The Statistical Properties of Large Scale Structure Alexander Szalay Department of Physics and...
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Transcript of The Statistical Properties of Large Scale Structure Alexander Szalay Department of Physics and...
The Statistical Properties of Large Scale
Structure
Alexander SzalayDepartment of Physics and Astronomy
The Johns Hopkins University
Outline
• Lecture 1: – Basic cosmological background
– Growth of fluctuations
– Parameters and observables
• Lecture 2: – Statistical concepts and definitions
– Practical approaches
– Statistical estimators
• Lecture 3: – Applications to the SDSS
– Angular correlations with photometric redshifts
– Real-space power spectrum
Lecture #2
• Statistical concepts and definitions– Correlation function
– Power spectrum
– Smoothing kernels
– Window functions
• Statistical estimators
Basic Statistical Tools
• Correlation functions
– N-point and Nth-order
– Defined in real space
– Easy to compute, direct physical meaning
– Easy to generalize to higher order
• Power spectrum
– Fourier space equivalent of correlation functions
– Directly related to linear theory
– Origins in the Big Bang
– Connects the CMB physics to redshift surveys
• Most common are the 2nd order functions:
– Variance 82
– Two-point correlation function (r)
– Power spectrum P(k)
32
21 321 21
The Galaxy Correlation Function
• First measured by Totsuji and Kihara, then Peebles etal
• Mostly angular correlations in the beginning
• Later more and more redshift space
• Power law is a good approximation
• Correlation length r0=5.4 h-1 Mpc
• Exponent is around =1.8
• Corresponding angular correlations
0
)(r
rr
1
0
)(rw
The Overdensity
• We can observe galaxy counts n(x), and compare to the expected counts
• Overdensity
• Fourier transform
• Inverse
• Wave number
1)(
)( n
n xx
)(
2
1)( 3
3 kkx kx
ied
)()( 3 xxk kx ied
2
k
n
)(xn
The Power Spectrum
• Consider the ensemble average
• Change the origin by R
• Translational invariance
• Power spectrum
• Rotational invariance
)()( 2*
1 kk
)()()(~
)(~
2*
1)(
2*
121 kkkk Rkk ie
2
121)3(3
2*
1 )()()2()()( kkkkk
)()( kPP k
2)()( kk P
Correlation Function
• Defined through the ensemble average
• Can be expressed through the Fourier transform
• Using the translational invariance in Fourier space
• The correlation function only depends on the distance
)()(),( 2121 xxxx
)()()2(
1),( 2
*1
)(2
31
3621
2211 kkkkxx xkxk
iedd
)()(),( 21)(3
2121 xxkxx xxk kPed i
)()()2(
1),( 121
)3()(2
31
3321
2211 kPedd i kkkkxx xkxk
P(k) vs (r)
• The Rayleigh expansion of a plane wave gives
• Using the rotational invariance of P(k)
• The power per logarithmic interval
• The power spectrum and correlation function form a Fourier transform pair
)()(4
1)()( 0
2221 kPkrjkdkr
xx
l
llli Pkrjlie )ˆˆ()()12( rkkr
)()(ln4
1)( 3
02kPkkrjkdr
Filtering the Density
• Effect of a smoothing kernel K(x), where
• Convolution theorem
• Filtered power spectrum
• Filtered correlation function
1)(3 xx Kd
)'()'(')( 3 xxxxx KdKs
)()()( kkk Ks
222)()()()()( kkkkk KPKPs
2302
)()()(ln4
1)( kKkPkkrjkdrs
Variance
• At r=0 separation we get the variance:
• Usual kernel is a ‘top-hat’ with an R=8h-1 Mpc radius
• The usual normalization of the power spectrum is using this window
232
2 )()(ln4
1kKkPkkd RR
kr
krjkK
Rr
RrrK RR
)()(
if,0
if,1)( 1
2
83
2
28 )()(ln
4
1kKkPkkd
Selection Window
• We always have an anisotropic selection window, both over the sky and along the redshift direction
• The observed overdensity is
• Using the convolution theorem
outside
insideW
if,0
if,0)(r
)()()( xxx Ww
)'()'(')( 3 kkkkk Wdw
The Effect of Window Shape
• The lines in the spectrum are at least as broad as the window – the PSF of measuring the power spectrum!
