class Workshop CANTATA Summer School

Transcription

1 class Workshop CANTATA Summer School Thomas Tram Aarhus University (Denmark) 12/9/2017 Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

2 Programme Programme of the class workshop 12/9 Overview of class and its Python wrapper. Compute and plot output. 13/9 Modifying class. The hiclass extension and exercises on modified gravity. 14/9 Cosmological parameter extraction using MontePython. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

3 Programme Programme of the class workshop 12/9 Overview of class and its Python wrapper. Compute and plot output. 13/9 Modifying class. The hiclass extension and exercises on modified gravity. 14/9 Cosmological parameter extraction using MontePython. Today Overview of class Installing software and starting exercises Coffee break Exercises. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

4 What is CLASS? An Einstein-Boltzmann code! Solves the coupled Einstein-Boltzmann equations for many types of matter in the Universe to first order in perturbation theory. Computes CMB observables such as temperature and polarisation correlations Cl TT, Cl TE, Cl EE, Cl BB. Computes LSS observables such as the total matter power spectrum P(k) and individual density and velocity transfer functions i δ (k, τ), i θ (k, τ). Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

5 Other Boltzmann codes A history of Boltzmann codes 1995 COSMICS by Ed Bertschinger in Fortran Seljak&Zaldarriaga adds a few functions to COSMICS for implementing the line-of-sight formalism. The new code is much faster and is called CMBFAST CMBFAST improved with RECFAST and nonzero curvature Antony Lewis translates CMBFAST into Fortran90, adds nonzero curvature and other improvements. Released as camb Similar restructuring and translation of CMBFAST done by Michael Doran in C++, called CMBEASY Only camb maintained class 1.0 released. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

6 The classy module classy, the class wrapper All the functionality of classy is found in the Python class called Class. Import Class in In Python by: from classy import Class Running class from Python from classy import Class import numpy as np import matplotlib. pyplot as plt cosmo = Class () cosmo. set ({ ' output ':'tcl,pcl, lcl ',' lensing ':' yes ','modes ':'s,t','r':'0.2 '}) cosmo. compute () Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

7 The Jupyter Notebook Jupyter Notebook Jupyter Notebook is a cell-based interface to IPython. Has Tab-completion of variables and function names. Nicely presents the documentation of each function. Easy way to get started on Python. Launching Jupyter Notebook Write the following command to launch the notebook: $> jupyter notebook Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

8 The notebook Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

9 Tab: Available class methods Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

10 Shift+Tab: Help on method Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

11 Shift+Tab: More help Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

12 What can it do? What is available in the wrapper? get_background() returns the information normally found in _background.dat. get_thermodynamics() returns the information of _thermodynamics.dat. get_primordial() corresponds to _primordial_pk.dat. get_perturbations() returns everything found in _perturbations*.dat get_transfer(z,format) returns the density and velocity transfer functions at any redshift z. (Format can be either 'camb' or 'class'). Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

13 And even more... What is available in the wrapper? raw_cl() returns unlensed C l. lensed_cl() returns lensed C l. density_cl() returns density C l. pk(k, z) returns the P(k) at redshift z. Many other small functions. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

14 Running a single instance of class Running class cosmo = Class()#Initiate one instance of the class Class cosmo.set({'output':'tcl lcl mpk','omega_cdm':0.25}) cosmo.compute()#run CLASS rawcl = cosmo.raw_cl()#get C_l^XY from CLASS print rawcl.keys()#print keys in dictionary Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

15 Ressources Useful links Exercises and slides for this course: GitHub repository for class: Help forum for class: Official homepage, lecture notes and online documentation: Some Jupyter Notebooks using class: Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

16 The CLASS directory Files and subdirectories The directory class/ contains subdirectories: include / # header files (*. h) source / # modules (*. c) main / # main CLASS function tools / # tools (*. c) output / # output files python / # python wrapper cpp / # C++ wrapper plus examples of input files,explanatory.ini, README.rst, Makefile, and few other directories containing data files (bbn/) or external code (hyrec/) Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

17 CLASS modules Structure of modules in class A module in class is: a file include/xxx.h containing some declarations a file source/xxx.c containing some functions a structure xx of the information that must be propagated to other modules Some fields in the structure are filled in the input.c module, and the rest are filled by a function xxx_init(...). executing a module calling xxx_init(...) Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

18 All modules in CLASS module structure ab. * main content input.c precision pr ppr prec. params. background.c background ba pba a(τ),... thermodynamics.c thermodynamics th pth x e (z),... perturbations.c perturbs pt ppt sources S(k, t) primordial.c primordial pm ppm prim. spectra P(k) nonlinear.c nonlinear nl pnl NL corr. α NL (k, τ) transfer.c transfers tr ptr trnsf. func. l (k) spectra.c spectra sp psp P(k, z), C l s lensing.c lensing le ple lensed C l s output.c output op pop output format Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

