Relativistic effects in large-scale structure

Transcription

1 Relativistic effects in large-scale structure Camille Bonvin University of Geneva, Switzerland COSMO August 2017

2 Galaxy survey The distribution of galaxies is sensitive to: Credit: M. Blanton, SDSS the initial conditions the theory of gravity the content of the universe The large-scale structure contains valuable information To interpret properly this information, we need to understand what we are measuring. COSMO 2017 Camille Bonvin p. 2 /39

3 Galaxy survey We count the number of galaxies per pixel: = N N N How is related to: the initial conditions, the theory of gravity and dark energy? Credit: M. Blanton, SDSS COSMO 2017 Camille Bonvin p. 3 /39

4 Galaxy distribution Simple picture: dark matter is not homogeneously distributed it creates gravitational potential wells baryons fall into them and form galaxies More dark matter Less dark matter = δρ ρ δ COSMO 2017 Camille Bonvin p. 4 /39

5 Complications Bias: the distribution of galaxies does not trace directly the distribution of dark matter = b δ We never observe directly the position of galaxies, we observe the redshift zand the direction of incoming photons n. In a homogeneous universe: we calculate the distance r(z) light propagates on straight lines (x 1,x 2,x 3 ) r(z) Observer n COSMO 2017 Camille Bonvin p. 5 /39

6 Complications Bias: the distribution of galaxies does not trace directly the distribution of dark matter = b δ We never observe directly the position of galaxies, we observe the redshift zand the direction of incoming photons n. In a inhomogeneous universe: the redshift is perturbed r(z) light propagates on perturbed geodesics (x 1,x 2,x 3 ) r(z) Observer n COSMO 2017 Camille Bonvin p. 6 /39

7 Complications Bias: the distribution of galaxies does not trace directly the distribution of dark matter = b δ We never observe directly the position of galaxies, we observe the redshift zand the direction of incoming photons n. In a inhomogeneous universe: the redshift is perturbed r(z) light propagates on perturbed geodesics n Observer (x 1,x 2,x 3 ) r(z) COSMO 2017 Camille Bonvin p. 7 /39

8 Galaxy distribution The structures seen on a galaxy map do not reflect directly the underlying dark matter structures. The observed position of galaxies are shifted radially and transversally. Credit: M. Blanton, SDSS To extract information from a galaxy map, we need to understand exactly which distortions there are. COSMO 2017 Camille Bonvin p. 8 /39

9 Outline Expression for encoding all distortions at linear order = density + redshift distortions + lensing + relativistic effects Impact of the different terms on the correlation function The distortions change the standard multipoles The distortions generate new multipoles COSMO 2017 Camille Bonvin p. 9 /39

10 Calculation of the distortions Perturbed Friedmann universe: Galaxy ds 2 = a 2[ ( 1+2Ψ ) dη 2 + ( 1 2Φ ) δ ij dx i dx j] n µ We calculate the propagation of photons, i.e. the null geodesics and infer: Observer the change in energy the change in direction COSMO 2017 Camille Bonvin p. 10/39

11 (z,n) = b δ 1 H r(v n) + r What we really observe dr r r 0 rr Ω (Φ+Ψ) ( ) 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) COSMO 2017 Camille Bonvin p. 11/39

12 What we really observe (z,n) = b δ 1 H r(v n) r dr r r 0 rr Ω (Φ+Ψ) ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r Velocities + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) COSMO 2017 Camille Bonvin p. 12/39

13 What we really observe (z,n) = b δ 1 H r(v n) r dr r r 0 rr Ω (Φ+Ψ) Lensing ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r Velocities + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) COSMO 2017 Camille Bonvin p. 13/39

14 What we really observe (z,n) = b δ 1 H r(v n) r dr r r 0 rr Ω (Φ+Ψ) Lensing ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r Velocities + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) Potentials COSMO 2017 Camille Bonvin p. 14/39

