Reheating mechanism of the curvaton with nonminimal derivative coupling to gravity

Abstract

In this paper, we continue our study of the curvaton model with nonminimal derivative coupling (NDC) to Einstein gravity proposed in our previous work (Feng in Phys Lett B 729:99, 2014; Feng and Qiu in Phys Rev D 90(12):123508, 2014), focusing on the reheating mechanism. We found that according to whether the curvaton has or has not dominated the background after the end of inflation, it will have two different behaviors of evolution, which should be the general property of the curvaton with nonminimal couplings. This will cause two different parts of reheating, which goes on via the parametric resonance process. The reheating temperature is estimated for both cases in which reheating completes before and after curvaton domination, and the constraints are quite loose compared to that of overproduction of gravitinos. Finally we investigated the evolution of curvature perturbation during reheating. We have shown both analytically and numerically that the curvature perturbation will not blow up during the resonance process.

1 Introduction

Fields in our universe may have various interactions with Einstein gravity, which are described by the non-minimal coupling terms in the action. Aside from the simplest direct ones, there can also be derivative couplings, e.g., $G_{\mu \nu }\partial ^\mu \phi \partial ^\nu \phi$, with $\phi$ denoting the scalar field and $G_{\mu \nu }$ the Einstein tensor, respectively. Being proposed by Amendola et al. in 1993 [1], it has lately been found that such a nonminimal derivative coupling (NDC) has the nice properties of the so-called “Horndeski theories” [2, 3], such as to keep the equation of motion second order. Therefore it has been paid more and more attention to in recent work; see [4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] for examples.

As an interesting application, in previous work [27, 28], we have proposed a curvaton model with such a NDC term. This term possesses a “self-modulation” mechanism that can phenomenologically make the power spectrum of perturbations scale-invariant independent of background evolutions. This is because in this case the kinetic term of the field couples directly to the geometric variables such as Hubble parameter, so that it can compensate the deviation of spacetime from de Sitter and make the field “feel” itself in the inflationary background, even if it actually is not. In [27], we calculated the background as well as both scalar and tensor perturbations generated by this model, and we provided several constraints on parameters considering its transfer to the curvature perturbations. In [28] we investigated the full types of non-Gaussianities up to the third order.

As a consequent work, in this paper we consider another region, namely the reheating process of this curvaton. Reheating is a very important region in the evolution of the universe, for it explains how particles and light elements can be generated, after the dilution of the inflation. The oldest discussions of reheating can be traced back to the last decades of last century; see [29,30,31,32] for original ideas and see [33,34,35,36] where the great breakthrough of parametric resonance mechanism was proposed for efficient reheating of single field inflation. As more and more models for early universe appear, more and more reheating mechanisms are also proposed, such as geometric reheating [37, 38], curvaton reheating [39], modulated reheating [40, 41], bounce reheating [42], and so on. Moreover, Recently there has been some discussions on the reheating mechanism for the inflaton with such non-minimal derivative coupling (NDC) terms [43,44,45,46,47,48,49]. These works raised a couple of questions and discussions as regards reheating in such models, which is interesting and related to the topic of this paper. For reviews of reheating mechanisms, see [50, 51].

In the following, we will investigate in detail the reheating process caused by the NDC curvaton model. For simplicity, we assume that the background is still given by inflation. We will mainly focus on the following questions:

• How will the curvaton evolve after the end of inflation?

• Can it give an efficient mechanism for particle creation; namely, does parametric resonance exist?

• How is the reheating temperature constrained?

• How will the curvature perturbation be affected by reheating process?

By our study, we wish to discover interesting properties of nonminimal coupling curvaton reheating and find the difference from other kinds of reheating processes, such as that caused by inflaton itself, or other minimal coupling curvatons.

Our paper is organized as follows: in Sect. 2 we briefly review the NDC curvaton model. In Sect. 3 we study the reheating process of our model. In Sect. 3.1 we show the background evolution both before and after curvaton domination, in Sect. 3.2 we study the process of parametric resonance, and in Sect. 3.3 we estimate the reheating temperature. In Sect. 4 we discuss the evolution of curvature perturbation in reheating process. Section 5 presents the conclusion.

2 The NDC curvaton model

Following the preceding work [27], one has the action including the NDC curvaton

$$\begin{aligned} \mathcal {S}=\int \mathrm {d}^4x\sqrt{g}\Big [\frac{R}{16\pi G}+\frac{\xi }{M^2}G_{\mu \nu }\partial ^\mu \varphi \partial ^\nu \varphi -V(\varphi )+\mathcal { L}_{\text {bg}}\Big ], \end{aligned}$$

(1)

where $\xi$ is the dimensionless coupling constant of the NDC term. $\mathcal { L}_{\text {bg}}$ is the Lagrangian of the background, which drives inflation, and we do not need to specify its detailed form. It is straightforward to write down the equation of motion for the curvaton field $\varphi$, such as

$$\begin{aligned} \frac{6\xi }{M^{2}}H^{2}\ddot{\varphi }+\frac{6\xi }{M^{2}}(2\dot{H}+3H^{2})H\dot{\varphi }+V_{\varphi }=0, \end{aligned}$$

(2)

and its energy density and pressure can be expressed as

$$\begin{aligned} \rho _\varphi= & {} \frac{9\xi }{M^2}H^2\dot{\varphi }^2+V(\varphi ), \end{aligned}$$

(3)

$$\begin{aligned} P_\varphi= & {} -\frac{\xi }{M^2}(3H^2\dot{\varphi }^2+2\dot{H}\dot{\varphi }^2+4H\dot{\varphi }\ddot{\varphi })-V(\varphi ), \end{aligned}$$

(4)

respectively.

