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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
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
5
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
| | ...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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
†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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
23
2
0
4( )
3( )gw hf f S f
H
22
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
27
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
( , ( ,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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
37
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
Principle of detection:
No geodesic deviation in 1st order, only “stretching” of coordinates (physical
distance travelled by a photon)
38
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008
Introduction
Basic Theory of GW’s
Cosmological GW’s
Production Processes
Experimental Bounds
Detection and Experiments
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
M. Agathos Student Seminar Universiteit Utrecht Dec. 2008