The formation of cosmic structures in non-Gaussian models Lauro Moscardini Dipartimento di Astronomia Università di Bologna, Italy Nonlinear cosmology program 2006, OCA Nice, January 2006

Testable predictions of standard models for inflation Cosmological aspects Critical density Universe Critical density Universe Almost scale-invariant and nearly Gaussian, adiabatic density fluctuations Almost scale-invariant and nearly Gaussian, adiabatic density fluctuations Almost scale-invariant stochastic background of relic gravitational waves Almost scale-invariant stochastic background of relic gravitational waves

Why (non-) Gaussian?  collection of independent harmonic oscillators (no mode-mode coupling)  the motivation for Gaussian initial conditions (the standard assumption) ranges from mere simplicity to the use of the Central Limit Theorem, to the property of inflation produced seeds (… see below) Gaussian free (i.e. non-interacting) field large-scale phase coherence non-linear gravitational dynamics

The phase information credits: Peter Coles

The present-day view on non- Gaussianity  Alternative structure formation models of the end of eighties considered strongly non-Gaussian primordial fluctuations (e.g. my PhD thesis  )  The increased accuracy in CMB and LSS observations has, however, excluded this extreme possibility.  The present-day challenge is either detect or constrain mild or even weak deviations from primordial Gaussianity.  Deviations of this type are not only possible but are unavoidably predicted in the standard perturbation generating mechanism provided by inflation.

N-body simulations of “old-generation” NG models from: Moscardini, Lucchin, Matarrese & Messina 1991 In the late late eighties and early nineties a variety of (mostly toy) models with strongly NG (e.g. χ 2 or lognormally distributed) primordial gravitational potential or density fields were adopted as initial conditions in N-body simulations (Moscardini et al. 1991; Weinberg & Cole 1992).

“Non-Gaussian=non-dog”  Need a model able to parametrize deviations from Gaussianity in a cosmological framework  A simple class of mildly non-Gaussian perturbations is described by a sort of Taylor expansion around the Gaussian case:   =  + f NL  2 + g NL  3 + … const. where  is the peculiar gravitational potential,  is a Gaussian field  f NL, g NL, etc. … are dimensionless parameters quantifying the non-Gaussianity (non- linearity) strength where  is the peculiar gravitational potential,  is a Gaussian field  f NL, g NL, etc. … are dimensionless parameters quantifying the non-Gaussianity (non- linearity) strength S. F. Shandarin

The non-Gaussian model     from WMAP Many primordial (inflationary) models of non-Gaussianity can be represented in configuration space by the general formula (e.g. Verde et al. 2000; Komatsu & Spergel 2001)  =  =  L + f NL * (  L 2 - )   where  is the large-scale gravitational potential,  L its linear Gaussian contribution and f NL is the dimensionless non- linearity parameter (or more generally non-linearity function). The percent of non-Gaussianity in CMB data implied by this model is     NG % ~ |f NL |

Where does large-scale non- Gaussianity come from? f NL   Falk et al. (1993) found f NL   (from non-linearity in the inflaton potential in a fixed de Sitter space) in the standard single-field slow-roll scenario Gangui et al. (1994), using stochastic inflation found f NL  f NL  (from second-order gravitational corrections during inflation). Acquaviva et al. (2003) and Maldacena (2003) confirmed this estimate (up to numerical factors and momentum-dependent terms) with a full second-order approach Bartolo et al. (2004) show that second-order corrections after inflation enhance the primordial signal leading to f NL ~ 1

Inflation models and f NL  2 D-cceleration post-inflation corrections not included  -3/5 ghost inflation second-order corrections not included typically warm inflation 13/12 – I - g(k 1, k 2 ) modulated reheating r ~ (   /  decay 2/3 - 5r/6 + 5/4r - g(k 1, k 2 ) curvaton scenario g(k 1, k 2 )= g(k 1, k 2 )=3(k 1 4 +k 2 4 )/2k 4 +(k 1. k 2 ). [4-3(k 1. k 2 )/k 2 ]/k 2, k=k 1 +k 2 7/3 – g(k 1, k 2 ) single-field inflation commentsf NL (k 1,k 2 ) post-inflation corrections not included “unconventional” inflation set-ups multi-field inflation order of magnitude estimate of the absolute value I = - 5/2 + 5   ) I = 0 (minimal case) model up to 10 2 up to 10 2 Compilation by S. Matarrese

