integrated Sachs-Wolfe effect

Transcription

1 integrated Sachs-Wolfe effect transregio 33 winter school on cosmology Astronomisches Recheninstitut Fakultät für Physik und Astronomie, Universität Heidelberg December 7, 2009

2 outline of this course 1 cosmology 2 cosmic microwave background 3 structure formation 4 nonlinearity 5 summary

3 Hubble expansion Hubble diagramme recession velocity of distant objects: need for dynamical cosmology redshift was originally interpreted as a galaxy evolution parameter!

4 expansion history of the universe expansion history of the universe

5 CDM-paradigm of structure formation Volker Springel cosmological standard model (ΛCDM): structures form by Newtonian gravitational instability of Gaussian, adiabatic inflationary initial fluctuations in the (collisionless) cold dark matter field on a flat accelerating expanding background open questions in structure formation determination of cosmological parameters from growth nonlinear structure formation and interplay with dark energy collective dynamics of dark matter particles (e.g. angular momenta)

6 cosmological standard model cosmology + structure formation are described by: dark energy density Ω ϕ cold dark matter density Ω m baryon density Ω b dark energy density equation of state parameter w Hubble parameter h primordial slope of the CDM spectrum n s, from inflation normalisation of the CDM spectrum σ 8 cosmological standard model: 7 parameters known to few percent accuracy, amazing predictive power

7 Friedmann eqns: evolution of a in FLWR-universes substitution of RW-line element into field equation yields: (ȧ ) 2 a = 8πG 3 ρ K a 2 (1) ä a = 4πG (ρ + 3p) 3 (2) for a homogeneous ideal fluid with density ρ and pressure p Hubble function H(a) and deceleration parameter H(a) H 0 = ȧ a = d ln a and q(a) = äa dt ȧ 2 (3) question the two Friedmann equations are equivalent, but why does curvature appear in the ȧ-equation, but not in the expression for ä?

8 cosmological fluids and equation of state adiabatic equation: combine the two Friedmann equations d ( a 3 ρ(a) ) p d ( ) a 3 = 0 or, equivalently 3 H(a) (p + ρ) + ρ = 0. da da H 0 (4) introduce equation of state parameter w p = wρ. (5) adiabatic equation describes the change of energy density in Hubble expansion question show that for a universe with no curvature the relation between deceleration and eos parameter is given by: q = 3(1+w) 2 1.

9 equation of state: overview fluid Ω(a) H(a) w q radiation a 4 a 2 +1/3 1 matter a 3 a 3/2 0 1/2 curvature a 2 a 1 1/3 0 dark energy a a / Λ = const = const 1 1 question fluids with w < 1 are called phantom dark energy. what is so weird about them?

10 negative equation of state fluids with negative eos w are very important (negative pressure) the cosmological constant Λ has w = 1 dark energy is constructed to have time-varying w = 1/ Hubble function for a multi-component universe with dark energy and matter, but critical density: ( 1 ) + Ω ϕ exp 3 d ln a [1 + w(a)] question H 2 (a) H 2 0 = Ω m a 3 show that w < 1/3 implies accelerated expansion and that w = 1 implies a constant Hubble function question simplify eqn. (6) to a constant dark energy eos w 0, and to ΛCDM a (6)

11 Hubble function H(a): expansion velocity Hubble function H0(a), derivative H1(a) scale factor a scaled Hubble function a 3/2 H(a)/H 0 and derivative a 5/2 dh(a)/da/h 0 Hubble function is monotonically decreasing and infinite at a = 0 representation: a 3/2 H(a)/H 0, because H(a) a 3/2 in Ω m = 1 question if dark energy dominates: at what redshift is q = 0 or Ω m = Ω Λ?

12 distance measures: comoving distance due to evolving metric, there is no unique way of defining a distance measure comoving distance χ is the distance on a spatial hypersurface between the world lines of a source and the observer moving with the Hubble flow photon geodesics are defined by ds = 0 (Fermat s principle) therefore cdt = adχ (from metric), dχ = cda/(a 2 H) χ = c aa a e da a 2 H(a) complete analogy to conformal time dη = da/(a 2 H), such that χ = cη question compute the comoving distance in ΛCDM to a high redshift quasar (z = 5), and to the CMB (z = 1098). compare to SCDM (7)

13 curvature curvature is a nonlinearity in the field equation formally w = 1/3, although curvature is not a physical substance! solutions (fully curved, empty universe, Ω k = 1) imply: deceleration vanishes, q = 0 Hubble expansion is constant, ȧ = const (but not H(a)!) distinguish carefully between geometry and dynamics an matter-underdense universe is hyperbolic and expands forever a matter-overdense universe is spherical and recollapses multicomponent fluids are more complicated! construction of critical universes is possible, with accelerating dynamics (ΛCDM) curvature is special: it is the only energy density, which can be negative Ω k < 0, in which case the curvature is hyperbolic

