Measuring the Hubble Constant with the Hubble Space Telescope

Ten years ago our team completed the Hubble Space Telescope Key Project on the extragalactic distance scale. Cepheids were detected in some 25 galaxies and used to calibrate four secondary distance indicators that reach out into the expansion field beyond the noise of galaxy peculiar velocities. The result was H_0 = 72 +/- 8 km/sec/Mpc and put an end to galaxy distances uncertain by a factor of two. This work has been awarded the Gruber Prize in Cosmology for 2009.Comment: Gruber Prize Lecture to be published in Transactions of the IA

On Tractable Exponential Sums

We consider the problem of evaluating certain exponential sums. These sums take the form $\sum_{x_1,...,x_n \in Z_N} e^{f(x_1,...,x_n) {2 \pi i / N}}$, where each x_i is summed over a ring Z_N, and f(x_1,...,x_n) is a multivariate polynomial with integer coefficients. We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \ge 3, we show the problem is #P-hard, even for any fixed prime or prime power modulus. This leads to a complexity dichotomy theorem - a complete classification of each problem to be either computable in polynomial time or #P-hard - for a class of exponential sums. These sums arise in the classifications of graph homomorphisms and some other counting CSP type problems, and these results lead to complexity dichotomy theorems. For the polynomial-time algorithm, Gauss sums form the basic building blocks. For the hardness results, we prove group-theoretic necessary conditions for tractability. These tests imply that the problem is #P-hard for even very restricted families of simple cubic polynomials over fixed modulus N

On the energy of gamma-ray bursts

We show that gamma-ray burst (GRB) afterglow observations strongly suggest, within the fireball model framework, that radiating electrons are shock accelerated to a power-law energy distribution, with universal index p \approx 2.2, and that the fraction of shock energy carried by electrons, \xi_e, is universal and close to equipartition, \xi_e ~ 1/3. For universal p and \xi_e, a single measurement of the X-ray afterglow flux on the time scale of a day provides a robust estimate of the fireball energy per unit solid angle, \epsilon, averaged over a conical section of the fireball of opening angle \theta ~ 0.1. Applying our analysis to BeppoSAX afterglow data we find that: (i) Fireball energies are in the range of 4\pi\epsilon=10^{51.5} to 10^{53.5} erg; (ii) The ratio of observed $\gamma$-ray to total fireball energy per unit solid angle, \epsilon_\gamma / \epsilon, is of order unity, satisfying abs[log10(\epsilon_\gamma/\epsilon)]<0.5; (iii) If fireballs are jet like, their opening angle should satisfy \theta>=0.1. Our results imply that if typical opening angles are \theta ~ 0.1, a value consistent with our analysis, the total energy associated with a GRB event is in the range of 10^{50} erg to 10^{51.5} erg.Comment: 16 pages; Submitted to Ap

The Measure of Cosmological Parameters

New, large, ground and space telescopes are contributing to an exciting and rapid period of growth in observational cosmology. The subject is now far from its earlier days of being data-starved and unconstrained, and new data are fueling a healthy interplay between observations and experiment and theory. I briefly review here the status of measurements of a number of quantities of interest in cosmology: the Hubble constant, the total mass-energy density, the matter density, the cosmological constant or dark energy component, and the total optical background light.Comment: 12 pages, 4 figures, to be published in "2001: A Spacetime Odyssey: Proceedings of the Inaugural Conference of the Michigan Center for Theoretical Physics", Michael J. Duff & James T. Liu, eds., (World Scientific, Singapore), in pres

Neutrino mass constraint from CMB and its degeneracy with other cosmological parameters

We show that the cosmic microwave background (CMB) data of WMAP can give subelectronvolt limit on the neutrino mass: m_nu < 0.63 eV (95% CL). We also investigate its degeneracy with other cosmological parameters. In particular, we show the Hubble constant derived from the WMAP data decreases considerably when the neutrino mass is a few times 0.1 eV.Comment: 3 pages, 2 figures, prepared for the TAUP2007 Proceeding

Cepheid and Tip of the Red Giant Branch Distances To the Dwarf Irregular Galaxy IC10

We present color-magnitude diagrams and luminosity functions of stars in the nearby galaxy IC 10, based on VI CCD photometry acquired with the COSMIC prime-focus camera on the Palomar 5m telescope. The apparent I-band luminosity function of stars in the halo of IC 10 shows an identifiable rise at I~21.7 mag. This is interpreted as being the tip of the red giant branch (TRGB) at M_V~-4 mag. Since IC 10 is at a very low Galactic latitude, its foreground extinction is expected to be high and the uncertainty associated with that correction is the largest contributor to the error associated with its distance determination. Multi-wavelength observations of Cepheid variable stars in IC 10 give a Population I distance modulus of 24.1 +- 0.2 mag, which corresponds to a linear distance of 660 +- 66 kpc for a total line-of-sight reddening of E(B-V) = 1.16 +- 0.08 mag, derived self-consistently from the Cepheid data alone. Applying this Population I reddening to the Population II halo stars gives a TRGB distance modulus of 23.5 +- 0.2 mag, corresponding to 500 +- 50 kpc. We consider this to be a lower limit on the TRGB distance. Reconciling the Cepheid and TRGB distances would require that the reddening to the halo is $\Delta$E(B-V) = 0.31 mag lower than that into the main body of the galaxy. This then suggests that the Galactic extinction in the direction of IC10 is (B-V) ~ 0.85

Computational equivalence of the two inequivalent spinor representations of the braid group in the Ising topological quantum computer

We demonstrate that the two inequivalent spinor representations of the braid group \B_{2n+2}, describing the exchanges of 2n+2 non-Abelian Ising anyons in the Pfaffian topological quantum computer, are equivalent from computational point of view, i.e., the sets of topologically protected quantum gates that could be implemented in both cases by braiding exactly coincide. We give the explicit matrices generating almost all braidings in the spinor representations of the 2n+2 Ising anyons, as well as important recurrence relations. Our detailed analysis allows us to understand better the physical difference between the two inequivalent representations and to propose a process that could determine the type of representation for any concrete physical realization of the Pfaffian quantum computer.Comment: 9 pages, 2 figures, published versio

Socioeconomic Status and Medical Care Expenditures in Medicare Managed Care

This study examined the effects of education, income, and wealth on medical care expenditures in two Medicare managed care plans. The study also sought to elucidate the pathways through which socioeconomic status (SES) affects expenditures, including preferences for health and medical care and ability to navigate the managed care system. We modeled the effect of SES on medical care expenditures using Generalized Linear Models, estimating separate models for each component of medical expenditures: inpatient, outpatient, physician, and other expenditures. We found that education, income, and wealth all affected medical care expenditures, although the effects of these variables differed across expenditure categories. Moreover, the effects of these SES variables were much smaller than the effects found in earlier studies of fee-for-service Medicare. The pathway variables also were associated with expenditures. Accounting for the pathways through which SES affects expenditures narrowed the effect of SES on expenditures; however, the change in the estimates was very small. Thus, although our measures of preferences and ability to navigate the system were associated with expenditures, they did not account for an appreciable share of the impact of SES on expenditures.