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Cosmology Population Monte Carlo

1. For N 1 subtract log E_min k KEY Use KEY string list instead of directory names default s SEP Use SEP as input separator for KEY list S SEP Use SEP as output separator default for both white space n Write number of model parameters L Use Laplace approximation reading file evidence fisher h This message e fisher to meanvar pl 38 fisher to meanvar pl OPTIONS file OPTIONS n No inverse m Marginal errors don t invert matrix X mixmvdens format default mvdens format k Keep temporary file fishtmp i h This message Options m and n exclude each other e get spar pl Usage get spar pl OPTIONS LANG PAR1 PAR2 OPTIONS c CONFIG Configuration file ONFIG default config_pmc i INDEX Returns only par INDEX P PATH Use PATH as CosmoPMC directory default environment variable COSMOPMC p Print p lt i gt for unknown parameters instead of input string LANG One of yorick gnuplot TeX R More languages can be defined in spar txt PAR1 Prameter strings e go fishing 121112 Usage go_fishing OPTIONS OPTIONS c CONFIG Configuration file default config_fish a Adaptive numerical differentiation default fixed difference f Force positive Fisher matrix q Quiet mode h This message Run in parallel on NP cpu s mpirun np NP go fishing OPTIONS Usage haloplot log18 M min log10 M1 log18 M8 sigma log M alpha halo halomodel par OPTIONS O
2. If there are points with very large weights they can dominate the other points whose normalised weights will be small Even a few sample points might dominate the sum over weights and result in a low perplexity The perplexity is the most sensitive quantity to those high weight points much more than e g the mean the confidence intervals or the evidence Effective number of proposal components enc The proposal q provides useful information about the performance of a PMC run For example the effective number of components defined in complete analogy to ESS D 1 ENC p Zi 5 d 1 is an indication of components with non zero weight If ENC is close to unity the number of remaining components to sample the posterior is likely to be too small to provide a represen tative sample For a badly chosen initial proposal this usually happens already at the first few iterations By monitoring the file enc which is updated each iteration an unsuccessful PMC run can be aborted 3 Running CosmoPMC The effective number of components can also be determined from any proposal file mix_mvdens format with the script neff proposal pl An additional diagnostic is the evolution of the proposal components with iteration This il lustrates whether the components spread out nicely across the high posterior region and reach a more or less stationary behaviour or whether they stay too concentrated at one point The the means variances as fu
3. only if nofz file Table 8 Redshift module file nofz par Nzbin N Number of redshift bins snzmode nz read from files File mode nzfile fi f fNzbin 27 5 The configuration file To create a config file of type max_post or go_fishing from a PMC config file the script Some flags are handled internally as integers enumerations but identified and set in the config file with strings The corresponding key word carries the same name as the internal variable preceded with an s e g the integer string pair lensdata slensdata The prior file indicated if desired with the flag sprior is a file in mvdens format It specifies a Gaussian prior with mean and covariance as given in the file Note that the covariance and not the inverse covariance is expected in the file 28 5 The configuration file Table 9 Basic common part of the configuration file version double Config file version Upwards compatibility config file version gt CosmoPMC version cannot be guaranteed Downwards compatibility config file version lt Cos MOPMC version is most likely ensured Parameter section npar integer Number of parameters n ded integer Number of deduced parameters The deduced parame ters are not sampled but deduced from the other param eters and written to the output files as well spar string Parameterisation type necessary for the wrapping into the individual poster
4. Navarro J F Frenk C S amp White S D M 1997 ApJ 490 493 21 Percival W J Cole S Eisenstein D J et al 2007 MNRAS 381 1053 45 A File formats Press W H amp Schechter P 1974 ApJ 187 425 Robert C P amp Wraith D 2009 in Proceedings of MaxEnt 2009 to be published by American Takada M amp Jain B 2003 MNRAS 340 580 21 Tinker J L Robertson B E Kravtsov A V et al 2010 ApJ 724 878 20 Tinker J L Weinberg D H Zheng Z amp Zehavi I 2005 ApJ 631 41 20 Wraith D Kilbinger M Benabed K et al 2009 Phys Rev D 80 023507 A File formats A 1 Data files A 1 1 Lensing For all lensdata t types the data format is the same Each line contains the data for a given angular scale and arbitrary many redshift bin pair combinations 46 A File formats The angular scales are defined as follows For lensformat angle_center the fist column contains the angular bin center in arc minutes For the cases lensformat angle mean angle wlinear and angle wquadr first two columns specify the lower and upper end of the angular bin Following the angular information are the data For N redshift bins N N 1 2 columns spec ify all pair combinations ij lt in lexical order that is 11 12 13 1N 22 23 NN Note that for lensdata xipm the first Ng lines of the data file contain for Ng angular scales the
5. DIR1 DIR2 OPTIONS i NITER Number of iterations needed if do proposal 2 c CONFIG FILE Configuration file default in order config mcmc config pmc t TITLE Title string for each panel default empty T TITLE Title string for all contour2d eps pdf default empty n No shade w WIDTH Line width WIDTH default 4 1 OPT Add 1d posterior plots OPT can contain the following letters m Plot line at mean position 123 Plot line at 68 95 99 7 density 56 B Syntax of all commands q h N FACTOR FACTOR NORM NUM KEY1 KEY2 FS FORMAT PAR PATH DIR1 e proposal mean pl t Write mean and 68 confidence intervals as text use with m and 1 n None of the above All contours with solid lines Outermost level is N sigma Aspect ratio 1 changes plot limits such that dx dy Gaussian smoothing of 2d histograms with variance box width FACTOR If FACTOR is negativ plots unsmoothed histogram in addition use with n Note For multiple contours use a list of values gl g2 Gaussian smoothing of 1d histograms default 2d factor Use covariance file covar fin for Gaussian smoothing Normalisation of 1d posterior m Maximum 1 default i Integral over posterior 1 Color scheme NUM 0 1 2 Add key to plots Key strings default directory names Font size FS default 24 Output file format FORMAT eps pdf default eps Writes the chi2 files in block format Pl
6. 3 is a common choice For y co a Gaussian distribution is reached asymptotically corresponding configuration files from the PMC config file for manual calls of max_post and go_fishing Updating the proposal The PMC algorithm automatically updates the proposal after each iteration no user interference is necessary 3 Running CosMoPMC The method to update the proposal is a variant of the Expectation Maximization algorithm EM l 7 It leads to an increase of the perplexity and an increase of ESS Detailed can be found in Cap Dead components A component can die during the updating if the number of points sam pled from that component is less than MINCOUNT 20 or its weight is smaller than the inverse total number of sample points 1 N There are two possibilities to proceed First the component is buried its weight set to zero so that no points are sampled from it in subsequent iterations Alternatively the component can be revived In this case it is placed near the component 4 which has maximum weight and it is given the same covariance as d The first case is the standard method used in Wraith et The second method tries to cure cases where the majority of components die This can happen if they start too far off from the high density posterior region Often only one component remains to the end not capable of sampling the posterior reliably Both options can be chosen using the config file Se
7. Do not force Fisher matrix F to be positiv If F is negative script exits with an error a Adaptive numerical differentiation for Fisher matrix s SEED Use SEED for random number generator If SEED 1 default the current time is used as seed S MIF Stops after maximum search M or Fisher matrix F A ylin Default answer to all questions on stdin P PATH Use PATH as CosmoPMC directory default environment variable COSMOPMC e Create essential plots p PRO Plotting scripts y yorick default R R or n none Combinations of letters are possible e g yR M MULT Output sample MULT times input default 1 Valid if plotting script is R 0 OPT Pass options OPT to plot contour2d pl q Quiet mode h This message e diag mvdens pl 38 Usage diag_mvdens pl IN Prints the mvdens file IN with the covariance replaced by its diagonal e essential cosmo pmc run pl Usage essential cosmo pmc pl OPTIONS OPTIONS c CONFIG Uses config file CONFIG default config pmc P PATH Use PATH as CosmoPMC directory default environment variable COSMOPMC k Keep temporary files V Verbose h This message e evidence pl 37 Usage evidence pl OPTIONS SAMPLE OPTIONS h This message SAMPLE PMC sample file e evidence list pl Usage evidence list pl OPTIONS DIR1 DIR2 OPTIONS rN Subtract log E from DIRN default no subtraction 32 B Syntax of all commands
8. NCPU to run CosmoPMC on NCPU CPUs See cosmo pmc pl h for more options Depending on the type of initial proposal Sect 3 2 a maximum search is started followed by a Fisher matrix calculation After that PMC is started Fig 1 shows a flow chart of the script s actions 3 Running CosmoPMC Diagnostics Check the files perplexity and enc If the perplexity reaches values of 0 8 or larger and if the effective number of components ENC is not smaller than 1 5 the posterior has very likely been explored sufficiently Those and other files are updated during run time and can be monitored while PMC is running See Sect 3 3 1 for more details Results The text file iter_ niter 1 mean contains mean and confidence levels The file iter_ niter 1 all_contour2d pdf shows the 1d and 2d marginals Plots can be redone or refined or created from other than the last iteration with plot contour2d pl Note that in the default setting the posterior plots are not smoothed See Sect for more details and for information on the alternative script 3 2 CosmoPMIC in detail This section describes in more detail how PMC is run and which decisions the user has to make before starting and after stopping a PMC run Initial proposal The choice of the initial proposal used during the first PMC iteration is of great importance for a successful PMC run The following options are implemented determined 1 sinitial fisher rshi