14 dark energy matter and radiation are physical fluids with w = 0 and w = +1/3 curvature and cosmological constant are GR phenomena with w = 1/3 and w = 1 is it possible to construct a fluid with varying negative eos? consider a scalar field ϕ with self-interaction V(ϕ) total energy ρ = ϕ 2 + V(ϕ) pressure p = ϕ 2 V(ϕ) w = p ρ = ϕ2 V(ϕ) ϕ 2 + V(ϕ) (8) slow roll: consider the limit ϕ 2 V(φ) w 1 + ɛ (9) fluids with low kinetic and high potential energy have negative w

15 why dark energy and Λ are two different things Λ is part of the gravitational theory slow-roll (w = 1) is perfectly fulfilled and holds always. dark energy is driven by V(φ) and naturally builds up ϕ, so that w moves away from 1 part of the vacuum equations, no external (scalar) field needed naturally appears when deriving the field equation from the Einstein-Hilbert action in a variational approach (see lecture of M. Bartelmann, Lovelock-theorem for constructing S grav ) any dark energy theory still would need to explain why Λ is zero never think Λ is just dark energy with w = 1! dark energy is necessarily dynamic and changes its eos w with time CPL-parameterisation: w(a) = w 0 + (1 a)w a (10)

16 thermal history of the universe: overview source: Addison-Wesley

17 inflationary fluctuations in the CMB source: WMAP

18 cosmic microwave background inflation has generated perturbations in the distribution of matter the hot baryon plasma feels fluctuations in the distribution of (dark) matter by gravity at the point of (re)combination: hydrogen atoms are formed photons can propagate freely perturbations can be observed by two effects: plasma was not at rest, but flowing towards a potential well Doppler-shift in photon temperature, depending to direction of motion plasma was residing in a potential well gravitational redshift between the end of inflation and the release of the CMB, the density field was growth homogeneously all statistical properties of the density field are conserved testing of inflationary scenarios is possible in CMB observations

19 formation of hydrogen: (re)combination temperature of the fluids drops during Hubble expansion eventually, the temperature is sufficiently low to allow the formation of hydrogen atoms but: high photon density (remember η B = 10 9 ) can easily reionise hydrogen Hubble-expansion does not cool photons fast enough between recombination and reionisation neat trick: recombination takes place by a 2-photon process question at what temperature would the hydrogen atoms form if they could recombine directly? what redshift would that be?

20 CMB motion dipole the most important structure on the microwave sky is a dipole CMB dipole is interpreted as a relative motion of the earth CMB dispole has an amplitude of 10 3 K, and the peculiar velocity is β = 371km/s c T(θ) = T 0 (1 + β cos θ) (11) question is the Planck-spectrum of the CMB photons conserved in a Lorentz-boost? question would it be possible to distinguish between a motion dipole and an intrinsic CMB dipole?

21 CMB dipole source: COBE

22 cosmology cosmic microwave background structure formation nonlinearity summary subtraction of motion dipole: primary anisotropies source: PLANCK simulation Bjo rn Malte Scha fer

23 CMB angular spectrum analysis of fluctuations on a sphere: decomposition in Y lm T(θ) = t lm Y lm (θ) t lm = dω T(θ)Ylm (θ) (12) l m spherical harmonics are an orthonormal basis system average fluctuation variance on a scale l π/θ C(l) = t lm 2 (13) averaging... is a hypothetical ensemble average. in reality, one computes an estimate of the variance, C(l) m=+l 1 t lm 2 (14) 2l + 1 m= l

24 parameter sensitivity of the CMB spectrum source: WMAP

25 secondary CMB anisotropies reconstructed isw map by B. Barreiro secondary anisotropies gravitational interaction: CMB lensing, int. Sachs-Wolfe effect, Rees-Sciama effect Compton scattering: Sunyaev-Zel dovich effect (thermal, kinetic), Ostriker-Vishniac effect isw-effect is special keeps Planckian photon spectrum cross correlation technique primary CMB fluctuations are a noise source total s/n 10

26 thermal Sunyaev-Zel dovich effect ecliptic latitude β [deg] Sunyaev-Zel dovich flux S Y and S W [Jy] ecliptic longitude λ [deg] thermal SZ sky map dimensionless frequency x = hν/(kbtcmb) CMB spectrum distortion Compton-interaction of CMB photons with thermal electrons in clusters of galaxies characteristic redistribution of photons in energy spectrum

27 kinetic Sunyaev-Zel dovich/ostriker-vishniac effect ecliptic latitude β [deg] Sunyaev-Zel dovich flux S Y and S W [Jy] ecliptic longitude λ [deg] thermal SZ sky map dimensionless frequency x = hν/(kbtcmb) CMB spectrum distortion Compton-interaction of CMB photons with electrons in bulk flows increase/decrease in CMB temperature according to direction of motion

