Gravitational Waves from Preheating in M-flation

Matrix inflation, or M-flation, is a string theory motivated inflationary model with three scalar field matrices and gauge fields in the adjoint representation of the $\mathbf{U}(N)$ gauge group. One of these $3N^2$ scalars appears as the effective inflaton while the rest of the fields (scalar and gauge fields) can play the role of isocurvature fields during inflation and preheat fields afterwards. There is a region in parameter space and initial field values,"the hilltop region,"where predictions of the model are quite compatible with the recent Planck data. We show that in this hilltop region, if the inflaton ends up in the supersymmetric vacuum, the model can have an embedded preheating mechanism. Couplings of the preheat modes are related to the inflaton self-couplings and therefore are known from the CMB data. Through lattice simulations performed using a symplectic integrator, we numerically compute the power spectra of gravitational waves produced during the preheating stage following M-flation. The preliminary numerical simulation of the spectrum from multi-preheat fields peaks in the GHz band with an amplitude $\Omega_{\mathrm{gw}}h^{2} \propto 10^{-16}$, suggesting that the model has concrete predictions for the ultra-high frequency gravity-wave probes. This signature could be used to distinguish the model from rival inflationary models


Introduction
Cosmological observations-evidenced, most notably, by recent data [1] from the Planck satellite-are best explained if we have a period of accelerated expansion, inflation [2], in the early Universe. Models of inflation usually involve one or more scalar fields coupled to (Einstein) gravity, though it is also possible that inflation is driven by gauge fields [3]. These models are specified by the form of their kinetic terms as well as the potential. It is more common to take the canonical kinetic term and define the model by its potential(s), even though inflationary models can be realized with non-canonical kinetic terms [4].
To explain the observed Universe, inflation should of course end and the energy stored in the inflationary sector should be transferred into the (beyond) Standard Model (SM) particles, an epoch known as the reheating era [2,5]. Perturbative decay of quantum fluctuations of an inflaton field (perturbative reheating) is usually not sufficiently fast and efficient, and leads to reheat temperatures that are too low to solve particle physics problems and hence to describe what we see 1 . One must therefore equip inflationary models with non-perturbative mechanisms of decay that yield sufficiently high reheat temperatures. In this context the inflaton field condensate can provide "time dependent mass terms" for other fields coupled to it, the preheat fields. This more efficient energy transfer mechanism to other (beyond Standard Model) fields, preheating, happens because of possible resonance bands [6]. The energy in the preheat fields will eventually equilibrate or thermalize through usual (perturbative) scattering processes [5].
Observable effects of inflationary models (in particular the observed CMB anisotropy [1]) are usually attributed to what happened during inflation and are related to super-horizon quantum fluctuations of inflaton fields that appear as classical background fluctuations long after inflation, and after the preheating and reheating eras [2]. The CMB data have hence been used to restrict inflationary models [1]. However the recent Planck mission data -in particular non-observation of non-Gaussianity -means that the CMB data does not provide sufficient constraints to specify the inflaton potential. Other sources of cosmic data must be sought out.
To this end, a more concrete understanding/modelling of reheating and preheating may be needed. Various inflationary models can be constrained by probing possible specific features they left during preheating or reheating. Previous analysis indicates that preheating may have detectable traces on CMB only for a specific class of exotic models [7]. If preheating occurs the turbulent, explosive and non-thermal energy transfer to the preheat sector can in principle have possible observable effects by producing a stochastic background of gravity waves typically in 10 7 − 10 9 Hz frequency band 2 [10].
The simplest scalar-driven slow-roll models (in particular, those with concave potentials 3 ) have so far passed the test very well insofar as Planck results are concerned, see e.g. [13]. Nevertheless there remain with these models a plethora of 1 Big Bang Nucleosynthesis (BBN) requires a temperature of order 1 − 10 MeV and baryogenesis requires a temperature of order 1 − 10 TeV. The reheat temperature should be at least bigger than these two temperatures. 2 Preheating can also lead to production of long-lived non-linear excitations of the scalar field which dominates the universe and can lead to stochastic gravitational wave background [8]. 3 Note that a choice of non-Bunch-Davies (excited) initial states for the cosmic perturbations can unresolved theoretical difficulties [14]. For instance, to have a successful slow-roll inflation we need to keep the inflaton mass hierarchically smaller than the Hubble scale H during inflation and quantum corrections to the inflaton potential should not spoil its flatness [15]. Moreover, in the class of large-field models there is also the problem of super-Planckian field excursions: that inflaton(s) in these scalar models typically have field displacements (in the last 60 e-folds) many times larger than M pl , in which case quantum (gravity) effects may become important [16].
