-Yang Zhu Xiao-Min Zhang



wangyy@mail.bnu.edu.cn
Department of Physics, Beijing Normal University, Beijing 100875, China

 zhangxm@mail.bnu.edu.cn
  School of Science, Qingdao University of Technology, Qingdao 266033, China

Corresponding author
 zhujy@bnu.edu.cn
Department of Physics, Beijing Normal University, Beijing 100875, China

We explore the homogeneous background dynamics and the evolution of generated perturbations of cosmological inflation that is driven by multiple scalar fields interacting with a perfect fluid.
Then we apply the method to warm inflation driven by two scalar fields and a radiation fluid, and present general results about the evolution of the inflaton and radiation. After decomposing the perturbations into adiabatic and entropy modes, we give the equation of motion of adiabatic and entropy perturbations on large scales. Then, we give numerical results of background and perturbation equations in a concrete model (the dissipative coefficient Γ∝ H). At last, we use the most recent observational data to constrain our models and give the observationally allowed regions of parameters. This work is a natural extension of warm inflation to multi-field cases.

98.80.Cq
Two-field Warm Inflation and Its Scalar Perturbations on Large Scales
    Jian-Yang Zhu
    December 30, 2023
=====================================================================





§ INTRODUCTION


Inflation has become one of the central paradigm in modern cosmology, because it solves many problems of standard cosmology and provides an origin of large-scale structure <cit.>. In inflation theory, a most common model is that cosmological inflation is driven by a scalar field whose potential dominates other forms of energy density. In standard inflation, cosmological expansion and reheating are two
distinguished periods and we still know little about the reheating
process. Warm inflation is an important inflationary model and it combine the cosmological expansion and the production of the radiation into one process, so the universe can become radiation-dominated smoothly <cit.>. In warm inflation, dissipative effects are important during the inflation period, so that radiation production occurs concurrently with cosmological expansion. Besides, recent observations imply that chaotic inflation with monomial quadratic potential and natural inflation are now disfavored for predicting too large tensor-to-scalar ratio r. In warm inflation, curvature perturbation are dominated by thermal fluctuation which is usually much stronger than quantum fluctuation, while tensor perturbation remain the same to the cold inflation results. Therefore, many inflationary models are in accordance with the observational data again for a decreased r in warm regime.

A different possible way to generate perturbations in agreement with observations is so-called multi-field inflationary model. Although single field inflation may seems appealing from the perspective of simplicity and economy, the microphysical origin of inflation still remains unclear and there is not theoretical reason to expect only one field to be important in the early Universe. In fact, fundamental physics, such as sting theory, commonly predicts the existence of multiple scalar fields <cit.>.

Compared to single-field inflation, a key feature of multi-field inflation is a relatively large non-Gaussianity. However, now the observations of non-Gaussianity is not precise enough to distinguish between inflationary models. Therefore it is important to study the effects of multi-field inflation and how they are constrained by observational data. In addition, the content of the Universe is commonly assumed to be a mixture of fluids and scalar fields, and there has been increasing interests focused on multi-component cosmology. In this paper we investigate cosmological inflation driven by multiple scalar fields and an interacting perfect fluid, and then apply the formalism to warm inflation in a two-field case. This work is a natural extension of warm inflation to multi-field cases.

This paper is organized as follows. In Sec. II we introduce the governing background and perturbation equations of multiple scalar fields interacting with a perfect fluid. In Sec. III we apply the formalism to warm inflation and obtain the evolution equations of curvature and entropy perturbation in a two-field case. In Sec. IV, we give numerical results in a representative case with the dissipative coefficient Γ∝ H, then we use the most up-to-data observational data to constrain our models and give the observationally allowed region of parameters. To conclude, we present some summaries and comments in Sec. IV. In this paper, we redefined some slow-roll parameters of the inflation and treat the radiation as a perfect fluid.



§ MULTI-COMPONENT INFLATION


Let us study the inflation in homogeneous and isotropic background. We consider a spatially flat Friedmann-Robertson-Walker (FRW) metric of the form 
    ds^2=-dt^2+a(t)^2δ_ijdx^idx^j,

where a(t) is the scale factor, and t is cosmic time. We use Planck Unit:
    8π G=k_B=ħ=c=1,
 where G is Newton's gravitational constant, k_B is Boltzmann's constant,
ħ is the reduced Planck's constant, and c is the speed of light. In this work, Greek indices μ, ν, λ denote spacetime dimensions, and Latin indices I, J, 𝒩 denote different scalar fields. Repeated spacetime indices are summed over.

First we consider a 𝒩-fields model with Lagrangian density <cit.>:

    ℒ= - 1/2Σ_I^𝒩 g^μν∇_μφ_I
    ∇_νφ_I - V (φ),

which is minimal coupled to gravity, where V(φ)=V(φ_1,
φ_2, …, φ_𝒩). We assume that there exists a perfect fluid
and the interaction between scalar fields and the fluid causes a
phenomenological dissipative term Γφ̇_̇İ in the equation of
motion. In general, Γ = Γ (φ_1, φ_2, …, φ_𝒩,
ρ_f), ρ_f is the energy density of the perfect fluid. From Eq. (<ref>) we can
get the equation of motion in the presence of a perfect fluid:

    φ̈_̈Ï + (3 H + Γ) φ̇_̇İ + V_φ_I = 0,

where H≡ȧ/a is the Hubble parameter, V_φ_I = ∂ V/∂φ_I, overdots represent derivatives with respect to cosmic time. In a spatially flat FRW
universe, H is determined by:

    H^2 = 8 π G/3( 1/2Σ_I^𝒩φ̇_̇İ^2 + V (φ) + ρ_f ),

and the continuity equation of the perfect fluid

    ρ̇_̇ḟ + 3 H (1 + ω) ρ_f = ΓΣ_I φ̇_̇İ^2,

where ω = p_f/ρ_f, p_f is the pressure of the fluid.