15 Distortions Velocities Kaiser (1987), Lilje & Efstathiou (1989), Hamilton (1992) Change in the bin size: Redshift distortions z Change in the bin position: Doppler effect z Observer Observer Lensing Gunn (1967), Schneider (1989), Broadhurst, Taylor & Peacock (1995) Change in the solid angle Observer Local terms: Potentials e.g. gravitational redshift Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) Observer Observer Integrated terms: e.g. Shapiro time-delay and Integrated Sachs-Wolfe COSMO 2017 Camille Bonvin p. 15/39

16 What we really observe (z,n) = b δ 1 H r(v n) r dr r r 0 rr Ω (Φ+Ψ) ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 Yoo et al (2010) CB and Durrer (2011) Challinor and Lewis (2011) COSMO 2017 Camille Bonvin p. 16/39

17 What we really observe (z,n) = b δ 1 H r(v n) current standard analysis r dr r r 0 rr Ω (Φ+Ψ) ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 COSMO 2017 Camille Bonvin p. 17/39

18 What we really observe (z,n) = b δ 1 H r(v n) current standard analysis r dr r r 0 rr Ω (Φ+Ψ) ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r lensing: important at high redshift + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 COSMO 2017 Camille Bonvin p. 18/39

19 What we really observe (z,n) = b δ 1 H r(v n) current standard analysis r dr r r 0 rr Ω (Φ+Ψ) ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r lensing: important at high redshift + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 relativistic effects: important at large separation COSMO 2017 Camille Bonvin p. 19/39

20 Correlation function Credit: M. Blanton, SDSS d Observer d d ξ = (x) (x ) The dark matter fluctuations generate isotropic correlations = b δ ξ(d) = C 0 (d) COSMO 2017 Camille Bonvin p. 20/39

21 Correlation function The distortions break the isotropy of the correlation function Credit: M. Blanton, SDSS β Observer n d ξ = (x) (x ) Redshift distortions generate a quadrupole and hexadecapole Kaiser (1987) Hamilton (1992) ξ =C 0 (d)+c 2 (d)p 2 (cosβ) +C 4 (d)p 4 (cosβ) Legendre polynomials COSMO 2017 Camille Bonvin p. 21/39

22 Correlation function The distortions break the isotropy of the correlation function Credit: M. Blanton, SDSS ij ij Observer n β d i j C 0 i j P 2 (cosβ ij ) C 2 i j P 4 (cosβ ij ) C 4 ξ = (x) (x ) Redshift distortions generate a quadrupole and hexadecapole ξ =C 0 (d)+c 2 (d)p 2 (cosβ) +C 4 (d)p 4 (cosβ) Kaiser (1987) Hamilton (1992) ij Legendre polynomials COSMO 2017 Camille Bonvin p. 22/39

23 Corrections to the standard multipoles How do the distortions change the amplitude of the monopole, quadrupole and hexadecapole? Fractional difference from local relativistic effects # "&!"" Tansella, CB, Durrer, Gosh and Sellentin (2017) z = 0.1 z = 1 "&#"" ξ l = ξrel l ξl st "&#"" "&"#" "&"!" "&"#" "&""! monopole quadrupole hexadecapole "&""# #"!" "&""#!" #"" #!" $"" $!" %"" %!" d #"!!!" #"" #!" $"" $!" %"" %!" d Dominant distortion studied in Papai and Szapudi (2008) Raccanelli, Samushia and Percival (2010) COSMO 2017 Camille Bonvin p. 23/39

24 Corrections to the standard multipoles Tansella, CB, Durrer, Gosh and Sellentin (2017) How do the distortions change the amplitude of the monopole, quadrupole and hexadecapole? Fractional difference from gravitational lensing ξ l = ξlens l ξl st z = 1 z = 2 "&#"" "&"!" # "&!" "&"#" "&""! "&#" "&"! monopole quadrupole "&""#!" #"" #!" $"" $!" %"" %!" "&"# hexadecapole!" #"" #!" $"" $!" %"" %!" d d COSMO 2017 Camille Bonvin p. 24/39

25 The distortions change the multipole expansion Lensing generates higher multipoles Tansella, CB, Durrer, Gosh and Sellentin (2017)!&!!!"'!!!!!!&!!!"!!&!!!!'!&!!!!!!!&!!!!' d!!!!!!! r = 100 Mpc/h r = 8 Mpc/h!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! dr = 200 Mpc/h! dr = 420 Mpc/h!!!!!!!!! d!!&!!!"!! "! #! $! %! COSMO 2017 Camille Bonvin p. 25/39

