The formation of The formation of cosmic structures in cosmic structures in non-Gaussian modelsnon-Gaussian models

Lauro MoscardiniLauro MoscardiniDipartimento di AstronomiaDipartimento di Astronomia

Università di Bologna, ItalyUniversità di Bologna, Italy

lauro.moscardini@unibo.itlauro.moscardini@unibo.it

Nonlinear cosmology program 2006, OCA Nice, January 2006

Testable predictions of standard Testable predictions of standard models for inflationmodels for inflation

Cosmological aspectsCosmological aspects

• Critical density UniverseCritical density Universe• Almost scale-invariant and nearly Almost scale-invariant and nearly

Gaussian, adiabatic density fluctuationsGaussian, adiabatic density fluctuations• Almost scale-invariant stochastic background Almost scale-invariant stochastic background

of relic gravitational wavesof relic gravitational waves

Why (non-) Gaussian?Why (non-) Gaussian?

collection of independent harmonic oscillators (no mode-mode collection of independent harmonic oscillators (no mode-mode coupling)coupling)

the motivation for Gaussian initial conditions the motivation for Gaussian initial conditions (the standard (the standard assumption)assumption) ranges from mere simplicity to the use of the Central ranges from mere simplicity to the use of the Central Limit Theorem, to the property of inflation produced seeds (… see Limit Theorem, to the property of inflation produced seeds (… see below) below)

Gaussianfree (i.e. non-interacting)field

large-scalephase coherence

non-linear gravitationaldynamics

The present-day view on non-The present-day view on non-Gaussianity Gaussianity

Alternative structure formation models of the end of eighties Alternative structure formation models of the end of eighties considered strongly non-Gaussian primordial fluctuations considered strongly non-Gaussian primordial fluctuations (e.g. my (e.g. my PhD thesisPhD thesis))

The increased accuracy in CMB and LSS observations has, The increased accuracy in CMB and LSS observations has, however, excluded this extreme possibility.however, excluded this extreme possibility.

The present-day challenge is either detect or constrain mild or The present-day challenge is either detect or constrain mild or even weak deviations from primordial Gaussianity. even weak deviations from primordial Gaussianity.

Deviations of this type are not only possible but are unavoidably Deviations of this type are not only possible but are unavoidably predicted in the standard perturbation generating mechanism predicted in the standard perturbation generating mechanism provided by inflation. provided by inflation.

N-body simulations of N-body simulations of “old-generation” NG models“old-generation” NG models

from: Moscardini, Lucchin, Matarrese & Messina 1991from: Moscardini, Lucchin, Matarrese & Messina 1991

In the late late eighties and early nineties a variety of (mostly toy) models with strongly NG (e.g. χ2 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”Non-Gaussian=non-dog”

Need a model able to parametrize deviations from Gaussianity in Need a model able to parametrize deviations from Gaussianity in a cosmological frameworka cosmological framework

A simple class of A simple class of mildlymildly non-Gaussian perturbations is described non-Gaussian perturbations is described by a sort of Taylor expansion around the Gaussian case: by a sort of Taylor expansion around the Gaussian case:

= = + f + fNLNL 22 + g + gNLNL 33 + … const. + … const.

where where is the peculiar gravitational potential, is the peculiar gravitational potential, is a Gaussian is a Gaussian fieldfieldffNL, NL, ggNL, NL, etc. … are dimensionless parameters quantifying etc. … are dimensionless parameters quantifying the non-Gaussianity (non-linearity) strengththe non-Gaussianity (non-linearity) strength

S. F. ShandarinS. F. Shandarin

The non-Gaussian modelThe non-Gaussian model

frofrommWMAWMA

PP

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 + fNL * ( L2 - <L

2>)

where is the large-scale gravitational potential, L its linear Gaussian contribution and fNL 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 % ~ 10-5 |fNL|

Where does large-scale non-Where does large-scale non-Gaussianity come from?Gaussianity come from?

Falk et al. (1993) found ffNL 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

ffNL 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

ffNLNL~ 1~ 1

Inflation models and Inflation models and ffNLNL

- 0.1 - 0.1 22D-ccelerationD-cceleration

post-inflation corrections not post-inflation corrections not includedincluded- 140 - 140 -3/5-3/5ghost inflationghost inflation

second-order corrections not second-order corrections not includedincludedtypically 10typically 10-1-1warm inflationwarm inflation

13/12 – I - g(13/12 – I - g(kk11, , kk22))modulated reheatingmodulated reheating

r ~ (r ~ (//decaydecay2/3 - 5r/6 + 5/4r2/3 - 5r/6 + 5/4r - g(- g(kk11, , kk22))curvaton scenariocurvaton scenario

g(g(kk11, , kk22)=)=3(k14+k2

4)/2k4+(k1. k2) .

