Giuseppina Coppola1 First predicted by the Russian scientists Sunayaev and Zel’dovich in Galaxy Clusters have hot gas that produce electrons by bremsstahlung (T gas ~ Kelvin). CMB photons are cold (T CMB ~ 2.7 Kelvin). Inverse Compton scattering occurs between CMB photons and the hot electrons of clustrer atmosphere. Energy will be transferred from the hot electrons to the low energy CMB photons changing the shape of their intensity vs. frequency plot : measuremnts made at low frequencies will have a lower intensity, since photons which originally had these energies were scattered to higher energies. This distorts the spectrum by ~0.1%.

Giuseppina Coppola2 SZ effect distorsion of the CMB signal Note the decrement on the low frequency side, and the increment at higher frequencies. The amplitude of the distorsion is proportional to T e, although shape is indipendent of T e. The relativistic equation has a slightly more complicated shape.

Giuseppina Coppola3 Overview 1.CMB 2.Radiation basic 3.Scattering by electron population 4.Kompaneets approximation 5.SZ and galaxy cluster 6.Struments

Giuseppina Coppola4 The Cosmic Microwave Background Radiation The CMBR is the dominant electromagnetic radiation field in the Universe. Isotropy Photon density: Peak brightness: at Specific intensity of the radiation: Energy density: Principal Properties T rad ~ 2.7K

Giuseppina Coppola5 Thermal history of the Universe and CMBR The origins of the CMBR lie in an early hot phase of the expansion of the Universe. Very high z Very high z: matter and radiation were in good thermal contact because of the abundance of free electrons. z of recombination z of recombination: most electrons have become bound to ions. z of decoupling z of decoupling: the interaction lenght of photons and electrons exceeds the scale of the Universe. z ~ z ~ : the Universe was becoming neutral, matter-dominated and transaprent to radiation. Most of the photons that are now in the CMBR were scatterated by electrons for the last time. After recombination…… Potential fluctuations grow to form Large Scale Structure overdensities collapse to form galaxies and galaxy cluster; underdensities expand into voids.

Giuseppina Coppola6 I. Radiation basics real space volume momentum space volume Distribution function Specific intensity photon frequency Number density of photons in the Universe Energy density of the radiation field

Giuseppina Coppola7 II. Radiation basics transport equation: In the presence of absorption, emission and scattering processes, and in a flat spacetime, I ν obeys a transport equation: emissivity absorption coefficient scattering coefficient scattering redistribution function Specific intensity may be changed by: redistributing photons to different directions and frequencies (e.g. scattering); absorbing or emitting radiation (e.g. thermal bremsstrahlung); making photon distribution function anisotropic (Doppler effect);

Giuseppina Coppola8 I. Single photon-electron scattering Compton scattering formula For classical Thomson cross-section formula The probability of a scattering with angle θ: v e = βc and μ = cosθ Redistribution function:

Giuseppina Coppola9 II. Single photon-electron scattering The scattered photon frequency: Introducing the logarithmic frequency shift: s=log( νʺ/ ν ), the probability that a single scattering of the photon causes a frequency shift s from an electron with speed βc is:

Giuseppina Coppola10 I. Photon Scattering by electron population Averaging over the electron β distribution If every photon is scattered once, then the resulting spectrum is given by: Probability that a scattering occurs from ν 0 to ν Since

Giuseppina Coppola11 II. Photon Scattering by electron population The probability of N scatterings: Optical depth Probability that a photon penetrates the electron cloud The full redistribution function is given by Raphaeli formula: In most situations the electron scattering medium is optically thin, then and

Giuseppina Coppola12 I. The Kompaneets approximation In the non-relativistic limit the scattering process may be described by the Kompaneets equation, which describes the change in the occupation number, by a diffusion process. For small x e, we have: Canonical form of the diffusion equation Solution

Giuseppina Coppola13 II. The Kompaneets approximation

Giuseppina Coppola14 III. The Kompaneets approximation At low y and for an incident photon spectrum of the form of CMBR, we can use the Approximation: The spectrum of the effect is given by a simple analytical function; the location of the spectral maxima, minima and zeros are indipendent of T e in the Kompaneets approximation; the amplitude of the intensity change depends only on y. Kompaneets vs. Raphaeli formula

Giuseppina Coppola15 1.Useful to determine the intrinsic three-dimensional shape of the cluster; 2.Useful to extract information on thermal structure in the intracluster gas; 3.Useful to measure the projected mass of gas in the cluster on the line of sight if the temperature structure of the cluster is simple; 4.Useful to detect clusters; 5.Useful to test the cosmology.

Giuseppina Coppola16 The Sunayaev-Zel’dovich effect from clusters of galaxies If a cluster atmosphere contains gas with electron concentration n e (r), then the scattering optical depth, Comptonization parameter and X-ray surface brightness are: There is no unique inversion of b x (E) to n e (r) and T e (r)

Giuseppina Coppola17 I. Parameterized model for gas cluster They use a parameterized model for the properties of the scattering gas in the cluster and they fit the values of these parameters to the X-ray data. Isothermal beta-model Isothermal beta-model: T e is constant and n e follows the spherical distribution

Giuseppina Coppola18 II. Parameterized model for gas cluster Hughes et al. (1998), on the basis of observations of the Coma cluster, indroduced a useful variation on beta-model Useful to describe the decrease of gas temperature at large radius

