Effect of weak lensing on GWs

Transcription

1 Effect of weak lensing on GWs Camille Bonvin Institute of Theoretical Physics CEA-Saclay, France LISA-France meeting June 2010

2 Outline Effect of large-scale structure on GWs emitted by a binary system. Impact on the redshift impact on the frequency and on the chirp mass. Impact on the luminosity distance. At small redshift, peculiar velocities dominate. At large redshift, the lensing term dominates. LISA-France Camille Bonvin May 2010 p. 2 /19

3 Binary system in an euclidien universe h + (t) = 2 ( )) 2/3(1 r M c 5/3 πf(t ret + cos 2 ϕ) cos Φ(t ret ) h (t) = 4 ( )) 2/3 r M c 5/3 πf(t ret cos ϕ sin Φ(tret ) t ret = t r Chirp mass M c = (m 1m 2 ) 3/5 (m 1 + m 2 ) 1/5 ϕ tret Wave form Φ(t ret ) = 2π dt f(t) Φ(τ) = 2(5M c ) 5/8 τ 5/8 + φ 0 τ = t coal t In a non-euclidien universe, both the frequency and the trajectory of the wave are affected by the geometry. LISA-France Camille Bonvin May 2010 p. 3 /19

4 Time and frenquency Time unit depends on the reference frame. The redshift is the quantity that relates the time unit in two reference frames dt S = dt O 1 + z The frequency of the wave is therefore modified f S = f O (1 + z) The wave form is also affected Φ(τ O ) = 2 ( 5M c (1 + z) ) 5/8τ 5/8 O + φ 0 LISA-France Camille Bonvin May 2010 p. 4 /19

5 Distance In an euclidien universe, propagation equation 1/r comes from the wave h + = h = 0 with = 2 r 2 t See e.g. Maggiore, Gravitational Waves, 2008 In a non-euclidien universe 1 r 1 + z d L = g µν D µ D ν Luminosity distance: d L L 4πF The luminosity distance tells us how the energy emitted by the source is spread during the propagation in a generic universe. LISA-France Camille Bonvin May 2010 p. 5 /19

6 Generic formula h + = 2 ( )) 2/3(1 (1 + z) 5/3 M 5/3 d c πf O (t ret + cos 2 ϕ) cos Φ O (t ret ) L h = 4 ( )) 2/3 (1 + z) 5/3 M 5/3 d c πf O (t ret cos ϕ sin ΦO (t ret ) L Redshifted chirp mass M c = (1 + z)m c We compute the impact of large-scale structure on gravitational waves at first order in perturbation theory. ds 2 = ( 1 + 2Ψ(t, x) ) dt 2 + a 2 (t) ( 1 2Ψ(t, x) ) δ ij dx i dx j Ψ(t, x) gravitational potential LISA-France Camille Bonvin May 2010 p. 6 /19

7 Generic formula h + = 2 ( )) 2/3(1 M 5/3 d c πf O (t ret + cos 2 ϕ) cos Φ O (t ret ) L h = 4 ( )) 2/3 M 5/3 d c πf O (t ret cos ϕ sin ΦO (t ret ) L Redshifted chirp mass M c = (1 + z)m c We compute the impact of large-scale structure on gravitational waves at first order in perturbation theory. ds 2 = ( 1 + 2Ψ(t, x) ) dt 2 + a 2 (t) ( 1 2Ψ(t, x) ) δ ij dx i dx j Ψ(t, x) gravitational potential LISA-France Camille Bonvin May 2010 p. 7 /19

8 Redshift We solve the null geodesic equation 1 + z S = a O a S ( 1 + z S = E S E O 1 + Ψ O Ψ S + n (v O v S ) + 2 χs 0 dχ Ψ ) Gravitational redshift Ψ O Ψ S Doppler effect v O v S Integrated effect χs 0 dχ Ψ vanishes in a CDM universe. Redshift perturbations affect the mass and the frequency LISA-France Camille Bonvin May 2010 p. 8 /19