The background evolution of the curvaton field with various types of potential as well as the linear perturbations during inflation has been classified and briefly analyzed in [27]. As has been shown there, if the curvaton field is massless, namely $V(\varphi )=0$, an exactly scale-invariant power spectrum will be obtained due to the nonminimally kinetic coupling. However, since exact scale invariance is not favored by today’s data, a non-zero potential is needed which could give a mass term to $\varphi$.¹ For the simplest choice, we choose

$$\begin{aligned} V(\varphi )=\frac{1}{2}m^2\varphi ^2~ \end{aligned}$$

(5)

as its potential. Then the equation of motion (2) reduces to

$$\begin{aligned} \ddot{\varphi }+(3-2\epsilon )H\dot{\varphi }+\left( \frac{mM}{\sqrt{6\xi }H}\right) ^2\varphi =0, \end{aligned}$$

(6)

where the slow-roll parameter $\epsilon$ is defined as $\epsilon \equiv -\dot{H}/H^2$. Moreover, one can define an effective mass of the curvaton field,

$$\begin{aligned} \bar{m}\equiv \frac{mM}{\sqrt{6\xi }H}. \end{aligned}$$

(7)

Moreover, according to [27], we have

$$\begin{aligned} \frac{a^2V_{\varphi \varphi }}{2z^2}=\frac{a^2m^2}{2z^2}=\frac{\Delta _1}{(\eta _*-\eta )^2}, \end{aligned}$$

(8)

where $z^2\simeq (3\xi /M^2)a^2H^2$ and $\Delta _1$ describes the deviation of spectral index from pure scale invariance, $\Delta _1\simeq 3(n_\mathrm{s}-1)/2$. From the central value of $n_\mathrm{s}$ given by the Planck observational constraint [52], one roughly has $\Delta _1\sim \mathcal { O}(0.01)$. Notice also that during inflation $aH\simeq -(\eta _*-\eta )^{-1}$; this furthermore gives

$$\begin{aligned} \bar{m}\sim 0.1H_{\text {inf}}~ \end{aligned}$$

(9)

during inflation.

3 Reheating mechanism

3.1 The behavior of the curvaton during oscillation

In order to investigate the reheating mechanism of the curvaton field $\varphi$, let us assume that after inflation ends, the background field (inflaton) decays rapidly without oscillation. A typical example is that the inflaton field is a canonical scalar field dominated by its kinetic energy, such as the so-called “quintessential inflation” [53]. Therefore one has

$$\begin{aligned} \rho _{\text {bg}}\propto a^{-6}(t),~~~w_{\text {bg}}=1. \end{aligned}$$

(10)

However, as the curvaton field has a potential (5), it will fall down and oscillate around its minimum, and if the energy density of the curvaton decreases slower than that of inflaton, the curvaton can dominate the universe, unless it reheats before domination. First of all, one can determine the averaged equation of state during its oscillation, which will be used in later analysis. To do this, one can parametrize the scale factor a(t) and Hubble parameter H(t) during the rapid deflation of the universe. One has

$$\begin{aligned} a(t)\sim t^p,~~H(t)=\frac{p}{t},~~p\equiv \frac{2}{3(1+w)}, \end{aligned}$$

(11)

and w should be equal to $w_{\text {bg}}$ before curvaton domination, and $w_{\varphi }$ after that. For convenience, we define $t_\mathrm{{eq}}$ as the time when the energy densities of inflaton and curvaton are equal. For the former case, $p=1/3$ as $w_{\text {bg}}=1$, while the latter case depends on the behavior of $\varphi$, which is the solution of the equation of motion (2). Considering (11), Eq. (2) can be solved:

$$\begin{aligned} \varphi =\Phi (t)\times \left\{ J_{\frac{3(1-p)}{4}}\left( \frac{mMt^2}{2\sqrt{6\xi }p}\right) ,~J_{\frac{-3(1-p)}{4}}\left( \frac{mMt^2}{2\sqrt{6\xi }p}\right) \right\} , \end{aligned}$$

(12)

where $\Phi (t)=\Phi _0 t^{(1-3p)/2}$ is the oscillation amplitude of $\varphi$ with the initial condition of $\Phi _0$, and J is the Bessel function. Note that in the reheating era where $\varphi$ begins to oscillate, one has $\bar{m}>H$, which roughly gives $\frac{mMt^2}{2\sqrt{6\xi }p}>1$. Thus the Bessel function can be mimicked by the trigonometric functions, namely

$$\begin{aligned} \varphi \approx \Phi (t)\cos \left( \frac{mMt^2}{2\sqrt{6\xi }p}\right) . \end{aligned}$$

(13)

Fig. 1

Plots of the value of $\varphi$ and its energy density $\rho _\varphi$ in terms of cosmic time t. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$

During rapid oscillation, the kinetic and potential terms in the energy density (3) are of the same order, therefore one roughly has

$$\begin{aligned} \rho _\varphi \sim \Phi (t)^2\propto t^{1-3p}. \end{aligned}$$

(14)

For $t<t_\mathrm{{eq}}$, $p=1/3$, so we have $\rho _\varphi \propto t^0\sim a^0$. This means that before curvaton domination, the energy density of the curvaton field will behave as a constant, giving its equation of motion to be $w_\varphi =-1$.

