The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger's theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov's condition. As a consequence, we obtain the analogue of Calderon's differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequalit

The Las Campanas/AAT Rich Cluster Survey III: Spectroscopic Studies of X-ray Bright Galaxy Clusters at z~0.1

[abridged] We present the analysis of the spectroscopic and photometric catalogues of 11 X-ray luminous clusters at z=0.07-0.16 from the Las Campanas / Anglo-Australian Telescope Rich Cluster Survey. Our spectroscopic dataset consists of over 1600 galaxy cluster members, of which two thirds are outside r_200. We assign cluster membership using a detailed mass model and expand on our previous work on the cluster colour-magnitude relation where membership was inferred statistically. We confirm that the modal colours of galaxies on the colour magnitude relation become progressively bluer with increasing radius and decreasing local galaxy density. Interpreted as an age effect, we hypothesize that these trends in galaxy colour should be reflected in mean Hdelta equivalent width. We confirm that passive galaxies in the cluster increase in Hdelta line strength as dHdelta / d r_p = 0.35 +/- 0.06. A variation of star formation rate, as measured by [OII], with increasing local density of the environment is discernible and is shown to be in broad agreement with previous studies from 2dFGRS and SDSS. We find that clusters at z~0.1 are less active than their higher redshift analogues. We also investigate unusual populations of blue and very red nonstarforming galaxies and we suggest that the former are likely to be the progenitors of galaxies which will lie on the colour-magnitude relation, while the colours of the latter possibly reflect dust reddening. The cluster galaxies at large radii consist of both backsplash ones and those that are infalling to the cluster for the first time. We make a comparison to the field population at z~0.1 and examine broad differences between the two populations. Individually, the clusters show significant variation in their galaxy populations which reflects their recent infall histories.Comment: 25 pages, 16 figures. Accepted for publication in MNRA

The Age of Ellipticals and the Color-Magnitude Relation

Using new narrowband color observations of early-type galaxies in clusters, we reconstruct the color-magnitude relation (CMR) with a higher degree of accuracy than previous work. We then use the spectroscopically determined ages and metallicities from three samples (Trager et al 2008, Thomas et al 2005, Gallazzi et al 2006), combined with multi-metallicity SED models, to compare predicted colors for galaxies with young ages (less than 8 Gyr) with the known CMR. We find that the CMR cannot by reproduced by the spectroscopically determined ages and metallicities in any of the samples despite the high internal accuracies to the spectroscopic indices. In contrast, using only the $$ index to determine [Fe/H], and assuming a mean age of 12 Gyr for a galaxy's stellar population, we derive colors that exactly match not only the color zeropoint of the CMR but also its slope. We consider the source of young age estimates, the H$\beta$ index, and examine the conflict between red continuum colors and large H$\beta$ values in galaxy spectra. We conclude that our current understanding of stellar populations is insufficient to correctly interpret H$\beta$ values and that the sum of our galaxy observations supports an old and monolithic scenario of galaxy formation. This result has a devastating impact on every study that has used the H$\beta$ index to calculate galaxy age, as the use of the H$\beta$ versus MgFe diagram will result in incorrectly deduced young ages.Comment: 29 pages, 9 figures. This version deviates from the version accepted by Ap

Estimation of absorption line indices of early-type galaxies using colours

Context. Absorption line indices are widely used to determine the stellar population parameters such as age and metallicity of galaxies, but it is not easy to obtain the line indices of some distant galaxies that have colours available. Aims. This paper investigates the correlations between absorption line indices and colours. Methods. A few statistical fitting methods are mainly used, via both the observational data of Sloan Digital Sky Survey and a widely used theoretical stellar population model. Results. Some correlations between widely used absorption line indices and ugriz colours are found from both observational data of early-type galaxies and a theoretical simple stellar population model. In particular, good correlations between colours and widely used absorption line indices such as Dn(4000), HgammaA, HgammaF, HdeltaA, Mg1, Mg2, and Mgb, are shown in this paper. Conclusions. Some important absorption line indices of early-type galaxies can be estimated from their colours using correlations between absorption line indices and colours. For example, age-sensitive absorption line indices can be estimated from (u-r) or (g-r) colours and metallicity-sensitive ones from (u - z) or (g - z). This is useful for studying the stellar populations of distant galaxies, especially for statistical investigations.Comment: 9 pages, 21 figures, will be shown in A&