Non-Gaussian CMB maps: Planck resolution Liguori, Matarrese & Moscardini ’ resolution l max = 3000, N side =2048 5’ resolution l max = 3000, N side =2048 f NL = 3000 f NL = 0 f NL =

Non-Gaussian CMB anisotropies: map making  assume mildly non-Gaussian large-scale potential fluctuations  account for radiative transfer radiation transfer functions harmonic transform:  lm (r) Liguori, Matarrese & Moscardini 2003, ApJ 597, 56

PDF of the NG CMB maps

Observational constraints on f NL  The strongest limits on non- Gaussianity so far come from WMAP data. Komatsu et al. (2003) find (at 95% cl)  According to Komatsu & Spergel (2001) using the angular bispectrum one can reach values as low as with WMAP & with WMAP & with Planck can be achieved with Planck can be achieved  Similar constraints have been obtained by various groups by applying different statistical techniques to WMAP data (e.g. Cabella et al. 2005, 2006, etc..) Komatsu et al < f NL < 134 |f NL | = 20 |f NL | = 5

Alternative probes for non-Gaussianity  Besides using standard statistical estimators, like bispectrum, trispectrum, three and four-point function, skewness, etc. …, one can look at the tails of the distribution, i.e. at rare events.  Rare events have the advantage that they often maximize deviations from what predicted by a Gaussian distribution, but have the obvious disadvantage of being … rare!  Matarrese, Verde & Jimenez (2000) and Verde, Jimenez, Kamionkowski & Matarrese have shown that clusters at high redshift (z>1) can probe NG down to f NL ~ 10 2 which is, however, not competitive with future CMB (Planck) constraints 1) Rare events

 Verde et al. (1999) and Scoccimarro et al. (2004) showed that constraints on primordial non- Gaussianity in the gravitational potential from large redshift-surveys like 2dF and SDSS are not competitive with CMB ones: f NL has to be larger than 10 2 – 10 3 in order to be detected as a sort of non-linear bias in the galaxy-to-dark matter density relation. However LSS gives complementary constraints, as it probes NG on different scales than CMB. Going to redshift z~1 could help (but one would surveys covering a large fraction of the sky). Alternative probes for non-Gaussianity 2) Local LSS

 Primordial non-Gaussianity would also strongly affect the abundance of the first non-linear objects in the Universe, thereby affecting the reionization epoch (Chen et al. 2003)  is the WMAP result more likely?  is the WMAP result more likely? Alternative probes for non-Gaussianity 3) LSS at high redshifts

The tool: N-body simulations of non-Gaussian models Margherita Grossi Margherita Grossi Università di Bologna Università di Bologna Enzo Branchini Enzo Branchini Università di Roma Tre Università di Roma Tre Klaus Dolag Klaus Dolag MPA, Garching MPA, Garching Sabino Matarrese Sabino Matarrese Università di Padova Università di Padova In collaboration with Expectations: Expectations: Weaker constraints w.r.t. CMB, but absolutely complementary  necessity of studying in much more detail structure formation in NG models

The N-body simulations Dark matter-only simulations, using the GADGET code (Springel 2005) Dark matter-only simulations, using the GADGET code (Springel 2005) Cosmological boxes: L=500 Mpc/h Cosmological boxes: L=500 Mpc/h particles, corresponding to a mass- resolution of m p  solar masses particles, corresponding to a mass- resolution of m p  solar masses (halo resolution m h  solar masses) (halo resolution m h  solar masses) CPU time per simulation: approx hours on the CINECA Supercomputing Centre (Bologna) CPU time per simulation: approx hours on the CINECA Supercomputing Centre (Bologna)

The models Standard Cold Dark Matter “concordance” power spectrum with  m0 =0.3,   0 =0.7, h=0.7,  8 =0.9, n=1 Standard Cold Dark Matter “concordance” power spectrum with  m0 =0.3,   0 =0.7, h=0.7,  8 =0.9, n=1 6 non-Gaussian models, same random phases, but with f_nl=-2000, -1000, -500, +500, +1000, +2000, plus 1 simulation with Gaussian initial conditions (for comparison) 6 non-Gaussian models, same random phases, but with f_nl=-2000, -1000, -500, +500, +1000, +2000, plus 1 simulation with Gaussian initial conditions (for comparison)

Main goals Redshift evolution of dark matter in NG models Redshift evolution of dark matter in NG models Redshift evolution of halo abundances in NG models Redshift evolution of halo abundances in NG models Biasing models in NG models Biasing models in NG models see Peacock & Dodds 1996 or Smith et al see Press-Schechter or Sheth & Tormen or Jenkins et al. see Mo & White or Sheth & Tormen Gauss. analogue as a function of f_nl, of course!