As in single-field inflation, we define some slow-roll parameters of the background quantities,

    ϵ = - Ḣ/H^2 = Σ_I ϵ_I + ϵ_f,   ϵ_I = 1/2φ̇_I^2/H^2,   ϵ_f = 2/3ρ_f/H^2, 
    η = - Ḧ/2 H Ḣ = Σ_I ϵ_I/ϵη_I + ϵ_f/ϵη_f, 
    η_I = - φ̈_̈Ï/H φ̇_̇İ,   η_f = - 1/2ρ̇_f/H ρ_f.


The slow-roll conditions are ϵ<1, |η|<1. In slow-roll approximation, we have

    (3 H + Γ) φ̇_̇İ + V_φ_I = 0, 
    
    3 H (1 + ω) ρ_f = ΓΣ_I φ̇_̇İ^2.

In order to study the evolution of the linear perturbations, we decompose
each of the scalar fields into a spatially homogenous background field and its
fluctuations φ_I (x, t) →φ_I (t) + δφ_I
(x, t). The line element of the FRW metric can be written as

    d s^2 = - (1 + 2 A) d t^2 + 2 a ∂_i B d x^i d t 
    
    + a^2 ((1 - 2 ψ) δ_ij + 2 ∂_i ∂_j E) d x^i d x^j,

and the gauge-invariant comoving curvature perturbation is given by <cit.>

    ℛ = ψ - H δ q/p + ρ,

where δ q = Σ_I δ q_I + δ q_f is the total momentum
density perturbation, and p and ρ are total pressure and energy
density <cit.>. The momentum perturbations of each components are given by

    δ q_I    =    - φ̇_̇İδφ_I, 
    δ q_f    =    a (p_f + ρ_f) (B + δ u),

where δ u is the scalar velocity potential of the fluid, and from above
definition we know δ u = δ q_f/a (p_f + ρ_f) - B. The
four-velocity of the fluid is defined by

    u^μ   =   1/a (1 - A, ∂^i δ u), 
    
    u_μ   =    a (- 1 - A, ∂_i δ u + ∂_i B).

The variation of the scalar field's equation of motion leads to:

    δ̈φ̈_̈Ï + (3 H + Γ) δ̇φ̇_̇İ +
    k^2/a^2δφ_I + Σ_J V_φ_I φ_Jδφ_J + φ̇_̇İδΓ
    
    = - (2 V_φ_I + Γφ̇_̇İ) A + φ̇_̇İ( Ȧ + 3 ψ̇ + k^2/a^2 (a^2 Ė - a B) ),

and the perturbation of energy and momentum conservation equation of the fluid is given by <cit.>

    δ̇ρ̇_̇ḟ + 3 H (δ p_f + δρ_f) - k^2/a^2δ q_f + k^2/a (p_f + ρ_f) B 
    
    - δΓΣ_I φ̇_̇İ^2 - 2 ΓΣ_I φ̇_̇İδ̇φ̇
    
    = k^2 Ė (p_f + ρ_f) - Γ A Σ_I φ̇_̇İ^2 + 3 (p +
    ρ) ψ̇,


    δ̇ ̇q̇_̇ḟ + 3 H δ q_r + δ p + ΓΣ_I
    φ̇_̇İδφ_I = - (p_f + ρ_f) A,

where k is the wave number in Fourier space.

The Einstein equation of the multi-component system is given by

    G_μν = T_μν^(φ)+T_μν^( f ),

where

    T_μν^(φ) = Σ_I^𝒩∂_μφ_I ∂_νφ_I - g_μν( 1/2Σ_I^𝒩∂^λφ_I ∂_λφ_I +V(φ) ),


    T_μν^(f) =  (ρ_f + p_f) u_μ u_ν + p_f g_μν,

where G_μν is the Einstein tensor, and T_μν^(φ), T_μν^(f) are the energy-momentum tensor of scalar fields and perfect fluid respectively.

The perturbation equations of Einstein's field equations are:

    δρ_f + Σ_I φ̇_̇İδφ_I + Σ_I
    V_φ_Iδφ_I = - 2 (V + ρ_f) A 
    
    + 2 k^2 H/a^2 (a B
    - a^2 Ė) - k^2/a^2ψ - 6 H ψ̇,


    δ q_f - Σ_I φ̇_̇İδφ_I = - 2 H A - 2 ψ̇,


    a^2 (Ë + 3 H Ė) - a (Ḃ - 2 H B) - A + ψ = 0,


    δ p_f + Σ_I φ̇_̇İ_̇δ̇φ̇_̇İ -
    V_φ_Iδφ_I 
    
    = - k^2/a^2 (a Ḃ - a^2 Ë)
    - k^2 H/a^2 (2 a B - a^2 Ė) - k^2/a^2 A 
    
    + 2 (V - p_f) A
    + 2 H Ȧ + k^2/a^2ψ + 6 H ψ̇ + 2 ψ̈.

Eqs. (<ref>)-(<ref>) are, respectively, the G_0^0 component of the field equation,
the G_i^0 component, the trace-free part of the G_i^j and the G_i^i
component <cit.>.



