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Pieter G. van Dokkum

Publications and source records attributed to Pieter G. van Dokkum.

At least 73 records · Page 4Linked to original sources

Tight Correlations Between Massive Galaxy Structural Properties and Dynamics: The Mass Fundamental Plane Was in Place by z~2

The Fundamental Plane (FP) is an empirical relation between the size, surface brightness, and velocity dispersion of early-type galaxies. This relation has been studied extensively for early-type galaxies in the local universe to constrain galaxy formation mechanisms. The evolution of the zeropoint of this plane has been extended to high redshifts to study the luminosity evolution of massive galaxies, under the assumption of structural homology. In this work, we assess this assumption by replacing surface brightness with stellar mass density and present the evolution of the "mass FP" for massive, quiescent galaxies since z~2. By accounting for stellar populations, we thereby isolate and trace structural and dynamical evolution. Despite the observed dramatic evolution in the sizes and morphologies of massive galaxies since z~3, we find that quiescent galaxies lie on the mass FP out to z~2. In contrast with ~1.4 dex evolution in the luminosity FP, average residuals from the z~0 mass FP are less than ~0.15 dex since z~2. Assuming the Hyde & Bernardi (2009) mass FP slope, we find that this minimal offset scales as (1+z)^{-0.095+/-0.043}. This result lends credence to previous studies that derived luminosity evolution from the FP. Therefore, despite their compact sizes and suggestions that massive galaxies are more disk-like at z~2, the relationship between their dynamics and structural properties are consistent with local early-type galaxies. Finally, we find no strong evidence for a tilt of the mass FP relative to the Virial plane, but emphasize the need for full models including selection biases to fully investigate this issue.

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Dynamical Versus Stellar Masses in Compact Early-Type Galaxies: Further Evidence for Systematic Variation in the Stellar Initial Mass Function

Several independent lines of evidence suggest that the stellar initial mass function (IMF) in early-type galaxies becomes increasingly `bottom-heavy' with increasing galaxy mass and/or velocity dispersion, sigma. Here we consider evidence for IMF variation in a sample of relatively compact early-type galaxies drawn from the Sloan Digital Sky Survey. These galaxies are of sufficiently high stellar density that a dark halo likely makes a minor contribution to the total dynamical mass, Mdyn, within one effective radius. We fit our detailed stellar population synthesis models to the stacked absorption line spectra of these galaxies in bins of sigma and find evidence from IMF-sensitive spectral features for a bottom-heavy IMF at high sigma. We also apply simple `mass-follows-light' dynamical models to the same data and find that Mdyn is significantly higher than what would be expected if these galaxies were stellar dominated and had a universal Milky Way IMF. Adopting Mdyn M_* therefore implies that the IMF is `heavier' at high sigma. Most importantly, the quantitative amount of inferred IMF variation is very similar between the two techniques, agreeing to within 0.1 dex in mass. The agreement between two independent techniques, when applied to the same data, provides compelling evidence for systematic variation in the IMF as a function of early-type galaxy velocity dispersion. Any alternative explanations must reproduce both the results from dynamical and stellar population-based techniques.

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The Assembly of Milky Way-like Galaxies Since z~2.5

Galaxies with the mass of the Milky Way dominate the stellar mass density of the Universe but it is uncertain how and when they were assembled. Here we study progenitors of these galaxies out to z=2.5, using data from the 3D-HST and CANDELS Treasury surveys. We find that galaxies with present-day stellar masses of log(M)~10.7 built ~90% of their stellar mass since z=2.5, with most of the star formation occurring before z=1. In marked contrast to the assembly history of massive elliptical galaxies, mass growth is not limited to large radii: the mass in the central 2 kpc of the galaxies increased by a factor of 3.2+-0.8 between z=2.5 and z=1. We therefore rule out simple models in which bulges were fully assembled at high redshift and disks gradually formed around them. Instead, bulges (and black holes) likely formed in lockstep with disks, through bar instabilities, migration, or other processes. We find that after z=1 the growth in the central regions gradually stopped and the disk continued to build, consistent with recent studies of the gas distributions in z~1 galaxies and the properties of many spiral galaxies today.

