NumCosmo Overview and examples - 200.145.112.249200.145.112.249/webcast/files/Sandro.pdf · NumCosmo Basic concepts Ncmpre x { stands forNumCosmoMathwhich contains the general codes

NumCosmo – Overview and examples

Sandro D. P. Vitenti1,2

Mariana Penna-Lima1,3

Cyrille Doux3

1CBPF - Centro Brasileiro de Pesquisas Fısicas

2 UEL - Universidade Estadual de Londrina

3AstroParticule et Cosmologie (APC) – Universite Paris Diderot

XIII Workshop on New Physics in Space - Sao Paulo – November 30,2016

Sandro D. P. Vitenti1,2 Mariana Penna-Lima1,3 Cyrille Doux3 NumCosmo – Overview and examples 1/29

NumCosmo – http://ascl.net/1408.013

Free software C library (GPL license);

Main goals:

Calculate multiple cosmological observables efficiently.Provide a set of tools to perform: cosmological (astrophysical)calculations (multiple probes) and data analyses.Collaborative tool: each object is developed separately,communicating through predefined interfaces.

GObject: object-oriented programming in C, reference basedgarbage collection, inheritance, etc.

GObjectIntrospection: automatic bindings between C, Python,Perl, Java, and others.

Serialization framework: saving/loading, sending throughnetwork.


NumCosmo

Basic concepts

Ncm prefix – stands for NumCosmoMath which contains thegeneral codes not specific to cosmology, e.g., NcmSpline2d,ncm sf sbessel.

Nc prefix – represents the cosmological codes written on topof Ncm.

NcmModel – is a GObject and an abstract class whichbasically defines a set of parameters.

NcmMSet – is a collection of models which can containmodels of the same and/or different families.

NcmData – is an abstract class which encapsulates functionssuch as resample, −2 ln L, bootstrap resample...

NcmDataSet – is a collection of NcmData (combined probes).


Example 1: Type Ia Supernova – Update all

NcmFit

NcmFitMCMC

NcmMSet

Model SetNcmLikelihood

NcmModel:pkey=0

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov

−2 ln(LSNIa) = ∆ ~mTC−1SNIa(α, β)∆ ~m,

∆mi = mBi−5 log10(DL(zheli , zcmb

i ))+αXi−βCi−Mhi +5 log10(c/H0)−25,

α and β are related to the stretch-luminosity and colour-luminosity,respectively, and Mhi are absolute magnitudes.



NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=0

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov







NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=1

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov







NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=1

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov







NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=1

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov







NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=1

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov






Example 2: Type Ia Supernova – Update cosmo only

NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=0

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov






NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=0

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov






NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov






NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov






NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov






NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=1

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcSNIADistCov

NcDistance

NcmDataSet

NcmDataGaussCov

NcDataSNIACov





Example 3: CMB TT only

NcmFit

NcmFitMCMC

NcmMSet


NcmModel:pkey=0

NcHIPrim

NcHIPrimPowerLaw

NcmModel:pkey=0

NcHIReion

NcHIReionCAMB

NcmModel:pkey=0

NcHICosmo

NcHICosmoDE

NcHICosmoDEXcdm

NcmModel:pkey=0

NcPlanckFI

NcPlanckFICorTT

NcHIPertBoltzmann

NcHIPertBoltzmannCBE

NcmDataSet

NcDataPlanckLKL

NcHIPrimPowerLaw: simple power law for the primordialpower spectrum.

NcHIReionCAMB: CAMB’s reionization scheme.

NcPlanckFICorTT: Planck foreground and instrumentsparameters.


Software Quality Assurance

Unit testing: Test file for each object to assure the correctbehavior of each part of NumCosmo.

NcmC (fundamental physical and mathematical constants):CODATA (2014), IAU (2015), NIST

NcmSplineFunc – Automatic determination of the splineknots, given a precision.

Autotools – Widely used build-system, support severaloperational systems and architectures, compilers, easilyintegrated with Linux distributions and MacPorts.Not reinventing the wheel – Uses well known and testedlibraries as back-end:

GLib (object system/portability);Cuba (multidimensional integration);NLopt (non-linear optimization);GSL (several scientific algorithms);cfitsio (fits file manipulation);Sundials (ODE integrator);


CMB status:

CLASS and the Planck’s likelihood (clik) are integrated in thelibrary building system.

Abstract Planck foreground and instrument models are defined byNcPlanckFI, NcPlanckFICorTT and NcPlanckFICorTTTEEE.

