Gravitational-Wave Mediated Preheating

We propose a new preheating mechanism through the coupling of the gravitational field to both the inflaton and matter fields, without direct inflaton-matter couplings. The inflaton transfers power to the matter fields through interactions with gravitational waves, which are exponentially enhanced due to an inflation-graviton coupling. One such coupling is the product of the inflaton to the Pontryagin density, as in dynamical Chern-Simons gravity. The energy scales involved are constrained by requiring that preheating happens fast during matter domination.

Introduction. Inflation is the paradigm wherein the universe undergoes exponential expansion, resolving the horizon, entropy and structure formation problems that plague the standard big bang scenario. It is usually believed that inflation ends once the inflaton reaches the bottom of its potential, at which point a new mechanism must act to transfer the inflaton's kinetic energy into a process that leads to particle creation. One such mechanism is preheating [1][2][3][4]: the inflaton enters a phase of parametric resonance, as it oscillates around the minimum of its potential, and through a direct matterinflaton coupling, it leads to particle creation. There is a large number of possible direct couplings between the inflaton and the standard model, and one must usually pick one somewhat arbitrarily.
But what if the coupling between the inflaton and the matter fields were indirect? Because of the equivalence principle, the graviton will interact with all matter fields and its coupling will be non-arbitrary. Let us then consider the inflaton coupling to matter fields through a graviton intermediary. That is, consider the inflaton at the end of inflation depositing its kinetic energy in the graviton, which due to a direct graviton-inflaton coupling becomes parametrically excited, and then deposits its energy in the matter fields. This can happen if there are new couplings between the inflation field and the graviton so that when the inflaton oscillates it causes resonances in the graviton's modulation. We will show that such couplings are possible via Chern-Simons modified general relativity. As we shall see, no direct inflaton-matter coupling will be necessary to obtain preheating in this gravitational-wave mediated scenario.
Action and Evolution Equations. Many inflationary paradigms exist, but for concreteness consider the following Chern-Simon extension to Chaotic Inflation [5,6] where φ is the inflaton, χ is the matter field, R is the Ricci scalar associated with the metric g µν , R R is the Pontryagin density, ie. the contraction of the Riemann tensor with its dual, and α is a coupling constant with dimensions of inverse mass (we work here in natural units c = 1 = h). Except for the interaction term L int , Eq. (1) is just a simple model for inflation with a quadratic inflaton potential, arising from a Taylor expansion about its minimum. Many possible graviton-inflaton couplings could be considered, but the one presented above, L int , is wellmotivated. Such a coupling arises naturally in a variety of frameworks: (i) in heterotic string theories upon 4-dimensional compactification and a low-energy expansion [7,8]; (ii) in loop quantum gravity when the Barbero Immirzi parameter is promoted to a field and coupled to fermions [9,10]; (iii) in effective field theories of inflation [11]; (iv) in dynamical Chern-Simons gravity [12]. Let us emphasize, however, that the gravitational wavemediated preheating mechanism proposed here does not depend on this particular coupling.
Regardless of the motivation, the theory described above should be considered effective, a truncated lowenergy expansion of a more fundamental theory that is thus valid only up to some energy cut-off Λ. The effective theory ceases to be a valid description when the interaction term L int becomes comparable to the Einstein-Hilbert term L EH . The former can be written as a total derivative if φ is constant. Therefore, to estimate its size we should first integrate by parts, moving a derivative from R R to φ. The interaction term then becomes comparable to L EH when αφ ∼ M 2 p (h 0 f ) −1 , where M p = G −1/2 is the Planck mass, f is the gravitational wave frequency and h 0 is the gravitational wave ampli-arXiv:1405.4288v1 [gr-qc] 16 May 2014 tude. Saturating α at Λ −1 ,φ at HM p , h 0 at unity and f at H, where H is the Hubble parameter, one finds Λ = (H/M p ) 2 M p , which of course satisfies Λ M p . Another consequence of the truncation of the effective theory at this order is that the terms neglected in the expansion, such as (∂φ) 4 /Λ 4 , are indeed small and ignorable. A consequence of all of this is that the interaction term L int acts as a small perturbation to whichever inflationary mechanism one wishes to consider, and thus, it does not spoil (or really affect) inflation, until inflation ends and the inflaton reaches the bottom of its potential.
Variation of the action in Eq. (1) with respect to all degrees of freedom leads to the field equations [12] where is the curved wave-operator, G µν is the Einstein tensor, T µν is the sum of the stress-energy tensors of the φ and χ fields, and with R β(µν)α the dual Riemann tensor with indices symmetrized.