§ APPLICATION TO WARM INFLATION


In warm inflation, the density perturbations are mainly sourced by thermal
noise <cit.>, and metric fluctuations has little effect on small scales <cit.>.
When k/a H≫ 1, inflaton fluctuations δφ_I are
described by a Langevin equation <cit.>

    δ̈φ̈_̈Ï (k, t) + (3 H + Γ) δ̇φ̇_̇İ (k, t) +
    k^2/a^2δφ_I = ξ_I (k, t),

where ξ_I (k, t) is a stochastic noise source and different components of ξ_I (k,t) is independent of each other. From the equation above, we know there is no direct coupling between different components of field perturbations when dropping out metric
fluctuations on small scales. If the temperature is sufficiently high, the noise source is
Markovian <cit.>,

    ⟨ξ_I (k, t) ξ_J (- k', t') ⟩ = 2 Γ T a^- 3δ_IJδ^3 (k -
    k') δ (t - t').


Thermal noise is transferred to inflation field mostly on small scales, and as the wavelength of perturbations expands, the thermal effects decrease until the fluctuation amplitude freezes out.

At horizon-crossing, for T-dependent dissipative coefficients the thermal fluctuations produce a power spectrum of perturbation <cit.>

    𝒫_φ_I = k^- 3√(π)/2 H^1 / 2 (3 H + Γ)^1 / 2 T,


After horizon-crossing, we have to take into account the influence of metric
perturbations <cit.>. For simplicity, we will work in spatially-flat gauge, in which
E = ψ = 0. Since there are only two degrees of freedom of metric
perturbation, only two of the equations of Eqs.(<ref>)-(<ref>) are independent. Working with Eq. (<ref>) and Eq. (<ref>), we can get A and B in terms of other
perturbation variables by solving these two equations algebraically

    A = - δ q_r + Σ_I φ̇_̇İδφ_I/2 H,


    B = - a/4 k^2 H^2 (24 H^2 δ q_r 
    
    + (4 ρ_r + Σ_I φ̇_̇İ^2) (- δ q_r + Σ_I φ̇_̇İδφ_I) 
    
    + 2 H (3 δ̇ ̇q̇_̇ṙ + 4 ΓΣ_I φ̇_̇İδφ_I
    + Σ_I φ̈_̈Ïδφ_I - Σ_I φ̇_̇İδ̇φ̇_̇İ)).


In warm inflation, we usually treat the radiation as a perfect fluid, so the
above results can be applied here. For the radiation fluid, p_r = 1/3ρ_r, δ p_r = 1/3δρ_r, where p_r and ρ_r are the pressure and energy density of the radiation, and δ p_r, δρ_r are their perturbations respectively. With these relations, we can substitute Eq. (<ref>) into Eq. (<ref>) and yield:

    δ̈ ̈q̈_̈r̈ + 7 H δ̇ ̇q̇_̇ṙ + 3 ( 7 H^2 + Ḣ +
    1/3k^2/a^2) δ q_f + 
    Σ_I ( 1/3φ̇_̇İ^2 δΓ+ (4 H Γ+Γ̇) φ̇_̇İδφ_I +Γφ̈_̈Ïδφ_I +5/3Γφ̇_̇İδ̇φ̇_̇İ) 
    
     = 1/3k^2/aρ_r B - 16/3 H ρ_r A + 1/3ΓΣ_I
    φ̇_̇İ^2 A - 4/3ρ_r Ȧ.

The relationship between energy density ρ_r and temperature of radiation
is ρ_r = π^2/30 g_∗ T^4, where g_∗ is the effective
particle number of radiation. Now we define two new parameters describing the φ_I dependence and temperature dependence of the damping term Γ(Γ = Γ (φ_1, φ_2, …, φ_𝒩, ρ_r)):

    β_I = Γ_φ_I V_φ_I/Γ V,   
    c = T
    Γ_T/Γ = 4 ρ_r Γ_ρ_r/Γ,

where β_I < 1+r, β_I are slow-roll parameters <cit.>, but c is not required to be small. In
order to go back to cold inflation (Γ = 0) when T = 0, we require
Γ_T > 0, so c is positive defined. Considering the consistency of
warm inflation <cit.>, we set 0 < c < 4.

Then Γ_φ_I = ∂Γ/∂φ_I,
Γ_ρ_r = ∂Γ/∂ρ_r can be denoted by
β_I, c and the corresponding background quantities. So we have <cit.>

    δΓ   =   Σ_I Γ_φ_Iδφ_I +
    Γ_ρ_rδρ_r, 
    Γ̇   =   Σ_I Γ_φ_Iφ̇_̇İ + Γ_ρ_rρ̇_̇ṙ.


Substitute δΓ, Γ̇, A, B into Eqs. (<ref>) and (<ref>), and
keep the leading order, we find

    δ̈ ̈q̈_̈r̈ + (7 - c) H δ̇ ̇q̇_̇ṙ + ( 12 - 3 c + 1/3k^2/a^2 H^2) H^2 δ q_r 
    
    = γΣ_I (5 H δ̇ ̇q̇_̇İ + (12 - 3 c) H^2 δ q_I),


    δ̈ ̈q̈_̈Ï + 3 H (1 + γ) δ̇ ̇q̇_̇İ + k^2/a^2δ q_I
    = 0.

where γ = Γ/3 H describes the dissipative strength in warm inflation.

We define a new variable z = k/a H, then

    d/d t   =    - k/a (1 - ϵ) d/d z, 
    d^2/d t^2   =    (1 - ϵ) k H/ad/d z + (1 - 2 ϵ) k^2/a^2d^2/d z^2.


