1. The Theory/Observation connection lecture 2 perturbations Will Percival The University of Portsmouth

2. Lecture outline  Describing perturbations – correlation function – power spectrum  Perturbations from Inflation  The evolution on perturbations before matter dominated epoch – Effects from matter-radiation equality – Effects from baryonic material  evolution during matter domination – linear growth – linear vs non-linear structure (introduction)

3. Perturbation statistics: correlation function definition of correlation function from statistical isotropy from statistical homogeneity can estimate correlation function using galaxy (DD) and random (RR) pair counts at separations ~r overdensity field

4. Perturbation statistics: power spectrum power spectrum is the Fourier analogue of the correlation function definition of power spectrum sometimes written in dimensionless form

5. The importance of 2-pt statistics Because the central limit theorem implies that a density distribution is asymptotically Gaussian in the limit where the density results from the average of many independent processes; and a Gaussian is completely characterised by its mean (overdensity=0) and variance (given by either the correlation function or the power spectrum)

6. Do 2-pt statistics tell us everything? Credit: Alex Szalay Same 2pt, different 3pt

7. Correlation function vs Power Spectrum The power spectrum has the advantage that different modes are uncorrelated (as a consequence of statistical homogeneity). The power spectrum and correlation function contain the same information; accurate measurement of each give the same constraints on cosmological models. Both power spectrum and correlation function can be measured relatively easily (and with amazing complexity) Models tend to focus on the power spectrum, so it is common for observations to do the same...

8. Phases of (linear) perturbation evolution Inflation linear Non-linear (next lecture) Transfer function Matter/Dark energy domination

9. Why is there structure? Inflation (a period of rapid growth of the early Universe driven by a scalar field) was postulated to solve some serious problems with “standard” cosmology:   why do causally disconnected regions appear to have the same properties? – they were connected in the past (discussed in last lecture)   why is the energy density of the Universe close to critical density? – driven there by inflation   what are the seeds of present-day structure? - Quantum fluctuations in the matter density are increased to significant levels

10. Driving inflation Every elementary particle (e.g. electron, neutrino, quarks, photons) is associated with a field. Simplest fields are scalar (e.g. Higgs field), and a similar field  (x,t) could drive inflation. Energy-momentum tensor for  (x,t) Acceleration requires field with For homogeneous (part of)  (x,t)

11. Driving inflation Acceleration requires field with more potential energy than kinetic energy Energy density is constant, so Einstein’s equation gives that the evolution of a is Gives exponential growth Problems: only way to escape growth is quantum tunneling, but has been shown not to work. Instead, think of particle slowly rolling down potential. Close to, but not perfectly stationary.

12. Inflation: perturbations Perturbations in the FRW metric Scalar potentials usually give rise to so can interchange these Spatial distribution of fluctuations can be written as a function of the power spectrum During exponential growth, there is no preferred scale, so (Gaussian) quantum fluctuations give rise to fluctuations in the metric with with n=1

13. Inflation: perturbations Poisson equation translates between perturbations in the gravitational potential and the overdensity In Fourier space So matter power spectrum has the formBecause quantum fluctuations are Gaussian distributed, so are resulting matter fluctuations, which form a Gaussian Random Field

14. Matter P(k) depends on inflation

15. Phases of perturbation evolution Inflation linearNon-linear Transfer function Matter/Dark energy domination

16. Jeans length After inflation, the evolution of density fluctuations depends on the scale and composition of the matter (CDM, baryons, neutrinos, etc.) An important scale is the Jeans Length which is the scale of fluctuation where pressure support equals gravitational collapse, where c s is the sound speed of the matter, and  is the density of matter. depends on Jeans scale “F=ma” for perturbation growth

17. Transfer function evolution in radiation dominated Universe, pressure support means that small perturbations cannot collapse (large Jeans scale). Jeans scale changes with time, leading to smooth turn-over of matter power spectrum. Cut-off dependent on matter density times the Hubble parameter W m h. scale factor of Universe a gravitational potential k 3/2 F large scale perturbation small scale

18. The power spectrum turn-over varying the matter density times the Hubble constant Can give a measurement of the matter density from galaxy surveys It is hard to disentangle this shape change from galaxy bias and non-linear effects

19. The effect of neutrinos   k         P(k) The existence of massive neutrinos can also introduce a suppression of T(k) on small scales relative to their Jeans length. Degenerate with the suppression caused by radiation epoch. Position depend on neutrino-mass equality scale.

