Relativistic effects in large-scale structure

Relativistic effects in large-scale structure Camille Bonvin Kavli Institute for Cosmology and DAMTP Cambridge Benasque August 2014

Outline How do relativistic effects distort our observables? Effect on: the galaxy number counts the convergence κ(or magnification) How can we measure relativistic effects? We can isolate these effects by looking at anti-symmetries in the correlation function. How can we use them to test gravity? Benasque Camille Bonvin p. 2 /32

Galaxy survey We want to measure fluctuations in the distribution of galaxies. We pixelise the map. We count the number of galaxies per pixel: = N N N Question: what are the effects contributing to? Credit: M. Blanton, SDSS Benasque Camille Bonvin p. 3 /32

Galaxy survey We want to measure fluctuations in the distribution of galaxies. We pixelise the map. We count the number of galaxies per pixel: = N N N Question: what are the effects contributing to? Credit: M. Blanton, SDSS Benasque Camille Bonvin p. 4 /32

Galaxy number counts We observe (z,n) = N(z,n) N(z) N(z) N(z,n) = ρ(z,n)v(z,n) and N = ρ(z) V(z) = ρ(z,n) V(z,n) ρ(z) V(z) ρ(z) V(z) (z,n) = b δ(z,n)+ δv(z,n) V 3 δz 1+z Benasque Camille Bonvin p. 5 /32

Galaxy number counts We observe (z,n) = N(z,n) N(z) N(z) N(z,n) = ρ(z,n)v(z,n) and N = ρ(z) V(z) ρ = ρ+δρ V = V +δv z = z +δz = ρ(z,n) V(z,n) ρ(z) V(z) ρ(z) V(z) (z,n) = b δ(z,n)+ δv(z,n) V 3 δz 1+z Benasque Camille Bonvin p. 6 /32

Galaxy number counts We observe (z,n) = N(z,n) N(z) N(z) N(z,n) = ρ(z,n)v(z,n) and same redshift bin different physical volume ρ = ρ+δρ = V = V +δv ρ(z,n) V(z,n) ρ(z) V(z) ρ(z) V(z) Observer dzdω (z,n) = b δ(z,n)+ δv(z,n) V 3 δz 1+z Benasque Camille Bonvin p. 7 /32

Galaxy number counts We observe (z,n) = N(z,n) N(z) N(z) N(z,n) = ρ(z,n)v(z,n) and same solid angle different physical volume ρ = ρ+δρ = V = V +δv ρ(z,n) V(z,n) ρ(z) V(z) ρ(z) V(z) Observer dzdω (z,n) = b δ(z,n)+ δv(z,n) V 3 δz 1+z Benasque Camille Bonvin p. 8 /32

Galaxy number counts We observe (z,n) = N(z,n) N(z) N(z) N(z,n) = ρ(z,n)v(z,n) and same radial bin different distance ρ = ρ+δρ = V = V +δv ρ(z,n) V(z,n) ρ(z) V(z) ρ(z) V(z) Observer dzdω (z,n) = b δ(z,n)+ δv(z,n) V 3 δz 1+z Benasque Camille Bonvin p. 9 /32

Fluctuations Perturbed Friedmann universe: ds 2 = a 2[ ( 1+2Ψ ) dη 2 + ( 1 2Φ ) δ ij dx i dx j] galaxy n µ We follow the propagation of photons from the galaxies to the observer and calculate: Changes in energy Observer Changes in direction Benasque Camille Bonvin p. 10/32

Result density redshift space distortion Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) (z,n) = b D 1 H r(v n) Doppler + r 0 ( dr r r rr Ω (Φ+Ψ) 1 Ḣ H 2 2 rh ) lensing V n+ 1 H V n+ 1 H rψ r +Ψ 2Φ+ 1 H Φ 3 H k V + 2 dr (Φ+Ψ) r 0 ( ) [ Ḣ + H 2 + 2 r ] Ψ+ dr ( Φ+ rh Ψ) 0 gravitational redshift potential Benasque Camille Bonvin p. 11/32

Result standard expression Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) (z,n) = b D 1 H r(v n) + r 0 ( dr r r rr Ω (Φ+Ψ) 1 Ḣ H 2 2 rh ) lensing: important at high z V n+ 1 H V n+ 1 H rψ r +Ψ 2Φ+ 1 H Φ 3 H k V + 2 dr (Φ+Ψ) r 0 ( ) [ Ḣ + H 2 + 2 r ] Ψ+ dr ( Φ+ rh Ψ) 0 relativistic contributions: important at large scale ( H k H k D ) 2 D Benasque Camille Bonvin p. 12/32

Convergence Galaxy surveys observe also the shape and the luminosity of galaxies measure of the convergence. The convergence κmeasures distortions in the size. The shear γmeasures distortions in the shape. Relativistic distortions affect the convergence at linear order. We solve Sachs equation D 2 δx α (λ) Dλ 2 = R α βµνk β k µ δx ν Benasque Camille Bonvin p. 13/32

