Gong-Bo Zhao ICG, Portsmouth May, 2010 Play with CAMB/CosmoMC.

Gong-Bo ZhaoICG, Portsmouth

May, 2010

Play with CAMB/CosmoMC

Why CAMB/CosmoMC?

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Very efficientWell-structured (modulised) -- easy to

tweak and hackWell-supported – cosmocoffee.infoPopular – mainstream numeric tool in

cosmology

ICG PhD lectures May, 2010

This 4.5-hr course is to …

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Show you how to run CAMB/CosmoMCLink CAMB equations to Ma&Bertschinger ‘96

Show several useful examples of hacking CAMB -- dynamical dark energy, modified gravity

Guide you how to modify CAMB/CosmoMC for your own research purpose

Show you useful numerical tricks extensively used in CAMB


This course is not on …

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Theory (see Kazuya, Rob and Cyril’s lectures)Fortran programming (See Numerical Recipe)


Warming-ups

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Download, compile and run CAMBcamb.infomake clean; make all./camb params.ini

Change model parameters in params.ini and make CMB, P(k) plots change cosmo. paramsadd neutrinosdo CMB lensingdo vector and tensor modescompare with Wayne Hu’s plots athttp://background.uchicago.edu/~whu/metaanim.html


Analyze the code using “understand for fortran”

http://www.scitools.com/download/(choose the free 15-day trial version)

Look into the code…

http://www.scitools.com/download/

Structure of CAMB

Messy? Not really!!

7 ICG PhD lectures May, 2010

Structure of CAMB

preparation

core

output results


Structure of CAMB_GetResults


Structure of CMBmain


CAMB language A,B,C

12€

Background : grho = 8πGρa2, adotoa =a'

a=

da /dτ

a, tau = conformal time

δT μν : dgrho = 8πGa2 ρ i

i

∑ δi, dgq = 8πGa2 (ρ i

i

∑ + pi)v i

clxc = δc, clxb = δb, clxq = δDE

δGμν : etak = ηk, z = h' /(2k), sigma =h'+6η '

2k


CAMB vs. M+B ’96 astro-ph/9506072

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η'k = dgq/2 η 'k 2 = 4πGa2(ρ + P )θ

clxcdot = − kz δ 'c = −1

2h'

z = (0.5dgrho/k +ηk)/adotoa k 2η −1

2

a'

ah' = 4πGa2δT0

0

sigma = z +1.5dgq/k 2 σ =h'+6η '

2k

Differential equations to evolve in CAMBDifferential equations to evolve in CAMB

Constraint equations (algebraic)Constraint equations (algebraic)

Hacking 1: Dynamical DE

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Task: Default CAMB only works for constant w.

Modify it to implement the CPL parametrisa-tion, i.e. w(a)=w0+wa(1-a)

and calculate the luminosity distance, CMB and P(k) for a model of {-1.5, +(-)0.4}. Compare your spectra with that of the LCDM model.


Start the game!

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Spot the dark energy stuff in CAMBWe need w(a), dw/da, Int {[1+w(a)]dlna} Files to hack: equations.f90, inidriver.F90Subroutine/functions to hack: dtauda, fderivs, outputChange the background and the DE perturbation equations!!


Background evolution

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For constant w (default CAMB) grhoa2=grhoa2+grhov*a2**2For functional w (hacked) grhoa2=grhoa2+grhov*Funcofw(a,..)*a2

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ρX = ρ X0 exp{−3

1

a

∫ [1+ w(a')]d ln a'}


function Funcofw(a,deriv) implicit none real(dl) a, Funcofw, a0 real(dl) w0, wa integer deriv if (a .lt. 1.d-8) then a0 = 1.d-8 else a0 = a end if

if (deriv==0) then Funcofw = w0+wa*(1.d0-a0) else if (deriv ==1) then Funcofw = -wa*a0 else if (deriv ==2) then Funcofw = a0**(2.d0 - 3.d0*(1.d0+w0+wa))*exp(3.d0*wa*(a0-1.d0)) end if end function Funcofw


Output

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In inidriver.F90

open(unit=50,file=‘dL_SN.dat’) do i=1, 1000 zz(i)=1.7d0+dble(i-1)/dble(1000-1) write(50,'(100e15.6)') zz(i),

