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
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