1. Craig Lawrie Advisor: Dr. John Ruhl Abstract Software is developed for the detection of galaxy clusters in data gathered by the South Pole Telescope (SPT). SPT has supplied a 170 deg 2 map of the intensities of the Cosmic Microwave Background (CMB) at 150 GHz, with resolution of approximately 0.25 arcminutes. In this data, galaxy clusters show up due to the Sunyaev-Zel’Dovich effect (SZE), causing a decrement in intensity independent of redshift. A Metropolis Hastings algorithm is applied to find and characterize the galaxy clusters amid a noisy background. Sample data from SPT Example of MH-MCMC Step Cross-Section of Covariance Matrices Source: pole.uchicago.edu Figure 1 (left): This 300 pixel by 300 pixel sample from the SPT data is centered around the most significant previously detected galaxy cluster. The dark spot in the center is the cluster; it is no brighter than the background noise but distinguishable because of its shape. Figure 2 (above): A simple set-up using fake data to test the Metropolis- Hastings Markov Chain Monte-Carlo algorithm. The small hump on the left is the form created by trial parameters, while the large hump on the right is the fake cluster that is being searched for. Figure 3 (right top): The pixel-pixel covariance matrix is used for this data set. Using the appropriate covariance matrix is vital for the calculation of the likelihood, which is the basis for the searching algorithm. Figure 4 (right bottom): As CMB photons pass through a galaxy cluster, photons gain energy. This shifts the ‘blackbody’ spectrum to the right; decreasing the intensity below about 212 GHz and incrementing it above. Figure 5 (below): A simple grid-search in position (right ascension and declination) provides a visual of how the likelihood varies with position. This tool is used to check that a previously known galaxy cluster is the best point in likelihood space, which is required for a successful search in all four parameters. A grid-search in other sets of parameters also shows a preference for the previously known galaxy cluster. References 1.Carlstrom, J. E., Holder, G. P., & Reese, E. D. (2002). COSMOLOGY WITH THE SUNYAEV-ZEL'DOVICH EFFECT. Annual Review of Astronomy & Astrophysics, 40(1), 643-680. 2.Hobson, M. P., & McLachlan, C. (2003). A Bayesian approach to discrete object detection in astronomical data sets. Monthly Notices of the Royal Astronomical Society, 338(3), 765-784. SZ Effect 1 The study of the CMB has been full of exciting new knowledge of the history of the universe. The SZE extends on this by allowing the detection of galaxy clusters within the observable universe independent of distance. The process is as follows: Inside of galaxy clusters, due to their enormous mass, is a very hot but very sparse electron gas. Photons from the CMB passing through galaxy clusters interact with those electrons via inverse Compton scattering, and gain energy. This results in a shift of the intensity spectrum (see figure 4). For this examination, the important feature is the decrement in intensity at 150 GHz. That decrement is created only within the galaxy cluster, and so is related to the distribution of electron gas. For this purpose, we approximate that shape as a 2D projection of the 3D isothermal beta model for electron gases. Result of Sunyaev-Zel’Dovich Effect Source: Carlstrom, Holder, & Reese. Cosmology with the Sunyaev-Zel’Dovich Effect MH-MCMC 2 A Monte-Carlo simulation refers to any method that randomly samples parameters to compare to data. A Markov Chain further specifies that each sample point depends only on the immediately preceding point. Lastly, the Metropolis Hastings algorithm is a type of Markov Chain Monte-Carlo (MCMC), with the requirement that a step in parameter space is taken with probability equal to the ratio of the likelihoods. That is, if L(first point) is twice as big as L(proposed step), then that step is taken with 50% probability. This type of sampling is chosen for one great benefit: after many steps have been taken, the density of trial points is a representation of the likelihood distribution. In cases such as this with many (five) variable parameters, this is an elegant solution to constrain all parameters at once. Figure 2 is a graphical representation of what a single step might look like. 1 2 3 4 Special Thanks to: Zak Staniszewski and the rest of the SPT team. Log-Likelihood Around a Galaxy Cluster 5 Department of Physics, CWRU