Cosmological surveys. Lecture 2. Will Percival

Transcription

1 Cosmological surveys Lecture 2 Will Percival

2 Physics from the linear galaxy power spectrum Projected clustering Galaxy clustering as a standard ruler BAO or full power spectrum Alcock-Paczynski effect Intrinsic power spectrum shape Matter density Baryon Acoustic Oscillations Neutrino mass Inflation fluctuation spectrum f NL P gal (k, µ, a) =k n T 2 (k)d 2 (a)[b(a)+f(a)µ 2 ] 2 k = comoving wavenumber μ = cos(angle to line-of-sight) a = cosmological scale factor b = galaxy bias factor D = linear growth rate f = dlnd/dlna Structure growth amplitude of power spectrum redshift-space distortions

3 Redshift-space distortions

4 Comoving velocities Locally, galaxies act as test particles in the flow of matter On large-scales, the distribution of galaxy velocities is unbiased if galaxies fully sample the velocity field expect a small peak velocity-bias due to motion of peaks in Gaussian random fields differing from that of the mass

5 Redshift-Space Distortions When making a 3D map of the Universe the radial distance is usually obtained from a redshift assuming Hubble s law; this differs from the real-space because of its peculiar velocity: s(r) =r v r (r) r r Where s and r are positions in redshift- and real-space and v r is the peculiar velocity in the radial direction

6 Two key regimes of interest linear flow non-linear structure Underdensity Overdensity Cluster Actual shape Underdensity Overdensity Cluster Apparent shape (viewed from below) Power is enhanced on large-scales Power is suppressed on small-scales

7 Fingers-of-God clearly visible in maps Image of SDSS, from U. Chicago

8 Linear plane-parallel redshift-space Transition from real to redshift space, with peculiar velocity v in units of the Hubble flow s = r + v losˆr los Jacobian for transformation d 3 s d 3 r = 1+ v 2 los 1+ dv los r los dr los Conservation of galaxy number µ = cos( ) = u μ=0 μ=1 n r (r)d 3 r = n s (s)d 3 s Trick to understand velocity field derivative los 2 r 2 = 1+ s g =(1+ r g) d3 r d 3 s k n r (r) n s (s) 2 = µ 2, = r v Gives to first order s g = r g µ 2 Kaiser 1987, MNRAS, 227, 1

9 what do linear z-space distortions measure? linear scales, µ = cos( ) = u μ=0 Galaxy-galaxy power Velocity-velocity power μ=1 Galaxy-velocity divergence cross power In linear regime g = b (mass), = f (mass), f d ln G d ln a So, the simplest model for the power spectrum is P s g (k, µ) = b + µ 2 f 2 Pmass (k) Linear growth rate Kaiser 1987, MNRAS, 227, 1

10 Including small-scale FOG Include model for linear and FOG terms P s g (k, µ) = P gg (k)+2µ 2 P g (k)+µ 4 P (k) F (k, µ 2 ) Note that non-linear model is not necessarily more accurate then the linear one. If we assume linear bias P s g (k, µ) =P r m(k) b 2 +2µ 2 fb+ µ 4 f 2 F (k, µ 2 ) On small scales, galaxies lose all knowledge of initial position. If pairwise velocity dispersion has an exponential distribution (superposition of Gaussians), then we get this damping term for the power spectrum. F (k, µ 2 )=(1+k 2 µ 2 2 p/2) 1

11 Modeling redshift space distortions Alternative for the data is to try to correct the data by collapsing the clusters Velocity dispersion of the Luminous Red Galaxies (LRGs) shifts them along the line of sight by 9 h 1 Mpc, and the distribution of intrahalo velocities has long tails. redshift space cylinder Use an asymmetric friends-offriends (FOF) finder to match galaxies in the same clusters, and collapse to spherical profile move galaxies back Parameters of FOF calculated by matching simulations Reid et al. 2008, arxiv: ; Reid et al. 2009, arxiv:

12 Cosmology improved with RSD Anisotropic clustering allows huge improvement on w! w = ± 0.25 (WMAP + D V (0.57)/r s ) w = ± (WMAP + anisotropic) Provided a number of GR tests Samushia et al. 2012; arxiv:

13 Measuring anisotropic clustering

14 Anisotropic clustering The Universe is (expected to be) statistically homogeneous and isotropic The observed Universe (when translated into a map using a fixed distanceredshift relation) is not it only has local transverse isotropy around the line-of-sight (LOS) Redshift-space distortions Alcock-Paczynsky effect Need to measure clustering relative to LOS Saw this previously with BAO, but let s consider this in more detail

