Two-field Warm Inflation and Its Scalar Perturbations on Large Scales

Two-field Warm Inflation and Its Scalar Perturbations on Large Scales

Yang-Yang Wang wangyy@mail.bnu.edu.cn Department of Physics, Beijing Normal University, Beijing 100875, China    Xiao-Min Zhang zhangxm@mail.bnu.edu.cn School of Science, Qingdao University of Technology, Qingdao 266033, China    Jian-Yang Zhu zhujy@bnu.edu.cn Department of Physics, Beijing Normal University, Beijing 100875, China
Abstract

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 ). 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.

pacs:
98.80.Cq
thanks: Corresponding author

I 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 the inflationary universe (); a new inflationary (). 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 warm inflation and (). 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 . 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 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 curvature and isocurvature (); cosmology ().

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 , 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.

Ii 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

(1)

where is the scale factor, and is cosmic time. We use Planck Unit:

where is Newton’s gravitational constant, is Boltzmann’s constant, is the reduced Planck’s constant, and is the speed of light. In this work, Greek indices , , denote spacetime dimensions, and Latin indices , , denote different scalar fields. Repeated spacetime indices are summed over.

First we consider a -fields model with Lagrangian density non-gaussianities in two-field ():

(2)

which is minimal coupled to gravity, where . We assume that there exists a perfect fluid and the interaction between scalar fields and the fluid causes a phenomenological dissipative term in the equation of motion. In general, , is the energy density of the perfect fluid. From Eq. (2) we can get the equation of motion in the presence of a perfect fluid:

(3)

where is the Hubble parameter, , overdots represent derivatives with respect to cosmic time. In a spatially flat FRW universe, is determined by:

(4)

and the continuity equation of the perfect fluid

(5)

where , is the pressure of the fluid.

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

(6)

The slow-roll conditions are , . In slow-roll approximation, we have

(7)
(8)

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 . The line element of the FRW metric can be written as

(9)

and the gauge-invariant comoving curvature perturbation is given by oscillatory power spectrum (); adiabatic and entropy2 ()

(10)

where is the total momentum density perturbation, and and are total pressure and energy density cosmological perturbations in (). The momentum perturbations of each components are given by

(11)
(12)

where is the scalar velocity potential of the fluid, and from above definition we know . The four-velocity of the fluid is defined by

(13)
(14)

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

(15)

and the perturbation of energy and momentum conservation equation of the fluid is given by xpand: an algorithm ()

(16)
(17)

where is the wave number in Fourier space.

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

(18)

where

(19)
(20)

where is the Einstein tensor, and , are the energy-momentum tensor of scalar fields and perfect fluid respectively.

The perturbation equations of Einstein’s field equations are:

(21)
(22)
(23)
(24)

Eqs. (21)-(24) are, respectively, the component of the field equation, the component, the trace-free part of the and the component perturbations in cosmologies ().

Iii Application to warm inflation

In warm inflation, the density perturbations are mainly sourced by thermal noise non-gaussianity in fluctuations (), and metric fluctuations has little effect on small scales scalar perturbation spectra (); a relativistic calculation (). When , inflaton fluctuations are described by a Langevin equation warm inflation and ()

(25)

where is a stochastic noise source and different components of 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 scalar perturbation spectra (),

(26)

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 density fluctuations from ()

(27)

After horizon-crossing, we have to take into account the influence of metric perturbations cosmological inflation and (). For simplicity, we will work in spatially-flat gauge, in which . Since there are only two degrees of freedom of metric perturbation, only two of the equations of Eqs.(21)-(24) are independent. Working with Eq. (21) and Eq. (22), we can get and in terms of other perturbation variables by solving these two equations algebraically

(28)
(29)

In warm inflation, we usually treat the radiation as a perfect fluid, so the above results can be applied here. For the radiation fluid, , , where and are the pressure and energy density of the radiation, and are their perturbations respectively. With these relations, we can substitute Eq. (16) into Eq. (17) and yield:

(30)

The relationship between energy density and temperature of radiation is , where is the effective particle number of radiation. Now we define two new parameters describing the dependence and temperature dependence of the damping term ():

(31)

where , are slow-roll parameters power spectrum for (), but is not required to be small. In order to go back to cold inflation when , we require , so is positive defined. Considering the consistency of warm inflation density fluctuations from (); consistency of the (), we set .

Then , can be denoted by , and the corresponding background quantities. So we have consistency of the2 ()

(32)
(33)

Substitute into Eqs. (15) and (30), and keep the leading order, we find

(34)
(35)

where describes the dissipative strength in warm inflation.

We define a new variable , then

(36)
(37)

Replace time variable with , and keep the leading order, Eqs. (34) and (35) can be put in the form

(38)
(39)

where a prime denotes a derivative with respect to .

From Eq. (39) we know that in the large-scale limit, i.e., , is a constant in the slow-roll approximation. So we drop out the term on the right-hand side of Eq. (38).

(40)

This is an inhomogeneous Bessel differential equation. The solution is given by adding the homogeneous solution to a particular solution. The homogeneous solution of (40) can be found in terms of Bessel functions

(41)

where , and , are two integral constants.