CLASS interface is described in the CLASS Backend object NcCBE.

Interface to Cl through NcHIPertBoltzmann and Planck’s likelihoodthrough NcDataPlanckLKL.

Currently fully working implementation of NcHIPertBoltzmann:CLASS based NcHIPertBoltzmannCBE.

Standard NumCosmo implementation NcHIPertBoltzmannStd, workin progress, support for seeds and GI implementation.


CMB status:

Alternative primordial power spectra interface:

NcHIPrimAtan: implements an inverse tangent modification ofthe power spectrum.NcHIPrimExpc: implements exponential cutoff powerspectrum.NcHIPrimBPL: implements broken power law power spectrum.

Thermodynamics through CLASS NcRecombCBE,NumCosmo implementation NcRecombSeager (provides asingle pass integration for the whole system, same equationsas RECFAST).

Linear matter power spectrum NcPowspecML, interface toCLASS Matter Transfer Function NcPowspecMLCBE

Matter power spectrum non-linear correctionsNcPowspecMNL through Halofit NcPowspecMNLHalofit(Halo model in the next release).


Observational Probes

1 SNeIa: NcDataSNIACov:NcmDataGaussCov(NcDataDistMu:NcmDataGaussDiag for diagonal errors)

2 Galaxy Cluster: NcDataClusterNCount:NcmData,NcDataClusterPoisson:NcmDataPoisson,NcDataClusterPseudoCounts:NcmData

3 BAO: NcDataBaoRDV:NcmDataGaussDiag,NcDataBaoEmpiricalFit:NcmDataDist1d . . .

4 H(z): NcDataHubble:NcmDataGaussDiag

5 CMB distance priors: NcDataCMBDistPriors:NcmDataGauss

6 CMB: NcDataPlanckLKL:NcmData7 Cross-Correlation: NcDataXcor:NcmDataGaussCov

Galaxies, QSO: NcXcorLimberGal:NcXcorLimberCMB lensing: NcXcorLimberLensing:NcXcorLimber


Matter power spectrum products

NcPowspecMLCBE: CLASS linear matter power spectrum;NcPowspecMLTransfer + NcTransferFuncEH +NcGrowthFunc: Eisenstein and Hu (EH);

10-5 10-4 10-3 10-2 10-1 100 101 102

k [Mpc−1]

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

k 3P(k, z)/(2π2)

CLASS z= 0. 00

EH z= 0. 00

CLASS z= 1. 00

EH z= 1. 00

CLASS z= 2. 00

EH z= 2. 00



CLASS vs EH linear matter power spectrum;

10-5 10-4 10-3 10-2 10-1 100 101 102

k [Mpc−1]

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1 |1−PEH(k, z)/PCLASS(k, z)|

CLASS EH cmp z= 0. 00







CLASS + NcPowspecMNLHaloFit;

10-5 10-4 10-3 10-2 10-1 100 101 102

k [Mpc−1]

10-15

10-13

10-11

10-9

10-7

10-5

10-3

10-1

101

103

105

k 3P(k, z)/(2π2)

CLASS z= 0. 00

CLASS+HaloFit z= 0. 00

CLASS z= 5. 00


CLASS z= 10. 00




NcmPowspecFilter + CLASS: σ8 = 0.846;NcmPowspecFilter + EH: σ8 = 0.853;NcmPowspecFilter uses NcmFftlog to compute∫∞

0 jl(kR)2P(k , z)k2dk efficiently.

10-4 10-3 10-2 10-1 100 101 102 103 104 105

R [Mpc]

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102 Tophat filtered σ(R)

σ CLASS z= 0. 00

σ EH z= 0. 00

σ CLASS z= 1. 00

σ EH z= 1. 00

σ CLASS z= 2. 00

σ EH z= 2. 00


Data Analysis Tools:

Best-fit finder: NcmFitGSLLS, NcmFitGSLMM,NcmFitGSLMMS, NcmFitLevmar and NcmFitNLOpt.

NcmFit – Fisher Matrix [ncm fit numdiff m2lnL hessian()].

NcmFitMC – Monte Carlo analysis (resample/bootstrap froma probability density function).

NcmFitMCBS – Monte Carlo and bootstrap analysis.Frequentist Analysis

NcmLHRatio1d – Profile likelihood for one-dimensionalparameter analysis.NcmLHRatio2d – Profile likelihood for bidimensionalparameter analysis.

