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
/ 60
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 2
/ 60
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?
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 3
/ 60
(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
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 4
/ 60
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
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 5
/ 60
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
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 6
/ 60
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
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 7
/ 60
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
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 8
/ 60
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
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 9
/ 60
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
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 10
/ 60
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)
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 11
/ 60
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)
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 12
/ 60
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
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 13
/ 60
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
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 15
/ 60
Development of Caustics
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 16
/ 60
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
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 17
/ 60
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
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 18
/ 60
Why ln(/3H2) : Gaussian in Position Space
Fluid variables have simple PDFs after initial transient
/
vx =aT x0+ P
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 19
/ 60
...and in Fourier Space...
Evolution of excess kurtosis for Fourier amplitudes.
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 20
/ 60
Lavf52.64.2
mompow_L10_n1024_kcut_log_new.mp4Media File (video/mp4)
...the PDFs are also Gaussian
0 < km < 20
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 21
/ 60
four_dist_real_bin1.aviMedia File (video/avi)
Power Spectrum Evolution ...
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 22
/ 60
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(
)
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 23
/ 60
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
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 24
/ 60
The Shock and Causticsln(a) structure set at the shock
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 25
/ 60
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)
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 26
/ 60
FNL on the Sky
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 27
/ 60
FNL on the Sky
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 28
/ 60
FNL on the Sky
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 29
/ 60
FNL on the Sky
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 30
/ 60
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/
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 31
/ 60
V (, ) = m2
2 2 + g
2
2 22
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 32
/ 60
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
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 33
/ 60
Modulation in the Ballistic Regime
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 34
/ 60
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
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 35
/ 60