Data Analysis of and Results from Observations of the Cosmic Microwave Background with the Cosmic Background

Much of the work in this thesis was done by his code, without which I would probably still be languishing in Robinson's basement. We present the results of observations of the cosmic microwave background (CMB) with the Cosmic Background Imager (CBI), a sensitive 13-element interferometer located high in the Chilean Andes.

Origin of the Microwave Background

The reason why the measurement of CMB anisotropies is of such importance is because the angular power spectrum of the anisotropies contains a wealth of detailed information about the properties and evolutionary history of the universe. Once the earliest spectrum is set (such as during inflation), the evolution of the fluctuations does not depend on exotic and uncertain physics.

Power Spectrum Basics

At Φℓm, ℓ more or less corresponds to the wavelength of the mode and is similar to its directivity. Under these assumptions, all the information contained in the CMB is contained in the set of coefficients Cℓ, so that.

Cosmological Effects on the Power Spectrum

The power at small scales is also reduced due to the finite thickness of the final scattering surface. The curvature of the universe does not affect the physical structure at the surface of the last distribution, since the universe was very matter + radiation dominated then.

Figure 1.1 Dependence of C ℓ on Ω k , the flatness of the universe while keeping the physical matter density fixed

Microwave Background Observations

DASI (Kovac et al., 2002) first measured the polarization power spectrum, although the spectrum is too noisy to have any cosmologically useful information. Shortly thereafter, WMAP measured the cross-correlation spectrum of the large-scale polarization and total intensity anisotropies (Kogut et al., 2003).

Figure 1.3 Dependence of C ℓ on τ c , the optical depth in the local universe to the surface of last scattering

Interferometers

For the case of a single dish, there will be a single-aperture diffraction pattern on the sky, which is the Fourier transform of the collecting aperture. The power pattern in the sky is called the primary beam and is the Fourier transform of the square of the dish's electric field response.

The Cosmic Background Imager

As a result, we have very dense UV coverage (see Figure 3.1 for a sample CBI UV coverage). The first two years were devoted to measurements of the total strength of the power spectrum, and in the fall of 2002 the CBI switched to predominantly polarization observations.

Figure 1.7 The CBI site, which is also the future ALMA site, has been touted by many others as one of the driest, highest places in the world

Uncorrelated Likelihood

In log-likelihood space, the joint log-likelihood is the sum of the individual probabilities. Note that the definition is χ2i x2i/V, so the maximum of the probability is the point at which the mean value of χ2 equals one.

Correlated Power Spectrum

This is the standard expression for the probability of a theory under a particular data set that starts most microwave background analysis papers. Thei, the jth element of the product of two matrices is the multiplication of the first times the jth column of the second.

Likelihood Gradient

Recall the formula for the derivative of the probability of uncorrelated data under these assumptions, Equation 2.8. The trace is the sum of the diagonal elements of the matrix and has a nice.

Likelihood Curvature

An early suggestion in Bond et al. (e.g. 1998) was to note that at the maximum probability the first term in (2.35) is approximately zero, and so we can approximate the curvature matrix by. With reasonably high accuracy, the error opqB for most experiments is simply that of the Gaussian approximation of the probability surface, FBB−1 (see, for example, Press et al., 1992).

Band Power Window Functions

If we are at the maximum, then the ∆∆T −C part of the gradient is zero, and we are left with the expected gradient as a result of the new signal. For example, we can calculate the expected contribution to the power spectrum from a population of weak radio signals. point sources that are statistically isotropic.

Early Observations

The ring configuration also provided a fairly uniform distribution of baseline lengths in the UV plane. As the CBI rotates the deck, a uniform distribution along the length also leads to fairly uniform sampling in the UV plane.

Figure 3.1 Antenna configuration for the commissioning run of the CBI. The dishes were placed in a ring around the outside for easy access

Ground Spillover

Since the observing pairs are observing the ground in identical ways, the ground signal should be identical in. Although critical to rejecting the ground signal, the cost of the difference is a factor of two in time.

