Two lectures on Primordial nonGaussianity Sabino Matarrese Dipartimento
Two lectures on Primordial non-Gaussianity Sabino Matarrese Dipartimento di Fisica Galileo Galilei Università degli Studi di Padova, ITALY and INFN, Sezione di Padova email: matarrese@pd. infn. it 9/9/2020 Galileo Galilei Institute, Firenze 1
Outline General ideas on NG from Inflation n NG and the CMB n NG and the LSS of the Universe n 9/9/2020 Galileo Galilei Institute, Firenze 2
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based on … (NG from Inflation) ü Acquaviva V. , Bartolo N. , Matarrese S. & Riotto A. 2003, Nucl. Phys. B 667 119 ü Bartolo N. , Matarrese S. & Riotto A. 2002, Phys. Rev. D 65 103505 ü Bartolo N. , Matarrese S. & Riotto A. 2004, Phys. Rev. D 69 043503 ü Bartolo N. , Matarrese S. & Riotto A. 2004, JCAP 0401 003 ü Bartolo N. , Matarrese S. & Riotto A. 2004, JHEP 0404 006 ü Gangui A. , Lucchin F. , Matarrese S. & Mollerach S. 1994, Ap. J. 430 447 ü Gupta S. , Berera A. , Heavens A. F. & Matarrese S. 2002, Phys. Rev. D 66 043510 9/9/2020 Galileo Galilei Institute, Firenze 4
and … (gauge-invariant formalism + NG in CMB) ü Bartolo N. , Matarrese S. & Riotto A. , 2004, Phys. Rev. Lett. 93 231301 ü Bartolo N. , Matarrese S. & Riotto A. 2005, JCAP 0508 010 ü Bartolo N. , Matarrese S. & Riotto A. 2006, JCAP 0605 010 ü Bartolo N. , Matarrese S. & Riotto A. 2006, JCAP 0606 024 ü Bartolo N. , D’Amico G. , Matarrese S. & Riotto A. 2006 in preparation ü Bartolo N. , Matarrese S. & Riotto A. 2006, preprint ü Bruni M. , Matarrese S. , Mollerach S. & Sonego S. 1997, CQG 14 2585 ü Cabella P. , Liguori M. , Hansen F. K. , Marinucci D. , Matarrese S. , Moscardini L. & Vittorio N. 2005, MNRAS 358 684 ü Cabella P. , Hansen F. K. , Liguori M. , Marinucci D. , Matarrese S. , Moscardini L. & Vittorio N. 2006, MNRAS 369 819 ü Liguori M. , Hansen F. K. , Komatsu E. , Matarrese S. & Riotto A. 2006, Phys. Rev. D 73 043505 ü Liguori M. , Matarrese S. & Moscardini L. 2003, Ap. J 597 56 ü Matarrese S. , Mollerach S. & Bruni M. 1998, Phys. Rev. D 58 043504 ü Mollerach S. , Gangui A. , Lucchin F. , Matarrese S. 1995, Ap. J 453 1 ü Mollerach S. & Matarrese S. 1997, Phys. Rev. D 56 4494 9/9/2020 Galileo Galilei Institute, Firenze 5
and … (primordial NG and LSS) ü Bartolo N. , Matarrese S. & Rotto A. , 2005, JCAP 0510 010 ü Catelan P. , Lucchin F. , Matarrese S. , 1988 Phys. Rev. Lett. , 61 267 ü Grossi M. , Branchini E. , Dolag K. , Matarrese S. & Moscardini L. , 2006 in preparation ü Lesgourgues J. , Liguori M. , Matarrese S. & Riotto A. 2005, Phys. Rev D. 71 103514 ü Lucchin F. & Matarrese S. 1988, Ap. J 330 535 ü Matarrese S. , Lucchin F. & Bonometto S. A. 1986, Ap. J 310 L 21 ü Matarrese S. , Verde L. & Jimenez R. 2000, Ap. J 541 10 ü Moscardini L. , Matarrese S. , Lucchin F. , & Messina A. 1990, MNRAS 245 244 ü Verde L. , Jimenez R. , Kamionkowski M. & Matarrese S. 2001, MNRAS 325 412 9/9/2020 Galileo Galilei Institute, Firenze 6
The phase information 9/9/2020 Galileo Galilei Institute, Firenze credits: Peter Coles 7
The microwave sky as seen by WMAP 9/9/2020 Galileo Galilei Institute, Firenze 8
The Large-Scale Structure of the Universe as described by the 2 d. FGRS The 2 d. FGRS contains ~250, 000 galaxies with measured redshifts The Anglo-Australian Telescope 9/9/2020 Galileo Galilei Institute, Firenze 9
Virgo Consortium simulation of a LCDM Universe … and the underlying Dark Matter distribution 9/9/2020 Galileo Galilei Institute, Firenze 10
Why (non-) Gaussian? Gaussian free (i. e. non-interacting) field Ø collection of independent harmonic oscillators (no mode-mode coupling) Ø the motivation for Gaussian initial conditions (the standard assumption) ranges from mere simplicity to the use of the Central Limit Theorem (e. g. Bardeen et al. 1986), to the property of inflation produced seeds (… see below) large-scale phase coherence 9/9/2020 non-linear gravitational dynamics Galileo Galilei Institute, Firenze 11
