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Two lectures on Primordial non-Gaussianity

Two lectures on Primordial non-Gaussianity. Sabino Matarrese Dipartimento di Fisica Galileo Galilei Universit à degli Studi di Padova, ITALY and INFN, Sezione di Padova email: matarrese@pd.infn.it. Outline. General ideas on NG NG from Inflation NG and the CMB

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Two lectures on Primordial non-Gaussianity

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  1. 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 Galileo Galilei Institute, Firenze

  2. Outline • General ideas on NG • NG from Inflation • NG and the CMB • NG and the LSS of the Universe Galileo Galilei Institute, Firenze

  3. based on … (NG from Inflation) • Acquaviva V., Bartolo N., Matarrese S. & Riotto A. 2003, Nucl. Phys. B 667 119 • Bartolo N., Matarrese S. & Riotto A. 2002, Phys. Rev. D 65 103505 • Bartolo N., Matarrese S. & Riotto A. 2004, Phys. Rev. D 69 043503 • Bartolo N., Matarrese S. & Riotto A. 2004, JCAP 0401 003 • Bartolo N., Matarrese S. & Riotto A. 2004, JHEP 0404 006 • Gangui A., Lucchin F., Matarrese S. & Mollerach S. 1994, Ap.J. 430 447 • Gupta S., Berera A., Heavens A.F. & Matarrese S. 2002, Phys. Rev. D 66 043510

  4. and … (gauge-invariant formalism + NG in CMB) • Bartolo N., Matarrese S. & Riotto A., 2004, Phys. Rev. Lett. 93 231301 • Bartolo N., Matarrese S. & Riotto A. 2005, JCAP 0508 010 • Bartolo N., Matarrese S. & Riotto A. 2006, JCAP 0605 010 • Bartolo N., Matarrese S. & Riotto A. 2006, JCAP 0606 024 • Bartolo N., D’Amico G., Matarrese S. & Riotto A. 2006 in preparation • Bartolo N., Matarrese S. & Riotto A. 2006, preprint • Bruni M., Matarrese S., Mollerach S. & Sonego S. 1997, CQG 14 2585 • Cabella P., Liguori M., Hansen F.K., Marinucci D., Matarrese S., Moscardini L. & Vittorio N. 2005, MNRAS 358 684 • Cabella P., Hansen F.K., Liguori M., Marinucci D., Matarrese S., Moscardini L. & Vittorio N. 2006, MNRAS 369 819 • Liguori M., Hansen F.K., Komatsu E., Matarrese S. & Riotto A. 2006, Phys. Rev. D 73 043505 • Liguori M., Matarrese S. & Moscardini L. 2003, ApJ 597 56 • Matarrese S., Mollerach S. & Bruni M. 1998, Phys. Rev. D 58 043504 • Mollerach S., Gangui A., Lucchin F., Matarrese S. 1995, ApJ 453 1 • Mollerach S. & Matarrese S. 1997, Phys. Rev. D 56 4494

  5. and … (primordial NG and LSS) • Bartolo N., Matarrese S. & Rotto A., 2005, JCAP 0510 010 • Catelan P., Lucchin F., Matarrese S., 1988 Phys. Rev. Lett., 61 267 • Grossi M., Branchini E., Dolag K., Matarrese S. & Moscardini L., 2006 in preparation • Lesgourgues J., Liguori M., Matarrese S. & Riotto A. 2005, Phys. Rev D. 71 103514 • Lucchin F. & Matarrese S. 1988, ApJ 330 535 • Matarrese S., Lucchin F. & Bonometto S.A. 1986, ApJ 310 L21 • Matarrese S., Verde L. & Jimenez R. 2000, ApJ 541 10 • Moscardini L., Matarrese S., Lucchin F., & Messina A. 1990, MNRAS 245 244 • Verde L., Jimenez R., Kamionkowski M. & Matarrese S. 2001, MNRAS 325 412