9. Quiet mode h This message e mean2eps pl 36 Usage mean2eps pl OPTIONS MEAN OPTIONS MEAN File containing mean and confidence levels output of cosmo pmc or histograms sample c CONFIG Uses config file CONFIG default config pmc P PATH Use PATH as CosmoPMC directory default environment variable COSMOPMC o BASE Outname BASE default lt MEAN gt vV Verbose 54 B Syntax of all commands h This message e meanvar2tab pl 36 Usage meanvar2tab pl OPTIONS file file2 Options s 123 68 1 95 2 or 99 7 3 errors default 1 p PREC Output with PREC digits PREC format string e Error s written to PREC significant digits use p PREC c CONFIG Uses config file CONFIG default config_pmc t TITLE Title table heading TITLE is string list with entries according to the number of input files S SEP Use SEP as input separator for TITLE list default white space P PATH Use PATH as CosmoPMC directory default environment variable COSMOPMC h This message e meanvar_sample 10 Usage meanvar_sample OPTIONS sample OPTIONS c CONFIG Configuration file default config_pmc W Ignore weights default weights first column of sample file C Write covariance and inverse covariance to files E Output evidence h This message sample PMC sample file e neff proposal pl 10 Usage neff_proposal pl PROP Calculates the effective number of components for
10. density parameters can not be mixed e g Q and wx on input causes the program to abort The parameter for massive neutrinos Qy mass is not contained in the matter density Qn Qe Op A parameter which is missing from the input list is assigned the default value found in the corresponding cosmology parameter file cosmo par unless there is an inconsistency with other input parameters E g if Que and Qx are input parameters Qm is assigned the value On 1 Qae AK Oy mass to keep the curvature consistent with Qg A flat Universe is assumed unless a both Om and Qae or b QX are given as input parameter 4 1 2 Matter power spectrum Usually models of the non linear power spectrum have a limited validity range in k and or redshift For small k each model falls back to the linear power spectrum which goes as P5 k oc See for more details on the models The Coyote emulator In the coyote10 case the power spectrum is zero for k gt kmax The same is true for redshifts larger than the maximum of Zmax 1 See Eifl for an alterna tive approach The Hubble constant h can not be treated as a free parameter For a given cosmology it has to be fixed to match the CMB first peak constraint 4 ztdi r 302 4 where dj is the distance to last scattering and rs is the sound horizon This can be done with the function set HO Coyote see Demo lensingdemo c for an example When doing sampling with non physical densi
11. last Ng lines contain _ where the angular scales first or first two columns are identical in both halfs The covariance matrix is in block format It consists of N lines and N columns where N NsNN 1 2 is the length of the data Usually Ns is the number of measured angular scales No unless there is more than one data point per scale e g for lensdata xipm M 2No A matrix element C equals d d di dj where d is the i data point In the counting over angular scale and redshift the former varies faster than the latter For example with two redshift bins and three angular scales the element C7 is the data variance for the redshift pair 11 and angular scale 6 starting counting at zero Or in other words the covariance matrix consists of N N 1 2 block sub matrices each of size N x Ns Each sub matrix corresponds to one redshift bin combination It is therefore easy to exclude some redshift bins by 1 setting the diagonal of a sub matrix to a very high value and 2 setting the off diagonal to zero see A 1 2 SNla The SNIa data file in SN SALT format starts with the following two lines QINTRINSIC DISPERSION double PECULIAR_VELOCITY double The peculiar velocity value is in units of km s This is followed by a list of supernovae one object on each line as follows name zm sc lt m gt lt s gt c ms mc lt sc gt A 1 3 BAO The BAO distance measures are modeled as Gaussian
12. mvdens file inverts the covariance matrix and prints the mean covariance diag_mvdens p1 replaces the covariance by its diagonal additional parameters and the initial proposal is chosen from a previous run with the reduced parameter set 38 7 Using and modifying the code 6 6 2 PMC simulation MCM chain utilities sample2fixpar pl reads a sample file and fixes a parameter by cutting off all points outside a given narrow range 6 6 3 PMC proposal diagnostics tity which is printed to the file enc 7 Using and modifying the code 7 1 Modifying the existing code Note Code to be used with MPI should not contain global variables and static variables 7 2 Creating a new module In this section the steps required to add a new cosmology module to CosmoPMC are described 1 Create the directory newmodule and create or copy files with the necessary code to deal with the data and likelihood Include files h should be in newnodule include source files c in newmodule src Edit the or create a new Makefile in newmodule and add the rules Libnewmodule so libnewmodule dylib and libnewmodule a as well as the rule clean 2 In wrappers include types h Define a new data type by extending the enumeration data t Add the corresponding string for identification of the module in the configuration file in the macro sdata t i and increase Ndata t by one 3 In wrappers include all wrappers h Add
13. plot contains the last plot command with all options This can be used to produce and modify a plot which has been generated automatically by other scripts e g osmo_pmc p iplot_confidence creates 1d and 2d marginals of the posterior from the re sample file sample Smoothing is done with a kernel density estimation using the R function kde2d The kernel width can be set with the option g The number of grid points relevant both for smoothing and filled contours is set with N Use both i and j options to only plot the 2D marginals of parameters iandj to save computation time 6 2 Mean and confidence intervals From a mean output file containing parameter means and confidence levels one can create a ps pdf file using the command mean2eps pl This is equivalent to the following steps see also essential cosmo pmc run pl meanvar2tab pl creates a table with parameter names and values formatted in TEX format itab2tex pl wraps a IATEX table header around the table 36 6 Post processing and auxiliary programs e txt2tex pl wraps a LATEX header around the file Example gt meanvar2tab pl s 1 p 2 e iter_9 mean gt mean tab gt tab2tex pl s 1 25 mean tab gt mean in tex gt txt2tex pl mean in tex gt mean tex 6 2 1 PMC proposal creates plots of the proposal component s means vari 6 3 Importance sampling A PMC simulation file pmcsim from an earlier PMC run correspon
14. see Table 7 Vc is computed between those two redshifts Second if both numbers are lt 0 Vc is weighted by the redshift distribution n z see e g eq 28 in Ross amp 1 2009 In this weighting the maximum value of n z is set to unity 4 6 BAO BAO constraints are implemented with two distance measures e smethod distance A The distance parameter A is defined in Eisenstein et _ Dy z VOm c Ho z 31 where i Dv z ziwe AG 32 is the spherically averaged distance to redshift z e smethod distance d_z The distance parameter d 1 f sound horizon r at drag epoch zg to spherically averaged distance e g Percival et a 2007 rs Za Dy z d z 33 We use the fitting formula for the drag redshift za from culate the sound horizon as the distance a sound wave can travel prior to za by numerical integration 4 7 Redshift distribution Some of the cosmology modules require a redshift distribution for example lensing and HOD Table 3 lists the implemented redshift distributions n z via the flag nofz Each redshift bin can have a different type The syntax for a redshift bin file is described in Appendix A 1 5 23 5 The configuration file Table 3 Redshift distribution types nofz Description n z parameter list hist Histogram NS Nea see text single Single redshift p z zo Z0 Z0 ludo Fitting function z zo exp z zo Zmin Zma
15. the line include newmodule h 4 In tools include par h If necessary add new parameter types p_newparameter to enumeration par t add the corresponding identifier strings to the macro spar t and increase Npar t by one Optional Add the parameter name and syntax for different programs e g gnuplot yorick TEX to bin spar txt 39 7 Using and modifying the code 5 In wrappers src wrappers c Add the corresponding case to the switch instruction in the function init_func_t This function sets the data type 6 Create the files wrappers include newmodule h and wrappers src newmodule c Those files need to have different names than the files in newmodule src include Write the following functions a init_function_newmodule b read_from_config_newmodule c init_newmodule d likeli_newmodule returning log L e special newmodule optional f print_newmodule optional To see what these functions are supposed to do have a look at already existing modules e g in bao c 7 In Makefile main a In the section Additional directories define the path to the new module s directory as NEWMODULE COSMOPMC newmodule b In the section Libraries define the library of the new module as LIBNEWMODULE libnewmodule EXT c In the section Combined cosmo include and linker flags add the following flags I NEWMODULE include to the variable IINCDIRS L NEWMODULE to LLIBDIRS l
16. variables the data files are in mvdens corresponding to where the distances are measured This was wrongly stated here until version 1 01 47 A File formats A 1 4 CMB The CMB data for WMAP are the ones released by the WMAP team They are not included in CosmoPMC and can be obtained e g from the LAMBDA site The SZ correction power spectrum file has two columns in each row containing and Cz re spectively The first line has to start with 2 A 1 5 Redshift distribution The first line of a file describing a the redshift distribution for a redshift bin contains the type see Sect nofz This is followed by the list of parameter values in the order given in Table 3 Each parameter value has to be in a new line with the exeption of the histogram nofz single In that case the parameter lines are as follows Zo No Z1 Ni Zn 1 Nh 1 Zn 0 N is the number of galaxies in the bin z z 1 The last line denotes the upper limit of the last histogram bin z Zmax followed by a zero For nofz single the file has to contain two identical lines with the value of zo in each line A 2 Output file names The default names of all output files are defined in stdnames h Edit this file and to make clean make to set user defined file names Note however that some of the pre processing scripts expect the default names A 3 Multi variate Gaussian Student t mvdens mixture models mix_mvdens Th
