Robust Simulations and Significant Separations

We define and study a new notion of "robust simulations" between complexity classes which is intermediate between the traditional notions of infinitely-often and almost-everywhere, as well as a corresponding notion of "significant separations". A language L has a robust simulation in a complexity class C if there is a language in C which agrees with L on arbitrarily large polynomial stretches of input lengths. There is a significant separation of L from C if there is no robust simulation of L in C. The new notion of simulation is a cleaner and more natural notion of simulation than the infinitely-often notion. We show that various implications in complexity theory such as the collapse of PH if NP = P and the Karp-Lipton theorem have analogues for robust simulations. We then use these results to prove that most known separations in complexity theory, such as hierarchy theorems, fixed polynomial circuit lower bounds, time-space tradeoffs, and the theorems of Allender and Williams, can be strengthened to significant separations, though in each case, an almost everywhere separation is unknown. Proving our results requires several new ideas, including a completely different proof of the hierarchy theorem for non-deterministic polynomial time than the ones previously known

On relaxation processes in collisionless mergers

We analyze N-body simulations of halo mergers to investigate the mechanisms responsible for driving mixing in phase-space and the evolution to dynamical equilibrium. We focus on mixing in energy and angular momentum and show that mixing occurs in step-like fashion following pericenter passages of the halos. This makes mixing during a merger unlike other well known mixing processes such as phase mixing and chaotic mixing whose rates scale with local dynamical time. We conclude that the mixing process that drives the system to equilibrium is primarily a response to energy and angular momentum redistribution that occurs due to impulsive tidal shocking and dynamical friction rather than a result of chaotic mixing in a continuously changing potential. We also analyze the merger remnants to determine the degree of mixing at various radii by monitoring changes in radius, energy and angular momentum of particles. We confirm previous findings that show that the majority of particles retain strong memory of their original kinetic energies and angular momenta but do experience changes in their potential energies owing to the tidal shocks they experience during pericenter passages. Finally, we show that a significant fraction of mass (~ 40%) in the merger remnant lies outside its formal virial radius and that this matter is ejected roughly uniformly from all radii outside the inner regions. This highlights the fact that mass, in its standard virial definition, is not additive in mergers. We discuss the implications of these results for our understanding of relaxation in collisionless dynamical systems.Comment: Version accepted for Publication in Astrophysical Journal, March 20, 2007, v685. Minor changes, latex, 14 figure

Close Pairs as Proxies for Galaxy Cluster Mergers

Galaxy cluster merger statistics are an important component in understanding the formation of large-scale structure. Unfortunately, it is difficult to study merger properties and evolution directly because the identification of cluster mergers in observations is problematic. We use large N-body simulations to study the statistical properties of massive halo mergers, specifically investigating the utility of close halo pairs as proxies for mergers. We examine the relationship between pairs and mergers for a wide range of merger timescales, halo masses, and redshifts (0<z<1). We also quantify the utility of pairs in measuring merger bias. While pairs at very small separations will reliably merge, these constitute a small fraction of the total merger population. Thus, pairs do not provide a reliable direct proxy to the total merger population. We do find an intriguing universality in the relation between close pairs and mergers, which in principle could allow for an estimate of the statistical merger rate from the pair fraction within a scaled separation, but including the effects of redshift space distortions strongly degrades this relation. We find similar behavior for galaxy-mass halos, making our results applicable to field galaxy mergers at high redshift. We investigate how the halo merger rate can be statistically described by the halo mass function via the merger kernel (coagulation), finding an interesting environmental dependence of merging: halos within the mass resolution of our simulations merge less efficiently in overdense environments. Specifically, halo pairs with separations less than a few Mpc/h are more likely to merge in underdense environments; at larger separations, pairs are more likely to merge in overdense environments.Comment: 12 pages, 9 figures; Accepted for publication in ApJ. Significant additions to text and two figures changed. Added new findings on the universality of pair mergers and added analysis of the effect of FoF linking length on halo merger

Comparing Simulations and Observations of the Lyman-Alpha Forest I. Methodology

We describe techniques for comparing spectra extracted from cosmological simulations and observational data, using the same methodology to link Lyman-alpha properties derived from the simulations with properties derived from observational data. The eventual goal is to measure the coherence or clustering properties of Lyman-alpha absorbers using observations of quasar pairs and groups. We quantify the systematic underestimate in opacity that is inherent in the continuum fitting process of observed spectra over a range of resolution and signal-to-noise ratio. We present an automated process for detecting and selecting absorption features over the range of resolution and signal-to-noise of typical observational data on the Lyman-alpha "forest". Using these techniques, we detect coherence over transverse scales out to 500 h^{-1}_{50} kpc in spectra extracted from a cosmological simulation at z = 2.Comment: 52 pages, includes 14 figures, to appear in ApJ v566 Feb 200