[4-3(k1. k2)/k2]/k2, k=k1+k2

7/3 – g(7/3 – g(kk11, , kk22))single-field inflationsingle-field inflation

commentscommentsfNL(k1,k2)

post-inflation corrections not post-inflation corrections not includedincluded

““unconventional” inflation set-upsunconventional” inflation set-ups

multi-field inflationmulti-field inflationorder of magnitude estimate of the order of magnitude estimate of the absolute valueabsolute value

I = - 5/2 + 5I = - 5/2 + 5))

I = 0 (I = 0 (minimal caseminimal case))

modelmodel

up to 10up to 1022

Compilation by S. MatarreseCompilation by S. Matarrese

Non-Gaussian CMB maps: Non-Gaussian CMB maps: Planck resolutionPlanck resolution

Liguori, Matarrese & Moscardini 2003Liguori, Matarrese & Moscardini 2003

5’ resolutionlmax = 3000, Nside=20485’ resolutionlmax = 3000, Nside=2048

ffNLNL = 3000 = 3000ffNLNL = 3000 = 3000

ffNLNL = 0 = 0ffNLNL = 0 = 0

ffNLNL = - 3000 = - 3000ffNLNL = - 3000 = - 3000

PDF of the NG CMB PDF of the NG CMB mapsmaps

Observational constraints on Observational constraints on ffNLNL The strongest limits on non- The strongest limits on non-

Gaussianity so far come from Gaussianity so far come from WMAP data. WMAP data. Komatsu et al.Komatsu et al. (2003)(2003) find (at 95% cl) find (at 95% cl)

According to According to Komatsu & SpergelKomatsu & Spergel (2001)(2001) using the angular using the angular bispectrum one can reach values as bispectrum one can reach values as low as low as

with with WMAPWMAP & & with with PlanckPlanck can be achieved can be achieved

Similar constraints have been Similar constraints have been obtained by various groups by obtained by various groups by applying different statistical applying different statistical techniques to WMAP data (e.g. techniques to WMAP data (e.g. Cabella et al. 2005, 2006, etc..)Cabella et al. 2005, 2006, etc..)

Komatsu et al. 2003

- 58 < - 58 < ffNLNL < 134 < 134

||ffNLNL| = 20| = 20

||ffNLNL| = 5| = 5

Alternative probes Alternative probes for non-Gaussianity for non-Gaussianity

Besides using standard statistical estimators, like Besides using standard statistical estimators, like bispectrum, trispectrum, three and four-point function, bispectrum, trispectrum, three and four-point function, skewness , etc. …, one can look at the tails of the skewness , etc. …, one can look at the tails of the distribution, i.e. at rare events. distribution, i.e. at rare events.

Rare events have the advantage that they often maximize Rare events have the advantage that they often maximize deviations from what predicted by a Gaussian distribution, deviations from what predicted by a Gaussian distribution, but have the obvious disadvantage of being … rare! but have the obvious disadvantage of being … rare!

Matarrese, Verde & Jimenez (2000) and Verde, Jimenez, Matarrese, Verde & Jimenez (2000) and Verde, Jimenez, Kamionkowski & Matarrese have shown that clusters at Kamionkowski & Matarrese have shown that clusters at high redshift (z>1) can probe NG down to fhigh redshift (z>1) can probe NG down to fNLNL ~ 10 ~ 1022 which which is, however, not competitive with future CMB (Planck) is, however, not competitive with future CMB (Planck) constraints constraints

1) Rare events1) Rare events

Verde et al. (1999) and Scoccimarro et al. (2004) showed Verde et al. (1999) and Scoccimarro et al. (2004) showed that constraints on primordial non-Gaussianity in the that constraints on primordial non-Gaussianity in the gravitational potential from large redshift-surveys like 2dF gravitational potential from large redshift-surveys like 2dF and SDSS are not competitive with CMB ones: and SDSS are not competitive with CMB ones: ffNLNL has to be has to be larger than larger than 101022 – 10 – 1033 in order to be detected as a sort of non-in order to be detected as a sort of non-linear bias in the galaxy-to-dark matter density relation. linear bias in the galaxy-to-dark matter density relation. However LSS gives complementary constraints, as it However LSS gives complementary constraints, as it probes NG on different scales than CMB.probes NG on different scales than CMB. Going to Going to redshift z~1 could help (but one would surveys covering a redshift z~1 could help (but one would surveys covering a large fraction of the sky). large fraction of the sky).

