Power Spectrum Super

Power Spectrum Super-Sample Covariance
Masahiro Takada !
(Kavli IPMU, U. Tokyo)!
Based on collaboration with Wayne Hu (KICP) !
and Yin Li (KICP)!
@ PASCOS 2013
Power spectrum and Sample variance
•  Data compression: 2D (or 3D) ⇒ Fourier mode decomposition ⇒ Power spectrum (1D under the
assumption of statistical isotropy)
0.4
0.3
r0.002
•  However, we can estimate PS from only
“one” realization – the sample variance
•  Theory of errors: For a Gaussian field (like
CMB), the sample variance is given as
⇤CDM+tensors
⇤CDM+running+tensors
N = 40
0.2
N = 50
N = 60
0.1
0.0
m2
0.94
0.96
ns
0.98
2
1.00
Large-scale structure (LSS): Nonlinearity
•  LSS arises from nonlinear gravitational instability
•  All Fourier modes become correlated with each
other
•  Long-wavelength modes of scales even beyond a
survey volume affect the dynamics of small-scale
modes
•  LSS perturbation theory predicts … Survey region
L
LSS and Super-Sample Covariance
MT & Hu 13
Mass map from CFHT lens survey
•  The previous formula of the PS
sample variance (Scoccimarro+
1999): 4-point function
•  An entire survey region may be embedded in a coherent, large-scale
density mode – super-survey mode that is unobservable
•  How does the super-survey mode affect the power spectrum estimation?
The effect is negligible? Super-Sample Covariance
•  Explicitly include the survey window
function in the covariance calculation
•  The observed field is given as
MT & Hu 13
Survey region
•  The Fourier-transformed field is
–  The width of W(k) is ~1/L
–  In this way, we can explicitly include
contributions of modes outside a survey region
•  The power spectrum estimator L
Super-Sample Covariance (contd.) MT & Hu 13
•  The following matter trispectrum (4pt function) contributes the power
spectrum covariance; here q<<k (q~1/L)
•  Our Conjecture: the trispectrum contribution for k>>q can be
approximated as follows, in analogy with the primordial NG
consistency relation (Maldacena 2003)
standard term
long-wavelength mode
The response of power spectrum to the
background density mode (the dynamics
of LSS determines the response)
Super-Sample Covariance (contd.) MT & Hu 13
•  The new formula for the power spectrum covariance for
any LSS probes New term: super-sample covariance
Here σb^2 is the variance of the long-wavelength density modes for the survey
volume
Perturbation theory and Halo model
predictions for SSC
MT & Hu 13
•  If we employ a model of the nonlinear clustering (or equivalently
the nonlinear mode coupling), we can explicitly compute
compute and therefore compute the SSC term
•  Here we use the halo model for such a model, and indeed found
that the SSC term has a form like the consistency relation form
we claimed (also see for an independent test using simulations)
Here
Also, Hamilton+06, MT & Bridle 07, Sato+09, MT & Jain 09 … Super-Sample Covariance
•  The new formula of PS covariance for any LSS
probes (WL, galaxy clustering, …) (MT & Hu 13)
halos
•  Physical picture of the SSC effect
–  Suppose that a survey region is
embedded in a large-scale overdensity
region
–  Growth of all the small-scale
fluctuations are accelerated
Survey region: V
δm>0 –  The power spectrum we can observe
tends to have greater amplitudes than
the ensemble average
Comparison with simulations
with SS cov. •  Super-sample variance
gives a dominant
contribution for almost
all scales of interest, for
any upcoming galaxy
surveys
•  Our formula shows an
excellent agreement
with the simulations
(used 1000 realizations)
w/o SS cov.
Also see"
MT & Bridle 07"
Sato+09"
Kayo, MT & Jain 13"
Kayo & MT 13"
MT & Spergel 13
Super Sample Covariance in Simulations – separate universe simulations
Yin Li, Wayne Hu & MT (in preparation)
N-body simulations:
powerful tool of LSS
Miyoshi & Kihara (1975): "
the first N-body simulation
Suto (2003: IAU proceeding)
Limitations of N-body simulations?
MICE simulations"
(up to ~8Gpc/h)
• 
• 
• 
• 
DEUS (Dark Energy
Universe Simulation)
project : up to ~10Gpc/h
N-body sim. now 40 yrs history
Employ Periodic boundary conditions
How large volume do we need to run?
If we run a very large-box simulation, most of the
computation time is for the linear or quasi-nonlinear
dynamics? This is against the aim of N-body simulations? •  How to include a super-box mode (DC mode)? •  Occasionally some papers have discussed the effect of DC
mode (e.g., Pen 99; Sirko 05), but has not really implemented
Separate universe simulation
initial redshift
later redshift
•  How can we include the DC mode in a simulation? •  We know that the DC mode grows according to the linear growth rate
–  For a sufficiently high redshift such as the initial redshift employed in a simulation
(say z~50 or 100), the amplitude is very small and the effect is negligible
Separate universe simulation (contd.)
The effect of such a DC mode can be treated by changing the background
cosmological model (an effective curvature parameter) (also, Sirko 05; Gnedin
+09; Baldauf et al. 12)
initial redshift
later redshift
Li, Hu & MT in prep.
The two simulations look
identical at sufficiently high
redshift
We can use the same
seeds of the initial density
fluctuations (which help to
reduce the stochasticity)
SSC simulations vs. Halo model
Halo model
Li, Hu & MT in prep.
SSC simulations
•  The halo model
fairly well
reproduces the
simulation
results!
Application of SSC simulations
n
^ h (~
x1 ) = Fh [±L (~
x1 ); ~
x1 ]
•  “halo bias” is given by the response of
the halo mass function to the
background density mode – peakbackground splitting (Mo & White 96)
•  Recently, Fabian et al. (2012) extended
the PB formula, and derived that the
halo bias is given by the response to
the background mass density
•  SS simulation method allows to reduce
the stochasticity – it can be a better
way of the halo bias calibration
±(~
x)
RL
~
x1
±L (~x)
±s (~
x) = ±(~
x) ¡ ±L (~
x)
~
x2
Summary
•  The gravitational instability of LSS causes a coupling between all the
Fourier modes
•  Three contributions to the sample variance for any LSS probe
–  Gaussian errors due to a finite number of the sampled modes
–  Non-Gaussian errors from the small-scale modes inside a survey volume
–  New term: Super-sample covariance arising from long-wavelength modes outside the
survey volume •  The SSC term can be given by the response of the power spectrum to
super-survey (long-wavelength) modes
–  Can be calibrated by a separate-universe simulation method
–  σW is computed from the Gaussian realizations, with any survey window function
•  The SSC term can be realized as a new signal of long-wavelength mode?
•  Developed a separate universe simulation method!