Adaptive hh-refinement for reduced-order models

This work presents a method to adaptively refine reduced-order models \emph{a posteriori} without requiring additional full-order-model solves. The technique is analogous to mesh-adaptive $h$-refinement: it enriches the reduced-basis space online by `splitting' a given basis vector into several vectors with disjoint support. The splitting scheme is defined by a tree structure constructed offline via recursive $k$-means clustering of the state variables using snapshot data. The method identifies the vectors to split online using a dual-weighted-residual approach that aims to reduce error in an output quantity of interest. The resulting method generates a hierarchy of subspaces online without requiring large-scale operations or full-order-model solves. Further, it enables the reduced-order model to satisfy \emph{any prescribed error tolerance} regardless of its original fidelity, as a completely refined reduced-order model is mathematically equivalent to the original full-order model. Experiments on a parameterized inviscid Burgers equation highlight the ability of the method to capture phenomena (e.g., moving shocks) not contained in the span of the original reduced basis.Comment: submitted to the International Journal for Numerical Methods in Engineering, Special Issue on Model Reductio

Cosmological Velocity Bias

Velocity bias is a reduction of the velocity dispersion of tracer galaxies in comparison to the velocity dispersion of the underlying mass field. There are two distinct forms of velocity bias. The single particle velocity reduction, $b_v(1)$, is the result of energy loss of a tracer population, and in virialized regions, such as galaxy clusters, is intimately associated with mass segregation which together lead to cluster mass underestimates. The pairwise velocity bias, $b_v(2)$, has an additional statistical reduction if the total mass per galaxy rises with velocity dispersion of the virialized cluster. Values of the velocity bias are estimated from n-body simulations, finding $b_v(1)\simeq 0.85\pm0.1$ and $b_v(2)\simeq 0.6\pm0.2$. The value of $b_v(1)$ is relatively secure and predicts that the virial radius of the cluster light is about 20\% of the cluster mass, which can be tested with observations of cluster mass profiles beyond the apparent virial radius. The value of $b_v(2)$ is sensitive to the formation efficiency of galaxies over environments ranging from voids to rich clusters, the latter of which are not yet well resolved in simulations. An $n=1$, $\Omega=1$, COBE normalized CDM spectrum requires b_v(2)\simeq0.20(15\mu K/Q)(\sigma_{12}/317 \kms) which is well below the measured range of $b_v(2)$. The pairwise velocities at 1 \hmpc\ allow $\Omega=1$ for a galaxy clustering bias near unity if $b_v(2)\simeq0.6$.Comment: 10pages (0.5Mbyte .ps.Z.uu file) in proceedings of IAP Colloq 9, Cosmological Velocity Fields ed F Bouchet & M. Lachieze-Rey (revised--typo in eq.4, numerical value correct

Modeling GD-1 Gaps in a Milky-Way Potential

The GD-1 star stream is currently the best available for identifying density fluctuations, "gaps", along its length as a test of the LCDM prediction of large numbers of dark matter sub-halos orbiting in the halo. Density variations of some form are present, since the variance of the density along the stream is three times that expected from the empirically estimated variation in the filtered mean star counts. The density variations are characterized with filters that approximate the shape of sub-halo gravitationally induced stream gaps. The filters locate gaps and measures their amplitude, leading to a measurement of the distribution of gap widths. To gain understanding of the factors influencing the gap width distribution, a suite of collisionless n-body simulations for a GD-1 like orbit in a Milky Way-like potential provides a dynamically realistic statistical prediction of the gap distribution. The simulations show that every location in the stream has been disturbed to some degree by a sub-halo. The small gaps found via the filtering are largely noise. Larger gaps, those longer than 1 kpc, or 10 degrees for GD-1, are the source of the excess variance. The suite of stream simulations shows that sub-halos at the predicted inner halo abundance or possibly somewhat higher can produce the required large sale density variations.Comment: ApJ accepte

Bulge Building with Mergers and Winds

The gravitational clustering hierarchy and dissipative gas processes are both involved in the formation of bulges. Here we present a simple empirical model in which bulge material is assembled via gravitational accretion of the visible companion galaxies. Assuming that merging leads to a starburst, we show that the resulting winds can be strong enough that they self-regulate the accretion. A quasi-equilibrium accretion process naturally leads to the Kormendy relation between bulge density and size. Whether or not the winds are sufficiently strong and long lived to create the quasi-equilibrium must be tested with observations. To illustrate the model we use it to predict representative parameter dependent star formation histories. We find that bulge building activity peaks around redshift two, with tails to both higher and lower redshifts.Comment: to appear in "The Formation of Bulges", eds. C.M. Carollo, H.C. Ferguson & R.F.G. Wyse, CU

Globular Clusters in a Cosmological N-body Simulation

Stellar dynamical model globular clusters are introduced into reconstituted versions of the dark matter halos of the Via-Lactea II (VL-2) simulation to follow the star cluster tidal mass loss and stellar stream formation. The clusters initially evolve within their local sub-galactic halo, later being accreted into the main halo. Stars are continually removed from the clusters, but those that emerged in the sub-galactic halos are dispersed in a wide stream when accreted into the main halo. Thin tidal streams that survive to the present can begin to form once a cluster is in the main halo. A higher redshift start places the star clusters in denser halos where they are subject to stronger tides leading to higher average mass loss rates. A z=3 start leads to a rich set of star streams with nearly all within 100 kpc having a remnant progenitor star cluster in the stream. In contrast, in a z=8 start, all star clusters that are accreted onto the main halo are completely dissolved. These results are compared to the available data on Milky-Way streams, where the majority of streams do not have clearly associated globular clusters. which, if generally true, suggests that there were at least twice as many massive globular clusters at high redshift.Comment: AAS submitted, revise