  6. The phase information Galileo Galilei Institute, Firenze credits: Peter Coles

  7. The microwave sky as seen by WMAP Galileo Galilei Institute, Firenze

  8. The Large-Scale Structure of the Universe as described by the 2dFGRS The 2dFGRS contains ~250,000 galaxies with measured redshifts The Anglo-Australian Telescope Galileo Galilei Institute, Firenze

  9. … and the underlying Dark Matter distribution Virgo Consortium simulation of a LCDM Universe Galileo Galilei Institute, Firenze

  10. Why (non-) Gaussian? free (i.e. non-interacting) field • 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 (e.g. Bardeen et al. 1986), to the property of inflation produced seeds (… see below) Gaussian large-scale phase coherence non-linear gravitational dynamics Galileo Galilei Institute, Firenze

  11. The view on Non-Gaussianity … circa 1990 Moscardini, Lucchin, Matarrese & Messina 1991 Galileo Galilei Institute, Firenze

  12. The present view on non-Gaussianity • Alternative structure formation models of the late eighties considered strongly non-Gaussian primordial fluctuations. • 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 generically predicted in the standard perturbation generating mechanism provided by inflation. Galileo Galilei Institute, Firenze

  13. “Non-Gaussian=non-dog” S. F. Shandarin • 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= f + fNLf2 + gNLf3 + … const. where F is the peculiar gravitational potential, f is a Gaussian field, fNL, gN, etc. … are dimensionless parameters quantifying the non-Gaussianity (non-linearity) strength Galileo Galilei Institute, Firenze

  14. The quadratic NG model • Many primordial (inflationary) models of non-Gaussianity can be represented in configuration space by the general formula F = fL + fNL * ( fL2 - <fL2>) • where Fis the large-scale gravitational potential, fLits linear Gaussian contribution and fNLis 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| • 10-3 fromWMAP Galileo Galilei Institute, Firenze

  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) t2 ,(Einstein-de Sitter case) with t the conformal time, the scale-invariant criterion requires < e(lk1) e(lk2) … e(lkn)>connected = l-n <e(k1) e(k2) … e(kn)>connected which extends the one implicit in the Harrison-Zel’dovich power- spectrum (can be further extended to more general scale-freedom) < e(lk1) e(lk2) > ~ k1d(3)(k1 + k2) 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 Galileo Galilei Institute, Firenze

  17. Classify Inflationary Models • The shape of the inflaton potential V (φ) determines the observables. • It is standard practice to use three “slow-roll” parameters to characterize it: ε“slope” of the potential ~ (V’/V)21 η“curvature” of the potential ~V’’/V~ ε1 ξ“jerk” of the potential ~(V’/V)(V’’’/V) ~ε2 slow-roll conditions Galileo Galilei Institute, Firenze

  18. Slow-roll parameters and observables Scalar (comoving curvature) perturbation power-spectrum Tensor (gravity-wave) perturbation power-spectrum Galileo Galilei Institute, Firenze

  19. “Generic” predictions of single field slow-roll models vs. WMAP 3 yr results from: Spergel et al. 2006 Galileo Galilei Institute, Firenze

  20. Where does large-scale non-Gaussianity come from (in standard inflation)? • Falk et al. (1993) foundfNL ~x ~ e2(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 fNL ~ 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 • Bartolo et al. (2004, 2005) showed that second-order corrections after inflation enhance the primordial signal leading to fNL~ 1 Galileo Galilei Institute, Firenze

  21. Non-Gaussianity requiresmore 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) Galileo Galilei Institute, Firenze

  22. First-order metric perturbations in the Newtonian gauge (dust case) scalar modes tensor modes vector modes Galileo Galilei Institute, Firenze

  23. Second-order metric perturbations in the Poisson gauge (dust case) scalar modes vector modes tensor modes Extended to fully non-linear scales by Carbone & Matarrese (2004) Galileo Galilei Institute, Firenze

