The Average Projected Area Theorem - Generalization to Higher Dimensions

In 3-d the average projected area of a convex solid is 1/4 the surface area, as Cauchy showed in the 19th century. In general, the ratio in n dimensions may be obtained from Cauchy's surface area formula, which is in turn a special case of Kubota's theorem. However, while these latter results are well-known to those working in integral geometry or the theory of convex bodies, the results are largely unknown to the physics community---so much so that even the 3-d result is sometimes said to have first been proven by an astronomer in the early 20th century! This is likely because the standard proofs in the mathematical literature are, by and large, couched in terms of concepts that are may not be familiar to many physicists. Therefore, in this work, we present a simple geometrical method of calculating the ratio of average projected area to surface area for convex bodies in arbitrary dimensions. We focus on a pedagogical, physically intuitive treatment that it is hoped will be useful to those in the physics community. We do discuss the mathematical background of the theorem as well, pointing those who may be interested to sources that offer the proofs that are standard in the fields of integral geometry and the theory of convex bodies. We also provide discussion of the applications of the theorem, especially noting that higher-dimensional ratios may be of use for constructing observational tests of string theory. Finally, we examine the limiting behavior of the ratio with the goal of offering intuition on its behavior by pointing out a suggestive connection with a well-known fact in statistics.Comment: 12 pages, 3 figures, submitted JGP after addition of discussion of previous work on this topi

Accelerating Computation of the Nonlinear Mass by an Order of Magnitude

The nonlinear mass is a characteristic scale in halo formation that has wide-ranging applications across cosmology. Naively, computing it requires repeated numerical integration to calculate the variance of the power spectrum on different scales and determine which scales exceed the threshold for nonlinear collapse. We accelerate this calculation by working in configuration space and approximating the correlation function as a polynomial at r <= 5 $h^{-1}$ Mpc. This enables an analytic rather than numerical solution, accurate across a variety of cosmologies to 0.1$-$1% (depending on redshift) and 10$-$20 times faster than the naive numerical method. We also present a further acceleration (40$-$80 times faster than the naive method) in which we determine the polynomial coefficients using a Taylor expansion in the cosmological parameters rather than re-fitting a polynomial to the correlation function. Our acceleration greatly reduces the cost of repeated calculation of the nonlinear mass. This will be useful for MCMC analyses to constrain cosmological parameters from the highly nonlinear regime, e.g. with data from upcoming surveys

Ruling Out Bosonic Repulsive Dark Matter in Thermal Equilibrium

Self-interacting dark matter (SIDM), especially bosonic, has been considered a promising candidate to replace cold dark matter (CDM) as it resolves some of the problems associated with CDM. Here, we rule out the possibility that dark matter is a repulsive boson in thermal equilibrium. We develop the model first proposed by Goodman (2000) and derive the equation of state at finite temperature. Isothermal spherical halo models indicate a Bose-Einstein condensed core surrounded by a non-degenerate envelope, with an abrupt density drop marking the boundary between the two phases. Comparing this feature with observed rotation curves constrains the interaction strength of our model's DM particle, and Bullet Cluster measurements constrain the scattering cross section. Both ultimately can be cast as constraints on the particle's mass. We find these two constraints cannot be satisfied simultaneously in any realistic halo model---and hence dark matter cannot be a repulsive boson in thermal equilibrium. It is still left open that DM may be a repulsive boson provided it is not in thermal equilibrium; this requires that the mass of the particle be significantly less than a millivolt.Comment: 13 pages, 3 figures, 1 table, accepted MNRAS August 9 201

Modeling the large-scale redshift-space 3-point correlation function of galaxies

We present a configuration-space model of the large-scale galaxy 3-point correlation function (3PCF) based on leading-order perturbation theory and including redshift space distortions (RSD). This model should be useful in extracting distance-scale information from the 3PCF via the Baryon Acoustic Oscillation (BAO) method. We include the first redshift-space treatment of biasing by the baryon-dark matter relative velocity. Overall, on large scales the effect of RSD is primarily a renormalization of the 3PCF that is roughly independent of both physical scale and triangle opening angle; for our adopted $\Omega_{\rm m}$ and bias values, the rescaling is a factor of $\sim 1.8$. We also present an efficient scheme for computing 3PCF predictions from our model, important for allowing fast exploration of the space of cosmological parameters in future analyses.Comment: 23 pages, 11 figures, submitted MNRA