Reheating mechanism of the curvaton with nonminimal derivative coupling to gravity

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.


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 μν ∂ μ φ∂ ν φ, with φ denoting the scalar field and G μν 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 a e-mail: qiutt@mail.ccnu.edu.cn b e-mail: fkaixi@itp.ac.cn motion second order. Therefore it has been paid more and more attention to in recent work; see  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.

The NDC curvaton model
Following the preceding work [27], one has the action including the NDC curvaton where ξ is the dimensionless coupling constant of the NDC term. L 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 ϕ, such as and its energy density and pressure can be expressed as 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 (ϕ) = 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 ϕ. 1 For the simplest choice, we choose as its potential. Then the equation of motion (2) reduces tö where the slow-roll parameter is defined as ≡ −Ḣ /H 2 . Moreover, one can define an effective mass of the curvaton field, Moreover, according to [27], we have where z 2 (3ξ/M 2 )a 2 H 2 and 1 describes the deviation of spectral index from pure scale invariance, 1 3(n s −1)/2. From the central value of n s given by the Planck observational constraint [52], one roughly has 1 ∼ O(0.01). Notice also that during inflation a H −(η * − η) −1 ; this furthermore gives m ∼ 0.1H inf (9) during inflation.

The behavior of the curvaton during oscillation
In order to investigate the reheating mechanism of the curvaton field ϕ, 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 ρ bg ∝ a −6 (t), w bg = 1.
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 and w should be equal to w bg before curvaton domination, and w ϕ after that. For convenience, we define t eq as the time when the energy densities of inflaton and curvaton are equal. For the former case, p = 1/3 as w bg = 1, while the latter case depends on the behavior of ϕ, which is the solution of the equation of motion (2). Considering (11), Eq. (2) can be solved: where (t) = 0 t (1−3 p)/2 is the oscillation amplitude of ϕ with the initial condition of 0 , and J is the Bessel function. Note that in the reheating era where ϕ begins to oscillate, one hasm > H , which roughly gives m Mt 2 2 √ 6ξ p > 1. Thus the Bessel function can be mimicked by the trigonometric functions, namely During rapid oscillation, the kinetic and potential terms in the energy density (3) are of the same order, therefore one roughly has For t < t eq , p = 1/3, so we have ρ ϕ ∝ t 0 ∼ 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 ϕ = −1.
Since the background energy density is decreasing, the curvaton will exceed the background (at the time t eq ) and dominate the universe. For t > t eq , the same solution of (13) Fig. 1 Plots of the value of ϕ and its energy density ρ ϕ in terms of cosmic time t. We choose the parameters as ξ = 1, M = 10 −4 M p , m = 10 −6 M p applied but with p be related to w ϕ , so one cannot determine p, or the behavior of ρ ϕ , solely by Eq. (14). However, since the curvaton dominates the universe, we have another equation, namely the Friedmann equation, This equation, together with (11), tells us that ρ ϕ ∝ t −2 .
Comparing it with Eq. (14), one gets p = 1, which gives 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 ϕ and its energy density ρ ϕ 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 p = H t. From the figure it is clearly seen that p 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 ϕ = t −3 p/2 cos(mt), and ρ ϕ ∼ t −3 p ∼ 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 ϕ = 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.