The Build-up of the Colour-Magnitude Relation as a Function of Environment

We discuss the environmental dependence of galaxy evolution based on deep panoramic imaging of two distant clusters taken with Suprime-Cam as part of the PISCES project. By combining with the SDSS data as a local counterpart for comparison, we construct a large sample of galaxies that spans wide ranges in environment, time, and stellar mass (or luminosity). We find that colours of galaxies, especially those of faint galaxies ($M_V>M_V^*+1$), change from blue to red at a break density as we go to denser regions. Based on local and global densities of galaxies, we classify three environments: field, groups, and clusters. We show that the cluster colour-magnitude relation is already built at $z=0.83$. In contrast to this, the bright-end of the field colour-magnitude relation has been vigorously built all the way down to the present-day and the build-up at the faint-end has not started yet. A possible interpretation of these results is that galaxies evolve in the 'down-sizing' fashion. That is, massive galaxies complete their star formation first and the truncation of star formation is propagated to smaller objects as time progresses. This trend is likely to depend on environment since the build-up of the colour-magnitude relation is delayed in lower-density environments. Therefore, we may suggest that the evolution of galaxies took place earliest in massive galaxies and in high density regions, and it is delayed in less massive galaxies and in lower density regions.Comment: 23pages, 19 figures, accepted for publication in MNRA

Exploring the Star Formation History of Elliptical Galaxies: Beyond Simple Stellar Populations with a New Estimator of Line Strengths

(Abridged) We study the stellar populations of 14 elliptical galaxies in the Virgo cluster. We propose an alternative approach to the standard side-band method to measure equivalent widths (EWs). Our Boosted Median Continuum maps the EWs more robustly than the side-band method, minimising the effect from neighbouring absorption lines and reducing the age-metallicity degeneracy. We concentrate on Balmer lines (Hbeta,Hgamma,Hdelta), the G band and the 4000A break as age-sensitive indicators, and on the combination [MgFe] as the main metallicity indicator. We go beyond the standard comparison of the observations with simple stellar populations (SSP) and consider various models to describe the star formation histories, either with a continuous star formation rate or with a mixture of two different SSPs. Composite models are found to give more consistent fits among individual line strengths and agree with an independent estimate using the spectral energy distribution. Our age and metallicity estimates correlate well with stellar mass or velocity dispersion, with a significant threshold around 5E10 Msun above which galaxies are uniformly old and metal rich. In a more speculative way, our models suggest that it is formation **epoch** and not formation timescale what drives the Mass-Age relationship of elliptical galaxies.Comment: 15 pages, 15 figures, 4 tables. Accepted for publication in MNRA

A multiscale approach to environment and its influence on the colour distribution of galaxies

We present a multiscale approach to measurements of galaxy density, applied to a volume-limited sample constructed from SDSS DR5. We populate a rich parameter space by obtaining independent measurements of density on different scales for each galaxy, avoiding the implicit assumptions involved, e.g., in the construction of group catalogues. As the first application of this method, we study how the bimodality in galaxy colour distribution (u-r) depends on multiscale density. The u-r galaxy colour distribution is described as the sum of two gaussians (red and blue) with five parameters: the fraction of red galaxies (f_r) and the position and width of the red and blue peaks (mu_r, mu_b, sigma_r and sigma_b). Galaxies mostly react to their smallest scale (< 0.5 Mpc) environments: in denser environments red galaxies are more common (larger f_r), redder (larger mu_r) and with a narrower distribution (smaller sigma_r), while blue galaxies are redder (larger mu_b) but with a broader distribution (larger sigma_b). There are residual correlations of f_r and mu_b with 0.5 - 1 Mpc scale density, which imply that total or partial truncation of star formation can relate to a galaxy's environment on these scales. Beyond 1 Mpc (0.5 Mpc for mu_r) there are no positive correlations with density. However f_r (mu_r) anti-correlates with density on >2 (1) Mpc scales at fixed density on smaller scales. We examine these trends qualitatively in the context of the halo model, utilizing the properties of haloes within which the galaxies are embedded, derived by Yang et al, 2007 and applied to a group catalogue. This yields an excellent description of the trends with multiscale density, including the anti-correlations on large scales, which map the region of accretion onto massive haloes. Thus we conclude that galaxies become red only once they have been accreted onto haloes of a certain mass.Comment: 22 pages, 14 figures. Accepted for publication in MNRAS