Gravitational Waves From the Post-Inflationary Universe

Richard Easther (Yale)

With: Peter Adshead, Mustafa Amin, Richard Anantua, Hal Finkel, Raphael Flauger, James Gilmore, Eugene Lim, Avi Loeb, Michael Mortonson, Hiranya Peiris, Jonathan Pritchard & Nathaniel Roth

Introduction...

• Gravitational waves from inflation well known (as an idea!)

• CMB + BBO / DECIGO style gravitational wave experiments

• Today: stochastic backgrounds generated after inflation

• Preheating and parametric resonance

• Prolonged matter dominated phase

• Decaying primordial black holes

Bigger picture...

• Gravitational wave source: frequency ~ physical timescale

• Example: Few x solar mass BH-BH merger

• IMSO: A few x 10 km

• Orbital period: a bit less than c / (2πr) ~ 1 kHz

• To go higher => need smaller IMSO => smaller mass

• M < 1.4 solar mass, physical radius >> IMSO

• Maybe primordial binary black hole inspirals?

Bigger Picture II

• GW Signal at frequencies >> 1 kHz: early universe

• Stochastic background (turbulence / bubbles), or PBH.

• Direct detection efforts focussed on “low” frequencies

• Very sensibly too: we know there are sources to see

• Inflationary background: Quantum mechanics during inflation

• There at any (almost) frequency; lower frequencies easier

• Post-inflationary signals: high frequency, many mechanisms

• Motivates technology development?

Big bang

Pre-inflation

1024

Inflation Matter domination

~1

BBNLHC

ν

Radiation domination

10-3106

mt CMB

1011Temp: eV 1024

LHC

GUT

Primordial dark age

????

Biggest Picture

• Range of (???) 1012 in energy between inflation and LHC

• Physics completely unknown

• Assuming inflationary phase, of course.

• Not wildly different from log(range) between LHC and today...

• Similar amounts of expansion, different amounts of time

• Could be radiation dominated (and no structure forms)

• Many ways not to be radiation dominated

• Look for signals visible today: gravitational waves

-3 -2 -1 0 1 2 30

20

40

60

80

100

Primordial PerturbationsV, V’, V’’,V’’’

Direct detection: BBOV and V’

(P)reheatingCouplings to other fields

0. Inflationary Observables and Post-Inflationary Expansion• Digression [not really gravitational waves; except for BBO]

• Inflationary sector in concordance model: As, ns, r, dns /dln(k)

• Function of inflaton field value at horizon exit

• But this depends on the number of e-folds we “need”

• And this depends on post-inflationary expansion history

• Planck: statistical error in ns ~ “reheating uncertainty” in ns

• Inflationary model coupled to post-inflationary historyPeiris and RE astro-ph/0609003, arXiv:0805.2154

Mortonson, Peiris and RE: arXiv:1007.4205

Adshead, RE, Pritchard and Loeb: arXiv:1007.3748

Martin, Ringeval and Trotta arXiv:1009.4157

Martin and Ringeval arXiv:1004.5525

I. Preheating

• Original work: Khlebnikov and Tkachev (1997)

• Revived: RE and Lim (2006) astro-ph/0601617

• Frequency correlated with inflationary scale

• RE, Giblin and Lim astro-ph/0612294 arXiv:0712.2991

• Dufaux et al., Garcia-Bellido et al., Price & Siemans

• Hard numerical problem; hard physical problem

• This meeting: already heard from Dufaux

Calculation of Gravitational Waves

• Compute tensor part of perturbation

• Full nonlinear equations for scalar fields (rigid metric)

• Evolve transverse traceless hμν (project source from Tμν)

• δρ/ρ large; metric perturbations small: ignore backreaction

• We solve for perturbation in Fourier space.

• Recently: released PSpectRe [RE, Finkel and Roth]

• PSpectRe at 2563 ~ finite differencing at 10243

• Gravitational wave code release to come [soon]

Plot: from Price & Siemens+ Us, Dufaux et al., Garcia-Bellido et al.

Different codes / techniques

Green dots -- PSpectRe + GW code @ 643

Pink - LatticeEasy + GW @ 5123

Spectral evolution Movie: Giblin

Where is it, what shape is it, and how high is it?

• Frequency: function of inflationary scale

• Causal process: has to fit inside post-inflationary horizon

• Lower scales, larger (comoving) horizon, lower frequency

• Shape: detailed function of physics

• Height: not dependent on inflationary scale

• d Ωgw/dlnk ~ 10-10 and can be higher (booming signal)

Current topics...

• Achieved “consensus” on overall properties of signal

• Would like to improve semi-analytic account...

• Better numerical methods (higher dynamic range)

• No real sense of whether preheating is generic

• Not overly tuned; more “likely” for low inflationary scales?

