BAO & RSD. Nikhil Padmanabhan Essential Cosmology for the Next Generation VII December 2017

Transcription

1 BAO & RSD Nikhil Padmanabhan Essential Cosmology for the Next Generation VII December 2017

2 Overview Introduction Standard rulers, a spherical collapse picture of BAO, the Kaiser formula, measuring distance scales with BAO Nonlinearities Beyond linear theory, reconstruction Observations Designing surveys If there are particular topics you d like me to address, ask!

3 Outline Introduction Standard rulers The acoustic feature in linear theory Redshift space distortions in linear theory Measuring BAO : I

4 Expansion Rate Time The expanding Universe Deceleration Acceleration

5 Dark Energy : The Big Questions Is cosmic expansion accelerating because of a breakdown of GR on cosmological scales or because of a new energy component that exerts repulsive gravity within GR? If the latter, is it consistent with a cosmological constant or does it evolve in time? Any answers to this will point to new physics!

6 Dark Energy Measure the expansion rate of the Universe The distance-redshift relations Directly measure H(z) Measure the rate at which structures grow in the Universe Growth function D(z), and its derivatives Two paradigms Dark Energy What is its equation of state? How does it evolve with redshift? Is it consistent with a cosmological constant? Modified gravity How do structures form in the Universe? Are matter and light affected the same way? This is a rich set of questions, and requires multiple probes.

7 Dark Energy vs Modified Gravity Expansion and growth probe different aspects From Eric Linder

8 Probes of Dark Energy Expansion +Growth Imaging Supernovae Lensing / Clusters Spectro BAO Redshift Distortions These are somewhat artificial distinctions Information/Robustness from combining different probes

9 Two point statistics Characterize the density field by its two-point correlations For a Gaussian random field, this is a complete description of the entire density field Probability of finding a pair of galaxies separated by For BAO, useful to think both in configuration space (correlation functions) and Fourier space (power spectra) Just FTs of one another We will consider the correlation function both perpendicular and parallel to the line of sight (exact definitions of which we can gloss over for now) There is more information in the density field beyond two point statistics.

10 Redshift space observed zspec = zcosmo + vp/ah peculiar velocities (vp) sourced by matter density fluctuations (δm) real space redshift space Beth Reid

11 Simulations : Real Space

12 Separation parallel to LOS Simulations : Redshift Space Separation perp to LOS NP et al, 2012

13 The BOSS 2D galaxy correlation fn Samushia et al, 2014

14 Notation Borrowed from M. White

15 Outline Introduction Standard rulers The acoustic feature in linear theory Redshift space distortions in linear theory Measuring BAO : I

16 Measuring two distances with standard rulers

17 Measuring d A (z) and H(z) Transverse scale measures angular diameter distance Radial scale measures the Hubble constant Internal consistency tests H(z) unique amongst dark energy probes H(z) important to constrain dark energy at high redshifts

18 Outline Introduction Standard rulers The acoustic feature in linear theory Redshift space distortions in linear theory Measuring BAO : I

19 Constructing a Standard Ruler The plasma of the early Universe supports sound waves Compton scattering between electrons and photons Coulomb interactions between electrons and protons Sound waves from the initial density perturbations expand outward Speed of sound ~ c/ 3 When the Universe cools below 0.3 ev, electrons and protons recombine Sound wave stalls, leaving imprint on density fluctuations. Characteristic scale of Mpc ~ 4.7e24 m

20 Sound Waves imprint a Standard Ruler Daniel Eisenstein

21 The Standard Ruler in the Galaxy Correlation Function Imprint of sound waves frozen In the early Universe Scale set by sound horizon Standard ruler analogous to standard candles

22 Observables Positions on the sky and redshifts 3D map of the Universe Precision redshifts require a spectroscopic survey Need to convert angular separations to physical distances Ruler oriented transverse to line of sight measures distance to the ruler. Distance as a function of redshift Integrated expansion rate Need to convert redshift separations to physical distances Ruler oriented parallel to the line of sight measures rate of change of distance with redshift. Expansion rate. Not possible with standard candles.