On the Clustering of Sub-millimeter Galaxies

We measure the angular two-point correlation function of sub-millimeter galaxies (SMGs) from 1.1-millimeter imaging of the COSMOS field with the AzTEC camera and ASTE 10-meter telescope. These data yields one of the largest contiguous samples of SMGs to date, covering an area of 0.72 degrees^2 down to a 1.26 mJy/beam (1-sigma) limit, including 189 (328) sources with S/N greater than 3.5 (3). We can only set upper limits to the correlation length r_0, modeling the correlation function as a power-law with pre-assigned slope. Assuming existing redshift distributions, we derive 68.3% confidence level upper limits of r_0 < 6-8 h^-1 Mpc at 3.7 mJy, and r_0 < 11-12 h^-1 Mpc at 4.2 mJy. Although consistent with most previous estimates, these upper limits imply that the real r_0 is likely smaller. This casts doubts on the robustness of claims that SMGs are characterized by significantly stronger spatial clustering, (and thus larger mass), than differently selected galaxies at high-redshift. Using Monte Carlo simulations we show that even strongly clustered distributions of galaxies can appear unclustered when sampled with limited sensitivity and coarse angular resolution common to current sub-millimeter surveys. The simulations, however, also show that unclustered distributions can appear strongly clustered under these circumstances. From the simulations, we predict that at our survey depth, a mapped area of two degrees^2 is needed to reconstruct the correlation function, assuming smaller beam sizes of future surveys (e.g. the Large Millimeter Telescope's 6" beam size). At present, robust measures of the clustering strength of bright SMGs appear to be below the reach of most observations.Comment: 23 pages, 8 figures, accepted for publication in The Astrophysical Journa

Not Alone: Tracing the Origins of Very Low Mass Stars and Brown Dwarfs Through Multiplicity Studies

The properties of multiple stellar systems have long provided important empirical constraints for star formation theories, enabling (along with several other lines of evidence) a concrete, qualitative picture of the birth and early evolution of normal stars. At very low masses (VLM; M <~ 0.1 M_sun), down to and below the hydrogen burning minimum mass, our understanding of formation processes is not as clear, with several competing theories now under consideration. One means of testing these theories is through the empirical characterization of VLM multiple systems. Here, we review the results of various VLM multiplicity studies to date. These systems can be generally characterized as closely separated (93% have projected separations Delta < 20 AU) and near equal-mass (77% have M_2/M_1 >= 0.8) occurring infrequently (perhaps 10-30%). Both the frequency and maximum separation of stellar and brown dwarf binaries steadily decrease for lower system masses, suggesting that VLM binary formation and/or evolution may be a mass-dependent process. There is evidence for a fairly rapid decline in the number of loosely-bound systems below ~0.3 M_sun, corresponding to a factor of 10-20 increase in the minimum binding energy of VLM binaries as compared to more massive stellar binaries. This wide-separation ``desert'' is present among both field (~1-5 Gyr) and older (> 100 Myr) cluster systems, while the youngest (<~10 Myr) VLM binaries, particularly those in nearby, low-density star forming regions, appear to have somewhat different systemic properties. We compare these empirical trends to predictions laid out by current formation theories, and outline future observational studies needed to probe the full parameter space of the lowest mass multiple systems.Comment: 16 pages, 7 figures, contributed chapter for Planets and Protostars V meeting (October 2005); full table of VLM binaries can be obtained at http://paperclip.as.arizona.edu/~nsiegler/VLM_binarie

Merger rates of double neutron stars and stellar origin black holes: The Impact of Initial Conditions on Binary Evolution Predictions

The initial mass function (IMF), binary fraction and distributions of binary parameters (mass ratios, separations and eccentricities) are indispensable input for simulations of stellar populations. It is often claimed that these are poorly constrained significantly affecting evolutionary predictions. Recently, dedicated observing campaigns provided new constraints on the initial conditions for massive stars. Findings include a larger close binary fraction and a stronger preference for very tight systems. We investigate the impact on the predicted merger rates of neutron stars and black holes. Despite the changes with previous assumptions, we only find an increase of less than a factor 2 (insignificant compared with evolutionary uncertainties of typically a factor 10-100). We further show that the uncertainties in the new initial binary properties do not significantly affect (within a factor of 2) our predictions of double compact object merger rates. An exception is the uncertainty in IMF (variations by a factor of 6 up and down). No significant changes in the distributions of final component masses, mass ratios, chirp masses and delay times are found. We conclude that the predictions are, for practical purposes, robust against uncertainties in the initial conditions concerning binary parameters with exception of the IMF. This eliminates an important layer of the many uncertain assumptions affecting the predictions of merger detection rates with the gravitational wave detectors aLIGO/aVirgo.Comment: Accepted for publication in Ap

The Formation of Low-Mass Binary Star Systems Via Turbulent Fragmentation

We characterize the infall rate onto protostellar systems forming in self-gravitating radiation-hydrodynamic simulations. Using two dimensionless parameters to determine disks' susceptability to gravitational fragmentation, we infer limits on protostellar system multiplicity and the mechanism of binary formation. We show that these parameters give robust predictions even in the case of marginally resolved protostellar disks. We find that protostellar systems with radiation feedback predominately form binaries via turbulent fragmentation, not disk instability, and we predict turbulent fragmentation is the dominant channel for binary formation for low-mass stars. We clearly demonstrate that systems forming in simulations including radiative feedback have fundamentally different parameters than those in purely hydrodynamic simulations.Comment: 11 pages, 10 figures, accepted to Ap