Clustering of dark matter and its tracers: renormalizing the bias

and mass power spectrum

Patrick McDonald

(CITA)

astro-ph/soonastro-ph/0606028

Non-linear power spectrum

• Galaxies/BAO• Lya forest• Cosmic shear• clusters/SZ• 21 cm(?)• Red non-linear

curves from Smith et al. simulation fits, not perfectly accurate

Observational Motivation:

Best short-term motivation: Galaxy clustering(Padmanabhan et al. 2006, SDSS LRGs, photo-z)

• Baryonic accoustic oscillations (dark energy)

• Shape of power spectrum (omega_m, neutrinos)

• Non-linearities• Bias• (Redshift-space

distortions)

Why not just use simulations?

• Incredibly slow and painful, to the point where no one has pushed through a complete, accurate (well-tested) mass power spectrum result, even though everyone knows it is just a matter of effort to do it.

• Galaxies, and other observable tracers of mass, can’t be simulated rigorously, so it is useful to use a variety of calculational methods to get an indication of the robustness of results (plus, again, speed).

General idea of renormalization• Not just absorbing infinite quantities into

parameters of a Lagrangian (or integrating out degrees of freedom).

• Quantities to be renormalized can be any not-directly-observable numbers in the problem, including those specifying the initial/boundary conditions (don’t have to be infinite).

• In the present context it is probably better to think of it as a rearrangement of a (supposedly) perturbative series to improve its convergence, although this is not explicit in the RG calculation.

Prelude: Bias of tracers (not RG) (base calculations in Heavens, Matarrese, & Verde, 1998, without the renormalization

interpretation)Naïve perturbation theory: tracer density is a Taylor series in mass density perturbation:

To make sense, higher order terms should decrease in size.Warm-up: compute the mean density of galaxies:

2nd term is ~divergent for linear mass power, and any renormalization of power (non-linearity) will make it infinite. Not a problem. Eliminate the “bare” Taylor series parameter in favor of a parameter for the observed mean density of galaxies.

The mean density is a trivial example, leads to nothing new.Now move to fluctuations:

Correlation function:

(assuming 4th order terms Gaussian):

Going to absorb divergent part into observable linear bias, but not yet because another piece comes from the cubic term.

Moving to the power spectrum, and using 2nd order perturbation theory for the cubic term:

The red integral has a ~divergent part. Constant as k->0 so it looks like shot-noise. Absorb the constant part into a free-parameter for the observable shot-noise power (preserve linear bias+shot noise model on large scales):

Final result:

Started with 4-5+ parameters:

Now have only 4, with much more cleanly separated effects:

(reiterate: Heavens, Matarrese, & Verde, 1998 did most of this calculation already, including recognizing the generation of “effective” bias and shot-noise)

Effect of 2nd order bias (in progress)

Renormalization group approach to dark matter clustering

astro-ph/0606028

Result: ratio of non-linear to linear mass power (thick, upward lines, or to PD96, thin lines)

• Red: RGPT (McDonald 2006)

• Black: simulations, PD96 (Peacock & Dodds 1996)

• Blue: simulations, HALOFIT

(Smith et al. 2003)• Green: standard PT

Standard perturbation theory:

Continuity

Euler

Poisson

Write density (and velocity) as a series of (ideally) increasingly small terms,

Solve evolution equations iteratively

Evolution equations:

Standard PT for P(k)

Renormalization group improvementRe-write PT power spectrum in more compact form:

Add and subtract a constant from the time evolution variable:

Absorb constant term into , defining

Renormalization group improvementObservable, , should not depend on the arbitrary constant

RG equation to 2nd order (in power):

Initial conditions for solution:

Desired result obtained by setting

What does it mean?

• Makes intuitive sense: slowly turn on non-linearities and feed them back into the calculation of subsequent non-linearities.

• Otherwise their effects would appear as higher order terms in the series (re-summation).

• Mathematical explanation exists in terms of the theory of envelopes. (Kunihiro and Tsumura).

Renormalization group for initial/boundary conditions of differential equations.

(Kunihiro and Tsumura, etc.)The basic idea is that solutions of differential equations are independent of the point where the initial or boundary conditions are specified:

The initial value is a function of the initial time, but the solutionis not:

This equation tells us how to change the initial value as we changethe initial time, to preserve the solution.

Can still use this when you have only approximate solutions which are only valid for a limited time.

Gives the “envelope” of the set of curves parameterized by t_i (the curve tangent to each at t_i - Kunihiro and Tsumura).

In other words: you try to string together a set of limited solutions into a globally valid solution.

How does this relate to my calculation?

If there were no decaying modes, the starred quantities would bethe initial power and the initial time (measured by the growth factor).

Ignoring the existence of decaying modes is an approximation,and not a very necessary one.

Back to the results: RGPT PD96 HALOFIT standard PT

Fixed-point slope

Coverge to n=-1.4 at high k

Gives:

Black: RGPTRed: standard PTGreen: linear

Smaller scales, more non-linear RGPT PD96 HALOFIT standard PT

Z=1RGPT PD96 HALOFIT standard PT

Z=3 (high-z fitting formula situation murky)

RGPT PD96 HALOFIT standard PT

Z=6RGPT PD96 HALOFIT standard PT

Issues & future directions• Missing vorticity and higher moments of the

velocity distribution function (e.g., dispersion)• Decaying modes (related)• More careful simulations• Bias (various observables) • Redshift-space distortions• Higher order in PT• Bispectrum, trispectrum, errors, etc.

Conclusions• RGPT is a significant improvement over

standard PT

• For k<0.3 h/Mpc it is very accurate

• Many comparisons limited by simulation-calibrated fitting formulas

• Many extensions are possible (see astro-ph/0606028)

• Can apply to any tracer of large-scale structure

• Or, more generally, any perturbation theory

Results from my day job:Seljak, Slosar, & McDonald (2006), astro-ph/0604335Cosmological parameter estimation using CMB, galaxy clustering,supernovas, and Lyman-alpha forest

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