17. 4 4 Cosmology Pure E B mode separating functions are chosen with slensdata decomp eb For the lack of analytical expressions for filter functions to obtain these real space statistics from the convergence power spectrum they are calculated by integrating over the 2PCF The integral is performed over the finite angular interval 9min Pmax The prediction for the E mode is max E al do 0 T 0 9 T 0 EO 11 f filter functions are implemented The optimized E B mode function for which the real space filter functions are Chebyshev polynomials of the N 1 20 max Umin 1 EMEN Y antro oy AA aD T T x i sin arccos x n 0 The coefficients a have been optimized with respect to signal to noise and the 2 0g Fisher matrix The function E is defined as a function of the lower angular limit Yin The ratio 7 of lower to upper limit 7 Omin Omax is fixed The second variant are the so called COSEBIs Complete Orthogonal Sets of E B mode Inte 10 We implement their logarithmic filter functions 9 n l ntl TEW ros e s Nn y ed Nn G ra 13 min j 0 j l The coefficients c are fixed by integral conditions that assure the E B mode decomposition of the 2PCF on finite angular integral They are given by a linear system of equations which 5 010 To solve this system a very high numerical accuracy is needed The MATHEMATICA
18. Additional prior one of none recom only if slensdata decomp_eb only if sdecom eb filter COSEBIs log only if sformat angle wquadr only if scov_scaling cov ESH99 only if Nexclude gt 0 31 mended unity de conservative 5 The configuration file Supernovae type la SNIa datname string Data file name datformat string Data format SNLS_firstyear schi2mode string X and distance modulus estimator type one of chi2 simple chi2 Theta2 denom fixed chi2_betaz chi2 dust chi2 residual Theta2 denom 2doubles Fixed a 8 in x denominator zAV name string File with Ay z table datname beta d string Prior file mvdens format on Ba if none add logdetCov integer 1 if 0 5 log det Cov is to be added to log likelihood 0 if not recommended see Sect 4 3 model file string Parameter file name e g cosmo SN sspecial string Additional prior one of none recommended unity de conservative only if schi2mode chi2 Theta2 denom fixed Ponly if schi2mode chi2 dust Table 11 Data specific entries in the configuration file s data section continued CMB anisotropies CMB scamb path string path to scamb data path string path to wmap data This path should contain the direc tory data with subdirectories healpix data highl lowlP lowlP ClSZ file string File with SZ correction angular power spectrum if none lmax integer Maximum for angul
19. Cosmology Population Monte Carlo CosmoPMC v1 2 User s manual Martin Kilbinger Karim Benabed Olivier Capp Jean Coupon Jean Francois Cardoso Gersende Fort Henry J McCracken Simon Prunet Christian P Robert Darren Wraith Contents ents i PMCLIB ee 2 CosMoPMC dnde 3 3 Output files METRI 44 CMB ADISOOPIES SSH LOUE RA eee A ate ec D 4 5 Galaxy clustering MED LC 4 9 Parameter files The configuration file Plotting and nice printi Analysis Modifying the existing code Mean and confidence intervals Bayesian evidence Bayes factor Reparameterisation di EA beo A e RO Error passing syste N e Rh amp D N o Ua OI 12 14 18 19 20 23 23 24 24 24 35 36 36 37 37 37 38 Acknowledgments 42 43 43 46 46 48 48 50 59 C i MCMC configuration de 40 0 sem ope e ds a m da 60 C 2 Proposal and starting point 60 3 Output A Wop deo der poe Rae p an 61 Di CS la bed dd de eee Boe eS iii he eee 61 1 What is CosmoPMC 1 What is CosmoPMC CosmoPMC Cosmology Population Monte Carlo is a Bayesian sampling method to explore the likelihood of various cosmological probes The sampling engine is implemented with the package PMCLIB It is called Po ulation Monte Carlo PMC which is a novel technique to s
20. D is a white space separated list in quotes e g 0 25 0 75 p FILE Fidcucial parameter from FILE e g maxlogP t TOLERANCE Tolerance for maximum search default 0 01 d Calculate only diagonal of Fisher matrix go fishing h This message One of M or F is obligatory The default starting point for maximum search is max min 2 For Fisher matrix F a fiducial parameter has to be indicated with f FID or p FILE e corr coeff sh 38 Usage corr coeff filename mvdens block e cosmo mcmc Usage cosmo_mcmc OPTIONS OPTIONS c CONFIG Configuration file default config_mcmc s SEED Use SEED for random number generator If SEED 1 default the current time is used as seed h This message e cosmo_pmc i Usage cosmo pmc OPTIONS OPTIONS c CONFIG Configuration file default config pmc s SEED Use SEED for random number generator If SEED 1 default the current time is used as seed q Quiet mode h This message Usage cosmo pmc pl OPTIONS 51 B Syntax of all commands OPTIONS n NCPU Run PMC in parallel on NPCU cpus using mpirun default 1 c CONFIG Configuration file for PMC default config_pmc f FID Fiducial starting point FID FID is a white space separated list in quotes e g 0 25 0 75 r Random starting point for maximum search default max min 2 m cla Maximum search method c cg a amoeba d Calculate only diagonal of Fisher matrix D
21. MC proposal OPTIONS c COL Adds parameter in column and row COL default last column m VAL Parameter mean VAL default 0 v VAL Parameter variance VAL default 1 X File is in mixmvdens format FILE File name h This message e bayes factor pl Usage bayes_factor pl OPTIONS DIR1 DIR2 Calculates the Bayes factor between models The corresponding evidence files from PMC have to be in the directories DIR1 and DIR2 OPTIONS i ITER1 ITER2 Use iteration ITER1 for DIR1 and ITER2 for DIR2 default all iterations f EVI1 EVI2 Use files DIR1 EVI1 and DIR2 EVI2 default evidence S Short output last iteration only 1 Laplace approx from Fisher matrix denoted with iter 1 h This message 50 B Syntax of all commands Usage cl_one_sided OPTIONS sample OPTIONS c CONFIG Configuration file default config_pmc i INDEX Parameter index d DIR Direction DIR 1 1 v VALUE Starting value w WHICH WHICH 0 68 95 99 7 c l default WHICH 1 68 90 95 c l sample PMC sample file The options i INDEX d DIR and v VALUE are required e config pmc to max and fish pl 7 28 Usage config pmc to max and fish pl OPTIONS OPTIONS M Create config file for maximum search max post F Create config file for Fisher matrix go fishing c CONFIG Input PMC config file CONFIG default config pmc r Random starting point for maximum search f FID Fiducial starting point FID FI
22. ample from the posterior C app eta PMC is an adaptive importance sampling method which iteratively improves the proposal to approximate the posterior This code has been introduced tested and applied to various cosmology data sets in raith et al 2009 Results on the Bayesian evidence using PMC are discussed in Kilbinger et al 2010 1 1 Importance sampling One of the main goals in Bayesian inference is to obtain integrals of the form mf f S x a x dx 1 over the posterior distribution x which depends on the p dimensional parameter x where f is an arbitrary function with finite expectation under z Of interest are for example the parameter mean f id or confidence regions S with f 1s being the indicator function of S The Bayesian evidence E used in model comparison techniques is obtained by setting f 1 but instead of x using the unnormalised posterior 7 L P in 1 with L being the likelihood and P the prior The evaluation of 1 is challenging because the posterior is in general not available analytically and the parameter space can be high dimensional Monte Carlo methods to approximate the above integrals consist in providing a sample x n 1 w under x and approximating 1 by the estimator 1 N AA v few 2 n 1 Markov Chain Monte Carlo MCMC produces a Markov chain of points for which x is the limiting distribution The popular and widely used package cosmomc http cosmolo
23. an be specified with prefix PREFIX default usr local http code google com p waf 2 Installing CosmoPMC 2 3 Patch PmcLiB For CosmoPMC v1 2 and pmclib v1 x a patch of the latter is necessary From http www cosmopmc info download patch pmclib 1 x 1 2 tar gz and follow the instructions in the readme file readme_patch_pmclib_1 x_1 2 txt 2 4 Download and install CosmoPMC First unpack the gzipped tar archive gt tar xzf CosmoPMC_v1 2 tar gz This creates the the CosmoPMC root directory CosmoPMC_v1 2 Change to that directory and run gt python configure py This poor man s configure script copies the file Makefile no_host to Makefile host and sets host specific variables and flags as given by the command line arguments For a complete list see configure py help Alternatively you can copy by hand the file Makefile no_host to Makefile host and edit it If the flags in this file are not sufficient to successfully compile the code you can add more flags by rerunning configure py or by manually editing Makefile main Note that a flag in Makefile main is overwritten if the same flag is present in Makefile host To compile the code run gt make make clean On success symbolic links to the binary executables in exec will be set in bin It is convenient to define the environment variable COSMOPMC and to set it to the main CosmoPMC directory For example in the C shell
24. ar power spectrum accurate 0 1 Accurate reionisation and polarisation calculations in camb model_file string Parameter file name e g cosmoDP par sspecial string Additional prior one of none recommended unity de_conservative WMAP distance priors CMBDistPrior datname string Data ML point and inverse covariance file model_file string Parameter file name e g cosmo_lens par sspecial string Additional prior one of none recommended unity de_conservative 32 5 The configuration file Galaxy clustering HOD GalCorr shalodata string Datatype woftheta shalomode string xX type one of galcorr_var galcorr_cov galcorr_log datname string Data variance file name covname string Covariance file name M corr invcov double Correction factor for inverse covariance ML estimate see Hart lap et al 2007 delta double Power law slope 6 for integral constraint intconst double Integral constant C area double Area deg sngal_fit_type string Likelihood type inclusion of galaxy number One of ngal lin fit ngal_log_fit ngal_no_fit ngal_lin_fit_only ngal double Number of observed galaxies ngalerr double Error on the number of observed galaxies model_file string Parameter file name sspecial string Not used for HOD set to none only if shalomode galcorr_cov galcorr log not if sngal_fit_type ngal no fit Table 11 Data specific entries in the configuration file s data section cont
25. bins only if sinitial only if sinitial only if sinitial fisher rshift or fisher eigen file random_position 30 5 The configuration file Table 11 Data specific entries in the configuration file s data section Weak gravitational lensing Lensing slensdata string Data type one of xipm xip xim map2poly map2gauss gsqr decomp_eb pkappa map3gauss map3gauss_diag map2gauss map3gauss map2gauss_map3gauss diag decomp eb map3gauss decomp eb map3gauss diag sdecomp eb filter string One of FK18 SN FK10_FoM etal0 FK10_FoM_eta50 COSEBIs log th min double Minimum angular scale th max double Maximum angular scale path double Path to COSEBIS files sformat string Data format of angular scales one of angle center angle mean angle wlinear angle wquadr al double Linear weight a2 double Quadratic weight w al O arcmin a2 0 arcmin datname string Data file name scov_scaling string One of cov_const cov_ESH09 covname string Covariance file name covname M4 string Covariance mixed term file name covname D string Covariance shot noise term file name corr invcov double Correction factor for inverse covariance ML estimate see Hartlap et al 2007 Nexclude integer Number of redshift bin pairs to be excluded from analysis exclude Nexclude integers Indices of redshift pairs to be excluded model file string Parameter file name e g cosmo_lens sspecial string