19 The main CLASS function int main () { input_init_..(.., ppr,pba,pth,ppt,ptr,ppm,psp,pnl,ple, pop ); background_init (ppr, pba ); thermodynamics_init (ppr,pba, pth ); perturb_init (ppr,pba,pth, ppt ); primordial_init (ppr,ppt, ppm ); nonlinear_init (ppr,pba,pth,ppt,ppm, pnl ); transfer_init (ppr,pba,pth,ppt,pnl, ptr ); spectra_init (ppr,pba,ppt,ppm,pnl,ptr, psp ); lensing_init (ppr,ppt,psp,pnl, ple ); output_init (pba,pth,ppt,ppm,ptr,psp,pnl,ple, pop ); lensing_free ( ple ); spectra_free ( psp ); transfer_free ( ptr ); nonlinear_free ( pnl ); primordial_free ( ppm ); perturb_free ( ppt ); thermodynamics_free ( pth ); background_free ( pba ); } Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

20 Background module Unit system We set = c = k B = 1, and all dimensionful quantities have unit Mpc n Friedmann equation a = a 2 H = a 2 α ρ α K a 2, where we have defined ρ α 8πG 3 ρphysical α. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

21 Background module background_functions() Most quantities can be immediately inferred from a given value of a without integrating any differential equations: ( ) ρ i = Ω 0 i H0 2 3(1+wi ) a a0 p i = w i ρ i ( ) 1/2 H = i ρ i K a ( 2 ) H = 3 2 i (ρ i + p i ) + K a a 2 ρ crit = H 2 Ω i = ρ i /ρ crit Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

22 Thermodynamics module Free electron fraction We must solve recombination and reionisation, to compute: the free electron fraction x e n e /n p. the optical depth κ(τ) where κ = σ T an p x e the visibility function g(τ) = κ e κ. x e Optical depth g Redshift z Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

23 Thermodynamics module Recombination Proper calculation requires out-of-equilibrium treatment of thousands of excited states in both hydrogen and helium. Fast computation by using effective 3-level atoms with fudged coefficients: RECFAST and HyRec. Priomordial helium fraction Y He Used to be a fixed input parameter for Boltzmann codes. BBN imposes a (non-analytic) relationship between N eff, ω b and Y He. By default class will find Y He by interpolation in a table computed by the BBN code Parthenope. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

24 Perturbations module Einstein and Boltzmann equations We must solve 2 of the 4 first order Einstein equations: k 2 η 1 a 2 a h = 4πGa 2 δρ, k 2 η = 4πGa 2 (ρ + p)θ, h + 2 a a h 2k 2 η = 24πGa 2 δp, h + 6η + 2 a ( h + 6η ) 2k 2 η = 24πGa 2 (ρ + p)σ a together with the Boltzmann equation for each species present in the Universe. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

25 Perturbations module The Boltzmann equation At an abstract level we can write: L [f α (τ, x, p)] = C [f i, f j ] (= 0). (1) The last equal sign is true for a collisionless species. We expand f α to first order: f α (τ, x, p) f 0 (q)(1 + Ψ(τ, x, q, ˆn)). (2) Plugging equation (2) into equation (1) gives a Boltzmann equation for Ψ in Fourier space: Ψ τ + i qk ɛ (k ˆn)Ψ + d ln f [ 0 η ḣ + 6 η ] (ˆk ˆn) 2 = C d ln q 2 Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

26 Perturbations module A few missing definitions We have defined the comoving momentum q and comoving energy ɛ by q p p T α and ɛ 2 +mα 2 T α. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

27 Perturbations module A few missing definitions We have defined the comoving momentum q and comoving energy ɛ by q p p T α and ɛ 2 +mα 2 T α. Why do we use Fourier-space? The Liouville operator L [f i (τ, x, p)] contains the gradient operator j......so in real space, the Boltzmann equation is a Partial Differential Equation (PDE). But since j e ik x = ik j e ik x, the PDE decouples and become a set of ordinary differential equations for each mode k. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

28 Perturbations module Legendre expansion of Ψ Since ˆk ˆn = cos θ, the equation from before Ψ τ + i qk ɛ (k ˆn)Ψ + d ln f [ 0 η ḣ + 6 η ] (ˆk ˆn) 2 = C d ln q 2 has no dependence on the angle φ. Thus we can expand the angular dependence of Ψ in Legendre multipoles: Ψ(τ, k, q, ˆk ˆn) = (2l + 1)Ψ l (τ, k, q)p l (ˆk ˆn) (3) l Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

29 Perturbations module The Boltzmann hierarchy Ψ 0 = qk ɛ + 1 6ḣ d ln f 0 d ln q + C 0, Ψ 1 = qk 3ɛ (Ψ 0 2Ψ 2 ) + C 1, Ψ 2 = qk 5ɛ (2Ψ 1 3Ψ 3 ) Ψ l = ( 1 15ḣ η ) d ln f0 d ln q + C 2, qk (2l + 1)ɛ (lψ l 1 (l + 1)Ψ l+1 ), l 3. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