It is a general belief that these and other theoretical issues regarding possible classical or quantum instabilities in an inflationary model can/should be addressed within a quantum gravity setup that is operative at some high energy (Planckian or sub-Planckian) scale. Despite providing a richer framework for inflationary model building and for addressing the above mentioned issues, being farther from SM physics, it becomes more challenging in the quantum gravity setups to make connections with physics after inflation and in particular to have a successful reheating scenario. Nonetheless, working within a string theoretic perspective, besides providing a framework to address questions about UV stability and completeness of inflationary models, usually brings another feature: there are many more fields besides the inflaton in the model. These fields can appear as isocurvature entropy modes, affecting the CMB directly, or can appear as preheat fields, affecting the production of primordial gravity waves in large frequency bands.
M-flation, which we will consider in this work, is one such model [17]. Although motivated from string theory (quantum gravity) M-flation, as we will show, has the advantage of having an embedded successful preheating mechanism in some regions of parameter space. Furthermore, the model is based on a gauge field theory, the same framework upon which beyond SM models operate, and is thus close to particle physics setups too.
In general and in a string theory/supergravity framework, depending on whether the inflaton field(s) is (are) coming from open string or closed string degrees of freedom, there are two venues for inflationary model building [18]. M-flation, in this sense, is an open string model. However it has its own specific features that may justify viewing it as a third venue. For example, as we will review in section 2, inflation in M-flation is not associated with a mobile brane, unlike all the other known open string models. M-flation is rather motivated by the dynamics of D3-branes subject to a proper RR six-form in a specific ten-dimensional type IIB supergravity background [17]. The inflaton fields of M-flation are three N ×N matrix valued scalar fields associated with the position of a stack of N D3-branes in this background. The action for M-flation, cf. section 2, will hence include U(N) gauge fields (and possibly readily change this conclusion [11]. Also if gravity is an inherently a classical theory, there will be no B-mode polarization in the CMB [12]. their spinorial counterparts in a supersymmetric setting). The model is compatible with the Planck data if inflation happens in the hilltop φ < µ region. In the symmetry breaking region, φ > µ, the model predicts a large tensor/scalar ratio, r ≃ 0.2, which is not compatible with the upper bound of 0.11 with 95% CL if one assumes that the perturbations start from a Bunch-Davies vacuum. The model could be still made compatible with Planck if we assume excited initial states for the scalar or tensor fluctuations, as pointed out in [11].
What renders M-flation theoretically appealing is not only its ability to naturally address and resolve the theoretical difficulties of standard inflationary scenarios raised above [19], but also the fact that it can connect to post-inflation physics: it comes with its own built-in preheating mechanism in some regions of parameter space with no extra parameters (compared to the inflationary background sector), and also it has the desirable form of a gauge theory (cf. discussions above).
While work on M-flation has so far been directed more toward exploring it during inflation [17,19,20], it is of appreciable importance to also address the question of its possible observable effects coming from its built-in preheating period. In particular, we focus our attention in this paper on gravity waves (GW) produced during the preheating phase following inflation, in some region of parameter space. Their observational signature is revealed by way of their power spectrum, which we numerically compute here with the help of the lattice simulator HLattice 2.0 [21].
The rest of this paper is organized as follows. In section 2, we review the basic setup of M-flation. In sections 3 and 4, we describe its embedded preheating mechanism. Then, in section 5, we proceed with computing the power spectra of GW thereby generated. Finally, section 6 presents some concluding remarks.