28 CMB lensing source: A. Lewis, A. Challinor gravitational deflection of CMB photons on potentials in the cosmic large-scale structure CMB spots get distorted, and their fluctuation statistics is changed, in particular the polarisation

29 source: B. Barreiro gravitational interaction of photons with time-evolving potentials higher-order effect on photon geodesics in general relativity

30 structure formation equations cosmic structure formation structure formation is a self gravitating, fluid mechanical phenomenon continuity equation: evolution of the density field due to fluxes ρ + div(ρ υ) = 0 (15) t Euler equation: evolution of the velocity field due to forces υ + υ υ = Φ (16) t Poisson equation: potential sourced by density field 3 quantities, 3 equations solvable Φ = 4πGρ (17) 2 nonlinearities: ρ υ in continuity and υ υ in Euler-equation

31 viscosity and pressure dynamics with dark matter dark matter is collisionless (no viscosity and pressure) and interacts gravitationally (non-saturating force) dark matter is collisionless no mechanism for microscopic elastic collisions between particles, only interaction by gravity derivation of the fluid mechancis equation from the Boltzmann-equation: moments method continuity equation Navier-Stokes equation energy equation system of coupled differential equations, and closure relation effective description of collisions: viscosity and pressure, but relaxation of objects if there is no viscosity? stabilisation of objects against gravity if there is no pressure? Navier-Stokes equation for inviscid fluids is called Euler-equation

32 collective dynamics: dynamical friction source: J. Schombert dynamical friction emulates viscosity: there is no microscopic model for viscosity, but collective processes generate an effective viscosity a particle moving through a cloud produces a wake behind the particle, there is a density enhancement density enhancement breaks down particle velocity kinetic energy of the incoming object is transformed to unordered random motion

33 Kelvin-Helmholtz instability shear flows become unstable if there are large perpendicular velocity gradients generation of vorticity in shear flows by the Kelvin-Helmholtz instability absent in the case of dark matter: flow is necessarily laminar

34 vorticity intuitive explanation of the nonlinearity of the Navier-Stokes eqn vorticity equation: ω rot υ p υ + υ υ = Φ + µ υ (18) t ρ ω + υ ω = ω υ ωdiv υ + 1 p ρ + µ ω t ρ2 } {{ }}{{}}{{}} {{ }}{{} material derivative tilting compression baroclinic diffusion (19) generation of vorticity by pressure gradients non-parallel to density gradients viscous stresses not present in the case of collisionless dark matter gravity as a conservative force is not able to induce vorticity vorticity equation is a nonlinear diffusion equation, vorticity is advected by its own induced velocity field

35 angular momentum of galaxies galaxy M81, HST image vorticity can t be generated in inviscid fluids flow is laminar initial vorticity decreases 1/a

36 angular momentum: tidal shearing Euler frame Lagrange frame non-constant displacement mapping across protogalactic cloud tidal forces i j Ψ set protogalactic cloud into rotation in addition: anisotropic deformation (not drawn!) gravitational collapse: non-simply connected fields

37 tidal shearing in Zel dovich-approximation current paradigm: galactic haloes acquire angular momentum by tidal shearing (White 1984) L ρ 0 a 5 V L d 3 q( q q) x (20) tidal shearing can be described in Zel dovich approximation x( q, t) = q D + (t) Ψ( q) x = Ḋ + Ψ (21) 2 relevant quantities: inertia I αβ and shear Ψ αβ L α = a 2 Ḋ + ɛ αβγ I βσ Ψ σγ (22) tidal shear Ψ αβ = α β Ψ, derived from Zel dovich displacement field Ψ Φ, solution to Ψ = δ

38 tidal interaction with the large-scale structure h 1 Mpc h 1 Mpc alignment of haloes with the tidal field, source: O. Hahn haloes interact with the large-scale structure with tidal forces decomposition IΨ = 1 2 [I, Ψ] {I, Ψ} commutator [I, Ψ]: angular momentum generation anticommutator {I, Ψ}: anisotropic deformation

39 regimes of structure formation look at overdensity field δ (ρ ρ)/ ρ, with ρ = Ω m ρ crit analytical calculations are possible in the regime of linear structure formation, δ 1 homogeneous growth, dependence on dark energy, number density of objects transition to non-linear structure growth can be treated in perturbation theory (difficult!), δ 1 first numerical approaches (Zel dovich approximation), directly solvable for geometrically simple cases (spherical collapse) non-linear structure formation at late times, δ > 1 higher order perturbation theory (even more difficult), ultimately: direct simulation with n-body codes

40 linearisation: perturbation theory for δ 1 move from physical to comoving frame, related by scale-factor a use density δ = ρ/ρ and comoving velocity u = υ/a question linearised continuity equation: t δ + div u = 0 linearised Euler equation: evolve momentum Φ u + 2H(a) u = t a 2 Poisson equation: generate potential Φ = 4πGρ 0 a 2 δ derive the linearised equations by subsituting a perturbative series ρ = ρ 0 (1 + δ) for all quantities, in the comoving frame