Bayesian AnalysisNcmFitMCMC – Markov Chain Monte Carlo(Metropolis-Hastings): NcmMSetTransKern.NcmFitESMCMC – Ensemble sampler Markov Chain MonteCarlo: NcmFitESMCMCWalker.NcmABC – Approximate Bayesian Computation (ABC).


ESMCMC example

gi.require_version(’NumCosmo’, ’1.0’)gi.require_version(’NumCosmoMath’, ’1.0’)

from gi.repository import NumCosmo as Ncfrom gi.repository import NumCosmoMath as Ncm

# Main library initializationNcm.cfg_init ()Ncm.func_eval_set_max_threads (40)

# New CLASS backend precision objectcbe_prec = Nc.CBEPrecision.new ()cbe = Nc.CBE.prec_new (cbe_prec)

# New Boltzmann CLASS backend objectbz_cbe = Nc.HIPertBoltzmannCBE.full_new (cbe)

# Loading MSet and setting Omega_k = 0.mset = Ncm.MSet.load ("model_set_file.mset", ser)cosmo = mset.peek (Nc.HICosmo.id ())

cosmo.param_set_by_name ("Omegak", 0.0)


ESMCMC example

# New Dataset objectdset = Ncm.Dataset.new ()

lkls = [’DataV2/commander_rc2_v1.1_l2_29_B.clik’,’DataV2/plik_dx11dr2_HM_v18_TT.clik’]

for lkl in lkls:plik = Nc.DataPlanckLKL.full_new (lkl, bz_cbe)dset.append_data (plik)

# Load SNIa data.fitsname = "jla_snls3_sdss_sys_stat_cmpl.fits"data_snia = Nc.DataSNIACov.new (False)data_snia.load (fitsname)

dset.append_data (data_snia)

# Creating a Likelihood from the Dataset.lh = Ncm.Likelihood (dataset = dset)Nc.PlanckFICorTT.add_all_default_priors (lh)


ESMCMC example

# Creating a Fit objectfit = Ncm.Fit.new (Ncm.FitType.NLOPT, algorithm,

lh, mset, Ncm.FitGradType.NUMDIFF_CENTRAL)

# Initial samplerinit_sampler = Ncm.MSetTransKernGauss.new (0)init_sampler.set_mset (mset)init_sampler.set_prior_from_mset ()init_sampler.set_cov_from_rescale (0.1)

# The ensemble sampler walker typenwalkers = 100stretch = Ncm.FitESMCMCWalkerStretch.new (nwalkers,

mset.fparams_len ())stretch.set_box_mset (mset)stretch.set_scale (1.2)

# ESMCMC objectesmcmc = Ncm.FitESMCMC.new (fit, nwalkers, init_sampler,

stretch, Ncm.FitRunMsgs.FULL)esmcmc.set_nthreads (40)


ESMCMC example

# Setting outputesmcmc.set_data_file ("output_chains.fits")

# Running ...esmcmc.start_run ()esmcmc.run_lre (1, 1.0e-3)esmcmc.end_run ()

esmcmc.mean_covar ()fit.log_covar ()


ESMCMC example

Snapshot of a ESMCMC run using Planck TT data and a modifiedprimordial power spectrum.

Object: NcmFitESMCMC,

Walker: NcmFitESMCMCWalkerStretch,

Output: a NcmMSetCatalog object serialized in a fits file.

# Elapsed time: 01 days, 21:57:54.8427910# degrees of freedom [002523]# m2lnL = 796.42768789029# Fit parameters:# 69.8108499055435 0.115986617072077 0.0226852361053738 3.23602243188527# NcmMSetCatalog: Current mean: 854.15 69.938 0.11588 0.022799# NcmMSetCatalog: Current msd: 0.54997 0.0049737 1.2071e-05 1.8152e-06# NcmMSetCatalog: Current sd: 511.73 0.90615 0.0020219 0.00036516# NcmMSetCatalog: Current var: 2.6187e+05 0.82111 4.0882e-06 1.3334e-07# NcmMSetCatalog: Current tau: 1 26.084 30.858 21.395# NcmMSetCatalog: Current skfac: 1.0029 1.1474 1.1388 1.0652# NcmMSetCatalog: Maximal Shrink factor = 1.25590891624521# NcmFitESMCMC:acceptance ratio 62.5521%, offboard ratio 0.0000%.# Task:NcmFitESMCMC, completed: 865800 of 10371900, elapsed time: 1 day, 21:57:54.8162# Task:NcmFitESMCMC, mean time: 00:00:00.1911 +/- 00:00:00.0036# Task:NcmFitESMCMC, time left: 21 days, 00:40:40.0570 +/- 09:22:42.2361# Task:NcmFitESMCMC, estimated to end at: Tue May 17 2016, 07:18:32 +/- 09:22:42.2361


Data Analysis Tools:

mcat analyze:

Compute 1− 3σconfidence intervals forthe best-fit, mean, modeor median of marginaldistributions.Compute covariancematrix.

de mc plot.py: make plot ofhistograms of marginalparameter distributions andthe bidimensional contours.