9 Luminosity distance d L L 4πF L F Luminosity Flux We need to know how a beam of GWs is distortated by the large-scale structure between the source and the observer. da O k α ζ α O Homogeneous and isotropic expanding universe S dω S d L (z S ) = a O a S zs 0 dz H(z) LISA-France Camille Bonvin May 2010 p. 9 /19

10 Luminosity distance perturbations Bonvin, Durrer and Gasparini, 2006 { d L = (1 + z S )χ S ( v O n H S χ S H S χ S ( ) ( ) Ψ S + 1 Ψ O H S χ S H S χ S + 4 χs dχψ + 2 ( ) χs 1 dχ χ Ψ χ S 0 χ S 0 H S } χs + dχ (χ χ S)χ Ψ χ S 0 ) v S n LISA-France Camille Bonvin May 2010 p. 10/19

11 Various contributions Local terms Ψ O Ψ S Doppler term v O v S Integrated terms along the trajectory Ψ Ψ Lensing term source χs 0 dχ (χ χ S)χ χ S Ψ mass observer LISA-France Camille Bonvin May 2010 p. 11/19

12 Angular power spectrum We compute the amplitude and scale-dependence of the perturbations. We assume a statistically homogeneous and isotropic gravitational potential, gaussian and with a flat spectrum. We compute the angular power spectrum. F (z S, n) = lm a F lm(z S )Y lm (n) F = d L, M c, f O F (z S, n)f (z S, n ) = l 2l + 1 4π CF l (z S )P l (n n ) At small redshift: peculiar velocities dominate. At large redshift: lensing dominates. LISA-France Camille Bonvin May 2010 p. 12/19

13 Peculiar velocities on the redshift l(l + 1) 2π δz S 1 + z S = v S n = δm c M c C (M c,f O ) l = δf O f O 0.1% effect on the mass and frequency z = 0.1 z = 0.3 z = l z = 1 z = 2 z = 4 LISA-France Camille Bonvin May 2010 p. 13/19

14 Peculiar velocities on the luminosity distance l(l + 1) 2π C (d L) l δd L d L = ( 1 1 H S χ S ) v S n 1% effect on the luminosity distance z = 0.1 z = z = z = z = z = l LISA-France Camille Bonvin May 2010 p. 14/19

15 Lensing on the luminosity distance l(l + 1) 2π C (d L) l δd L d L = χs 0 dχ (χ χ S)χ χ S Ψ 3% effect on the luminosity distance z = 10 z = 5 z = z = z = z = l LISA-France Camille Bonvin May 2010 p. 15/19

16 Lensing on the luminosity distance With non-linearities: distance. 5 10% Holz and Hughes, 2005 effect on the luminosity It can be reduced by a factor 2-3, using the nongaussian feature of the distribution. Hirata, Holz and Cutler, z = 10 z = 5 z = 2 z = 1 z = z = l LISA-France Camille Bonvin May 2010 p. 16/19

17 Lensing on the luminosity distance With non-linearities: distance. 5 10% Holz and Hughes, 2005 effect on the luminosity Modelling the magnification from galaxy distribution 3 can reduce the error by a factor Gunnarsson et al., z = 10 z = 5 z = 2 z = 1 z = z = l LISA-France Camille Bonvin May 2010 p. 17/19

18 Other effects The deflection has an impact on the position of the source in the sky Ψ The deflection has an impact on the orientation of the binary system ϕ = π 2 with respect to us: even if we see the system edge-on, we will observe a small h component. LISA-France Camille Bonvin May 2010 p. 18/19

19 Conclusion Large-scale structure affect both the redshift and the luminosity distance they affect the amplitude and the frequency of the wave, and the redshifted chirp mass. At small redshift, the dominant contribution is due to peculiar velocities. They have an impact on the redshift and on the luminosity distance. At large redshift, the dominant contribution is due to gravitational lensing, that affects only the luminosity distance. LISA-France Camille Bonvin May 2010 p. 19/19