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,Ω,w,w’) or C(κ,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’)=<P(k)P(k’)>
• Depends on two 3-D vectors: 4-pt function with
two points in one place.
• In general, F(k,k’,cos(θ)): for Gaussian fields, is δ
function in θ
• 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
N-1.
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.