Nonparametric Methods in Astronomy: Think, Regress, Observe -- Pick Any Three

Telescopes are much more expensive than astronomers, so it is essential to minimize required sample sizes by using the most data-efficient statistical methods possible. However, the most commonly used model-independent techniques for finding the relationship between two variables in astronomy are flawed. In the worst case they can lead without warning to subtly yet catastrophically wrong results, and even in the best case they require more data than necessary. Unfortunately, there is no single best technique for nonparametric regression. Instead, we provide a guide for how astronomers can choose the best method for their specific problem and provide a python library with both wrappers for the most useful existing algorithms and implementations of two new algorithms developed here.Comment: 19 pages, PAS

Testing Inflation: A Bootstrap Approach

We note that the essential idea of inflation, that the universe underwent a brief period of accelerated expansion followed by a long period of decelerated expansion, can be encapsulated in a "closure condition" which relates the amount of accelerated expansion during inflation to the amount of decelerated expansion afterward. We present a protocol for systematically testing the validity of this condition observationally.Comment: 4 pages, 2 figures, matches Phys. Rev. Lett. versio

The Cosmic Microwave Background and the Stellar Initial Mass Function

We argue that an increased temperature in star-forming clouds alters the stellar initial mass function to be more bottom-light than in the Milky Way. At redshifts $z \gtrsim 6$, heating from the cosmic microwave background radiation produces this effect in all galaxies, and it is also present at lower redshifts in galaxies with very high star formation rates (SFRs). A failure to account for it means that at present, photometric template fitting likely overestimates stellar masses and star formation rates for the highest-redshift and highest-SFR galaxies. In addition this may resolve several outstanding problems in the chemical evolution of galactic halos.Comment: 9 pages, 5 figures. Published in MNRAS. Added further reference

Local growth of icosahedral quasicrystalline tilings

Icosahedral quasicrystals (IQCs) with extremely high degrees of translational order have been produced in the laboratory and found in naturally occurring minerals, yet questions remain about how IQCs form. In particular, the fundamental question of how locally determined additions to a growing cluster can lead to the intricate long-range correlations in IQCs remains open. In answer to this question, we have developed an algorithm that is capable of producing a perfectly ordered IQC, yet relies exclusively on local rules for sequential, face-to-face addition of tiles to a cluster. When the algorithm is seeded with a special type of cluster containing a defect, we find that growth is forced to infinity with high probability and that the resultant IQC has a vanishing density of defects. The geometric features underlying this algorithm can inform analyses of experimental systems and numerical models that generate highly ordered quasicrystals.Comment: 13 pages, 15 figures, 1 tabl

Interacting dark energy, holographic principle and coincidence problem

The interacting and holographic dark energy models involve two important quantities. One is the characteristic size of the holographic bound and the other is the coupling term of the interaction between dark energy and dark matter. Rather than fixing either of them, we present a detailed study of theoretical relationships among these quantities and cosmological parameters as well as observational constraints in a very general formalism. In particular, we argue that the ratio of dark matter to dark energy density depends on the choice of these two quantities, thus providing a mechanism to change the evolution history of the ratio from that in standard cosmology such that the coincidence problem may be solved. We investigate this problem in detail and construct explicit models to demonstrate that it may be alleviated provided that the interacting term and the characteristic size of holographic bound are appropriately specified. Furthermore, these models are well fitted with the current observation at least in the low red-shift region.Comment: 20 pages, 3 figure

Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations

We derive the complete spectrum of gravitational waves induced by primordial scalar perturbations ranging over all observable wavelengths. This scalar-induced contribution can be computed directly from the observed scalar perturbations and general relativity and is, in this sense, independent of the cosmological model for generating the perturbations. The spectrum is scale-invariant on small scales, but has an interesting scale-dependence on large and intermediate scales, where scalar-induced gravitational waves do not redshift and are hence enhanced relative to the background density of the Universe. This contribution to the tensor spectrum is significantly different in form from the direct model-dependent primordial tensor spectrum and, although small in magnitude, it dominates the primordial signal for some cosmological models. We confirm our analytical results by direct numerical integration of the equations of motion.Comment: 19 pages, 5 figure

Hyperuniformity of Quasicrystals

Hyperuniform systems, which include crystals, quasicrystals and special disordered systems, have attracted considerable recent attention, but rigorous analyses of the hyperuniformity of quasicrystals have been lacking because the support of the spectral intensity is dense and discontinuous. We employ the integrated spectral intensity, $Z(k)$, to quantitatively characterize the hyperuniformity of quasicrystalline point sets generated by projection methods. The scaling of $Z(k)$ as $k$ tends to zero is computed for one-dimensional quasicrystals and shown to be consistent with independent calculations of the variance, $\sigma^2(R)$, in the number of points contained in an interval of length $2R$. We find that one-dimensional quasicrystals produced by projection from a two-dimensional lattice onto a line of slope $1/\tau$ fall into distinct classes determined by the width of the projection window. For a countable dense set of widths, $Z(k) \sim k^4$; for all others, $Z(k)\sim k^2$. This distinction suggests that measures of hyperuniformity define new classes of quasicrystals in higher dimensions as well.Comment: 12 pages, 14 figure

A Highly Consistent Framework for the Evolution of the Star-Forming "Main Sequence" from z~0-6

Using a compilation of 25 studies from the literature, we investigate the evolution of the star-forming galaxy (SFG) Main Sequence (MS) in stellar mass and star formation rate (SFR) out to $z \sim 6$. After converting all observations to a common set of calibrations, we find a remarkable consensus among MS observations ($\sim 0.1$ dex 1$\sigma$ interpublication scatter). By fitting for time evolution of the MS in bins of constant mass, we deconvolve the observed scatter about the MS within each observed redshift bins. After accounting for observed scatter between different SFR indicators, we find the width of the MS distribution is $\sim 0.2$ dex and remains constant over cosmic time. Our best fits indicate the slope of the MS is likely time-dependent, with our best fit $\log\textrm{SFR}(M_*,t) = \left(0.84 \pm 0.02 - 0.026 \pm 0.003 \times t\right) \log M_* - \left(6.51 \pm 0.24 - 0.11 \pm 0.03 \times t\right)$, with $t$ the age of the Universe in Gyr. We use our fits to create empirical evolutionary tracks in order to constrain MS galaxy star formation histories (SFHs), finding that (1) the most accurate representations of MS SFHs are given by delayed-$\tau$ models, (2) the decline in fractional stellar mass growth for a "typical" MS galaxy today is approximately linear for most of its lifetime, and (3) scatter about the MS can be generated by galaxies evolving along identical evolutionary tracks assuming an initial $1\sigma$ spread in formation times of $\sim 1.4$ Gyr.Comment: 59 pages, 10 tables, 12 figures, accepted to ApJS; v2, slight changes to text, added new figure and fit