ROE-JSPS workshop 2007 @ Edinburgh 26 th October 2007 Shun Saito Theoretical Astrophysics Group

1. The impact of non-linear evolution of the cosmological matter power spectrum on the measurement of neutrino masses ROE-JSPS workshop 2007 @ Edinburgh 26th October 2007 Shun Saito Theoretical Astrophysics Group University of Tokyo, JAPAN collaborators：Masahiro Takada (Tohoku Univ. ) Atsushi Taruya (Tokyo Univ.)

2. 2 (c.f.) WMAP3+SDSS 1 Massive Neutrinos & Cosmology Constraints of total neutrino masses are given by cosmology. background expansion history affect the CMB angular power spectra or SN Ia , etc. (c.f.) WMAP3 only Spergel et al (2006) suppression of matter perturbation affect matter power spectrum of large-scale structure Massive neutrinos contribute naturally to matter perturbation, but do not contribute below the free-streaming scale due to large velocity dispersion. Spergel et al (2006) Introduction ①

3. Suppression of matter perturbation dimensionless power suppression !! Baryon Acoustic Oscillations (BAOs) calculated by Linear theory Non-linear regime !! Introduction ②

4. Non-linear theory improvement? ★ The planned galaxy high-redshift surveys such as WFMOS aim the Baryon Acoustic Oscillations (BAOs). ★ Neutrinos’ free-streaming scale is comparable to BAO scale, which is around non-linear regime. ★ If we have Non-linear Theory, the constraint on neutrino masses may be improved over linear theory. Our Work ◆ calculate the next-to-leading order correction based on perturbation theory for mixed dark matter model for the first time ◆ present the parameter forecast using Fisher matrix analysis Introduction ③

5. Formalism: Perturbative Approach baryon + mixed dark matter (CDM + Neutrino) fluid approximation The continuity & Euler equation for and are non-linear equations ⇒ Perturbative expansion of Continuity & Euler equation Assuming that neutrino’s density contrast stay at linear level, we calculate the next-to-leading order, one-loop correction , only for CDM plus baryon. The effect of neutrinos appears only in scale-dependent growth rate of linear . Perturbative Approach ①

6. One-loop corrections From standard perturbation theory Makino,Sasaki,Suto (1992) The one-loop correction is roughly proportional to the square of . Note that the effect of neutrinos is entirely encoded in . Perturbative Approach ②

7. fiducial cosmology linear Non-linear power spectrum dimensionless power one-loop The amplitude is enhanced by non-linear gravitational evolution. Non-linear P(k) ①

8. linear non-linear Suppression factor valid range @ z = 3 : 0.18 hMpc^-1 : 0.69 hMpc^-1 Jeong,Komatsu(2006) error linear ★More suppressed than linear theory ★ Reduced shot noise error non-linear Non-linear P(k) ②

9. 0.195eV 0.085eV 40% improve ! Fisher matrix analysis lower limit for inverted mass hierarchy lower limit for normal mass hierarchy Parameter forecast

10. Parameter degeneracy correlation coefficient | r | ~ 1 : strong degeneracy | r | ~ 0 : independent ● Degeneracy between w0 & f_nu is resolved by BAOs ● Strong degeneracy with b1 ⇒ necessity of non-linear biasing analysis Parameter forecast ②

11. Summary & Future Work Summary • We derive the non-linear power spectrum including the effect of neutrinos for the first time. • We found that the signal and the suppression are enhanced by the non-linear gravitational evolution. • The constraint on neutrino masses may be improved from our non-linear theory for future galaxy redshift surveys. Future Work • Discuss the validity of our theory, treatment of neutrinos, valid range of PT, linear bias, Gaussian statistics etc. • For a more realistic forecast, MCMC method should be adopted. • The improvement of constraint on neutrino masses by SDSS data. • Comparison with numerical simulation including massive neutrinos. Summary