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Stellar Kinematics of z~2 Galaxies and the Inside-Out Growth of Quiescent Galaxies

Using stellar kinematics measurements, we investigate the growth of massive, quiescent galaxies from z~2 to today. We present X-Shooter spectra from the UV to NIR and dynamical mass measurements of 5 quiescent massive (>10^11 Msun) galaxies at z~2. This triples the sample of z>1.5 galaxies with well constrained (dsigma <100 km/s) velocity dispersion measurements. From spectral population synthesis modeling we find that these galaxies have stellar ages that range from 0.5-2 Gyr, with no signs of ongoing star formation. We measure velocity dispersions (290-450 km/s) from absorption lines and find that they are 1.6-2.1 times higher than those of galaxies in the SDSS at the same mass. Sizes are measured using GALFIT from HST-WFC3 H160 and UDS K-band images. The dynamical masses correspond well to the SED based stellar masses, with dynamical masses that are ~15% higher. We find that M_*/M_dyn may decrease slightly with time, which could reflect the increase of the dark matter fraction within an increasing effective radius. We combine different stellar kinematic studies from the literature, and examine the structural evolution from z~2 to z~0: we confirm that at fixed dynamical mass, the effective radius increases by a factor of ~2.8, and the velocity dispersion decreases by a factor of ~1.7. The mass density within one effective radius decreases by a factor of ~20, while within a fixed physical radius (1 kpc) it decreases only mildly (~2). When we allow for an evolving mass limit by selecting a population of galaxies at fixed number density, a stronger size growth with time is found (~4), velocity dispersion decreases by a factor of ~1.4, and interestingly, the mass density within 1 kpc is consistent with no evolution. This finding suggests that massive quiescent galaxies at ~2 grow inside-out, consistent with the expectations from minor mergers.

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Quiescent Galaxies in the 3D-HST Survey: Spectroscopic Confirmation of a Large Number of Galaxies with Relatively Old Stellar Populations at z~2

Quiescent galaxies at z~2 have been identified in large numbers based on rest-frame colors, but only a small number of these galaxies have been spectroscopically confirmed to show that their rest-frame optical spectra show either strong Balmer or metal absorption lines. Here, we median stack the rest-frame optical spectra for 171 photometrically-quiescent galaxies at 1.4 < z < 2.2 from the 3D-HST grism survey. In addition to Hbeta (4861A), we unambiguously identify metal absorption lines in the stacked spectrum, including the G-band (4304A), Mg I (5175A), and Na I (5894A). This finding demonstrates that galaxies with relatively old stellar populations already existed when the universe was ~3 Gyr old, and that rest-frame color selection techniques can efficiently select them. We find an average age of 1.3^0.1_0.3 Gyr when fitting a simple stellar population to the entire stack. We confirm our previous result from medium-band photometry that the stellar age varies with the colors of quiescent galaxies: the reddest 80% of galaxies are dominated by metal lines and have a relatively old mean age of 1.6^0.5_0.4 Gyr, whereas the bluest (and brightest) galaxies have strong Balmer lines and a spectroscopic age of 0.9^0.2_0.1 Gyr. Although the spectrum is dominated by an evolved stellar population, we also find [OIII] and Hbeta emission. Interestingly, this emission is more centrally concentrated than the continuum with L_[OIII] = 1.7 +/- 0.3 x 10^40 erg s^-1, indicating residual central star formation or nuclear activity.

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HST/WFC3 Confirmation of the Inside-Out Growth of Massive Galaxies at 0<z<2 and Identification of their Star Forming Progenitors at z~3

We study the structural evolution of massive galaxies by linking progenitors and descendants at a constant cumulative number density of n_c=1.4x10^{-4} Mpc^{-3} to z~3. Structural parameters were measured by fitting Sersic profiles to high resolution CANDELS HST WFC3 J_{125} and H_{160} imaging in the UKIDSS-UDS at 1<z<3 and ACS I_{814} imaging in COSMOS at 0.25<z<1. At a given redshift, we selected the HST band that most closely samples a common rest-frame wavelength so as to minimize systematics from color gradients in galaxies. At fixed n_c, galaxies grow in stellar mass by a factor of ~3 from z~3 to z~0. The size evolution is complex: galaxies appear roughly constant in size from z~3 to z~2 and then grow rapidly to lower redshifts. The evolution in the surface mass density profiles indicates that most of the mass at r<2 kpc was in place by z~2, and that most of the new mass growth occurred at larger radii. This inside-out mass growth is therefore responsible for the larger sizes and higher Sersic indices of the descendants toward low redshift. At z<2, the effective radius evolves with the stellar mass as r_e M^{2.0}, consistent with scenarios that find dissipationless minor mergers to be a key driver of size evolution. The progenitors at z~3 were likely star-forming disks with r_e~2 kpc, based on their low Sersic index of n~1, low median axis ratio of b/a~0.52, and typical location in the star-forming region of the U-V versus V-J diagram. By z~1.5, many of these star-forming disks disappeared, giving rise to compact quiescent galaxies. Toward lower redshifts, these galaxies continued to assemble mass at larger radii and became the local ellipticals that dominate the high mass end of the mass function at the present epoch.

