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

2. Massive Neutrinos & Cosmology
Constraints of total neutrino masses are given by cosmology. 1
background expansion history
affect the CMB angular power spectra or SN Ia , etc.
(c.f.) WMAP3 only
Spergel et al (2006) 2
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.
(c.f.) WMAP3+SDSS
Spergel et al (2006)
Introduction ①
April 16, 2019

3. Suppression of matter perturbation
dimensionless power
suppression !!
Baryon Acoustic Oscillations
(BAOs)
calculated by Linear theory
Non-linear regime !!
Introduction ②
April 16, 2019

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
April 16, 2019

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
April 16, 2019

6. From standard perturbation theory
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
April 16, 2019

7. Non-linear power spectrum
dimensionless power
fiducial cosmology
one-loop
The amplitude is enhanced
by non-linear gravitational
evolution.
linear
Non-linear P(k) ①
April 16, 2019

8. Suppression factor @ z = 3 error linear non-linear linear
valid range
@ z = 3
: 0.18 hMpc^-1
: 0.69 hMpc^-1
Jeong,Komatsu(2006)
error
linear
non-linear
linear
★ More suppressed than linear theory
★ Reduced shot noise error
non-linear
Non-linear P(k) ②
April 16, 2019

9. Fisher matrix analysis
0.195eV
40% improve !
lower limit for inverted mass hierarchy
0.085eV
lower limit for normal mass hierarchy
Parameter forecast
April 16, 2019

10. correlation coefficient
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
April 16, 2019

11. Summary & Future Work Summary Future Work
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
April 16, 2019