Intermittent Density Perturbations from Preheating Caustics and the

Nonlinear Dynamics in the Early Universe :Intermittent Density Perturbations from Preheating

Caustics and the Shock-in-Time

Jonathan Braden

CITA / U. Toronto

Primordial Cosmology, KITP, UCSB, April 9, 2013

w/ J. Richard Bond, Andrei Frolov, and Zhiqi Huang

Jonathan Braden (CITA / U. Toronto) Nonlinear Dynamics in the Early Universe : Intermittent Density Perturbations from Preheating Caustics and the Shock-in-TimePrimordial Cosmology, KITP, UCSB, April 9, 2013 1

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Starting the Big Bang

Inflation

Cold (T 0)Few active d.o.f.

Big Bang

Hot (T > MeV )

Many active d.o.f.

Huge entropy production

But how does it happen?

Does it leave observable signatures?


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(Toy) Model for the Transition

Basic Requirements

Inflation must end

Must produce Standard Model particles couplings to other particles

Model

V (, ) =1

2m22 +

1

2g222

Background Evolution

During inflation, H m

0

3

2mt constant

After inflation H m

endsin(mt)

a3/2 oscillating


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Preheating: Linear Resonance[Kofman, Linde, Starobinski 94, 97]

Linear Fluctuations

k + 3Hk +

(k2

m2a2+2endg

2

m2a3sin2(mt)

)k = 0

0 5 10 15 20

0

2

4

6

8

10

2

f+(2 +sin2 (t))f


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Existence of Parametric Instability

Dynamical field with oscillating mass

Background field to create this oscillating mass

This is not special to this model

Model Requirments

Inflation must end inflaton oscillates about minimumInflaton must decay time-dependent parameters for coupled fields

Rich zoo of instabilities

tachyonic preheating M2eff = k2 + V () < 0 (e.g. hybrid inflation)

tachyonic resonance ( 2, F F )resonance persists with multiple inflatons


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Linear growth of fluctuations must end

Expansion weakens resonance

or

Nonlinear effects and backreaction become important

Numerical Simulations

Massively parallel

High order (up to O(dt9))symplectic integrator

I Preserve Hubble constraint

Quantum fluctuations realization of random field


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Assumed metric and Lagrangian

ds2 = a(t)2(d2 + dx2)

Lmat =

i

1

2i

i V (~)

Equations of Motion Follow from Hamiltonian

i = i/a2

i = a22i a4iV

a = a/6.

a = a3

i

(2ia6

+(i2i )

2a2

) 4V

= T 00 =

i

i2

2+

(i )2

2a2+ V (~)

Initial Spectrum

Gaussian Random Field: P(k) =1

2k, P(k) =

k2


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Evolution of ln(/)

V (, ) =

44 +

g2

222

0 5 10 15 20

0

1

2

3

4

5

6

7

8

2

f+(2 +cn2 (t|21/2 ))f


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Lavf53.3.0

lnrho_phi4.mp4Media File (video/mp4)

Can this Leave Density Perturbations Behind?

= ln(a)|H = ln(a)|Spatially modulate expansion history curvature perturbations


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Example I : V (, ) = 44 + g

2

2 22

[Bond, Frolov, Huang, Kofman 09][Bond, JB, Frolov, Huang 13]

Superhorizon isocurvature fluctuations in are converted into adiabaticfluctuations.

-2.0

0.0

2.0

4.0

6.0

8.0

10.0

12.0

0.1 1 10 100

N =

ln(a

end/a

ref)

* 10

5

(ini/mpl) * 107

raw Monte-Carlo samplesquadratic fit for small valuesbroadened transfer function

fluctuations

0T periodicity and its harmonics

Obtained from many highly accurate symplectic lattice simulations.

= G + FNL()

: decay field for inflaton (or field modulating couplings)


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Where do the spikes come from?

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

-10 -5 0 5 10

a/M

P

a/MP

-3.0

-2.0

-1.0

0.0

1.0

2.0

3.0

a/M

P

initial growth, magnified 2x105

initial growth, magnified 2x105

Veff =

44 +

1

2(2 + 2)2

In arm and out-of-armtrajectories have differentexpansion historiesTrajectories in arms pile-up focusing and caustic formation

(t + T ) = e0T(t)


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Ballistic Dynamics After Inflation

Post-Inflation N

We can effectively extend the independent trajectory approach past = 1until site-to-site coupling (ie. gradients) are important.