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Galaxy environments over cosmic time: the non-evolving radial galaxy distributions around massive galaxies since z=1.6

We present a statistical study of the environments of massive galaxies in four redshift bins between z=0.04 and z=1.6, using data from the Sloan Digital Sky Survey (SDSS) and the NEWFIRM Medium Band Survey (NMBS). We measure the projected radial distribution of galaxies in cylinders around a constant number density selected sample of massive galaxies and utilize a statistical subtraction of contaminating sources. Our analysis shows that massive primary galaxies typically live in group halos and are surrounded by 2 to 3 satellites with masses more than one-tenth of the primary galaxy mass. The cumulative stellar mass in these satellites roughly equals the mass of the primary galaxy itself. We further find that the radial number density profile of galaxies around massive primaries has not evolved significantly in either slope or overall normalization in the past 9.5 Gyr. A simplistic interpretation of this result can be taken as evidence for a lack of mergers in the studied groups and as support for a static evolution model of halos containing massive primaries. Alternatively, there exists a tight balance between mergers and accretion of new satellites such that the overall distribution of galaxies in and around the halo is preserved. The latter interpretation is supported by a comparison to a semi-analytic model, which shows a similar constant average satellite distribution over the same redshift range.

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A Tentative Detection of an Emission Line at 1.6 μm for the z~12 Candidate UDFj-39546284

We present deep WFC3 grism observations of the candidate z~12 galaxy UDFj-39546284 in the HST Ultra Deep Field (UDF), by combining spectroscopic data from the 3D-HST and CANDELS surveys. The total exposure time is 40.5 ks and the spectrum covers 1.10 < λ < 1.65μm. We search for faint emission lines by cross-correlating the 2D G141 spectrum with the observed H160 morphology, a technique that is unique to slitless spectroscopy at HST resolution. We find a 2.7σ detection of an emission line at 1.599 μm---just redward of the JH140 filter---with flux 3.5 +/- 1.3x10-18 erg/s/cm2. Assuming the line is real, it contributes 110 +/- 40% of the observed H160 flux and has an observed equivalent width > 7300 Å. If the line is confirmed, it could be Ly-α at z = 12.12. However, a more plausible interpretation, given current results, could be a lower redshift feature such as [O III]λ4959,5007 at z = 2.19. We find two other 3D-HST [O III] emitters within 1000 km/s of that redshift in the GOODS-South field. Additional support for this interpretation comes from the discovery of a bright "[O III] blob" with a secure G141 grism redshift of z = 1.605. This object has a strikingly large observed equivalent width of nearly 9000 Å that results in similar "dropout" colors as UDFj-39546284.

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The stellar mass structure of massive galaxies from z=0 to z=2.5; surface density profiles and half-mass radii

We present stellar mass surface density profiles of a mass-selected sample of 177 galaxies at 0.5 < z < 2.5, obtained using very deep HST optical and near-infrared data over the GOODS-South field, including recent CANDELS data. Accurate stellar mass surface density profiles have been measured for the first time for a complete sample of high-redshift galaxies more massive than 10^10.7 M_sun. The key advantage of this study compared to previous work is that the surface brightness profiles are deconvolved for PSF smoothing, allowing accurate measurements of the structure of the galaxies. The surface brightness profiles account for contributions from complex galaxy structures such as rings and faint outer disks. Mass profiles are derived using radial rest-frame u-g color profiles and a well-established empirical relation between these colors and the stellar mass-to-light ratio. We derive stellar half-mass radii from the mass profiles, and find that these are on average ~25% smaller than rest-frame g band half-light radii. This average size difference of 25% is the same at all redshifts, and does not correlate with stellar mass, specific star formation rate, effective surface density, Sersic index, or galaxy size. Although on average the difference between half-mass size and half-light size is modest, for approximately 10% of massive galaxies this difference is more than a factor two. These extreme galaxies are mostly extended, disk-like systems with large central bulges. These results are robust, but could be impacted if the central dust extinction becomes high. ALMA observations can be used to explore this possibility. These results provide added support for galaxy growth scenarios wherein massive galaxies at these epochs grow by accretion onto their outer regions.