The Dynamics of Star Stream Gaps

When a massive object crosses a star stream velocity changes are induced both along and transverse to the stream which can lead to the development of a visible gap. For a stream narrow relative to its orbital radius the time of stream crossing is sufficiently short that the impact approximation can be used to derive the changes in angular momenta and radial actions along the star stream. The epicyclic approximation is used to calculate the evolution of the density of the stream as it orbits around in a galactic potential. Analytic expressions are available for a point mass, however, the general expressions are easily numerically evaluated for perturbing objects with arbitrary density profiles. With a simple allowance for the velocity dispersion of the stream, moderately warm streams can be modeled. The predicted evolution agrees well with the outcome of simulations of stellar streams for streams with widths up to 1% of the orbital radius of the stream. The angular momentum distribution within the stream shears out gaps with time, further reducing their visibility, although the size of the shear effect requires more detailed simulations. An illustrative model indicates that shear will limit the persistent gaps to a minimum length of a few times the stream width. In general the equations are useful for dynamical insight into the development of stream gaps and their measurement.Comment: ApJ submitted after revisio

Star Stream Folding by Dark Galactic Sub-Halos

Star streams in galactic halos are long, thin, unbound structures that will be disturbed by the thousands dark matter sub-halos that are predicted to be orbiting within the main halo. A sub-halo generally induces a localized wave in the stream which often evolves into a "z-fold" as an initially trailing innermost part rotates faster than an initially leading outermost part. The folding, which becomes increasingly complex with time, leads to an apparent velocity dispersion increase and thickening of the stream. We measure the equivalent velocity dispersion around the local mean in the simulations, finding that it rises to about 10 km/s after 5 Gyr and 20 km/s after 13 Gyr. The currently available measurements of the velocity dispersion of halo star streams range from as small as 2 km/s to slightly over 20 km/s. The streams with velocity dispersions of 15-20 km/s are compatible with what sub-halo heating would produce. A dynamical understanding of the low velocity dispersion streams depends on the time since the progenitor's tidal disruption into a thin stream. If the streams are nearly as old as their stars then sub-halos cannot be present with the predicted numbers and masses. However, the dynamical age of the streams can be significantly less than the stars. If the three lowest velocity streams are assigned ages of 3 Gyr, they are in conflict with the sub-halo heating. The main conclusion is that star stream heating is a powerful and simple test for sub-halo structure.Comment: revised version submitted to ApJ

Counting Dark Sub-halos with Star Stream Gaps

The Cold Dark Matter paradigm predicts vast numbers of dark matter sub-halos to be orbiting in galactic halos. The sub-halos are detectable through the gaps they create gaps in stellar streams. The gap-rate is an integral over the density of sub-halos, their mass function, velocity distribution and the dynamical age of the stream. The rate of visible gap creation is a function of the width of the stream. The available data for four streams: the NW stream of M31, the Pal~5 stream, the Orphan Stream and the Eastern Banded Structure, are compared to the LCDM predicted relation. We find a remarkably good agreement, although there remains much to be done to improve the quality of the result. The narrower streams require that there is a total population of order 10^5 sub-halos above 10^5 M_sun to create the gaps.Comment: contribution to the Third Subaru Conference, Galactic Archaeology: Near Field Cosmology and the Formation of the Milky Way ed. Wako Aok

Rotational and Radial Velocities of 1.3-2.2 M_Sun Red Giants in Open Clusters

This study presents the rotational distribution of red giant stars (RGs) in eleven old to intermediate age open clusters. The masses of these stars are all above the Kraft break, so that they lose negligible amounts of their birth angular momentum (AM) during the main sequence evolution. However, they do span a mass range with quite different AM distributions imparted during formation, with the stars less massive than ~1.6 M_Sun arriving on the main sequence with lower rotation rates than the more massive stars. The majority of RGs in this study are slow rotators across the entire red giant branch regardless of mass, supporting the picture that intermediate mass stars rapidly spin down when they evolve off the main sequence and develop convection zones capable of driving a magnetic dynamo. Nevertheless, a small fraction of RGs in open clusters show some level of enhanced rotation, and faster rotators are as common in these clusters as in the field red giant population. Most of these enhanced rotators appear to be red clump stars, which is also true of the underlying stellar sample, while others are clearly RGs that are above or below the clump. In addition to rotational velocities, the radial velocities and membership probabilities of individual stars are also presented. Cluster heliocentric radial velocities for NGC 6005 and Pismis 18 are reported for the first time.Comment: 17 pages, 7 figures, 7 tables. Accepted for publication in The Astronomical Journal. Machine readable versions of Tables 2 & 5 are provided as supplementary materia

Milky Way Halo Vibrations and Incommensurate Stream Velocities

Collisionless dark matter galactic halos are expected to exhibit damped oscillations as a result of ongoing late time accretion. An n-body model of the cosmological assembly of a Milky Way-like halo is used to quantify the time dependence of its gravitational field. The simulation contains stellar streams whose incommensurate perpendicular velocities are found to have an approximately exponential distribution with a scale of 10-20\kms, depending on how the stars are selected, comparable to those reported for the Orphan stream. The fluctuations in the quadrupole moment of the dark matter halo are sufficient to largely explain the tangential velocities. If velocity measurements of a larger sample of Milky Way streams finds (or does not find) the expected distribution of transverse velocities it will lead to limits on the cross-section of self-interacting dark matter, in which kinetic viscosity can damp the oscillations more rapidly than the mixing processes of collisionless dark matter alone.Comment: ApJ revise