• Want “end to end” calculations in well-specified theories

• Inflaton potential + couplings (maybe string or SUSY?)

• How fast is thermalization after resonance? [Transfer function]

II. “Structure” in the Early Universe

• Minimal post-inflationary universe

• Oscillations at the bottom of potential

• Matter dominated on average (mean pressure is zero)

• Long modes grow [~a(t)], short modes decay

• Consequence: nonlinear before 1010 GeV, without resonance!

• With resonance: short nonlinear phase

• Does the universe thermalize efficiently after resonance?

• Can form coherent structures: oscillons (scalar field matter)RE, Flauger and Gilmore: arXiv:1003.3011

Oscillons: post resonanceMatter dominated phase

Amin, RE and Finkel: arXiv:1003.3011

Formation /existence function of potential shape

Oscillons: post resonanceMatter dominated phase

Thermalization and gravitational waves

• Naively: matter dominated phase dilutes gravitational waves

• But statistics for merger, interactions and structure unknown

• And can be worked out

• Basic argument

• Matter dominated phase leads to structure

• Structure leads to gravitational waves

• Weak: second order production [other talks here]

• Strong: gravitational collapse & black hole formation

III. Primordial black holes

• Formed after inflation if:

• Power spectrum rises at (very) short scales?

• Via nonlinear dynamics in post-inflationary universe?

• Contribute mass and radiation to universe

• Constraints: BBN, microlensing, x-ray background, dark matter abundance....

• But very small PBH (~1gram) decay before BBN

• Leaves no trace behind??

Hawking radiation

• Black hole radiates all “massless” states: m << TBH

• Including gravitons: quantum source of gravitational waves

• Rough guess: g states; power is GW ~ 2/g

• Grey body corrections in paper [not here].

Anantua, RE and Giblin arXiv:0812.0825

TBH =M2

p

8πMBH

dMBH

dt= − g

30, 720π

M4p

M2BH

How many PBH?

• Mass of the PBH is determined by Energy scale: Einit

• Mass contained in Hubble volume ~ Einit-2

• Some fraction of universe (initially) in PBH

• Assumes horizon mass black holes.

• Need decay within ~100 seconds to protect nucleosynthesis

• Bounds initial mass

ρBH = βρ

τ =10, 240

g

M3BH

M4p

=240g

32π

M5p

E6init

Computational Strategy

• The evolution equations for the major constituents are

• The number density of gravitons / unit frequency / unit time

dρBH

dt= −3 a

aρBH + ρBHMBH

dρrad

dt= −4 a

aρrad − ρBHMBH

a

a=

8π

3M2p

(ρBH + ρrad)1/2

dN(k)dt

=2g

π

M2BH

M4P

k2

a2

1ek/(aT ) − 1

An Example

Einit = 1014GeV, g = 103, β = 10−3

ΩBH

Ωrad

1010 1012 1014 1016 1018 1020f !Hz"10!17

10!15

10!13

10!11

10!9

"gwh2

Energy Scale [Initial mass] Left to right: Einit = 1016 to 1012 GeV g=1000, β=10-3

Lower the inflationary energy scale, peak moves to higher frequencies

107 1010 1013 1016 1019f !Hz"10!27

10!23

10!19

10!15

10!11

10!7"gwh2

Number of channels From top down: Einit = 1015 GeV g=102 to 1010, β=10-3

Total number of decay channels

10 12 14 16 18 2010!19

10!17

10!15

10!13

10!11

10!9

10!7

Log!f"Hz#

h2"gw

Initial Fraction

• PBH-driven matter dominated phase

• Matter perturbations will grow

• PBH live for many Hubble times; can cluster...

• PBH radiate: “Hawking stars”

• Clustering statistics unknown: open problem

• GW from PBH: Speculative; new recipe from old ingredients...

• Could have been worked out ages ago (but wasn’t)

Matter dominated PHase

Primordial Preheating PBH

Quantum + weak gravity

ClassicalQuantum + strong

gravity

Scale InvariantPeaked near “human”

rangesHigh frequency (THz)

Low Hinf : Low amplitude

Low Hinf : redder peak High Hinf only

Always generatedModel dependent, but

common Strongly model

dependent

Amplitude bounded by CMB

Amplitude possibly large

Amplitude high

Ωgw,inf h2 < 10-14 Ωgw h2 ≲ 10-10 (10-7) Ωgw h2 ≲ 10-7

Remarks

• GW from resonance / structure: model dependent

• One message: need end-to-end description of cosmology

• Cannot specify inflationary phase on its own

• Gravitational waves vital probe of primordial dark age

• Theorists:

• Post-inflation, pre-LHC cosmology: we should do more of it!

• Experimentalists:

• GW detectors above 10 kHz?? Should/can you do it?

Here be dragons...