Prethermalization in early Universe
D. Podolsky, G. Felder, L. Kofman, M. Peloso
CITA (Toronto), Landau ITP (Moscow), University of Minnesota
What happens between reheating and radiation dominated
stage?
• 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
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Setup
φ – inflaton, χ – matter field, m~10-6 M P
End of inflationary stage
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Reheating
In terms of
equation of motion for modes of matter field is
, number
density
There are growing solutions
at some k (param. resonance):
Driving parameter is
At large q, instability
zones overlap
Resonance structure for k=0 in flat spacetime
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Prethermalization: results of numerical simulations
1. Effective equation of state w = p/ε
Corresp. regime of expansion is
1. Sharp change of equation of state
2. Moment of transition is non-monotonic function of g2
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2. Various components of energy density (why particle
-7
description is valid), g 2 =2.5 . 10
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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
-7
-5
g²=10
Variances can be estimated as
and
They are trivially related with effective masses (Hartree approx.):
and
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7. Intermediate regime of expansion: kinematic explanation
Contributions 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
Numerics
• 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
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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Notes
1. Instability bands for Mathieu equation:
,
2. Moment of transition
as function of coupling g:
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