Gravitational Waves in Cosmology - Universiteit Utrecht

“The weakest link”

Michael Agathos

Student Seminar ‟08

Universiteit Utrecht1

Introduction

Basic Theory of GW’s

Cosmological GW’s

Candidate Processes for Production

Experimental Bounds

GW Detection and Upcoming Experiments

2

M. Agathos Student Seminar Universiteit Utrecht Dec. 2008

Radiative Solutions are predicted by General Relativity

(tiny ripples of spacetime)

Because of extremely weak coupling, observation of

such waves requires highly sophisticated technology

The only available laboratory is the Universe itself. We

obviously have no control over the conditions of the

experiments.

Sources can be of Astrophysical (massive bodies) or Cosmological (early Universe) nature

3


Introduction


Cosmological GW’s

Production Processes

Experimental Bounds

Detection and Experiments

Why are we so interested in Gravitational

Radiation?

◦ Detection of Gravitational waves will not only be

undisputed support for GR but will also open a new

window to the cosmos

◦ Study of Astrophysical systems that comprise

sources of GW‟s

◦ Probe for a snapshot of the early Universe and

validate or exclude cosmological models

◦ Challenge to push current technology to its extreme

limits

4


Introduction


Cosmological GW’s


Experimental Bounds


Why are we so interested in Gravitational

Radiation?

◦ Detection of Gravitational waves will not only be

undisputed support for GR but will also open a new

window to the cosmos

◦ Study of Astrophysical systems that comprise

sources of GW‟s

◦ Probe for a snapshot of the early Universe and

validate or exclude cosmological models

◦ Challenge to push current technology to its extreme

limits

5


Introduction


Cosmological GW’s


Experimental Bounds


GR provides us with a set of dynamical equations for the

geometry of spacetime.

12

8R g R GT

These equations are highly nonlinear wrt the metric and

cannot be solved analytically in the generic case.

One has to resort to a linearized version of the above

equations in order to come up with radiative solutions for

small field perturbations.

Interactions?

6


Introduction


Cosmological GW’s


Experimental Bounds


Take metric perturbations around Minkowski spacetime

and evaluate all quantities up to 1st order in the perturbation

hg

The Christoffel connection reads :

And the Ricci tensor :

)(][

])[(

2

21

21

hhhh

hhhh

v

v

12

(1)12

[ ]

[ ]

v v

v

R

h h h h h h

h h h h R

1h

7


Introduction


Cosmological GW’s


Experimental Bounds


The 10 Einstein equations are not functionally independent due to the 4 Bianchi identities

the metric is underdetermined

General covariance imposes diffeomorphism invariance on the theory (via coordinate xfms). This is the gauge freedom of GR :

0G

( )x x x x

So from the 6 remaining d.o.f. one can choose to “fix the

gauge” by imposing certain properties on the metric

and get rid of 4 more non-physical d.o.f.

Remember EM gauge invariance

8


Introduction


Cosmological GW’s


Experimental Bounds


So the Einstein equation can be recast in the simple

form

We can always transform to a gauge that is

convenient, here the Lorenz gauge :

Now the perturbation will transform as :

and

trace reverse :

h h

2h h

0h

12

h h h

16 NTh G

9


Introduction


Cosmological GW’s


Experimental Bounds


Vanishing source term

This allows us to bring the metric to the transverse

traceless form :

and for a single plane wave

Generic way of finding TT :

0h 0

00 00

0 0h h 0 0ih 0h

12

TT

ij ik jl ij kl klh P PP P h

ij ij i jnP n

n̂0

0

0 0 0

TT

ij

h h

h h h

10


Introduction


Cosmological GW’s


Experimental Bounds


Minkowski is boring for cosmology.

g hg

The Christoffel connection reads :

12

1 12 2

212

( )[ ]

[ ] [ ]

[ ] ( )

[ ]1

2

v v

v v

v

v

g g g g

g g g g g

h h h h

h

h h h h

g g

g

g gh h h h

and we can define :

g gh

1[ ]

2vh h hg

11


Introduction


Cosmological GW’s


Experimental Bounds


And the Riemann tensor :

where we used the Riemann normal coord.