0.5 0.6 0.7 0.8 0.9 1.0 1.1αSZ

0.2

0.4

0.6

0.8

1−b SZ

0.6

83

0.9

55

0.99

3

de mc corner plot.py: makecorner plot.

0.4

0.6

0.8

1.0

1−b SZ

0.0

0.3

0.6

0.9

σSZ

0.0

0.2

0.4

0.6

σL

0.6

0.9

1.2

1.5

αSZ

0.6

0.0

0.6

r

0.4

0.6

0.8

1.0

1− bSZ

0.0

0.3

0.6

0.9

σSZ

0.0

0.2

0.4

0.6

σL

0.6

0.0

0.6

r


Example: Chains evolution

-12

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 500 1000 1500 2000 2500

H0OcObAsnsc1c2c3zre


Example: Parameter distribution evolution

0 500 1000 1500 2000 2500

66

67

68

69

70

71

72

73

74

10-3

10-2

10-1

100


Conclusions

NumCosmo is a suitable framework for collaborations /collaborative work.Further NumCosmo objects can be directly implemented inPython or in any other bound language.github.com/NumCosmo/NumCosmowww.nongnu.org/numcosmo/

References :M. Penna-Lima et al. [JCAP05(2014)039], arXiv:1312.4430 – Cluster number counts, spline 2D, ProfileLikelikood, MC (resample), Fisher matrix.E. Ishida, S. Vitenti, M. Penna-Lima, et al. [Astronomy and Computing, Vol. 13 (2015) p. 1-11],arXiv:1504.06129 – Approximate Bayesian Computation.S. Vitenti and M. Penna-Lima [JCAP09(2015)045], arxiv:1505.01883 – SNeIa, BAO, H(z), spline 1D, MC,resample - bootstrap, ESMCMC.M. Penna-Lima, J. Bartlett, et al. (submitted to A&A): Cluster pseudo counts, ESMCMC.P. Peter, A. Valentini, S. Vitenti: Non-equilibrium primordial powerspectrum in prep. – NcHIPrimAtan,NcDataPlanckLKL, NcPlanckFICorTT, ESMCMC.S. Vitenti, MPL, C. Doux: NumCosmo in prep.C. Doux, MPL, S. Vitenti: NcXCor in prep.

Obrigado!


Conclusions

NumCosmo is a suitable framework for collaborations /collaborative work.Further NumCosmo objects can be directly implemented inPython or in any other bound language.github.com/NumCosmo/NumCosmowww.nongnu.org/numcosmo/

References :M. Penna-Lima et al. [JCAP05(2014)039], arXiv:1312.4430 – Cluster number counts, spline 2D, ProfileLikelikood, MC (resample), Fisher matrix.E. Ishida, S. Vitenti, M. Penna-Lima, et al. [Astronomy and Computing, Vol. 13 (2015) p. 1-11],arXiv:1504.06129 – Approximate Bayesian Computation.S. Vitenti and M. Penna-Lima [JCAP09(2015)045], arxiv:1505.01883 – SNeIa, BAO, H(z), spline 1D, MC,resample - bootstrap, ESMCMC.M. Penna-Lima, J. Bartlett, et al. (submitted to A&A): Cluster pseudo counts, ESMCMC.P. Peter, A. Valentini, S. Vitenti: Non-equilibrium primordial powerspectrum in prep. – NcHIPrimAtan,NcDataPlanckLKL, NcPlanckFICorTT, ESMCMC.S. Vitenti, MPL, C. Doux: NumCosmo in prep.C. Doux, MPL, S. Vitenti: NcXCor in prep.