Replace time variable t with z, and keep the leading order, Eqs. (<ref>) and (<ref>) can be put in the form

    z^2 δ q_r” - (6 - c) z δ q_r' + ( 12 - 3 c + 1/3 z^2
    ) δ q_r 
    
    = γΣ_I (- 5 z δ q_I' + (12 - 3 c) δ q_I),


    z^δ q_I” - (2 + γ) δ q_I' + z δ q_I = 0,

where a prime denotes a derivative with respect to z.

From Eq. (<ref>) we know that in the large-scale limit, i.e., z = k/a H→
0, δ q_I is a constant in the slow-roll approximation. So we drop out
the δ q_I' term on the right-hand side of Eq. (<ref>).

    z^2 δ q_r” - (6 - c) z δ q_r' + ( 12 - 3 c + 1/3 z^2
    ) δ q_r 
    
    = γΣ_I ((12 - 3 c) δ q_I).


This is an inhomogeneous Bessel differential equation. The solution is given
by adding the homogeneous solution to a particular solution. The homogeneous
solution of (<ref>) can be found in terms of Bessel functions

    δ q_r^h = C_1 z^7 - c/2 J_ν̃(
    z/ √(3)) + C_2 z^7 - c/2 Y_ν̃( z/ √(3)),

where ν̃=(- 1+ c)/2, and C_1, C_2 are two integral constants.

Then, we try to find a particular solution for this equation. Because z
→ 0 rapidly after horizon-crossing, we drop out the z^2 term in
the coefficient of δ q_r. In this case, a particular solution is given
by

    δ q_r^p ≈γΣ_I ((12 - 3 c) δ q_I)/12 - 3 c =
    γΣ_I δ q_I.


Then we can get the general solution of Eq. (<ref>)

    δ q_r = C_1 z^7 - c/2 J_ν̃(
    z/ √(3)) + C_2 z^7 - c/2 Y_ν̃( z/ √(3)) + γΣ_I δ q_I.


In case of z ≪ 1, the Bessel functions in the above equation can be
approximated by

    z^7 - c/2 J_ν̃(
    z/ √(3)) ∼12^1 - c/4/Γ_R ( 1 + c/2) z^3,


    z^7 - c/2 Y_ν̃(
    z/ √(3)) ∼ - 1/π 12^- 1 + c/4Γ_R ( - 1 + c/2 ) z^4 - c
    
    - 1/π 12^1 - c/4Γ_R (
    1 - c/2) sin( c π/2) z^3,

where Γ_R is the Gamma function. From the above approximation we
know these two terms tend to zero rapidly after horizon-crossing, so

    δ q_r ≈γΣ_I δ q_I = - γΣ_I φ̇_̇İδφ_I.


From Eq. (<ref>) we know 4/3ρ_r = γΣ_I φ̇_̇İ^2, so in spatially-flat gauge the comoving curvature perturbation is given by <cit.>

    ℛ = - H δ q/p + ρ = - H Σ_I δ q_I + δ
    q_r/Σ_I φ̇_̇İ^2 + 4/3ρ_r = - H Σ_I
    δ q_I/Σ_I φ̇_̇İ^2.


Curvature perturbation ℛ has a same form to that in cold inflation. In one field case, ℛ
reduces to our familiar form ℛ = H δφ/φ̇.

Now, we consider a two-field model, φ_1 = ϕ, φ_2 =
χ. In this case, the perturbation equation of scalar field is given by

    δ̈ϕ̈ + (3 H + Γ) δ̇ϕ̇ + ( V_ϕϕ
    + k^2/a^2) δϕ + V_ϕχδχ + ϕ̇δΓ
    
    = ϕ̇( - k^2/a B + Ȧ), 
    δ̈χ̈ + (3 H + Γ) δ̇χ̇ + ( V_χχ
    + k^2/a^2) δχ + V_ϕχδϕ + χ̇δΓ
    
    = χ̇( - k^2/a B + Ȧ).


Substituting Eq. (<ref>) into Eqs. (<ref>) and (<ref>), we can express the metric perturbation A and B in terms of field perturbation δϕ, δχ. In previous section, we have expressed δΓ in terms of field perturbations, therefore now we get two closed differential equations for the variables δϕ, δχ after replacing A and B in above two equations with these results.

As in cold two-field inflation, we define two new adiabatic field σ
and entropy field s by a rotation in field space. d σ is tangent to
the background trajectory and d s is normal to it <cit.>.

    ( [ d σ; d s ])=( [  cosθ  sinθ; -sinθ  cosθ ]) ( [ d ϕ; d χ ]),

where cosθ = ϕ̇/√(ϕ̇^2 + χ̇^2),
sinθ = χ̇/√(ϕ̇^2 + χ̇^2)

Using this definition, the equation of motion can be described in terms of
σ, s is given by

    σ̈ + (3 H + Γ) σ̇ + V_σ = 0,


    θ̇σ̇ + V_s = 0,

where V_σ = cosθ V_ϕ + sinθ V_χ, V_s = - sinθ V_ϕ + cosθ V_χ.







Similarly, it is useful to decompose the field perturbations into an adiabatic
δσ and entropy δ s component as illustrated in Fig. <ref>, δσ is parallel to the background trajectory and δ
s is orthogonal to it.

    ( [  δσ; δ s ])=( [  cosθ  sinθ; -sinθ  cosθ ]) ( [ δϕ; δχ ]).