Fig. 2

Plots of the value of $\varphi$, $\dot{\varphi }$, the parameter p in Eq. (11) and $\dot{H}$ in terms of cosmic time t. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$

Since the background energy density is decreasing, the curvaton will exceed the background (at the time $t_\mathrm{{eq}}$) and dominate the universe. For $t>t_\mathrm{\mathrm{{eq}}}$, the same solution of (13) applied but with p be related to $w_{\varphi }$, so one cannot determine p, or the behavior of $\rho _\varphi$, solely by Eq. (14). However, since the curvaton dominates the universe, we have another equation, namely the Friedmann equation,

$$\begin{aligned} H^2=\frac{8\pi G}{3}\rho _\varphi . \end{aligned}$$

(15)

This equation, together with (11), tells us that $\rho _\varphi \propto t^{-2}$. Comparing it with Eq. (14), one gets $p=1$, which gives $w_\varphi =-2/3$.²

In order to confirm our analysis, we performed numerical calculations for the two cases. One can see that in Fig. 1, both the field value $\varphi$ and its energy density $\rho _\varphi$ oscillate with a stable amplitude, giving a constant averaged value. However, in Fig. 2, the field value oscillates with a damping amplitude. The Hubble parameter is therefore also oscillating, so the value of p is also an oscillating function with its averaged value $\langle p\rangle =\langle H\rangle t$. From the figure it is clearly seen that $\langle p\rangle$ is at the position of about unity. We also plot the velocities of $\varphi$ and H for later use.

As a side remark, we remind the reader that our result is different from that for minimal coupling curvatons [39]. In the latter case, if the curvaton oscillates (let us still take a mass-squared potential for the curvaton as an example), the solution of the curvaton will be $\varphi =t^{-3p/2}\cos (mt)$, and $\rho _\varphi \sim t^{-3p}\sim a^{-3}$, accordingly. One can see that, no matter what value p will be given, or no matter whether the curvaton is dominant or not, the curvaton will behave like ordinary matter, with $w_\varphi =0$ on average, as long as the mass-squared potential is applied. This means the energy density will evolve synchronically with the scale factor, in terms of t. This result is consistent with [39]. However, when the curvaton is nonminimally coupled, this synchronicity will be violated, and the behavior before and after domination will be different. As will be seen later, this difference will directly cause a different rate of particle creation. To the best of our knowledge, this has not been given much notice in previous studies.

3.2 Parametric resonance

In this section, we begin to discuss how the oscillating behavior of the curvaton $\varphi$, as shown above, can be responsible for creating particles (we denote this $\chi$), namely, the preheating process. One assumes that the free-field Lagrangian of $\chi$ has a canonical form, and that $\varphi$ can interact with $\chi$ through some interacting term, such as $g\varphi ^2\chi ^2$, with a dimensionless coefficient of g. It gives the total form of the Lagrangian of $\chi$ field:

$$\begin{aligned} \mathcal { L}_\chi =-\frac{1}{2}\partial _\mu \chi \partial ^\mu \chi -\frac{1}{2}m_\chi ^2-\frac{1}{2}g\varphi ^2\chi ^2. \end{aligned}$$

(16)

It is true that one can also have the interaction term of $\chi$ and the background field (the inflaton), however, here we drop this for simplicity. Actually, as the inflaton decays quickly after inflation ends, it will have much less effects on $\chi$ than $\varphi$. Since $\chi$ is a quantum field, one can write down the Heisenberg presentation of $\chi$ [33, 34]:

$$\begin{aligned} \chi (t,\overrightarrow{x})= & {} \frac{1}{(2\pi )^{3/2}}\int d^3k(\hat{a}_k\chi _k(t)\mathrm{e}^{-i\overrightarrow{k}\cdot \overrightarrow{x}}\nonumber \\&+\,\hat{a}_k^\dagger \chi _k^*(t)\mathrm{e}^{i\overrightarrow{k}\cdot \overrightarrow{x}}) \end{aligned}$$

(17)

where $\hat{a}_k$ and $\hat{a}_k^\dagger$ are the annihilation and creation operators, respectively, and k is the comoving wavenumber. Using this, one can also get the equation of motion for $\chi$ by simply varying the Lagrangian (16):

$$\begin{aligned} \ddot{\chi }_k+3H\dot{\chi }_k+\left( \frac{k^2}{a^2}+m_\chi ^2+g\varphi ^2\right) \chi _k=0. \end{aligned}$$

(18)

In the rest of this paper, we assume $m_\chi =0$ for simplicity.