Obrigado!Sandro D. P. Vitenti1,2 Mariana Penna-Lima1,3 Cyrille Doux3 NumCosmo – Overview and examples 24/29

COSMOMC + (CAMB or PICO)

.ini files

Program driven: one tool (mainly MCMC)Parameters: range, scale, fixed

Parameter priors: Gaussian onlyRun options: likelihoods, sampler, models

cosmomc

Creates Ini object from .ini filesBuild the likelihood using flags in Ini

Calls TCosmologyCalculator(CAMB Calculator or PICO Calculator)

CAMBPICO

Calculates the cosmological observ-able necessary for the likelihoods

cosmomcCalculates the next proposal points and calls

the Calculator to evaluate the likelihoodaccept/reject and repeat . . .

getdist analysis of the chains and post processing

many text files, emulatesa script language

Ini object holdsall definitions

All observables must bedefined in Ini matching

CAMB/PICO parameters

Creates text filescontaining the chains,

one for each chain

Reads from text files,separated from cos-

momc, cannot performcosmological calculations


NumCosmo

Scripts orprograms

Library driven: many toolsCreates/Loads: models, priors, data

Extensible: new priors, samplers, modelsAvoids flags: one script/program per task

NumCosmoNcmLikelihood

NcmMSet

Handles each object separatelyNew likelihoods without modifying sources

Can use different models and back-endsModels grouped together in NcmMSet

NumCosmoNcmDataset

NcmLikelihoodNcmFit?

Different analysis tools:NcmFit: best-fit and Fisher matrix

NcmLHRatio1/2d: frequentist confidence regionsNcmFitMC: MC resampling or bootstrapingNcmFitMCMC: MCMC NcmMSetTransKern

NcmFitESMCMC: Ensemble sam-pler MCMC NcmFitESMCMCWalker

NumCosmoNcmMSetCatalog

Handles the output of all samplersChains’ checkpoint

MCMC diagnostics: Shrink factor, au-tocorrelation time, visual diagnostics

mcat analyzeanalysis of the chains and post processing

can perform cosmological calcula-tions for each point in the chains

programming lan-guage, easier debugging

and error checking

All objects com-municate throughcommon interfaces

Independent tools,common output

Sync to a single fits(cfitsio) file containingall chains/checkpoints

Fast to read binaryfiles, part of NumCosmo


Theory vs Observations

Likelihood: Probability of measuring “Data” given a “Model”

L(Data|Model)

Alternatives: Approximated Bayesian computation, simulations

Theory (Models)

Modeling observablesModeling uncertainties

Observations (Data)

Data pointsUncertainties

Probability Distribution (Statistics)

Hypothesis about the uncertainties ⇒ Likelihood form (Bayesiananalysis, posterior, priors)

Data Analysis

Methodology to extract information from the Data: parameterconstraints, model testing, model comparison, posterior exploration




L(Data|Model)


Theory (Models)


Observations (Data)




Data Analysis





L(Data|Model)


Theory (Models)


Observations (Data)




Data Analysis





L(Data|Model)


Theory (Models)


Observations (Data)




Data Analysis





L(Data|Model)


Theory (Models)


Observations (Data)




Data Analysis



Cross-correlations with NumCosmo I

For all observables that are linear functionals of the density field

a(θ) =

∫dzW (z)δ

(θχ(z), z

)(1)

Cross-spectra in the Limber approximation

C ab` =

∫ z∗

0dz

H(z)

c

1

χ(z)2∆a`(z)∆b

` (z)P

(k =

l + 1/2

χ(z), z

)(2)

with kernels

∆κ` (z) =

3Ωm

2

H20

H(z)(1 + z)χ(z)

χ∗ − χ(z)

χ∗(3)

∆g` (z) = b(z)

dn

dz+

3Ωm

2

H20

H(z)(1 + z)χ(z)

∫ z∗

zdz ′(

1−χ(z)

χ(z ′)

)(α(z ′)− 1

) dndz

(z ′)

(4)

Valid for CMB lensing, SZ, CIB, ISW, galaxy lensing, galaxydensity...


Cross-correlations with NumCosmo III

Goal is to extract as much information as available : use bothcross- AND auto-spectra Cκκ` , C gg

` , Cκg`Gaussian likelihood ready for correlation of n observables withpseudo-spectra

Constraints on biases + cosmological parameters

Statistical tools already available in NumCosmo and pluggable: MCMC, bootstrap, ABC, etc. !

Stable version available on GitHub

NumCosmo paper : Vitenti et al. in prep.

xcor paper : Doux et al. in prep.


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NumCosmo Overview and examples - 200.145.112.249200.145.112.249/webcast/files/Sandro.pdf · NumCosmo Basic concepts Ncmpre x { stands forNumCosmoMathwhich contains the general codes