gravitational waves from the early...

Gravitational waves from the early Universe

Sachiko Kuroyanagi (Nagoya University)

26 Aug 2017

Summer Institute 2017

Part 2

GWs from inflation

InflationAccelerated expansion in the early Universe

Solves horizon/flatness/monopole problem

log(a) →

H-1

radiation → matter-dominated

today

H-1∝a3/2 or a2

aλaλ: physical scale

H-1: Hubble horizon(region of causality = the distance light can travel )

Horizon problem

aλpast

now

horizon grows faster than physical scale

no causal relation

H-1

log(

Scal

e) →

log(a) →

H-1

radiation → matter-dominated

today

H-1∝a3/2 or a2

Inflation phase

aλexponential growth of the scale factor

H-1～const

inside the causal region↑

a exp(Ht)→

Horizon problem

ϕ

VInflation is driven by a scalar field slowly rolling down in its potential

slow-‐roll

Friedmann equation

Exponential expansion

Standard scenario

a exp(Ht)

H = const.

What drives inflation?

→ new particle? modification of gravity?

energy density of a scalar field Equation of Motion

slow-roll approximation

ϕ

V

Oscillation

scalar field

radiation :decay rate

→ matter-dominated like Universe for φ2 potential（radiation-dominated for φ4 potential）

ReheatingStandard scenarioThe scalar field decays into standard model particles

Equation of Motion

Friedmann equation

Reheating temperature

GWs from inflation

GW Power ∝ H2inf ∝ Vinf

wide range

Generation mechanism

InflationH-1～const

a/k

H-1

log(a) →

log(

Scal

e) →

wavelength of GW

Hubble horizon

scale factor of the Universe

Today

quantum fluctuation in space-time

→ becomes GWs

stretched over the horizon


log(a) →

log(

Scal

e) →

Direct detection

CMB

long wavelength

short wavelength

a/k

H-1

wavelength of GW


→ GWs are produced for all wavelength

Hubble horizon

hij + 3Hhij 1a22hij = 16Gij

・H>k/a

・H<k/a

Let us neglect the matter contribution and perform Fourier transform

expansion of the Universe effect from matter

Equation for GWs in the expanding Universe

oscillation

→ behavior is determined by the balance between H (Hubble) and k (wavenumber = f/2π)

MD

RDInflationReheatingH-1～const

H-1∝a3/2

H-1∝a2

H-1∝a3/2

log(a) →

log(

Scal

e) → in the conventional cosmological scenario

Hubble expansion history after inflation

a/k

H-1

Hubble horizon size

wavelength of GW

ϕ

Vϕ

Vdriven by a scalar field

slow-roll

decay of the scalar field

oscillations

MD


H-1∝a3/2

H-1∝a2

H-1∝a3/2

log(a) →

log(

Scal

e) → in the conventional cosmological scenario


a/k

H-1

Hubble horizon size

wavelength of GW

2nd term

3rd term

MD


H-1∝a3/2

H-1∝a2

H-1∝a3/2

outside the horizon

log(a) →

log(

Scal

e) →

inside the horizon

a/k

H-1

Hubble horizon size

wavelength of GW

Evolution of GWs

decay


H-1∝a3/2

H-1∝a2

Each mode experiences different evolution = different amplitude for different frequency

outside the horizon

log(a) →

log(

Scal

e) →

inside the horizon


Direct detection

CMB

a/k

H-1

Hubble horizon size

wavelength of GW

GW amplitude


a/k

H-1

log(a) →

log(

Scal

e) →

Hubble horizon size


Today


wavelength of GW

initial condition

|hk| =1

apkMpl

decay


a/k

H-1

log(a) →

log(

Scal

e) →


Today


low energy scale inflation

wavelength of GW

GW amplitude

initial condition

decay

|hk| =1

apkMpl


a/k

H-1

log(a) →

log(

Scal

e) →


Today


wavelength of GW

aout,1 aout,2<

Horizon out at k=aH

GW amplitude

initial condition

decay

|hk| =1

apkMpl

→ a-1∝H


a/k

H-1

log(a) →

log(

Scal

e) →


Today


wavelength of GW

aout,high aout,low<hk,high hk,low>→GW Power ∝ H2inf

GW amplitude

low-scale inflationinitial condition

decay

|hk| =1

apkMpl

Inflation

reheating

radiationdominant

matterdominant

a ∝exp(Ht)

a ∝t2/3

a∝t1/2

a∝t2/3

Outside the horizon

Inside the horizon

Amplitude of GW background

k-‐2

k-2

k0

k

ΩGW

scale invariant spectrum

small scale modes begin to enter the horizon and damp with ∝a-1

k-2

k0

k-2

the expansion decelerates = damping ∝ a-1 becomes smaller

PT∝k0

Spectral shape

scale invariant =

→ ΩGW∝k2

spec

tral

am

plit

ude

frequency = k/2π

∝ k-2

∝ k0

∝ k-2

Spectral shape

spec

tral

am

plit

ude

frequency = k/2π

Spectral shape

∝ k0 Power spectrum PT(k)