41 growth equation structure formation is homogeneous in the linear regime, all spatial derivatives drop out combine continuity, Jeans- and Poisson-eqn. for differential equation for the temporal evolution of δ: d 2 δ da + 1 ( 3 + d ln H ) dδ 2 a d ln a da = 3Ω M(a) δ 2a 2 (23) growth function D + (a) δ(a)/δ(a = 1) (growing mode) position and time dependence separated: δ( x, a) = D + (a)δ 0 ( x) in Fourier-space modes grows independently: δ( k, a) = D + (a)δ 0 ( k) for standard gravity, the growth function is determined by H(a) question derive H(a) as a function of D + (a)

42 terms in the growth equation 3 source S (a) and dissipation Q(a) scale factor a source (thin line) and dissipation (thick line) two terms in growth equation: source Q(a) = Ω m (a): large Ω m (a) make the grav. fields strong dissipation S(a) = 3 + d ln H/d ln a: structures grow if their dynamical time scale is smaller than the Hubble time scale 1/H(a)

43 growth function growth function D+(a) scale factor a D + (a) for Ω m = 1 (dash-dotted), for Ω Λ = 0.7 (solid) and Ω k = 0.7 (dashed) density field grows a in Ω m = 1 universes, faster if w < 0 question derive growth equation, use scale-factor a as time variable, and show that D + (a) = a is a solution for Ω m = 1

44 Gaussian random fields in cosmology fluctuations in the density field are a Gaussian random process sufficient to measure the variance ergodicity: postulated (theorem by Adler) volume averages are equivalent to ensemble averages δ n = 1 d 3 x δ n ( x)p(δ( x)) (24) V homogeneity: statistical properties independent of position x V p(δ( x)) p(δ( x + x)) (25) the density field is a 3d random field isotropy p(δ( x)) = p(δ(r x)), for all rotation matrices R (26) finite correlation length: amplitudes of δ at two positions x 1 and x 2 are not independent: covariance needed for Gaussian distribution p(δ( x 1 ), δ( x 2 )) measurement of cross variance/covariance δ( x 1 )δ( x 2 ) δ( x 1 )δ( x 2 ) is called correlation function ξ

45 Gaussian random field isodensity surfaces, threshold 2.5σ, shading local curvature, CDM power spectrum, smoothed on 8 Mpc/h-scales

46 statistics: correlation function and spectrum finite correlation length zero correlation length correlation function quantification of fluctuations: correlation function ξ( r) = δ( x 1 )δ( x 2 ), r = x 2 x 1 for Gaussian, homogeneous fluctuations, ξ( r) = ξ(r) for isotropic fields

47 statistics: correlation function and spectrum Fourier transform of the density field d 3 k δ( x) = (2π) δ( k) exp(i k x) δ( k) = 3 d 3 x δ( x) exp( i k x) (27) variance δ( k 1 )δ ( k 2 ) : use homogeneity x 2 = x 1 + r and d 3 x 2 = d 3 r δ( k 1 )δ ( k 2 ) = d 3 r δ( x 1 )δ( x 1 + r) exp( i k 2 r)(2π) 3 δ D ( k 1 k 2 ) (28) question definition spectrum P( k) = d 3 r δ( x 1 )δ( x 1 + r) exp( i k r) spectrum P( k) is the Fourier transform of the correlation function ξ( r) homogeneous fields: Fourier modes are mutually uncorrelated isotropic fields: P( k) = P(k) show that the unit of the spectrum P(k) is L 3! what s the relation between ξ(r) and P(k) in an isotropic field?

48 why correlation functions? a proof for climate change and global warming please be careful: we measure the correlation function because it characterises the random process generating a realisation of the density field, not because there is a badly understood mechanism relating amplitudes at different points! (PS: don t extrapolate to 2009)

49 CDM spectrum P(k) and the transfer function T(k) 10 4 power spectrum δ(k)δ (k) [ Mpc/h ] comoving wave vector k [ Mpc/h ] 1 ansatz for the CDM power spectrum: P(k) = k n s T(k) 2 small scales suppressed by radiation driven expansion Meszaros-effect asymptotics: P(k) k on large scales, and k 3 on small scales

50 nonlinear density fields ΛCDM SCDM (Ω m = 1) source: Virgo consortium dark energy influences nonlinear structure formation how does nonlinear structure formation change the statistics of the density field?