The view on Non-Gaussianity … circa 1990 Moscardini, Lucchin, Matarrese & Messina 1991 9/9/2020 Galileo Galilei Institute, Firenze 12
The present view on non. Gaussianity Ø Alternative structure formation models of the late eighties considered strongly non-Gaussian primordial fluctuations. Ø The increased accuracy in CMB and LSS observations has, however, excluded this extreme possibility. Ø The present-day challenge is either detect or constrain mild or even weak deviations from primordial Gaussianity. Ø Deviations of this type are not only possible but are generically predicted in the standard perturbation generating mechanism provided by inflation. 9/9/2020 Galileo Galilei Institute, Firenze 13
“Non-Gaussian=non-dog” S. F. Shandarin Ø Need a model able to parametrize deviations from Gaussianity in a cosmological framework Ø A simple class of mildly non-Gaussian perturbations is described by a sort of Taylor expansion around the Gaussian case F= f + f. NL f 2 + g. NL f 3 + … const. where F is the peculiar gravitational potential, f is a Gaussian field, f. NL, g. N, etc. … are dimensionless parameters quantifying the non-Gaussianity (non-linearity) strength 9/9/2020 Galileo Galilei Institute, Firenze 14
The quadratic NG model ü Many primordial (inflationary) models of non-Gaussianity can be represented in configuration space by the general formula F = f. L + f. NL * ( f. L 2 - <f. L 2>) where F is the large-scale gravitational potential, f. 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 % ~ 10 -5 |f. NL| 9/9/2020 Galileo Galilei Institute, Firenze < 10 -3 from WMAP 15
Non-Gaussianity and scale-invariance According Otto, Politzer, Preskill & Wise (1986) and Grinstein & Wise (1986), if the scales of the perturbation-generating process are negligible w. r. t. astrophysically relevant scales, then a generalized scale-invariance criterion should apply: given the density fluctuation field d ( k, t ) = e ( k) t 2 , (Einstein-de Sitter case) with t the conformal time, the scale-invariant criterion requires < e(lk 1) e(lk 2) … e(lkn)>connected = l-n <e(k 1) e(k 2) … e(kn)>connected which extends the one implicit in the Harrison-Zel’dovich powerspectrum (can be further extended to more general scale-freedom) < e(lk 1) e(lk 2) > ~ k 1 d(3)(k 1 + k 2) 9/9/2020 Galileo Galilei Institute, Firenze 16
Testable predictions of inflation • (Almost) critical density Universe • Almost scale-invariant and nearly Gaussian, adiabatic density fluctuations • Almost scale-invariant stochastic background of relic gravitational waves 9/9/2020 Galileo Galilei Institute, Firenze 17
Classify Inflationary Models ü The shape of the inflaton potential V (φ) determines the observables. slow-roll conditions ü It is standard practice to use three “slow-roll” parameters to characterize it: ε “slope” of the potential ~ (V’/V)2 η “curvature” of the potential ~V’’/V ~ ε 1 ξ “jerk” of the potential ~(V’/V)(V’’’/V) ~ ε 2 9/9/2020 Galileo Galilei Institute, Firenze 1 18
Slow-roll parameters and observables Scalar (comoving curvature) perturbation power-spectrum Tensor (gravity-wave) perturbation power-spectrum 9/9/2020 Galileo Galilei Institute, Firenze 19
“Generic” predictions of single field slow-roll models vs. WMAP 3 yr results from: Spergel et al. 2006 ns 0. 951 ± 0. 022 r < 0. 55 Spectral index < 0. 28 (+ SDSS data) 9/9/2020 Tensor to scalar ratio Galilei Institute, Firenze 20
Where does large-scale non. Gaussianity come from (in standard inflation)? q Falk et al. (1993) found f. NL ~ x ~ e 2 (from non-linearity in the inflaton potential in a fixed de Sitter space) in the standard single-field slow-roll scenario q Gangui et al. (1994), using stochastic inflation found f. NL ~ e (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 q Bartolo et al. (2004, 2005) showed that second-order corrections after inflation enhance the primordial signal leading to f. NL~ 1 9/9/2020 Galileo Galilei Institute, Firenze 21