Gravitational lensing κ = 1 2r 1 r + r 0 r 0 Convergence Doppler lensing ( ) dr r r 1 r Ω (Φ+Ψ)+ rh 1 V n ( dr (Φ+Ψ)+ 1 1 ) r dr ( Φ+ rh Ψ) 0 ) Ψ+Φ ( 1 1 rh Sachs Wolfe CB (2008) Bolejko et al (2013) Bacon et al (2014) Integrated terms Doppler lensing The moving galaxy is further away it looks smaller, i.e. demagnified Observer Benasque Camille Bonvin p. 14/32

Observations Due to relativistic effects, and κcontain additional information. δ,v,φ,ψ (Φ+Ψ),V,Φ,Ψ This can help testing gravity by probing the relation between density, velocity and gravitational potentials. Two difficulties: The relativistic effects are small: we need to go to large scales. We always measure the sum of all the effects. We need a way of isolating relativistic effects look for anti-symmetries in the correlation function. Benasque Camille Bonvin p. 15/32

Density The density contribution = b δ, generates an isotropic correlation function. x x s ξ(s) = (x) (x ) the separation depends only on s = x x n Observer ξ(s) = Ab2 D 2 1 2π 2 dk k ( k H 0 ) ns 1 T 2 δ(k)j 0 (k s) Benasque Camille Bonvin p. 16/32

Density The density contribution = b δ, generates an isotropic correlation function. ξ(s) = (x) (x ) the separation depends only on s = x x s x x n Observer ξ(s) = Ab2 D 2 1 2π 2 dk k ( k H 0 ) ns 1 T 2 δ(k)j 0 (k s) Benasque Camille Bonvin p. 17/32

Redshift distortions Redshift distortions break the isotropy of the correlation function. Real space Redshift space Quadrupole = b δ 1 H r(v n) Hamilton (1992) n Observer n Observer ξ 2 = D 2 1 µ 2 (s) = A 2π 2 ( ) 4fb 3 + 4f2 µ 2 (s)p 2 (cosβ) 7 dk k ( k H 0 ) ns 1 T 2 δ(k)j 2 (k s) P2 cosβ P 2 (cosβ) = 3 2 cos2 β 1 2 1.0 0.5 0.0 0.5 0 45 90 135 180 Β Benasque Camille Bonvin p. 18/32

Redshift distortions Redshift distortions break the isotropy of the correlation function. x β s x = b δ 1 H r(v n) Quadrupole Hamilton (1992) Observer n ξ 2 = D 2 1 µ 2 (s) = A 2π 2 ( ) 4fb 3 + 4f2 µ 2 (s)p 2 (cosβ) 7 dk k ( k H 0 ) ns 1 T 2 δ(k)j 2 (k s) P2 cosβ P 2 (cosβ) = 3 2 cos2 β 1 2 1.0 0.5 0.0 0.5 0 45 90 135 180 Β Benasque Camille Bonvin p. 19/32

Redshift distortions Redshift distortions break the isotropy of the correlation function. x β s x = b δ 1 H r(v n) Hexadecapole ξ 4 = D 2 1 µ 4 (s) = A 2π 2 Hamilton (1992) 8f 2 35 µ 4(s)P 4 (cosβ) dk k ( k H 0 ) ns 1 T 2 δ(k)j 4 (k s) Observer n P 4 (cosβ) = 1 8 P4 cosβ 1.0 0.5 0.0 0.5 [ ] 35cos 4 β 30cos 2 β +3 0 45 90 135 180 Β Benasque Camille Bonvin p. 20/32

Redshift distortions ξ 0 (s) = 1 2 = b δ 1 H r(v n) ξ 2 (s) = 5 dµ ξ(s,µ)p 2 (µ) 2 ξ 4 (s) = 9 2 1 1 1 1 1 Hamilton (1992) 1 dµ ξ(s,µ) dµ ξ(s,µ)p 4 (µ) 1.0 x β Observer n s x Measure separately: and b σ 8 f σ 8 f = dlnd 1 dlna µ = cosβ µ 4 (s) = A 2π 2 dk k ( k H 0 ) ns 1 T 2 δ(k)j 4 (k s) P4 cosβ 0.5 0.0 0.5 0 45 90 135 180 Β Benasque Camille Bonvin p. 21/32

Relativistic effects The relativistic effects break the symmetry of the correlation function. x x The correlation function differs for galaxies behind or in front of the central one. Observer n This differs from the breaking of isotropy, due to redshift distortions, which is symmetric. To measure the asymmetry, we need two populations of galaxies: faint and bright. Benasque Camille Bonvin p. 22/32

Relativistic effects The relativistic effects break the symmetry of the correlation function. The correlation function differs for galaxies behind or in front of the central one. x x Observer n This differs from the breaking of isotropy, due to redshift distortions, which is symmetric. To measure the asymmetry, we need two populations of galaxies: faint and bright. Benasque Camille Bonvin p. 23/32