Now we can write Eqs. (<ref>) and (<ref>) in the form

    δ̈σ̈ + (3 H + Γ) δ̇σ̇ + (
    k^2/a^2 + V_σσ - θ̇^2 ) δσ - 2 θ̇δ̇ ̇ṡ
    
     + 2 ( 2 θ̇
    V_σ/σ̇ - θ̈) δ s + σ̇δΓ
    
     = - k^2/aσ̇ B - (σ̇Γ + 2
    V_σ) A + σ̇Ȧ,


    δ̈ ̈s̈ + (3 H + Γ) δ̇ ̇ṡ + ( k^2/a^2 +
    V_ss - θ̇^2 ) δ s 
    
    +2 θ̇δ̇σ̇
    - 2 θ̇σ̈/σ̇δσ = 2
    θ̇σ̇ A,

where

    V_σσ   =   cos^2 θ V_ϕϕ + sin 2 θ V_ϕχ + sin^2 θ V_χχ, 
    
    V_s s   =   sin^2 θ V_ϕϕ - sin 2 θ V_ϕχ + cos^2 θ V_χχ.


Since Γ = Γ (ϕ, χ, ρ_r) in previous section, now we can
treat Γ as Γ = Γ (σ, s, ρ_r). For simplicity, we
redefine some slow-roll parameters

    ϵ_σ = σ̇^2/2 H^2,   η_σ = -
    σ̈/H σ̇,   η_r = - 1/2ρ̇_̇ṙ/H ρ_r, 
    β_σ = V_σΓ_σ/V Γ,   β_s = V_σΓ_s/V Γ.

Then, the metric perturbations A, B in Eqs. (<ref>), (<ref>) and δΓ = Γ_σδσ + Γ_s δ s + Γ_ρ_rδρ_r can be expressed in term of δσ, δ s (According to Eqs. (<ref>), (<ref>) and (<ref>), we can rewrite δρ_r in terms of δσ,
δ s in spatially-flat gauge).

Substituting δΓ, A, B into Eqs. (<ref>) and (<ref>), we get

    δ̈σ̈ + ( 3 H + Γ + Γσ̇^2/3
    H^2 + Γσ̇^2 Γ_ρ_r/H) δ̇σ̇ +  ( k^2/a^2- θ̇^2 - 3 σ̇^2 - . 
    . 3 Γσ̇^2/2 H -Γ^2 σ̇^2/6 H^2+ V_σσ -ḢΓσ̇^2 Γ_ρ_r/H^2-2 ρ_r σ̇^2 Γ_ρ_r/H -2 Γρ_r σ̇^2 Γ_ρ_r/3 H^2. 
    . + Γσ̇σ̈Γ_ρ_r/H + ρ̇_̇ṙσ̇^2 Γ_ρ_r^2/H + σ̇Γ_σ + σ̇^3 Γ_σ/3 H^2 + σ̇^3 Γ_ρ_rΓ_σ/H ) δσ
    
      = 2 θ̇δ̇ ̇ṡ + ( θ̇σ̇^2/H - 2 θ̇ V_σ/σ̇ + 2 θ̈ - σ̇Γ_s ) δ s,

 
    δ̈ ̈s̈ + (3 H + Γ) δ̇ ̇ṡ + ( k^2/a^2 +
    V_ss - θ̇^2 ) δ s 
    
    = - 2 θ̇δ̇σ̇ + ( (3 H + Γ) σ̇^2/3 H^2 + 2
    σ̈/σ̇) θ̇δσ.


The comoving curvature perturbation is given by

    ℛ = H ϕ̇δϕ + χ̇δχ/ϕ̇^2 + χ̇^2 = H δσ/σ̇.

Replacing δσ in Eq. (<ref>) with ℛ, we can rewrite Eqs. (<ref>) and (<ref>) as two coupled differential equations of ℛ, δ s. If we keep only the leading order in the slow-roll approximation, the equations are given by

    ℛ̈ + 3 (1 + γ + c γ) H ℛ̇ + ( k^2/a^2 -
    θ̇^2  ) ℛ
    
    = 2 H θ̇/σ̇δ̇ ̇ṡ
    +  ( 2 H θ̇/σ̇ (3 + 3 γ) H + 2 H θ̈/σ̇ ) δ s,


    δ̈ ̈s̈ + 3 (1 + γ) H δ̇ ̇ṡ + ( k^2/a^2 + M_ eff^2 ) δ s = -
    2 θ̇σ̇/Hℛ̇,

where M_ eff^2=V_s s- θ̇^2, M_ eff is the effective mass of δ s.

According to Eqs. (<ref>) and (<ref>), we know when we neglect the curvature of background trajectory in field space (θ̇ = 0), δ s behave like a free field, and when r=0, these equations can go back to cold inflation.

Now we define the isocurvature perturbation 𝒮 = H/σ̇δ s <cit.>,
and the power spectrum of curvature and isocurvature perturbation <cit.>

    𝒫_ℛ   =   k^3/2 π^2| ℛ|^2, 
    𝒫_𝒮   =   k^3/2 π^2| 𝒮|^2.

The spectral index of the curvature perturbation is

    n_s-1 = d ln𝒫_ℛ/d ln k.


The tensor modes of perturbations are not affected by the thermal noises, so the tensor power spectrum and tensor-to-scalar ratio at the pivot scale are given by <cit.>

    𝒫_T    =    8 ( H_*/2 π)^2, 
    
    r    =   𝒫_ℛ/𝒫_T.


In this work, we are mainly concerned with the large scale evolution of curvature perturbation ℛ, because the value of ℛ at the end of inflation seeds the observed CMB temperature anisotropies, corresponding to the variance of inhomogeneities' distribution.



