1. Matter Power Spectrum Zhaoming Ma June 2, 2007

2. Linear v.s. Nonlinear P(k) Theory: linear Data: nonlinear Simulation Higher order pert. theory? OR Tegmark et al 2003

3. 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

4. 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?

5. 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)

6. 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 k NL with no k dependency HKLM scaling would predict the peak being mapped to a particular k NL

7. Halo model prediction xThe peak is not smeared The peak boosts power at all nonlinear scales ≈Slight scale dependency

8. Does P(k) depend on growth history?

9. 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)

10. Matching growth histories

11. 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

12. 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)

13. Effect of substructure Hagan, Ma & Kravtsov 05  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.

14. Effect of Baryons Rudd, Zentner & Kravtsov 07

15. 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.

16. 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.