Giuseppina Coppola19 z=0.5455; D A =760 h -1 Mpc X-ray emission mapped by ROSAT PSPC Structural parameters by isothermal beta-model β = 0.73 ∓ 0.02 Θ c = 0.69 ∓ 0.04 arcmin r c = (150 ∓ 10) h-1 kpc b 0 = ∓ counts s -1 arcmin -2 ΔT 0c ≈ h-1 mK at low frequency These values are consistent with the results obtained using X-ray spectrum

Giuseppina Coppola20 III. Parameterized model for gas cluster Ellipsoidal model Ellipsoidal model: M encodes the orientation and relative sizes of semi-major axes of the cluster. β = ∓ Θ c = ∓ arcmin ΔT 0c ≈ h-1 mK SZ model X-ray surface brightness model

Giuseppina Coppola21 Mass of the gas For an isothermal model, the surface mass density in gas is: Mean mass of gas per electron If the electron temperature of the gas is constant: This quantity can be compared with mas estimates produced by lensing studies.

Giuseppina Coppola22 Sz effect in cosmological terms Method Method: comparison of SZ effect predicted from the model with the measured effect by X-ray data. Since the predicted effect is proportional to h -1/2 via the dependence on D A, this comparison measures the value of H 0 and other cosmological parameters 1.Measuring the CMB decrement from a cluster 2.Mesuring X-ray emission from a cluster Measuring the size of a cluster

Giuseppina Coppola23 Measuring the CMB decrement from a cluster Consider simplest model of cluster  Spherical with radius R  Constant gas number density n  Constant temperature T e SZ effect decrement ΔT  Directly related to density  Directly related to the cluster path length  Directly related to the temperature of the gas, T e R n TeTe Temperature Decrement ΔT = -T rad 2y or ΔT ≈ T rad 2Rn

Giuseppina Coppola24 Measuring X-ray emission from a cluster Model of cluster  Sphere of radius R  Central number density of electron gas, n  Temperature of the gas, T e X-ray surface brightness b X  Directly related to square of density  Directly related to the cluster path length  Temperature of the gas, T e X-ray brightness b X ≈2Rn 2 R n TeTe

Giuseppina Coppola25 Measuring Size and Distance of the cluster Combined observations of b X and ΔT measure the path length along the line of sight Use the radius of the cluster and the angular size to make an estimate at the cluster distance. Remember, we assumed that cluster was spherical ΔT/T rad = 2Rn b X ≈ 2Rn 2 R = ( ΔT/T rad ) 2 /2b X D A ≈ R/θ H 0 is obtained from the measured z of the cluster and the value of D A under some assumption about q 0. R DADA θ

Giuseppina Coppola26 Result of SZ Distance Measurements SZ distance vs. z SZ effect distances are direct (rather than relative); SZ effect distances possible ar very large lookback times; can see the theoretical angular diameter distance relation; Comments H 0 = 63 ∓ 3 km/s/Mpc for Ω M =0.3 and Ω Λ =0.7 But…. selection effect, which caues the value of H 0 to be biased low the value of H 0 depends by cluster model unknown intrinsic shape of cluster atmospheres uncertainties in the parameters of the model

Giuseppina Coppola27 Cluster Detectability The total flux from the cluster that is requested: Angular position on the sky Any SZE clusters survey has some fixed angular resolution, which will not allow to spatially resolve low mass cluster. Therefore a background y b parameter will be present. If the gas temperature profile is isothermal, the integrated flux SZE cay be related to the cluster temperature weighted mass divided by D A 2 : If the temperature profile is isothermal only in the inner regions (Cardone, Piedipalumbo, Tortora (2005))

Giuseppina Coppola28 Interferometers used to measure the SZ effect Cosmic Background Imager (CBI) Located at the ALMA cite in Chajantor, Chile. These 13 antennae operate at GHz Degree Angular Scale Interferometer (DASI) A sister project to CBI, located at South Pole. These interferometers are suited to measure nearby clusters

Giuseppina Coppola29 X-ray telescopes used to measure the SZ effect ROSAT X-ray satellite in operation between 1990 and Mainly, its data has been used in conjunction with the radio observations to make estimates of H 0 and Ω b. Uncertainties of the X-ray intensity are ~ 10%. Chandra X-ray Observatory Provides X-ray observations of the clusters to make etimates of the gas temperature. Chandra currently has the best resolution of all X-ray observatories. XMM-Newton ESA’s X-ray telescope. Has 3 European Photon Imaging Cameras (EPIC)

Giuseppina Coppola30 All-sky project used to measure SZ effect Microwave Anisotropy Probe Measures temperature fluctuations in the CMB. Planck satellite ESA project designed to image the entire sky at CMB wavelengths. Its wide frequency coverage will be used to measure the SZ decrement and increment to the CMB photons.

Giuseppina Coppola31 Systematic Uncertainties in current SZ effect measurements SZ effect calibration ( ∓8%) X-ray calibration (∓ 10% ) galactic absorption column density (∓ 5% ) unresoved point sources still contaminate measurement of the temperature decrement (∓ 16%) Clusters that are prolate or oblate along the line of sight will be affected. Reese et al. 2001

Giuseppina Coppola32 References 1.Birkinshaw astro-ph/ Bernstein & Dodelson Physical Review, 41, Cardone et al. A&A 429, (2005) 4.Carlstrom et al. astro-ph/

Giuseppina Coppola33 CL H 0 = 68 km s -1 Mpc -1 if the cluster is modeled with a sphere isothermal