31 Oct 2015 @ Koube Univ. Progress of code development 2nd-order Einstein-Boltzmann solver for CMB anisotropy Takashi Hiramatsu Yukawa Institute for Theoretical Physics (YITP) Kyoto University Collaboration with Ryo Saito (APC), Atsushi Naruko (TITech), Misao Sasaki (YITP) Introduction : Inflation paradigm Takashi Hiramatsu A natural (?) way to give a seed of observed large-scale structure. http://www.sciops.esa.int Quantum fluctuations of inflationary space-time = inhomogeneity of gravitational field in the Universe Inhomogeneity of dark matter and baryons and photons 2/42 Introduction : CMB Takashi Hiramatsu Cosmic Microwave Background (CMB) http://www.sciops.esa.int Temperature fluctuations of embedded into the Planck distribution with A probe for the extraordinarily deep Universe. 3/42 Introduction : Non-Gaussianity Takashi Hiramatsu Focus on the statistical property of fluctuations... Pure de Sitter inflation provides Gaussian fluctuations, Non-gaussianity (NG) indicates the deviation from de Sitter space-time (slow-roll inflation, multi-field inflation, etc.) ... parameterised by Observed NG has a possibility to screen enormous kinds of inflation models But, non-linear evolution of fluctuations can also generate NG... To specify the total amount of intrinsic is an important task. 4/42 Introduction : current status Takashi Hiramatsu Linear Boltzmann solvers (CAMB, CMBfast, CLASS, etc. ) have been available, giving familiar angular power spectrum of temperature fluctuations. http://www.sciops.esa.int 3 Boltzmann solvers for 2nd-order perturbations are available, but the resultant is not converged... Do it ourselves ! (At least, my code may be the first domestically-produced CMB code in Japan...?) 5/42 Introduction : current status of my code Takashi Hiramatsu 1st-order perturbations 2nd-order perturbations Implementing basic equations History of electron density line-of-sight integral +angular power spectrum Implementing basic equations 1 Qualitative check Quantitative check with CAMB NOW 2nd-order line-of-sight formula NOW R.Saito, Naruko, Hiramatsu, Sasaki, JCAP10(2014)051 Bispectrum estimator [arXiv:1409.2464] Implementing basic equations 2 Speed-up + Optimisation 1st-order 95% 2nd-order 35% 6/42 Takashi Hiramatsu 1st-order 7/42 /42 8 Takashi Hiramatsu Einstein equation MATTER RADIATION (Bose-Einstein dist.) (Fermi-Dirac dist.) Photon temperature/polarisation fluc. Neutrino temperature fluc. CDM/baryon density fluc. CDM/baryon velocity fluc. Gravity field fluc. last scattering surface initial fluctuations Euler/continuity equation reheating Boltzmann equation spatial fluctuation spatial potential of gravity DARK-ENERGY T I b INFLATION S i A I S L curvature curvature B F S o perturbation Evolution offluctuations Evolution B M m A A M L s C C CosmoLib : Huang, JCAP 1206 CosmoLib:(2012) 012Huang, JCAP CLASS II : Blas,1107 CLASS Lesgourgues, Tram, (2011) 034 JCAP CAMB :Lewis, (2000)Challinor, 473 APJ538 CAMB C o : Seljak, Zaldarriaga,(1996) 437 CMBFAST APJ469 C Photons/Neutrinos Takashi Hiramatsu They are described by the Bose-Einstein/Fermi-Dirac distribution functions: Temperature fluctuation has three independent variables Fourier trans. Directional cosine Multipole exp. http://spud.spa.umn.edu/~pryke Multipole expansion 9/42 Photons/Neutrinos Takashi Hiramatsu Distribution function satisfies the Boltzmann equation : Liouville term Collision term S. Dodelson, “Modern Cosmology” The efficiency of collision term is controlled by , time-derivative of optical depth : Thomson cross-section (cf. Rayleigh scattering adds ~1% contribution) Alipour, Sigurdson, Hirata, arXiv:1410.6484 Deep in radiation dominant epoch Extremely large tight-coupling Last-scattering surface suddenly decays Late-time Universe No collision, free-streaming Reionisation is revived, but not so significant 10/42 Photons/Neutrinos Takashi Hiramatsu Polarisation According to Dodelson's textbook, polarisation strength is given by , relating with the Stokes parameter Q and U. Furthermore, they relates with E and B modes as Lin, Wandelt, arXiv:astro-ph/0409734 ( : spin raising/lowerling operators) Here we treat as nothing but another intrinsic degree-of-freedom of photons in solving Boltzmann equations for . Massive neutrino Don't take care, but it'd be possible to implement. 