Non-Gaussianity requires more than linear theory … The leading contribution to higher-order statistics (such as the bispectrum, i. e. the FT of the three-point function) comes from second-order metric perturbations around the RW background, unless the primordial non-Gaussianity is very strong “… the linear perturbations are so surprisingly simple that a perturbation analysis accurate to second order may be feasible …” (Sachs & Wolfe 1967) 9/9/2020 Galileo Galilei Institute, Firenze 22
First-order metric perturbations in the Newtonian gauge (dust case) scalar modes vector modes 9/9/2020 Galileo Galilei Institute, Firenze tensor modes 23
Second-order metric perturbations in the Poisson gauge (dust case) scalar modes vector modes tensor modes 9/9/2020 Galileo Galilei Institute, Firenze Extended to fully non-linear scales by 24 Carbone & Matarrese (2004)
Second-order cosmological perturbations Second-order gauge-invariant curvature perturbation (Malik & Wands 2003) See also Lyth & Wands, 2003; Bartolo, Matarrese & Riotto 2002; Rigopoulous & Shellard 2003 Such a quantity is related to the analougous non-perturbative quantity defined by Salopek & Bond (1990) which expanded at second order is Firenze et al. 2003, Kolb et al. 2004, 25 For recent 9/9/2020 non-perturbative definitions of. Galileo , Galilei see Institute, Rigopoulos Lyth et al. 2004, Langlois & Vernizzi 2005
From the energy-continuity equation on super-horizon scales to z conservation The key point is that (2) remains constant on superhorizon scales after it has been generated and possible isocurvature (entropy) perturbations are no longer present. Thus (2 ) provides all the information about the primodial level of NG generated during inflation, as in the standard scenario, or after inflation, as in the curvaton scenario. 9/9/2020 Galileo Galilei Institute, Firenze 26
Evaluating non-Gaussianity: from inflation to the present universe Ø Evaluate non-Gaussianity during inflation by a self-consistent secondorder calculation. Ø Evolve scalar (vector and tensor) perturbations to second order after inflation outside the horizon, matching conserved second-order gauge -invariant variable, such as the comoving curvature perturbation z(2) defined by Malik & Wands (2004), or the similar quantity defined by Salopek & Bond (1990), z(2)SB = z(2) - 2 (z(1))2 (or non-linear generalizations of it), to its value at the end of inflation (accurately accounting for reheating after inflation) Ø Evolve them consistently inside the horizon this necessarily involves the calculation of the second-order radiation transfer function (Bartolo, Matarrese & Riotto 2005, 2006) for CMB and second-order matter transfer function for LSS (preliminary results in Bartolo, Matarrese & Riotto 2005) 9/9/2020 Galileo Galilei Institute, Firenze 27
Non-Gaussianity from Inflation: results DT/T = -1/3 (f. L + f. NL) f. NL = f. NL * f. L 2 + const. Sachs-Wolfe limit; replaced by full transfer function in true CMB maps f. NL = f NL 0 – K (k 1, k 2) model-dependent term 9/9/2020 Universal (gravitational) term going to zero in the squeezed limit Galileo Galilei Institute, Firenze 28
Sachs-Wolfe effect: a compact relation Post-inflation non-linear evolution of gravity Initial conditions set during or after inflation standard scenario curvaton scenario Bartolo, Matarrese & Riotto 2004 9/9/2020 Galileo Galilei Institute, Firenze 30
Extracting the non-linearity parameter f. NL Connection between theory and observations This is the proper quantity measurable by CMB experiments, via the phenomenological analysis by Komatsu and Spergel 2001 k = | k 1 + k 2 | 9/9/2020 Galileo Galilei Institute, Firenze 31
Non-Gaussianity in the standard scenario (I) Non-Gaussianity generated during inflation: Accounting for the inflaton self-interactions and metric fluctuations at second-order in the perturbations brings Acquaviva et al. 2003; Maldacena 2003 Non-Gaussianity for single-field models of slow-roll inflation is tiny during inflation: 9/9/2020 Galileo Galilei Institute, Firenze 32
