Pre-heating in the framework of massive gravity

In this paper we propose a mechanism of natural preheating of our universe by introducing an inflaton field dependent mass term for the gravitational wave for a specific class of massive gravity theory. For any single field inflationary model, the inflaton must go through the oscillatory phase after the end of inflation. As has recently been pointed out, if the gravitational fluctuation has inflaton dependent mass term, there will be a resonant amplification of the amplitude of the gravitational wave during the oscillatory phase of inflaton. Because of this large enhancement of the amplitude of the gravitational wave due to parametric resonance, we show that universe can naturally go through the pre-reheated phase with minimally coupled matter field. Therefore, during the reheating phase, there is no need to introduce any arbitrary coupling between the matter field and the inflaton .

During the last decade, observational cosmology [1] has reached to the label, where we can go backward in time and see our universe with a great precession almost near the beginning of our universe. We have understood how the observed very tiny fluctuation (δT /T ≃ 10 −5 ) in temperature in the Cosmic Micro-wave Background (CMB) sky play the seed of the structure formation of the present universe. So far inflation, which is believed to have happened within a very shot span of time after the big-bang, is believed to be the best theoretical explanation of the origin of those tiny fluctuation. The inflation was first proposed [2][3][4] to explain some important conceptual problems in the standard hot big-bang cosmology. In spite of its simplicity, it was soon realised that constructing such a model within the known theory is very difficult. The main problems in constructing a inflation model are the origin of inflaton, the mechanism of how it has happend, the mechanism of ending the inflation, and the mechanism of reheating the universe after such a super cooling phenomena. In addition to those, there generally exist many issues within a particular model under consideration. Living with may fundamental questions, in this letter we will ask the question as to how reheating happen after the end of inflation. There exist lot of studies on this particular mechanism [5,6], which we generally call pre-heating. In this letter we will introduce a mechanism which will be very generic in the framework of general relativity. This will be generic for those class of inflationary models where after the end of inflation, some fields (may not be just inflaton itself) must oscillate, and the gravitational fluctuation has those oscillating field dependent mass term. The general physical mechanism to transfer the energy is to couple the probe field with background oscillating source field. In the usual preheating scenario, one generally couples the matter fields with the inflaton field in an ad hoc manner, and try to tune those unknown coupling constant to have explosive particle production through parametric resonance. In this letter we will show that * debu.imsc@gmail.com how the universal minimal coupling of the matter field with the gravity is sufficient to reheat our universal after the end of inflation. The idea was first put forward in the reference [7]. In that paper the inflaton field is assumed to be coupled with the higher derivative gravitational Chern-Simons term. However, as we have explained in [8], it is very difficult to have successful preheating with the Chern-Simons coupling. It was shown to be crucially dependent upon the large initial amplitude of the gravitational fluctuation during reheating, and this large value of initial amplitude is difficult to achieve. Thanks to the reference [9], following the aforementioned points, the large amplitude of the gravitational wave can be easily generated through parametric resonance if one considers the inflaton field dependent mass term for the graviton. Purpose of the proposal was to create large scalar to tensor ratio without violating Lyth bound [10]. We are interested in preheating. Therefore, to achieve that we will have sequence of parametric resonance phenomena. We will see how the parametric resonance of the gravitational wave naturally leads to the parametric resonance in the matter sector though gravitational coupling. In this letter we will try to construct a minimal model to realise this mechanism.
We will start with the following action where L φ is the inflaton field Lagrangian. For the sake of our numerical computation we will consider the model of inflation from our previous paper [8], where we have considered the following Lagrangian for a single inflaton with higher derivative term which we call modified natural inflation, where X = (1/2)∂ µ φ∂ µ φ. s is the dimensionful parameter. M (φ) is the some specific dimensionless function of the inflaton field. f is the axion decay constant. λ is the known to be Pecci-Quinn symmetry breaking scale.