Figure 3.4 The 14 hour deep field. Same as Figure 3.3 for the 14 hour deep field.

Analysis

Interferometer Response to a Random Temperature Field

A is the square of a receiver's response to the electric field (the primary beam), and the axis position on the sky relative to the pointing center. Therefore, we need to understand the response of the interferometer to a plane wave on the sky, which is most conveniently done by taking the Fourier transform of (3.5).

Figure 3.10 Comparison of CBI fit beam to the Gaussian approximation to it. This is the same CBI fit beam as in Figure 3.9, and the Gaussian has a FWHM of 45.1’

Visibility Window Functions

The scaling of the primary beam in the coefficient is a bit unusual at first because we expect the total variance to be proportional to the total area of the beam, namely σp−2. We are now in a position to choose a parameterization of the power spectrum, which specifies S(w).

Complex Visibilities

This is a very reasonable expression - basically, a two-element interferometer is sensitive to modes in the sky that have the same wavelength as the separation of the elements, and the sensitivity from this peak drops off as the primary beam Fourier transform. This is a set of four relations, since each of the two equations must hold for both its real and imaginary parts.

Power Spectrum

The set of relations can be solved for the covariances between the real and imaginary parts of the visibilities as follows: However, if both Ci∗j and Cij are nonzero, then the symmetry is broken and the real and imaginary parts of the visibilities are no longer statistically equivalent, and therefore should be treated separately.

Interpretation and Importance of Spectrum

My main contribution to the first year papers was extracting the power spectrum from the mosaics using CBIGRIDR/MLIKELY. Finally, in Section 4.6, I describe the final power spectrum of the first-year mosaics and cosmological results from the spectrum.

Noise Statistics

Fast Fourier Transform Integrals

For the first term, we can convolve all wi fori > 1 to get a new variable, sayq. The values plotted are x, where the correction factor applied to the variance is of the form 1 + d.o.f.x, where the d.o.f.

Noise Correction Using Monte Carlo

The reason is that individual UV points are not independent, but rather are correlated due to the primary beam. As such, maximum likelihood combines, with weights, several different UV points to create independent estimators of the CMB.

GRIDR/MLIKELY Speedups

In this case we regularized to the value from the unregularized spectrum, so the only effect is the small error bar. The acceleration of the hybrid mesh occurs both in CBIGRIDR, as each visibility is gridded onto fewer estimators, and in the linear algebra part of the pipeline, MLIKELY.

Figure 4.2 Comparison between spectra using a fine mesh in CBIGRIDR and a hybrid mesh with coarser sampling at ℓ > 800

Source Effects in CBI Data

Source Effects on Low-ℓ-Spectrum

This leaves significant uncertainties in the residual flux from point sources that are difficult to estimate (since the statistics of weak sources at 30 GHz are poorly known), which can add a significant amount of energy. However, the question remains as to what the maximum likelihood is actually doing when it projects the sources and what the expected effects are in the spectrum.

Two Visibility Experiment

In this simple case, it would also be correct to think of maximum likelihood using a long baseline to measure the current from the source and subtract it. This works because the long baseline is only sensitive to the current from the source, which is also true for a pure source brightness measurement.

Sources in a Single Field

In the general case, however, there is no such pure measurement, and thus there is no estimate of the source flux to subtract. The short baseline is sensitive to both the CMB and a foreground point source, while the long baseline is only sensitive to the source.

Figure 4.3 Relative efficiency of a two visibility experiment with one long baseline and one short baseline

Source Effects in the First-Year Mosaics

The total signal available is just the sum of the eigenvalues in the window matrix after a matrix transformation that transfers the noise matrix to the identity matrix. So the source will not really be projected from the mosaic spectrum even though it has disappeared from the deep spectrum.

Figure 4.4 Expected behavior of total signal available and signal lost due to sources as the ℓ range of the data is varied.

