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