15 Legendre moments power spectrum Remember that, in linear theory and the plane-parallel limit, we have a angular dependence P s g (k, µ) =P gg (k)+2µ 2 P g (k)+µ 4 P (k) P s` (k) Then we can consider the orthogonal Legendre multipoles of P where 2` +1 2 Such that Z +1 1 dµ P s g (k, µ)l`(µ) P s 0 (k) P s 2 (k) P s 4 (k) 1 A = L 0 (µ) = 1 L 2 (µ) = (3µ 2 1)/2 L 4 (µ) = (35µ 4 30µ 2 + 3)/8 1 2/3 1/5 0 4/3 4/ / P gg (k) P g (k) P (k) P s (k) =P s 0 (k)l 0 (µ)+p s 2 (k)l 2 (µ)+p s 4 (k)l 4 (µ) 1 A

16 Legendre moments power spectrum These can then be manipulated leading to cosmological information From the monopole, quadrupole and hexadecapole we obtain: P gg (k) P g (k) P (k) 1 A = 1 1/2 3/8 0 3/4 15/ /8 1 0 P s 0 (k) P s 2 (k) P s 4 (k) 1 A The ratio of quadrupole to monopole gives: P (k) = 7 48 P2 s 4 (k) P0 s(k) = 3 bf f 2 b fb+ 1 5 f 2 A more complicated formula can be used to eliminate bias: h 5(7P s 0 + P s 2 ) p 35[35(P s 0 ) P s 0 P s 2 7(P s 2 ) 2 ] 1/2i e.g. Percival & White 2009; MNRAS 393, 297

17 Legendre moments correlation function The correlation function can similarly be decomposed into Legendre moments Z +1 s` (2` + 1) (r) = dµ s (r, µ)l`(µ) 2 1 s (r, µ) = X L`(µ) s` (r) ` even The first three even moments ξ 0, ξ 2, ξ 4 allow the full linear theory to be recovered s 0(r) = (b bf f 2 ) r (r) s 2(r) = ( 4 3 bf f 2 )[ r (r) r (r)] s 4(r) = 8 35 f 2 [ r (r)+ 5 2 r (r) r (r) 3r 3 Z r 0 r (r 0 )r 02 dr r (r)] r (r) 5r 5 Z r r (r 0 )r 04 dr 0 0 Hamilton 1992; ApJ 385, L5

18 The LOS varies across a survey LOS LOS

19 Wide-angle effects LOS to pair LOS to gal 1 LOS to gal 2 First order RSD model allowing for different LOS for each galaxy is possible, but messy! (e.g. Szalay et al ApJ 498, L1, Szapudi 2004, PRD 70, )

20 Wide-angle effects are small Thus can calculate correlation function by pair counting, allowing each pair to have a different LOS - that to the pair centre (e.g. Landy & Szalay 1993; ApJ 412, 64) But methods scale as N 2, where N is the number of galaxies number of randoms size of grid (depending on method) Samushia, Percival & Raccanelli 2012, MNRAS, 420, 2102

21 The correlation function wrt LOS DD = number of galaxy-galaxy pairs DR = number of galaxy-random pairs RR = number of random-random pairs All calculated as a function of separation and direction to LOS = DD RR 1 = DD DR 1 DD RR = DR 2 1 DD 2DR = +1 RR Survey volume Galaxies Landy & Szalay (1993) considered noise from these estimators, and showed that this has the best noise properties Randoms Landy & Szalay 1993; ApJ 412, 64

22 FFTs mix clustering wrt LOS LOS LOS

23 Measuring the anisotropic power spectrum l=2,m=1 Spherical Harmonics (θ,φ) + l=2,m=0 2d Fourier basis (x,y) + k x,k y Spherical Bessel function (r) n=2 r 1d Fourier basis (z) k z z advantage: radial/angular split more matched to survey geometry, easily model redshift space distortions advantage: simplicity, speed e.g. Heavens & Taylor 1995; MNRAS, 275, 483

24 Yamamoto et al. method (astro-ph/ ) Define the overdensity field Multipole power spectra can be written as a integral over pairs The clever part is defining the LOS to the pair as LOS to one galaxy

25 Wide-wide-angle effects are small Samushia, Branchini & Percival 2014, arxiv:

26 Writing this in terms of FFTs The unit to be solved is We can expand the dot product on a Cartesian basis So that (for example) A 2 is decomposed (similarly for n>2) Where B ij can be solved with FFTs Bianchi et al. 2015; arxiv: , Scoccimarro arxiv:

27 Same result as sums Bianchi et al. 2015; arxiv:

28 Large-scale RSD & AP measurements

29 AP effect on monopole & quadrupole Varying F=D A H by 10%, while keeping peak position in monopole fixed Linear RSD shift is scale-independent for both AP moves ξ(r) in scale (left-right). Movement of BAO bump is clear. Shape of ξ(r) close to power law, so AP is very similar to amplitude shift (as RSD). Allows measurements of F & fσ 8 to be separated Reid et al. 2012; arxiv:

30 RSD and AP amplitude shift strongly correlated We should allow for the coupling between the redshift-space distortions and the geometrical squashing caused by getting the geometry wrong. Effects are not perfectly degenerate Linear redshiftspace distortions Geometric squashing Fit to redshift-space distortions cannot mimic geometric squashing Ballinger, Peacock & Heavens 1999, MNRAS, 282, 877

31 Degradation of RSD measurements by AP effect Samushia et al 2011, MNRAS 410, 1993

32 BOSS AP & RSD measurement degeneracy Dotted: free growth, geometry, ΛCDM prior on large-scale linear P(k) shape at z=0.57 Solid: F forced to match ΛCDM model Dashed: WMAP ΛCDM+GR prediction Reid et al. 2012; arxiv:

33 BOSS F measurements in context Samushia et al. 2012; arxiv:

34 BOSS RSD measurements in context Samushia et al. 2012; arxiv:

35 The effect of AP uncertainty on RSD Howlett et al. 2014; arxiv:

36 Primordial non-gaussianity

37 Measuring primordial non-gaussianity: f NL g NL δ is sourced from a potential field Φ, whose form might not be Gaussian r 2 (x) =4 G (x) (x) (x)+f NL 2 (x)+... ϕ is a Gaussian field. the non-linear terms in Φ make Φ non-gaussian. This map completely specifies Φ statistics. skewness ~ f NL kurtosis ~ f NL 2... Salopek and Bond 1990; Gangui, Lucchin, Matarrese, Mollerach 1994; Komatsu and Spergel 2001 (x) (x)+g NL 3 (x)+... f NL is not the only option for local potential fluctuations you can go even further down this route skewness ~ 0 kurtosis ~ g NL Okamoto and Hu 2002; Enqvist and Nurmi 2005 Non-local models introduce non-trivial higher order correlations in Φ

38 Measuring non-gaussianity: halo abundance Dark matter halos form in the peaks of the density field δ δ c Non-Gaussianity changes the number density of the peaks This in turn affects the halo mass function x

39 Measuring non-gaussianity: halo abundance number of haloes / number with f nl =0 Largest effect is seen at highest masses Insensitive to shape of bispectrum But difficult to observe relies on cluster masses being precisely known LoVerde & Smith 2011, arxiv:

40 Peak-background split bias model Halo formation much easier with additional long-wavelength fluctuation δ c l Number density of halos n! n dn d c l Leads to a revised density To first order, this leads to a bias new = l + n n (1 + new )= b = new l 1+ n (1 + l ) n =1+ n n l =1 Directly from the large-scale mode From the change in Number of haloes d ln n d c

41 Peak-background split galaxy bias model Sheth & Tormen 1999, arxiv:

42 δ This is altered by f NL signal Now split non-gaussian potential into long and short wavelength components 2 2 (x) = l + f NL l +(1+2f NL l ) s + f NL s +cnst l(k) = (k) (k) small Link between potential and overdensity field shows how changing long wavelength potential component changes critical density (k) = 2c2 k 2 T (k)d(z) 3 m H 2 0 c l 2f NL l = c l 1+ 2f NL (k)

43 Peak-background split for non-gaussianity Halo formation much easier with additional long-wavelength fluctuation δ c l 2f NL l = c l 1+ 2f NL (k) Number density of halos n! n 1+ 2f NL dn (k) d c l Leads to a revised bias b =1 1+ 2f NL (k) d ln n d c

44 K 2 dependence in simulations Dalal, Dore, Huterer, Shirokov 2007 ; Smith, LoVerde 2010; Smith, Ferraro, LoVerde 2011; Pillepich, Porciani, Hahn 2008; Desjacques, Seljak, Iliev 2008; Grossi et al 2009; Shandera, Dalal, Huterer 2010; Hamaus et al. 2011