First-Year Data

First-Year Results

Power Spectrum

The band power window functions, which describe the sensitivity of the CBI bands to the CMB power at a givenℓ, are shown in Figure 4.12. Also note how much the CBI expands the range over which the CMB power spectrum is measured.

Figure 4.8 Map of the 02 hour mosaic. The left half shows the image before source subtraction, the right half shows the same image with the sources measured by the OVRO 40 meter subtracted.

Cosmology with the CBI Spectrum

-h - measurement of the Hubble cash from the HST key project of 72±8, as found in Freedman et al. The CBI is not very sensitive to the spectrum below ℓ∼400, so these cosmological results are basically independent of the first acoustic peak.

Table 4.2. Parameter Grid for Likelihood Analysis. From Sievers et al. (2003)

Compression

To calculate the compression matrix, we need to diagonalize the blocks along the diagonal of the covariance matrix. Another useful feature of compression is that it (usually) only needs to be done once.

Mosaic Window Functions

General Mosaic Window Functions

Because only the data itself changes between different realizations and not the statistical properties, the only compression required is again that of the data vector.

Gaussian Beam

The terms involving A and B do not have any θ dependence, so they can be drawn from the integral. So instead of making separate calls to different Bessel functions, we can add the sum of the products of the Bessel functions and normalize them at the end, making the whole integral only marginally more work than two calls to Bessel routines from high order.

Comparisons with Other Methods

BOOMERANG (described in Hivon et al., 2002), as they require a fast spherical harmonic transformation of the data to calculate the window matrices. See Table 5.2 for a summary of the comparison statistics, and Figures 5.8 and 5.9 for CBISPEC and CBIGRIDR fits to the first (high signal) and last (high noise) bins, respectively.

Foreground with CBISPEC

Measuring the Spectral Index

The red dots are the expected variances for the same Cℓ spectrum, with an applied frequency spectral index, forα= 0. The addition of the 125 cm baseline has broken the degeneracy between the flat,ν0 spectrum and the ℓ−6.4, Planck spectrum.

Figure 5.10 Figure showing the degeneracy for a single baseline between a tilt in the power spectrum ( C ℓ ∝ ℓ γ ) and a flat power spectrum with a non-Black Body spectrum

The Spectral Index Measured by CBI

See Figure 5.12 for the histogram of the best fit values for α for the individual simulations. The blue crosses are the four contaminated spots at the northern end of the 02 hour mosaic.

Figure 5.12 Histogram of spectral index fits to a flat band power CMB model, made using simulations based on the 02 hour mosaic

Future Improvements

The CBI is a highly sensitive interferometer operating at 30 GHz optimized for observations of the CMB in the multipole range 500 < ℓ <3500. Initially, I helped build the CBI, including mounting and testing the CBI receivers.

Figure 5.13 Figure showing the distribution of spectral indices of the individual 3 by 3 chunks of the CBI data, plotted against their low-ℓ power levels

Statistical Basics

Variance of a Product

We need to know the variance of the product of two independent random variables (not necessarily identically distributed). It is also worth noting explicitly that if the expected values of the variables are zero, the variance of the product is the product of the variances: Var(xy) = Var(x)Var(y).

Expectation of f(x)

We can break the expectation into different terms as the sum expectation is the sum of the expectations. If we set the reference valuex0 to be the expectation ex, then the second term goes to zero, since hx−x0i=hxi −x0= 0 ifx0=hxi.

Some Relevant Distributions

The general expectation relationship is established by integrating parts and comparing the resulting integral with the expectation value of the underlying order. It is the distribution of the ratio of two empirically determined variance estimates when they come from samples with the same intrinsic variance.

Combining Two Identical Data Points

However, if the points are widely spaced, they must actually give a better estimate of the mean. So if we want the expected variance of V to be the same as the expectation of our estimate, we need to scale by a factor.

Combining Many Identical Data Points

As cluster size decreases, longer and longer baselines are expected to persist. Essentially, exploring a larger region allows us to better characterize the behavior of the CMB below the cluster.