30 Perturbations module Simplifications for different species Massive neutrinos and Dark Matter usually have no interactions, so C = 0. Photons, massless neutrinos, baryons and Cold Dark Matter have no momentum dependence, so we can integrate out q. Baryons and Cold Dark Matter have negligible shear, so we only need to consider l = 0 and l = 1. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

31 Perturbations module Simplifications for different species Massive neutrinos and Dark Matter usually have no interactions, so C = 0. Photons, massless neutrinos, baryons and Cold Dark Matter have no momentum dependence, so we can integrate out q. Baryons and Cold Dark Matter have negligible shear, so we only need to consider l = 0 and l = 1. Complications for photons Photons are not completely described by a single distribution function but the three Stoke s parameters Θ, Q and U. (Note that Θ Tγ T γ = 1 4 δ γ) Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

32 Perturbations module The line-of-sight formalism The Boltzmann equation has a formal solution in terms of an integral along the line-of-sight: Θ l (τ 0, k) = τ0 τ ini dτ S T (τ, k) j l (k(τ 0 τ)) ) S T (τ, k) g (Θ 0 + ψ) + (g k 2 θ b + e κ (φ + ψ ) + pol.. }{{}}{{}}{{} SW Doppler ISW Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

33 Perturbations module Relevant sources sources for CMB temperature (decomposed in 1 to 3 terms) sources for CMB polarisation (only 1 term) metric perturbations φ and ψ and derivatives, used for lensing and galaxy number counts. density perturbations of all components {δ i } velocity perturbations of all components {θ i } Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

34 Transfer module Purpose of the transfer module The goal is to compute harmonic transfer functions by performing several integrals of the type X l (q) = dτ S X (k(q), τ) φ X l (q, (τ 0 τ)) for each mode, initial conditions, and several types of source functions. In flat space k = q. Sparse l-sampling Calculation done for few values of l (controlled by precision parameters). C l s are interpolated later. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

35 Transfer module Overview of transfer_init interpolate all sources along k. (k grid is finer in transfer module than in perturbation module.) compute all flat Bessel functions loop over q (or k in flat space). : if non-flat, compute hyperspherical Bessels for this q : loop over modes, initial conditions, types : : loop over l : : : integrate transfer function (or Limber approx.) Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

36 Primordial module The primordial power spectrum class have a number of different ways to infer the primordial power spectrum of perturbations: Analytic parametrisation: ( P(k) = A exp (n 1) log(k/k pivot ) + α log(k/k pivot ) 2). Taylor expansion of inflationary potential V (φ φ ) or H (φ φ ). (See astro-ph/ and astro-ph/ for details.) Parametrisation of V (φ) in the whole region between the observable part and the end of inflation. Primordial powerspectra from external code. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

37 Spectra module Linear matter power spectra P(k, z) = (δ m (k, τ(z))) 2 P(k) or in the case of several initial conditions, P(k, z) = ij δ i m(k, τ(z)) δ j m(k, τ(z))p ij (k), Angular power spectra C XY l = 4π ij dk k X l (k) Y l (k)p(k) ( or its generalisation to several ICs). We have: { TT, TE, EE, BB, PP, TP, EP, XY N i N j, TN i, PN i, L i L j, TL i, N i L j }. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

38 Installing class and Jupyter Notebook Obtaining the codes $> git clone class If git is not installed: download from class-code.net or github.com/lesgourg/class_public. Download Anaconda Python 2.7: Installing class Should be as easy as writing make, since class does not require any external libraries. Note that the builtin C-compiler in OSX does not support OpenMP. class is compatible with Python3, but not yet MontePython. Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

39 Installation details Obtaining the codes This PDF is available at in the Day1 folder. $>git clone https :// github. com / lesgourg / class_ public class $>cd class $ > which python # Is it Anaconda Python? $ > which gcc # Is it the correct compiler? $ > make clean ; make -j $ > python -c ' from classy import Class ' Problems and solutions Not Anaconda Python: $>source ANACONDA_FOLDER/bin/activate undefined symbol: _ZGVbN2v_sin Follow the instructions at Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37

40 Exercise 0: Plot the CMB temperature power spectrum Launch the Jupyter Notebook: $> jupyter notebook Run these lines in the notebook: % matplotlib inline # Plots inside cells % config InlineBackend. figure_ format = ' svg ' import matplotlib import matplotlib. pyplot as plt import numpy as np from classy import Class cosmo = Class () cosmo. set ({ ' output ':'tcl,pcl, lcl '}) cosmo. compute () cl = cosmo. raw_cl () l = cl['ell '] plt. loglog (l, l*(l+1) /(2.* np.pi)*cl['tt ']) Thomas Tram (AU) class Workshop CANTATA Summer School 12/9/ / 37