26. bove equation ngal log fit logarithmical ngal no fit no inclusion second term is omitted e ngal lin fit only exclusive first term is omitted 4 5 2 Deduced parameters The following deduced parameters can be computed e Mean galaxy bias baa dM MONDE 25 Ngai Z where bp is the halo bias and Ngal Z T NM n M z dM 26 is the total number of galaxies e Mean halo mass Moro am amar a7 27 Ngal Z e Fraction of satellite galaxies M amp O i Aex fo fama O 28 Ngal Z Use the program add deduced halomodel to add those deduced parameters to a PMC sample See the example config file config_pmc_ded in Demo MC_Demo HOD CFHTLS T06 4 5 3 Clustering data The angular two point correlation function w is implemented with the flag shalodata woftheta The measured input data Wmes is corrected for the integral constraint via W 8 Wmes 0 wc 29 assuming that the measured correlation function can be fit by a power law Wmes 9 Ay 6 C 30 outputs the correlation functions w 0 and r the HOD function N M for given HOD input parameters 22 4 Cosmology 4 5 4 Comoving volume The comoving volume is needed to calculate the comoving number density of galaxies follow ing from the halomodel and the HOD parameters There are two possibilities to calculate the comoving volume Vc First if Zmin and Zmax are larger than zero in the HOD parameter file halomodel par
27. camb and the WMAP likelihood library which are called by CosmoPMC for CMB anisotropies Further CosmoPMC contains a wrapper layer to communi cate between the PMC sampling and the cosmology modules 4 1 Basic calculations A number of routines to calculate cosmological quantities are included in the code These are e Background cosmology Hubble parameter distances geometry e Linear perturbations growth factor transfer function cluster mass function linear 3D power spectra e Non linear evolution fitting formulae for non linear power spectra 11996 Si 03 emulators Heitmant halo model e Galaxy clustering HOD model e Cosmic shear convergence power spectrum second order correlation functions and de rived second order quantities third order aperture mass skewness e CMB anisotropies via camb http www2 iap fr users kilbinge ni caea 12 4 Cosmology Table 1 Extrapolation of the power spectra snonlinear Kmax Next linear 333 6hMpc nm 4 pd96 3336hMpc 25 0 smith03 smith83 de 333 65 Mpc Eq 61 Smith et al 2003 coyotelg 2416Mpc no extrapolation 4 1 1 Density parameters Both the density parameters Qx px pc and the physical density parameters wx Qh are valid input parameters for sampling with PMC Internally the code uses non physical density parameters Qx All following rules hold equivalently for both classes of parameters Note that physical and non physical
28. ct 5 key sdead comp bury revive Errors If an error occurs during the calculation of the likelihood the error is intercepted and the likelihood is set to zero Thus the parameter vector for which the error occurs is attributed a zero importance weight and does not contribute to the final sample An error message is printed to stderr unless CosMoPMC is run with the option q and PMC continues with the next point An error can be due to cosmological reasons e g a redshift is probed which is larger than the maximum redshift in a loitering Universe Further a parameter could be outside the range of a fitting formulae e g a very small scalar spectral index in the dark matter transfer function Usually the errors printed to stderr during PMC sampling can be ignored Random numbers The GSL random number generator is used to generate random variables It is initialised with a seed reading the current time to produce different pseudo random numbers at each call The seed is written to the log file Using the option s SEED a user specified seed can be defined This is helpful if a run is to be repeated with identical results 3 3 Output files Each iteration i produces a number of output files which are stored in subdirectories iter_i of the CosmoPMC starting directory Files which are not specific to a single iteration are placed in the starting directory 3 Running CosmoPMC 3 3 1 Diagnostics Unlike in MCMC with a
29. daptive importance sampling one does not have to worry about con vergence In principle the updating process can be stopped at any time There are however diagnostics to indicate the quality and effectiveness of the sampling Perplexity and effective sample size perplexity The perplexity p is defined in eq 18 of Wraith et al 2009 The range of p is 0 1 and will approach unity if the proposal and posterior distribution are close together as measured by the Kullback Leibler divergence The initial perplexity is typically very low lt 0 1 and should increase from iteration to iteration Final values of 0 99 and larger are not uncommon but also for p of about 0 6 0 8 very accurate results can be obtained If p is smaller than say 0 1 the PMC sample is most likely not representative of the posterior Intermediate values for p are not straight forward to interpret Closely related to the perplexity is the effective sample size ESS which lies in the range 1 N It is interpreted as the number of sample point with zero weight Liu amp Chen 1995 A large perplexity is usually accompanied by a high ESS For a successful PMC run ESS is much higher than the acceptance rate of a Monte Carlo Markov chain which is typically between 0 15 and 0 25 The file perplexity contains the iteration i perplexity p ESS for that iteration and the total ESS This file is updated after each iteration and can therefore be used to monitor a PMC run
30. ding to a sample from posterior p can be used to do importance sampling with another posterior p2 For that simply replace the data section of the earlier config file with the corresponding data section of posterior sample under the posterior product p po 6 4 Bayesian evidence Bayes factor mean and covariances are remapped This is useful if different runs are to be reduced to a common parameter set for comparison or joint plotting The removal of parameters is equivalent to marginalisation over the corresponding parameter subspace For example suppose there is a SNIa run in directory Sn and a lensing run in Lensing SNla has the following parameters Omegam Omegade w0de M alpha beta 37 6 Post processing and auxiliary programs Lensing has the parameters Omegam sigma8 w de Omegade h100 In Sn create the file remap dat with the line In Lensing create the file remap dat with the line 032 In both directories run the command gt remap sh i iter_ lt niter 1 gt which creates sub directories remap containing symbolic links and or copies of histogram files to from iter_ niter 1 mean covariance files and updated configuration files To create joint marginal plots simply run gt plot_contours2d pl c path to Sn remap config_pmc n path to Sn remap path to Lensing remap 6 6 Analysis 6 6 1 mvdens mix_mvdens format utilities for a description of the mvdens and mix mvdens formats reads a
31. e mvdens file format is as follows The first header line contains four integers http lambda gsfc nasa gov 48 A File formats p v B c Here p is the number of dimensions v the degrees of freedom For a multi variate Gaussian choose v 1 and v gt 0 for Student t B indicates the number of secondary diagonal of the covariance matrix which are updated during the PMC iterations For most purposes B can be set equal to p which corresponds to the whole matrix being updated Finally c is 1 if the matrix is Cholesky decomposed and 0 otherwise This is followed by p doubles indicating the mean followed by p lines with p doubles each giving the symmetric covariance matrix Here is an example of a 5 dimensional multi variate Gaussian not Cholesky decomposed 5 158 0 38559 1 5238 19 338 1 3692 2 4358 0 0053677 0 025608 0 00066748 0 0011893 0 00087517 0 025608 0 16837 0 0079163 0 0027364 0 0035709 0 00066748 0 0079163 0 0011077 0 0010986 0 00067815 0 0011893 0 0027364 0 0010986 0 016716 0 0026266 0 00087517 0 0035709 0 00067815 0 0026266 0 014881 The mix_mvdens format has two doubles as the header D p where D is the number of components of the mixture and ndim the dimension This is followed by D blocks specifying the weights wg doubles and data mg in mvdens format of the D multi variate densities of the mixtures The weights should be normalised p Wa 1 In many cases an mvdens file
32. etical model of galaxy clustering is the one used in C for details of the model and further references A basis to describe galaxy clustering we implement the halo model as reviewed in Cooray eth 2002 which accounts for the clustering of dark matter halos On top of that a halo oc ion distribution HOD function Berlind amp Weinber 2 Kravtsov et al 200 eta al 2005 i is the prescription of how galaxies populate those halos This function is the number of g es N in a halo of mass M With the flag hod berwein82 excl this number is expressed as the sum of central No plus satellite N galaxies N M NAM x 1 Ns M 16 with 1 M 1 Mmi n M gt 1 e Se eed 2 17 gt Clog M M MoY ao V aene 18 0 else We further compute the galaxy two point correlation function r and its angular projection w 0 using the redshift distribution provided by the user as well as the galaxy number density for a full description of the model see Co 2 To prevent haloes from overlapping we implement the halo exclusion formalism as described in Tinke For the halo bias three options are available e shalo bias bias sc Bias expansion from the spherical collapse model see e g eq 68 from 2002 e shalo bias bias_tinker05 Bias calibrated with numerical simulations Ti e shalo bias bias_tinker10 The mass function describes the number of halos for a
33. ft The Fisher matrix is used as the covariance of a multi variate Gaussian Student t distribution g A mixture model is constructed by creating D copies of g Each copy is displaced from the ML point by a random uniform shift and its variance is stretched by random uniform factor 2 sinitial fisher eigen A mixture model is constructed in a similar way as the first case with the difference that the shift from the ML point is now performed along the major axes of the Fisher ellipsoid Note that if the Fisher matrix is diagonal the shift of each component only concerns one parameter 3 sinitial file The initial proposal is read from a file of mix mvdens format e g from a previous PMC run 4 sinitial random pos Mixture model components with random variance up to half the box size and random positions This case should only be used if the posterior is suspected to be multi modal or the calculation of the Fisher matrix fails In many cases a mixture of multi variate Gaussians as the proposal is the best choice For that set the degrees of freedom v parameter df to 1 For a posterior with heavy tails a Student t 3 Running CosmoPMC CO Fisher matrix needed config_fish exists maxlogP exists create config_fish config_max exists create config_max Figure 1 Flow chart for cosmo_pmc pl distribution might be more suited The degrees of freedom v can be chosen freely v