M-flation
Our setting is a 10-dimensional type IIB supergravity background, 4 which is probed by a stack of N D3-branes endowed with Yang-Mills gauge fields. Thus, there exist 6 spatial dimensions perpendicular to the D3-branes, whose positions within this subspace are represented by 6 N × N matrices. The role of the inflaton, according to the original M-flation setup, is assumed by 3 out of 6 matrix degrees of freedom, 5 which we henceforth denote as Φ i , i = 1, 2, 3. The inflaton matrices are, by construction, in the adjoint representation of the U(N) gauge group; therefore they are non-commutative as well as Hermitian.
In principle, the dynamics of these matrices is very complicated (increasingly so with larger N), as one has any number of possible configurations of the D3-branes within the chosen background. However, there is a way to simplify the situation and make it computationally tractable. As we will elaborate in the next subsection, the classical dynamics of this model can be consistently truncated to a solution where the N D3-branes are uniformly distributed along the surface of a 2-sphere (within the 6-dimensional orthogonal subspace), and their positions on this sphere do not change during inflation. What instead changes is the sphere's radius, which thereby plays the role of an effective scalar inflaton.
Aside from the above, many other solutions-that make use of more of the available (classical) degrees of freedom-are of course possible. This possibility was considered in [20] and generically appears as a multi field inflationary model. In this work, however, we focus on the single field model where the other "unused" degrees of freedom in this particular solution will be identified with preheat fields after inflation ends.

Action and equations of motion
We work in the (−, +, +, +) metric signature, and use boldface to denote matrices of dimension N. The effective (3 + 1)-dimensional action of M-flation [19] comprises Einstein gravity, minimally coupled to a Yang-Mills gauge field A µ and the three inflaton matrices Φ i , is the gauge covariant derivative. Moreover, the potential is given by where in (2.1) and (2.2) there is a sum on repeated i, j, k indices and the three coupling constants have various stringy meanings: λ = 8πg s = 2g 2 YM is related to the string coupling g s , κ =κg s √ 8πg s is related to the Ramond-Ramond antisymmetric form strengthκ, and m is a parameter that multiples the three spatial coordinates along the D3-branes in the metric of the background SUGRA theory [17]. To ensure a constant dilaton therein, we must also impose the constraint λm 2 = 4κ 2 /9 [17].
The equations of motion for the scalar and gauge fields that follow from the action (2.1) are

Truncation to the SU(2) sector
The dynamics determined by the equations of motion (2.3) and (2.4) can generically be quite complicated, but this may be simplified considerably as follows. Let we decompose the inflaton matrices into two parts, one parallel and one perpendicular to the N ×N representation of SU (2), respectively (that is Tr(J i Ψ i ) = 0). It was shown in [17] that if Ψ i =Ψ i = 0 initially, then (2.3) implies that Ψ i will remain vanishing for all time. Analogously, if A µ is also initially turned off, then the commutator in (2.4) will not source F µν , and therefore the gauge field always stays turned off as well.
Hence, it is possible to consistently restrict the classical dynamics of this model to a sector where Ψ i = A µ = 0, so that the inflationary trajectory is determined solely byφ, the length of the inflaton matrices along the direction of SU(2). This realizes precisely the picture described earlier of the D3-branes fixed upon the surface of a 2-sphere with variable radius, now identified with the value of effective inflaton fieldφ.
Concordantly, the vanishing Ψ i and A µ fields are referred to as spectators. Upon setting them to zero, the action (2.1) simplifies propitiously to where TrJ 2 i = N(N 2 − 1)/4, using the properties of SU(2). Performing a field redefinition φ = TrJ 2 iφ brings the inflaton to a canonically normalized form, yielding which is the familiar single scalar field inflationary action.
Defining effective couplings λ eff ≡ 8λ/N(N 2 − 1) and κ eff ≡ 2κ/ N(N 2 − 1) and then using the constraint that the background is a solution to the supergravity equations of motion with constant dilaton, λm 2 = 4κ 2 /9, the effective potential can be written, with µ ≡ √ 2m/ √ λ eff , simply as Thus, in the SU(2) sector, the inflationary potential of M-flation assumes the form of a symmetry-breaking potential. It has two global minima: one at φ = µ (corresponding to a supersymmetric vacuum, when the N D3-branes blow up into a giant D5-brane wrapping a fuzzy two sphere) and one at φ = 0 (corresponding to the trivial solution, when the matrices become commutative). For typical inflationary trajectories determined by this potential, all necessary parameters can be obtained by demanding certain standard requirements (namely, 60 e-foldings of inflation, together with a COBE normalization of δ H ≃ 2.41 × 10 −5 and a spectral index of n s = 0.96). The resultant numerical values are as follows. Further details about this analysis and the corresponding slow-roll trajectories in M-flation may be found in [17,20]; here we just quote the results.
Taking n S ≃ 0.96, the tensor/scalar ratio turns out to be 0.2 which is outside the 2σ allowed region of Planck in the n S − r plane. One can render this region of M-flationary phase space compatible with the data by assuming the modes start from a non-Bunch-Davies vacuum [11].
To fit the observational constraints we find In this case we obtain Due to the φ → µ − φ symmetry of the background, the curvature perturbations in regions (b) and (c) turn out to have the same spectral tilt n S = 0.96 and tensor-toscalar ratio r = 0.048. These predictions are within the 1σ region of Planck -allowed parameter space. The two regions (b) and (c), however, could be distinguished by their predictions for the amplitude of isocurvature perturbations at the Hubble scale [17], as the masses of isocurvature modes do not satisfy the symmetry φ → µ − φ, which the classical background enjoys. As we will see, in region (c) the model has an embedded preheating mechanism that leads to observable gravity waves in the high frequency region.
In this way, M-flation resolves all of the problems raised earlier vis-à-vis single scalar field inflation for several reasons. First, its effective couplings can easily be made naturally small, provided N is chosen to be sufficiently large. For example, N ≈ 48 000 D3-branes turn out to suffice in this case for ameliorating any hierarchy problem. Second, the total amount of field displacement during M-flation has been argued [19] to be less than the UV cutoff of this model, so there is no trans-Planckian problem. Finally, this approach suggests a clear physical meaning for the inflaton, namely the radius of the two-sphere on which D3-branes live.
Despite its theoretical successes, M-flation has not been up to now extensively exploited in terms of deriving observationally testable predictions that may help set it aside from rival inflationary models. This is what we turn our attention to next, in the context of preheating.