11/42 CDM/baryons Takashi Hiramatsu Boltzmann equations for Relativistic Fluid equations CDM/baryons for CDM and baryons + photon coupling. Fluid approx. with source terms. = Taking momenta of dist.func. Fourier transform CDM baryon 12/42 Gravity Takashi Hiramatsu Conformal Newton gauge : Perturbed Einstein equations read... (non-dynamical) NOTE : CAMB, CMBFAST use sychronous gauge : 13/42 1st-order perturbation equations Takashi Hiramatsu Photon temperature Photon polarisation Massless neutrino temperature CDM, baryon Gravity (conformal Newtonian gauge) 14/42 Line-of-sight integral Takashi Hiramatsu Large Boltzmann hierarchy, say , is required, but it is too hard to calculate... Up to last-scattering-surface, superhorizon scale tight-coupling So, to guarantee the accuracy of After LSS, we use the integral representation, line-of-sight formula. Seljak, Zaldarriaga, APJ 469 (1996) 437 15/42 Line-of-sight integral Takashi Hiramatsu visibility function optical depth optically thick suppressing early-time contributions 16/42 Line-of-sight integral Sourced by fluctuations at LSS ( including Sachs-Wolfe effect ) Takashi Hiramatsu Monopole Dipole Quadrupole Integrated Sachs-Wolfe effect LoS integral directly solving 17/42 Initial conditions : relations between fluctuations Takashi Hiramatsu Deep in radiation dominant epoch where all modes are larger than horizon scale. We set h t g n e (cf. which crosses the horizon at ) l From the Einstein equation, time unchanged potential, superhorizon, radiation dominant similarly fluctuated radiation dominant 18/42 Initial conditions : curvature perturbation Takashi Hiramatsu Primordial curvature perturbation (preserved on superhorizon scales) after inflation during inflation = almost flat spectrum 19/42 Initial conditions : Eventually, ... Takashi Hiramatsu We impose neutrino fraction : @ Separating primordial (quantum) curvature perturbation, we focus on the transfer functions, quantum classical 20/42 Miscellaneous : Background geometry Takashi Hiramatsu Flat FLRW model fiducial parameters It can be easily extended to include non-flat case. NOTE : we use as the time variable instead of in solving EB equations. Then we don't have to solve the Friedmann equation. 21/42 Miscellaneous : Recombination/Ionisation Takashi Hiramatsu Number density of free electrons - Strength of Photon-Baryon coupling controls ... cf. - Opacity of the Universe (affects ISW) Weinberg : Peebles, APJ 153 (1968) 1; Weinberg, “Cosmology” 2-stage reionisation Peebles-Weinberg scheme Interpolated by He2+ He+ tanh() function. He He+ He+ He2+ He+ He H H+ H+ H Planck : 22/42 Miscellaneous : Spherical Bessel functions Takashi Hiramatsu Use three different methods to maintain an accuracy of for Descending reccurence : Debye's expansion : Taylor expansion : [Debye] [Descending] exponentially small [Debye] [Descending] ] r o oscillatory behaviour l [Taylor] y a T [ Once storing for , with arbitrary argument is given by the partitioned polynomial interpolation. 23/42 Takashi Hiramatsu Preliminary Results 24/42 Angular power spectrum (1st-order) Takashi Hiramatsu Angular power spectrum PRELIMINARY still has 14% undesired overestimation... CAMB 25/42 Angular power spectrum (1st-order) Takashi Hiramatsu 2012/04/24 2012/06/12 2014/06/20 2014/08/18 2014/06/28 2014/07/30 26/42 Takashi Hiramatsu 2nd-order 27/42 Go to 2nd-order Takashi Hiramatsu 2nd-order contributions appear in... Einstein-Boltzmann equations for 2nd-order quantities sourced by [1st-order]2 CDM+Baryon+Gravity have been implemented, but Baryon-Photon/Gravity-Photon couplings are not considered yet. Line-of-sight integral sourced by [1st-order]2 Formulations have been completed by R.Saito. R.Saito, Naruko, Hiramatsu, Sasaki, JCAP10(2014)051 [arXiv:1409.2464] (cf. Fidler, Koyama, Pettinari, arXiv:1409.2461) Existing 2nd-order Boltzmann solver CMBquick CMBquick : Creminelli, Pitrou, Vernizzi, arXiv:1109.1822 SONG SONG : Pettinari, arXiv:1405.2280 (thesis) CosmoLib2nd CosmoLib2nd : Huang, Vernizzi, arXiv:1212.3573 28/42 Non-Gaussianity Takashi Hiramatsu Bispectrum Bartolo et al., Phys.Rep.402(2004)103 [arXiv:astro-ph/0406398] Significance and shape of non-Gaussianity Significance : parametrised by local-type equilateral-type orthogonal-type folded-type + a variety of non-separable types Planck collaboration, A&A 571(2014) A24 [arXiv:1303.5084] 29/42 Non-Gaussianity Takashi Hiramatsu Results not well converged CMBquick Pitrou, Uzan, Bernardeau, JCAP07(2010)003 [arXiv:1103.0481] SONG Pettinari, Fidler, Crittenden, Koyama, Wands, JCAP04(2013)003 [arXiv:1302.0832] CosmoLib2nd Huang, Vernizzi, PRD89(2014)021302 [arXiv:1311.6105] Observational constraints by Planck (68% confidence level) Planck collaboration, A&A 571(2014) A24 [arXiv:1303.5084] 30/42 Expansion up to 2nd-order Takashi Hiramatsu Poisson gauge + neglecting 1st-order and Expanding up to 2and-order 31/42 2nd-order perturbation equtaions Takashi Hiramatsu CDM + gravity (contribution from radiation is work in progress....) 'Q'uadratic terms of 1st-order quantities is a fractional expression like 32/42 2nd-order perturbation equtaions Takashi Hiramatsu CDM + gravity (contribution from radiation is work in progress....) CDM+Baryon+Gravity have been implemented, but Baryon-Photon/Gravity-Photon