Effects of primordial non Gaussianity on large scale structure - PowerPoint PPT Presentation

竹原理論物理学研究会@竹原 Effects of primordial non ‐ Gaussianity on large scale structure Shuichiro Yokoyama (Nagoya Univ.) Shuichiro Yokoyama (Nagoya Univ.) in collaboration with N. Sugiyama(Nagoya U.), S. Zaroubi(U. of Groningen) and J. Silk (Oxford U.) arXiv:1103.2586 and J. Gong (CERN) in progress

Primordial non Gaussianity Primordial non ‐ Gaussianity

How to parameterize ? How to parameterize ? • Local type non ‐ Gaussianities L l G i i i Komatsu & Spergel (2001), … 9 g NL ð 3 3 f NL ( ð 2 G à h ð 2 G + á á á ð = ð G + 5 G i ) + 25 non ‐ linear parameters Non ‐ zero higher order spectra h h ( higher order correlation functions ) Leadingly Leadingly, … f NL • Bispectrum (3 ‐ point corr. func.) … • Trispectrum (4 ‐ point corr. func.)

fNL vs tauNL fNL vs tauNL • Trispectrum (“local ‐ type”) (“l l ”)  2 parameters  2 parameters Byrnes, et al, arXiv:0705.4096 cubic term  gNL SY, T.Suyama and T.Tanaka, arXiv:0810.3053 quadratic term x quadratic term  tauNL Consistency relation

fNL vs tauNL fNL vs tauNL • “Local ‐ type” inequality “L l t ” i lit In general, for local ‐ type non ‐ Gaussianity we have T. Suyama and M. Yamaguchi, arXiv:0709.2545 T. Suyama and M. Yamaguchi, arXiv:0709.2545 e.g. e.g. (mixed inflaton and curvaton case) Note that it is important to consider Note that it is important to consider tauNL independently of fNL !!

Current observational limits Current observational limits • CMB observations (temperature bi ‐ ,tri ‐ spectra (WMAP 7yr)) (temperature bi ,tri spectra (WMAP 7yr)) also, Komatsu et al.(2010) Smidt et al.(2010) ( ) Fergusson Regan and Shellard (2010)

Effect on the structure formation Effect on the structure formation

How NG affect the LSS formation? How NG affect the LSS formation? • Probability Density Function (PDF) • Probability Density Function (PDF) Gaussian fluctuation variance mean characterized by mean and variance

How NG affect the LSS formation? How NG affect the LSS formation? • Moments for the given distribution function • Moments for the given distribution function Gaussian Fourier space Fourier space mean; variance; variance skewness; skewness; ) ) fNL fNL kurtosis; gNL, τ NL These parameters characterize the non ‐ Gaussianities !!

• PDF of ζ skewness Kurtosis Red; Gaussian Red; Gaussian R d G Red; Gaussian i Blue ; non ‐ zero kurtosis Blue ; non ‐ zero skewness  Sharp peak / smooth peak  Peak shift  Peak shift p p / p However, … if we consider …

• PDF of ζ Skewness (fNL = 100) Kurtosis (gNL = 10^6) difficult to see the differences… the differences… F ( ð ) /F G ( ð ) F ( ð ) /F G ( ð ) large effect on the tails of distribution !!!

How NG affect the LSS formation? How NG affect the LSS formation? • Primordial non Gaussianity • Primordial non ‐ Gaussianity  large effect on the tails of PDF  large effect on the tails of PDF primordial curvature fluctuations  density fluctuations primordial curvature fluctuations  density fluctuations • In the context of LSS formation,… Large effect on the rare event!! e g massive clusters large voids e.g., massive clusters, large voids, high ‐ redshift objects, …

How NG affect the LSS formation? How NG affect the LSS formation? • The effect of fNL (skewness)  observational constraints CMB level ‐ halo mass function ‐ halo mass function (analytically , N ‐ body simulation) ‐ scale ‐ dependent bias ‐ scale ‐ dependent bias ‐ matter power spectrum, bispectrum, … Reviews; Verde (2010), Reviews; Verde (2010), There are a lot of works … h l f k Desjacques and Seljak (2010), … We focus on the kurtosis We focus on the kurtosis ‐ type type especially, non especially, non ‐ zero (large) especially non zero (large) especially non zero (large) τ NL zero (large) τ NL τ NL case τ NL case. case case.