Relativistic effects The relativistic effects break the symmetry of the correlation function. B F The correlation function differs for galaxies behind or in front of the central one. F Observer n This differs from the breaking of isotropy, due to redshift distortions, which is symmetric. To measure the asymmetry, we need two populations of galaxies: faint and bright. Benasque Camille Bonvin p. 24/32

Cross-correlation The following terms break the symmetry: rel = 1 H rψ+ ( 1 Ḣ H 2 2 rh ) V n+ 1 H V n Benasque Camille Bonvin p. 25/32

Dipole in the correlation function CB, Hui and Gaztanaga (2013) ξ(s,β) = D 2 1f H H 0 ( Ḣ H 2 + 2 rh ) (b B b F )ν 1 (s) cos(β) B β F ν 1 (s) = A 2π 2 dk k ( k H 0 ) ns 1 T δ (k)t Ψ (k)j 1 (k s) Observer n By fitting for a dipole in the correlation function, we can measure relativistic effects, and separate them from the density and redshift space distortions. ξ 1 (s) = 3 2 1 1 dµ ξ(s,µ) µ µ = cosβ Benasque Camille Bonvin p. 26/32

120 z = 0.25 Monopole Multipoles 0 Quadrupole 100 80 20 s 2 Ξ 0 60 40 s 2 Ξ 2 40 20 60 0 20 80 50 100 150 200 50 100 150 200 s Mpc h s Mpc h 8 Hexadecapole 2.0 Dipole 6 1.5 s 2 Ξ 4 4 s 2 Ξ 1 1.0 2 0.5 0 50 100 150 200 0.0 50 100 150 200 s Mpc h s Mpc h Benasque Camille Bonvin p. 27/32

κ g = 1 2r r 0 Convergence dr r r r Ω (Φ+Ψ) κ v = ( ) 1 rh 1 V n We can isolate the Doppler lensing by looking for anti-symmetries in κ redshift space κ κ real space κ κ demagnified magnified Observer n Observer n Benasque Camille Bonvin p. 28/32

Dipole CB, Bacon, Clarkson, Andrianomena and Maartens (in preparation) β κ s 2 Ξ 1 ξ(s,β) = 2A 9π 2 Ω 2 D1 2 m 0 2 4 6 8 10 H H 0 f ( 1 1 )( b+ 3f Hr 5 z = 0.1 z = 0.5 Gravitational lensing Doppler lensing 50 100 150 200 s Mpc h ) ν 1 (s)cos(β) Observer n 50 100 150 200 The dipole due to gravitational lensing is completely subdominant. s 2 Ξ 1 0.5 0.0 0.5 1.0 1.5 2.0 Gravitational lensing Doppler lensing s Mpc h Benasque Camille Bonvin p. 29/32

Testing Euler equation The monopole and quadrupole in allow to measure V The dipole allows to measure: rel = 1 H rψ+ ( 1 Ḣ H 2 2 rh ) V n+ 1 H V n If Euler equation is valid: rel = V n+hv n+ r Ψ = 0 ( Ḣ H 2 + 2 rh ) V n With the dipole, we can test Euler equation. Benasque Camille Bonvin p. 30/32

Measuring the anisotropic stress The dipole in the convergence is sensitive to: The standard part κ v = ( ) 1 rh 1 κ g = 1 2r r can be measure through κκ and 0 V n dr r r r Ω (Φ+Ψ) γγ Assuming Euler equation, we can test the relation between the two metric potentials Φand Ψ. Benasque Camille Bonvin p. 31/32

Conclusion Our observables are affected by relativistic effects. These effects have a different signature in the correlation function: they induce anti-symmetries. By measuring these anti-symmetries we can isolate the relativistic effects and use them to test the relations between the density, velocity and gravitational potential. Benasque Camille Bonvin p. 32/32

Contamination The density and velocity evolve with time: the density of the faint galaxies in front of the bright is larger than the density behind. This also induces a dipole in the correlation function. F Larger redshift smaller density 2 1 relativistic B s 2 Ξ 1 0 total F Observer n 1 2 evolution 50 100 150 200 s Mpc h Benasque Camille Bonvin p. 33/32

Dipole in the correlation function ξ(s,β) = D 2 1f H H 0 ( Ḣ H 2 + 2 rh ) (b B b F )ν 1 (s) cos(β) B β F ν 1 (s) = A 2π 2 dk k ( k H 0 ) ns 1 T δ (k)t Ψ (k)j 1 (k s) 2.0 Observer n 1.5 z = 0.25 s 2 Ξ 1 1.0 0.5 z = 0.5 z = 1 0.0 CB, Hui and Gaztanaga (2013) 50 100 150 200 s Mpc h b B b F 0.5 Benasque Camille Bonvin p. 34/32