How to make maps from CMB data without losing information

The next generation of CMB experiments can measure cosmological parameters with unprecedented accuracy - in principle. To achieve this in practice when faced with such gigantic data sets, elaborate data analysis methods are needed to make it computationally feasible. An important step in the data pipeline is to make a map, which typically reduces the size of the data set my orders of magnitude. We compare ten map-making methods, and find that for the Gaussian case, both the method used by the COBE DMR team and various variants of Wiener filtering are optimal in the sense that the map retains all cosmological information that was present in the time-ordered data (TOD). Specifically, one obtains just as small error bars on cosmological parameters when estimating them from the map as one could have obtained by estimating them directly from the TOD. The method of simply averaging the observations of each pixel (for total-power detectors), on the contrary, is found to generally destroy information, as does the maximum entropy method and most other non-linear map-making techniques. Since it is also numerically feasible, the COBE method is the natural choice for large data sets. Other lossless (e.g. Wiener-filtered) maps can then be computed directly from the COBE method map.Comment: Minor revisions to match published version. 12 pages, with 1 figure included. Color figure and links at http://www.sns.ias.edu/~max/mapmaking.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/mapmaking.html (faster from Europe) or from [email protected]

Consciousness as a State of Matter

We examine the hypothesis that consciousness can be understood as a state of matter, "perceptronium", with distinctive information processing abilities. We explore five basic principles that may distinguish conscious matter from other physical systems such as solids, liquids and gases: the information, integration, independence, dynamics and utility principles. If such principles can identify conscious entities, then they can help solve the quantum factorization problem: why do conscious observers like us perceive the particular Hilbert space factorization corresponding to classical space (rather than Fourier space, say), and more generally, why do we perceive the world around us as a dynamic hierarchy of objects that are strongly integrated and relatively independent? Tensor factorization of matrices is found to play a central role, and our technical results include a theorem about Hamiltonian separability (defined using Hilbert-Schmidt superoperators) being maximized in the energy eigenbasis. Our approach generalizes Giulio Tononi's integrated information framework for neural-network-based consciousness to arbitrary quantum systems, and we find interesting links to error-correcting codes, condensed matter criticality, and the Quantum Darwinism program, as well as an interesting connection between the emergence of consciousness and the emergence of time.Comment: Replaced to match accepted CSF version; discussion improved, typos corrected. 36 pages, 15 fig

Doppler peaks and all that: CMB anisotropies and what they can tell us

The power spectrum of fluctuations in the cosmic microwave background (CMB) depends on most of the key cosmological parameters. Accurate future measurements of this power spectrum might therefore allow us to determine h, Omega, Omega_b, Lambda, n, T/S, etc, with hitherto unprecedented accuracy. In these lecture notes, which are intended to be readable without much prior CMB knowledge, I review the various physical processes that generate CMB fluctuations, focusing on how changes in the parameters alters the shape of the power spectrum. I also discuss foregrounds and real-world data analysis issues and how these affect the accuracy with which the parameters can be measured.Comment: 40 pages, including 12 figures. Postscript. Latest version available from http://astro.berkeley.edu/~max/power.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/power.html (faster from Europe) or from [email protected]

A method for extracting maximum resolution power spectra from microwave sky maps

A method for extracting maximal resolution power spectra from microwave sky maps is presented and applied to the 2 year COBE data, yielding a power spectrum that is consistent with a standard n=1, Q=20 micro-Kelvin model. By using weight functions that fall off smoothly near the galactic cut, it is found that the spectral resolution \Delta l can be more than doubled at l=15 and more than tripled at l=20 compared to simply using galaxy-cut spherical harmonics. For a future high-resolution experiment with reasonable sky coverage, the resolution around the CDM Doppler peaks would be enhanced by a factor of about 100, down to \Delta l\approx 1, thus allowing spectral features such as the locations of the peaks to be determined with great accuracy. The reason that the improvement is so large is basically that functions with a sharp edge at the galaxy cut exhibit considerable "ringing" in the Fourier domain, whereas smooth functions do not. The method presented here is applicable to any survey geometry, chopping strategy and exposure pattern whatsoever. The so called signal-to-noise eigenfunction technique is found to be a special case, corresponding to ignoring the width of the window functions.Comment: 25 pages, including 4 figures. Postscript. Substantially revised, more than twice original length, matches accepted version. Latest version available from http://www.sns.ias.edu/~max/window.html (faster from the US), from http://www.mpa-garching.mpg.de/~max/window.html (faster from Europe) or from [email protected]