Non-Gaussianity in the standard scenario (II) What about the post-inflationary evolution ? (2) is conserved during the reheating stage and during radiation/matter phases Use second-order evolution of the gravitational potentials (in Poisson gauge) k = | k 1 + k 2 | f. NL~ O(1) Bartolo, Matarrese & Riotto 2003 Thus the main contribution to the non-Gaussian signal comes from the non-linear gravitational dynamics in the post-inflationary stages 9/9/2020 Galileo Galilei Institute, Firenze 33
Inflation models and f. NL See review: Bartolo, Komatsu, Matarrese & Riotto, 2004, Phys. Rept. 404, 103 model single-field inflation curvaton scenario modulated reheating multi-field inflation f. NL(k 1, k 2) 4/3 – g(k 1, k 2) comments g(k 1, k 2)=3(k )= 14+k 24)/2 k 4+(k 1. k 2) [4 -3(k 1. k 2)/k 2]/k 2, k=k 1+k 2 -1/3 - 5 r/6 + 5/4 r - g(k 1, k 2) r ~ (rs/r)decay I = - 5/2 + 5 G / (12 a. G 1) I=0 ( minimal case) 1/12 – I - g(k 1, k 2) up to 102 . order of magnitude estimate of the absolute value “unconventional” inflation set-ups warm inflation ghost inflation D-cceleration 9/9/2020 typically 10 -1 - 140 b a-3/5 - 0. 1 g 2 Galileo Galilei Institute, Firenze second-order corrections not included post-inflation corrections not included 34
More about (standard singlefield slow-roll) inflation SW limit leading contribution to the bispectrum: q Quadratic non-linearity on large-scales (up to ISW and 2 -nd order tensor modes). Standard slow-roll inflation yields a. NL~ b. NL~ 1 additional contribution to trispectrum (together with f. NL 2 terms): q Cubic non-linearity on large-scales (up to ISW and 2 -nd order tensor modes Bartolo, D’Amico, Matarrese & Riotto, in prep. ) 9/9/2020 Galileo Galilei Institute, Firenze 35
Second-order transfer function Improve treatment of radiation transfer function, going to second order. Crucial ingredient for |f. NL| ~ 1 (standard inflation). For large NG (|f. NL| > ~10) the standard procedure is O. K. . q (Bartolo, Matarrese & Riotto, 2006 JCAP 0605 010): calculation of the full 2 -nd order radiation transfer function on large scales (low-l), which includes: NG initial conditions non-linear evolution of gravitational potentials on large scales second-order SW effect (and second-order temperature fluctuations on the lastscattering surface) ü second-order ISW effect, both early and late ü ISW from second-order tensor modes (unavoidably arising from non-linear evolution of scalar modes), also accounting for second-order tensor modes produced during inflation ü ü ü q (Bartolo, Matarrese & Riotto, 2006 JCAP 0606 024): Boltzmann equation at 2 -nd order for the photon, baryon and CDM fluids allows to follow CMB anisotropies at 2 -nd order on all scales; includes both scattering and gravitational secondaries, like: ü ü Thermal and Kinetic Sunyaev-Zel’dovich effect Ostriker-Vishniac effect Inhomogeneous recombination and reionization Further gravitational terms: gravitational lensing (by scalar and tensor modes), Rees. Sciama effect, Shapiro time-delay, effects from second-order vector (i. e. rotational) 9/9/2020 etc. … Galileo Galilei Institute, Firenze 36 modes,
Non-Gaussian CMB anisotropies: map making Liguori, Matarrese & Moscardini 2003 q assume mildly non-Gaussian large-scale potential fluctuations q account for radiative transfer radiation transfer functions 9/9/2020 harmonic transform: Flm(r) Galileo Galilei Institute, Firenze 37
Spherical coordinates in real space (I) Work directly with multipoles in real space (to avoid Bessel transform and Cartesian coordinates) 1. generate white noise coefficients nlm(r) 2. cross-correlate different nlm(r) by a convolution with suitable filters Wl(r, r 1) 9/9/2020 Galileo Galilei Institute, Firenze 38
Spherical coordinates in real space (II) band-pass filters Wl(r, r 1) linear gravitational potential power-spectrum Wl(r, r 1) does not oscillate as fast as jl(kr) 9/9/2020 Galileo Galilei Institute, Firenze 39
