Zeldovich-90, 21.12.04 1

Prethermalization in early Prethermalization in early UniverseUniverse

D. PodolskyD. Podolsky, G. Felder, L. Kofman, M. , G. Felder, L. Kofman, M. PelosoPelosoCITA (Toronto), Landau ITP (Moscow), University of Minnesota

• Sharp change of equation of state from non-relativistic to relativistic: reheating• Prethermalization: soon after the end of reheating spectrum is close to thermal at interesting energy scales• “Intermediate” regime of expansion: effective equation of state w=1/4, due to non-trivial time dependence of effective masses

What happens between reheating and radiation dominated stage?

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Setup

End of inflationary stageφ – inflaton, χ – matter field, m~10 M-6

P

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ReheatingIn terms of equation of motion for modes of matter field is

, number density

Driving parameter is

Resonance structure for k=0 in flat spacetime

There are growing solutionsat some k (param. resonance):

At large q, instabilityzones overlap

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Prethermalization: results of numerical simulations1. Effective equation of state w = p/ε

Corresp. regime of expansion is

1. Sharp change of equation of state2. Moment of transition is non-monotonic function of g2

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2. Various components of energy density (why particle description is valid), g =2.5 102 . -7

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3. Total (comoving) number densities

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4. Energy densities per mode k

Various curves correspond to various moments of time, separation is Δt = 4π/m;curve in the box is Rayleigh spectrum k³, corresponding to

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5. Spectra of number densities (in log scale)

• Direct cascade – possibly, this regime can be described in terms of weak turbulence (Tkachev, Micha (2004), KP (work in progress))• Tendency for creation of χ-condensate• Prethermalization – at interesting scales spectrum is close to thermal soon enough after the end of reheating

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6. Means and variances

g²=2.5 10 g²=10

Variances can be estimated as and

They are trivially related with effective masses (Hartree approx.):

and

-7. -5

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7. Intermediate regime of expansion: kinematic explanationContributions of particles into energy density may be estimated as

Variances are:

and

Since number densities are slow functions of time, one has

0-0 component of Einstein equations gives

i.e., w=1/4 in the beginning of evolution

Physical reason is non-trivial change of effective masses with time in expanding Universe

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Conclusions

• Sharp change of equation of state due to preheating• Prethermalization – soon after the end of reheating spectrum is close to thermal at all physically interesting scales

Numerics

Kinematics

• Intermediate regime of expansion after reheating, corresponds to effective equation of state w ≈ ¼. Explanation: non-trivial dependence of effective masses on time in expanding universe

Kinetics

• Much work to be done – turbulent thermalization of quantum scalar fields, relation to weak turbulence theory (in preparation, 2005)

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Notes1. Instability bands for Mathieu equation: ,

2. Moment of transition as function of coupling g: