Matter Power Spectrum
Zhaoming Ma
June 2, 2007
Linear v.s. Nonlinear P(k)
Data: nonlinear
Theory: linear
D2 (a)
PLin (k , a)  PLin (k , aini ) 2
D (aini )
Simulation
D (a)   (k , a)  (k , a  1)
 growth function
 a for CDM
D (a)
G (a)  
a
 growth suppressio n
OR
Higher order
pert. theory?
Tegmark et al 2003
Fitting formulas
• Simulation is expensive,
so fitting formulas are
developed.
• HKLM relation
Hamilton et al 1991
Peacock & Dodds 1996
• Halo model
• Smith et al 2003 (10%)
i) translinear regime: HKLM
ii) deep nonlinear regime:
halo model fit
Foundations of fitting formulas
• HKLM relation or Halo model.
• Nonlinear power is determined by linear
power at the same epoch; history of
linear power spectrum doesn’t matter.
Q: are these physically sound assumptions?
Tools to test these assumptions
Use the public PM code developed by
Anatoly Klypin & Jon Holtzman
Modified to take arbitrary initial input
power spectrum
Modified to handle dark energy models
with arbitrary equation of state w(z)
The difference a spike makes
• Compare P(k) from
simulations w/ and w/o a
spike in the initial power
• Peak is smeared by
nonlinear evolution
• More nonlinear power at all
kNL with no k dependency
• HKLM scaling would
predict the peak being
mapped to a particular kNL
Halo model prediction
x The peak is not smeared
 The peak boosts power at
all nonlinear scales
≈ Slight scale dependency
Does P(k) depend on growth history?
History does matter
• Linear part of the power
spectra are consistent (by
construction)
• Nonlinear power spectra
differ by about 2% simply
due to the differences in the
linear growth histories
• This is not the maximum
effect, but already at the
level that future surveys care
(1% Huterer et al 2005)
Matching growth histories
Same growth histories <==> same P(k)
• Linear part of the power
spectra are consistent with the
differences in the linear growth
• Nonlinear part of the power
spectra are also consistent given
the differences in the linear part
• Result validates the conventional
wisdom that the same linear
growth histories produce the
same nonlinear power spectra
Expansion histories
• Is there any hope of telling apart
the two dark energy models with
degenerated growth histories?
• Their expansion histories differ
by about 0.6%
• BAO could potentially achieve
such precision. But hinges on
systematics (See Smith’s talk)
Effect of substructure
 Plotted is the ratio of the
original spectrum to that w/
a subset of the substructures
smoothed out.
 Halo model has to take
substructures into account.
Hagan, Ma & Kravtsov 05
Effect of Baryons
Rudd, Zentner & Kravtsov 07
Summary
 We test the building blocks of the nonlinear power
spectrum fitting formulas.
i) We find that HKLM scaling relations should be
abandoned, halo model could be kept but with
further work necessary.
ii) We propose that linear growth history should
be included in the next generation fitting
formula.
iii) We also validate the conventional wisdom
that linear power spectrum together with the
linear growth history uniquely determines the
nonlinear power spectrum.
Acknowledgement
Results presented here are part of my Ph.D thesis work which
is supervised under Prof. Wayne Hu. I own a great deal to him.
I am also thankful to my thesis committee members Prof.
Scott Dodelson, Josh Frieman and Andrey Kravtsov for their
invaluable help and suggestions. Many thanks to all my fellow
graduate students, especially Eduardo Rozo and Doug Rudd.