The Sample Complexity of Search over Multiple Populations

This paper studies the sample complexity of searching over multiple populations. We consider a large number of populations, each corresponding to either distribution P0 or P1. The goal of the search problem studied here is to find one population corresponding to distribution P1 with as few samples as possible. The main contribution is to quantify the number of samples needed to correctly find one such population. We consider two general approaches: non-adaptive sampling methods, which sample each population a predetermined number of times until a population following P1 is found, and adaptive sampling methods, which employ sequential sampling schemes for each population. We first derive a lower bound on the number of samples required by any sampling scheme. We then consider an adaptive procedure consisting of a series of sequential probability ratio tests, and show it comes within a constant factor of the lower bound. We give explicit expressions for this constant when samples of the populations follow Gaussian and Bernoulli distributions. An alternative adaptive scheme is discussed which does not require full knowledge of P1, and comes within a constant factor of the optimal scheme. For comparison, a lower bound on the sampling requirements of any non-adaptive scheme is presented.Comment: To appear, IEEE Transactions on Information Theor

Formation of Multiple Populations in Globular Clusters: Another Possible Scenario

While chemical composition spreads are now believed to be a universal characteristic of globular clusters (GCs), not all of them present multiple populations in their color-magnitude diagrams (CMDs). Here we present a new scenario for the formation of GCs, in an attempt to qualitatively explain this otherwise intriguing observational framework. Our scenario divides GCs into three groups, depending on the initial mass (M_I) of the progenitor structure (PS), as follows. i) Massive PSs can retain the gas ejected by massive stars, including the ejecta of core-collapse SNe. ii) Intermediate-mass PSs can retain at least a fraction of the fast winds of massive stars, but none of the core-collapse SNe ejecta. iii) Low-mass PSs can only retain the slow winds of intermediate-mass stars. Members of the first group would include omega Centauri (NGC 5139), M54 (NGC 6715), M22 (NGC 6656), and Terzan 5, whereas NGC 2808 (and possibly NGC 2419) would be members of the second group. The remaining GCs which only present a spread in light elements, such as O and Na, would be members of the third group. According to our scenario, the different components in omega Cen should not display a sizeable spread in age. We argue that this is consistent with the available observations. We give other simple arguments in favor of our scenario, which can be described in terms of two main analytical relations: i) Between the actual observed ratio between first and second generation stars (R_SG^FG) and the fraction of first generation stars that have been lost by the GC (S_L); and ii) Between S_L and M_I. We also suggest a series of future improvements and empirical tests that may help decide whether the proposed scenario properly describes the chemical evolution of GCs.Comment: Accepted for publication in Astronomy and Astrophysic

Mining SDSS in search of Multiple Populations in Globular Clusters

Several recent studies have reported the detection of an anomalous color spread along the red giant branch (RGB) of some globular clusters (GC) that appears only when color indices including a near ultraviolet band (such as Johnson U or Stromgren u) are considered. This anomalous spread in color indexes such as U-B or c_{y} has been shown to correlate with variations in the abundances of light elements such as C, N, O, Na, etc., which, in turn, are generally believed to be associated with subsequent star formation episodes that occurred in the earliest few 10^{8} yr of the cluster's life. Here we use publicly available u, g, r Sloan Digital Sky Survey photometry to search for anomalous u-g spreads in the RGBs of nine Galactic GCs. In seven of them (M 2, M 3, M 5, M 13, M 15, M 92 and M 53), we find evidence of a statistically significant spread in the u-g color, not seen in g-r and not accounted for by observational effects. In the case of M 5, we demonstrate that the observed u-g color spread correlates with the observed abundances of Na, the redder stars being richer in Na than the bluer ones. In all the seven clusters displaying a significant u-g color spread, we find that the stars on the red and blue sides of the RGB, in (g, u-g) color magnitude diagrams, have significantly different radial distributions. In particular, the red stars (generally identified with the second generation of cluster stars, in the current scenario) are always more centrally concentrated than blue stars (generally identified with the first generation) over the range sampled by the data (0.5r_{h} < r < 5r_{h}), in qualitative agreement with the predictions of some recent models of the formation and chemical evolution of GCs. Our results suggest that the difference in the radial distribution between first and second generation stars may be a general characteristic of GCs.Comment: 11 pages, 5 figures, typos adde