General covariance implies:

The Ricci tensor :

And Einstein eqn :

in the gauge and

R RR

R

1 1 1( )2 2 2

R

h h Rh

12

G R Rg

0h 0h

12

0G h R h

transversality tracelessness

12


Residual symmetry TT gauge

In a spacetime with a nonvanishing source

term one also gets „spurious‟ l-l and l-t

contributions Φ, Θ, Ξi

“They are not objective and are not detectable by any

conceivable experiment. They are merely sinuosities in

the co-ordinate system, and the only speed of

propagation relevant to them is the speed of thought.”

A. S. Eddington

TT part is gauge invariant observable and

obeys a wave-like equation

0

16TT

ij ijh ,ij ij lm lmT

13


Introduction


Cosmological GW’s


Experimental Bounds


Consider a spacetime with a source :

Introducing the Green‟s function :

For r = |x| >> L remember quadrupole

formula ( )

8 Nh G T

L

1 1

| |4 | |

( ) t x x tx x

G x x

(4)( ) ( )G x x x x flat

3 ( ) )8 (Nh G x G x x Td x

x

x

14


Introduction


Cosmological GW’s


Experimental Bounds


| | ...x rx

23

002

2,( )( ) i j

ijh d xT t r x xr t

x x

Linearized Einstein eqns no self-interactions

Metric perturbations can be idealized as plane waves :

As a free field theory of a relativistic spin-2 particle

2,

3

3/8 ) () (

(( )

2 )

P P

ij

P

ikx

N ij k

dh k h e

kG

3

3/ 2,

† *1 ˆ8 [ )(

( ) ( ) ( ) ( ) ( ) ] (2

ˆ)

ˆ2

ikx ikP P x

N k k ij

P P P

ij

P

b bh hd

h k e kk

keG

15


Introduction


Cosmological GW’s


Experimental Bounds


†ˆ ˆ, ( )k k

b b k k

† †ˆ 0ˆ ˆ ˆ, ,k k k k

b b b b

In Cosmology we are looking for the Stochastic background and Ωgw

Astro vs. Cosmo

Thermal history of the Universe

Graviton decoupling

Redshift and cooling of background radiation

CMBR correspondence

16



Ignorance

Blindness

Suspicion

Myopia

17


Ignorance

Blindness

Suspicion

Myopia

18

Consider the case of CMB

• Photons decouple at z=1089 and T=3000K

• Equilibrium gives a black body spectrum

• Photons carry energy

• Photons cool down with expansion

2

/ 32

1 4

1hf kTd

Vf df

e cE hf

3

3 /

8

1em hf kT

dE h f df

V cd

e

19


Introduction


Cosmological GW’s


Experimental Bounds


Introduction


Cosmological GW’s


Experimental Bounds


Recall neutrino decoupling and temperature relation with SM

relativistic d.o.f.

Gravitons decouple below Planck temperature

when (at least!) :

And of course the physical wavelength is redshifted

Assuming adiabatic evolution since emission :

*0 *

02f

akf f

a a

23210Pl

PlTM c

Kk

* ( ) 106.75S decg T

3 3 3

0* * 0

3

*S Sg T a g T a

So

and

1/3

13

0 *

* *

101

0 10

S

GeV

gf f

T

0

1/3

0

3.91

106.750.9gT T K

20


Gravitational radiation carries energy

Express graviton density spectrum by a useful

quantity 1( )

ln

gw

gw

cr

fd

d f

frequency dependence

(spectrum, peak, amplitude?)

2

03,

8cr

H

G

This includes a measuring uncertainty of H0 .

Better use 2

0 gwh

2 2

0016 16

1 1( ) ( )ij ij gw

N N

t xh xh h hG G

21


Introduction


Cosmological GW’s


Experimental Bounds


23

2

0

4( )

3( )gw hf f S f

H

22


Introduction


Cosmological GW’s


Experimental Bounds


Stochastic isotropic background :

Spectral density :

2

,

ˆ ˆ ) ˆ) )(( ,( ift P

ij ij

P

Ph fh t df d e

* * 2( , ) (1ˆ ˆ, ˆ ˆ( ) ( , ( )))

8hP PQQ f Sfh f ff h

In order to have a background of gravitational radiation from the early Universe, which is strong enough for us to detect in the present time, we have to look for extremely violent phenomena that occur throughout the Universe.

