1. What you need to know about large scale structure Licia Verde University of Pennsylvania www.physics.upenn.edu/~lverde

2. Outline 1)Motivation and basics Large Scale Structure probes 2) Real world effects 3) Measuring P(k) & (Statistics) (spherical cows) (less spherical cows)

3. The standard cosmological model 96% of the Universe is missing!!!

4. Major questions : 2) What makes the Universe accelerate? 1)What created the primordial perturbations? These questions may not be unrelated The standard cosmological model Questions that can be addressed exclusively by looking up at the sky 96% of the Universe is missing!!!

5. CMB is great and told us a lot, but large scale structures are still useful: Check consistency of the model We will concentrate on dark energy and inflation If this test is passed Combine to reduce the degeneracies

6. On blackboard: Power spectrum (for DM) definitions Gaussian random fields Linear perturbations growth Transfer function

7. Primordial power spectrum=A k n Amplitude of the power law slope ln k ln P(k) A (convention dependent) !

8. Primordial power spectrum=A k n(k) Amplitude of the power law slope ln k ln P(k) A,n (convention dependent) !  =dn/dlnk

9. CONSTRAINTS ON NEUTRINO MASS

10. CDM density Neutrino mass WMAP II WMAP+high l experiments 2dFGRS SDSS main LRG SDSS CMB+SDSS LRG 0.9eV (95% CL) Tegmark et al ‘07 Spergel et al ‘07 2dFGRS SDSS main WMAP II

11. Flatness SN1A Riess et al 04 2dfGRS ‘02 WMAPII WMAPII + H From Sperget et al 07

12. How about dark energy?

13. Planck scale (At EW scale it’s only 56 orders of magnitude) If it dominated earlier, structures would not have formed

14. And it’s moving fast

15. What’s going on? Non exhaustive list of possibilities: We just got lucky “landscape” there are many other vacuum energies out there with more reasonable values It is a slowly varying dynamical component (quintessence) Einstein was wrong (we still do not understand gravity)

16. Quintessence If dark energy properties are time dependent, so are other basic physical parameters Equation of state parameter w= p/  w=-1 is cosmological constant what other options to consider? clustering? Couplings?

17. Varying fine structure constant alpha Oklo Natural reactor: 1.8 billion yr ago there was a natural water-moderated fission reactor in Gabon. Isotopic abundances contrain 149 Sm neutron capture cross section ad thus alpha

18. Dark energy 2dfGRS H prior WMAPII SN With DE clustering

19. Why so weak dark energy constraints from CMB? The limitation of the CMB in constraining dark energy is that the CMB is located at z=1090. What if one could see the peaks pattern also at lower redshifts? We need to look at the expansion history (I.e. at least two snapshots of the Universe)

20. Baryonic Acoustic Oscillations Courtesy of D. Eisenstein For those of you who think in Real space Evolution of a single perturbation, Imagine a superposition

21. Fore those of you who think in Fourier space If baryons are ~1/6 of the dark matter these baryonic oscillations should leave some imprint in the dark matter distribution

22. Data from Tegmark et al 2006 Matter-radn equality Acoustic horizon at last scattering

23. from Percival et al 2006 DR5 Robust and insensitive to many systematics

24. THE SYMPTOMS Or OBSERVATIONAL EFFECTS of DARK ENERGY Recession velocity vs brightness of standard candles: dL(z) CMB acoustic peaks: Da to last scattering LSS: perturbations amplitude today, to be compared with CMB Da to z survey Perturbation amplitude at z survey

25. Galaxy clusters number counts Galaxy clusters are rare events: P(M,z) oc exp(-  2 /  (M,z) 2 ) In here there is the growth of structure Beware of systematics!“What’s the mass of that cluster?” x 

26. Galaxy clusters number counts Galaxy clusters are rare events: P(M,z) oc exp(-  2 /  (M,z) 2 ) In here there is the growth of structure Beware of systematics!“What’s the mass of that cluster?” x 

27. Inflation  V(  ) H ~ const Solves cosmological problems (Horizon, flatness). Cosmological perturbations arise from quantum fluctuations, evolve classically. Guth (1981), Linde (1982), Albrecht & Steinhardt (1982), Sato (1981), Mukhanov & Chibisov (1981), Hawking (1982), Guth & Pi (1982), Starobinsky (1982), J. Bardeen, P.J. Steinhardt, M. Turner (1983), Mukhanov et al. 1992), Parker (1969), Birrell and Davies (1982)

28. Flatness problem Horizon problem Structure Problem

29. Information about the shape of the inflaton potential is enclosed in the shape and amplitude of the primordial power spectrum of the perturbations. Information about the energy scale of inflation (the height of the potential) can be obtained by the addition of B modes polarization amplitude. In general the observational constraints of Nefold>50 requires the potential to be flat (not every scalar field can be the inflaton). But detailed measurements of the shape of the power spectrum can rule in or out different potentials. Seeing (indirectly) z>>1100

30. But the spacing of the fluctuations (their power as a function of scale) depend on how fast they exited the horizon (H) Which in turns depend on the inflaton potential The shape of the primordial power spectrum encloses information on the shape of the inflaton potential!