Preheating in M-flation
The preheating mechanism after inflation in typical models of inflation necessitates the introduction of one or more extra matter fields, or preheat fields, into which the inflaton presumably ought to decay [6]. M-flation comes with this feature tacitly built-in, by way of its spectators Ψ i and A µ . Although, as discussed, these are assumed to be turned off classically, they can nevertheless be excited quantum mechanically. During inflation, these quantum fluctuations can cross the horizon and can become observable as isocurvature perturbations. The amplitude of the largest modes in each inflationary region was computed in [17], and shown to be generically too small to have observable effects. After inflation, however, they appear as preheat fields which can have observable effects on the GWs produced in this era.
To this end, we need to study the equations of motion and quantize their solution. We hence start withΨ i andÂ µ as perturbations in the action (2.1)-with the hats denoting "quantumness"-and deduce the resulting equations of motion. As usual in inflationary cosmic perturbation theory we assume these perturbations to be of the same order and both be much smaller than the background field values and hence keep only the first order terms in these perturbations in the equations of motion. In either case, these will take the expected form of Mathieu equations suitable for preheat fields. We discuss each case separately.

Scalar preheat fields
SettingÂ µ = 0 and expanding (2.1) to quadratic order inΨ i , we get [19]: where there are two solutions for the scalar spectator masses, dubbed α-modes and β-modes respectively: 2) with degeneracy 2j + 1 for each mode. The above φ−dependent masses, besides a bare mass, induce both types of φ 2 χ 2 and φχ 2 interactions for the preheat fields χ.
It can be easily shown that if inflation happens in the region (c), the above masses for α− and β− modes become tachyonic for an interval during the preheating era, if j > j min . For α-modes, j min = 94 and for β− modes j min = 16. For these modes, we have to alleviate the problem by including the corrections up to quartic order in Ψ i 6 . We get: and where and (:::) and {:::} respectively denote Wigner 3j and 6j symbols [23].
We remark that the cubic couplings (3.4) are linearly dependent on the inflaton, whereas the quartic ones (3.5) are manifestly independent (i.e. they are constants for a given j). Moreover, for reasonable values of φ, it is plain to see that in virtue of the fact that the left-hand side is proportional to products of Wigner symbols, while the right-hand side is proportional to large sums of products of squares of Wigner symbols 7 . Consequently, we can treat the cubic terms as negligible. Λ Ψ in general is mode dependent, however, one can show that for large j it becomes j-independent and is Λ Ψ ≃ 1.0069 × 10 11 λ eff 4 . (3.8) One can therefore take the potential of any scalar (α or β) modeχ to be Performing the usual Fourier decomposition the corresponding equation of motion can then be written as As we will see, this has the familiar form of a Mathieu equation in the regime where φ is oscillating about the vacuum (modulo the last term which, as discussed, was included to keep the potential bounded from below), and can therefore lead to parametric resonance.