23 Why BAO? Simple measurement Only requires positions Underlying theory is simple Mostly linear physics (fluctuations are 1 part in 10 4 ) Exquisitely calibrated by the CMB (~1% with WMAP, much better with Planck) 3D feature (hard to mimic) Very large scales >> scales of astrophysical complications Can be treated perturbatively

24 BOSS measures the BAO standard ruler >7 sigma 1.0% distance

25 BAO : A Spherical Collapse Picture Slepian & Eisenstein 2016 Build up intuition for the various terms behind the BAO feature See ref for the detailed calculation

26 Sound waves Expand the Euler equation in the Compton mean-free path. At lowest order, obtain the tight-coupling limit Driven harmonic oscillator Speed of sound : c s2 = 1/3(1+R) R = 3 4 ρ b ρ γ

27 Scales Matter radiation equality : a eq y = a/a eq Sound horizon

28 Spherical Shells

29 Three cases Shell outside the sound horizon Shell inside the sound horizon massless baryons Shell inside the sound horizon massive baryons

30 Background Taylor expand to linear order in C

31 Perturbed shell outside the horizon Existence of the photon wave does not matter Need to determine curvature C due to perturbed shell Enforce that time is the same for background and perturbation Taylor expand to determine beta

32 Relate shell to overdensity We now must relate the shells to overdensities Adiabaticity relates photons to matter

33 Inside the horizon Treat the photon term as a forcing function

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35 Massive baryons -- decoupling After decoupling, baryons just behave like dark matter. However, we need to set the initial conditions. CDM is set by previous; baryons set by photons.

36 Massive baryons -- decoupling But the baryon derivative is undefined at the sound horizon

37 A BAO feature

38 A final wrinkle Silk damping

39 A final wrinkle Silk damping

40 Outline Introduction Standard rulers The acoustic feature in linear theory Redshift space distortions in linear theory Measuring BAO : I

41 Velocities ~ potential In linear theory, rotational component drops off as 1/a Velocity completely specified by its divergence Percival & White, 2009

42 Relating density and velocity : the continuity equation To linear order

43 Borrowed from M. White

44 To power spectra Velocities probe the matter density, even when considering galaxy bias Percival & White, 2009

45 Legendre Expansion : Data compression You can measure both b \sigma_8 and f \sigma_8? Percival & White, 2009

46 Outline Introduction Standard rulers The acoustic feature in linear theory Redshift space distortions in linear theory Measuring BAO : I

47 Measuring BAO : I The data consist of angles and redshifts. To convert to comoving coordinates, we need to assume a fiducial cosmology. Is the BAO feature at the correct place in this cosmology? That determines the distance scale We can do this perpendicular and parallel to the LOS. Measure DA and H An alternative parametrization : dilations and warping

48 Definitions Define an angle averaged distance. Measure shifts in the BAO scale. Note the scaling with the sound horizon

49 The Alcock-Paczynski (AP) effect Corrections to the cosmology involve alpha (dilations) and warping (epsilon). Alpha=1, epsilon=0 => true cosmology How do these effect the BAO feature?

50 Relative importance of dilations and warping Xu et al, 2012

51 Relative importance of dilations and warping Xu et al, 2012

52 Multipole expansions In redshift space, in linear theory, only l=0,2,4 exist Xu et al, 2012

53 The Anisotropic BAO signal Primed coordinates represent true cosmology, unprimed are fiducial Xu et al, 2012

54 Effects on correlation function multipoles Mixing of monopole and quadrupole These are leading order corrections; for large corrections, need to numerically evaluate the corrections Xu et al, 2012

55 Effects on the power spectrum 2% warp 5% warp 10% warp

56 Effects on the correlation function

57 Effects on the correlation function

58 Effects on the correlation function Xu et al, 2012

59 Including nonlinearities and RSD Xu et al, 2012

60 Final comments This is not a unique treatment; can be done many ways eg. divide the correlation function into angular wedges, weighted correlation functions to directly measure the distance-redshift relatin The scaling with alpha and Dv depends on a survey where perpendicular and parallel separations are in the normal proportion. (eg. not true for Lymanalpha measurements)

61 Measuring BAO : More comments In principle, any point in the correlation function may be used as a standard ruler Except if the correlation function is a pure power law This can be exploited to great effect eg. if you have a model for redshift space distortions BAO are special A 3D feature in the matter distribution Difficult to mimic Broadband features may be affected by variations in cosmology, survey systematics etc. BAO measurements marginalize out shape information Loss in information, gain in robustness

62 Marginalizing broadband Xu et al, 2012

63 Building a template Redshift space FoG Nonlinear evolution; no-wiggle correlation function from Eisenstein & Hu 98

64 Degeneracies? Note that alpha and epsilon look like shifts of the correlation function (see previous slide) Xu et al, 2012

65 Xu et al, 2012 Degeneracies? Note the different shape