Cosmological Information Ue-Li Pen Tingting Lu Olivier Dore Cosmological Errors
How do we know errors on measurement and theory? For Gaussian field, P(k) is 2-PCF, its error (Fisher) is 4-PCF, and the error on Fisher is 8PCF. LSS data is non-linear, need to measure 8-PCF!
Cosmological Precision Goal: measure parameters precisely: e.g. Omega, w, etc Observable: Gaussian random field, N random variables, only variance is useful Theory: map observables/variance to theory
Precision: bounded by 1/N, assuming perfect theory, and independence of modes. We consider information in non-linear matter, e.g. lensing, 21cm. Fisher Information
For N modes, variance on variance (e.g. power spectrum) is 2/N Fisher information I=N/2 Will decrease if points are correlated. Basis of Dark Energy Task force Figure of Merit (FoM)
Fisher Information Cosmic Limits Information in CMB is limited: I=106 (modes). 3-D information in principle unbounded,
galaxy surveys with 109 redshifts, I=109? Rimes and Hamilton showed information saturation of dark matter: how useful is weak lensing? 21cm information potentially unbounded Measure cosmological parameters to 10-8
accuracy? Applications CMB is linear and clean, but only a 2-D map on the sky, with N=l(2l+1)~1,000,000 modes. Limiting accuracy is 1/sqrt(N) ~ 10-3.
WMAP5 uses only CMB+BAO+SN: tiny fraction of the information in SDSS/2dF. 3-D structures (galaxies, lensing, 21cm, etc): many more modes, but how many are useful? Optimal searches with non-Gaussianity: lensing, BAO, strings.
Non-Gaussian Case Studies Non-Gaussian information saturation: Rimes and Hamilton Lensing: Information in the dark matter field Lensing of non-Gaussian sources: changes 2pt/4-pt statistics
Information Propagation Measure some 2 pt statistic C(x,y) Find the dependence of C on your favorite parameters: P(k,,w,w) or C(,l,m) Taylor expand around pivot point, and find
minimum variance estimator. Fisher matrix gives errors and information. For Gaussian random fields and certain Baysian priors, this can be equivalent to max likelihood Non-Gaussian Sources
Still have 2-pt statistics Minimum variance estimators are no longer derived from 2-pt+Wick. Need full covariance matrix of C, e.g. from simulation or data. May seem like daunting 4-pt function too
complicated? Error on covariance is 8-PCF! There may be more information in even higher point statistics, but that is even more daunting. Just need to know N
How much information? F(k,k)=
Depends on two 3-D vectors: 4-pt function with two points in one place. In general, F(k,k,cos()): for Gaussian fields, is )): for Gaussian fields, is
function in )): for Gaussian fields, is Legendre transform to diagonalize theta dependence -> F(k,k, l) : for Gaussian, independent of l Measure from simulations in 3-D, and propagate! Most observations (lensing, BAO) only need l=0,2
Rimes and Hamilton 2005 Rimes and Hamilton (2005): The cumulative Fisher
information has a translinear plateau. Cosmic Shear Direct measurement of dark matter power
spectrum. Several existing and proposed dark energy shear surveys: CFHTLS, DES, SNAP, DUNE, LSST Also possible with magnification (Zhang and Pen 2006)
Non-Gaussian Lensing Info Hu and White (2000), Semboloni et al (2008): stacked images. Improved accuracy by Limber projection of slices. Further improved by covariance projection
from 3-D power (Hernois-Deraps, in progress). 21cm/CMB Lensing First detection in WMAP (Smith et al) Quadratic estimator on source field (Pen 2004, Zahn and Zaldarriaga 2006)
Potentially huge (1018) number of sources at high z with measured redshifts Non-linear saturation: Lu and Pen 2008 Pre-reionization 21cm Pen 2004, Lewis & Challinor 2007, Loeb & Zaldarriaga
2004: up to 1018 modes. Kim & Pen in prep Intensity Mapping Stars get fainter with distance: hard to see individually at cosmological distance. Galaxies still visible.
Galaxies get fainter with distance: hard to see in HI. Large scale structure still visible? Large scale structure is LARGE: degree scale. High resolution not needed. Modest size, monolithic radio telescopes needed. (CPPM 2008, Wyithe&Loeb 2008)
From: talk by O. Lahav Keck-DEEP2 GBT z=0.9 cross correlation Chang et al 2009, submitted
CMU cylinder under construction: U. Seljak, J. Peterson, K. Bandura, K. Sigurdson DRAO Penticton CLAR Site
Extrinsic Lensing Noise Lu et al, in prep Optimal Non-Gaussian Estimator Optimal estimation of parameters (e.g. kappa)
from 2pt fcn. Standard map-making approach, but with power spectrum as map. Weigh power spectrum by its inverse Fisher N1 .
Summary Optimal non-Gaussian quadratic estimation solvable: 4-pt statistics. Applicable to lensing (intrinsic+extrinsic noise), BAO, etc. Information saturation: standard 2-pt has much less information for non-Gaussian sources. Impact on lensing,
BAO. Intensity mapping allows LSS/lensing measurements down to Fisher saturation limit. Future redshift/21cm surveys contain a lot of information, with exciting cosmology limited only by Fisher information saturation.
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