Since in our case, $\varphi (t)$ behaves as in Eq. (13), in the static limit where $a=\text {const.}$, $H=0$, Eq. (18) will reduce to the Mathieu-like equation, the solution of which is the well-known Floquet solution [33, 34]. In an expanding phase such as inflation, however, one can define a new variable: $X_k(t)\equiv a^{3/2}(t)\chi _k$, and Eq. (18) can be rewritten as

$$\begin{aligned}&\ddot{X}_k+\omega _k^2X_k=0, \nonumber \\&\omega _k^2\equiv \frac{k^2}{a^2}+\frac{g\Phi _0^2}{t^{3p-1}}\cos ^2\left( \frac{mMt^2}{2\sqrt{6\xi }p}\right) -\frac{3}{2}\dot{H}-\frac{9}{4}H^2. \end{aligned}$$

(19)

One can apply the WKB approximation to get the solution of Eq. (19):

$$\begin{aligned} X_k(t)=\frac{\alpha _k(t)}{\sqrt{2\omega _k}}\mathrm{e}^{-i\int ^t\omega _k\mathrm{d}t}+\frac{\beta _k(t)}{\sqrt{2\omega _k}}\mathrm{e}^{i\int ^t\omega _k\mathrm{d}t}, \end{aligned}$$

(20)

with $\alpha _k$ and $\beta _k$ being the time-dependent coefficient of positive and negative frequency parts, respectively. Moreover, $\alpha _k$ and $\beta _k$ satisfy the equations

$$\begin{aligned} \dot{\alpha }_k=\frac{\dot{\omega }_k}{2\omega _k}\mathrm{e}^{2i\int ^t\omega _k\mathrm{d}t}\beta _k,~~~\dot{\beta }_k=\frac{\dot{\omega }_k}{2\omega _k}\mathrm{e}^{-2i\int ^t\omega _k\mathrm{d}t}\alpha _k~ \end{aligned}$$

(21)

with the additional normalization relation

$$\begin{aligned} |\alpha _k|^2-|\beta _k|^2=1. \end{aligned}$$

(22)

Furthermore, the comoving occupation number of the $\chi$ particles in the mode k is defined by

$$\begin{aligned} n_k\equiv \frac{\omega _k}{2}\left( \frac{|\dot{X}_k|^2}{\omega _k^2}+|X_k|^2\right) -\frac{1}{2}. \end{aligned}$$

(23)

Using Eqs. (20), (21) and (22), one gets a simple expression of $n_k=|\beta _k|^2$, and the vacuum expectation value of particle number density of $\chi$ per comoving volume is

$$\begin{aligned} \langle n_\chi \rangle \equiv \int _0^\infty \frac{\mathrm{d}^3k}{(2\pi a)^3}n_k=\frac{1}{2\pi ^2a^3}\int _0^\infty \mathrm{d}k k^2|\beta _k|^2. \end{aligned}$$

(24)

It is straightforward to find how the $\chi$ particles increase with time just by integrating $|\beta _k|^2$, which is the solution of Eq. (21). However, it is rather difficult, if not impossible, to have an analytical solution of (21) due to the complication of the dispersion relation in (20). In order to solve the problem, Refs. [33, 34] introduced a method, which is to consider the solution of each period around the time point at which the source field of particle creation is zero, namely $t_j$, $j=1,2,3,\ldots$ where $\varphi (t_j)=0$ in our case. This is because only around $t_j$ the particle number can be dramatically changed. If we Taylor expand the function $g\varphi ^2(t)$ with $\varphi$ given in (13) around $t_j$, we get

$$\begin{aligned} g\varphi ^2(t)=g\Phi _0^2\frac{m^2M^2t_j^{3(1-p)}}{6p^2\xi }(t-t_j)^2+\mathcal { O}(t-t_j)^3. \end{aligned}$$

(25)

So the leading order of $g\varphi ^2(t)$ gives a parabolic potential for $\chi$, and the particle creation at $t_j$ could be viewed as the scattering of $\chi$ particles through this potential for the jth time. We assume the solution of $X_k$ at the period of $t_{j-1}<t<t_j$ is

$$\begin{aligned} X^j_k(t)=\frac{\alpha _k^j}{\sqrt{2\omega _k}}\mathrm{e}^{-i\int ^t\omega _k\mathrm{d}t}+\frac{\beta _k^j}{\sqrt{2\omega _k}}\mathrm{e}^{-i\int ^t\omega _k\mathrm{d}t}, \end{aligned}$$

(26)

with $\alpha _k^j$ and $\beta _k^j$ approximately constant. Following [33, 34] and noticing the relation of $n_k=|\beta |^2$, one can get a recursive relation of $n_k^j$ and $n_k^{j+1}$:

$$\begin{aligned} n_k^{j+1}=n_k^j\mathrm{e}^{2\pi \mu _k^j} \end{aligned}$$

(27)

with $\mu _k^j\equiv (2\pi )^{-1}\ln (1+2\mathrm{e}^{-\pi \kappa ^2}+\ldots )$, where the ellipsis denotes the random (stochastic) terms. Here $\kappa \equiv k/(ak_*)$ and $k_*=\sqrt{\sqrt{g/(6\xi )}mMt_j^{3(1-p)/2}/p}$. In [33, 34], $k_*$ is independent of $t_j$, but as mentioned before, in our case there are two differences. One is that p does not have the same value before and after curvaton domination, and the other is that the scaling of $\varphi$ on $t_j$ is different from the minimal coupling case, as was shown in (13). This causes $k_*\propto t_j^{1/2}$ for $t_j<t_\mathrm{{eq}}$, while $k_*\propto t_j^0$ for $t>t_\mathrm{{eq}}$. Considering also the dependence of a on t, namely $a\propto t^{1/3}$ before $t_\mathrm{{eq}}$ and $a\propto t$ after, one gets $\kappa \propto t^{-5/6}$ before $t_\mathrm{{eq}}$ and $\kappa \propto t^{-1}$ after. Therefore one gets

$$\begin{aligned} \begin{aligned} \mu _k^j&\equiv \frac{1}{2\pi }\ln \left( 1+2\mathrm{e}^{-\frac{\pi }{t^{5/3}}}+\cdots \right) ~{\text{ with }}~\kappa \propto t^{-5/6},\\ \mu _k^j&\equiv \frac{1}{2\pi }\ln \left( 1+2\mathrm{e}^{-\frac{\pi }{t^{2}}}+\cdots \right) ~{\text{ with }}~\kappa \propto t^{-1}. \end{aligned} \end{aligned}$$

