Cosmological Information Ue-Li Pen Tingting Lu Olivier
Dore
Slide 2
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 8- PCF. LSS data is non-linear,
need to measure 8-PCF!
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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.
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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)
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Fisher Information
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Cosmic Limits Information in CMB is limited: I=10 6 (modes).
3-D information in principle unbounded, galaxy surveys with 10 9
redshifts, I=10 9 ? 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?
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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.
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Non-Gaussian Case Studies Non-Gaussian information saturation:
Rimes and Hamilton Lensing: Information in the dark matter field
Lensing of non-Gaussian sources: changes 2- pt/4-pt statistics
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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
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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.
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Just need to know N
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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 function in 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
Slide 13
Rimes and Hamilton 2005
Slide 14
Rimes and Hamilton (2005): The cumulative Fisher information
has a translinear plateau.
Slide 15
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)
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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).
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21cm/CMB Lensing First detection in WMAP (Smith et al)
Quadratic estimator on source field (Pen 2004, Zahn and Zaldarriaga
2006) Potentially huge (10 18 ) number of sources at high z with
measured redshifts Non-linear saturation: Lu and Pen 2008
Slide 21
Pre-reionization 21cm Pen 2004, Lewis & Challinor 2007,
Loeb & Zaldarriaga 2004: up to 10 18 modes. Kim & Pen in
prep
Slide 22
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)
Slide 23
From: talk by O. Lahav
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Keck-DEEP2 GBT z=0.9 cross correlation Chang et al 2009,
submitted
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CMU cylinder under construction: U. Seljak, J. Peterson, K.
Bandura, K. Sigurdson
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DRAO Penticton CLAR Site
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Extrinsic Lensing Noise Lu et al, in prep
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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.
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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.