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Thompson Scattering
Consider a beam of radiation moving in the direction, and being scattered by an electron through an angle in the plane. The electron experiences the following electric fields which give rise to accelerations:

The incident energy per unit area on the electron is given by the Poynting vector,

Using Larmor's formula, and averaging over time gives the intensity of radiation scattered into the solid angle

It is clear that the scattered radiation is therefore polarised polarised even for unpolarised incident radiation, rising to at . Also note that as much radiation is scattered backwards as forwards.

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Compton and Inverse-Compton Scattering
In Compton scattering high energy photons scatter off stationary electrons. They transfer energy and momentum to these electrons and so their wavelength increases. It is often succifient to transform into the zero-momentum frame, consider the collision as Thompson scattering and then transform back into the lab frame. This gives

However, if in the zero-momentum frame then the relativistic cross-section must be used, as given by the Klein-Nishina formula:

The process of inverse Compton scattering considers collisions between high speed electrons moving at speed and low energy photons. Again it is often possible to transfer into the zero-momentum frame and consider the collision as Thompson scattering (if ). Energy is lost by the electrons to the photons during this collision

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The Cosmic Microwave Background
After the Big Bang the Universe expanded and cooled.

Until the Universe was 300,000 years old the temperature was so high that all baryonic matter was ionised. Therefore the mean free path for a photon was very small, and so the optical depth . The radiation and matter were is almost perfect thermal equilibrium The radiation had a blackbody spectrum. At at redshift the electrons and ions in the plasma combined to form atoms the optical depth dropped dramatically and their mean free path became much greater than the size of the Universe. The photons have continued to cool and are now at a temperature of (microwaves). Observing this CMB radiation allows us to directly observe the Universe at the epoch of recombination.

Detecting these anisotropies allows calculation of the geometry of the Universe, determination of mass densities, etc. Other radiation sources also lie along the line of sight and contaminate our measurements.

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Foregrounds
Galactic
There are many sources of emission from the galaxy. This emission is generally on large scales similar to the interesting scales of CMB anisotropies. Remove contamination by observing away from the galactic plane and observing at many frequencies. Can then use spectral discrimination, ( ). Dust The spectral index of optically thin dust is Free-free Optically thin so . Synchrotron Optically thin, fairly young sources so .

Extragalactic sources
Extra galactic sources may be optically thick or thin, free-free or synchrotron sources A wide variety of different spectra, so it is not possible to remove contamination spectrally. However, extragalactic sources generally have small angular sizes, so can remove them with high resolution imaging.

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The high energy electrons will inverse-Compton scatter CMB photons passing through the gas.

The SunyaevZel'dovich effect

Even for a very rich cluster , so can be considered as the fraction of photons that undergo a scattering.

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Clusters of galaxies have a hot ( of ionised gas. Densities are of the order of

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The spectrum is shifted from a blackbody to higher energy, conserving photon number. At low frequency observe a decrease in intensity towards cluster; at high frequency observe an increase in intensity. The frequency where there is no intensity change is approximately 220 GHZ.

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Intensity

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On average the photons will be scattered to higher energies

We can describe this intensity change as a change in brightness temperature, , which in the Rayleigh-Jeans region

Outside the Rayleigh-Jeans region

NB The above expression is independent of redshift. This means that the SZ effect is a very impor tant tool for investigating early structure formation.

All the above neglects relativistic effects (approximately a effect for a cluster at 10 keV.)

As well as this thermal SZ effect there is also a kinematic SZ effect due to cluster velocity along the line of sight. In the Rayleigh-Jeans region

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Compare SZ and X-ray signals

Assuming that (ie that cluster is spherical) and a geometry for the Universe ( ) can calculate a value for

In theory can determine the geometry of the Universe from clusters at a range of different redshifts. Average over many randomly orientated clusters to reduce error due to aspherical clusters.

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The hot gas also emits Bremsstrahlung radiation.

Source Subtraction
Background sources will mask the SZ effect and their contamination must be removed. These confusing sources are generally much smaller than the size of a cluster, so can determine their flux by high resolution mapping. With an interferometer (eg Ryle) get this high resolution imaging from the long baselines which are insensitive to the large angular scale SZ effect. Subtract these sources from the shor t baseline data which is sensitive to the SZ.

Figure 2: Simulated flux observed by the RT against baseline for both a spatially extended SZ effect and for a point-like confusing source.

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Glossary
Absorbtion Coefficient, : Fraction of incident radiation absorbed per unit length of absorber. Includes the effect of stimulated emission. Blackbody Radiation: Radiation produced when the emitters and the radiation are in prefect thermodynamic equilibrium. Intensity predicted by Planck's law. Brightness Temperature, : The temperature that a blackbody would have to emit radiation of the observed intensity at a given frequency. Compton Scattering: The collision of a high energy photon with an electron. The photon looses energy and momentum to the electron and so its wavelength increases.

Extinction: Attenuation (of starlight) due to absorbtion and scattering by the Ear th's atmosphere and by interstellar dust. Flux Density: The power in radiation incident per area unit area per unit frequency. ). Often measured in Jy (

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Emissivity, frequency.

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Gaunt Factor: A quantum mechanical correction factor applied to formula for Bremsstrahulung radiation. Often . Inverse-Compton Scattering: The collision of a photon and a high energy electron. The electron looses energy to the photon whose frequency therefore increases. Kirchoff's Law: For a material in thernal equilibrium the absorbtion and spontaneous emission coefficients are linked by Larmor's Formula: Equation relating the total radiation rate and the acceleration of a charged par ticle

Luminosity: Power emitted in radiation. : The total cross-section .

Optical Depth, : A measure of the integrated opacity along a path through a layer of material at a par ticular frequency . Planck Function: Formula that determines the distribution of intensity of radiation under conditions of thermal equilibrium at temperature . Rayleigh-Jeans Approximation: Low frequency approxi. mation to the Planck formula, appropriate when .

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Mass Absorption Coefficient, per unit mass of material

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Source Function, : The ratio of emissivity to the opacity of a material. For the case of thermodynamic equilibrium .

Spontaneous Emission Coefficient, : Power emitted spontaneously per unit volume per unit frequency per unit solid angle. Stoke's Parameters: Four parameters (I, Q, U, V) which characterise a beam of polarised light.

Surface Brightness: Emergent specific intensity.

Synchrotron-Self-Compton: Radiation emitted through the synchrotron process which is then inverse-Compton scattered to higher frequency by the same population of electrons responsible for the original emission. Thermal Radiation: Radiation from emitters in thermal equilibrium. If the radition and emitters are also both in equilibrium then the radiation is blackbody.

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