Alternative probes Alternative probes for non-Gaussianity for non-Gaussianity

2) Local LSS2) Local LSS

Primordial non-Gaussianity would also strongly Primordial non-Gaussianity would also strongly affect the abundance of the first non-linear affect the abundance of the first non-linear objects in the Universe, thereby affecting the objects in the Universe, thereby affecting the reionization epoch (Chen et al. 2003) reionization epoch (Chen et al. 2003)

is the WMAP result more likely? is the WMAP result more likely?

Alternative probes Alternative probes for non-Gaussianity for non-Gaussianity

3) LSS at high redshifts3) LSS at high redshifts

The tool: N-body simulations of The tool: N-body simulations of non-Gaussian modelsnon-Gaussian models

Margherita GrossiMargherita Grossi Università di BolognaUniversità di Bologna Enzo BranchiniEnzo Branchini Università di Roma TreUniversità di Roma Tre Klaus DolagKlaus Dolag MPA, GarchingMPA, Garching Sabino MatarreseSabino Matarrese Università di PadovaUniversità 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 simulationsThe N-body simulations

Dark matter-only simulations, using the Dark matter-only simulations, using the GADGET code (Springel 2005)GADGET code (Springel 2005)

Cosmological boxes: Cosmological boxes: L=500 Mpc/hL=500 Mpc/h 80080033 particles, particles, corresponding to a mass- corresponding to a mass-

resolution of mresolution of mpp 2 10 2 10 10 10 solar massessolar masses

(halo resolution m(halo resolution mhh5 10 5 10 11 11 solar masses)solar masses) CPU time per simulation: approx. 7000 hours CPU time per simulation: approx. 7000 hours

on the SP5 @ CINECA Supercomputing on the SP5 @ CINECA Supercomputing Centre (Bologna)Centre (Bologna)

The modelsThe models

Standard Cold Dark Matter “concordance” Standard Cold Dark Matter “concordance” power spectrum with power spectrum with m0m0=0.3, =0.3, 00=0.7, =0.7,

h=0.7, h=0.7, 88=0.9, n=1=0.9, n=1

6 non-Gaussian models, same random 6 non-Gaussian models, same random phases, but with phases, but with f_nl=-2000, -1000, -500, f_nl=-2000, -1000, -500, +500, +1000, +2000,+500, +1000, +2000, plus 1 simulation plus 1 simulation with Gaussian initial conditions (for with Gaussian initial conditions (for comparison)comparison)

Main goalsMain goals

Redshift evolution of Redshift evolution of dark matter in NG dark matter in NG modelsmodels

Redshift evolution of Redshift evolution of halo abundances in halo abundances in NG modelsNG models

Biasing models in NG Biasing models in NG modelsmodels

see Peacock & Dodds see Peacock & Dodds 1996 or Smith et al. 1996 or Smith et al. 20032003

see Press-Schechter or see Press-Schechter or Sheth & Tormen or Sheth & Tormen or Jenkins et al.Jenkins et al.

see Mo & White or see Mo & White or Sheth & TormenSheth & Tormen

Gauss. analogueGauss. analogue

as a function of f_nl, of course!as a function of f_nl, of course!

Possible ApplicationsPossible Applications Analytic models for object (galaxies, clusters, etc.) Analytic models for object (galaxies, clusters, etc.)

clustering; application to lensingclustering; application to lensing (X-ray/SZ) galaxy cluster abundances(X-ray/SZ) galaxy cluster abundances Statistics of rare events; formation time for first Statistics of rare events; formation time for first

objects, its implication for reionizationobjects, its implication for reionization Cosmic velocity fields; reconstruction problemsCosmic velocity fields; reconstruction problems Calibration of statistical tests for non-Gaussianity: Calibration of statistical tests for non-Gaussianity:

high-order moments and correlations, topology, high-order moments and correlations, topology, Minkowski functionals, etc…. Minkowski functionals, etc….