20. Configuration space description W m =0.3, W v =0.7, h=0.7, W b / W m =0.16 position-space description: Bashinsky & Bertschinger astro-ph/0012153 & astro-ph/02022153 plots by Dan Eisenstein

21. Configuration space description W m =0.3, W v =0.7, h=0.7, W b / W m =0.16 position-space description: Bashinsky & Bertschinger astro-ph/0012153 & astro-ph/02022153 plots by Dan Eisenstein

22. Configuration space description W m =0.3, W v =0.7, h=0.7, W b / W m =0.16 position-space description: Bashinsky & Bertschinger astro-ph/0012153 & astro-ph/02022153 plots by Dan Eisenstein

23. Configuration space description W m =0.3, W v =0.7, h=0.7, W b / W m =0.16 position-space description: Bashinsky & Bertschinger astro-ph/0012153 & astro-ph/02022153 plots by Dan Eisenstein

24. Configuration space description W m =0.3, W v =0.7, h=0.7, W b / W m =0.16 position-space description: Bashinsky & Bertschinger astro-ph/0012153 & astro-ph/02022153 plots by Dan Eisenstein

25. Configuration space description W m =0.3, W v =0.7, h=0.7, W b / W m =0.16 position-space description: Bashinsky & Bertschinger astro-ph/0012153 & astro-ph/02022153 plots by Dan Eisenstein

26. Configuration space description W m =0.3, W v =0.7, h=0.7, W b / W m =0.16 position-space description: Bashinsky & Bertschinger astro-ph/0012153 & astro-ph/02022153 plots by Dan Eisenstein

27. Baryon Oscillations in the power spectrum varying the baryon fraction “Wavelength” of baryonic acoustic oscillations is determined by the comoving sound horizon at recombination At early times can ignore dark energy, so comoving sound horizon is given by Sound speed c s Gives the comoving sound horizon ~110h -1 Mpc, and BAO wavelength 0.06hMpc -1

28. The matter power spectrum Overall shape of matter power spectrum is given by Current best estimate of the galaxy power spectrum from SDSS: no sign of turn-over yet

29. Link with CMB

30. Link with N-body simulations LCDMOCDM SCDMtCDM

31. Phases of perturbation evolution Inflation linearNon-linear Transfer function Matter/Dark energy domination

32. Spherical perturbation leading to linear growth homogeneous dark energy means that this term depends on scale factor of background “perfectly” clustering dark energy – replace a with a p cosmology equation Consider homogeneous spherical perturbation – evolution is “same” as “mini-universe” Overdense perturbation Radius a p Background Radius a

33. Spherical perturbation leading to linear growth definition of d to first order in perturbation radius (linear approximation) gives can also be derived using the Jeans equation only has this form if the dark energy does not cluster – derivation of equation relies on cancellation in dark energy terms in perturbation and background cosmology equation

34. Linear growth: EdS model For flat matter dominated model, this has solution Remember that the gravitational potential and the overdensity are related by Poisson’s equation Then the potential is constant: there is a delicate balance between structure growth and expansion Not true if dark energy or neutrinos

35. Linear growth: general models For general models, denote linear growth parameter (solution to this differential equation) For lambda models, can use the approximation of Carroll, Press & Turner (1992) For general dark energy models, need to solve the differential equation numerically

36. Linear growth factor Present day linear growth factor relative to EdS value

37. Linear vs Non-linear behaviour z=0 z=1 z=2 z=3 z=4 z=5 linear growth non-linear evolution z=0 z=1 z=2 z=3 z=4 z=5 large scale power is lost as fluctuations move to smaller scales P(k) calculated from Smith et al. 2003, MNRAS, 341,1311 fitting formulae

38. Further reading  Dodelson, “Modern Cosmology”, Academic Press  Peacock, “Cosmological Physics”, Cambridge University Press  Liddle & Lyth, “Cosmological Inflation and Large-Scale Structure”, Cambridge University Press  Eisenstein et al. 2006, astro-ph/0604361 (configuration space description of perturbation evolution)  Percival 2005, astro-ph/0508156 (linear growth in general dark energy models)