(28)

In the limit of $\kappa ^2\ll \pi ^{-1}$, $\mu _k$ reduces to the trivial case in [33, 34]: $\mu _k^j\simeq (2\pi )^{-1}\ln {3}\simeq 0.18$ where we neglected the random terms.

Fig. 3

The numerical plot of $X_k$ and $\ln n_k$ in terms of cosmic time t before curvaton domination. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$, $g=4.8\times 10^{-3}$

Fig. 4

The numerical plot of $X_k$ and $\ln n_k$ in terms of the cosmic time t after curvaton domination. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$, $g=4.8\times 10^{-3}$

The total number during the whole process of resonance yields

$$\begin{aligned} n_k=\frac{1}{2}\mathrm{e}^{2\pi \sum _j\mu _k^j}\simeq \mathrm{e}^{2m\int ^t\mathrm{d}t\mu _k(t)}, \end{aligned}$$

(29)

in mode k. In principle one can get $n_k$ by integrating $\mu _k$ according to the expressions (28), however, unfortunately it might not be analytical, so we will refer to numerical calculations later on. The particle number density in real space is

$$\begin{aligned} \langle n_\chi \rangle= & {} \frac{1}{2\pi ^2a^3}\int _0^\infty \mathrm{d}k k^2n_k \nonumber \\\sim & {} \frac{k_*^3\mathrm{e}^{2\mu mt}}{64\pi ^2a^3\sqrt{\pi \mu mt}}, \end{aligned}$$

(30)

while the increasing rate of the particle number in the comoving volume is roughly estimated to be

$$\begin{aligned} \frac{\mathrm{d}\left( a^3\langle n_\chi \rangle \right) }{\mathrm{d}t}\sim \left\{ \begin{array}{l} \left( \frac{3g\Phi _0^2m^2M^2}{2\xi }\right) ^{\frac{3}{4}}\frac{\mathrm{e}^{2\mu m t}(2\mu mt+1)}{64\pi ^{5/2}\sqrt{\mu m}}~\mathrm{for}~t<t_\mathrm{{eq}},\\ \left( \frac{g\Phi _0^2m^2M^2}{6\xi }\right) ^{\frac{3}{4}}\frac{\mathrm{e}^{2\mu m t}(4\mu mt-1)}{128\pi ^{5/2}t\sqrt{\mu m t}}~\mathrm{for}~t>t_\mathrm{{eq}}. \end{array}\right. \nonumber \\ \end{aligned}$$

(31)

In Figs. 3 and 4 we numerically calculated the creation of particles before and after curvaton domination, and we plot the values of $X_k$ and $\ln n_k$ in terms of t. The numerical results are in good agreement with the theoretical analysis. In our plot we can see that in both cases $X_k$ oscillates rapidly due to the parametric resonance, and with the amplitude getting higher and higher. The number density $n_k$ (in logarithm) of both cases increases rapidly, though in some instances it decreases, due to the stochastic random process. Nevertheless, one can see that there are quantitative differences in the shapes as well as the increasing rates of $X_k$ and $n_k$ in the two cases, which is due to the different behavior of the source field $\varphi$ before and after curvaton domination.

As a side remark, we roughly estimate the time when back-reaction of the produced particle $\chi$ becomes important, which can prevent a further reheating process. To see this, one could define the energy density of the $\chi$ by

$$\begin{aligned} \rho _\chi \approx \langle n_\chi \rangle m_\chi , \end{aligned}$$

(32)

where $\langle n_\chi \rangle$ is calculated in (30), while the effective mass of the $\chi$ particle is

$$\begin{aligned} m_\chi \approx \sqrt{\frac{g\Phi _0^2}{t^{3p-1}}\cos ^2\left( \frac{mMt^2}{2\sqrt{6\xi }p}\right) -\frac{3}{2}\dot{H}-\frac{9}{4}H^2} \end{aligned}$$

(33)

from Eq. (19). The condition of back-reaction becoming important is $\rho _\chi \approx \rho _\varphi$, where $\rho _\varphi \approx m^2\varphi ^2$. Since it is difficult to have an analytical solution of the time t, we perform a numerical calculation, and plot both $\rho _\chi$ and $\rho _\varphi$ (after neglecting the oscillation/resonance effect) in Fig. 5. From the plot we can see that the time when back-reaction becomes important is far after the curvaton domination.

Fig. 5

The numerical plot of $\rho _\chi$ and $\rho _\varphi$ in terms of the cosmic time t. The time point when the two lines cross over corresponds to the time when the back-reaction becomes important. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$, $g=4.8\times 10^{-3}$

3.3 Constraint on reheating temperature

When reheating process completed, the curvaton will decay into the relativistic products, which scales as $T^4$ where T is the temperature. Therefore the reheating temperature will be related to the final state of the curvaton field. Knowing the initial state of the curvaton field at where inflation had just ended, and the scaling of the curvaton field during reheating, one can estimate the reheating temperature and compare it with various data constraints. This procedure is initially done for the normal curvaton mechanism given in [39]. Now we also use this method to estimate the reheating temperature in our model.