34. gist info cosmomc Lewis amp Bridle 2002 implements MCMC exploration of the cosmological parameter space Importance sampling on the other hand uses the identity n f fora o BD as 3 2 Installing CosmoPMC where q is any probability density function with support including the support of 7 A sample xn under q is then used to obtain the estimator T Xn q Xn N AA D fos 4 n 1 The function q is called the proposal or importance function the quantities w are the importance weights Population Monte Carlo PMC produces a sequence q of importance functions t 1 T to approximate the posterior 7 Details of this algorithm are discussed in Wraith et al 2009 The package CosmoPMC provides a C code for sampling and exploring the cosmological pa rameter space using Population Monte Carlo The code uses MPI to parallelize the calculation of the likelihood function There is very little overhead and on a massive cluster the reduc tion in wall clock time can be enormous Included in the package are post processing plotting and various other analysis scripts and programs It also provides a Markov Chain Monte Carlo sampler 1 2 This manual This manual describes the code CosmoPMC and can be obtained from www cosmopmc info CosmoPMC is the cosmology interface to the Population Monte Carlo PMC engine PMCLIB Documentation on the PMC ia can be found at the same url The cosmology module of War
35. given mass and redshift It is defined as dn _ povf v dv dinM M v dinM 19 20 4 Cosmology where v M z 6 z D z o M is a measure of the overdensity with c M being the rms matter fluctuation in a top hat window containing the mass M py Qmpc o is the mean density of matter at the present day The following mass functions are implemented via the flag smassfct e From the spherical eliptical collapse model vf v AJ meer d Jar 7 20 st p Da 0 75 iS st2 p 0 3 q 0 707 e From numerical simulations vf v f a 0 315 exp In o 0 6115 21 The dark matter halos have the density profile a 3 a pr ps EA nm 22 For slopes unequal to the value of 1 closed expressions for the Fourier transform of p do not exist and the code will be slower The concentration parameter is given by c M z p i 23 Mx 003 The parameters co and can be chosen freely in the halomodel parameter file halomodel par The log likelihood function is the sum of the contribution from the angular correlation function and the galaxy number density ngal ugs _ nmodel 2 S 1 l Lj Ngal where n The number of galaxies second term in eq config flag sngal fit type is estimated at the mean redshift of the sample can be included in the following way with the 21 4 Cosmology e ngal lin fit linear standard according to the a
36. gt setenv COSMOPMC path to CosmoPMC_v1 2 This command can be placed into the startup file e g cshrc for the C shell One can also add COSMOPMC bin to the PATH environment variable 3 Running CosMoPMC 3 Running CosmoPMC 3 1 Quick reference guide Examples To get familiar with CosmoPMC use the examples which are contained in the package Simply change to one of the subdirectories in COSMOPMC Demo MC Demo and proceed on to the point Run below User defined runs To run different likelihood combinations or your own data the following two steps are necessary to set up a CosmoPMC run 1 Data and parameter files Create new directory with newdir pmc sh When asked enter the likelihood data type More than one type can be chosen by adding the corresponding bit coded type id s Symbolic links to corresponding files in COSMOPMC data are set and parameter files from COSMOPMC par_files are copied to the new directory on request If necessary copy different or additional data and or parameter files to the present direc tory 2 Configuration file Create the PMC configuration file config pmc Examples for existing data modules can be found in COSMOPMC Demo MC Demo see also Sect 5 for details In some cases information about the galaxy redshift distribution s have to be provided and the corresponding files copied see COSMOPMC Demo for example files nofz Run Type gt COSMOPMC bin cosmo pmc pl n
37. hinning out the chain The results produced by cosmo_mcmc mean errors histograms covariance are based on this file The chains are ASCII files in the same format as the PMC sample files All weights are 1 and the second column contains the log likelihood only in chain all The parameter mean and confidence intervals are printed to the file mean The names of files containing the histograms and parameter covariances are the same as for PMC C 4 Diagnostics In general it is not straight forward to diagnose an MCM chain There exists tests but no formal proofs for convergence e g Gellman Rubin which in addition require very long or multiple chains We have not implemented such tests in the code However there are a few rather hand waving diagnostic tools to check the reliability of an MCMC run Firstly the acceptance rate 7 should be in the range between 15 and 25 A larger 7 most probably corresponds to a chain which stayed mainly in the high density region and strongly under sampled the lower density posterior regions In that case the error bars will be underesti mated A very small 7 means probably an under sampling of the posterior since only few points are accepted However this need not cause a bias for the parameters and errors if the chain has been run long enough C 5 Resuming an interrupted run Sometimes a MCMC run is interrupted before finishing or one wishes a previous run to be extended for example because
38. ile of parameter vectors sample which is resampled from the PMC simulation pmcsim taking into account the importance weights The resampled points all have unit weight Resampling is a post processing steop it is performed by calling the R script sample from pmcsimu R from cosmo pmc p1 this can also be done manually with any pmscim simulation 10 3 Running CosmoPMC Histograms iter_i chi_j iter_i chi_j_k One and two dimensional histograms are written at each iteration to the text files chi_j and chi_j_k respectively where j and k j lt k are parameter indices Those histograms can be used to create 1d and 2d marginals using the script plot contour2d pl The bin number is set by the config entry nbinhist In post processing use histograms sample to produce histograms from a PMC sample This can be useful if deduced parameters have been added to the sample Covariance iter i covar fin The parameter covariance and inverse covariance are printed to the files covar fin and respec tively covarinv fin The addition ded in the file name indicates the inclusion of deduced parameters The covariance matrices are in mvdens format see Sect A 3 Evidence evidence This file contains the Bayesian evidence as a function of iteration Before the first iteration the Laplace approximation using the Fisher matrix is printed to evidence fisher if the file fisher exists At each iteration i iter i evidence covarinv con
39. indicates a parameter covariance matrix for example to be used as Gaussian prior using the config file flag sprior In some cases the inverse covariance matrix is expected as in the case of the Fisher matrix 49 B Syntax of all commands B Syntax of all commands All following scripts are located in COSMOPMC bin All programs executables are located in COSMOPMC exec and linked from COSMOPMC bin after running make in COSMOPMC Usage add_deduced_halomodel OPTIONS PSIM PAR_1 PAR_2 OPTIONS c CONFIG Configuration file default config_pmc o OUTNAME Ouput pmcsim name default psim ded PSIM pmc simulation file pmcsim_iter PAR_i String for deduced parameter i If not given deduced parameters are read from the config file default e add par from prior pl 38 Usage add par from prior pl OPTIONS sample Adds a new random parameter to a PMC sample file drawn under a distribution OPTIONS o OUT Output sample file OUT default lt sample gt out p DIST Prior distribution DIST one of Flat default Gauss P ARG Prior arguments white spaced list if more than one For DIST Flat ARG min max defaut 1 1 Gauss ARG mean sigma C COL Column COL of new parameter default last s STR Name string STR of new parameter h This message e add par to mvdens pl 38 add par to mvdens pl MIX MVDENS OPTIONS Adds a parameter to a mix mvdens file e g Fisher matrix P
40. inued Baryonic acoustic oscillations BAO smethod string BAO method one of distance A distance d z datname string Data covariance file name mvdens format model file string Parameter file name e g cosmoDP par sspecial string Additional prior one of none recommended unity de_conservative Table 12 contains a list of input parameters which can be given as strings to the spar key in the config file 33 5 The configuration file Table 12 Input parameters Name Symbol Description Basic cosmology some of them given in cosmo par Omega_m omega_m Omega_b omega_b 198 omega b Omega de omega de Omega nu mass omega nu mass Omega c omega c Omega K omega K wG de w1 de h 108 N eff nu mass sigma 8 Delta 2 R n spec alpha s nt r Inr tau A_SZ Om Wm O Wb 100 x wp Ode Wde C mass Wy mass O We Ok WK Wo w1 h Neff y mass 08 2 AR Ms Matter density cold dark matter baryons Baryon density Dark energy density if w 1 corresponds to O4 Massive neutrino density so far only for CMB Cold dark matter Curvature density parameter Dark energy equation of state parameter constant term Dark energy equation of state parameter linear term see sde_param Dimensionless Hubble parameter Effective number of massive neutrinos so far only for CMB Power spectrum normalisation at small scales Power spectrum normali
41. ior parameters and for plotting see Table 12 for possible parameters min npar n ded doubles Parameter minima max npar n ded doubles Parameter maxima Data section ndata integer Number of data sets sdata string Data set 1 sdata string Data set ndata Prior section sprior string Prior file name for no prior nprior integer If sprior Number of parameters to which prior applies indprior npar x 0 1 If sprior Indicator flags for prior parameters 29 5 The configuration file Table 10 PMC part of the configuration file nsample integer Sample size per iteration niter integer Number of iterations fsfinal integer Sample size of final iteration is fsfinal x nsample niter integer Number of iterations importance runs nclipw integer The nclipw points with the largest weights are discarded Proposal section df double Degrees of freedom df 1 is Gaussian df 3 is typical Student t ncomp integer Number of components sdead_comp string One of bury revive sinitial string Proposal type one of fisher_rshift fisher_eigen file random_position fshift double Random shift from ML point U r r r fshift max min fvar double Random multiplier of Fisher matrix prop_ininame string File name of initial proposal fmin double Components have variance U fmin max min 2 Histogram section nbinhist integer Number of density histogram
42. its convergence is doubted The MCMC program allows to read in and extend a previous chain To that end rename the file chain acc into chain pre The proposal for the resumed run can but need not be calculated from the previous chain to be 61 C MCMC controlled in the config file see Sect C 2 In the config file the number of desired sample points has to be larger than the previous chain 62
