This summer, I worked with Professor Peter Timbie and the Observational Cosmology group at the University of Wisconsin, Madison to simulate instrument performance for a space based mission to examine the polarization of the cosmic microwave background. This was done in prepartion for the EPIC (Einstein Polarization Interferometer for Cosmology) mission to examine possible array configurations and instrument tolerances. I tested a series of array configurations on simulated CMB maps and examined the recovered power spectra for a variety of instrument beams and frequency bandwidths.

The Cosmic Microwave Background is the leftover radaition of the primordial plasma which was emitted at recombination. As such it contains a wealth of information about the early universe. The primary temperature anisotropy shown above as been well studied and has revelead considerable information about the constituent make up of the early universe. This temperature anistropy occurs at a scale of tens of micro Kelvin and is due to variations in the local density at recombination. It is found by simply measuring the intensity (i.e. the square of the electic field) of incoming cmb radiation in a particular direction.

On the other hand the Polarization of the CMB is far less well studied and no less useful. It is common practice to separate out the polarization anisotropy into two components or modes; its curlless but divergenceful component known as "E-Modes" for thier resemblance to an electric field and its curlful but divergenceless component known as the "B-Modes" for thier resemblance to a magnetic field. These Modes can be detected by measuring the electric field along two independent axes and then creating a map of stokes parameters.

The E-modes, which have been somewhat measured, are due to bulk motions of the primordial plasma and occur at a scale of 1 micro Kelvin. The B-modes are particularly interesting since they are believed to be caused by gravitational waves from inflation and so provide a link to the first instants of the big bang; however, these B-modes are particular difficult to measure since they occur at a scale of only .1 micro Kelvin and they have not yet been detected.

The Einstein Polarization Interferometer for Cosmology or EPIC is a proposed space probe to measure the B-modes of the CMB Polarization. It consists of a series of interferometric arrays tuned to various frequencies to precisely measure the CMB and to remove galactic foregrounds from infrared dust emission and syncrotron radiation. Each array as shown below has a set of 64 conical horns to detect the microwave signal each attached to a series of amplifiers and filters, the signal from these horns is then interfered using mirrors in a Cassegrain telescope design. The interference pattern is then measured in the focal plane with an array of extremely sensitive bolometers. Specifically it is planned to use Transition-Edge Hot-Electron Micro-Bolometers (THMs) with Transition Edge Sensors (TES) and read out by Superconducting Quantum Interference Devices (SQUIDs). I tested array configurations for the 64 horns and simulated the recovered power spectra for an array at one of the primary science frequencies of 90 GHZ.

In an interferometer array the distance between two horns defines a particular baseline and that baseline in turn corresponds to a particular angular scale on the sky and a particular mutlipole moment. Since the power spectra is a plot of anisotropy vs. multipole moment, I wished to produce as many unique baselines as possible so as to maximize the number of data points. For an interferometer with n horns there are n*(n-1)/2 possible distinct pairs of horns and therefore baselines. For EPIC with its 64 horns this corresponds to 2016 possible baselines. My goal then was to find an array configuration with a smooth distribution of baselines across an order of magnitude in distance in which none of the 2016 baselines were repeated while at the same time also being sufficiently compact to go into space and allowing room for the secondary mirror. I found that the most efficient way to do this was to arrange the horns along points of a Archimedes spiral beginning at the secondary mirror and spiralling outwards to near the edge of the primary mirror while slowly varying the angular distance between points. When taking into account the need to prevent horns and mirrors from overlapping I arrived at the configuration shown below. For the primary science frequency at 90 GHz each horn is 2.5 cm wide, the secondary mirror shown in center is 7.5 cm wide, and the width of the whole array is 30.2 cm. The array dimensions can then be scaled accordingly for each of the other frequencies of interest.

Finally with an array configuration the power spectra can be found, this done most simply by taking the Fourier transform of the image. In Fourier space, also known as the uv plane, each baseline or multipole moment defines a ring at a particular radius from the center. When the frequency bandwidth is taken in account this ring becomes an anulus since the different frequencies mix in different multipole moments, which has the effect of introducing horizontal error bars into the power spectra. The anisotropies or Cls as they are called is then simply the average variance in the anulus. Finally the effect of the instrument beam is taken into account by modeling is a 2d gaussian beam, for an interferometer where a wider beam is desired in order to take in more of the sky. This effect is accounted for simply by appying a 2d gaussian to the 20 by 20 degree section of simulated CMB sky and results in a slightly reduced signal in the power spectra but in no way alters the shape.

With all the the simulated process described I can now show you the results. Shown below is the temperature power spectra (used here because it better demonstrates the virtues of the configuration the B mode spectra) obtained with a 15 degree instrument beam and a 10 percent bandwidth. The array shown above is used on 4 20 by 20 degree taken from the equator of the simulated CMB map shown above and the results are averaged together. The crosses are the recovered spectrum while the dots show the actual power spectrum and its inherent error.

While the second and third peaks are washed out by the bandwidth, the first peak is clearly visible, demonstrating that this configuration is capable of making useful measurements of the anisotropies.