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3D-HST Grism Spectroscopy of a Gravitationally Lensed, Low-metallicity Starburst Galaxy at z=1.847

We present Hubble Space Telescope (HST) imaging and spectroscopy of the gravitational lens SL2SJ02176-0513, a cusp arc at z=1.847. The UV continuum of the lensed galaxy is very blue, which is seemingly at odds with its redder optical colors. The 3D-HST WFC3/G141 near-infrared spectrum of the lens reveals the source of this discrepancy to be extremely strong [OIII]5007 and H-beta emission lines with rest-frame equivalent widths of 2000 +/- 100 and 520 +/- 40 Angstroms, respectively. The source has a stellar mass ~10^8 Msun, sSFR \sim 100/Gyr, and detection of [OIII]4363 yields a metallicity of 12 + log(O/H) = 7.5 +/- 0.2. We identify local blue compact dwarf analogs to SL2SJ02176-0513, which are among the most metal-poor galaxies in the SDSS. The local analogs resemble the lensed galaxy in many ways, including UV/optical SED, spatial morphology and emission line equivalent widths and ratios. Common to SL2SJ02176-0513 and its local counterparts is an upturn at mid-IR wavelengths likely arising from hot dust heated by starbursts. The emission lines of SL2SJ02176-0513 are spatially resolved owing to the combination of the lens and the high spatial resolution of HST. The lensed galaxy is composed of two clumps with combined size r_e \sim 300 pc, and we resolve significant differences in UV color and emission line equivalent width between them. Though it has characteristics occasionally attributed to active galactic nuclei, we conclude that SL2SJ02176-0513 is a low-metallicity star-bursting dwarf galaxy. Such galaxies will be found in significant numbers in the full 3D-HST grism survey.

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The Star Formation Rate Function for Redshift z~4-7 Galaxies: Evidence for a Uniform Build-Up of Star-Forming Galaxies During the First 3 Gyr of Cosmic Time

We combine recent estimates of dust extinction at z~4-7 with UV luminosity function (LF) determinations to derive star formation rate (SFR) functions at z~4, 5, 6 and 7. SFR functions provide a more physical description of galaxy build-up at high redshift and allow for direct comparisons to SFRs at lower redshifts determined by a variety of techniques. Our SFR functions are derived from well-established z~4-7 UV LFs, UV-continuum slope trends with redshift and luminosity, and IRX-beta relations. They are well-described by Schechter relations. We extend the comparison baseline for SFR functions to z~2 by considering recent determinations of the Hα and mid-IR luminosity functions. The low-end slopes of the SFR functions are flatter than for the UV LFs, Δα\sim+0.13, and show no clear evolution with cosmic time (z~0-7). In addition, we find that the characteristic value SFR* from the Schechter fit to SFR function exhibits consistent, and substantial, linear growth as a function of redshift from ~5 M_sun/yr at z~8, 650 Myr after the Big Bang, to ~100 M_sun/yr at z~2, ~2.5 Gyr later. Recent results at z~10, close to the onset of galaxy formation, are consistent with this trend. The uniformity of this evolution is even greater than seen in the UV LF over the redshift range z~2-8, providing validation for our dust corrections. These results provide strong evidence that galaxies build up uniformly over the first 3 Gyr of cosmic time.

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The Star-formation Mass Sequence out to z=2.5

We study the star formation rate (SFR) - stellar mass (M*) relation in a self-consistent manner from 0 < z < 2.5 with a sample of galaxies selected from the NEWFIRM Medium-Band Survey. We find a significant non-linear slope of the relation, SFR \propto M*^0.6, and a constant observed scatter of 0.34 dex, independent of redshift and M*. However, if we select only blue galaxies we find a linear relation SFR \propto M*, similar to previous results at z = 0 by Peng et al. (2010). This selection excludes red, dusty, star-forming galaxies with higher masses, which brings down the slope. By selecting on L_IR/L_UV (a proxy for dust obscuration) and the rest-frame U-V colors, we show that star-forming galaxies fall in three distinct regions of the log(SFR)-log(M*) plane: 1) actively star-forming galaxies with "normal" dust obscuration and associated colors (54% for log(M*) > 10 at 1 < z < 1.5), 2) red star-forming galaxies with low levels of dust obscuration and low specific SFRs (11%), and 3) dusty, blue star-forming galaxies with high specific SFRs (7%). The remaining 28% comprises quiescent galaxies. Galaxies on the "normal" star formation sequence show strong trends of increasing dust attenuation with stellar mass and a decreasing specific SFR, with an observed scatter of 0.25 dex (0.17 dex intrinsic scatter). The dusty, blue galaxies reside in the upper envelope of the star formation sequence with remarkably similar spectral shapes at all masses, suggesting that the same physical process is dominating the stellar light. The red, low-dust star-forming galaxies may be in the process of shutting off and migrating to the quiescent population.