BUT, BUT, SORRY, SIMULATIONS ARE STILL RUNNINGSORRY, SIMULATIONS ARE STILL RUNNING

ON THE SUPERCOMPUTERON THE SUPERCOMPUTER

The test simulationsThe test simulations

The same 6 NG models (with f_nl between The same 6 NG models (with f_nl between –2000 and +2000) plus the gaussian one–2000 and +2000) plus the gaussian one

The same CDM power spectrum withThe same CDM power spectrum with

m0m0=0.3, =0.3, 00=0.7, h=0.7, =0.7, h=0.7, 88=0.9, n=1=0.9, n=1

But only 200But only 20033 particles in a box of particles in a box of 250 250 Mpc/hMpc/h: the mass particle is m: the mass particle is mpp 4 10 4 10 10 10

solar massessolar massesVERY VERY PRELIMINARY RESULTSVERY VERY PRELIMINARY RESULTS

Initial density distributionsInitial density distributions

Models have both Models have both positive and negative positive and negative

skewness in the skewness in the primordial density primordial density

distributiondistribution

f_nl=-2000 f_nl=+2000

Gaussian

model:

f_nl=0

10 Mpc/h 10 Mpc/h

slice atslice at

z=10z=10

f_nl=-2000 f_nl=+2000

Gaussian

model:

f_nl=0

10 Mpc/h 10 Mpc/h

slice atslice at

z=3.1z=3.1

f_nl=-2000 f_nl=+2000

Gaussian

model:

f_nl=0

10 Mpc/h 10 Mpc/h

slice atslice at

z=1.1z=1.1

10 Mpc/h 10 Mpc/h

slice atslice at

z=0z=0

f_nl=-2000 f_nl=+2000

Gaussian

model:

f_nl=0

Redshift Redshift

evolutionevolution

of density of density

distributiondistribution

Largest Largest differences are differences are expected in the expected in the

tails: high-tails: high-density regions density regions

and voidsand voids

Redshift evolution of skewnessRedshift evolution of skewness

The growth of The growth of skewness via skewness via gravitational gravitational

instability is quite instability is quite regular: no evident regular: no evident

differnces with differnces with respect to the respect to the

gaussian modelsgaussian models

Redshift evolution of Redshift evolution of power spectrum power spectrum

Gaussian model: f_nl=0Gaussian model: f_nl=0

f_nl=+1000f_nl=+1000f_nl=-1000f_nl=-1000

Redshift evolution of Redshift evolution of

Power spectrumPower spectrum

f_nl=-1000f_nl=-1000 f_nl=+1000f_nl=+1000

Redshift evolution of Redshift evolution of

Power spectrumPower spectrum

Comparing different models:Comparing different models:present time (z=0)present time (z=0)

Differences Differences are relatively are relatively

small and small and affect only affect only

small scalessmall scales

Comparing different models:Comparing different models:intermediate redshiftsintermediate redshifts

Comparing different models:Comparing different models:higher redshiftshigher redshifts

22

22

22

22

0.70.7 0.70.7

0.70.70.70.7

Power spectrum Power spectrum

ratio w.r.t. ratio w.r.t.

gaussian model:gaussian model:

High redshiftsHigh redshifts

Power Power spectrum spectrum

ratio w.r.t. ratio w.r.t.

gaussian gaussian model:model:

Low redshiftsLow redshifts

22 22

2222

0.70.7

0.70.70.70.7

0.70.7

Halo definitionHalo definition

Friends-of-Friends (FoF) technique with a Friends-of-Friends (FoF) technique with a linking parameter of b=0.2linking parameter of b=0.2

Spherical overdensity (SO) criterion by Spherical overdensity (SO) criterion by assuming a treshold of 200 times the critical assuming a treshold of 200 times the critical densitydensity

We applied two different methods:We applied two different methods:

Rare event statistics: Rare event statistics: the redshift evolution of the redshift evolution of

the mass of the largest objectthe mass of the largest object

44 33 22 11 00z

Formation times:Formation times:

for cluster- or for cluster- or group-like objects group-like objects it can be changed it can be changed

by one unity in by one unity in redshift! redshift!

Evolution Evolution of (SO) of (SO) mass mass

functionfunction

Comparison with Comparison with theoretical modelstheoretical models

JenkinsJenkins PS74PS74

Present timePresent time

Comparison @ higher redshiftsComparison @ higher redshifts

ST model works ST model works reasonably well reasonably well at all redshift, at all redshift,

but attention to but attention to the high-density the high-density

tail!tail!

Preliminary ConclusionsPreliminary Conclusions

Effects of non-Gaussianity are more evident Effects of non-Gaussianity are more evident at intermediate redshifts (1<z<5) and are at intermediate redshifts (1<z<5) and are affecting both halo abundances and affecting both halo abundances and clustering evolutionclustering evolution

Obtaining weaker but complementary Obtaining weaker but complementary constraints on f_nl w.r.t. CMB can be constraints on f_nl w.r.t. CMB can be certainly possible by using future (SZ /X-certainly possible by using future (SZ /X-ray) clusters, high-z galaxy clustering, ray) clusters, high-z galaxy clustering, reionization epoch.reionization epoch.