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Outline of the code 1. precompute transfer functions (extracted from CMBfast) for a given model 2. precompute filters Wl(r, r 1) 3. generate white-noise coefficients nlm(r) 4. correlate white-noise coefficients to find multipoles FLlm(r) and then extract FNLlm(r) 5. obtain CMB multipoles by convolving with 6. radiation transfer function 9/9/2020 Galileo Galilei Institute, Firenze 41
CPU time and memory requirements ℓmax 500 3000 nside 256 1024 2048 CPU time 40 min 10 hours 10 days WMAP Planck (on a dec alpha, 400 Mhz CPU) nlmax = 3000 100 Mbyte RAM Parallelized version of the code (with F. Hansen): 6 hours on 100 processors at Planck angular resolution 9/9/2020 Galileo Galilei Institute, Firenze 42
The simulations q A parallel version of the LMM code has been produced (with F. Hansen): it allows to obtain NG maps at the maximum Planck resolution (n_side = 2048, lmax = 3000). Each map requires ~ 3 hours over 100 processors (20 minutes for a single n_side = 1024, lmax = 2000). q A large sample (300 maps) of Planck LFI resolution (n_side = 1024, lmax = 2000) non-Gaussian CMB maps in HEALPIX format with free f. NL parameter (is already available (simulations have been run in Oslo; the maps are presently in Oslo + DPC/OATS). q Soon upgraded to include NG polarization maps (M. Liguori, P. Cabella, F. Hansen, E. Komatsu, S. Matarrese & B. Wandelt) at WMAP resolution. q An important issue to be considered for future applications is the flexibility of the code to changes in the input power-spectrum. 9/9/2020 Galileo Galilei Institute, Firenze 43
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Non-Gaussian CMB maps: Planck resolution 5’ resolution f. NL = 3000 lmax = 3000, Nside=2048 Liguori, Matarrese & Moscardini 2003, Ap. J 597, 56 f. NL = 0 9/9/2020 Galileo Galilei Institute, Firenze f. NL = - 3000 45
PDF of NG CMB maps 9/9/2020 Galileo Galilei Institute, Firenze 46
Observational constraints on f. NL ü The strongest limits on non-Gaussianity come from 3 -yr WMAP data. Spergel et al. (2006) find (95% CL): - 54 < f. NL < 114 ü According to Komatsu & Spergel (2001) using the angular bispectrum one can reach values as low as |f. NL| < 20 with WMAP & |f. NL| < 5 with Planck can be achieved ü The role of the f. NL momentum-dependent is a characteristic inflation signature that can enhance the S/N for NG detection (Liguori, Hansen, Komatsu, Matarrese & Riotto 2006), possibly making NG from single-field inflation detectable. Note that perfect quadratic NG should never be used to approximate inflationary NG even for very high values of f. NL !! 9/9/2020 Galileo Galilei Institute, Firenze Komatsu et al. 2003 47
Statistical analyses of NG CMB maps q The maps have been analyzed with a bunch of statistics and the results have been presented in Santander last September q Roughly speaking it seems that bispectrum-based statistics are most powerful (bet very slow!) in detecting quadratic NG. Upper bounds on |f. NL| down to ~18 (2 sigmas) appear achievable with Planck (lmax=3000). (cf. Komatsu & Spergel 2001 |f. NL|<5). Creminelli et al. (2005) from 1 yr data obtain - 27 < f. NL < 121 q The analysis of WMAP 3 years data (Spergel et al. 2006) yields: - 54 < f. NL < 114 q More promising technique uses “integrated bispectrum” (Marinucci 2005; Cabella et al. 2006): for no galactic cuts achieves up to 107 speed-up compared with full bispectrum with comparable accuracy. Method tested on presently available NG maps, but needs extension to presence of galactic cut!. Application to 1 st year WMAP data yields: - 160 < f. NL < 160 Cabella et al. 2006) q Expected improvements with combined analysis of polarization maps and full exploitation of specific angular dependence of f. NL are extremely important to probe NG of standard single-field slow-roll inflation models 9/9/2020 Galileo Galilei Institute, Firenze 48