§ NUMERICAL EXAMPLES AND CONSTRAINTS FROM OBSERVATIONS




 §.§  Numerical Examples


When dealing with multi-component systems, a numerical method is almost essential. In this section we use the formalism introduced above to investigate a toy model, in which massive scalar fields ϕ and χ are coupled through an interaction term 1/2 g^2 ϕ^2 χ^2 <cit.>.

    V (ϕ, χ) = 1/2 m_ϕ^2 ϕ^2 + 1/2 m_χ^2
    χ^2 + 1/2 g^2 ϕ^2 χ^2.


The background equations are:

    ϕ̈+(3H + Γ)ϕ̇ + ∂ V(ϕ,χ)/∂φ=0 
    χ̈+(3H + Γ)χ̇ +∂ V(ϕ,χ)/∂χ = 0 
    ρ̇_̇ṙ+4 H ρ_r = Γ (ϕ̇^2 + χ̇^2)
         H^2 = 1/3 ( 1/2ϕ̇^2 + 1/2χ̇^2 + V(ϕ,χ) + ρ_r  )


There are five free parameters associated with the initial conditions of the
equation of motion, ϕ_0, χ_0, ϕ̇_̇0̇, χ̇_̇0̇, ρ_r 0. After making use of slow-roll approximation, ϕ̇ = - 1/3 H + Γ∂ V(ϕ,χ)/∂ϕ,
χ̇ = - 1/3 H + Γ∂ V(ϕ,χ)/∂χ, ρ_r = Γ/4H (ϕ̇^2 + χ̇^2), the initial conditions
are given by ϕ_0, χ_0. We choose the parameters associated with
potential to be m_ϕ = 2 × 10^- 7, m_χ = 10^- 6, g = 2
× 10^- 8, and give a numerical result below.

In our numerical calculations, we set γ = Γ/3 H to be a constant. In order to get a clear picture of the evolution of background and
perturbation variables, we integrate the exact background equations (<ref>)-(<ref>) first until horizon-crossing. After horizon-crossing, we integrate the background and perturbation equations (<ref>)-(<ref>) simultaneously to the end of inflation. We set the effective particle number of radiation g_∗ = 228.75 <cit.>, and choose the initial values of curvature perturbation ℛ and entropy perturbation δ s at horizon-crossing according to Eq. (<ref>). In addition, we take the number of e-foldings from horizon-crossing to the end of inflation to be Δ N =60 to make definite calculations.


According to the top left panel of Fig. <ref> we know that the heavy field χ decrease faster than the light field ϕ, and after a period of time, χ reaches zero and then inflation will be driven by one single field ϕ. the bottom two graphs of Fig. <ref> show that the potential dominates the total energy in inflationary period, which is consistent with slow-roll condition. However, radiation density will increase rapidly at the end of inflation and then become dominated, at the same time slow-roll conditions break down.

The left panel of Fig. <ref> shows after horizon-crossing k/a H→ 0 quickly and ℛ, δ s tend to a constant value and they are weakly correlated. From Eqs. (<ref>) and (<ref>) we know θ̇ plays an important role in the interaction of curvature perturbation ℛ and entropy perturbation δ s. The upper left panel of Fig. <ref> shows at around N = 80 the heavy field χ decays to zero and this will cause a bump in θ̇. At around N = 80, θ̇ increases suddenly and there is a strong interaction between ℛ and δ s, and they all change significantly. After that, the entropy perturbation decay to zero and the curvature perturbation ℛ become a nearly constant value again.




















 §.§  Constraints from Observations


The most recent measurements of the cosmic microwave background (CMB) provides narrow constraints on cosmological parameters, ruling out large classes of models. Having established representative examples in previous subsection, we now turn our attention to the compatibility with observational data. We will use the most recent Plank data to constrain our models, finding the allowed regions of parameter space consistent with the observational values of n_s and r. For simplicity, we neglect the interaction between ϕ and χ (set g=0), and introduce the mass ratio defined as R_m = m_χ / m_ϕ.
When we fix the value of R_m, for a given (γ, m_ϕ) when Γ∝ H, every set of initial condition (ϕ_∗, χ_∗) will produce a corresponding e-folding N, and a corresponding curvature power spectrum 𝒫_ℛ
at the end of inflation (ϵ = 1), just like the numerical examples shown in last subsection. Then we use the condition N = 60 and 𝒫_ℛ = 2.207 × 10^- 9(68%CL, Plank
TT,TE,EE+LowP) to constrain the parameter space and pick out the exact initial condition (ϕ_∗, χ_∗) for each value of (γ, m_ϕ). Next we give a numerical result of (r,n_s) at the end of inflation with the initial condition we obtained. That is to say, we have a set of (r, n_s) for every (γ, m_ϕ), so we can determine the range of our parameters in face of the observational results of (r, n_s).

We use the observational data n_s = 0.9645 ± 0.0049, r < 0.1 (68% CL, Planck TT,TE,EE+LowP) to constrain our models in case of R_m = 3 and R_m=10. Our results are given in parameter space of (γ, m_ϕ) and (r, n_s) plotted in Fig. <ref>. First, we neglect the points (in white areas) which do not have a corresponding initial condition of (ϕ_∗, χ_∗) to produce expected e-foldings and power spectrum at the same time. Then we proceed to search for the observationally allowed regions of parameters from the remaining points. In the plot, the light-gray shaded regions indicate areas for r > 0.1 while the dark-gray shaded regions indicate areas for r < 0.1. The regions highlighted in red are for 0.9596 < n_s < 0.9694, so the intersections of dark-gray shaded areas and red shaded areas give the regions of parameter space consistent with the observational data.