Modern cosmological theories provide us with such scenarios. Some of them are:◦ Fluctuation amplification during inflation◦ Phase transitions◦ Topological defects (cosmic strings)◦ Brane world scenarios◦ etc

23


Quantum fluctuations are amplified in an inflating

Universe [Grishchuck-Starobinsky]

Linearized eqns of TT perturbations give:

In FLRW :

33 2 2 2

3 2 2

1 3 ((

) 1) ( ) ( ) 0

( ) ( )t t ij t ij

a a ta h x x

a a a t a th

2 2 2[ 1, ( ), ( ) ( )],g diag a t a t a t 3( )tg a

1( ) ( ) ) ( ) 0ij ijh x g g h x

g

2 2

2

3 ( ) 1( ) 0

( ) ( )

ikx

t k

a te

a t ah

t

24


Introduction


Cosmological GW’s


Experimental Bounds


In conformal time :

And the box equation becomes

where we used the wave zone expansions

22 0k k k

ah h hk

a

( (

2

) )k k

k k k

k k k k

ah

a h ah

a ha h ah

2) ( 0( )k kUk

( )

dtd

a t

25


Introduction


Cosmological GW’s


Experimental Bounds


2 2 2( ) )( i

ij

jg dx dx a d da dx x

4 ( )tg a

End up with an effective “potential” which gives a time varying harmonic oscillator

Simple case : de Sitter inflation

Here we have to distinguish between sub-Hubble

and super-Hubble modes depending on value of k.

)(a

aU

1

I

aH

1

1

2

1 1

2

,

,ik

k I

k k k I

h

h A

ae H

k a k

d aH

a kB

26


Introduction


Cosmological GW’s


Experimental Bounds


27


Introduction


Cosmological GW’s


Experimental Bounds


( , ( ,1

) )32

gw ij ijV

N

th x h xG

t

2 2

2 2 2 2 4

4

1 1 12

1

ij ij ij ij ij ijh h aH a Ha a a a a

a

a a

3 3 3*

2/3 2/34

3 *

4

1) )

32 2 2( , (

1)

,

( , ( , )32

ikx ik x

gw V ij ij

N

V ij ij

N

k kk e k e

G

k k kG

d x d d

a V

da V

Recall :

After approximating for sub-Hubble modes with the help of a

few assumptions we work out the Green‟s function solution :

Matching this solution with the one without a source

will give us the amplitudes that will propagate to today

28


Introduction


Cosmological GW’s


Experimental Bounds


2 ( ( ) 1( , ) ) ( ), 6 TT

N iij ij jUk k kTk G

16

( (, ) sin ( ) ( ) , )

i

TT

ij ij

Gk d k a

kT k

3

, 3/2ˆ( ) ( ) ( ) ( )

(2 )

TT

ij ij lm l m a a

d pT k k p p p k p

, ) ( )sin ( )( ( )cos ( )ij ij f ij fk A k k B k k

12

( )a a a aT g g V

After approximating for sub-Hubble modes with the help of a

few assumptions we work out the Green‟s function solution :

Matching this solution with the one without a source

will give us the amplitudes that will propagate to today

29


Introduction


Cosmological GW’s


Experimental Bounds


2 ( ( ) 1( , ) ) ( ), 6 TT

N iij ij jUk k kTk G

16

( (, ) sin ( ) ( ) , )

i

TT

ij ij

Gk d k a

kT k

3

, 3/2ˆ( ) ( ) ( ) ( )

(2 )

TT

ij ij lm l m a a

d pT k k p p p k p

, ) ( )sin ( )( ( )cos ( )ij ij f ij fk A k k B k k

12

( )a a a aT g g V

2 23

,

4

4( ) cos( ) ( ) ( , ) sin( ) ( ) ( , )

( )( )

ln

f f

i i

f

g

TT TT

w k f

Nk f ij ij

i j

gw

G kS d d k a T k d k a T

d Sk

d a

V

k

k

In Standard Inflation spectrum is flat or descending and

COBE bound sets it too weak to be detected ~10-13 even for

LISA

Bound is at Ultra low frequencies, so a theory with ascending

spectrum could be detectable

There are such theories :

• Pre-big-bang scenario (super-inflation) [Veneziano-1991]

spectral slope nT=3

• Bouncing Universe includes

o a slow contraction phase

o a superinflation phase

o Radiation dominating era

and nT ~ 2+2ε or nT ~ 3

• Quintessential Inflation [Peebles-Vilenkin 1999]

o non standard equation of state

o and nT ~ 1

(1 3)/2)(a

1/2)(Ea

( )Ea

1/a a 30


Introduction


Cosmological GW’s


Experimental Bounds


Phase transitions in the early Universe

can produce GW‟s. More specifically EW

Primordial Universe is initially in a false

vacuum state in high temperatures

As T drops below the Higgs mass scale,

the broken true vacuum shows but is still

hidden by an energy barrier

Tunneling takes action

31


Introduction


Cosmological GW’s


Experimental Bounds


Introduction


Cosmological GW’s


Experimental Bounds



Bubbles of true vacuum are nucleated and

expanding

The latent heat left over by the transition

becomes kinetic energy transferred to the

bubble walls (and also reheats the plasma)

[Kosowsky, Kamionkowski, Turner]

Isotropy is spontaneously lost

0

te H

. .