Gauge preheat fields
The story here proceeds along similar, albeit slightly simpler lines. SettingΨ i = 0 and expanding (2.1) to quadratic order inÂ µ yields [19]: where the mass spectrum is given by The degeneracy for j = 0 is 2 (corresponding to massless gauge fields) while for j ≥ 1 is 3(2j + 1), the factor of three corresponding to the three polarizations of a four dimensional massive vector field. Unlike the scalar case, though, because (3.12) only contains a φ 2 term, we need not worry about the danger of acquiring tachyonic masses and the higher order corrections will always remain small compared the leading quadratic terms. 8 We can therefore safely ignore all higher-order corrections and write the equation of motion for the Fourier modes A k of the gauge preheat fields as Despite the fact that the Hubble friction term appears with a different coefficient than in the scalar case (3.10), we still get a Mathieu equation when the inflaton φ oscillates around its minimum toward the end of inflation.
The next question to ask is then what the parametric resonance idiosyncratic to (3.10) and (3.13) can give us. A potentially rich and predictive product thereof is GW production.

Parametric resonance 4.1 SUSY-breaking vacuum
If the initial condition is such that inflation happens in regions (a) or (b), the inflaton will finally end up oscillating around the SUSY-breaking vacuum, φ = µ. It might be thought the inflaton oscillations around the vacuum, φ = µ, and its couplings to different preheat fields can create parametric resonance. However, it can be shown that the rest masses of α and β modes in this region are so large that non-adiabatic particle production is suppressed. To be specific, let us focus on α−modes and β−modes. A similar analysis and argument could be repeated for the gauge modes as well.
The mass functions for the α and β modes can be unified in the following form where Expanding the interaction term around the SUSY-breaking vacuum φ = µ and introducing the variable ϕ ≡ φ − µ, the interaction term between the inflaton and spectators looks like 9 where and ϕ varies between zero and Φ = µ − φ f ≃ 1M pl . Despite the existence of interactions like ϕχ 2 , since the rest masses of all theχ fields are larger or equal to the mass of the inflaton, perturbative decay of the inflaton to none of theχ fields is possible. 10 Around the SUSY-breaking vacuum, the inflaton potential to a large extent resembles 1 2 m 2 ϕ 2 . Therefore, the inflaton has an oscillatory behavior ϕ(t) ≈ Φ sin(mt) [6] around the SUSY-breaking vacuum. It can be shown that the contribution of the g 2 4 φ 2χ2 interaction is subdominant with respect to the g 3 φχ 2 for all ω's. The ratio of two interactions is (4.5) 9 In the rest of the analysis we will drop the quartic Λ Ψχ 4 term. As we will see in the next subsection presence of this term weakens the particle production and thus strengthens our results. 10 For the same reason the tachyonic resonance of [24] does not occur in our case.
For all values of ω > 0 this ratio is less than one 11 , since the ratio Φ/µ 0.04 in both the (a) and (b) regions. Thus we will drop this quartic interaction term in comparison with the cubic one in the rest of the analysis.
Let us analyze (3.10) in a non-expanding background where a = 1. Dropping the contribution of the quartic interaction, for an oscillating inflaton the approximated equation takes the form Introducing the new variable z ≡ mt 2 + π 4 and ′ ≡ d dz , the equation takes the form of a Mathieu equation [25]χ It is known [26] that equation (4.7) has solutions with an exponential instabilitŷ χ ∝ exp(µ (n) k z) that represent a burst of particle production. The solutions have resonance bands with the width ∆k (l) ≃ q l . If q ≪ 1, what is known as narrow resonance band, the resonance occurs in bands near A k ≃ l 2 , where l is a nonzero integer. Hence the widest band is the first instability band. Imposing the condition q < 1 for the inflationary region (a) where µ ≃ 26 M pl , only 0 ≤ ω ≤ 2 (1 ≤ j ≤ 3 β modes) lead to narrow resonance. In the region (b), where µ < 36, besides the aforementioned modes, ω = −3 (j = 1 α mode) can also lead to narrow resonance. The factor µ k , the Floquet index, for the first instability band is given by [6] where the resonance happens for the narrow momentum k range 1 − q 2 ≤ 2k m ≤ 1 + q 2 . It obtains its maximum at µ k = q/2 at k = m/2.
In an expanding background the redshift of momentum k from the resonance band can prevent the resonance. As pointed out in [6], the condition for the first band to be effective during expansion is q 2 m H. The inequality is not satisfied for the modes that can undergo parametric resonance in flat space-time. This is because during preheating H ≃ 0.1m [6] and Φ 2 /µ 2 1.5× 10 −3 . Thus narrow parametric resonance for these modes cannot lead to preheating.
For larger values of ω, the resonance is broad. However, one can show that the large rest mass of these modes, mχ = m(ω + 1), and the smallness of the amplitude of oscillations with respect to the supersymmetry-breaking vacuum µ, shuts off the particle production. To see this, let us note that the time-dependent frequency in the equation of motion forχ in an expanding background is given by The condition for the adiabaticity violation is that a condition that cannot be satisfied for large values of ω due to the smallness of Φ/µ. Similar arguments can be given for the gauge spectator modes.
Recapitulating our results, it is not possible to reheat M-flation around the SUSY-breaking minimum via any of the α, β or gauge spectators modes. The supersymmetric model is equipped with fermionic spectators that might contribute to this process. Nonetheless, due to Pauli exclusion, resonances cannot happen for fermionic modes and considering them will not change the above result.

