The sigma - L correlation in Nearby Early-Type Galaxies

Early-type galaxy velocity dispersions and luminosities are correlated. The correlation estimated in local samples (< 100 Mpc) differs from that measured more recently in the SDSS. This is true even when systematics in the SDSS photometric and spectroscopic parameters have been accounted-for. We show that this is also true for the ENEAR sample if galaxy luminosities are estimated using distances which have been corrected for peculiar motions. We then show that, because the estimate of the `true' distance is derived from a correlation with velocity dispersion, in this case the D_n-sigma relation, using it in the sigma-L relation leads to an artificially tight relation with a biased slope. Making no correction for peculiar velocities results in a sigma-L relation which is very similar to that of the SDSS, although with larger scatter. We also measure the sigma-L correlation in a mock ENEAR catalog, in which the underlying galaxy sample has the same sigma-L correlation as seen in the SDSS. The mock catalog produces the same D_n-sigma relation as the data, the same biased slope when D_n-sigma distances are used to estimate luminosities, and good agreement with the input sigma-L relation when redshift is used as the distance indicator. This provides further evidence that the true sigma-L relation of ENEAR galaxies is indeed very similar to that of SDSS early-types. Our results suggest that local sigma-L relations which are based on Fundamental Plane distances should also be re-evaluated. Our findings also have important implications for black hole demographics; the best direct estimates of the masses of supermassive black holes come from local galaxies, so estimates of the black hole mass function are more safely made by working with the Mbh-sigma correlation than with Mbh-L.Comment: 9 pages, 9 figures. Accepted by AJ. A new appendix describes systematics effects we found in the SDSS velocity dispersion measurements (sigmas < 150 km/s are biased towards larger values; this bias was not present in the Bernardi et al. 2003 sample) and luminosity measurement

Plain fundamentals of Fundamental Planes: Analytics and algorithms

The coefficients a and b of the Fundamental Plane relation R ~ Sigma^a I^b depend on whether one minimizes the scatter in the R direction or orthogonal to the Plane. We provide explicit expressions for a and b (and confidence limits) in terms of the covariances between logR, logSigma and logI. Our analysis is more generally applicable to any other correlations between three variables: e.g., the color-magnitude-Sigma relation, the L-Sigma-Mbh relation, or the relation between the X-ray luminosity, Sunyaev-Zeldovich decrement and optical richness of a cluster, so we provide IDL code which implements these ideas, and we show how our analysis generalizes further to correlations between more than three variables. We show how to account for correlated errors and selection effects, and quantify the difference between the direct, inverse and orthogonal fit coefficients. We show that the three vectors associated with the Fundamental Plane can all be written as simple combinations of a and b because the distribution of I is much broader than that of Sigma, and Sigma and I are only weakly correlated. Why this should be so for galaxies is a fundamental open question about the physics of early-type galaxy formation. If luminosity evolution is differential, and Rs and Sigmas do not evolve, then this is just an accident: Sigma and I must have been correlated in the past. On the other hand, if the (lack of) correlation is similar to that at the present time, then differential luminosity evolution must have been accompanied by structural evolution. A model in which the luminosities of low-L galaxies evolve more rapidly than do those of higher-L galaxies is able to produce the observed decrease in a (by a factor of 2 at z~1) while having b decrease by only about 20 percent. In such a model, the Mdyn/L ratio is a steeper function of Mdyn at higher z.Comment: 11 pages, 1 figure, associated IDL code, MNRAS accepte

