The probability distribution for non-Gaussianity estimators constructed from the CMB trispectrum

Considerable recent attention has focussed on the prospects to use the cosmic microwave background (CMB) trispectrum to probe the physics of the early universe. Here we evaluate the probability distribution function (PDF) for the standard estimator tau_nle for the amplitude tau_nl of the CMB trispectrum both for the null-hypothesis (i.e., for Gaussian maps with tau_nl = 0) and for maps with a non-vanishing trispectrum (|tau_nl|>0). We find these PDFs to be highly non-Gaussian in both cases. We also evaluate the variance with which the trispectrum amplitude can be measured, , as a function of its underlying value, tau_nl. We find a strong dependence of this variance on tau_nl. We also find that the variance does not, given the highly non-Gaussian nature of the PDF, effectively characterize the distribution. Detailed knowledge of these PDFs will therefore be imperative in order to properly interpret the implications of any given trispectrum measurement. For example, if a CMB experiment with a maximum multipole of lmax = 1500 (such as the Planck satellite) measures tau_nle = 0 then at the 95% confidence our calculations show that we can conclude tau_nl < 1005; assuming a Gaussian PDF but with the correct tau_nl-dependent variance we would incorrectly conclude tau_nl < 4225; further neglecting the tau_nl-dependence in the variance we would incorrectly conclude tau_nl < 361.Comment: 9 pages, 5 figure

Precautionary saving and portfolio allocation: DP by GMM

There is much research on consumption-savings problems with risky labor income and a constant interest rate and also on portfolio allocation with risky returns but nonstochastic labor income. Less is known quantitatively about the interaction between the two forms of risk. Under CRRA utility, undiversifiable income risk should be reflected in both savings rates and portfolio allocations. To quantify these effects in a model of consumption and portfolio choice, we adopt a semi-parametric projection method for solving dynamic programmes, based on generalized method of moments estimation of the parameters of approximate decision rules. We find that background income risk does affect optimal portfolios but that this effect may be difficult to detect empirically.portfolio theory, precautionary saving

Inflationary gravitational-wave background and measurements of the scalar spectral index

Inflation predicts a stochastic background of gravitational waves over a broad range of frequencies, from those accessible with cosmic microwave background (CMB) measurements, to those accessible directly with gravitational-wave detectors, like NASA's Big-Bang Observer (BBO), currently under study. In a previous paper [Phys. Rev. D 73, 023504 (2006)] we connected CMB constraints to the amplitude and tensor spectral tilt of the inflationary gravitational-wave background (IGWB) at BBO frequencies for four classes of models of inflation by directly solving the inflationary equations of motion. Here we extend that analysis by including results obtained in the Wilkinson Microwave Anisotropy Probe third-year data release as well as by considering two additional classes of inflationary models. As often noted in the literature, the recent indication that the primordial density power spectrum has a red spectral index implies (with some caveats) that the amplitude of the IGWB may be large enough to be observable in the CMB polarization. Here we also explore the implications for the direct detection of the IGWB

Household Formations

Between 1960 and 1980, the number of households in the U.S. increased by 50 percent and the proportion of the population that were household heads rose from 29.5 to 36.3. While some of this increase was due to the maturing of the"baby boom" population, over half was caused by rising age-specific headship rates. In contrast, between 1980 and 1983, headship rates fell sharply for the under 34 population. This paper explains household formations due to changes in headship rates in terms of changes in real income and the price of privacy.

Direct detection of the inflationary gravitational-wave background

Inflation generically predicts a stochastic background of gravitational waves over a broad range of frequencies, from those accessible with cosmic microwave background (CMB) measurements, to those accessible directly with gravitational-wave detectors, like NASA's Big-Bang Observer (BBO) or Japan's Deci-Hertz Interferometer Gravitational-wave Observer (DECIGO), both currently under study. Here we investigate the detectability of the inflationary gravitational-wave background at BBO/DECIGO frequencies. To do so, we survey a range of slow-roll inflationary models consistent with constraints from the CMB and large-scale structure (LSS). We go beyond the usual assumption of power-law power spectra, which may break down given the 16 orders of magnitude in frequency between the CMB and direct detection, and solve instead the inflationary dynamics for four classes of inflaton potentials. Direct detection is possible in a variety of inflationary models, although probably not in any in which the gravitational-wave signal does not appear in the CMB polarization. However, direct detection by BBO/DECIGO can help discriminate between inflationary models that have the same slow-roll parameters at CMB/LSS scales

An improved estimator for non-Gaussianity in cosmic microwave background observations

An improved estimator for the amplitude fnl of local-type non-Gaussianity from the cosmic microwave background (CMB) bispectrum is discussed. The standard estimator is constructed to be optimal in the zero-signal (i.e., Gaussian) limit. When applied to CMB maps which have a detectable level of non-Gaussianity the standard estimator is no longer optimal, possibly limiting the sensitivity of future observations to a non-Gaussian signal. Previous studies have proposed an improved estimator by using a realization-dependent normalization. Under the approximations of a flat sky and a vanishingly thin last-scattering surface, these studies showed that the variance of this improved estimator can be significantly smaller than the variance of the standard estimator when applied to non-Gaussian CMB maps. Here this technique is generalized to the full sky and to include the full radiation transfer function, yielding expressions for the improved estimator that can be directly applied to CMB maps. The ability of this estimator to reduce the variance as compared to the standard estimator in the face of a significant non-Gaussian signal is re-assessed using the full CMB transfer function. As a result of the late time integrated Sachs-Wolfe effect, the performance of the improved estimator is degraded. If CMB maps are first cleaned of the late-time ISW effect using a tracer of foreground structure, such as a galaxy survey or a measurement of CMB weak lensing, the new estimator does remove a majority of the excess variance, allowing a higher significance detection of fnl.Comment: 21 pages, 7 figure

The Probability Distribution for Non-Gaussianity Estimators

One of the principle efforts in cosmic microwave background (CMB) research is measurement of the parameter fnl that quantifies the departure from Gaussianity in a large class of non-minimal inflationary (and other) models. Estimators for fnl are composed of a sum of products of the temperatures in three different pixels in the CMB map. Since the number ~Npix^2 of terms in this sum exceeds the number Npix of measurements, these ~Npix^2 terms cannot be statistically independent. Therefore, the central-limit theorem does not necessarily apply, and the probability distribution function (PDF) for the fnl estimator does not necessarily approach a Gaussian distribution for N_pix >> 1. Although the variance of the estimators is known, the significance of a measurement of fnl depends on knowledge of the full shape of its PDF. Here we use Monte Carlo realizations of CMB maps to determine the PDF for two minimum-variance estimators: the standard estimator, constructed under the null hypothesis (fnl=0), and an improved estimator with a smaller variance for |fnl| > 0. While the PDF for the null-hypothesis estimator is very nearly Gaussian when the true value of fnl is zero, the PDF becomes significantly non-Gaussian when |fnl| > 0. In this case we find that the PDF for the null-hypothesis estimator fnl_hat is skewed, with a long non-Gaussian tail at fnl_hat > |fnl| and less probability at fnl_hat < |fnl| than in the Gaussian case. We provide an analytic fit to these PDFs. On the other hand, we find that the PDF for the improved estimator is nearly Gaussian for observationally allowed values of fnl. We discuss briefly the implications for trispectrum (and other higher-order correlation) estimators.Comment: 10 pages, 6 figures, comments welcom