Enhanced constraints from multi-tracer surveys

Transcription

1 Enhanced constraints from multi-tracer surveys or How to beat cosmic variance Raul Abramo Physics Institute, USP & USP & J-PAS / Pau-Brasil Collaboration J-PAS

2 Galaxy surveys are evolving We used to live in an era of shot noise We are now starting an era of cosmic variance [Finding galaxies was the limiting factor] [Gaining volume is the limiting factor]

3 Cosmology from galaxy clustering g (~x) =n g (~x)[1+ g (~x)]! n g (~x)[1+b g (~x) (~x)] Clustering in position space BAOs Clustering in Fourier space BAOs Bias: 1 (r) =b 2 1 (r)! P 1 (k) =b 2 1 P (k) 2 (r) =b 2 2 (r)! P 2 (k) =b 2 2 P (k)

4 Fisher information matrix of galaxy surveys FKP - Feldman, Kaiser & Peacock (1994) Tegmark et al. (1997), R.A. (2012) The galaxy power spectrum in redshift space, for any galaxy survey, can be expressed in units of its shot noise (1/ng). This defines the survey s effective power spectrum (which is adimensional): P g ( ~ k,~x)! n g (~x) b g (~x)+f(~x) µ 2 k 2 G 2 (~x) P ( ~ k) f = µ k = k k d log G d log z average density of galaxies in the survey galaxy bias (function of z, at least) matter growth rate = redshift distortion parameter angle of Fourier mode w.r.t. lineof-sight matter growth function matter power spectrum The Fisher information for the (log of the) effective power spectrum is: F [log P g ]= 1 2 Pg 1+P g 2 FKP Fisher matrix

5 Fisher information in phase space On each cell of phase space volume there is a certain amount of information about the spectrum (and other quantities), given by: k F [log P g ] V x V k (2 ) 3 = 1 2 Pg 1+P g 2 V x V k (2 ) 3 V phase space density of information < 1/2 phase space volume =. V x The precision with which we can estimate the effective power spectrum from the information in each cell of phase space is: (P g ) P g = 1 p F [log Pg ] V = 1+P g P g r 2 V The Fisher information is additive, so integrating over the phase space volume gives the total information.

6 Fisher information, effective volume and cosmic variance p a! i : F ij [ ] = X i j p a = log P g! i ) F ij = Z d 3 kd 3 x (2 ) 3 d log P g d i 1 2 Pg 1+P g 2 d log P g d j Power of a survey: ~ effective volume V eff ( ~ k)= Z d 3 x Pg 1+P g 2 <V P 2 H1 + PL shot noise P COSMIC VARIANCE P g n g (z) P (k)

7 Why? Much of the cosmological information resides in the power spectrum h ( ~ k) ( ~ k 0 )i =(2 ) 3 ( ~ k ~ k 0 )P ( ~ k) In a finite volume, even if we map a huge number of tracers, the precision with which we can measure the modes, and P(k), is limited P (k) P (k) / 1 p # of modes P (k) P (k) P (k) P (k) COSMIC VARIANCE

8 Implications for cosmology Example: k=0.1 h Mpc -1 PR, PE, PQ Cosmic variance dominates here, so a colossal waste of galaxies? J-PAS is cosmic variance-limited up to z~1, and shot noise-limited above that sweet spot 0.1 LRGs ELGs QSOs z Are these many millions of z 1 galaxies really wasted, from the point of view of cosmology?

9 NO!

10 How to beat cosmic variance By comparing the clustering of the different tracers of large-scale structure (LRGs, ELGs, etc.), we can measure with arbitrary accuracy* the physical parameters that determine their different clustering amplitudes Seljak 2008; McDonald & Seljak 2008 Gil-Marín et al Hamaus, Seljak & Desjacques 2011,2012 Cai & Bernstein 2011 R.A. & K. Leonard, MNRAS 2013 (arxiv: ) P 1 = n 1 (b 1 + fµ 2 k) 2 P (k; z) P 2 = n 2 (b 2 + fµ 2 k) 2 P (k; z) P 1 = n 1 (b 1 + fµ 2 k )2 P 2 n 2 (b 2 + fµ 2 k )2 Cosmic Variance does not apply!