Selection bias in the M_BH-sigma and M_BH-L correlations and its consequences

It is common to estimate black hole abundances by using a measured correlation between black hole mass and another more easily measured observable such as the velocity dispersion or luminosity of the surrounding bulge. The correlation is used to transform the distribution of the observable into an estimate of the distribution of black hole masses. However, different observables provide different estimates: the Mbh-sigma relation predicts fewer massive black holes than does the Mbh-L relation. This is because the sigma-L relation in black hole samples currently available is inconsistent with that in the SDSS sample, from which the distributions of L or sigma are based: the black hole samples have smaller L for a given sigma or have larger sigma for a given L. This is true whether L is estimated in the optical or in the NIR. If this is a selection rather than physical effect, then the Mbh-sigma and Mbh-L relations currently in the literature are also biased from their true values. We provide a framework for describing the effect of this bias. We then combine it with a model of the bias to make an estimate of the true intrinsic relations. While we do not claim to have understood the source of the bias, our simple model is able to reproduce the observed trends. If we have correctly modeled the selection effect, then our analysis suggests that the bias in the relation is likely to be small, whereas the relation is biased towards predicting more massive black holes for a given luminosity. In addition, it is likely that the Mbh-L relation is entirely a consequence of more fundamental relations between Mbh and sigma, and between sigma and L. The intrinsic relation we find suggests that at fixed luminosity, older galaxies tend to host more massive black holes.Comment: 12 pages, 7 figures. Accepted by ApJ. We have added a figure showing that a similar bias is also seen in the K-band. A new appendix describes the BH samples as well as the fits used in the main tex

The Evolution of the M_BH-sigma relation Inferred from the Age Distribution of Local Early-Type Galaxies and AGN Evolution

We utilize the local velocity dispersion function (VDF) of spheroids, together with their inferred age--distributions, to predict the VDF at higher redshifts (0<z<6), under the assumption that (i) most of the stars in each nearby spheroid formed in a single episode, and (ii) the velocity dispersion sigma remained nearly constant afterward. We assume further that a supermassive black hole (BH) forms concurrently with the stars, and within ~1 Gyr of the formation of the potential well of the spheroid, and that the relation between the mass of the BH and host velocity dispersion maintains the form M_BH ~ sigma^{beta} with beta~4, but with the normalization allowed to evolve with redshift as ~(1+z)^{alpha}. We compute the BH mass function associated with the VDF at each redshift, and compare the accumulated total BH mass density with that inferred from the integrated quasar luminosity function (LF; the so--called Soltan argument). This comparison is insensitive to the assumed duty cycle or Eddington ratio of quasar activity, and we find that the match between the two BH mass densities favors a relatively mild redshift evolution, with alpha ~ 0.26, with a positive evolution as strong as alpha>1.3 excluded at the 99% confidence level. A direct match between the characteristic BH mass in the VDF--based and quasar LF--based BH mass functions also yields a mean Eddington ratio of lambda ~ 0.5-1 that is roughly constant within 0<z<3. A strong positive evolution in the M_BH-sigma relation is still allowed by the data if galaxies increase, on average, their velocity dispersions since the moment of formation, due to dissipative processes. If we assume that the mean velocity dispersion of the host galaxies evolves as sigma(z)=sigma(0)*(1+z)^{-gamma}, we find a lower limit of gamma>0.23 for alpha>1.5. abridgedComment: Accepted for publication on ApJ; 10 pages, 9 Figure

The Role of Environment on the Formation of Early-Type Galaxies

(Abridged) We present a detailed study of the stellar populations of a volume-limited sample of early-type galaxies from SDSS, across a range of environments -- defined as the mass of the host dark matter halo. The stellar populations are explored through the SDSS spectra, via projection onto a set of two spectral vectors determined from Principal Component Analysis. We find the velocity dispersion of the galaxy to be the main driver behind the different star formation histories of early-type galaxies. However, environmental effects are seen to play a role (although minor). Galaxies populating the lowest mass halos have stellar populations on average ~1Gyr younger than the rest of the sample. The fraction of galaxies with small amounts of recent star formation is also seen to be truncated when occupying halos more massive than 3E13Msun. The sample is split into satellite and central galaxies for a further analysis of environment. Satellites are younger than central galaxies of the same stellar mass. The younger satellite galaxies in 6E12Msun halos have stellar populations consistent with the central galaxies found in the lowest mass halos of our sample (i.e. 1E12Msun). This result is indicative of galaxies in lower mass halos being accreted into larger halos.Comment: 11 pages, 10 figures. Accepted for publication in MNRA