Preheating after N-flationShinsuke Kawai (Helsinki Institute of Physics, University of Helsinki)

PRD77(2008)123507 [arXiv:0803:0321] with D. Battefeld

Abstract: We study preheating in N-flation, assuming the Marcenko-Pastur mass distribution, equal-energy initial conditions at the beginning of inflation and equal axion-matter couplings, where matter is taken to be a single, massless bosonic field. By numerical analysis we find that preheating via parametric resonance is suppressed, indicating that the old theory of perturbative preheating is applicable. While the tensor-to-scalar ratio, the non-Gaussianity parameters and the scalar spectral index computed for N-flation are similar to those in single-field inflation (at least within an observationally viable parameter region), our results suggest that the physics of preheating can differ significantly from the single-field case.

Abstract: We study preheating in N-flation, assuming the Marcenko-Pastur mass distribution, equal-energy initial conditions at the beginning of inflation and equal axion-matter couplings, where matter is taken to be a single, massless bosonic field. By numerical analysis we find that preheating via parametric resonance is suppressed, indicating that the old theory of perturbative preheating is applicable. While the tensor-to-scalar ratio, the non-Gaussianity parameters and the scalar spectral index computed for N-flation are similar to those in single-field inflation (at least within an observationally viable parameter region), our results suggest that the physics of preheating can differ significantly from the single-field case.

IntroductionIntroductionIntroductionIntroduction

Mass spectrum of inflatonsMass spectrum of inflatonsMass spectrum of inflatonsMass spectrum of inflatons

Dynamics of N-flationDynamics of N-flationDynamics of N-flationDynamics of N-flationN-flation[1] is a specific realization of multi-field inflation motivated by KKLT compactification of type IIB string theory. Advantages of this model are that the initial values of inflaton fields can be chosen to be sub-Planckian, and that the mass spectrum of inflatons is calculable under some assumptions. Observable cosmological imprints (the tensor-to-scalar ratio, the non-Gaussianity parameter and the spectral index) have been computed and are found to be similar to single-field inflation. While this suggests that N-flation is compatible with current observational data, it also means one cannot distinguish N-flation from standard inflation by these data alone.

• Slow-roll conditions

We assume equal-energy initial conditions at the onset of inflation:

An advantage of N-flation is possibility to avoid super-Planckian initial conditions. So we shall set

• Light axion dominance

We study the process of preheating for this model. Result: it is very different from the single-field case.

• Analytic solutions

The Lagrangian

where : hundreds ~ thousands Easther and McAllister[2] proposed Marcenko-Pastur distribution of inflaton masses (based on random matrix):

(from renormalization of Newton’s const)(COBE normalization)

We use rescaled mass parameter

Inflatons (axions) interact only through gravity

• Initial conditions

(1) for each inflaton and

(2) for the whole system, where

The conditions (1) are violated for heavy inflatons first, even when the whole system still undergoes inflation.

The MP law prescribes the majority of the axions to be distributed around the lightest mass

By the time the slow-roll parameter for the whole system (2) becomes of order unity, heavy axions are settled down at the bottom of the potential and become negligible. This is due to:

Heavy fields start from smaller values due to

Heavy fields undergo overdamped evolution as the Hubble parameter is still large when they exit slow-roll regime

the equal-energy initial conditions

The slow-roll approximation yields analytic solutions for the field evolution:which together with the initial conditions describe the axions during N-flation. Because of the light axion dominance this approximation is valid before preheating commences, provided the heavy fields are ignored.

Preheating after inflationPreheating after inflationPreheating after inflationPreheating after inflation

ReferencesReferencesReferencesReferences

• The model of preheating

When the masses of the axions are all equal, the multi-field model reduces to a single-field model, by setting ; Eqn (3) becomes

• N-flation with MP mass distribution

• Equal-mass = single-field model (for comparison)

It is well known that this system exhibits parametric resonance[3], driven by the oscillating inflaton.

Ignoring backreaction (justified below), the EOM are

We will model preheating with a single massless bosonic field coupled to the axions via the same coupling strength. The Lagrangian is

Choose (sufficient e-folding) and use only the lightest 10%, , for the preheating computation (justified by the light axion dominance)

(3)

(a) The time evolution of the matter field. Cosmic expansion causes occasional decreases of the amplitude (stochastic resonance).

(b) The time evolution of the comoving particle number defined by

where and

Increasing the number of inflatons (with the same masses) does not change the resonance effect since it is equivalent to the single-field model of inflation.

(the bottom panel) The oscillating part ofthe mass term that drives the resonance.

In N-flation with MP-mass distribution the collective behavior of axions is not coherent (the oscillation of each field being dephased due to mass differences), leading to suppression of resonance.

In short time scales

In long time scales

(a) The time evolution of the matter field. While there is some enhancement due to parametric resonance, the overall amplitude decreases as the enhancement is not strong enough to overcome the dilution due to cosmic expansion. The parameters are chosen so that the single-filed model (the left column) is recovered if we set all the masses to be equal.

(b) The oscillating part of the matter mass term (to be compared with the bottom panel of the left column). The oscillations are blunt due to the relative mass differences of the axions.

(c) The adiabaticity parameter which indicates the strength of resonance effect (when its modulus is large the system is less adiabatic and particles may be produced due to the resonance effect). In the equal-mass case it oscillates with amplitude larger than 1, so we can see the resonance effect is extremely weak in the MP case.

(a) The time evolution of the matter field, in longer scales (time is measured in units of ). There is a burst around

(b) The comoving occupation number computed for the comoving matter field X. The particle number increases at the burst, but such bursts are not frequent enough to overcome the dilution due to cosmic expansion.

(c) The oscillating part of the mass term, showing beats due to oscillating axions.

(d) The adiabaticity parameter, in longer time scales than above. The burst occurs at one of the peaks, but not at every peak (note that inclusion of cosmic expansion renders the resonance effect stochastic).

• Summary of the results

In N-flation with MP mass distribution, the resonance effect is weak in short time scales and not frequent in long time scales. Consequently, the old theory of perturbative reheating applies to this scenario.

[1] Dimopoulos, Kachru, McGreevy, Wacker, arXiv:hep-th/0507205[2] Easther, McAllister, JCAP 05 (2006) 018[3] Kofman, Linde, Starobinsky, Phys. Rev. D 56 (1997) 3258

See the paper for the complete list.