26 The distortions change the multipole expansion Relativistic effects generate odd multipoles CB, Hui & Gaztanaga (2014) Credit: M. Blanton, SDSS Observer n COSMO 2017 Camille Bonvin p. 26/39

27 The distortions change the multipole expansion Relativistic effects generate odd multipoles CB, Hui & Gaztanaga (2014) Credit: M. Blanton, SDSS Observer n COSMO 2017 Camille Bonvin p. 27/39

28 The distortions change the multipole expansion Relativistic effects generate odd multipoles CB, Hui & Gaztanaga (2014) Credit: M. Blanton, SDSS Observer n COSMO 2017 Camille Bonvin p. 28/39

29 The distortions change the multipole expansion Relativistic effects generate odd multipoles CB, Hui & Gaztanaga (2014) Credit: M. Blanton, SDSS F F B Observer n COSMO 2017 Camille Bonvin p. 29/39

30 Breaking of symmetry from gravitational redshift CB, Hui & Gaztanaga (2014) = 1 H rψ shift in position due to gravitational redshift COSMO 2017 Camille Bonvin p. 30/39

31 The number count at linear order (z,n) = b δ 1 H r(v n) r dr r r 0 rr Ω (Φ+Ψ) ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 COSMO 2017 Camille Bonvin p. 31/39

32 The number count at linear order (z,n) = b δ 1 H r(v n) r dr r r i j cosβ ij 0 rr Ω (Φ+Ψ) ij ( ) + 1 Ḣ H 2 2 V n+ 1 rh H V n+ 1 H rψ r Dipole in the correlation function + 2 dr (Φ+Ψ)+3H 2 ( V)+Ψ 2Φ r 0 ( ) [ + 1 H Φ+ Ḣ H r ] Ψ+ dr ( Φ+ rh Ψ) 0 COSMO 2017 Camille Bonvin p. 32/39

33 Measuring the dipole in BOSS Gaztanaga, CB & Hui (2015) We split the LOWz and CMASS samples into 2 populations and measure the dipole. Dipole LOWz S N LOWz CMASS d Mpc h d Mpc h COSMO 2017 Camille Bonvin p. 33/39

34 Improvements Measure the dipole at lower redshift to increase the signal. Use a sample with diverse populations to increase the bias difference. Divide the sample into more than 2 populations to gain in statistics. Use an optimal estimator: weight each pair by the bias difference. COSMO 2017 Camille Bonvin p. 34/39

35 Forecasts CB, Hui & Gaztanaga, (2015) Main sample of SDSS: galaxies in total, 6 populations with bias from 0.96 to 2.16 Percival et al (2007) S N d Mpc h Cumulative signal-to-noise of 2.4 COSMO 2017 Camille Bonvin p. 35/39

36 Forecasts CB, Hui & Gaztanaga, (2015) DESI Bright Sample: 10 million galaxies, 6 populations with bias from 0.96 to S N d Mpc h Cumulative signal-to-noise of 7.4 COSMO 2017 Camille Bonvin p. 36/39

37 Forecasts with 21cm intensity mapping from SKA Hall and CB (2016) S/N Euclid DESI BOSS eboss LRGs eboss ELGs eboss GQSs SKA z COSMO 2017 Camille Bonvin p. 37/39

38 Interest The dipole is sensitive to the gravitational potential. rel = 1 H rψ+ ( 1 Ḣ H 2 2 rh ) V n+ 1 H V n Combining the dipole with the quadrupole, we can test Euler equation. CB and Fleury, in preparation V n+hv n+ r Ψ = 0 COSMO 2017 Camille Bonvin p. 38/39

39 Conclusion The fluctuations in the number of galaxies is affected by many effects besides the matter density fluctuations. These effects have a different signature in the correlation function: density monopole redshift distortions quadrupole and hexadecapole lensing higher multipoles relativistic effects dipole The dipole should be detectable in the near future and allow for new tests of gravity. COSMO 2017 Camille Bonvin p. 39/39