Frontiers: Sunyaev-Zeldovich effectAn effect predicted more than three decades ago, the S-Z effect is coming into its ownnow as a probe of cosmological conditions, due to instrumental advances and a certainamount of cleverness! Here we’ll talk about the basic effect itself, with a little discussion ofsome of the subtleties. We’ll then discuss a number of the implications that observations ofthe S-Z effect has for estimation of cosmological parameters, such as the Hubble constant.Finally, we’ll see the current state of the observations themselves.Basics of the S-Z effectWe’ll take much of the following from the ARA&A articles by Yoel Rephaeli (1995, 33,541) and Carlstrom et al. (2002, 40, 643).The idea is that clusters are in the foreground of the cosmic microwave background, sothe CMB is affected by clusters. In its essence, the S-Z effect is simply Compton scattering.Photons in the cosmic microwave background scatter off of the hot gas (T ∼ 108K) inclusters. Ask class: what is the typical direction of change of photon energy as a result(i.e., does the photon become more or less energetic)? More energetic, because the electronshave a high temperature. Therefore, photons that had been at low energies become higherenergy. As a result, if the initial CMB spectrum was a perfect blackbody, after scatteringthe spectrum is distorted. Scattering does not create or destroy photons, so the net resultis transfer of photons from the low-energy (Rayleigh-Jeans) portion of the spectrum tothe high-energy (Wien) portion of the spectrum. The CMB is in radio wavelengths, soradio observations are best at detecting both the decrement and the increment. Since thefractional frequency shift in a single Compton scattering is ∼ kT/mec2, the signal alonga given line of sight is proportional to the integrated pressure,RnT dx, where x is thecoordinate of location in the cluster. Note that the magnitude of this effect is entirelyindependent of the redshift; it depends only on the cluster properties. This makes ispractically unique as a cosmological probe. Virtually anything else you could imagineobserving has serious surface brightness effects, usually like (1 + z)−4or at best (1 + z)−2for the flux of point objects, which make observations at high redshift very difficult. TheS-Z effect therefore has substantial future promise.This effect was first considered by Sunyaev and Zeldovich in the context of ahypothesized hot gas that pervades the universe, but it is now most often considered inrelation to clusters. To estimate the significance of this effect, let’s first compute theoptical depth to scattering through the cluster. Ask class: for photons from a 2.7 Kthermal bath scattering off of 108K electrons, what is the appropriate limit of scattering?Thomson, because in the rest frame of the electrons the photon energy is much less thanmec2. So, we can calculate the typical optical depth through a cluster. If the numberdensity of electrons is 10−3cm−3, and the size of the cluster is 1 Mpc, then the opticaldepth is τ = nσd ≈ 10−3× 6 × 10−25× 3 × 1024≈ 2 × 10−3. This is an extremely smalloptical depth, so the majority of photons from the CMB never scatter at all in a givencluster. This also means that one cannot treat the Comptonization process in the diffusionlimit, as was done initially. Moreover, the speed of electrons at cluster temperatures isv ≈ (kT/511 keV)1/2c ≈ 0.1c, so the motion is mildly relativistic and corrections have to bemade. Another point is that in addition to the thermal Comptonization featured above, thedirected movement of clusters relative to the Hubble flow can also produce a ”kinematic S-Zeffect”. The kinematic S-Z effect is usually a factor of 10 or more smaller than the thermalS-Z effect. An exception comes with measurements near where the modified spectrumintersects the original spectrum (it has to, to go from a decrement to an increment). To firstorder, this “null” in the change occurs at hν = 3.83kT, or about 217 GHz for T = 2.726 K.Other subtleties that people have considered include (1) the incident spectrum mightnot be quite a blackbody, e.g., if the hot intercluster gas has already changed the spectrum,(2) as a variant, cluster positions are correlated with each other, so Comptonization ina supercluster is a possibility, and (3) if the cluster is dynamically collapsing, then thegravitational potential of the cluster might change significantly during the radiation crossingtime. All of these tend to be small effects compared to the dominant S-Z effect.S-Z effect as probe of clusters and cosmologyCluster properties.—The thermal S-Z effect depends on the integrated product ofnumber density n and temperature T , which is the integrated pressure along the line ofsight. The kinematic S-Z effect depends on the product of the peculiar velocity of thecluster and the integrated column depth. Therefore, when combined with X-ray measuresof the cluster (the temperature directly, from the spectrum; and the luminosity, whichdepends on n2T1/2), the variety of dependences on number density and temperature allowredundant checks of many different cluster properties. This gives a great deal of informationabout the gas clustering properties.Determination of H0.—A long-anticipated (and now realized) benefit of the S-Z effect isthat one can use it to estimate the Hubble constant in a way that is completely independentof all other estimates of H0. Consider for simplicity a cluster that has a characteristic radiusR, a characteristic number density n, and a characteristic temperature T. The magnitudeof the thermal S-Z effect depends on nT R. The temperature comes directly from the X-rayspectrum, and the bremsstrahlung surface brightness depends on n2T1/2R. Measurement ofthe S-Z effect, the spectrum, and the luminosity therefore give independent determinationof n, T , and R. Now, suppose that the redshift and apparent angular size of the cluster havebeen measured (in the best case, from an X-ray image, but it could also be optically). Ifyou make the further assumption that the cluster is spherical, then from R and the angularsize of the cluster you know the angular diameter distance.At low redshift, the angular diameter distance depends only on H0. One can, therefore,use this combination of observations to determine the Hubble constant. At higher redshift,the angular diameter distance also depends on Ωmand ΩΛ, so one could in principle use itto