For slow-roll inflation

→GW Power ∝ H2inf ∝ Vinf

ϕ

Vslow-‐roll

spec

tral

am

plit

ude

frequency = k/2π

ΩGW, today∝ k-2

reheating temperature TRH

low high

Effect of reheating

→ matter-dominated Universe

Reheating after inflation

for Φ2 potential, H ∝a-3/2ϕ

V

spec

tral

am

plit

ude

frequency = k/2π

→ radiation-dominated Universe

ΩGW, today∝ k0

for Φ4 potential, H ∝a-2ϕ

VEffect of reheatingReheating after inflation

spec

tral

am

plit

ude

frequency = k/2π

ΩGW, today∝ k1

reheating temperature TRH

low high

Effect of reheating

→ kination-dominated Universe

Reheating after inflation

kinatic energy > potential energy → H ∝a-6

Inflation (m2Φ2 potential)matter

dominatedradiation

dominatedreheating

MD

RD

MD

tilt of the spectrum+

deviation from the slow-roll

ϕ

V(Φ)

Spectral shape from numerical calculation

S. Kuroyanagi, T. Chiba and N. Sugiyama, Phys. Rev. D 79, 103501 (2009)

S. Kuroyanagi, T. Chiba and N. Sugiyama, Phys. Rev. D 79, 103501 (2009)

Spectral shape from numerical calculation

primordial spectrum with tilt

Damping due to the changes in effective number of degrees of freedom g*

log( T [MeV] )

log(

g* )

As the temperature of the universe decreases, relativistic matter particles become non-relativistic.

temperature decreases→ contribution to ρ and s decreases→ step-like changes in H→ step shape in ΩGW becomes non-relativistic when T~m

Effective number of degrees of freedom

=0anisotropic stress term

Damping due to the neutrino anisotropic stress

Before neutrino decoupling (T>2MeV)Anisotropic stress is suppressed by the coupling with matter (e±)

Neutrino anisotropic stress


initially =0

gives energy→ Damping only when H~k/a


After neutrino decoupling (T<2MeV)Neutrino anisotropic stress affects GWs as a viscosity when they enter the horizon


anisotropic stress term ∝ρν ~0



After the Universe becomes matter-dominatedThe energy density of radiation becomes negligible




After the Universe becomes matter-dominatedThe energy density of radiation becomes negligible

anisotropic stress term


① Neutrino decoupling

② Start of matter domination Damping due to the neutrino anisotropic stress


→ ②

→ ①

“Planck 2015 results. XX. Constraints on inflation”, Planck Collaboration, A&A, 594 (2016) A20

Constraint on inflation from Planck

Dependence on inflation models

Dependence on inflation modelsKuroyanagi et al. PRD 79, 103501 (2009)Kuroyanagi et al. PRD 90, 063513 (2014)

-19

-18

-17

-16

-15

-14

-13

-12

-11

-15 -10 -5 0 5

TRH=109GeV

log 1

0 ΩG

W

log10 f[Hz]

DECIGOUpgraded DECIGOUltimate DECIGO

-19

-18

-17

-16

-15

-14

-13

-12

-11

-15 -10 -5 0 5

TRH=109GeV

log 1

0 ΩG

W

log10 f[Hz]

λφ4/4m2φ2/2Natural inflation: f=7Mpl

R2 inflation ← quantum limit

no MD↓

GWs from cosmic strings

Cosmological history

inflation now

Log(time)

cosmic phase transitions

Theories

We are here

reheatingcosmic string

cosmic superstring

Phase transition in the Universe

Φ

potential VΦ1Φ2

Φ=0

vacuumΦ=Φ2

vacuumΦ=Φ1

height = energyhigher energy

low energy low energy

In 3-dimension space,it looks like a wall

2- dimensional potential in 3-dimension space

becomes a string

→ Cosmic string

Φ=0high energy in a string

Phase transition in the Universe

Line Density:

Φ=σ

Z(@i)

2d

2x

Z

r

2d

2x

2

2: Cosmic superstrings

Important notes

1: Phase transition

“The remarkable fact is that, although many possibilities remain, every one of them predicts the formation of topological or embedded cosmic strings at the end of inflation. So it seems that cosmic strings are almost unavoidable.”