51 mode coupling linear regime structure formation: homogeneous growth δ( x, a) = D + (a)δ 0 ( x) δ( k, a) = D + (a)δ 0 ( k) (29) separation fails if the growth is nonlinear, because a void can t get more empty than δ = 1, but a cluster can grow to δ 200 δ( x, a) = D + (a, x)δ 0 ( x) (30) product of two x-dependent quantities in real space convolution in Fourier space: δ( k, a) = d 3 k D + (a, k k )δ 0 ( k ) (31) k-modes do not evolve independently: mode coupling correlation produces a non-gaussian field (central limit theorem)

52 perturbation theory perturbative series in density field: δ( x, a) = D + (a)δ (1) ( x) + D 2 +(a)δ (2) ( x) + D 3 +(a)δ (3) ( x) +... (32) lowest order: δ (2) ( k) = with mode coupling d 3 p (2π) 3 M 2( k p, p)δ( p)δ( k p ) (33) M 2 ( p, q) = p q pq properties: time-independent, no scale p 0 strongest coupling if p = q some coupling of modes p q no coupling if p = q ( p q + q ) + 4 p 7 ( ) 2 p q (34) pq

53 homogeneity, linearity and Gaussianity homogeneity, linearity and Gaussianity...almost the same thing in structure formation! linearity eqns can be linearised: δ 1 linearisation fails: δ 1 homogeneity homogeneous: δ( x, a) = D + (a)δ( x, a = 1) inhomogeneous: δ( x, a) = D + ( x, a)δ( x, a = 1) Gaussianity (with central limit theorem) Gaussian amplitude distribution p(δ)dδ non-gaussian (lognormal) distribution p(δ)dδ mode coupling easiest way to visualise: resonance phenomenon

54 nonlinearity triangle linearity, homogeneity and Gaussianity imply each other nonlinear structure formation breaks homogeneity and produces non-gaussian statistics mode coupling - can be described in perturbation theory barrier at delta= 1 linearity SF equations can be linearised delta <<1 barrier at delta= 1 homogeneity position independent growth central limit theorem independent Fourier modes Gaussianity Gaussian amplitude distribution delta(x,a) = D+(a) delta(x) delta(k,a) = D+(a) delta(k) p(delta)d delta

55 link between dynamics and statistics nonlinear structure formation couples modes superposition of various k-modes (not independent anymore) generate a non-gaussian density field non-gaussian density field: odd moments are not necessarily zero even moments are not powers of the variance finite correlation length: n-point correlation functions 3-point-function: bispectrum 4-point-function: trispectrum higher order correlations quickly become unpractical, and are really difficult to determine

56 nonlinear CDM spectrum P(k) 10 5 CDM spectrum P(k,a) and P(k,a)/P lin (k) [(Mpc/h) 3 ] wave numer k [(Mpc/h) 1 ] fit to numerical data, z = 9, 4, 1, 0, normalised on large scales extra power on large scales, time dependent, saturates on top of scaling P(k, a) D 2 +(a)

57 quantification of non-gaussianities: bispectrum (l1, l2) 10 0 configuration dependence Rl multipole order l1 500 multipole order l bispectrum (3-point function) quantifies nonlinearity to lowest order configuration dependence: compare arbitrary triangle to equilateral triangle, keeping base fixed: R l3 (l 1, l 2 ) = l 1l 2 B(l 1, l 2, l 3 ) l3 2 B(l 3, l 3, l 3 ) (35)

58 n-body simulations of structure formation basic issue: gravity is long-ranged, for each particle the gravitational force of all other particle needs to be summed up, complexity n 2 algorithmic challenge to break down n 2 -scaling particle-mesh particle 3 -mesh tree-codes tree-particle mesh

59 on largest scales: filaments and sheets time sequence of structure formation in a dark energy cosmology formation of sheets and filaments no relaxation (collapsing sphere would reexpand to orginial radius)

60 gravitational interaction of CMB photons with time-varying potentials sensitive to the growth of structures secondary anisotropy in the CMB, large angular scales

61 isw-derivation grav. interaction of CMB photons with time-evolving potentials temperature perturbation τ, conformal time η τ = T = 2 dη ϕ T CMB c 2 η = 2 dχ a 2 H(a) Φ c 3 a reformulation: use comoving distance χ as a distance measure: dχ = cdη scale factor a as a time variable: d dη = a2 H(a) d da generate potential from density field with comoving Poisson equation Φ = 3H2 Ω m δ Φ 2a c = 3Ω m 1 δ 2 2a χ 2 H isw-effect measures d/da(d + /a)

62 cross correlation techique isw-perturbation have the same spectrum as the CMB use a tracer (i.e. galaxy density) which marks the potential wells cross-correlation between the CMB and the tracer (τ isw + τ CMB ) γ tracer = τ isw γ tracer tracer is uncorrelated with primary CMB tracer picks out isw-perturbations tracer density: redshift distribution p(z), bias b γ = dχ p(z) dz dχ b D + δ careful: isw-effect measures φ, but tracers follow δ different scales

