gravitational waves from primordial density perturbations kishore n. ananda university of cape town...
DESCRIPTION
Calculate the GW’s produced non-linearly during radiation era. Compute as power series in (perturbation parameter) Carryout the standard SVT decomposition Work in Fourier space. Calculate EFE’s at each order –Linear order - modes decouple and evolve independently –Higher order – mode-mode coupling Calculation overview - ITRANSCRIPT
Gravitational Waves from primordial density perturbations
Kishore N. AnandaUniversity of Cape Town
In collaboration withChris Clarkson and David Wands
PRD 75 123518 (2007) arXiv:gr-qc/0612013
Cosmo 07, 22nd August 2007
The cosmological standard model
• GW’s are inevitable consequence of GR.
• Studying linear perturbations during Inflation:– Large-scale GW’s are produced– Amplitude depends on the energy scale.– Current observations allow power up to 30% of scalars
• What is the minimum (guaranteed) background of tensor modes?– Density perturbations do exist.– Density perturbations will produce GW’s via non-linear evolution.– We have detailed information on scalars.– What does the power spectrum look like?– What about in the frequency range of direct detectors?
• Calculate the GW’s produced non-linearly during radiation era.
• Compute as power series in (perturbation parameter)
• Carryout the standard SVT decomposition
• Work in Fourier space.
• Calculate EFE’s at each order– Linear order - modes decouple and evolve independently– Higher order – mode-mode coupling
Calculation overview - I
(1) 2 (2)g g g g
, , , , , ,i i ijB E F S h
• The metric can be written as (longitudinal gauge)
• The EMT – perfect fluid description of radiation.
Calculation overview - II
2 (1)
2 (1)
0
(2
0
0
)
1 2
0
1 2
,
,
1 .2ij
i
ij ijh
g a
g
g a
scalarsO 2 tensorsO
0 and 0qi ijP
Linear modes - I
• The background equations are
• The standard first order equations for scalars
2 2 2'' (1 ) 0s sc c k H
2 283
a Gaa
H 3 P H\
Linear modes - II
• The radiation solutions
3
( ), cos sin( ) 3 3 3rA k k kk
kk
10log k
10lo
g,
/r
rA
k
k
1k
Linear modes - III
• The power spectra definition
• The curvature perturbation
2*
1 2 1 23
2( ) ( ) ( ) , ( ), ,kk k k k k P
22 2
3
216( ) ( ),A k kk
R
2 9 1 at a scale 0.002 Mpc2.4 10 CMBk R
Tensor modes - I
• The tensor wave equation
• The source
• The solution is given via Green’s function method
2'' '2 4lmijij ij ij lmh Hh h T S
2| | | | | | | | |2 2
3 ' ' '4 2 2ij ij i j i j i j i ja S H H
0
, , ,1kh d G a
a
k kS
2
33/2
( )( , ) 12 ( , ) ( , ) ( , ) ( , ) ( , ) .(2 ) 2
ij
i jq d k k k
kk k k k k k k k k
S
Tensor modes - II
• The solution is given via Green’s function method• Calculate the tensor power spectrum
• After much simplification
• The input PS
– Delta function
– Power-law
31 2 1 2, ' ' , '~ , ,,h k d k d kd Fk k k k P P P
2 2 /ln49 CMB ink k k k RP A
2
1 2 1 23
2( ) ( ) ( ) ( ,, , )hh h kk k k k k P
124 /9
snCMB CMBk k k k
RP
The delta function case
2010~ 1 Hz today
xf
210
width
amp
1/
(lo. )g
x
x
3 tailk
24 / , ,h in ink kk xkP A F
The power-law case - I
1sn
2 1
4s
s
n
h nCMB
k xk
RP F
The power-law case - II
Baumann, Ichiki, Steinhardt, & Takahashi, hep-th/0703290
The GW spectrum today
• For the power-law case
• For the delta function case, the amplitude of the resonance peak
– Advance LIGO could constrain A~100 at Tent~108 GeV.– BBO could constrain A~1 at Tent ~100 TeV
2( 1)20 34( 1)
1
1.86 10 3.2Hz
s
ss
s
nnn
GWn
f
FF
17 410( ) 4.5 10 1 0.09log
1GeVent
GW peakTf
A
The GW spectrum today III
Baumann, Ichiki, Steinhardt, & Takahashi, hep-th/0703290
• Calculated the background of GW’s generated from the scalar power spectrum during the radiation era.
– Exists independently of the inflationary model.
– Spectrum is scale-invariant at small scales with r~10-6.
– GW’s can be used to look for features in scalar PS at scales much smaller than those probed by CMB+LSS.
Conclusions