According to Fig. <ref>, we know all models in our analysis have some observational allowed regions in parameter plane. As the plots show, thermal fluctuations are much stronger than quantum fluctuations in warm inflation, so we need smaller masses of the scalar field to produce observationally allowed power spectrum. In upper two panels, we show when Γ∝ H, different R_m leads to different permitted range of dissipative strength γ, and inflation can happen in both weak and strong regime of warm inflation. In R_m = 3 case, γ takes values of 0.38 ≲γ≲ 1.0, and when R_m = 10, 0.4 ≲γ≲ 19. The lower two panels give the lower bound of tensor-to-scalar ratio r in the observationally allowed range of spectral index n_s, depending on the value of R_m. As illustrated, observational data favour large value of R_m in two-field cases. For R_m=3, r gets a lower bound r ≈ 0.068, this is a rather large value and may become disfavored by the observations in the near future. In case of R_m=10, we obtain a much smaller bound of r ≈ 0.007, which is in good agreement with the observational constraints.


§ CONCLUSIONS


In this paper, we have studied inflation driven by multiple scalar fields and
an interacting perfect fluid. We defined some new parameters and perform a full analysis of perturbation equations, including field perturbation, fluid perturbation and metric perturbation.  Then we apply the theory to warm inflation, and give the evolution equations of curvature perturbation ℛ and isocurvature perturbation δ s in a two-field case. Next, we perform numerical calculations in our representative examples and give the main features of the evolution of background and perturbation variables. Finally, in order to check the compatibility of our models with observations, we use the most up-to-date observational data to constrain our model and give the observational permitted regions of the parameters. In the calculation of perturbations, we have used the slow-roll approximation for simplicity.

According to the numerical results above, the correlation between curvature and entropy perturbations can change 𝒫_ℛ significantly on large scales, and the change mainly occurs simultaneously to the turning of the background trajectory. However, the change cannot be observed if it happens much more than 60 e-foldings before the end of inflation <cit.>, in which case the effects of multi-field are negligible. The damping term in background equation can slow the decrease of scalar fields, so warm inflation will produce more e-foldings after the turning of trajectory. Consequently, the multi-field effect is more likely to be observed when the damping effect is not too strong. Fortunately, the Planck satellite has put a tight upper bound on the primordial non-Gaussianity, and warm inflationary models tend to produce large non-Gaussianity, so the observations seems not to be compatible with the very strong version of warm inflation <cit.>.

Our results also show that the inflaton field may starts to oscillate after the end of inflation, just as the reheating phase in cold inflation. This is a common phenomenon in weak regime of warm inflation, and different forms of the damping coefficient can lead to different dynamical features of warm inflation. In fact, many models lie between warm inflation and standard inflation, and there may be some general framework to describe them <cit.>. The existence of radiation will not alter the main features of super-horizon evolution of scalar perturbations during slow-roll regime, compared with standard inflation, which is consistent with observations. However, the value of comoving curvature perturbation at the end of inflation depends on the form of damping coefficient Γ.

To study the compatibility with observations, we compared the predictions of our models with the most up-to data observational data, and show the results in Fig. <ref>. In all the cases considered, we find some observationally allowed regions in parameter space and the mass ration R_m has a significant impact on the allow regions of parameters. As illustrated by Fig. <ref>, the models with a smaller value of R_m tend to be more constrained by observational data. In our results, the observationally permitted range of the mass of the lighter field m_ϕ are all less than 5 × 10^- 8 M_p (M_p is the reduced Planck mass), which is much smaller than the double inflation models in cold inflation <cit.>. This is easy to understand because in warm inflation thermal fluctuations are much stronger than quantum fluctuation, and we do not need large m_ϕ to produce the expected value of scalar power spectrum. Note that if we set γ = 0 in our models, we go back to the cold inflation regime. However, according to upper two panels Fig. <ref> we know this case is not observational allowed for it predicts too large scalar-to-tensor ratio. Therefore, we can conclude that warm inflation effects can reduce the value of r in multi-field cases, making more inflationary models fit the observations.

In our numerical analysis, we take a T-independent dissipative coefficient as an example, which may not be a realistic case <cit.>. As shown in <cit.>, the T-dependent dissipative coefficient leads to a growing mode in the fluctuations before horizon-crossing in case of γ > 1 <cit.>. This is an important effect in warm inflation and has to be taken into account. In T-dependent cases the perturbations need to be computed numerically, and we leave this for our future work. The effective mass of isocurvature mode M_ eff in the models we studied here is large enough so that the isocurvature perturbation will decay to zero before the end of inflation. However, this is not always the case <cit.>, and further research should be done on this topic. Besides, when dealing with 𝒩≫ 1 scalar fields in inflationary models, the method of constructing random potentials is worth considering <cit.>.