Parametric resonance
In this section, we begin to discuss how the oscillating behavior of the curvaton ϕ, as shown above, can be responsible for creating particles (we denote this χ ), namely, the preheating process. One assumes that the free-field Lagrangian of χ has a canonical form, and that ϕ can interact with χ through some interacting term, such as gϕ 2 χ 2 , with a dimensionless coefficient of g. It gives the total form of the Lagrangian of χ field: It is true that one can also have the interaction term of χ 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 χ than ϕ. Since χ is a quantum field, one can write down the Heisenberg presentation of χ [33,34]: whereâ k andâ † k are the annihilation and creation operators, respectively, and k is the comoving wavenumber. Using this, one can also get the equation of motion for χ by simply varying the Lagrangian (16): In the rest of this paper, we assume m χ = 0 for simplicity. Since in our case, ϕ(t) behaves as in Eq. (13), in the static limit where a = const., H = 0, Eq. (18) will reduce to the Mathieu-like equation, the solution of which is the wellknown Floquet solution [33,34]. In an expanding phase such as inflation, however, one can define a new variable: X k (t) ≡ a 3/2 (t)χ k , and Eq. (18) can be rewritten as One can apply the WKB approximation to get the solution of Eq. (19): with α k and β k being the time-dependent coefficient of positive and negative frequency parts, respectively. Moreover, α k and β k satisfy the equationṡ with the additional normalization relation Furthermore, the comoving occupation number of the χ particles in the mode k is defined by Using Eqs. (20), (21) and (22), one gets a simple expression of n k = |β k | 2 , and the vacuum expectation value of particle number density of χ per comoving volume is It is straightforward to find how the χ particles increase with time just by integrating |β 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, . . . where ϕ(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 gϕ 2 (t) with ϕ given in (13) around t j , we get So the leading order of gϕ 2 (t) gives a parabolic potential for χ , and the particle creation at t j could be viewed as the scattering of χ particles through this potential for the jth time. We assume the solution of X k at the period of and [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 ϕ on t j is different from the minimal coupling case, as was shown in (13). This causes k * ∝ t 1/2 j for t j < t eq , while k * ∝ t 0 j for t > t eq . Considering also the dependence of a on t, namely a ∝ t 1/3 before t eq and a ∝ t after, one gets κ ∝ t −5/6 before t eq and κ ∝ t −1 after. Therefore one gets In the limit of κ 2 π −1 , μ k reduces to the trivial case in [33,34]: μ j k (2π) −1 ln 3 0.18 where we neglected the random terms.
The total number during the whole process of resonance yields in mode k. In principle one can get n k by integrating μ 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 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 ξ = 1, while the increasing rate of the particle number in the comoving volume is roughly estimated to be 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 ϕ before and after curvaton domination.
As a side remark, we roughly estimate the time when backreaction of the produced particle χ becomes important, which can prevent a further reheating process. To see this, one could define the energy density of the χ by where n χ is calculated in (30), while the effective mass of the χ particle is from Eq. (19). The condition of back-reaction becoming important is ρ χ ≈ ρ ϕ , where ρ ϕ ≈ m 2 ϕ 2 . Since it is difficult to have an analytical solution of the time t, we perform a numerical calculation, and plot both ρ χ and ρ ϕ (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.

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 where the subscripts osc and eq denote the value when ϕ 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 Therefore, if the curvaton reheats after its domination, then we have ρ eq ϕ > ρ rh ϕ , and the amount of difference depends on how long reheating will last after curvaton domination. From Eqs. (34) and (35) we have At the time when ϕ ends slow-rolling and begins oscillating, the kinetic and potential term of ϕ will be of the same order, namely 9ξ H 2φ2 /M 2 ∼ m 2 ϕ 2 /2, therefore where y osc is the value of y(≡ ξφ 2 /M 2 ) at the oscillation time. So the reheating temperature is T rh < 540H 2 osc y osc gπ 2 1 4 .
The constraints on the Hubble parameter and y can be obtained from the Planck constraints on the scalar spectrum: |P ζ | 2.2065 +0.0763 −0.0738 × 10 −9 (TT, TE, EE+lowP, 68% CL) and the tensor/scalar ratio: r 0.10 (TT, TE, EE+lowP, 95% CL) [52]. In our model, from [27] we have for this case, where * denotes the value evaluated at horizon crossing. These first give the constraints: from which one can see, if we set * ∼ O(10 −2 ), that y * is as low as 10 −7 M 2 p . As y and H vary slowly during inflation, one can roughly write H * H osc , y * y osc ; therefore the reheating temperature can be set to which actually gives a very loose constraint on the reheating temperature, comparing with that from the gravitino producing [54][55][56][57].
For a perturbation in this case, we have where r ≡ ρ ϕ /ρ r is the ratio of energy densities of ϕ field and radiation. From the Planck constraints, we have so we have 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 rh < 5×10 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)

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 ofφ = 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 in spatial flat gauge. Assuming that the perturbations of the background field (inflaton) are negligible, we have δρ bg ≈ 0. is equal to zero whenφ = 0. This can also be seen in Fig.  2. So we still haveζ = 0, and ζ will not diverge. Note that a similar conclusion has been obtained in the early work of canonical single field inflation [60]. 5 We performed numerical calculations for the two cases and plot the evolutions of ζ in Figs. 6 and 7. One can see that, in Fig. 6, ζ 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 ζ . However, in Fig. 7, ζ nearly behaves as a constant. This is because when the curvaton dominates and the isocurvature perturbation has been transformed into 5 However, since the sound speed squared parameter c 2 s contains ∼Ḣ [27,48], in the reheating era when H is oscillating,Ḣ will also oscillate between positive and negative values, which causes negative c 2 s and a gradient instability for large-k modes during reheating [48]. Since now we are interested in the fluctuations outside the horizon, the instability will not be too influential. We thank the anonymous referee for pointing this out to us. the curvature one, ζ will be a conserved quantity. Although there will be some issues on ζ , 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 plotζ in Fig. 8, so one can see more clearly that, in the range of the typical amplitude of ζ (∼ 10 −5 ), the variation of ζ can hardly be seen.

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 reheat-ing 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 whenφ 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 χ 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.