Order Reduction and Perturbation Theory. Let us expand the equation for the metric tensor and the inflaton about a fixed background: g µν = g µν + λh µν and φ(t, x) = φ(t) + λδφ(t, x), where λ is an ordercounting parameter. The background g µν and φ(t) will be taken to be the flat Friedmann-Lemaître-Robertson-Walker (FLRW) metric and a homogeneous and isotropic background field respectively, while h µν and δφ(t, x) are first-order perturbations.
The FLRW metric satisfies the background field equations exactly for any homogeneous and isotropic background inflaton field. The Hubble parameter is sourced by the energy density and pressure of this background field and the matter fields (for reasons that will become clear later, we do not decompose χ). The background inflaton field satisfies the homogeneous and isotropic wave equation on an FLRW background with a mass potential. The Pontryagin density does not contribute to the background evolution of the inflaton, because this quantity vanishes exactly when evaluated for any spherically symmetric metric.
To first-order in λ, the equations for the metric tensor perturbation become [13] while, neglecting scalar metric perturbations (we are looking at modes shorter than the Hubble scale), the equation for the inflaton perturbation is where ∇ 2 and are the Laplacian and wave operators in a homogeneous and isotropic FLRW background, p χ is the pressure of the χ field and h ij ≡ a −2 h ij . Notice again that the Pontryagin density does not enter the evolution equation of the inflaton perturbation, since it also vanishes identically to linear order in λ.
We can simplify the evolution equation for the metric perturbation through order reduction. As discussed in [14][15][16], we decompose the metric perturbation into a general relativistic piece h GR µν and a deformation δh µν , namely h µν = h GR µν + α 2 δh µν . Note that the deformation is proportional to α 2 because Eq. (4) is proportional to αC µν , C µν is proportional to φ, and φ is proportional to α due to Eq. (5). Using this decomposition, we can order reduce Eq. (8): the left-hand side is proportional to δh µν , while the right-hand side is proportional to a differential operator acting on h GR µν . This differential operator will contain one term of the form , which automatically vanishes when acting on h GR µν because R µν [h GR αβ ] = 0. Using this and going to the transverse-traceless (TT) gauge [17] and in the left/right-circular polarization basis for the gravitational wave perturbation, Eq. (8) becomes The equation for h L can be obtained by taking i → −i and h L/R → h R/L . Notice that Eqs. (8) and (9) are not coupled and can thus be solved independently, once the evolution of the background fields is obtained. Let us now discuss the evolution of the matter fields. We anticipate that the matter occupation number will be generated through parametric resonance, so even a small perturbation of size O(|h µν |), may have a large effect. We therefore treat the metric exactly, without a perturbative decomposition, to obtain In the TT gauge and in a circular GW polarization basis, this equation becomes where Re[x] is the real part of x and we have defined ∂ L,R ≡ (∂/∂x ∓ i∂/∂y)/ √ 2. Behavior of Solutions. Before attempting to solve the equations of motion, let us make some approximations. On the one hand, as the φ field oscillates, it will amplify and cause the gravitational waves to modulate, which in turn will drive the production of χ particles. The latter will therefore occur on the timescale τ φ = 1/m φ . On the other hand, the expansion of space is governed by the Hubble parameter and the scale factor changes on the timescale τ H = 1/H. When τ φ τ H , or equivalently H m φ , preheating occurs much faster than the expansion of space and one is justified in setting a = 1 and H = 0. Using that the Hubble parameter for a simple quadratic potential satisfies M 2 p H 2 = 4π 3 (m φ φ) 2 , the requirement H m φ implies φ 0 M p , where φ 0 is the value of φ at the end of inflation. Even though it works well, this approximation can not be claimed to be excellent, as we know that φ 0 < M p . Nevertheless, the approximation is correct, since the timescale involved in the growth of the particle number is smaller than the oscillation time.
With this approximation, the equation of motion for the background inflaton field greatly simplifies to a that of a simple harmonic oscillator with frequency m φ , so where φ 0 and δ are constants of integration, ie. the amplitude of the background inflaton and a phase shift. The background inflaton sources the evolution of the metric perturbation through Eq. (10). Using the approximation described above, and transforming to the Fourier domain, this equation becomes (14) where the overhead tilde stands for the Fourier transform and we have defined which controls the strength of the oscillatory antidamping term. The above equation of motion admits an exact solution through a linear combination of Heun functions times an oscillatory term, which we confirm by numerically solving Eq. (14). We could obtain a similar expression for the left-polarized mode, but we do not need to. As found in [7,17,18], Eq. (10) leads to exponential amplification/damping of right-/left-handed gravitational waves during the inflationary epoch, so by the end of inflation, the left-handed gravitational waves are negligible and can be neglected during preheating.