Supersymmetric vacuum
Unlike the supersymmetry breaking vacuum, parametric resonance around φ = 0 (supersymmetric vacuum) can be quite effective through the spectator modes. We first focus on the scalar preheat fields. The equation of motion for the perturbations Ψ i can be decomposed into the equation of motion for the α and β spectator modes which in Fourier space takes the form The bare masses of the spectator modes are equal to the inflaton mass m 2 and in principle for large values of ω, the adiabatic condition may be broken violently. However, as we will see, self-interactions of theχ particles, incorporated in the last term of the equation of motion, slows down the parametric resonance.
In terms of the dimensionless time variablez, defined as z ≡ mt, (4.15) the equations of motion for the inflaton and the background are The equation of motion for the Fourier mode, X k ≡ a 3/2χ k , is where (4.20) Eq. (4.19) can be solved imposing the Bunch-Davies vacuum on the mode X k at the beginning of preheating. The number density for the produced particles is [6] n To demonstrate the contribution of the cubic term to the comoving number density, we have numerically solved the equations for perturbation in the presence and absence of the cubic contribution to the equations of motion (4.19) for k = 0 for the largest j β−mode. As can be seen in the L.H.S. graph of Fig. 1, in the absence of the cubic term, the number density of the produced particles exhibits stochastic resonance behavior [6], i.e. it typically increases at some specific moments but it may decrease as well. In between these instants, the number density remains approximately constant (sharp oscillations on the plateaus are only numerical artifacts). The interval between the kicks in n k is roughly about π, which is the small interval in which the mode becomes massless and tachyonic. However, once the cubic term (from the quartic self-coupling term) is added to the equation of motion (4.19), n k ceases to exhibit resonance behaviour initially, its value being highly suppressed. This continues until the cubic term in the equation of motion of the scalar spectator redshifts and the mode revert to resonance behaviour.  For the gauge mode the equation of motion is given by (3.13). Introducing the new variable the equation takes the following form Again (4.24) can be solved numerically imposing the Bunch-Davies vacuum in infinite past for the A k .
We have numerically solved (4.24) for k = 0. As it can be seen in Fig. 2 the gauge mode number density of produced particles, which is given by [6] also demonstrates stochastic resonance behaviour. Note that the 1/a 2 factor in n A k will in principle cause the gauge mode particles to dilute. The comoving number density of the particles overall increases more slowly due to the expansion of the universe. The production of gauge modes happens in the brane-antibrane inflation too [9].