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

(42)

Then we can get the general solution of Eq. (40)

(43)

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

(44)
(45)

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

(46)

From Eq. (8) we know , so in spatially-flat gauge the comoving curvature perturbation is given by evolution of the ()

(47)

Curvature perturbation has a same form to that in cold inflation. In one field case, reduces to our familiar form .

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

(48)
(49)

Substituting Eq. (46) into Eqs. (28) and (29), we can express the metric perturbation and in terms of field perturbation , . In previous section, we have expressed in terms of field perturbations, therefore now we get two closed differential equations for the variables , after replacing and in above two equations with these results.

As in cold two-field inflation, we define two new adiabatic field and entropy field by a rotation in field space. is tangent to the background trajectory and is normal to it adiabatic and entropy ().

(50)

where ,

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

(51)
(52)

where , .

Figure 1: An illustration of the decomposition of an arbitrary perturbation in field space. , represent fluctuations with respect to a fixed local frame, and , represent fluctuations parallel and normal to the background path.

Similarly, it is useful to decompose the field perturbations into an adiabatic and entropy component as illustrated in Fig. 1, is parallel to the background trajectory and is orthogonal to it.

(53)

Now we can write Eqs. (48) and (49) in the form

(54)
(55)

where

(56)
(57)

Since in previous section, now we can treat as . For simplicity, we redefine some slow-roll parameters

(58)

Then, the metric perturbations , in Eqs. (28), (29) and can be expressed in term of (According to Eqs. (17), (28) and (46), we can rewrite in terms of in spatially-flat gauge).

Substituting into Eqs. (54) and (55), we get

(59)
(60)

The comoving curvature perturbation is given by

(61)

Replacing in Eq. (59) with , we can rewrite Eqs. (59) and (60) as two coupled differential equations of , . If we keep only the leading order in the slow-roll approximation, the equations are given by

(62)
(63)

where , is the effective mass of .

According to Eqs. (62) and (63), we know when we neglect the curvature of background trajectory in field space (), behave like a free field, and when , these equations can go back to cold inflation.

Now we define the isocurvature perturbation spectra running and (), and the power spectrum of curvature and isocurvature perturbation on the importance ()

(64)
(65)

The spectral index of the curvature perturbation is

(66)

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 inflation and the ()

(67)
(68)

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

Iv Numerical Examples and Constraints from Observations

iv.1 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 and are coupled through an interaction term correlation-consistency cartography of ().

(69)

The background equations are:

(70)
(71)
(72)
(73)

There are five free parameters associated with the initial conditions of the equation of motion, , , , , . After making use of slow-roll approximation, , , , the initial conditions are given by , . We choose the parameters associated with potential to be , , , and give a numerical result below.

In our numerical calculations, we set 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 (70)-(73) first until horizon-crossing. After horizon-crossing, we integrate the background and perturbation equations (62)-(63) simultaneously to the end of inflation. We set the effective particle number of radiation observational constraints on (), and choose the initial values of curvature perturbation and entropy perturbation at horizon-crossing according to Eq. (27). In addition, we take the number of e-foldings from horizon-crossing to the end of inflation to be to make definite calculations.

According to the top left panel of Fig. 2 we know that the heavy field decrease faster than the light field , and after a period of time, reaches zero and then inflation will be driven by one single field . the bottom two graphs of Fig. 2 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. 3 shows after horizon-crossing quickly and , tend to a constant value and they are weakly correlated. From Eqs. (62) and (63) we know plays an important role in the interaction of curvature perturbation and entropy perturbation . The upper left panel of Fig. 2 shows at around the heavy field decays to zero and this will cause a bump in . At around , increases suddenly and there is a strong interaction between and , and they all change significantly. After that, the entropy perturbation decay to zero and the curvature perturbation become a nearly constant value again.

Figure 2: The evolution of different background variables are shown against e-foldings () in case of , , and we have set at the initial time. Top left and top right panel: the evolution of scalar fields and the slow-roll parameter , . Bottom left and bottom right panel show the evolution of kinetic energy density (), potential and radiation energy density , and all the energy is scaled by total energy density in the graph. In bottom right panel, we zoom in to the few e-foldings around the end of inflation to give more details, and the dot-dashed vertical line indicates the end of inflation (). In this case inflation ends at about .
Figure 3: The evolution of curvature perturbation , entropy perturbation and spectral index in case. The initial value of , are chosen based on Eq. (27). The spectral index is calculated using finite-difference method.

iv.2 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 and . For simplicity, we neglect the interaction between and (set ), and introduce the mass ratio defined as . When we fix the value of , for a given when , every set of initial condition will produce a corresponding e-folding , and a corresponding curvature power spectrum at the end of inflation (), just like the numerical examples shown in last subsection. Then we use the condition and (68%CL, Plank TT,TE,EE+LowP) to constrain the parameter space and pick out the exact initial condition for each value of . Next we give a numerical result of at the end of inflation with the initial condition we obtained. That is to say, we have a set of