45 Model testing with data

46 Bayes theory Modern observational cosmology relies on Bayes theory I Things assumed to be true (e.g. model) H Hypothesis to be tested, often a vector of parameters θ d Data sampling distribution of data often called the Likelihood L(H) p(d H, I) prior p(h d, I) = p(d H, I)p(H I) p(d I) posterior probability normalisation Assuming θ=(ϕ,ψ), with ϕ interesting and ψ not, parameter inference is performed as Z p( d, I) / L(, )p(, I)d e.g. review by Trotta 2008; arxiv:

47 Model parameters (describing LSS & CMB) content of the Universe total energy density Ω tot (=1?) matter density Ω m baryon density Ω b neutrino density Ω n (=0?) Neutrino species f n dark energy eq n of state w(a) (=-1?) or w 0,w 1 perturbations after inflation scalar spectral index n s (=1?) normalisation σ 8 running a = dn s /dk (=0?) tensor spectral index n t (=0?) tensor/scalar ratio r (=0?) evolution to present day Hubble parameter h Optical depth to CMB τ parameters usually marginalised and ignored galaxy bias model b(k) (=cst?) or b,q CMB beam error B CMB calibration error C Assume Gaussian, adiabatic fluctuations

48 Multi-parameter fits to multiple data sets Given CMB data, other data are used to help break degeneracies (although CMB is now doing a pretty good job by itself) and understand dark energy Main problem is keeping a handle on what is being constrained and why difficult to allow for systematics you have to believe all of the data! Have two sets of parameters those you fix (part of the prior) those you vary Need to define a prior what set of models what prior assumptions to make on them (usual to use uniform priors on physically motivated variables) Need a sampling method for exploring multi-dimensional parameter space

49 degeneracies: CMB Planck collaboration 2013; arxiv:

50 degeneracies: CMB Planck collaboration 2013; arxiv:

51 degeneracies: CMB Planck collaboration 2013; arxiv:

52 Bayesian model selection A lot of the big questions to be faced by future experiments can be reduced to: Is the most simple model (e.g. Λ) correct? Bayes theory can test the balance between quality of fit and predictivity A model with more parameters will always fit the data better than a model with less parameters (provided it replicates the original model). But does the improvement show the parameter is needed? p(h d, I) = p(d H, I)p(H I) p(d I) Bayesian evidence Bayesian evidence is average of Likelihood under the prior. Splitting into model and parameters p(d I) = Z p(d,m)p( M)d e.g. review by Trotta 2008; arxiv:

53 Bayesian model selection Bayes factor is ratio of evidence for 2 models Jeffries scale B 01 = p(d I 0) p(d I 1 ) Problem: depends on prior on new parameter. More stringent criteria can be set Other options are available about a lecture s worth! e.g. review by Trotta 2008; arxiv:

54 Future surveys: next 4-6 years

55 Dark Energy Survey (DES) New wide-field camera on the 4m Blanco telescope Survey started, with first data in hand Ω = 5,000deg 2 multi-colour optical imaging (g,r,i,z) with link to IR data from VISTA hemisphere survey 300,000,000 galaxies Aim is to constrain dark energy using 4 probes LSS/BAO, weak lensing, supernovae cluster number density Redshifts based on photometry weak radial measurements weak redshift-space distortions See also: Pan-STARRS, VST-VISTA, SkyMapper

56 eboss / SDSS-IV The new cosmology project with SDSS Use the Sloan telescope and MOS to observe to higher redshift Basic parameters Ω = 1,500deg 2 7,500deg 2 ~ 1,000,000 galaxies (direct BAO) ~ 60,000 quasars (BAO from Ly-α forest) Distance measurements 0.9% at z=0.8 (LRGs) 1.8% at z=0.9 (ELGs) 2.0% at z=1.5 (QSOs) 1.1% at z=2.5 (Ly-α forest, inc. BOSS) Survey will start 2014, lasting 6 years Received $10M from Sloan foundation and significant funding from partners

57 Future surveys: > 4 years

58 MOS on 4m-telescope New fibre-fed spectroscopes proposed for 4m telescopes Mayall (BigBOSS) Blanco (DESpec) WHT (WEAVE) VISTA (4MOST) Various stages of planning & funding DESI has just passed DOE CD-1, 2019 start 4MOST chosen by ESO, 2020 start? WEAVE, 2018 start All capable of observing Ω =5--14,000deg 2 DESI 2--40,000,000 galaxies (direct BAO) ,000 quasars (BAO from Ly-α forest) Cosmic variance limited to z ~ 1.4