Gravity Waves from preheating around the SUSY vacuum
Effective preheating can lead to explosive particle creation and, consequently, the production of stochastic Gravitational Waves (GWs) [10]. The latter arise from the tensor modes h ij of perturbations to the FRW metric, and are linked to the former via the perturbed Einstein equations, where δS TT ij is the transverse-traceless part of the stress tensor perturbation δS ij = δT ij − 1 3 δ ij δT k k which depends by construction on the number density and energy of the preheat fields. This stress-tensor perturbations are receiving contribution from the particles produced during the preheating era discussed in the previous section, which in turn source the gravity waves through (5.1).
Recalling that the Landau-Lifshitz pseudotensor [27] associated with gravitational radiation is T µν = h ij,µ h ij ,ν /32πG, we can write the ratio between the spectral energy density thereof and the present-day total energy density as where f denotes the GW frequency. Using this, it is in principle possible to compute the power spectrum, Ω gw h 2 .
Of course, the dynamics involved are highly nonlinear and far too complicated to render this task analytically tractable; instead, we resort to numerics. Thus, to determine the power spectrum of GW generated during preheating after inflation by the various scalar and gauge modes described in the previous section, we employ the lattice simulator HLattice 2.0 [21].
HLattice is generically designed to solve equations of motion via a numerical scheme known as symplectic integration, which is typically very stable and often used for long-term many-body simulations in astronomy and particle physics. The basic idea of how it works is as follows (for a detailed overview, the reader is referred to [21]). Spatial coordinates are discretized on a three-dimensional lattice -in our cases, with 64 grid points along each edge-and time evolution is achieved by considering the Hamiltonian H of the system which, in lieu of a spatial integral, can be written as a sum over all of the lattice points. Then, any arbitrary function F evolves via where {·, ·} is the Poisson bracket andĤ is the corresponding functional operator. The solution is thus An n-th order symplectic integrator is constructed by factorizing exp(Ĥdt) into a product of exponentials of the constituent (kinetic and potential) terms of the Hamiltonian up to O(dt n+1 ). While HLattice 2.0 is in principle able to implement this up to sixth order (using a fourth order Runge-Kutta subintegrator, with a time step much smaller than dt, to solve the resulting equations of motion), we simply used its second order symplectic integrator in obtaining all of the results that follow, for the sake of keeping computational times manageable.

GW from scalar modes
The power spectra of GW due to the most massivei.e. highest j-scalar modes (both α and β) are shown in Figure 3. The scale factor is normalized to a = 1 at the end of inflation/beginning of preheating, and we carry out the computation up to a = 14, when the spectrum becomes UV dominated. Indeed, after preheating, field energies typically cascade towards the UV, 12 and in HLattice this renders all further (higher a) computations non-physical because of the finite resolution of the simulator as well as its lacking treatment of quantum effects at very high wavenumbers [29].
To illustrate this, we plot the kinetic energy spectrum of the highest j α mode in Fig. 4 and observe that it starts to be dominated at the UV end for a ≥ 14.
12 Note that all simulations start out (small a) "UV dominated" and have larger energies at larger wave lengths. However, they do not remain so. But, there at a later time (larger a) which become UV dominated again. We remark that, as is seen in Fig. 3, the two α and β types of scalar preheat fields produce very similar GW spectra, as may well be expected from inspecting their masses (3.2) and quartic couplings (3.5): For large j, both α and β type preheat fields have an approximate mass of and quartic coupling of In producing these graphs we have assumed that N = 48000. We have also taken the largest j α and β modes individually, i.e. j = 48000 single β and α mode.

GW from gauge modes
The GW power spectrum due to the most massive gauge mode, up to a = 7, before the UV domination kicks in, is shown in Figure 5. Again we have focused on the That is, the plot shows, largest j gauge mode, j = 47999. As in the scalar mode case, the amplitude grows with increasing scale factor under the clear effect of parametric resonance. However the growth is much faster: amplitudes become as large as 10 −11 by a = 7, at which point the computations become UV dominated. The difference between gauge and scalar modes is essentially coming from the difference in their corresponding equations, and in particular the difference between Ω k (4.20) andΩ k (4.25). The delay in the enhancement of GW spectrum from scalar modes could be traced back to the fact that the presence of cubic coupling term in their equations of motion generically slows down the resonance. To compare the contributions to the total GW spectrum from the scalar and gauge modes, they are plotted together in Figure 6. The spectrum from a single gauge mode is also flatter in comparison with its scalar counterpart, but still a double hump feature of the gravity profile from preheating can be distinguished.
Thus, we see that the spectrum of GW produced by preheating following Mflation is dominated by the gauge preheat fields, which give rise to GW amplitudes more than 10 orders of magnitude greater (at a=7) than those due to (either type of) their scalar counterpart.