• The shape of the window in Fourier space is the conjugate to the shape in real space
• The larger the survey volume, the sharper the k-space window => survey design
)()()()()()( ''''*'''''*'''''3''3'* kkkkkkkkkk WWdd wwww
)()()()2()()( '''*''''''33'* kkkkkkkk WWPdww
1-Point Probability Distribution
• Overdensity is a superposition from Fourier space
• Each depends on a large number of modes
• Variance (usually filtered at some scale R)
• Central limit theorem: Gaussian distribution
)(
2
1)( 3
3 kkx kx
ied
2
2
2)(
2
2
e
P
22
N-point Probability Distribution
• Many ‘soft’ pixels, smoothed with a kernel
• Raw dataset x
• Parameter vector
• Joint probability distribution is a multivariate Gaussian
• M is the mean, C is the correlation matrix
• C depends on the parameters
)()(
2
1exp)2(),( 12/12/ mxCmxCΘx Tnf
xm jijiij mmxxC
Fisher Information Matrix
• Measures the sensitivity of the probability distribution to the parameters
• Kramer-Rao theorem:one cannot measure a parameter more accurately than
jiij
f
ln2
F
iiF
1
Higher Order Correlations
• One can define higher order correlation functions
• Irreducible correlations represented by connected graphs
• For Gaussian fields only 2-point, all other =0
• Peebles conjecture: only tree graphs are present
• Hierarchical expansion
• Important on small scales
321
4321
terms) 1-N(... nijkijNN Q
Correlation Estimators
• Expectation value
• Rewritten with the density as
• Often also written as
• Probability of finding objects in excess of random
• The two estimators above are NOT EQUIVALENT
)()(),( 2121 xxxx
21
221112
1121
2112
RR
DD
Edge Effects
• The objects close to the edge are different
• The estimator has an excessvariance (Ripley)
• If one is using the firstestimator, these cancelin first order (Landy and Szalay 1996)
21
2211
21
221112
))((
RR
RDRD
RR
RRDRDD
212
Discrete Counts
• We can measure discrete galaxy counts
• The expected density is a known <n>, fractional
• The overdensity is
• If we define cells with counts Ni
)()( rrr Dn
n
nn )(r
i
iii N
NN )(r
Power Spectrum
• Naïve estimator for a discrete density field is
• FKP (Feldman, Kaiser and Peacock) estimatorThe Fourier space equivalent to LS
n
i neN
f krk1
)(ˆ
Ne
Ne
NfkP
nn
i
nn
i nnnn111
)(ˆ)(ˆ'
)(2
',
)(2
''
rrkrrkk
)()()(ˆ krk kr wefn
in
n
)()(1
)()(
kPrn
rn
r
Wish List
• Almost lossless for the parameters of interest
• Easy to compute, and test hypotheses(uncorrelated errors)
• Be computationally feasible
• Be able to include systematic effects
The Karhunen-Loeve Transform
• Subdivide survey volume into thousands of cells
• Compute correlation matrix of galaxy counts among cells from fiducial P(k)+ noise model
• Diagonalize matrix– Eigenvalues
– Eigenmodes
• Expand data over KL basis
• Iterate over parameter values:– Compute new correlation matrix
– Invert, then compute log likelihood
Vogeley and Szalay (1996)
Eigenmodes
Eigenmodes
• Optimal weighting of cells to extract signal represented by the mode
• Eigenvalues measure S/N
• Eigenmodes orthogonal
• In k-space their shape is close to window function
• Orthogonality = repulsion
• Dense packing of k-space => filling a ‘Fermi sphere’
Truncated expansion
• Use less than all the modes: truncation
• Best representation in the rms sense
• Optimal subspace filtering, throw away modes which contain only noise
NMbfM
iii
1
,ˆ
N
Miiff
1
2)ˆ(
Correlation Matrix
• The mean correlation between cells
• Uses a fiducial power spectrum
• Iterate during the analysis
)()(),( 2121)(
23
13 rWrWrrrdrd ji
sij
Whitening Transform
• Remove expected count ni
• ‘Whitened’ counts
• Can be extended to other types of noise=> systematic effects
• Diagonalization: overdensity eigenmodes
• Truncation optimizes the overdensity
i
iii n
ngd
ji
ij
i
ijijij nnn
R
Truncation
• Truncate at 30 Mpc/h – avoid most non-linear effects
– keep decent number of modes