11 Multi-tracer Fisher information matrix R.A., MNRAS 2012 ( ) R.A. & K. Leonard, MNRAS 2013 ( ) Let s say we have several (α = 1,2,... N) different types of tracers of large-scale structure: α=1 (LRGs), α=2 (ELGs), α=3 (quasars), etc. The multi-tracer Fisher matrix is: F = F (log P, log P ) = 1 4 apple P P 1+P + P P (1 P), P = X (1 + P) 2 P Or, in terms of the usual parameters: F ij = F ( i, j )= X Z d 3 kd 3 x (2 ) 3 d log P d log P d i F d j

12 Multi-tracer Fisher information matrix The multi-tracer Fisher information is unbounded! F = 1 4 apple P P 1+P + P P (1 P) (1 + P) 2 But this still does not necessarily mean that the total information is unbounded... In fact, there is no clear meaning to total information for nondiagonal Fisher matrices But if we diagonalize the Fisher matrix, then each eigenvalue adds positive, independent amounts of information! k x

13 Multi-tracer Fisher information matrix R.A. & K. Leonard, MNRAS 2013 (arxiv: ) We can diagonalize the multi-tracer Fisher matrix by a change of variables: P! Q a First, we define the aggregate effective spectra as: S a = NX =a P The variables which diagonalize the multi-tracer Fisher matrix (eigenvectors) are: Q 1 = S 1 = P Q 1 is the total effective spectrum of the survey. Only it involves P(k) CVlimited Q a =. Q N = S a P a 1 (a>1) S N P N 1 = P N P N 1 Q a (a>1) are relational variables (ratios of spectra between the different tracers). They do not involve P(k) NOT CV-limited

14 Diagonalized multi-tracer Fisher matrix In terms of the relational power spectra Q a the Fisher matrix is diagonal! F ab = F a ab F 1 = 1 2 P 1+P 2 < 1/2 CV [FKP] codifies info about P(k) F a = 1 4 P 1+P S a P a 1 S a 1 unbounded but no info about P(k) ) F ij = Z d 3 kd 3 x (2 ) 3 X a d log Q a d i F a d log Q a d j = X a F a ij

15 Example: 2-tracer survey Includes P(k) Q 1 = P = P 1 + P 2 Q 2 = P 2 P 1 F 1 = 1 2 P1 + P P 1 + P 2 F 2 = 1 P 1 P P 1 + P 2 Does not include P(k) 1, P 2 =8 P 2 =4 P 2 =2 P 2 =1 P 2 =

16 Constraints on the RSD parameter f(z) [J-PAS] P g = n g (b g + fµ 2 k) 2 P (k; z) FHfs L x boost! F ( ) = 1 2 c ( ) ELGs only LRGs only QSOs only All, FKP Relat. inform. Total Relat.êTotal Information from the relative clusterings can improve the constraints on f(z) by up to ~3 at low z s! z

17 Redshift-space distortions & modified gravity G µ + G µ =8 GT µ +8 G T µ r 2 = 16 G 1 Matter growth: 3 6 R(f R) =) G(z) f =d LogHGLêd LogHaL f(z) = d ln G d ln a = CDM m(z) ELGs only LRGs only QSOs only All 0.5 DGP z * Marginalized against shape of P(k) & cosmological parameters * Assumed biases were fixed, linear and not stochastic lensing would help a lot!

18 Constraints in local non-gaussianity parameter f NL P g = n g (b g + fµ 2 k) 2 P (k; z) b g! b g + b g (f NL,k) Prediction (and discriminator) of inflation models F ( ) = 1 2 c ( ) Log FHf NL L D Total ELGs only LRGs only QSOs only Relat.êTotal Information from relative clustering can improve constraints on f NL by ~5 at low-z! z

19 Conclusions: Cosmic variance is a fundamental limitation only for measurements of the power spectrum Multi-tracer strategies are able to optimally explore the new era of volumelimited surveys of large-scale structure In particular, the multi-tracer approach can enhance dramatically the constraints on: modified gravity (through the RSD function f) inflation (f NL ) Even BAOs can benefit: both indirectly (through marginalizations), and also directly, via enhancements of the constraints from AP tests Biggest challenge is covariance of biases and shot noise between tracers; we are studying those covariances with the help of N-body simulations

20 Forecasted distance constraints from BAOs in J-PAS * RSDs and shape of P(k) were marginalized * Equivalent to Seo/Eisenstein BAOs-only method 0.04 w 0 = w 0 = shhlêh shda LêDa CDM CDM z z ELGs only LRGs only QSOs only All

21 Figure of Merit for J-PAS wa w 0 FoM=@Det F -1 Hw 0,wa LD -1ê z ELGs only LRGs only QSOs only All * With Planck priors * Marginalized against shape of P(k) & other parameters * RSD information was marginalized, but not projected into final set of cosmological parameters * Same methods/criteria as used for EUCLID papers