GW from several gauge modes
As noted above the spectrum of GWs from the gauge modes dominate the scalar modes by a factor of 10 orders of magnitude. This suggests that if all three modes are run together as the preheat fields, the gauge modes are more effective in the production of GWs. However this was done for a single scalar or gauge mode and at large j there are ∼ 2j (for scalars) and ∼ 6j (for vectors) such modes for a given j. In principle one should consider the effects of all the degenerate modes. It may seem from (5.1) and (5.2) that the GW power spectrum should grow like j 2 ∼ N 2 . However, given the highly nonlinear character of these equations this expectation can only hold for a very short time in the very low frequency region where the nonlinear effects are negligible. The larger the degeneracy, the earlier the UV domination, and hence modes have a shorter growth time. This is compatible with the analysis of [28]. However one should note that in the study of [28] the preheat modes are scalar fields, whereas the ones in our simulations are gauge modes, i.e. they appear with the friction term proportional to H, instead of 3H in the equations of motion. Given the fact that for large j gauge modes have a 6j degeneracy, to check the effects of degeneracies in our setup we should simulate the effect of 3 × 95999 = 287997 gauge mode as preheat fields. This number is quite huge and cannot be handled without substantial computational resources. To get an idea of the effects of degeneracy, we tried three and six gauge modes 13 .
To explore the degeneracy effects more clearly we have shown the spectrum of GWs from one, three and six largest j gauge modes in the same plot, Fig. 7. Although these data are not enough for making a very sharp deduction, they still exhibit the following features: • Time dependence. At the beginning of preheating, low a up to a = 3, the amplitude of the GW spectrum resulting from the three and six gauge preheat modes, is larger than that of single mode. As pointed out in [28], this is the 13 We should note that the simulation of a single mode with highest j-number up to the onset of UV domination took a week to perform on the Sharcnet cluster of the University of Waterloo. In comparison with previous studies on gravitational wave production from preheating, this is due to the large value of coupling of the inflaton to the preheat field which is of order λ eff N 2 ∼ 1.6 × 10 −4 . stage the inflaton is coherently oscillating around its minimum and non-linear effects have not kicked in yet. However, as the inhomogeneities of the inflaton grow, gravitational radiation is counteracted by the backreaction and the model with multiple preheat fields stops being efficient; nonlinear effects suppress the degeneracy effects and we see no large degeneracy effect. Moreover, UV domination happens earlier (at lower a) for larger degeneracy such that the amplitude of GWs is almost degeneracy independent.
• Frequency dependence. Besides the amplitude of the produced GWs, frequency is the distinctive observational feature in our model. Our current data with six gauge preheat modes already shows that the GWs of our model are in the 1 − 3 GHz band and they are almost flat with amplitudes around 10 −16 .
Revealing the exact amplitude of the GW spectrum and its finer features in this range needs an analysis with a larger number of modes.

Concluding remarks
In this work we extended the analysis of [17,15,20] on the M-flation model. As discussed, M-flation helps with the resolution of many of the principal theoretical difficulties endemic to standard scalar field inflationary models. Moreover, M-flation is also able to furnish concrete observational predictions courtesy of its built-in preheating mechanism around the φ = 0 vacuum. In search for possible, beyond CMB, observational signatures of M-flation we have analyzed the power spectra of gravitational waves produced in this model due to the different types of its preheat fields. We have found that the gauge preheat fields contribute overwhelmingly to this process as compared to their scalar counterparts, producing a large amplitude spectrum in the few GHz band with an amplitude of order 10 −16 . It is hoped that such a spectrum could be observed by ultra-high frequency GW detectors that may be able to probe the GHz band, such as the Birmingham HFGW resonant antenna [30] or the one at Chongqin University [31]. The Birmingham detector works based on the detection of the rotation of the polarization vector of an electromagnetic wave induced by the interaction between a gravitational wave and the polarization vector of the electromagnetic wave. The sensitive frequency range is at 10 8 HZ. The Chongqing detector exploits the electromagnetic interaction of a Gaussian beam propagating through a static magnetic field. These detectors work based on different principles from the phase measurement with the laser interferometry developed in the groundbased large-scale interferometers around few hundred Hz.
One should note that the GW spectrum we discussed in this paper is in the high frequency range, and is in addition to the spectrum of gravity waves (tensor modes) that the model produces at the CMB scales, with the tensor-to-scalar ratio r ≃ 0.048 [17]. In addition the lightest spectator mode in this inflationary region will create a substantial amplitude of isocurvature perturbations with amplitude P S /P R ≃ 5×10 −3 which has a degeneracy of three [17]. 14 These features could be used to distinguish M-flation in this region from other inflationary models.
acknowledges the hospitality of Uppsala Institute for Theoretical Physics during the completion of this work.