§ ACKNOWLEDGEMENTS

This work was supported by the National Natural Science Foundation of China (Grants No. 11575270, No. 11175019, No. 11235003 and No.11605100).

unsrt
the inflationary universe A. H. Guth, Inflationary universe: A possible solution to the horizon and flatness problems, Phys. Rev. D 23, 347 (1981).
a new inflationary A. D. Linde, A new inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems, Phys. Lett. B 108, 389 (1982).
warm inflation and A. Berera, I. G. Moss and R. O. Ramos, Warm inflation and its microphysical basis, Rep. Prog. Phys. 72, 026901 (2009).
curvature and isocurvature  Z. Lalak, D. Langlois, S. Pokorski and K. Turzynski, Curvature and isocurvature perturbations in two-field inflation, J. Cosmol.Astropart.Phys. 07, 014 (2007).
cosmology S. Weinberg, Cosmology. (Oxford University Press,New York, 2008).
non-gaussianities in two-field F. Vernizzi and D. Wands, Non-Gaussianities in two-field inflation, J. Cosmol.Astropart.Phys. 05, 019 (2006).
oscillatory power spectrum R Kabir and A Mukherjee, Oscillatory power spectrum and strongly k-dependent r in hybrid inflation. arXiv:1602.01221.
adiabatic and entropy2 K. A. Malik and D. Wands, Adiabatic and entropy perturbations with interacting fluids and fields, J. Cosmol.Astropart.Phys. 02, 007 (2005).
cosmological perturbations in K. A. Malik, Cosmological perturbations in an inflationary universe. arXiv:0101563.
xpand: an algorithm C. Pitrou, X. Roy and O. Umeh, xPand: An algorithm for perturbing homogeneous cosmologies, Class. Quant. Grav. 30, 165002 (2013).
perturbations in cosmologies N. Bartolo, P. Corasaniti, A. Liddle, and M. Malquarti, Perturbations in cosmologies with a scalar field and a perfect fluid, Phys. Rev. D 70, 043532 (2004). arXiv:/0311503 [astro-ph].
non-gaussianity in fluctuations I. G. Moss and C. Xiong, Non-Gaussianity in fluctuations from warm inflation, J. Cosmol.Astropart.Phys. 04, 007 (2007).
scalar perturbation spectra L. M. H. Hall and I. G. Moss, Scalar perturbation spectra from warm inflation, Phys. Rev. D 69, 083525 (2004).
a relativistic calculation W. L. Lee and L. Z. Fang, A relativistic calculation of super-Hubble suppression of inflation with thermal dissipation, Class. Quant. Grav., 17, 4467 (2000).
density fluctuations from C. Graham and I. G. Moss, Density fluctuations from warm inflation, J. Cosmol.Astropart.Phys. 07, 013 (2009).
cosmological inflation and A. R. Liddle and D. H. Lyth, Cosmological Inflation and Large-Scale Structure. (Cambridge University Press,Cambridge,England, 2000).
power spectrum for R. O. Ramos and L. A. da Silva, Power spectrum for inflation models with quantum and thermal noises, J. Cosmol.Astropart.Phys. 03, 032 (2013).
consistency of the X. M. Zhang, J. Y. Zhu, Consistency of the tachyon warm inflationary universe models, J. Cosmol.Astropart.Phys. 02, 005 (2014).
consistency of the2 Z. P. Peng, J. N. Yu, X. M. Zhang and J. Y. Zhu, Consistency of warm k-inflation, Phys. Rev. D 94, 103531 (2016).
evolution of the T. Matsuda, Evolution of the curvature perturbations during warm inflation, J. Cosmol.Astropart.Phys. 06, 002 (2009).
adiabatic and entropy C. Gordon, D. Wands, B. A. Bassett and R. Maartens, Adiabatic and entropy perturbations from inflation, Phys. Rev. D 63, 023506 (2001).
spectra running and K. Y. Choi, L. M. H. Hall and C. van de Bruck, Spectral running and non-Gaussianity from slow-roll inflation in generalized two-field models, J. Cosmol.Astropart.Phys. 02, 029 (2007).
on the importance S. Cespedes, V. Atal, and G. A. Palma, On the importance of heavy fields during inflation, J. Cosmol. Astropart. Phys. 05, 008 (2012).
inflation and the A. Riotto, Inflation and the theory of cosmological perturbations. arXiv:0210162.
correlation-consistency cartography of S. Tsujikawa, D. Parkinson and B. A. Bassett, Correlation-consistency cartography of the double-inflation landscape, Phys. Rev. D 67, 083516 (2003).
observational constraints on L. Visinelli, Observational constraints on monomial warm inflation, J. Cosmol.Astropart.Phys. 07, 054 (2016).
computing observables in M. Dias, J. Frazer and D. Seery, Computing observables in curved multifield models of inflation–A guide (with code) to the transport method, J. Cosmol.Astropart.Phys. 12, 030 (2015).
extended warm inflation  J. M. F. Maia and J. A. S. Lima, Extended warm inflation, Phys. Rev. D 60, 101301 (1999).
double inflation and B. Feng and X. Zhang, Double inflation and the low CMB quadrupole, Phys. Lett. B 570, 145 (2003).
warm little inflation M. Bastero-Gil, A. Berera, R. O. Ramos and J. G. Rosa, Warm little inflaton, Phys. Rev. Lett. 117, 151301 (2016).
cosmological fluctuations of M. Bastero-Gil, A. Berera, I. G. Moss and R. O. Ramos, Cosmological fluctuations of a random field and radiation fluid, J. Cosmol.Astropart.Phys. 05, 004 (2014).
shear viscous effects M. Bastero-Gil, A. Berera, and R. O. Ramos, Shear viscous effects on the primordial power spectrum from warm inflation, J. Cosmol.Astropart.Phys. 07, 030 (2011).
isocurvature perturbations and I. Huston and A. J. Christopherson, Isocurvature perturbations and reheating in multi-field inflation, arXiv:1302.4298.
charting an inflationary M. C. D. Marsh, L. McAllister, E. Pajer, and T. Wrase, Charting an inflationary landscape with random matrix theory, J. Cosmol.Astropart.Phys. 11 040 (2013).




