Theoretical Cosmology

Transcription

1 Theoretica Cosmoogy Ruth Durrer, Roy Maartens, Costas Skordis Geneva, Capetown, Nottingham Benasque, February Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

2 Theoretica Cosmoogy What is the observationa basis of homogeneity and isotropy in cosmoogy? Isotropy CMB Homogeneity RM What are very arge scae gaaxy cataogs reay measuring? What are very arge scae N-body simuations simuating? RD How can we test genera reativity in cosmoogy? Is dark energy a manifestation of deviations from GR? CS Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

3 Theoretica Cosmoogy What is the observationa basis of homogeneity and isotropy in cosmoogy? Isotropy CMB Homogeneity RM What are very arge scae gaaxy cataogs reay measuring? What are very arge scae N-body simuations simuating? RD How can we test genera reativity in cosmoogy? Is dark energy a manifestation of deviations from GR? CS Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

4 Theoretica Cosmoogy What is the observationa basis of homogeneity and isotropy in cosmoogy? Isotropy CMB Homogeneity RM What are very arge scae gaaxy cataogs reay measuring? What are very arge scae N-body simuations simuating? RD How can we test genera reativity in cosmoogy? Is dark energy a manifestation of deviations from GR? CS Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

5 (See: Yoo et a. 09, Yoo 10, Bonvin and RD, in preparation 11) For each gaaxy in a cataog we measure (z, θ, φ) = (z, n) + info about mass, spectra type... We can count the gaaxies inside a redshift bin and sma soid ange, N(z, n) and measure its fuctuation, N(z, n) N(z) (z, n) =. N(z) This quantity is directy measurabe. On sma scaes where fuctuations in the spacetime geometry can be negected it is simpy reated to the density contrast δ = (ρ(x, t) ρ(t))/ ρ(t). On arge scaes, however, we have to take into account that the measured redshift is not simpy the background redshift z, not ony the number of gaaxies but aso the voume is distorted the anges we are ooking into are not the ones into which the photons from a given gaaxy arriving at our position have been emitted. Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

6 (See: Yoo et a. 09, Yoo 10, Bonvin and RD, in preparation 11) For each gaaxy in a cataog we measure (z, θ, φ) = (z, n) + info about mass, spectra type... We can count the gaaxies inside a redshift bin and sma soid ange, N(z, n) and measure its fuctuation, N(z, n) N(z) (z, n) =. N(z) This quantity is directy measurabe. On sma scaes where fuctuations in the spacetime geometry can be negected it is simpy reated to the density contrast δ = (ρ(x, t) ρ(t))/ ρ(t). On arge scaes, however, we have to take into account that the measured redshift is not simpy the background redshift z, not ony the number of gaaxies but aso the voume is distorted the anges we are ooking into are not the ones into which the photons from a given gaaxy arriving at our position have been emitted. Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

7 (See: Yoo et a. 09, Yoo 10, Bonvin and RD, in preparation 11) For each gaaxy in a cataog we measure (z, θ, φ) = (z, n) + info about mass, spectra type... We can count the gaaxies inside a redshift bin and sma soid ange, N(z, n) and measure its fuctuation, N(z, n) N(z) (z, n) =. N(z) This quantity is directy measurabe. On sma scaes where fuctuations in the spacetime geometry can be negected it is simpy reated to the density contrast δ = (ρ(x, t) ρ(t))/ ρ(t). On arge scaes, however, we have to take into account that the measured redshift is not simpy the background redshift z, not ony the number of gaaxies but aso the voume is distorted the anges we are ooking into are not the ones into which the photons from a given gaaxy arriving at our position have been emitted. Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

8 (See: Yoo et a. 09, Yoo 10, Bonvin and RD, in preparation 11) For each gaaxy in a cataog we measure (z, θ, φ) = (z, n) + info about mass, spectra type... We can count the gaaxies inside a redshift bin and sma soid ange, N(z, n) and measure its fuctuation, N(z, n) N(z) (z, n) =. N(z) This quantity is directy measurabe. On sma scaes where fuctuations in the spacetime geometry can be negected it is simpy reated to the density contrast δ = (ρ(x, t) ρ(t))/ ρ(t). On arge scaes, however, we have to take into account that the measured redshift is not simpy the background redshift z, not ony the number of gaaxies but aso the voume is distorted the anges we are ooking into are not the ones into which the photons from a given gaaxy arriving at our position have been emitted. Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