Bispectrum analysis Liguori, Hansen, Komatsu, Matarrese & Riotto, 2005 Usual parametrization for primordial non-Gaussianity: f. NL is constant A full second order perturbative approach for single-field yields a momentum-dependent f. NL The momentum-dependent part accounts for the growth of non-Gaussianity due to post-inflationary non-linear evolution Model dependent (intrinsic NG) NG 9/9/2020 Model independent: post-inflationary evolution Galileo Galilei Institute, Firenze 49
CMB angular bispectrum (I) The expression for the CMB angular bispectrum in the standard case has been derived by Komatsu & Spergel (2001). The shape of the reduced bispectrum is determined by the line-of-sight integral: radiation transfer function The average bispectrum is obtained from the reduced bispectrum 9/9/2020 Galileo Galilei Institute, Firenze 50
CMB angular bispectrum (II) In the full second order treatment the averaged bispectrum becomes: new l. o. s. integral Combination of Wigner 3 j and 6 j symbols Now the line of sight Integral has a different expression 9/9/2020 Galileo Galilei Institute, Firenze 51
CMB angular bispectrum (III) We performed a Fisher analysis at WMAP angular resolution considering Standard single-field slow-roll inflation Scenarios leading to higher levels of NG, such as curvaton, inhomegeneous reheating, etc. … Standard single-field slow-roll inflation: q The signal is still undetectable at WMAP angular resolution q S/N grows faster than in the standard parametrization q If S/N keeps growing at lmax > 500 Planck could detect non-Gaussianity arising from standard single-field inflation 9/9/2020 Galileo Galilei Institute, Firenze 52
Planck detection threshold as calculated in Komatsu and Spergel (2001) f. NL required in the standard parametrization in order to reproduce S/N obtained at a given lmax from the momentum-dependent parametrization Liguori et al. 2005 9/9/2020 Galileo Galilei Institute, Firenze 53
Effective f. NL: Curvaton scenario 9/9/2020 Galileo Galilei Institute, Firenze 54
NG effects in LSS Bartolo, Matarrese & Riotto (2005) have computed the effects of NG in the dark matter density fluctuations in a matter-dominated universe. Only for high values of f. NL (>~10) the standard parameterization is valid. On small scales stagnation effects during radiation dominance should be taken into account up to secondorder. Going to higher redshift enhances the relative weight of the NG term 9/9/2020 Galileo Galilei Institute, Firenze 55
Searching for primordial non-Gaussianity with LSS ü 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 2 d. F and SDSS are not competitive with CMB ones: f. NL has to be as large as 102 – 103 in order to be detected as a sort of nonlinear bias in the galaxy-to-dark matter density relation. However LSS gives complementary constraints, as it probes NG on different scales than CMB. Hint: use reconstruction of initial data (e. g. Mohayaee et al. 2004) to detect primordial NG. ü Going to redshift z~1 helps (but one would need surveys covering a large fraction of the sky). Going to higher redshifts (e. g. through SZ cluster surveys or via 21 -cm background anisotropies) may largely help, as the effective NG strength in the underlying CDM overdensity scales like (1+z) (Pillepich, Porciani & Matarrese, 2006, in prep. ). ü Primordial non-Gaussianity also strongly affects the abundance of the first non-linear objects in the Universe, thereby modifying the reionization epoch (Chen et al. 2003). Also: primordial black-hole abundance. 9/9/2020 Galileo Galilei Institute, Firenze 56
Searching for non-Gaussianity with rare events Komatsu et al. 2003 Mass function of massive halos ü 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. ü 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! Number counts of massive halos ü Matarrese, Verde & Jimenez (2000) and Verde, Jimenez, Kamionkowski & Matarrese showed that clusters at high redshift (z>1) can probe NG down to f. NL ~ 102 which is, however, not competitive with future CMB (Planck) constraints. Sefusatti et al. (2006) have studied effects of NG on maximum likelihood estimates of σ8 9/9/2020 Galileo Galilei Institute, Firenze 58