5*log10((1+zz(i)) & **2*AngularDiameterDistance(zz(i)))+25 end doclose(50)


DE perturbation equations

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ayprime(EV%w_ix)= -3*adotoa*(cs2_lam-Funcofw(a,0))*(clxq+3*adotoa*& (1+Funcofw(a,0))*vq/k) -(1+Funcofw(a,0))*k*vq -& (1+Funcofw(a,0))*k*z -3*adotoa*adotoa*Funcofw(a,1)*vq/k

ayprime(EV%w_ix+1) = -adotoa*(1-3*cs2_lam)*vq + k*cs2_lam*clxq/(1+Funcofw(a,0))


Output Dark Energy perturbation DE(), vDE() at k=10-3 Mpc-1

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Note that in CAMB language DE = y(EV%w_ix) vDE = y(EV%w_ix+1)

Uncomment the stuff in cmbmain.f90,

EV%q=1.d-3; EV%q2=EV%q**2 tol1=tol/exp(AccuracyBoost-1) do j=1,6000 tauend = taustart * exp(j/6000._dl*log(CP%tau0/taustart)) call

GaugeInterface_EvolveScal(EV,tau,y,tauend,tol1,ind,c,w) write (*,’(4E15.5)') tauend,1/y(1)-1, y(EV%w_ix) ,y(EV

%w_ix+1) end do stop


Weller and Lewis, 2003

Singularity?!

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Output CMB Cl’s for the model w0=-1.5, wa=+1.0 to see what happens.

This singularity stems from the fact that w crosses -1. Technically, it’s due to the term of dw/dz/(1+w).

No-Go Theorem: The w for single fluid (scalar field) cannot cross -1 in GR! For proof, see Appendix of astro-ph/0703202.

Solution: astro-ph/0507482 (quintom scenario),

or arXiv:0808.3125 (PPF implementation) ICG PhD lectures May, 2010

Generalize our hacked code to make it work for an

arbitrary w(z).

One further step?

Leave it to you guys to play with it for fun!!

Hacking 2: TO BE OR NOT TO BE ??-- MODIFY CAMB TO WORK IN MODIFIED GRAVITY

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Task

Default CAMB only works in GR. Modify it to implement MG. Reproduce the plots on my website.

http://userweb.port.ac.uk/~zhaog/MGCAMB.html


Universe is Accelerating

Scalar-tensor gravity(e.g. Chameleon, f(R)…)

DGP, Degravitation

…

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Dark Matter

Atoms

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Λ ?!

Indistinguishable at the background levelNeed to study the structure growth to break the degeneracy

Linear perturbation in FRW universe

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General RelativityModified Gravity

What to do technically?

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Given (a,k) and (a,k)we need to solve

k2(a,k) 4G a2(a,k)

in the synchronous gauge, which is used in CAMB.Note that



So we have

Finally, we have (for details, see Appendix of 0809.3791),

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Note that in default CAMB, overdensity for different speciesand metric perturbation in synchronous gauge are actually evolved, and other Einstein equations are used as constraint equations (see this slide).

In MG, we need to evolve and and get from constraint equations.

To add one more variable to evolve in CAMB, we need to define it, set initial condition for it, and change fderivs and output properly.

See how to modify the code in the lecture.



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μ(a,k) =1+ α 1k

2as

1+ α 2k2as

, γ(a,k) =1+ β1k

2as

1+ β 2k2as

Fit your own parameters using CosmoMC

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CosmoMC fits your model parameters to data by exploring the parameter space in a smart way (Metropolis-Hastings algorithm). It calls CAMB to calculate the theoretical spectra of a model, and compare to data.


How to add your own parameter to CosmoMC?

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Tell CosmoMC your parameters’ names in CMB_Cls_simple.f90

Define them in cmbtype.f90Map them in params_CMB.f90Change the # of parameters in settings.f90Done!!See the example in the lecture!!



Astro-ph/0511625

Useful references

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Camb.info; cosmologist.info/cosmomc/Cosmocoffee.infoM+B: astro-ph/9506072Jussi’s ICG lectures: www.icg.port.ac.uk/~valiviij/Numerical Recipe: www.nr.comPlotting software: OriginPro, Matlab, IDL, gnuplot


Have fun!!


Gong-Bo Zhao ICG, Portsmouth May, 2010 Play with CAMB/CosmoMC.

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