The curvaton may reheat the universe in two ways. Since we know from the above that the energy density of the curvaton scales as $a^0$ before domination, we have

$$\begin{aligned} \rho _{\varphi }^{\text {osc}}\simeq \rho _{\varphi }^\mathrm{{eq}}, \end{aligned}$$

(34)

where the subscripts osc and eq denote the value when $\varphi$ begins to oscillate and when it has the same amount of energy density as the background, respectively. On the other hand, at the time when reheating completes we have

$$\begin{aligned} \rho _{\varphi }^{\text {rh}}=\frac{g\pi ^2}{30}T_{\text {rh}}^4. \end{aligned}$$

(35)

Therefore, if the curvaton reheats after its domination, then we have $\rho _{\varphi }^\mathrm{{eq}}>\rho _{\varphi }^{\text {rh}}$, and the amount of difference depends on how long reheating will last after curvaton domination. From Eqs. (34) and (35) we have

$$\begin{aligned} T_{\text {rh}}<\left( \frac{30}{g\pi ^2}\rho _{\varphi }^{\text {osc}}\right) ^\frac{1}{4}. \end{aligned}$$

(36)

At the time when $\varphi$ ends slow-rolling and begins oscillating, the kinetic and potential term of $\varphi$ will be of the same order, namely $9\xi H^2\dot{\varphi }^2/M^2\sim m^2\varphi ^2/2$, therefore

$$\begin{aligned} \rho _{\varphi }^{\text {osc}}\simeq \frac{18\xi }{M^2}H^2\dot{\varphi }^2=18H_{\text {osc}}^2y_{\text {osc}}, \end{aligned}$$

(37)

where $y_{\text {osc}}$ is the value of $y (\equiv \xi \dot{\varphi }^2/M^2)$ at the oscillation time. So the reheating temperature is

$$\begin{aligned} T_{\text {rh}}<\left( \frac{540H_{\text {osc}}^2y_{\text {osc}}}{g\pi ^{2}}\right) ^{\frac{1}{4}}. \end{aligned}$$

(38)

The constraints on the Hubble parameter and y can be obtained from the Planck constraints on the scalar spectrum: $|\mathcal { P}_\zeta |\simeq 2.2065^{+0.0763}_{-0.0738}\times 10^{-9}$ (TT, TE, EE+lowP, $68\%$ CL) and the tensor/scalar ratio: $r\lesssim 0.10$ (TT, TE, EE+lowP, $95\%$ CL) [52]. In our model, from [27] we have

$$\begin{aligned} \mathcal {P}_\zeta \simeq \frac{3\sqrt{3}H_*^{2}\epsilon _*^2}{196\sqrt{7}\pi ^{2}|y_*|},~~~\mathcal { P}_T\simeq \frac{2H_*^2}{\pi ^2M_p^2} \end{aligned}$$

(39)

for this case, where $*$ denotes the value evaluated at horizon crossing. These first give the constraints:

$$\begin{aligned} \frac{\epsilon _*^2M_p^2}{|y_*|}\gtrsim 2\times 10^3,~~~H_*\lesssim 10^{-5}\pi M_p, \end{aligned}$$

(40)

from which one can see, if we set $\epsilon _*\sim \mathcal { O}(10^{-2})$, that $y_*$ is as low as $10^{-7}M_p^2$. As y and H vary slowly during inflation, one can roughly write $H_*\simeq H_{\text {osc}}$, $y_*\simeq y_{\text {osc}}$; therefore the reheating temperature can be set to

$$\begin{aligned} T_{\text {rh}}\lesssim & {} 0.72\times 10^{-4}\sqrt{\left( \frac{\epsilon }{0.01}\right) }M_p \nonumber \\\approx & {} 0.879\times 10^{15}\sqrt{\left( \frac{\epsilon }{0.01}\right) }\text {GeV}, \end{aligned}$$

(41)

which actually gives a very loose constraint on the reheating temperature, comparing with that from the gravitino producing [54,55,56,57].

On the other hand, if the curvaton decays and reheats before dominating the universe, one has $\rho _{\varphi }^{\text {rh}}\simeq \rho _{\varphi }^{\text {osc}}\simeq \rho _{\varphi }^\mathrm{{eq}}$. From Eqs. (35), (37), and (38) we have

$$\begin{aligned} T_{\text {rh}}\simeq \left( \frac{540H_{\text {osc}}^2y_{\text {osc}}}{g\pi ^{2}}\right) ^{\frac{1}{4}}. \end{aligned}$$

(42)

For a perturbation in this case, we have

$$\begin{aligned} \mathcal {P}_\zeta \simeq \frac{3\sqrt{3}H_*^{2}\epsilon _*^2r^2}{784\sqrt{7}\pi ^{2}|y_*|},~~~\mathcal { P}_T\simeq \frac{2H_*^2}{\pi ^2M_p^2}, \end{aligned}$$