Possible Applications Analytic models for object (galaxies, clusters, etc.) clustering; application to lensing Analytic models for object (galaxies, clusters, etc.) clustering; application to lensing (X-ray/SZ) galaxy cluster abundances (X-ray/SZ) galaxy cluster abundances Statistics of rare events; formation time for first objects, its implication for reionization Statistics of rare events; formation time for first objects, its implication for reionization Cosmic velocity fields; reconstruction problems Cosmic velocity fields; reconstruction problems Calibration of statistical tests for non-Gaussianity: high-order moments and correlations, topology, Minkowski functionals, etc…. Calibration of statistical tests for non-Gaussianity: high-order moments and correlations, topology, Minkowski functionals, etc…. BUT, SORRY, SIMULATIONS ARE STILL RUNNING ON THE SUPERCOMPUTER ON THE SUPERCOMPUTER

The test simulations The same 6 NG models (with f_nl between –2000 and +2000) plus the gaussian one The same 6 NG models (with f_nl between –2000 and +2000) plus the gaussian one The same CDM power spectrum with The same CDM power spectrum with  m0 =0.3,   0 =0.7, h=0.7,  8 =0.9, n=1  m0 =0.3,   0 =0.7, h=0.7,  8 =0.9, n=1 But only particles in a box of 250 Mpc/h: the mass particle is m p  But only particles in a box of 250 Mpc/h: the mass particle is m p  solar masses solar masses VERY VERY PRELIMINARY RESULTS

Initial density distributions Models have both positive and negative skewness in the primordial density distribution

f_nl=-2000f_nl=+2000 Gaussian model: f_nl=0 10 Mpc/h slice at z=10

f_nl=-2000f_nl=+2000 Gaussian model: f_nl=0 10 Mpc/h slice at z=3.1

f_nl=-2000f_nl=+2000 Gaussian model: f_nl=0 10 Mpc/h slice at z=1.1

10 Mpc/h slice at z=0 f_nl=-2000f_nl=+2000 Gaussian model: f_nl=0

Redshiftevolution of density distribution Largest differences are expected in the tails: high-density regions and voids

Redshift evolution of skewness The growth of skewness via gravitational instability is quite regular: no evident differnces with respect to the gaussian models

Redshift evolution of kurtosis Again, similar evolution produced by gravitational instability

Redshift evolution of power spectrum Gaussian model: f_nl=0

f_nl=+1000f_nl=-1000 Redshift evolution of Power spectrum

f_nl=-1000 f_nl=+1000 Redshift evolution of Power spectrum

Comparing different models: present time (z=0) Differences are relatively small and affect only small scales

Comparing different models: intermediate redshifts

Comparing different models: higher redshifts

Power spectrum ratio w.r.t. gaussian model: High redshifts

Power spectrum ratio w.r.t. gaussian model: Low redshifts

Halo definition Friends-of-Friends (FoF) technique with a linking parameter of b=0.2 Friends-of-Friends (FoF) technique with a linking parameter of b=0.2 Spherical overdensity (SO) criterion by assuming a treshold of 200 times the critical density Spherical overdensity (SO) criterion by assuming a treshold of 200 times the critical density We applied two different methods:

Rare event statistics: the redshift evolution of the mass of the largest object z Formation times: for cluster- or group-like objects it can be changed by one unity in redshift!

Evolution of (SO) mass function

Comparison with theoretical models JenkinsPS74 Present time

higher redshifts ST model works reasonably well at all redshift, but attention to the high-density tail!

Preliminary Conclusions Effects of non-Gaussianity are more evident at intermediate redshifts (1<z<5) and are affecting both halo abundances and clustering evolution Effects of non-Gaussianity are more evident at intermediate redshifts (1<z<5) and are affecting both halo abundances and clustering evolution Obtaining weaker but complementary constraints on f_nl w.r.t. CMB can be certainly possible by using future (SZ /X- ray) clusters, high-z galaxy clustering, reionization epoch. Obtaining weaker but complementary constraints on f_nl w.r.t. CMB can be certainly possible by using future (SZ /X- ray) clusters, high-z galaxy clustering, reionization epoch.