59 MOS on 10m-telescope New fibre-fed spectroscopes proposed for 10m telescopes Hobby-Eberly (HETDEX) Subaru (PFS) Different baseline strategies HETDEX 420deg 2 Ly-alpha emitters 800,000 galaxies 1.9<z<3.5 Greig, Komatsu & Wyithe, 2012, arxiv: PFS 1400deg 2 ELGs 3,000,000 galaxies 0.6<z<2.4 Ellis et al., 2012, arxiv:

60 The ESA Euclid Mission

61 The Euclid spectroscopic survey Wide survey 15,000deg 2 4 dithers NIR Photometry Y, J, H 24mag, 5σ point source NIR slitless spectroscopy nm ergcm -2 s σ line flux 3 or 4 dispersion directions, 1 broad wavebands 0.9<z<1.8 45M galaxies Deep survey 40deg 2 48 dithers 12 passes, as for wide survey dispersion directions for 12 passes >10deg apart

62 BAO measurements for future surveys using the code of Seo & Eisenstein 2007, arxiv:

63 BOSS CMASS DR9 galaxy clustering BOSS CMASS galaxies at z~0.57 Total effective volume V eff = 2.2 Gpc 3 Anderson et al. 2012; arxiv:

64 Predicted Euclid galaxy clustering Redshift slice 0.9 < z < 1.1 Total effective volume (of Euclid) V eff = 57.4 Gpc 3

65 Improvement in precision factor of 30 improvement in statistics! but what about systematics?

66 Testing with subsamples

67 Testing with blue / red subsamples Ross et al. 2013, in prep

68 Testing with blue / red subsamples Ross et al. 2014, MNRAS 437, 1109

69 Getting the likelihood right

70 Getting the likelihood calculation 100% correct The Likelihood under the standard assumption of a set of data drawn from a multi-variate Gaussian distribution is given by L(x p, t )= t apple 1 p exp (x, p, t ), where 2 (x, p, t ) X ij x d i x i (p) t ij x d j x j (p). now suppose that the covariance matrix (size n b x n b ) has been calculated from n s simulations µ i = 1 n s X s x s i C ij = 1 n s 1 then an unbiased estimator of the inverse covariance matrix is X (x s i µ i )(x s j µ j ) s = n s n b 2 n s 1 C 1 Hartlap J., Simon P., Schneider P., 2007, A&A, 464, 399

71 Errors in the covariance matrix Simply providing an unbiased estimator of the inverse covariance matrix is not enough The inverse covariance matrix also has its own error h ij i 0 j 0 i = A ij i 0 j 0 + B( ii 0 jj 0 + ij 0 ji 0 ), A = 2 (n s n b 1)(n s n b 4) B = (n s n b 2) (n s n b 1)(n s n b 4) Strictly, we should form a joint likelihood L(x, p, t )=L(x p, )L( t ), If we don t, this leads to an additional error on the n p parameters being fitted hp p i s.o. = B(n b n p )F 1, Taylor et al., 2012, arxiv: ; Dodelson & Schneider 2007, arxiv:

72 Errors in likelihood calculations Given a set of mocks, we can form two possible estimates of the errors: 1. From the individual likelihood surface from each mock 2. From the distribution of recovered measurements from the set of mocks These should agree! The estimates from each are biased in subtly different ways gives errors in the covariance matrix Percival et al., 2013: arxiv:

73 Getting the model right

74 BAO from simulations Real space Redshift space Seo et al., 2010, arxiv:

75 BAO from simulations Seo et al., 2010, arxiv:

76 What will you be showing in 15 years time?

77 At the same time as my PhD

78 SDSS-II LRG BAO vs other data ΛCDM models with curvature flat wcdm models Union supernovae WMAP 5year SDSS-II BAO Constraint on r s (z d )/D V (0.2) & r s (z d )/D V (0.35) Percival et al. 2009; arxiv:

79 Euclid BAO predictions ΛCDM models with curvature flat wcdm models Union supernovae WMAP 5year SDSS-II BAO Constraint on r s (z d )/D V (0.2) & r s (z d )/D V (0.35)

80 Cosmology from surveys What is the expansion rate of the Universe? What is the expansion rate of the Universe? Understanding Dark Energy Galaxy Survey Redshift-Space Distortions How does structure form within this background? Weak lensing What is a combination of the expansion rate of the Universe and the growth rate? What are the neutrino masses, matter density? Understanding energy-density What is f nl, which quantifies non- Gaussianity? Understanding Inflation