With the solution to the background inflaton and the metric perturbation, we can now solve for the evolution of the matter fields. Working in the Heisenberg picture, we promote the matter field to a quantum operatorχ(x, t) and expand the latter in Fourier modes with raising and lowering operators, as done routinely in quantum field theory [19]. Using the flat-space approximation described above, the evolution of the Fourier mode functions obeys where k is the magnitude of the 3-momentum k i -vector and h.c. stands for Hermitian conjugate. Let us contrast our matter production equation with the usual way particle creation occurs via the Mathieu equation. In that case, the matter fields obey the following evolution equation where g is the coupling constant between the χ and φ fields. Parametric resonance occurs because of the purely temporal oscillations of the inflaton. In our case, however, the graviton couples through spatial gradients of the inflaton, and thus, parametric resonance occurs due to non-linear mode couplings and temporal oscillations in the gravitational wave's source term. Parametric Resonance and Particle Creation. The evolution equations for the gravitational wave and the mode functions, Eqs. (14) and (16), are those of a parametric oscillator, ie. a harmonic oscillator with parameters, like the damping coefficient or the natural frequency, that oscillate in time at some other frequency. For example, the damping coefficient in Eq. (14) is a function of time with frequency m φ , while the natural frequency in Eq. (16) is also an oscillatory function of time.
For the oscillatory damping to have an effect in the evolution of the metric perturbation, the quantity γk/m φ must be close to unity. The quantity γk is compared to m φ since the damping must occur on timescales comparable to the oscillation of φ. This implies that the energy scale M * = 1/α should satisfy We can compare this with the condition that L int remains small compared to L EH . By takingφ ∼ m φ φ 0 and identifying f and k we can write this condition as We can satisfy this equation if the gravitational wave amplitude satisfies h 0 1, which is simply equivalent to the condition that h µν is a small metric perturbation.
Before presenting a solution to the evolution equations, let us define a good measure of particle production: the occupation number. Following [4], this quantity in a given mode function is given by where as usual ω k = k 2 + m 2 χ . . We here choose γ = 0.062, as this is the smallest value of γ for which particle production occurs when k = 200m φ is the largest wavenumber used. We assume an initial scale-invariant power-spectrum for hR. Right: Total particle number for χ, calculated from thehR in the left-panel and summing over all wavenumbers. Most of the contribution to the sum comes from the larget wavenumbers, i.e. χ particles with large momenta are preferentially produced. Calculations have been done for qz from m φ to 100 m φ , qx = 0, qy = 100m φ .
We now have all the ingredients we need to explore particle production due to mediated parametric resonance. We solve Eqs. (14) and (16) numerically through the Dormand-Prince Runge-Kutta method [20]. We ignore the back-reaction of χ onh, and thus first solve Eq. (14) and then use this solution to solve Eq. (16). The integral term in Eq. (16) is treated through the convolution theorem, using a fast Fourier transform. The numerical code is gridded in momentum from a minimum value of m φ to 200m φ . Any mode with momentum k < m φ will not have time to oscillate much during the period of preheating, and thus can be ignored. The gravitational waves are assumed to start with a scale-invariant power-spectrum, but with large amplitude, since the right-handed waves are exponentially amplified during inflation. The χ field starts with Gaussian fluctuations with variance proportional to 1/k. The right panel shows the corresponding growth in particle number, calculated by summing over wavenumber in Eq. (20). Observe that there are large jumps in particle number when the amplitude of the gravitational waves becomes large. Energy must be conserved in the production of χ particles, and this energy must come from that contained in gravitational waves. The stored gravitational wave energy is proportional to the square of their amplitude, so the energy carried by them peaks just as the χ particles are being created. If we include the effect of χ back on h, the peaks in amplitude will be much smaller.
Conclusion and Discussion. We have presented a grav-itational wave-mediated preheating scenario where during its oscillatory phase, the inflaton deposits its energy to the graviton and the latter then excites matter production through parametric resonance. This mediated mechanism eliminates the need for any ad hoc direct interaction between the inflation and the matter fields, obviating issues of fine tuning the inflaton's interactions with those of the standard model. In our case, the minimal coupling between the standard model and gravity is non-arbitrary. We have provided a concrete realization of the mechanism in the context of Chaotic Inflation with a Chern-Simons coupling to the inflaton field. These models are promising because they have already been implemented to generate lepton asymmetry during the inflationary epoch, through parity-violating gravitational waves [7]. Furthermore, the notion that gravitational waves can generate preheating is generic so long as there are couplings of the inflaton to curvature invariants such as in Higgs-Inflation [21], which we leave for future investigations.