63 Crittenden (2003): discovery 0.4 <XT> (TOT cnts s!1 µk) 0.2 0!0.2! ! (degrees) Figure 1: The X-ray intensity measured by HEAO-A1 is correlated with the microwave sky measured by WMAP at a higher level than would be expected by chance correlations. Here we plot the cross correlation between the X-ray intensity fluctuations and the CMB temperature fluctuations along with the theoretical predictions for the ISW effect in a cosmological constant (ΩΛ = 0.72), the best fit WMAP model for scale invariant fluctuations. To give an idea of the level of accidental correlations, the green curves show the result of correlating the X-ray map with 100 independent Monte Carlo realized CMB maps with the same power spectrum as the WMAP data. The variance increases at smaller angular separations, where there are fewer pairs of pixels contributing to the correlation and one can see that the signals in neighboring bins are highly correlated for a given realization. Due to the shape of the expected correlation, the signal to noise is greatest at smaller angular separations. For θ = 0, 1.3, and 2.6, the Monte Carlo trials exceed the amplitude of the actual X-ray/CMB correlation only 0.3%, 0.8%, and 0.3% of the time respectively. These correspond to 2.4 to 2.8 σ. At larger angular separations, the observed correlations appear to fall faster than predicted by theory. The blue line shows the theoretical predictions if the quadrupole and octupole modes are suppressed cross correlation function with X-ray catalogue

64 isw-spectra line of sight expressions, ϕ = 1 δ/χ 2 H, χ H = c/h 0 τ = 3Ω m dχ a 2 H(a) d D + c da a ϕ = dχ W τ (χ)ϕ γ = dχ p(z) dz dχ D +b δ = dχ W γ (χ)δ Limber-equation: project 3d spectrum to 2d spectrum, flat-sky approximation angular power spectra: fluctuation on angular scale l = π/ θ C ττ (l) = dχ W2 τ (χ) P(k) χ 2 (χ H k) 4 k=l/χ C τγ (l) = dχ W τ(χ)w γ (χ) P(k) χ 2 (χ H k) 2 k=l/χ Poisson-equation in Fourier-space: Φ δ ( k 2 )Φ δ

65 angular isw-spectra isw auto spectrum l(l+1) Cττ(l) 2π isw cross spectrum l(l+1) Cτγ(l) 2π multipole order l multipole order l most signal at low l, cosmic variance limitations easy to remember: C τγ (l) l 2 C ττ (l) l 4

66 which redshift contributes most to isw? 0.5 evolution Q(a) and derivative dq/da scale factor a rewrite line of sight integral τ τ = 3H2 0 c 3 χh special redshift: Ω m (z) = Ω DE (z) 0 dχ a 2 H(a) dq da 1 δ,

67 parameter sensitivity C τγ (l) = 3Ω m c dχ χ 2 [ D + bp(z) dz ] [ a 2 H(a) d dχ da ] D + a P(k) (χ H k) 2 k=l/χ prefers intermediate values for Ω m signature for dark energy: SCDM: D + (a) = a, d/da(d + /a) vanishes σ 8 is completely degenerate with bias b external prior on σ 8 combination of C τγ (l) with C γγ (l) minor dependency on n s and h (via shape parameter) sensitivity to w, from growth and cosmology compare to lensing: very similar, D + /a

68 dark energy sensitivity source: Afshordi (2004) ideal measurment 10% accuracy on Ω DE, 20% accuracy on w

69 parameter sensitivity Ωm σ σ h h n s 0.24 ns ideal measurement: PLANCK combined with EUCLID w CMB priors on Ω m, σ 8, n s and h from PLANCK 10% accuracy on Ω DE, 20% accuracy on w

70 constraints: covariance cross correlation technique: C τγ (l) does not CMB-fluctuations tracer is uncorrelated with CMB primary CMB fluctuations enter as correlation noise! 2 1 [ cov[c τγ ] = C 2l + 1 f τγ(l) 2 + C γγ (l) C ττ (l) ] sky C τγ (l) = C τγ (l), cross correlation! C γγ (l) = C γγ (l) + C Poisson (l), Poissonian error C ττ (l) = C ττ (l) + C CMB (l) + C noise (l), primary CMB cosmic variance: important, highest amplitudes at low l isw-effect much weaker (10σ) than gravitational lensing (> 100σ) weaker constraints, but useful for degeneracy breaking weird models can be investigated

71 Giannantonio (2008): cross correlation functions 10 cross correlation with 6 different galaxy catalogues combined significance of 4.5σ