We would like to emphasize that in this work we are only interested in the regime where the inflaton field coherently oscillates after the end of inflation. To have successful reheating this is one of the important requirements for any model of inflation. Therefore, our final conclusion of this work is valid for any model of inflation.
The new ingredient that we have here is the extra term in the total Lagrangian L lv which is assumed to contribute only to the mass of the gravitational fluctuation. For the time being we ignore the detail origin of the mass term for the gravitational fluctuation. We assume it is coming from some Lorenz violating theory. Thanks to the reference [9], which provide us an explicit example of a such model in the framework of massive gravity [12]. We also consider the minimally coupled matter Lagrangian during reheating For simplicity of our calculation, the added term L lv is such that at the label of background expansion, it will not contribute [9]. At the linear order in perturbation, we assume that the term only contributes to the mass of the gravitational fluctuation. Similar situation happens for gravitational Chern-Simons term coupled with the inflaton. The crucial difference is that the Chern-Simon term provides oscillating anti-dissipation term for the dynamics of the graviton. On the other hand here we have the oscillating mass term.
The background evolution is governed by the inflaton field φ with the following metric After the end of inflation, the inflaton start to oscillate which will lead to reheating of our universe. To study the reheating we will consider the solution of the oscillating inflaton from our reference [8]. Of course, our final conclusion does not depend on the particular model under consideration. As long as we have oscillating background inflaton, we will have parametric resonance for the gravitational fluctuation.
Since we are only interested in the gravitational fluctuation, the tensor perturbation is As usual, we have the transverse and traceless condition for the tensor perturbation The fourier mode function of h ij defined as To the linear order in cosmological perturbation theory, the Fourier transformed gravitational wave equation turns out to bë and the corresponding Fourier transformed equation for the matter field θ which is minimally coupled with the gravity will take the following from As we see, the mass term of the graviton fluctuation m(φ) is coming from the extra Lorentz violating term in the Lagrangian. We also assume [9], in order not to have large power suppression of the gravitational wave during inflation, where φ * is the value of φ at the end of inflation. Such a functional form will lead to very small mass of the gravitational wave during inflation, as inflaton field will have very large constant vacuum expectation value. Furthermore, during it oscillation period, since inflaton undergoes damped oscillation around the minimum of it effective potential, its amplitude will again be very small compared to φ * . Therefore, during oscillating phase, we can approximate where λ is dimensionless coupling constant. As one can see from the above expression of the inflaton dependent mass term, one would expect to get back the usual massless gravity theory at the very late time. For a class of massive gravity theory [11], it has been shown that the theory reproduces usual gravity theory in the massless gaviton limit. In this letter we will not be restricted to a particular class of models. , the matter field θ k and particle number density n k in time t which is measured in unit of s. We have plotted for kx = 0, ky = 170, kz = 130 in unit of s. We chosen the dimensionless graviton mass parameter λ = 1. We also checked that as we decrease the initial amplitude of the gravitational wave, the resonant production of the gravitational wave happens at later time which leads to the particle production at late time.
However, in the above eq.(16), the backreaction of the matter field is ignored. Assuming the above evolution equations for a particular mode k of the quantum fields h and θ, We numerically solve those in the oscillatory background of the inflaton field φ. Finally we computed the particle number density n k of θ with a definite momentum k. The use the following expression for the particle number density is [6] where, ω k is the time dependent frequency of a particular mode k.
where k phy is the physical momentum. For simplicity we fix the value of k x , k y and do the numerical integration for the mode equation of θ k along the k z .