9 We first define the redshift density fuctuation δ z(n, z) by δ z(n, z) = ρ(n, z) ρ(z) ρ(z) = N(n,z) N(z) V(n,z) V(z) N(z) V(z) This together with the voume fuctuations, resuts in the directy observed number fuctuations δv(n, z) (n, z) = δ z(n, z) + V(z) Both these terms are in principe measurabe and therefore gauge invariant. The cacuation, especiay of the second term is however quite invoved. The first term is obtained easiy by considering With δ z(n, z) = = ρ( z) + δρ(n, z) ρ(z) ρ(z) dρ δz δρ ρ dz ρ = = δ(n, z) 3 δz 1 + z δz 1 + z = `n V + Ψ (n, z) ρ(z δz) + δρ(n, z) ρ(z) ρ(z) Z to t S ( Φ + Ψ)dt Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

10 We first define the redshift density fuctuation δ z(n, z) by δ z(n, z) = ρ(n, z) ρ(z) ρ(z) = N(n,z) N(z) V(n,z) V(z) N(z) V(z) This together with the voume fuctuations, resuts in the directy observed number fuctuations δv(n, z) (n, z) = δ z(n, z) + V(z) Both these terms are in principe measurabe and therefore gauge invariant. The cacuation, especiay of the second term is however quite invoved. The first term is obtained easiy by considering With δ z(n, z) = = ρ( z) + δρ(n, z) ρ(z) ρ(z) dρ δz δρ ρ dz ρ = = δ(n, z) 3 δz 1 + z δz 1 + z = `n V + Ψ (n, z) ρ(z δz) + δρ(n, z) ρ(z) ρ(z) Z to t S ( Φ + Ψ)dt Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

11 We first define the redshift density fuctuation δ z(n, z) by δ z(n, z) = ρ(n, z) ρ(z) ρ(z) = N(n,z) N(z) V(n,z) V(z) N(z) V(z) This together with the voume fuctuations, resuts in the directy observed number fuctuations δv(n, z) (n, z) = δ z(n, z) + V(z) Both these terms are in principe measurabe and therefore gauge invariant. The cacuation, especiay of the second term is however quite invoved. The first term is obtained easiy by considering With δ z(n, z) = = ρ( z) + δρ(n, z) ρ(z) ρ(z) dρ δz δρ ρ dz ρ = = δ(n, z) 3 δz 1 + z δz 1 + z = `n V + Ψ (n, z) ρ(z δz) + δρ(n, z) ρ(z) ρ(z) Z to t S ( Φ + Ψ)dt Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

12 We first define the redshift density fuctuation δ z(n, z) by δ z(n, z) = ρ(n, z) ρ(z) ρ(z) = N(n,z) N(z) V(n,z) V(z) N(z) V(z) This together with the voume fuctuations, resuts in the directy observed number fuctuations δv(n, z) (n, z) = δ z(n, z) + V(z) Both these terms are in principe measurabe and therefore gauge invariant. The cacuation, especiay of the second term is however quite invoved. The first term is obtained easiy by considering With δ z(n, z) = = ρ( z) + δρ(n, z) ρ(z) ρ(z) dρ δz δρ ρ dz ρ = = δ(n, z) 3 δz 1 + z δz 1 + z = `n V + Ψ (n, z) ρ(z δz) + δρ(n, z) ρ(z) ρ(z) Z to t S ( Φ + Ψ)dt Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

13 we arrive at δ z(n, z) = δ(n, z) + 3(V n)(n, z) + 3Ψ(n, z) + 3 For the voume we use Z to t S ( Ψ + Φ)(n, z(t))dt. dv = p g ǫ abcd u a dx b dx c dx d = p g ǫ abcd u a x b x c x d z θ S ϕ S (θ S, ϕ S ) (θ O, ϕ O ) dzdθ Odϕ O = vdzdθ O dϕ O To first order in the perturbations one finds v = a3 r 2 sin θ O H» 1 3φ + (cot θ O + δφ )δθ + θ φ v n + 2δr dδr + 1 r dt H The engthy cacuation of δθ, δφ δr aong the perturbed geodesic finay yieds» d(v n) Φ + rψ + (n, z) = δ 2Φ + Ψ V n + 1 H Ḣ H + 2 «Z! to Ψ + V n + dt( 2 rh Φ + Ψ) + 1 t S r Z to t S dt dt dδz dt» 2 t t S (t O t) S 2 (Φ + Ψ). Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

14 we arrive at δ z(n, z) = δ(n, z) + 3(V n)(n, z) + 3Ψ(n, z) + 3 For the voume we use Z to t S ( Ψ + Φ)(n, z(t))dt. dv = p g ǫ abcd u a dx b dx c dx d = p g ǫ abcd u a x b x c x d z θ S ϕ S (θ S, ϕ S ) (θ O, ϕ O ) dzdθ Odϕ O = vdzdθ O dϕ O To first order in the perturbations one finds v = a3 r 2 sin θ O H» 1 3φ + (cot θ O + δφ )δθ + θ φ v n + 2δr dδr + 1 r dt H The engthy cacuation of δθ, δφ δr aong the perturbed geodesic finay yieds» d(v n) Φ + rψ + (n, z) = δ 2Φ + Ψ V n + 1 H Ḣ H + 2 «Z! to Ψ + V n + dt( 2 rh Φ + Ψ) + 1 t S r Z to t S dt dt dδz dt» 2 t t S (t O t) S 2 (Φ + Ψ). Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