72 Giannantonio (2008): constraints w w k 0.1 flat k Log m 2 c s 0.5 FIG. 12: Likelihood for flat models with dynamical dark energy as a function of the sound speed, where we fix c 2 the matter s 0 density based on the equation of state, assuming the CMB shift constraint. 1 and 2 σ intervals are shown. No constraint is 1 possible for the cosmological constant limit (w = 1). give negative ISW, and can cancel the effect of increasing Björn Malte 1.5 the Schäfer cosmological constant, while the opposite happens for open models (below the flat line). w m FIG. 13: Likelihood for curved models with varying Ωm and ΩΛ from the MC2 errors. The shaded areas represent 1 and 2 σ intervals. ΛCDM is a good fit to the data. constraints on dark energy cosmologies from the isw-spectrum only constraints on 2 parameters (low s/n) Finally, for the baryon oscillation (BAO) measurements [6] we use the constraint on the volume distance measure defined as

73 Vielva (2004): parameter likelihoods 10 Vielva et al P P P ! DE! DE! DE P P P !2!1.8!1.6!1.4!1.2!1!0.8!0.6!0.4!0.2 0 w 0!2!1.8!1.6!1.4!1.2!1!0.8!0.6!0.4!0.2 0 w 0!2!1.8!1.6!1.4!1.2!1!0.8!0.6!0.4!0.2 0 w ! DE ! DE ! DE !2!1.8!1.6!1.4!1.2!1!0.8!0.6!0.4!0.2 0 w 0!2!1.8!1.6!1.4!1.2!1!0.8!0.6!0.4!0.2 0 w 0!2!1.8!1.6!1.4!1.2!1!0.8!0.6!0.4!0.2 0 w Figure 7. The logarithm of the marginalized pdf for ΩDE (upper) and w (middle) are presented. The 2D likelihood is also provided (bottom). The errors in the parameter estimation at 1σ, 2σ and 3σ are also plotted. First column shows the limits given by the CAPS, the second ones is for the CSMHW and the last one is for the CCF. wavelet-transformation (filtered spectrum) likelihoods for individual parameters of them can be expressed as a linear combination of any of the other two. 5.3 Constraints in ΩDE and w The limits in the dark energy density are compatible with the vacuum energy density calculated in the previous section. The constraints in w are less restrictive. The standard inflationary concordance model with ΩDE 0.71 and w = 1 is perfectly compatible within the 1σ CL. As ex-

74 application: coupled fluids growth function D+(a) isw cross spectrum l(l+1) Cτγ(l) 2π scale factor a multipole order l recent flurry in the literature: coupled DM/DE construct cosmologies with very similar growth functions isw-effect can still distinguish them! other field: modified gravity theories, DGP-gravity

75 systematics: biasing models 10 7 l(l+1) Cτγ(l), Cτγ(l)/ b 2π 10 8 isw-spectra l(l+1) 2π multipole order l isw-spectra with evolving bias parameter estimation error systematical error in parameter estimation statistical error other systematics: 1 redshift-distortions due to peculiar motion (Anais Rassat) 2 magnification bias (Marilena LoVerde) 3 nonlinearities (Robert Schmidt)

76 Angelos Kalovidouris: bias-free isw measurements 10 2 noisy and noiseless spectra for EUCLID l*(l+1)/(2*pi)*cyz(l) gg kk tg tk tt noisy gg noisy kk noisy tg noisy tk noisy tt l isw-spectra (incl. noise) for the lensing correlation bias-free measurement of the isw-effect weak cosmic shear as a tracer of the density field comparable s/n, but sadly only marginal improvement of constraints

77 isw-effect: pros and cons maps structure growth, compares D + to a signature of dark energy, vanishes in SCDM sensitivity for non-standard Poisson equation DE/CDM coupling or modified gravity weak constraints (CMB noise), total significance 10σ can access information hidden to geometrical probes, gravitational analogy to lensing: lensing κ dχ D + /aδ isw-effect τ dχ d(d + /a)/daϕ strongest for intermediate Ω m : coupling vs. growth uncertainties related to bias bias decreases with time: db/da < 0, different for every tracer scale dependence b(k), different for every tracer

78 Rees-Sciama effect RS-effect: isw-effect from nonlinear structures distinction a bit artificial (similar to kin. Sunyaev-Zeldovich-effect vs. Ostriker-Vishniac-effect) two different approaches in perturbation theory perturbed density field, solve for potential δ(a) = D + (a)δ (1) + D 2 +(a)δ (2) +... continuity equation: velocity-density products, get Φ δ = div(δ v) Poisson-equation first approach: 2 nd order, second approach: 1 st order both involve computation of 4-point correlation functions

79 isw-effect vs. Rees-Sciama effect perturbation: δ(a) = D + δ (1) + D 2 +δ (2) +... first order second order τ (1) = 3Ω m c τ (2) = 3Ω m c dark energy sensitivity: χh 0 χh 0 dχ a 2 H(a) d da dχ a 2 H(a) d da ( D+ a ) 1 δ (1) χ 2 H ( ) D δ (2) a χ 2 H linear isw-effect: vanishes in SCDM, D + = a, nonzero in DE cosmologies nonlinear isw-effect: largest in SCDM, smaller in DE