  24. Such a quantity is related to the analougous non-perturbative quantity defined by Salopek & Bond (1990) which expanded at second order is For recent non-perturbative definitions of , see Rigopoulos et al. 2003, Kolb et al. 2004, Lyth et al. 2004, Langlois & Vernizzi 2005 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

  25. 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. Galileo Galilei Institute, Firenze

  26. Evaluating non-Gaussianity: from inflation to the present universe • Evaluate non-Gaussianity during inflation by a self-consistent second-order calculation. • 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 perturbationz(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) • Evolve them consistently inside the horizon  this necessarily involves the calculation of thesecond-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) Galileo Galilei Institute, Firenze

  27. Non-Gaussianity from Inflation: results DT/T = -1/3 (fL + fNL) fNL = fNL*fL2 + const. fNL=f NL0– K (k1,k2) Sachs-Wolfe limit; replaced by full transfer function in true CMB maps Universal (gravitational) term going to zero in the squeezed limit model-dependent term Galileo Galilei Institute, Firenze

  28. Sachs-Wolfe effect: a compact relation Initial conditions set during or after inflation Post-inflation non-linear evolution of gravity standard scenario curvaton scenario Bartolo, Matarrese & Riotto 2004 Galileo Galilei Institute, Firenze

  29. Extracting the non-linearity parameter fNL Connection between theory and observations This is the proper quantity measurable by CMB experiments, via the phenomenological analysis by Komatsu and Spergel 2001 k = | k1 + k2 | Galileo Galilei Institute, Firenze

  30. Non-Gaussianity in the standard scenario (I) Non-Gaussianity generatedduring inflation: Accounting for the inflaton self-interactions and metric fluctuations at second-order in the perturbationsbrings Acquaviva et al. 2003; Maldacena 2003 Non-Gaussianity for single-field models of slow-roll inflation is tiny during inflation: Galileo Galilei Institute, Firenze

  31. 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 = | k1 + k2 | fNL~ 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 Galileo Galilei Institute, Firenze

  32. Inflation models and fNL See review: Bartolo, Komatsu, Matarrese & Riotto, 2004, Phys. Rept. 404, 103 model fNL(k1,k2) comments single-field inflation g(k1, k2)=3(k14+k24)/2k4+(k1. k2) . [4-3(k1. k2)/k2]/k2, k=k1+k2 4/3 – g(k1, k2) curvaton scenario -1/3 - 5r/6 + 5/4r- g(k1, k2) r ~ (rs/r)decay I = - 5/2 + 5G / (12 aG1) I = 0 (minimal case) modulated reheating 1/12 – I - g(k1, k2) order of magnitude estimate of the absolute value multi-field inflation up to 102 “unconventional” inflation set-ups second-order corrections not included warm inflation typically 10-1 post-inflation corrections not included ghost inflation - 140 b a-3/5 post-inflation corrections not included D-cceleration - 0.1 g2 Galileo Galilei Institute, Firenze

  33. More about (standard single-field slow-roll) inflation SW limit • Quadratic non-linearity on large-scales (up to ISW and 2-nd order tensor modes). Standard slow-roll inflation yields aNL~ bNL~ 1 • Cubic non-linearity on large-scales (up to ISW and 2-nd order tensor modes  Bartolo, D’Amico, Matarrese & Riotto, in prep.) leading contribution to the bispectrum: additional contribution to trispectrum (together with fNL2 terms): Galileo Galilei Institute, Firenze

  34. Second-order transfer function Improve treatment of radiation transfer function, going to second order. Crucial ingredient for |fNL| ~ 1 (standard inflation). For large NG (|fNL| > ~10) the standard procedure is O.K.. • (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 last-scattering surface) • second-order ISW effect, both early and late • 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 • (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: • 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) modes, etc. … Galileo Galilei Institute, Firenze

  35. Non-Gaussian CMB anisotropies: map making Liguori, Matarrese & Moscardini 2003 • assume mildly non-Gaussian large-scale potential fluctuations • account for radiative transfer radiation transfer functions harmonic transform:Flm(r) Galileo Galilei Institute, Firenze