43. ly sreduced none K10 q mag size q Basic cosmology file name cosmo par Redshift distribution master file Tomography correlations All correlations Only auto correlations ii Only cross correlations i j Reduced shear treatment Fitting formulae from Kilbinge 2010 Magnification bias coefficient q 2 a B 1 see Kilbinger 2010 eq 16 only if nofz_file only if sreduced K10 Table 6 SNla parameter file cosmo_SN par cosmo_file Theta2 M a ff Basic cosmology file name cosmo par Distance modulus parameters 26 5 The configuration file Table 7 HOD parameter file halomodel par cosmo_file nofz_file redshift module zmin zmax alpha_NFW cO beta NFW smassfct M min M1 MO sigma log M alpha shod Zmin Zmax C0 log M Qa berwein02_hexcl Basic cosmology file name cosmo par Redshift distribution master file Minimum redshift 1 if read from nzfile Maximum redshift 1 if read from nzfile Halo density profile slope a 1 for NFW Concentration parameter at z 0 Concentration parameter slope of mass depen dence en 1999 p 0 3 q 0 707 Minimal mass for central galaxies h Mo Scale mass for satellites h Mo Minimum mass for satellites A Mo Logarithmic dispersion for central galaxies Slope for satellite mass dependence HOD type Berlind amp Weinbei ith halo exclusion
44. nction of iteration t 3 3 2 Results PMC samples iter i pmcsim This file contains the sample points The first column is the unnormalised importance weight log the second column denotes the component number from which the corresponding point was sampled Note that the rai points with highest weights are not considered in subsequent calculations of moments perplexity evidence etc The next p columns are the p dimensional parameter vector Optionally ngea numbers of deduced parameters follow Proposals iter i proposal The proposal used for the importance sampling in iteration i is in mix mvdens format Sect A 3 The final proposal updated from the sample of the last iteration is proposal fin Mean and confidence intervals iter i mean This file contains mean and one dimensional left and right sided confidence levels c 1 A c l of p is calculated by integrating the one dimensional normalised marginal posterior starting from the mean in positive or negative direction until a density of p 2 i hed PMC outputs c l s for p 63 27 95 45 and 99 73 With the program c l one sided c l s can be obtained For post processing the program meanvar sample outputs the same information mean and c l from an existing PMC sample including possible deduced parameters Resampled PMC simulations iter niter 1 sample If cosmo_pmc p1 has been run with the option p the directory of the final iteration contains the f
45. newmodule to LLIBS 8 In exec Makefile Define the new rule LIBNEWMODULE cd CNEWMODULE amp amp MAKE The second line has to start with a TAB and not with spaces 9 Optional Extend 40 7 Using and modifying the code 7 3 Error passing system Most of the situations where an error occurs are intercepted by the program In such a case a variable err of type error is set via the macros err addError error type message err __LINE__ Or err addErrorVA error type formatted message err __LINE__ VA LIST printing the current line and function in the code a message and the error type negative integer With testErrorRet test error type message err LINE return value or testErrorRetVA test error type formatted message err __LINE__ return value VA LIST a conditional error is produced if the Boolean expression test is true The error is transported up the stack to the calling function with the macro forwardError err LINE return value RE Omit return value in case of a void function This can be used as diagnostics even for errors deep in the hierarchy of functions During the calculation of the importance weights any error is intercepted and the corresponding point does not contribute to the final sample See Sect 2 for more details Therefore in the routines which calculate the importance weights the following is used forwardErr
46. ning Use undocumented features of the code at your own risk 2 Installing CosmoPMC 2 1 Software requirements CosmoPMC has been developed on GNU Linux and Darwin FreeBSD systems and should run on those architectures Required are C compiler e g gcc icc e PMCLIB Sect e GSL http www gnu org software gsl version 1 15 or higher e FFTW t e MESSAGE PARSING INTERFACE MPI 1 calculations i for parallel 2 Installing CosmoPMC Optional csh for post processing auxiliary scripts recommended for post processing auxiliary scripts recommended python http www python org for running the configuration script R http www r project org post processing To produce 1D and 2D marginal posterior plots with scripts that come with CosmoPMC either yorick or R are required Necessary for CMB anisotropies support e Fortran compiler e g ifort After downloading unpack the gzipped tar archive gt tar xzf pmclib_x y tar gz This creates the PMcLIB root directory pmclib_x y PMcum uses waf instead of configure make to compile and build the software Change to that directory and type gt waf local configure See waf help for options The packages lua hd 5 and lapack are optionally linked with PMCLIB but are not necessary to run CosMoPMC Corresponding warnings of missing files can be ignored Instead of a local installation indicated by 10cal a install prefix c
47. notebook file COSMOPMC par_files COSEBIs cosebi nb adapted from can be run to obtain the coefficients for a given min and Pmax An output text file is created with the zeros r and amplitudes N The file name is cosebi_tplog_rN_ Nmax _ thmin _ thmax where Nmax is the number of COSEBI modes thmin and thmax are the minimum and maximum angular scale Vmin and Pmax respectively For a given 24 and max specified with the config entries th min and th max CosmoPMC reads the corresponding text file from a directory that is specified by path A sample of files with various scales are provided in COSMOPMC par files COSEBIs The COSEBIs are discrete numbers they are specified by an integer mode number n In both cases of pure E B mode separating statistics the function T is calculated from T4 according to Schneider et al 2002 The additional flag decomp eb fi leer decides between different filter functions 15 4 Cosmology decomp_eb_filter Reference Filter function typ 0 FK10_SN optimized Signal to noise 1 50 FK10_FoM_etal0 optimized Fisher matrix 1 10 FK10_FoM_eta50 optimized Fisher matrix 1 50 COSEBIs_log logarithmic The convergence power spectrum P with covariance matrix can be used with the flag slensdata pkappa 4 2 2 Third order vis et al 2004 Schneid We implement the aperture mass skewness Pen e j with the Gaussian filter eq There are two cases e sle
48. nsdata map3gauss The generalised skewness M gt A 01 2 03 Schneider eta e slensdata map3gauss diag 0 with three filter scales The diagonal skewness Mil 0 using a single aperture filter scale TODO equations 4 2 3 Second plus third order A joint data vector of second and third order observables can be used in CosmoPMC The covariance is interpreted as a joint block matrix with the second order and third order auto covariances on the diagonal and the cross correlation on the off diagonal blocks The possible scenarios are e slensdata map2gauss_map3gauss Gaussian aperture mass dispersion and generalised skewness e slensdata map2gauss_map3gauss_diag Gaussian aperture mass dispersion and diagonal skewness e slensdata decomp eb map3gauss Log COSEBIs and generalised aperture mass skewness The flag decomp_eb_filter has to be set to COSEBIs_log e slensdata decomp_eb_map3gauss_diag Log COSEBIs and diagonal aperture mass skewness The flag decomp_eb_filter has to be set to COSEBIs_log 16 4 Cosmology The first two cases use the same filter for second and third order and provide therefore a con sistent measure for both orders The last two cases use the optimal E B mode function known for second order 4 2 4 Covariance The covariance matrix is read from a file and the inverse is calculated in CosmoPMC The matrix has to be positive definite An Anderson Hartlap debiasing facto
49. ntries in the 24 5 The configuration file Table 4 Basic cosmology parameter file cosmo par Omega_m Omega_de w0_de w1_de h_100 Omega_b Omega_nu_mass N_eff_nu_mass normalization n_spec snonlinear stransfer sgrowth sde_param normmode amin Oy Oy mass Neff y mass T8 Ms linear pd96 smith03 smith03_de coyote10 bbks eisenhu eisenhu osc heath growth de jassal linder Amin Matter density cold dark matter baryons Dark energy density if w 1 corresponds to O4 Dark energy equation of state parameter constant term Dark energy equation of state parameter linear term see sde_param Dimensionless Hubble parameter Baryon density Massive neutrino density so far only for CMB Effective number of massive neutrinos so far only for CMB Power spectrum normalisation at small scales for normmode 0 see below Scalar power spectrum index Power spectrum prescription Linear power spectrum _ icosmo org Coyote Universe 977 fitting formula Numerical integration of differential equation for rec ommended Dark energy parameterisation w a wo wia l a w a wo w1 1l a Normalization mode 0 normalization c g Minimum scale factor 25 5 The configuration file Table 5 Weak lensing parameter file cosmo lens par cosmo_file nofz_file redshift module stomo tomo_all tomo_auto_only tomo_cross on
50. orNoReturn err LINE return value ParameterErrorVerb err param quiet ndim In case of an error the first line forwards the error but does not return from the current rou tine The second line prints the ndim dimensional parameter param to stderr if quiet 1 and purges the error To exit on an error use 41 7 Using and modifying the code quitOnError err LINE FILE y 3 This is usually done only from the main program More macros and functions regarding error communication and handling can be found in the files errorlist h errorlist c which are part of PMCLIB Acknowledgements CosmoPMC was developed with support of the CNRS ANR ECOSSTAT contract number ANR 05 BLAN 0283 04 ANR ECOSSTAT We thank P Astier F Beaujean J Guy L Fu A Halkola J Hartlap B Joachimi J Lui K Markovi P Schneider F Simpson R E Smith M Takada and I Tereno for discussions and insights which helped to develop the cosmology code We thank L Fu for helping with and testing the lensing E B mode decomposition and the third order lensing code We acknowledge R E Smith and J A Peacock for making public their code halofit which we implemented into CosmoPMC The people from the Coyote project are thanked for making their code public An adapted version of their emulator is part of this code The following people are thanked for providing data or simulations covariance computed using