4

*

f vE

aT

4 3/ 2 10 5 10peakf Hz

GW‟s are produced on collision surface or even

by turbulent areas in plasma

ˆ ˆ2

01

0

1ˆ, ( , ))2

( n

n

Ri kxi t i kx

ij S ij

N

n

T dte e d drr Tek r t

32

Introduction


Cosmological GW’s


Experimental Bounds



Strongly 1st order transition is needed

SM predicts smooth crossover (mH>100GeV)

MSSM needs light stop for 1st order

NMSSM can provide strong 1st order (also gives baryon

asymmetry

33

Introduction


Cosmological GW’s


Experimental Bounds



Strongly 1st order transition is needed

SM predicts smooth crossover (mH>100GeV)

MSSM needs light stop for 1st order

NMSSM can provide strong 1st order (also gives baryon

asymmetry

34

Topological defects are produced after a phase

transition via the Kibble mechanism

Cosmic strings are stringy formations of enormous

mass densities that extend throughout the universe

Quantity of interest

Loops can be created

Pulsate relativistically and decay emitting GW‟s

35


Introduction


Cosmological GW’s


Experimental Bounds


dm

dl

Direct and indirect experimental evidence restrict

amplitude of GW‟s

Currently operating GW experiments :

• LIGO (2x) (Luisiana-Washington, USA)

• VIRGO (Pisa, Italy)

• GEO 600 (Hannover, Germany)

Important indirect bounds also given by

BBN

COBE and the Sachs-Wolfe effect

msec Pulsars

36


37


Introduction


Cosmological GW’s


Experimental Bounds


Principle of detection:

No geodesic deviation in 1st order, only “stretching” of coordinates (physical

distance travelled by a photon)

38


2 different types of detection (none of which work yet)

• Resonant bar detectors [J. Weber ‟60s]

• MiniGRAIL spherical cryogenic antenna (Leiden)

• Laser interferometers 10 to 104 Hz

LIGO

VIRGO

GEO 600

TAMA 300

2110dL

L

LIGO LIGO

VIRGO

3,000km

39


Introduction


Cosmological GW’s


Experimental Bounds


Consider detector as an input/output system

The signal could be smaller than the noise.

Correlate

Optimal filtering [Michelson-1987]

Observing for a period of time T :

/2 /2

1 2/2 /

12

2 ( ) ( ) ( )T T

T TS dt dt S Q t tt S t

( ) ( ) ( )i i iS t s t n t

40


Introduction


Cosmological GW’s


Experimental Bounds


1 2 1 2 1 2, ,,S S s s nS n

2

1 2

2

1 2

, ( )

, ( )

fTs

fT

f

n n n f

s h

2( )

min

n f

fT

Laser Interferometer Space Antenna

Joint project planned by NASA and ESA

41


Introduction


Cosmological GW’s


Experimental Bounds


2005

2010

2015

2020

2025

2030

1995 2000 2005 2010 2015

3 spacecrafts equipped with same instrumentation

armlength

Low frequency band with peak sensitivity

at mHz ( )

Possible because of no ground noise

20o wrt earth‟s position, with a 60o tilt

Proof mass! (cool)

510 1Hz Hf z

65 10L km

42


Introduction


Cosmological GW’s


Experimental Bounds


1210

We overviewed the basic mechanisms of the early Universe that are expected (if they ever happened) to produce a stochastic gravitational wave spectrum.

Considering the predicted numerical restrictions some mechanisms are not strong enough to be detected even by close future experiments.

If some signal shows up it will provide a snapshot of the early Universe and will feed more arguing around cosmological models

But detecting a tiny signal and extracting a stochastic background from it is still a great challenge

Hopefully LISA will give some useful data

43


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Gravitational Waves in Cosmology - Universiteit Utrecht