Following bundles of phase-space trajectories focussing/defocussingleads to caustics


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caustic_pond.aviMedia File (video/avi)

Caustics and PerturbationsFor this model, a2Htime = const. encodes = ln a|H .

11 10 9 8 7ln(0/0 )/0T

0

10

20

30

40

50

(a2H)

initJonathan Braden (CITA / U. Toronto) Nonlinear Dynamics in the Early Universe : Intermittent Density Perturbations from Preheating Caustics and the Shock-in-Time

Primordial Cosmology, KITP, UCSB, April 9, 2013 14/ 60

Spikes from Caustics

10.5 10.0 9.5 9.0 8.5 8.0 7.5ln(0/0 )/0T

0

10

20

30

40

50

60

Phase space string of ICs (more generally a sheet)

Treat homogeneous evolution along each trajectory

Horizon scale evolution obtained by ensemble averaging trajectories


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Development of Caustics


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caustic.mp4Media File (video/mp4)

End of Ballistics : Entropy Production and theShock-in-Time

(Nonequilibrium) Shannon Entropy

Sshannon = TrP[f ] lnP[f ]

P[f ] : Probability density functional

P[f] = ?

Coarse-Graining Procedure

Take all of our knowledge of the system

Maximize S subject to constraints imposed by this knowledge


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Application to Preheating

Constraint = Measured Power Spectrum

Power Spectrum P(k) Covariance matrix C (homogeneous field)Sshannon maximized by multivariate-Gaussian with spectrum P(k)

S

N=

1

2+

1

2ln(2) +

1

2NlndetC

=1

2+

1

2ln(2) +

1

2N

k

ln(P(k))

N : number of modes (or volume in case of continuum field)

Dynamical Field

Choose : f = ln((x)/) phonons (sound waves)Measurable quantity (regardless of model choice)

cannot be Gaussian


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Why ln(/3H2) : Gaussian in Position Space

Fluid variables have simple PDFs after initial transient

/

vx =aT x0+ P


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...and in Fourier Space...

Evolution of excess kurtosis for Fourier amplitudes.


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Lavf52.64.2

mompow_L10_n1024_kcut_log_new.mp4Media File (video/mp4)

...the PDFs are also Gaussian

0 < km < 20


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four_dist_real_bin1.aviMedia File (video/avi)

Power Spectrum Evolution ...


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Lavf52.64.2

PSD_g100mov.mp4Media File (video/mp4)

... leads to a Shock-in-Time[Bond, JB (in preparation)]

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6

0.0

0.5

1.0

S

Neff

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6100

101

102

1

| ln(/) |

2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6ln(a/aend)

1.00.50.00.51.01.52.02.5

3(1

+w

)ln(a

)+

ln(

)


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View on a Single Spike

1 2 3 4 5 6ln(a/aend)

0.03

0.02

0.01

0.00

0.01

0.02

0.03

a4

a4 (raw)a4 (smoothed)103 a

103 a


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The Shock and Causticsln(a) structure set at the shock


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FNL on the Sky

-1.5e-05

-1e-05

-5e-06

0

5e-06

1e-05

1.5e-05

0 2e-06 4e-06 6e-06 8e-06 1e-05

Fnl(0) Fnl(0)

2 Fnl(0)


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FNL on the Sky


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Modulation of Couplings

11 10 9 8 7ln(0/0 )/0T

1.4

1.6

1.8

2.0

2.2

2.4

2.6

g2/


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V (, ) = m2

2 2 + g

2

2 22


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Lavf53.3.0

lnrho_volray.mp4Media File (video/mp4)

Modulation via Shock Timing

0 500 1000 1500 2000time (mt)

1.0

0.5

0.0

0.5

1.0

P/

wpreshock 6= wpostshock ln(ashock)

but...

wpostshock 6= 13wpostshock depends on g

2


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Modulation in the Ballistic Regime


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Conclusions

Local Nonlinear Conversion of Isocurvature to Adiabatic FluctuationsI Presence of random field in addition to inflatonI Caustic formation seems to be quite general for motion in the bottom

of a potentialI Can also be mediated by shock-in-time surface directly

Can create localized features on the sky

Information theoretic measures of field entropyI Shock-in-Time surface with rapid production of entropy


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Documents

Intermittent Density Perturbations from Preheating Caustics and the