51. ots a mark at position PAR e g best fit PAR is white space separated list use quotes or e g 0 3 0 8 Use PATH as CosmoPMC root directory default environment variable COSMOPMC Run quietly no verbose This message List of directories containing histogram files chi2_ _ Usage proposal mean pl OPTIONS OPTIONS d C e proposal var pl DIR CONFIG Directory DIR containing the sub directories iter with the proposal files default Configuration file CONFIG default DIR config pmc No plotting only creates gnu file x and y axes inverted x and y labels on top right Use PATH as CosmoPMC root directory default environment variable COSMOPMC This message Usage proposal var pl OPTIONS OPTIONS d DIR C P CONFIG PATH Directory DIR containing the sub directories iter with the proposal files default Configuration file CONFIG default DIR config pmc Use PATH as CosmoPMC root directory default environment 57 B Syntax of all commands variable COSMOPMC h This message remap sh 37 Usage remap sh OPTIONS OPTIONS c CONFIG Input PMC configuration file default config pmc i INPUT Input directory INPUT default s PMCSIM Sample PMC simulation file PMCSIM o OUTPUT Output directory OUTPUT default remap r REMAP Remap file REMAP default remap dat n NPAR Number of parameters NPAR default read f
52. r P Guy J Regnault N et al 2006 A amp A 447 31 19 Bardeen J M Bond J R Kaiser N amp Szalay A S 1986 ApJ 304 15 25 Beaujean E Bobeth C van Dyk D amp Wacker C 2012 Journal of High Energy Physics 8 Benjamin J Van Waerbeke L Heymans C Kilbinger M et al 2012 submitted to MNRAS also arXiv 1212 3327 43 Coupon J Kilbinger M McCracken H J et al 2012 A amp A 542 A5 20 43 Crittenden R G Natarajan P Pen U L amp Theuns T 2002 ApJ 568 20 14 Dempster A Laird N amp Rubin D 1977 J Royal Statist Society Series B 39 1 8 Hartlap J Simon P amp Schneider P 2007 A amp A 464 399 Heath D J 1977 MNRAS 179 351 25 44 References Kaiser N Kilbinger Kilbinger M Fu L Heymans C et al 2012 submitted to MNRAS also arXiv 1212 3338 Kilbinger M Wraith D Robert C P et al 2010 MNRAS 405 2381 Komatsu E Smith K M Dunkley J et al 2011 ApJS 192 18 19 Kowalski M Rubin D Aldering G et al 2008 ApJ 686 749 Kravtsov A V Berlind A A Wechsler R H et al 2004 ApJ 609 35 20 Lawrence E Heitmann K White M et al 2010 ApJ 713 1322 12 Lewis A amp Bridle S 2002 Physical Review D 66 103511 Liu J amp Chen R 1995 JASA 90 567 9 M nard B Kilbinger M amp Scranton R 2010 MNRAS 406 1815 1
53. r a supernova with index i is MB mpi M ats 1 Bei 14 where the quantities measured from the light curve fit are the rest frame B band magnitude m the shape or stretch parameter s and the color c The universal absolute SNIa magnitude is M the linear response parameters to stretch and color are and respectively The y7 function is Pc y a p 5logio 46m 2 2 2 o UBi Oyi el 15 where di is the luminosity distance and z the redshift of object i The contributions to the total error for object i are 1 The light curve parameter variance o up 0 W 0 with the 3P Simon private communication 18 4 Cosmology parameter vector 05 1 a B and the covariance W of the data vector mp is sj cj 2 The peculiar velocity uncertainty pv 5 1n10 vp cz 3 The intrinsic absolute magnitude scatter Cint The Hubble parameter is absorbed into the absolute magnitude which we define as M M The form of this log likelihood function has been used in Astier et al The following variations of the distance modulus and log likelihood are piste e schi2mode chi2 Thetal The x is extended to include photometric zero point un certainties se e schi2mode chi2 no sc The stretch and color parameters are ignored the distance modulus is ug Mz M e schi2mode chi2 betaz Instead of a single parameter the color response is redshift dependent 6 6 z
54. r is multiplied to the inverse Anderson 2003 Hartlap et al 2007 which is specified with the config entry corr_invcov This can also be used to rescale the covariance e g to take into account a different survey area Set this value to unity if no correction is desired The covariance is either taken to be constant and not dependent on cosmology In that case set scov_scaling to cov_const Or the approximated schemes from Eifler et al 2009 are adopted see j for the implementation In that scheme the shot noise term D is constant the mixed term M is modulated with Qn and cg using fitting formluae and the cosmic variance term V is proportional to the square of the shear correlation function This scheme is available for slensdata xipm The three covariance terms have to be read individually The entry covname which for scov scaling cov const corresponds to the total covariance matrix now specified the file name of cosmic variance term covname M the name of the mixed term and covname D the name of the shot noise term 4 2 5 Reduced shear The fact that not the shear y but the reduced shear g y 1 K is observable leads to corrections to the shear power spectrum of a few percent mainly on small scales These corrections are either ignored or modelled to first order according to 0 This is controlled in the lensing parameter file cosmo_lens par The parameter range where the reduced shear corrections are valid a
55. ray tracing through the Millennium Simulation S 2010 J Hartlap and S Hilbert for the Lensing et al 42 7 Using and modifying the code PMC references Population Monte Carlo Adaptive importance sampling in general mixture classes Estimation of cosmological parameters using adaptive impor tance sampling Bayesian model comparison in cosmology with Population Monte Carlo Schrabback et al 2010 Evidence of the accelerated expansion of the Universe from weak lensing tomography with COSMOS On the impact of intergalactic dust on cosmology with Type la supernovae TEASING a fast and accurate approximation for the low mul tipole likelihood of the cosmic microwave background tempera ture Galaxy clustering in the CFHTLS Wide the changing relation ship between galaxies and haloes since z 1 2x CFHTLenS Combined probe cosmological model comparison using 2D weak gravitational lensing CFHTLenS tomographic weak lensing Quantifying accurate redshift distributions CFHTLenS Testing the Laws of Gravity with Tomographic Weak Lensing and Redshift Space Distortions Forecasts of non Gaussian parameter spaces using Box Cox transformations Bayesian fit of exclusive b s decays the standard model operator basis 43 References References Anderson T W 2003 An introduction to multivariate statistical analysis 3rd edn Wiley Interscience 17 Astie
56. re 2 Flow chart of the MCMC implementation V Verbose h This message C MCMC We provide a Metropolis Hastings Monte Carlo Markov Chain sampler which is included in the CosmoPMC package This MCMC implementation has been used in Wraith et al 2009 in 59 C MCMC Table 13 MCMC section of the configuration file nchain integer Chain Length ncov integer Interval between updates of the proposal covariance fburnin double Burn in phase are the first ncovxncor points ndecorr double De correlation thinning out one in ndec points is kept in the final chain fudge double Proposal covariance is multiplied by fudge n par sinitial string Initial proposal type one of Fisher inv Fisher Fisher previous Hessian Hessian diag diag boxdiv double Diagonal of proposal covariance is max min boxdiv sstart string Starting point type one of ran fid min max nul fid npar doubles Starting parameter Histogram section nbinhist integer Number of density histogram bins only if sinitial diag Ponly if sstart fid C 1 MCMC configuration file C 2 Proposal and starting point The proposal for the Metropolis Hastings algorithm is a multi variate Gaussian distribution After choosing an initial proposal a new proposal can optionally be re calculated after a number of ncov accepted steps The covariance of this new proposal is the chain covariance from steps up to this point This proposal is then updated af
57. re indicated in Table 2 4 2 6 Angular scales The flag sformat describes the mapping of angular scales given in the data file and effective scales where the model predictions of the shear functions are evaluated 1 sformat angle center The effective scale is the same as given in the data file 0er 0 2 sformat angle mean The model is averaged over a range of scales 6 01 given in the data file 17 4 Cosmology a Parameter lower upper 1 On 0 22 0 35 2 Ode 0 33 1 03 3 w 1 6 0 6 4 Ob 0 005 0 085 5 h 0 61 1 11 6 og 0 65 0 93 Tin 0 86 1 16 3 sformat angle wlinear The model is the weighted average over a range of scales 60 01 where the weight is w 0 arcmin 4 sformat angle_wquadr The model is the weighted average over a range of scales 0 01 where the weight is w a arcmin a 0 arcmin The first mode angle center should be used for aperture mass shear rms and ring statistics since those quantities are not binned but instead are integrals up to some angular scale 0 For the correlation functions in particular for wide angular bins one of the last three modes is preferred The quadratic weighting angle wquadr corresponds to a weighting of i i by the number of pairs This mode was used in the COSMOS analysis 4 3 SNIa The standard distance modulus schi2mode chi2_simple fo
58. rom remap file d N DED Number of deduced parameters N DED default 0 h This message sample2fixpar pl 39 Usage sample2fixpar pl SAMPLE IN COL MIN MAX SAMPLE IN Input sample PMC simulation or MCM chain COL Column number of fixed parameter Note that par Zi is in column i 2 MIN MAX Minimum and maximum values for fixed parameter tab2tex pl 36 Usage tab2tex pl OPTIONS file OPTIONS a Produce tex array not tex table b Bare output no table array header s STRETCH Set arraystretch to STRETCH m Add around entries tex inline math mode 1 MODE Print vertical lines between rows according to MODE a all lines default n no lines h header lines L MODE Print horizontal lines between columns according to Mode a all lines default n no lines h This message test suite cosmo pmc pl Usage test suite cosmo pmc pl OPTIONS OPTIONS r Do PMC test runs R Only do PMC test runs n NCPU Run PMC in parallel on NCPU cpus default 1 C Include CMB tests P PATH Use PATH as CosmoPMC root directory default environment variable COSMOPMC sS Short without time taking PMC runs e g Lensing COSMOS S10 k Keep temporary files X Clean previous run and exit 58 C MCMC de no chain fin chain acc chain pre exists exists exists yes yes initial proposal V read chain fin read chain acc remove burn in decorrelate iste Figu
59. tains the Laplace approx imation of the evidence from the inverse covariance matrix of the sample iter i pmcsim 3 3 3 Deduced parameters Deduced parameters can be part of a PMC simulation These parameters are not sampling pa rameters but they are deduced from the main parameters For example if Qu and Qa are sampling parameters of a non flat model the curvature Og Om O4 can be a deduced param eter In most cases deduced parameters are ignored while running CosmoPMC They are usually added to the PMC simulation after the sampling for example using a script In the case of galaxy clustering adds deduced parameters which depend on the sampling parameters but also on the underlying cosmology and halo model A PMC simulation with deduced parameters added can be used as input to hi stograms sample to create the histogram files now including the deduced parameters These can then in turn be read by and to produce 1d and 2d marginals including the deduced pa rameters Alternatively the PMC simulation with added parameters can be resampled using from which plots can be created by p 11 4 Cosmology 3 3 4 Other files Maximum posterior parameter max_logP Fisher matrix fisher Log files log max post log fish log pmc mc each produce their corresponding log file 4 Cosmology The cosmology part of CosMoPMC is essentially the same as the stand alone package NICAEA This excludes the external program
