Prac%cal  lessons  in  extrac%ng  cosmology  from  the  CMB  

Benedikt  Diemer,  ASTRO  448,  12/05/2012  

Outline  

  Part  I:  Use  CosmoMC  to  extract  cosmological  parameters  from  WMAP7  data  

  Part  II:  Inves%gate  extra  terms  into  the  cl  spectrum   Run  CosmoMC  on  cl  only   Create  mock  cl  based  on  WMAP7    Introduce  terms  such  as  warp  features  into  the  cl,  see  how  they  change  the  likelihood  

Extrac%ng  cosmology  from  the  WMAP7  data  with  CosmoMC  

Part  I  

CosmoMC  

  Step  1:  Download  and  compile  CosmoMC   Comes  with  its  own  version  of  CAMB   Uses  the  WMAP7  Likelihood  code  

  Step  2:  set  parameters:   CosmoMC  can  vary  Ωbh2,  Ωdmh2,  θ100,  τ,  Ωk,  fν,  w,  ns,  nt,  nrun,  ln(1010AS),  r,  ASZ  

 Derived  parameters  t0,  zre,  H0  

 Here  I  vary  only  Ωbh2,  Ωdmh2,  θ100,  τ,  ns,  log10(1010AS),  ASZ  

 Fit  with  no  priors  except  flat  age  prior  

CosmoMC:  astro-­‐ph/0205436;  CAMB:  Lewis  et  al.  2000  

WMAP7  Likelihood  Code  

  Step  3:  Install  WMAP7  likelihood  code  

  Large  due  to  complicated  data  set  

  10  possible  contribu%ons  to  likelihood:  

  TTTT  χ2  

  TTTT  Low-­‐l  χ2  

  TTTT  Low-­‐l  determinant    TTTT  beam  correc%on  

  TETE  χ2  

  TETE  determinant  

  TE/EE/BB  Low-­‐l  χ2  

  TE/EE/BB  Low-­‐l  Determinant    TBTB  χ2  

  TBTB  determinant  

hdp://lambda.gsfc.nasa.gov/product/map/dr4/likelihood_get.cfm  

WMAP7  Likelihood  Code  Test  

  Step  4:  Compile  and  test  likelihood  code  

Running  CosmoMC  

  Each  chain  ran  on  8  threads  (CPUs),  4  chains  total  

  1000  samples  ≈  9  minutes,  200,000  ≈  28  hours  

Understanding  the  output  

2,000  samples  

Understanding  the  output  

200,000  samples  

Analyzing  the  output  

10

  Use  getdist  to  analyze  and  combine  chains  

  The  Gelman-­‐Rubin  diagnos%c  (devia%on  between  chains)  is  OK  (<  0.01)  

  Acceptance  rate  =  1  /  mul%plicity  =  32%  200,000  samples   2,000  samples  

Covariance  matrix  

  Original  covariance  matrix  which  comes  with  CosmoMC  

  Plomng  the  log  of  the  absolute  values  

  Note  the  small  values    good  parameter  choices  

Covariance  matrix  

  Covariance  matrix  derived  in  the  run  with  2,000  samples  

Covariance  matrix  

  Original  again  

Covariance  matrix  

  Covariance  matrix  derived  in  the  run  with  200,000  samples  

  Differences  mainly  in  poorly  constrained  ASZ    

   This  looks  OK  

Likelihood  distribu%ons  

Marginalized  

Mean  

2,000  samples  

Likelihood  distribu%ons  

200,000  samples  

Marginalized  

Mean  

Results  –  Comparison  to  official  

Parameter   This  computa0on   Official  WMAP7   Devia0on  (σ)  

Ωbh2   0.02252  ±  0.57×10-­‐3   0.02258  ±  0.57×10-­‐3   0.10  

Ωdmh2   0.1118  ±  0.55×10-­‐2   0.1109  ±  0.56×10-­‐2   0.16  

100  θ   1.0392  ±  0.27×10-­‐2   1.0388  ±  0.27×10-­‐2   0.14  

τ   0.0883  ±  0.015   0.088  ±  0.015   0.02  

ns   0.9687  ±  0.014   0.963  ±  0.014   0.41  

ln(1010AS)   3.0832  ±  0.035   3.190  +0.044  -­‐0.046  (?)   2.32  (?)  

ASZ   1.0017  ±  0.58   0.97  +0.68  -­‐0.97   0.05  

ΩΛ   0.7288  ±  0.029   0.734  ±  0.029   0.17  

t0  (Gyr)   13.756  ±  0.13   13.75  ±  0.13   0.05  

Ωm   0.2712  ±  0.029   0.266  ±  0.029   0.18  

zre   10.588  ±  1.21   10.5  ±  1.2   0.07  

H0   70.581  ±  2.57   71.0  ±  2.5   0.17  

hdp://lambda.gsfc.nasa.gov/product/map/dr4/params/lcdm_sz_lens_wmap7.cfm  

Results  –  2D  distribu%ons  

Results  –  All  combina%ons  

Crea%ng  and  analyzing  mock  cl  realiza%ons  based  on  WMAP7  

Part  II  

Outline  of  Part  II  

  Generate  mock  realiza%ons  of  cl  based  on  a  WMAP7  cosmology,  and  WMAP7-­‐like  noise  

  Run  CosmoMC  on  WMAP7  cl  only  (ignoring  the  actual  sky  map,  pixel  likelihood  etc.)  

  Run  CosmoMC  on  the  mock  cl,  recover  the  same  cosmology  as  with  the  real  cl  data  

  Insert  other  terms  into  the  underlying  cosmology  (e.g.  warp  features),  check  whether  they  give  systema%cally  beder  likelihood  

Cosmology  from  cl  only  

  10  likelihood  terms  for  WMAP7,  some  nega%ve  

  Hacked  CosmoMC  to  only  use  “Master  TTTT”  

  Get  much  beder  results  when  adding  correc%on  terms  (beam  correc%on  etc.)  

Results  for  cl  only  

Marginalized  

Mean  

30,000  samples  

Covariance  for  cl  only  

cl  only   Original  

2D  contours  from  cl  only  

25

Finding  the  correct  best  fit  cl  

Komatsu  et  al.  2009,  Table  1  hdp://lambda.gsfc.nasa.gov/product/map/dr4/params/lcdm_sz_lens_wmap7.cfm  

WMAP7  best  fit  cl  (logarithmic)  

  Dominated  by  cosmic  variance  at  low  l,  and  beam  noise  at  high  l  

  Note  the  low  quadrupole  in  WMAP  

WMAP7  best  fit  cl  (linear)  

  Note  the  nega%ve  data  points!  

Mock  cl  

  Extract  noise  Nl  from  likelihood  code  data  file,  since  WMAP7  data  files  contain  very  strange  noise  quan%%es  

  Try  to  generate  mock  cl  using:  

Mock  cl  

Mock  cl  

  Variance  does  not  agree    there  must  be  something  wrong  with  the  mock  cl  

Conclusion  

  Extracted  cosmological  parameters  from  WMAP7  data  and  recovered  their  values  well  

  Modified  CosmoMC  to  use  cl  only,  and  showed  how  that  degrades  the  results  

  Adempted  to  create  mock  cl  realiza%ons