80 Rees-Sciama effect 1.8 time evolution D n +(a), d(d n +/a)/da scale factor a evolution of density field + potential simulated RS-field (V.Springel) isw-effect from nonlinear structures nonlinear corrections to isw-spectrum at high l 3rd order perturbation theory needed, δ( x, a) = n D n +(a)δ (n) ( x)

81 spectrum of the Rees-Sciama effect use mode coupling for δ (2), decompose with Wick-theorem alternative approach: via velocity field v, coupled to density field source: Cooray + Sheth 2002 RS-spectrum much flatter than isw-spectrum, cross-over at l = 100

82 gravitomagnetic potentials change of photon energy in time-variable potential wells: τ = T T = 2 c dχ Φ 3 η with conformal time η connection to gravitomagnetic potentials: continuity η Φ = G d 3 r ρ( r ) r r = +G d 3 r j( r ) r r integration by parts (ignore boundary terms)... = G d 3 r j( r ) 1 use identity (... = 1 r r ( d 3 r j( r ) r r r r ) ( ) 1 =, pull out : r r ) τ = 2 c dχ div A 3 interpretation: isw-effect is due to 1. formation of objects: ρ > 0 Φ > 0, or equivalently 2. converging matter streams: div j < 0 div A < 0 A is called gravitomagnetic potential, sourced by j

83 cosmology cosmic microwave background structure formation nonlinearity summary analogies: RS-effect and lensing P δ (k) density δ( r) flux j( r) P j (k) Φ( r)= G d 3 r δ( r ) r r potential Φ( r) potential A( r) A( r)= G d 3 r j( r ) r r C α (l) α= Φ τ=div A deflection α( θ) isw effect τ( θ) C τ (l) κ= 1 2 div α χ= τ C κ (l) gradient convergence κ( θ) χ( θ) C χ (l) Φ conserves energy k, rotates direction k/k A does not influence k/k, but stretches wave length

84 RS-effect: visual impression credit: V.Springel (MPA), Millenium simulation non-gaussian fluctuations, sharp features in the temperature field Birkinshaw-Gull effect from haloes

85 Rees-Sciama effect: non-gaussianities 10 2 (l1, l2) configuration Rq,l multipole l1 500 multipole l2 configuration dependence signal to noise ratio Σq multipole l signal/noise ratio increase with l mixed bispectra between isw-effect and galaxy overdensity τ q γ 3 q cross-correlation of PLANCK with EUCLID yields 0.6σ extention to trispectrum? very technical!

86 Granett (2008): RS-effect from voids 2 FIG. 1. Stacked regions on the CMB corresponding to supervoid and supercluster structures identified in the SDSS LRG catalog. We averaged CMB cut-outs around 50 supervoids (left) and 50 superclusters (center), and the combined sample (right). The cut-outs are rotated, to align each structure s major axis with the vertical direction. Our statistical analysis uses the raw images, but for this figure we smooth them with a Gaussian kernel with FWHM 1.4. Hot and cold spots appear in the and void stacks, respectively, with a characteristic radius stacks of 4, corresponding from to identified spatial scales of 100 voids h 1 Mpc. and The inner superclusters circle (4 radius) and equal-area identified outer ring mark in the extent SDSS of the compensated filter used in our analysis. Given the uncertainty in void and cluster orientations, small-scale features should be interpreted cautiously. high significance of > 4σ with previous results (Giannantonio et al. 2008), we measured a cross-correlation amplitude between our two data sets on 1 scales of 0.7µK. To find supervoids in the galaxy sample, we used the parameter-free, publicly available ZOBOV (ZOnes Bordering On Voidness; Neyrinck 2008) algorithm. For each galaxy, ZOBOV estimates the density and set of neighbors using the Björn Malte parameter-free Schäfer Voronoi tessellation (Okabe et al. 2000; van de Weygaert & Schaap 2007). Then, around each density minicluster and void lists, we discarded any structures that overlapped LRG survey holes by 10%, that were 2.5 (the stripe width) from the footprint boundary, that were centered on a WMAP point source, or that otherwise fell outside the boundaries of the WMAP mask. We found 631 voids and 2836 clusters above a 2σ significance level, evaluated by comparing their density contrasts to those of voids and clusters in a uniform integrated PoissonSachs-Wolfe point sample. effect There are so many structures because of the high sensitivity

87 open questions isw-effect needs precise understanding of biasing models bias time evolution, scale-dependence, non-locality, stochasticity? usage of unbiased tracers such as weak cosmic shear? isw-effect is so far only detected at low redshift correlation of isw with CMB lensing? isw-effect from individual objects accretion rates or void dynamics Rees-Sciama effect spectrum too weak, bispectrum too small, other non-gaussian quantifiers? distinguishing with the isw-effect between curved cosmologies and dark energy is possible, in combination with other structure formation probes with Patricio Vielva (IFCA/Santander): review on the isw-effect for Physics Reports being written