Numerical solution and particle production: Since we are only interested in the oscillatory regime of the inflaton, our final conclusion does not really depend on the specific model of inflation. As mentioned earlier, in order to get the background of oscillatory inflaton for our numerical calculation, we consider our specific model of modified natural inflation [8]. We choose the following parameters M (φ) = sin 7 x 1 − cos x sin 2 x 14 , where x = φ/f . For this specific form of the function we had the number of e-folding N = 50, and the spectral index to be n s = 0.96. In order to achieve those cosmological quantities which are in agreement with observation, we find the values of the other parameters to be f = 0.84M p , s = 2.25 × 10 −6 M p , Λ = 0.011M p . For our numerical purpose we set t = 0 as the beginning of the inflation. After the end of inflation at around t ∼ 4 cosmic time in unit of particular model parameter s, the inflaton field will start to have coherent damped oscillation as shown in the fig-1. At this point, we are again emphasizing the fact that our analysis and the final result is confined within the oscillatory regime of the inflaton field. The oscillation after inflation is generic to any inflationary model. Therefore, our qualitative results are insensitive to the specific model under consideration. For the sake of our numerical analysis, at time t = 6 we normalize the cosmological scale factor a(6) = 1. For a wide range of initial amplitude of the gravitational wave, we found the resonant production of the gravitational wave. More over this resonant production of gravitational wave will trigger the reheating phase of the universe through the matter fields with universal gravitational coupling.
In the inflationary background, to quantize a field we choose usual Bunch-Davis vacuum, We consider approximately the de-Sitter vacuum. As one can see, lower the value of a particular mode k, larger the initial amplitude in the quantum vacuum, leads to the earlier production of matter particle of that particular modes after the end of inflation. As a particular mode starts evolving from its quantum vacuum defined in the far past and deep inside the Hubble volume during inflation, its amplitude will start to decrease due to Hubble friction and finally gets frozen after the horizon exit. Therefore, during reheating the amplitude at the horizon exit will set the initial condition for a particular mode of the gravitational wave. From the aforementioned argument we see that the parametric resonance for the matter field will happen mostly for the long wavelength modes compared to the Hubble length. We can clearly see from the fig-2 that because of the resonant gravitational wave production triggered by its oscillatory mass coming from the oscillating inflaton, minimal coupling of the matter field with the gravity is sufficient to reheat the universe.
We have tried to constructed a phenomenological model of reheating by introducing a inflaton field dependent mass term for the gravitational wave. We assume that the term is coming from some Lorentz violation in very high energy regime at least above the inflation energy scale. During inflation, gravitational mass term will suppress the power in the gravitational wave spectrum. In order to avoid the suppression, we parametrize the mass function in such a way that during inflation, its value is very small. On the other hand, after the end of inflation, the amplitude of the gravitational wave will be amplified due to parametric resonance because of mass term [9]. Now the question naturally arises is why have not we observed this large amplitude gravitational wave yet? Natural answer would be that this large amplitude gravitational wave may contribute largely to reheat the universe. In this letter we showed that how this resonant gravitational wave production mechanism can naturally lead to the reheating phase of the universe. Interestingly enough, to reheat the universe we do not need to tune the coupling between the visible sector matter field and the inflaton field in an ad hoc manner. The natural universal gravitation coupling is sufficient to reheat the universe.
Using the same mechanism that we discussed so far, one can study the production of chiral gravitational wave during preheating if we further introduce the Chern-Simons (CS) higher derivative term [13,14] coupled with the inflaton field in the action, i.e.
The inflaton field is now like a pseudo scalar axion, where ǫ µναβ is the usual 4-dimensional Levi-Civita tensor. g is the free coupling parameter. Since in the background of inflaton this term violates parity, the left and right handed gravitational wave propagate differently according to the following equation where A = L, R are left and right circular polarization mode of the gravitational wave. Therefore, we will have resonant chiral gravity wave production. This effect may be very important to produce the leptogenesis in the early universe. As is well know [15], if we couple gravity with the leptons, the total lepton number current will no longer be conserved because of gravitational anomaly, where, < RR > is the quantum expectation value of the Chern-Simons term. Therefore, non-zero expectation value of this Chern-Simons operator will produce the leptogenesis. This mechanism has already been discussed in [13]. However, in our model the resonant production of chiral gravitational wave will happen during oscillation period of the inflaton. Therefore, the maximum lepton number violating contribution will be coming during the preheating period. Currently we are looking into this issue in greater detail.