(43)

where $r\equiv \rho _\varphi /\rho _r$ is the ratio of energy densities of $\varphi$ field and radiation. From the Planck constraints, we have

$$\begin{aligned} \frac{\epsilon _*^2M_p^2}{|y_*|}\gtrsim \frac{8\times 10^3}{r^2},~~~H_*\lesssim 10^{-5}\pi M_p, \end{aligned}$$

(44)

so we have

$$\begin{aligned} T_{\text {rh}}\simeq 0.623\times 10^{15}\sqrt{\frac{\epsilon r}{0.01}}\text {GeV}, \end{aligned}$$

(45)

which gives a high reheating temperature. So if the constraints on gravitino are trustworthy, then this case will be ruled out.³ Note that recently, in Ref. [58], the authors obtained a more stringent upper bound for the reheating temperature of $T_{\text {rh}}<5\times 10^{4}$ GeV for minimal coupling curvaton models by a Bayesian inference method.⁴

Moreover, one can also constrain some of the parameters for the model, using that on the Hubble parameter. From Eq. (9), one has $H_*\sim \sqrt{10mM/\sqrt{6\xi }}$ during inflation, while $H_{\text {osc}}\simeq (m^2M^2/6\xi )^{1/4}$ when $\varphi$ begins to oscillate, which is given by $H_{\text {osc}}\simeq \bar{m}$. Since $H_*\simeq H_{\text {osc}}$, these two conditions actually do not differ much. From either (40) or (44), one can get the relation of $mM/\sqrt{6\xi }\lesssim 10^{-10}M_p$. That means, if we choose m to be of $10^{-6}M_p$, that we will get $M/\sqrt{6\xi }\lesssim 10^{-4}M_p$, which is consistent with our numerical calculations.

4 Effects on curvature perturbations

In this section, we discuss how the resonance process could affect the final curvature perturbations in our model, especially whether the points of $\dot{\varphi }=0$ will make the curvature perturbations diverge. This issue has been in debate in early work of canonical inflation reheating [59, 60] (see also [61,62,63] for further discussions), and has been revisited recently for NDC inflation reheating [49]. As is well known, since there is more than one component in the universe, the perturbations generated by the curvaton are basically isocurvature, which will transfer to curvature perturbation after curvaton domination. According to the original analysis of the curvaton in [64], the curvature perturbation is defined as

$$\begin{aligned} \zeta =-H\frac{\delta \rho _{\text {bg}}+\delta \rho _\varphi }{\dot{\rho }_{\text {bg}}+\dot{\rho }_\varphi } \end{aligned}$$

(46)

in spatial flat gauge. Assuming that the perturbations of the background field (inflaton) are negligible, we have $\delta \rho _{\text {bg}}\approx 0$.

From the continuity equations for both components, $\dot{\rho }=-3H(\rho +P)$. For a background whose equation of state is assumed to be unity, we have $\dot{\rho }_{\text {bg}}=-6H\rho _{\text {bg}}$. Therefore for the time before the curvaton is dominating, one roughly has

$$\begin{aligned} \zeta =-H\frac{\delta \rho _\varphi }{\dot{\rho }_{\text {bg}}}=\frac{\rho _{\varphi ,\varphi }\delta \varphi }{6\rho _{\text {bg}}}. \end{aligned}$$

(47)

Since $\rho _{\text {bg}}$ are non-zero, one can easily see that $\zeta$ will not diverge.

On the other hand, for the time after curvaton dominating, one has

$$\begin{aligned} \zeta =-H\frac{\delta \rho _\varphi }{\dot{\rho _\varphi }}=\frac{\rho _{\varphi ,\varphi }\delta \varphi }{3H(\rho _\varphi +P_\varphi )}=-\frac{H}{\dot{\varphi }}\delta \varphi , \end{aligned}$$

(48)

where we have made use of Eqs. (3), (4) as well as the relation

$$\begin{aligned} \rho _{\varphi ,\varphi }=\frac{\dot{\rho }_{\varphi }}{\dot{\varphi }}=\frac{6\xi }{M^2}(2H^2\ddot{\varphi }+H\dot{H}\dot{\varphi }-3H^3\dot{\varphi }). \end{aligned}$$

(49)

So it seems that when $\dot{\varphi }$ passes through 0 during oscillation, $\zeta$ might diverge. However, this is not true. From the equation of motion for $\delta \varphi$ [27]

$$\begin{aligned} (az^2)^\cdot \dot{\delta \varphi }+az^2\ddot{\delta \varphi }-aQ\partial _i^2\delta \varphi +a^3m_{\text {eff}}^2\delta \varphi =0, \end{aligned}$$

(50)

where $z=a\sqrt{Q}/c_\mathrm{s}\sim aH$, and Q, $c_\mathrm{s}$ and $m_{\text {eff}}$ have all been defined in [27]. Making use of (48), we have

$$\begin{aligned} (az^2)^\cdot \left( \frac{\dot{\varphi }}{H}\zeta \right) ^\cdot +\,az^2 \left( \frac{\dot{\varphi }}{H}\zeta \right) ^{\cdot \cdot }-\, aQ\frac{\dot{\varphi }}{H}\partial _i^2\zeta \,+\,a^3m_{\text {eff}}^2\frac{\dot{\varphi }}{H}\zeta =0. \end{aligned}$$

(51)