15 We can now expand (n, z) in spherica harmonics, (n, z) = X m a m (z)y m (n), C (z) = a m 2 (z). a m (z)a m (z ) = δ δ mm C (z, z ). The transversa power spectrum at redshift z is now given by thec (z, z) and the ongitudina power spectrum can be obtained from C (z, z ) which probaby can be approximated by C (z)f( r) where r = r(z) r(z ). Z P ong(k) = e ik r f( r)d r. Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

16 Theoretica power spectra in synchronous and Newtonian gauge (from Yoo et a. 09) Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

17 Contributions to the transverse power spectrum at redshift z = 0.1 (from Bonvin & RD 11) Π C C DD C zv (red), C zz (back), C DV (green), C Dz (yeow). (bue), C LL (magenta), C VV (cyan), Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

18 Contributions to the transverse power spectrum at redshift z = 2 (from Bonvin & RD 11) Π C C DD C zv (red), C zz (back), C DV (green), C Dz (yeow). (bue), C LL (magenta), C VV (cyan), Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

19 Contributions to the transverse power spectrum at redshift = 10 and = 50 (from Bonvin & RD 11) 1 1 2Π C Π C z z Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

20 The observabe matter power spectrum δ z (from Yoo 10) Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

21 What are Newtonian N-body simuations simuating? At present we have Hubbe size N-body simuations which go out to redshifz z 2. What are such simuations reay cacuating? (Exampe: sice through MareNostrum, by Gottöber et a. 06) Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

22 What are Newtonian N-body simuations simuating? In principe, Newtonian N-body simuations are soving the Poisson equation in a cever way (initia conditions from inear perturbation theory, Ze dovich approximation), φ = 4πGδρ Interestingy enough in inear perturbation theory (which is sufficient on arge scaes, where reativistic effects are most reevant), this is exacty the 00-constraint equation if we interpret δ as the matter density fuctuations in comoving gauge and φ as the Bardeen potentia Φ = Ψ (in absence of anisotropic stresses). Hence the power spectrum obtained from Newtonian N-body simuation, agrees with the one of the density fuctuations in comoving gauge (see aso Chisari & Zadarriaga 11). This is reated to the density fuctuation in ongitudina (or Newtonian) gauge via δ cm = δ Newt + 3Hk 1 V The veocity is obtained from the non-reativistic continuity equation, and from the eqn. of motion δ = v, and v + H v = φ Interestingy, for pressureess matter this equation is exacty equa to the energy conservation equation if δ = δ cm and v is the veocity in Newtonian gauge, v = iˆkv in Fourier space. Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

23 What are Newtonian N-body simuations simuating? In principe, Newtonian N-body simuations are soving the Poisson equation in a cever way (initia conditions from inear perturbation theory, Ze dovich approximation), φ = 4πGδρ Interestingy enough in inear perturbation theory (which is sufficient on arge scaes, where reativistic effects are most reevant), this is exacty the 00-constraint equation if we interpret δ as the matter density fuctuations in comoving gauge and φ as the Bardeen potentia Φ = Ψ (in absence of anisotropic stresses). Hence the power spectrum obtained from Newtonian N-body simuation, agrees with the one of the density fuctuations in comoving gauge (see aso Chisari & Zadarriaga 11). This is reated to the density fuctuation in ongitudina (or Newtonian) gauge via δ cm = δ Newt + 3Hk 1 V The veocity is obtained from the non-reativistic continuity equation, and from the eqn. of motion δ = v, and v + H v = φ Interestingy, for pressureess matter this equation is exacty equa to the energy conservation equation if δ = δ cm and v is the veocity in Newtonian gauge, v = iˆkv in Fourier space. Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14

24 Questions What do arge gaaxy surveys reay measure: Is it possibe to isoate some of terms in the formua for (z, n), e.g. with compementary measurements? How can we measure pure voume distortions? Info in transversa vs. ongitudina power? Is C (z, z ) usefu or shoud we stay with P(k ) and P(k )? What do arge N-body simuations reay cacuate: Is it surprising that 1st order scaar reativistic perturbations agree with Newtonian gravity? Is this sufficient or do we need more? What happens at second order? Ruth Durrer (Université de Genève) Theoretica Cosmoogy Benasque / 14