60. ted in COSMOPMC bin 35 6 Post processing and auxiliary programs 6 1 Plotting and nice printing 6 1 1 Posterior marginal plots Marginals in 1d and 2d can be plotted in two ways using 1 plot_contour2d pl or 2 plot confidence R The first is a perl script calling yorick for plotting the second is an R script The second option produces nicer plots in general in particular smoothing workes better without producing over smoothed contours Further filled contours with more than one data set are only possible with the R option yorick can only combine several plots with empty contours The computation time of the R script is however much longer plot contour2d pl creates 1d and 2d marginals of the posterior from the histogram files chi2_j and chi2 j k To smooth 1d and 2d posteriors with a Gaussian use plot_contour2d pl n g FACTOR The width of the Gaussian is equal to the box size divided by FACTOR It is rec ommended to test the smoothing width FACTOR by setting it to a negative number which causes both smoothed and unsmoothed curves being plotted This can reveal cases of over smoothing If contours have very different width in different dimension the addition option C uses the PMC sample covariance from the file covar ded fin as the covari ance for the Gaussian For the final plot replace FACTOR with FACTOR to remove the unsmoothed curves Remove the option n to add color shades to the 2d contours The file log
61. ter each ncov accepted steps using all previous accepted points There are several options for the initial proposal 1 sinitial diag A diagonal covariance with width being a fraction of the box size 2 sinitial Fisher The Hessian at a given point in parameter space If this point is the maximum likelihood point the Hessian corresponds to the Fisher matrix 3 sinitial Fisher_inv The inverse Hessian Fisher matrix e g the covariance from a previous chain This can be useful for ill conditioned matrices which are difficult to invert numerically 4 sinitial previous A proposal read from a file e g from a previous MCMC run The starting point is either chosen randomly or specified in the config file The second case might be convenient if the prior volume is very large and a very long burn in phase is to be 60 C MCMC avoided For example the ML point or best fit value from a previous experiment can be chosen C 3 Output files The MCMC output files have the same format as their PMC counterparts see Sect 3 3 2 A complete run of cosmo_mcmc produces three files containing the points of the Markov chain 1 chain all containing all accepted and rejected sample points This is the only chain file will not be read or used in subsequent calls of cosmo mcmc 2 chain acc containing the accepted points 3 chain fin containing the accepted points after removal of the burn in phase and after de correlating t
62. the mix_mvdens file PROP e newdir_pmc sh 5 40 Usage newdir_pmc sh DIR Directory DIR default read on input is created Links are set to data files in COSMOPMC data Parameter files are copied on request from COSMOPMC par_files Usage plot_confidence R options Options h help Show this help message and exit N NGRID Ngrid NGRID Number of grid points for smoothing kde2d default 100 Use lt 30 for fast but dirty plots g GSMOOTH gsmooth GSMOOTH Smoothing kernel width with respect to box size default 30 In case of more 55 B Syntax of all commands than one sample use list separated with _ for more than value S solid All contours with solid lines w WIDTH width WIDTH Line width default 1 k with_keys Add key to plots K KEYSTRING keystring KEYSTRING Key strings separate items with _ L no_key_line Do not add a line to the keys in the legend c CONFIG config CONFIG Config file default config_pmc t TITLE title TITLE Title string for each panel default empty i INDEX_I index_i INDEX_I Only create plots with i th parameter on x axis j INDEX_J index_j INDEX_J Only create plots with j th parameter on y axis s SIGMA sigma SIGMA Plot SIGMA confidence levels default 3 F COLOR_SCHEME color_scheme COLOR_SCHEME Color scheme 0 1 default 0 plot_contour2d pl Usage plot contour2d pl OPTIONS
63. ty 13 4 Cosmology parameters h has to be set at each sample point Alternatively the physical density parameters can be sampled where h is set internally to match the CMB peak 4 1 3 Likelihood Each cosmological probe has its own log likelihood function The log likelihood function is called from a wrapping routine which is the interface to the PMC sampler In general within this function the model vector is computed using the corresponding cosmology routine The exception are the WMAP modules where the C s are calculated using camb and handed over to the log likelihood function as input 4 2 Cosmic shear CosmoPMC implements second and third order weak lensing observables 4 2 1 Second order The basic second order quantities in real space for weak gravitational lensing are the two point correlation functions 2PCF e g I e 1 OO E amp O 2 d P Jo 4 0 6 7T Jo Data corresponding to both functions slensdata xipm as well as only one of them xip xim can be used The aperture mass dispersion Schneider MO d CP O07 60 7 T Jo is supported for two filter functions Ug u 9 0 0 9 1 polynomial map2poly u x me x E e Hd x 8 1 x 2 Gaussian map2gauss u x l e 7 9 271 2 The top hat shear dispersion Kaiser 1992 1 f 4J 0 2 1 0 d PL 10 Pisa 7 S ero 777 10 is used with slensdata gsar 1
64. utputs HOD derived quantities OPTIONS o OUT Output file name t TYPE Output type TYPE in wtheta wp xi xihalo deltaSig nofm halo pk default wtheta nbins Number of bins 53 B Syntax of all commands range Range linear scale min max Z Z Used fixed redshift Z no w theta output Mhalo log10M log10 Halo mass for deltaSig and xihalo in M_sol h c CONFIG PMC config file to calculate chi 2 h This message M_min M1 and MO are in units of M_ sol h e histograms_sample 11 Usage histograms_sample OPTIONS sample OPTIONS c CONFIG Configuration file default config_pmc 1 Only 1d histograms 2 Only 2d histograms sample PMC sample file h This message e importance sample 37 Usage importance_sample OPTIONS INSAMPLE Performs an importance run on a PMC sample Run in parallel with MPI use mpirun OPTIONS c CONFIG Configuration file default config_pmc o OUTSAMPLE Output sample name default insample out q Quiet mode h This message INSAMPLE Input sample name e max post 2 Usage max post OPTIONS OPTIONS c CONFIG Configuration file default config max m clal n Maximum search method a amoeba default c cg n none print posterior for fiducial parameter and exit t Test maximum at the end s SEED Use SEED for random number generator If SEED 1 default the current time is used as seed p Prints the maximum posterior model to the file model maxlog q
65. x D Zo jonben Er 2 c Zmin gt Zmax 4 b C ymmk 2 2 2 c Smin Zmar Hot All redshift distributions are internally normalised as Zmax f dzn z 1 34 Zmin 4 8 CMB and the power spectrum normalisation parameter The power spectrum normalisation parameter taken as input for CAMB is A2 which is the ampli tude of curvature perturbations at the pivot scale k 0 002 Mpc For lower redshift probes such as lensing or HOD the normalisation is described by cs the rms fluctuation of matter in spheres of 8 Mpc h To combine those probes in a PMC run AR has to be an input parameter and og a deduced parameter CMB has to come first in the list of data sets so that cAMB can calculate cs which in turn is handed over to the lensing likelihood 4 9 Parameter files Tables 4 6 list the contents of the parameter files for basic cosmology lensing SNIa and HOD Proto types can be found in COSMOPMC par files These files specify the default values of parameters and flags These default values are over written if any of those parameter is used for Monte Carlo sampling 5 The configuration file The programs max post go fishing cosmo_pmc and cosmo_mcmc read a configuration file on startup Each configuration file consist of two parts The first basic part is common to all four config file types Table 9 It consists of 1 the parameter section 2 the data section and 3 the prior section The data specific e
66. zation at large scales CMB Scalar power spectrum index Running spectral index so far only for CMB Tensor power spectrum index Tensor to scalar ratio Optical depth for reionisation SZ power spectrum amplitude 34 6 Post processing and auxiliary programs Table 12 Input parameters continued SNIa specific some of them given in cosmo_SN par M M logigh7 Universal SNIa magnitude alpha a Linear response factor to stretch beta B Linear response factor to color betaz f Redshift dependent linear response to color beta d Ba Linear response to the color component due to intergalactic dust Galaxy clustering specific some of them given in halomodel par M min Mmgin Minimum halo mass for central galaxies Moh log18 M min logio Mmin Moh M_1 Mi Scale mass for satellite galaxies Moh log10_M_1 log p M1 Moh M Mo Minimum halo mass for satellite galaxies Moh log10 MO 10810 Mo Moh sigma_log_M log M Dispersion for central galaxies alpha halo Uh Slope of satellite occupation distribution M_halo_av Mp Average halo mass Mah logi18 M halo av logio Mn Moh b halo av by Average halo bias N gal av Ng Average galaxy number per halo fr_sat fs Fraction of satellite galaxies to total ngal den Ng Comoving galaxy number density Mpc log10ngal_den logio Ng 6 Post processing and auxiliary programs All scripts described in this section are loca
67. zi lactic dust absorption is taken into account in the distance modulus The covariance matrix W of the data vector m7 sj ci depends on the parameters e and f In Bi a Bayesian framework this leads to an additional term 5 log det W2 in the log likelihood func tion Taking into account this parameter dependent term leads however to a biased maximum likelihood estimator in particular for and 64 Therefore it is recommended to not include this term Use the flag add_logdetCov 0 1 in the configuration file to disable enable this term 4 4 CMB anisotropies The full CMB anisotropies are handled externally The C s are calculated by calling camb the WMAP likelihood function 3 4_ 5t and 7 year is computed using the WMAP public code Dunkley et al 009 The maximum up to which the C s are calculated and used in the likelihood can ermined in the configuration file An Fmax 2000 is recommended for high precision calculations The power spectrum from the Sunyaev Zel dovich SZ effect can be added to the C s multi plied with an amplitude A as free parameter The predicted SZ power spectrum is taken from l 2 This model has been used in the 3 5 and 7 year analyses of the WMAP data J Guy private communication ETE camb info 19 4 Cosmology Alternatively the WMAP distance priors can be employed 4 5 Galaxy clustering 4 5 1 Halomodel and HOD The theor

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