31. Specific models critically tested nn rr dn s /dlnk=0 Models like V(  )~  p dn s /dlnk=0 HZ p=4p=2 For 50 and 60 e-foldings p fix, Ne varies p varies, Ne fix

32. Possible probes of large scale structure Galaxy surveys Clusters surveys (SZ, thermal and Kinetic) Lyman alpha surveys Weak lensing surveys (***) H21surveys (far future)

33. Weak lensing (cosmic shear)

34. Very near future: Atacama Cosmology telescope High resolution map of the CMB Use the CMB as a background light to “illuminate” the growth of foreground cosmological structures Thermal Sunyaev-Zeldovich Kinetic SZ CMB gravitational Lensing e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- e-e- Coma ClusterT electron = 10 8 K (& South Pole telescope, & Planck)

35. Summary Large-scale structure (LSS) (in combination with CMB) Can be used to test the consistency of the model (LCDM) and if that holds, to better constrain cosmology So far we have seen the basic theory behind LSS In the future expect an avalanche of LSS data (and acronyms) 2 problems: dark energy, inflation can be addressed exclusively by looking up at the sky Next time: real world effects

36. Fingers-of -God Great walls Redshift space distortions

37. In linear theory : enhancement of P(k) along the line of sight Kaiser (1987) P(k) => P(k)(1+2/3f+1/5f^2)

38. Redshift-space distortions ( Kaiser 1987 ) z obs = z true +  v / c  v prop. to  m 0.6  m 0.6 b -1  n/n (bias) linear Non-linear Fourier space shells

40. Fingers-of -God Great walls

41. What’s bias?

42. ? Measured for 2dFGRS (Verde et al. 2002)

43. “If tortured sufficiently, data will confess to almost anything” Fred Menger

44. Treat your data with respect (Licia Verde)

45. Interpretation: Likelihood analysis CMBFAST or CAMB to get P(K) Bayes Theorem: LikelihoodPriorWhat you really want (Posterior) You should not forget

46. Likelihood: Gaussian vs non-gaussian Likelihood analysis Best fit parameters  Maximize the likelihood Central Limit Theorem  distribution will converge to Gaussian What is the probability distribution of your data? Examples: Cl, alm,, etc.. L Gaussian likelihood: If data uncorrelated… much simpler

47. Error bars and Confidence Levels Why errors? Joint or marginalized? Errors Cosmic variance noise ( ignore approximations, mistakes etc..)

48. Errors From: “Numerical recipes” Ch. 15

49. Example: for multi-variate Gaussian L Errors From: “Numerical recipes” Ch. 15 If likelihood is Gaussian and Covariance is constant

50. There is a BIG difference between  reduced & 2 2  Only for multi varaite Gaussian with constant covariance

51. Statistical and systematic errors As you add more data points (or improve the S/N) the statistical errors become smaller but the systematic errors do not. Errors Examples of statistical (random) errors: cosmic variance, instrumental noise, roundoff (!)….. Examples of systematic errors: approximations, incomplete modeling, numerics, …. (introduce biases)

52. Grid-based approach What if you have (say) 7 parameters? You’ve got a problem ! Operationally: e.g., 2 params: 10 x 10

53. Markov Chain Monte Carlo (MCMC) Random walk in parameter space At each step, sample one point in parameter space The density of sampled pointsposterior distribution marginalization is easy: just project points and recompute their density FAST: before likelihood evaluations, now Adding external data sets is often very easy

54. Operationally: 1. Start at a random location in parameter space: old i  L 2. Try to take a random step in parameter space: new i  L 3a. If L new L old Accept (take and save) the step, “new”  “old” and go to 2. 3b. If new L old L Draw a random number x uniform in 0,1 If x L new L old do not take the step (i.e. save “old”) and go to 2. L new L old If x do as in 3a. KEEP GOING….

55. “Take a random step” The probability distribution of the step is the “proposal distribution”, which you should not change once the chain has started. The proposal distribution (the step-size) is crucial to the MCMC efficiency. Steps too small  poor mixing Steps too big  poor acceptance rate MCMC

56. When the MCMC has forgotten about the starting location and has well explored the parameter space you’re ready to do parameter estimation. Burn-in USE a MIXING and CONVERGENCE criterion!!! (From Verde et al 2003)

57. Beware of DEGENERACIES Reparameterization. h e.g., Kososwsky et al. 2002

58. Once you have the MCMC output: The density of points in parameter space gives you the posterior distribution To obtain the marginalized distribution, just project the points To obtain confidence intervals, - integrate the “likelihood” surface -compute where e.g. 68.3% of points lie To add to the analysis another dataset (that does not require extra parameters) renormalize the weight by the “likelihood” of the new data set. To each point in parameter space sampled by the MCMC give a weight proportional to the number of times it was saved in the chain No need to re-run cmbfast! warning: if new data set is not consistent with the old one  nonsense

59. Thermal Sunyaev Zeldovich effect

60. Expansion rate of the universe a(t) ds 2 =  dt 2 +a 2 (t)[dr 2 /(1-kr 2 )+r 2 d  2 ] Einstein equation (å/a) 2 = H 2 = (8  /3)  m +  H 2 (z) = (8  /3)  m + C exp{  dlna [1+w(z)]} Growth rate of density fluctuations g(z) = (  m /  m )/a Our Tools Poisson equation  2  (a)=4  Ga 2  m = 4  G  m (0) g(a) Second oder diff eqn, here.