Halo mass function and primordial NG Press-Schechter type halo mass-function accounting for non-Gaussian initial conditions skewness of the primordial density field, being non-zero only in case of NG initial data ~ f. NL Matarrese, Verde & Jimenez 2000 9/9/2020 Galileo Galilei Institute, Firenze 59
Impact of NG in maximum likelihood determination of cosmological parameters from: Sefusatti et al. 2006 9/9/2020 Galileo Galilei Institute, Firenze 60
NG N-Body simulations M. Grossi, E. Branchini, K. Dolag, S. Matarrese & L. Moscardini, 2006 q Standard CDM “concordance” model with: m 0=0. 3, 0=0. 7, h=0. 7, 8=0. 9, n=1 q 6 non-Gaussian models with f_nl=-2000, -1000, -500, +1000, +2000, q 1 simulation with Gaussian initial conditions 7 Dark Matter-only simulations Code: GADGET-2 (Springel 2005) 9/9/2020 Galileo Galilei Institute, Firenze 61
The simulations n 8003 particles, corresponding to a mass-resolution of mp 2 *10 10 solar masses n Cosmological boxes: L=5003 (Mpc/h)3 n Computations performed at CINECA Supercomputing Centre (Bologna) on a 3 k (only initial conditions) and sp 5 machines: about 7000 hours of CPU time per simulation (55 hours on 128 parallel processors with 1. 6 Gb memory each). A second set of simulations has run at MPA (Garching) 9/9/2020 Galileo Galilei Institute, Firenze 62
Primordial density PDF The considered models have both positive and negative skewness in the primordial density PDF 9/9/2020 Galileo Galilei Institute, Firenze 63
z = 5. 2 5 10 Mpc/h Slices f. NL= 0 Gaussian model f. NL = -1000 9/9/2020 Galileo Galilei Institute, Firenze f. NL= +1000 64
z = 3. 1 3 10 Mpc/h Slices f. NL= 0 Gaussian model f. NL = -1000 9/9/2020 Galileo Galilei Institute, Firenze f. NL= +1000 65
z = 2. 1 10 Mpc/h Slices f. NL= 0 Gaussian model f. NL = -1000 9/9/2020 Galileo Galilei Institute, Firenze f. NL= +1000 66
z = 1. 1 10 Mpc/h Slices f. NL= 0 Gaussian model f. NL = -1000 9/9/2020 Galileo Galilei Institute, Firenze f. NL= +1000 67
z = 0. 5 0 10 Mpc/h Slices f. NL= 0 Gaussian model f. NL = -1000 9/9/2020 Galileo Galilei Institute, Firenze f. NL= +1000 68
z = 0. 0 0 10 Mpc/h Slices f. NL= 0 Gaussian model f. NL = -1000 9/9/2020 Galileo Galilei Institute, Firenze f. NL= +1000 69
Redshift evolution of the PDF Differences are in the tails 9/9/2020 Galileo Galilei Institute, Firenze 70
Abundance of massive DM halos vs. redshift 9/9/2020 Galileo Galilei Institute, Firenze 71
Rare event statistics: redshift evolution of the mass of the largest object The formation time can change up to 1 in Dz 9/9/2020 Galileo Galilei Institute, Firenze 72
Conclusions & future prospects ü Contrary to earlier naive expectations, some level of non-Gaussianity is generically present in all inflation models ü The level of non-Gaussianity predicted in the simplest inflation models is slightly below the minimum value detectable by Planck, but the predicted angular dependence of f. NL, extensive use of simulated NG CMB maps, measurements of polarization and use of alternative statistical estimators might help non-Gaussianity detection down to f. NL ~ 1 ü Constraining/detecting non-Gaussianity is a powerful tool to discriminate among competing scenarios for perturbation generation (standard slow-roll inflation, curvaton, modulated-reheating, multi-field, etc. …) some of which imply large non-Gaussianity ü Accounting for the presence of sizeable non-Gaussianity in maximum likelihood analyses might change the estimated value of cosmological parameters ü Constraining non-Gaussianity in LSS allows to put independent limits on NG and on a different range of scales: high-redshift samples appear most promising in this respect ü Predicting/constraining non-Gaussianity has become a branch of Precision Cosmology 9/9/2020 Galileo Galilei Institute, Firenze 73
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