On the other hand, from Eq. (2), one has

$$\begin{aligned} \ddot{\varphi }=-2\frac{\dot{H}}{H}\dot{\varphi }-3H\varphi -\frac{M^2V_{,\varphi }}{6\xi H^2}~ \end{aligned}$$

(52)

where $V_{,\varphi }=m^2\varphi$, and taking one more time derivative gives

$$\begin{aligned} \dddot{\varphi }= & {} -2\frac{\ddot{H}}{H}\dot{\varphi }-2\frac{\dot{H}}{H}\ddot{\varphi }+2\frac{\dot{H}^2}{H^2}\dot{\varphi }-3\dot{H}\dot{\varphi }-3H\ddot{\varphi }\nonumber \\&-\frac{M^2m^2}{6\xi H^2}\dot{\varphi }+\frac{M^2m^2\dot{H}}{3\xi H^3}\varphi . \end{aligned}$$

(53)

Expanding Eq. (51) and making use of the above two equations, one finally has

$$\begin{aligned} \frac{a^3M^2m^2\varphi }{3\xi H}\left( \frac{2\dot{H}}{H}\zeta -\dot{\zeta }\right) =0, \end{aligned}$$

(54)

when $\dot{\varphi }=0$. It has the unique solution $\dot{\zeta }=(2\dot{H}/H)\zeta$. However, since now the curvaton is dominant, we have $\dot{H}=-(\rho _\varphi +P_\varphi )/2=(\xi /M^2)(\dot{H}\dot{\varphi }^2+2H\dot{\varphi }\ddot{\varphi }-3H^2\dot{\varphi }^2)$, which is equal to zero when $\dot{\varphi }=0$. This can also be seen in Fig. 2. So we still have $\dot{\zeta }=0$, and $\zeta$ will not diverge. Note that a similar conclusion has been obtained in the early work of canonical single field inflation [60].⁵

Fig. 6

The numerical plot of $\zeta$ in terms of the cosmic time t before curvaton domination. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$. The initial conditions for $\delta \varphi$ are $\delta \varphi _i=10^{-4}M_p$, $\dot{\delta \varphi }_i=0$

Fig. 7

The numerical plot of $|\zeta |$ in terms of the cosmic time t after curvaton domination. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$. The initial conditions for $\delta \varphi$ are $\delta \varphi _i=10^{-4}M_p$, $\dot{\delta \varphi }_i=0$

We performed numerical calculations for the two cases and plot the evolutions of $\zeta$ in Figs. 6 and 7. One can see that, in Fig. 6, $\zeta$ oscillates with an increasing amplitude. This can be explained as follows: the perturbation generated before curvaton domination is of isocurvature type, which can be a source of $\zeta$. However, in Fig. 7, $\zeta$ nearly behaves as a constant. This is because when the curvaton dominates and the isocurvature perturbation has been transformed into the curvature one, $\zeta$ will be a conserved quantity. Although there will be some issues on $\zeta$, which may be due to the secondary effects of the resonance, numerical fluctuation or other unknown reasons, things will not be so bad to make it diverge. We also plot $\dot{\zeta }$ in Fig. 8, so one can see more clearly that, in the range of the typical amplitude of $\zeta$ ($\sim 10^{-5}$), the variation of $\zeta$ can hardly be seen.

Fig. 8

The numerical plot of $\dot{\zeta }$ in terms of the cosmic time t after curvaton domination. We choose the parameters as $\xi =1$, $M=10^{-4}M_p$, $m=10^{-6}M_p$. The initial conditions for $\delta \varphi$ are $\delta \varphi _i=10^{-4}M_p$, $\dot{\delta \varphi }_i=0$

5 Conclusions

In this paper, we investigated the reheating mechanism of the curvaton model with non-minimally derivative coupled to gravity. We assume that the curvaton has a mass-squared potential term, which cannot only explain the observationally required tilt of the perturbation spectrum from scale invariance but also provide a minimum around which the curvaton can oscillate. This gives an efficient way of particle creation, namely a parametric resonance process. Different from the reheating process of the inflaton is the fact that the background evolutions before and after curvaton domination are not the same, and if the curvaton is nonminimally coupled, the averaged decaying rate of the curvaton may also not be the same, which can affect the rate of particle creation. Though we did not provide a rigid proof (which will be left for future work), this may be a general phenomenon.

For our model, as a specific case, we found that the averaged value of the curvaton scales as a constant before curvaton domination, while it is proportional to the time inverse after that. The particles, whose field interact with the curvaton, will indeed be effectively created due to the resonance process. To confirm our analysis, we also numerically calculated the background evolution for the curvaton field and the increase of created particle numbers by the resonance.

Moreover, we also estimated the constraints on the reheating temperature of our model, for both cases which the reheating completes before and after curvaton domination. We only got an upper limit on the temperature, which is looser than that given by the overproduction of gravitinos. Finally, we investigated the curvature perturbation generated in our model. We showed that the curvature perturbation will be regular, although it is apparently divergent when $\dot{\varphi }$ goes to zero in each oscillation. Unlike that the amplitude of curvature perturbation is increasing before curvaton domination, which should be due to the source of isocurvature perturbation, after the curvaton domination it will remain approximately constant.

One can also investigate the back-reaction of the created particle $\chi$ on the reheating process, which we leave for a future study. As an application, our work can be used either to compare cases of single and multiple field theories nonminimally coupled to gravity, or to compare cases of minimal and nonminimal coupled theories in fixed degrees of freedom.