  36. 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,r1) Galileo Galilei Institute, Firenze

  37. Spherical coordinates in real space (II) band-pass filters Wl(r,r1) linear gravitational potential power-spectrum Wl(r,r1) does not oscillate as fast as jl(kr) Galileo Galilei Institute, Firenze

  38. Galileo Galilei Institute, Firenze

  39. Outline of the code • precompute transfer functions (extracted from CMBfast) for a given model • precompute filters Wl(r,r1) 3.generate white-noise coefficients nlm(r) 4. correlate white-noise coefficients to find multipoles FLlm(r) and then extract FNLlm(r) • obtain CMB multipoles by convolving with • radiation transfer function Galileo Galilei Institute, Firenze

  40. CPU time and memory requirements 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 Galileo Galilei Institute, Firenze

  41. The simulations • 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). • A large sample (300 maps) of Planck LFI resolution (n_side = 1024, lmax = 2000) non-Gaussian CMB maps in HEALPIX format with free fNL parameter (is already available (simulations have been run in Oslo; the maps are presently in Oslo + DPC/OATS). • Soon upgraded to include NG polarization maps (M. Liguori, P. Cabella, F. Hansen, E. Komatsu, S. Matarrese & B. Wandelt) at WMAP resolution. • An important issue to be considered for future applications is the flexibility of the code to changes in the input power-spectrum. Galileo Galilei Institute, Firenze

  42. Non-Gaussian CMB maps: Planck resolution 5’ resolution lmax = 3000, Nside=2048 fNL = 3000 Liguori, Matarrese & Moscardini 2003, ApJ 597, 56 fNL = 0 fNL = -3000 Galileo Galilei Institute, Firenze

  43. PDF of NG CMB maps Galileo Galilei Institute, Firenze

  44. Observational constraints on fNL • The strongest limits on non-Gaussianity come from 3-yr WMAP data. Spergel et al.(2006) find (95% CL): - 54 < fNL < 114 • According to Komatsu & Spergel(2001) using the angular bispectrum one can reach values as low as |fNL| < 20 with WMAP & |fNL|< 5 with Planck can be achieved • The role of the fNL 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 fNL !! Komatsu et al. 2003 Galileo Galilei Institute, Firenze

  45. Statistical analyses of NG CMB maps • The maps have been analyzed with a bunch of statistics and the results have been presented in Santander last September • Roughly speaking it seems that bispectrum-based statistics are most powerful (bet very slow!) in detecting quadratic NG. Upper bounds on |fNL| down to ~18 (2 sigmas) appear achievable with Planck (lmax=3000). (cf. Komatsu & Spergel 2001 |fNL|<5). Creminelli et al. (2005) from 1yr data obtain - 27 < fNL < 121 • The analysis of WMAP 3 years data (Spergel et al. 2006) yields: - 54 < fNL < 114 • 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 1st year WMAP data yields: - 160 < fNL < 160 Cabella et al. 2006) • Expected improvements with combined analysis of polarization maps and full exploitation of specific angular dependence of fNL are extremely important to probe NG of standard single-field slow-roll inflation models Galileo Galilei Institute, Firenze

  46. Bispectrum analysis Liguori, Hansen, Komatsu, Matarrese & Riotto, 2005 Usual parametrization for primordial non-Gaussianity: fNL is constant A full second order perturbative approach for single-field yields a momentum-dependent fNL The momentum-dependent part accounts for the growth of non-Gaussianity due to post-inflationary non-linear evolution Model dependent (intrinsic NG) Model independent: post-inflationary evolution Galileo Galilei Institute, Firenze

  47. 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 Galileo Galilei Institute, Firenze

  48. CMB angular bispectrum (II) In the full second order treatment the averaged bispectrum becomes: new l.o.s. integral Combination of Wigner 3j and 6j symbols Now the line of sight Integral has a different expression Galileo Galilei Institute, Firenze

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