Jeannerot et al., PRD 68 103514 (2003)

“In all acceptable spontaneous symmetry breaking schemes, cosmic string formation is unavoidable”

Kibble, Lecture at COSLAB 2004, arXiv:astro-ph/0410073

Cosmic string network

loop

infinite string

infinite string becomes a loop by reconnection

strings emit gravitational waves especially from singular structures

kink cusp

loops lose energy and shrink by emitting GWs and eventually evaporate

kinks on infinite strings

cusps on loops

GWs from cosmic strings

→ GW from cusps are dominant

3 main parameters to characterize cosmic string

・Gμ：tension = line density

・α：initial loop size L～αH-1

・p：reconnection probabilityPhase transition origin: p=1Cosmic superstring: p<<1


Network evolution

What determines the GW amplitude?

The energy density of cosmic strings

∝a-2 ～ (line density × length)/volume

energy density

a: scale factor

∝a-4

∝a-3

∝a-2 Cosmic strings

radiation

matter

X

a: scale factor of the Universe

Evolution of cosmic string network

∝a1 ∝a3

L∝a1 V∝a3

→ becomes dominant component of the Universe?

The energy density of cosmic strings

∝a-2 ～ (line density × length)/volume

energy density

a: scale factor

∝a-4

∝a-2 X

Cosmic strings

radiation

X

a: scale factor of the Universe

Evolution of cosmic string network

∝a1 ∝a3

L∝a1 V∝a3

→ follows the total energy density of the Universe

→ GWs

horizon

Strings can communicate each other inside the horizon

reconnection

→ Loops disappear after continuous GW emission

Evolution of cosmic string networkScaling law The Universe always has O(1-10) strings per horizon

String network keeps producing loops

i

GW power

Initial loop length ＝

（initial loop energy）（energy release rate per time）Lifetime of the loop ＝

Γ: numerical constant ～50-100

＝

Loop length at time t

（energy of loop at time t =μl）＝（initial energy of the loop =μαti）ー（enegry released to GWs =PΔt）

From the energy conservation law

ti： time when the loop formed

i

depends on Gμ and αEvolution of loops

Gravitational waves from cosmic string loops

Gravitational waves coming from different directions overlap each other and form gravitational wave background

Search for bursts & stochastic background are both important

Wide range GW background spectrum

horizon

Early Universe Late Universe

GW frequency ~ 1 / (loop size)

→ high frequency GWs → low frequency GWs

Scaling law The Universe always has O(1-10) strings per horizon

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104

Gμ=10-8, 10-10, 10-12, 10-14, 10-16

α=10-1, p=1

oldnewRadiation dominant

Matter dominant

ΩGW

frequency [Hz]

dependence on Gμ

large Gμ

small Gμ

GW power from cusps

h2∝(Gμ)2

life time of loops∝(Gμ)-1

Spectrum of the GWB

loop size directly corresponds to the frequency of the GW

Gμ=10-8, p=1 α=10-1, 10-5, 10-9, 10-13, 10-17

10-14

10-13

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104

ΩGW

small α

Spectrum of the GWBdependence

on α

frequency [Hz]

ΩGW

Gμ=10-12, α=10-1

p=1,10-1, 10-2, 10-3

10-18

10-16

10-14

10-12

10-10

10-8

10-6

10-4

10-2

10-18 10-16 10-14 10-12 10-10 10-8 10-6 10-4 10-2 100 102 104

small p increases the number density of loopssmall p

large p

dependence on p

frequency [Hz]

Spectrum of the GWB

Accessible cosmic string parameter space

G

α

点線：バースト実線：背景重力波

DECIGO

Adv-LIGO

eLISASKA

Kuroyanagi et al. PRD 87, 023522 (2013)

Loop size α

Line

den

sity

Gμ

reconnection probability p=1

Dotted: burstSolid: background



DECIGOAdv-LIGO

eLISASKA


Loop size α

Line

den

sity

Gμ

reconnection probability p=10-2



DECIGOAdv-LIGO

eLISASKA


Loop size α

Line

den

sity

Gμ

reconnection probability p=10-2

Gμ>10-12

Theoretical lower limit ofcosmic superstring theory

Summary

→ It may provide information not only on inflation but also on the thermal history after inflation (eg. reheating)

・GWs from inflation are promising

(but the amplitude may be small)

・GWs from cosmic strings has relatively large

amplitude and testable by near-future experiments

GWs can become a powerful probe of the very early Universe

→ It will be useful constraint for the model building of the early Universe

gravitational waves from the early...

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