Formulation for the halo mass function Formulation for the halo mass function

Formula for halo mass function Formula for halo mass function • number density of collapsed structures (halos) • number density of collapsed structures (halos) with the mass between M and M + dM Based on the spirit of Press ‐ Schechter formula Based on the spirit of Press Schechter formula, î c /û M (including non ‐ Gaussian features) (including non Gaussian features) ÷ ñ î M /û M ; smoothed density field on a mass scale M û M ; variance of î M ; M ú ö ; background energy density of matter î î c Collapsed structures are formed in the overdensity region (> Collapsed structures are formed in the overdensity region (> ) ) î c ; critical density ( = 1.69 for spherical collapse)

Formula for halo mass function Formula for halo mass function • Non Gaussian PDF of the density field î M î M • Non ‐ Gaussian PDF of the density field smoothed q î ð ð î Poisson eq. î M î on scale M on scale M matter Primordial Including the information of density fluctuations y curvature perturbations curvature perturbations primordial non ‐ Gaussianity primordial non Gaussianity with non ‐ Gaussianity Based on Edgeworth expansion (Hermite polynomials expansion), M M M M non ‐ Gaussian corrections Hermite polynomials; ; skewness k ; kurtosis

Non Gaussian Halo mass function Non ‐ Gaussian Halo mass function non ‐ Gaussian corrections i I In general, skewness and kurtosis include the multiple integrations. .. l k d k t i i l d th lti l i t ti ( skewness  3, kurtosis  6)  some simple formulae For local type non ‐ Gaussianities (in the squeezed limit ), we obtain F l l t G i iti (i th d li it ) bt i new term De Simone et al.(2010), Enqvist et al(2010), Chongchitnan and Silk(2010)

Results Following discussion, we mainly consider fNL Following discussion, we mainly consider fNL = 100 case and τ NL = 10^6 case, which are based on ...

halo mass function halo mass function enhancement Due to the positive primordial non Due to the positive primordial non ‐ Gaussianities Gaussianities, we can see the , we can see the enhancement of the halo mass function for more massive objects. enhancement of the halo mass function for more massive objects.

fNL vs tauNL fNL vs tauNL ; ratio between non ‐ Gaussian mass func. and Gaussian one form of correction terms; Skewness Kurtosis Kurtosis for larger mass for larger mass some difference of the enhancement behavior ??  can we distinguish ?

Redshift dependence Redshift dependence Here, we change the value of τ NL with fixing mass. form of correction terms; Kurtosis Kurtosis with increasing z D(z);growth function during matter ‐ dominant era, Due to the positive Due to the positive τ NL τ NL (also (also fNL fNL ), we can see the enhancement ), we can see the enhancement of the halo mass function at higher of the halo mass function at higher redshift redshift. .

Massive and high redshift objects !! Massive and high redshift objects !! • Effects on reionization history of the Universe Effects on reionization history of the Universe ( z > 10 ) ; Cumulative photon number density emitted from the pop III stars per neutral hydrogen density from the pop III stars per neutral hydrogen density Ref.) Somerville et al (2003) Around z ~ 10 Around z 10, the primordial NG is not so effective. In the early stage ( z ~ 20 ), y g ( ), the NG effect becomes large.

Massive and high redshift objects !! Massive and high redshift objects !! • High redshift massive clusters High redshift massive clusters Weak lensing analysis of the galaxy cluster XMMU J2235 ‐ 2557 presented by Jee et al(2009) and Rosati et al(2009) presented by Jee, et al(2009) and Rosati, et al(2009) (~ 0.4 Mpc^ ‐ 1) In Λ CDM (+ Gaussian) universe, such a massive cluster at this redshift would be a rare event (at least 3 σ ) redshift would be a rare event (at least 3 σ ). In order to explain the existence of such a cluster naturally (at least 2 σ ), Cayon, et al.(2010) found (at least 2 σ ), Cayon, et al.(2010) found Scale ‐ dependent fNL ?? (Ref.) Takahashi ‐ san’s talk and Tasinato ‐ san’s talk) On the other hand we find On the other hand, we find For gNL, Enqvist et al.(2010)

Massive objects  large scale voids? Massive objects  large scale voids? • Abundance of voids (underdensity region ( < δ v )) Abundance of voids (underdensity region ( < δ v )) Positive τ NL  enhancement !!  enhancement !! (same in cluster abundance) Positive fNL  damping  d (opposite to in cluster abundance) ref. Kamionkowski et al(2009) By comparing the observations of clusters and that of void abundance, By comparing the observations of clusters and that of void abundance, we could distinguish we could distinguish skewness skewness ‐ type and kurtosis type and kurtosis ‐ type ?? type ??

Scale ‐ dependent bias work in progress with Jinn ‐ Ouk Gong Takeuchi ‐ kun’s talk

Scale dependent bias Scale ‐ dependent bias • High peak limit (Matarresse, Lucchin and Bonometto(1986)) Power spectrum of halos  power spectrum of density field  P t f h l t f d it fi ld Bias Bias , , ; form factors

tauNL vs fNL ? tauNL vs fNL ? enhanced on large scales g and at high redshift

tauNL vs fNL ? tauNL vs fNL ? On large scales T R, F R  1 !! g _ , _ This term goes to 0 on large scales

Scale dep bias from trispectrum Scale ‐ dep. bias from trispectrum