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Observations of dark and luminous matter: the radial distribution of satellite galaxies around massive red galaxies

We study the projected radial distribution of satellite galaxies around more than 28,000 Luminous Red Galaxies (LRGs) at 0.28 25 kpc whereas baryons account for more than 50% of the mass at smaller radii. We calculate the total dark-to-baryonic mass ratio and show that it is consistent with measurements from weak lensing for environments dominated by massive early type galaxies. Finally, we divide the satellite galaxies in our sample into three luminosity bins and show that the satellite light profiles of all brightness levels are consistent with each other outside of roughly 25 kpc. At smaller radii we find evidence for a mild mass segregation with an increasing fraction of bright satellites close to the central LRG.

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Sizes and surface brightness profiles of quiescent galaxies at z ~ 2

We use deep Hubble Space Telescope Wide Field Camera 3 near-infrared imaging obtained of the GOODS-South field as part of the CANDELS survey to investigate a stellar mass-limited sample of quiescent galaxies at 1.5 < z < 2.5. We measure surface brightness profiles for these galaxies using a method that properly measures low surface brightness flux at large radii. We find that quiescent galaxies at z ~ 2 very closely follow Sersic profiles, with n_{median} = 3.7, and have no excess flux at large radii. Their effective radii are a factor ~ 4 smaller than those of low-redshift quiescent galaxies of similar mass. However, there is significant spread in sizes (sigma_{log r_e} = 0.24), with the largest z ~ 2 galaxies lying close to the z = 0 mass-size relation. We compare the stellar mass surface density profiles with those of massive elliptical galaxies in the Virgo cluster and confirm that most of the mass-growth which occurs between z ~ 2 and z = 0 must be due to accretion of material onto the outer regions of the galaxies. Additionally, we investigate the evolution in the size distribution of massive quiescent galaxies. We find that the minimum size growth required for z ~ 2 quiescent galaxies to fall within the z = 0 size distribution is a factor ~ 2 smaller than the total median size growth between z ~ 2 and z = 0.

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Spatially resolved Halpha maps and sizes of 57 strongly star-forming galaxies at z~1 from 3D-HST: evidence for rapid inside-out assembly of disk galaxies

We investigate the build-up of galaxies at z~1 using maps of Halpha and stellar continuum emission for a sample of 57 galaxies with rest-frame Halpha equivalent widths >100 Angstroms in the 3D-HST grism survey. We find that the Halpha emission broadly follows the rest-frame R-band light but that it is typically somewhat more extended and clumpy. We quantify the spatial distribution with the half-light radius. The median Halpha effective radius r_e(Halpha) is 4.2+-0.1 kpc but the sizes span a large range, from compact objects with r_e(Halpha) ~ 1.0 kpc to extended disks with r_e(Halpha) ~ 15 kpc. Comparing Halpha sizes to continuum sizes, we find =1.3+-0.1 for the full sample. That is, star formation, as traced by Halpha, typically occurs out to larger radii than the rest-frame R-band stellar continuum; galaxies are growing their radii and building up from the inside out. This effect appears to be somewhat more pronounced for the largest galaxies. Using the measured Halpha sizes, we derive star formation rate surface densities. We find that they range from ~0.05 Msun yr^{-1} kpc^{-2} for the largest galaxies to ~5 Msun yr^{-1} kpc^{-2} for the smallest galaxies, implying a large range in physical conditions in rapidly star-forming z~1 galaxies. Finally, we infer that all galaxies in the sample have very high gas mass fractions and stellar mass doubling times < 500 Myr. Although other explanations are also possible, a straightforward interpretation is that we are simultaneously witnessing the rapid formation of compact bulges and large disks at z~1.

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Mass growth and mergers: direct observations of the luminosity function of LRG satellite galaxies out to z=0.7 from SDSS and BOSS images

We present a statistical study of the luminosity functions of galaxies surrounding luminous red galaxies (LRGs) at average redshifts =0.34 and =0.65. The luminosity functions are derived by extracting source photometry around more than 40,000 LRGs and subtracting foreground and background contamination using randomly selected control fields. We show that at both studied redshifts the average luminosity functions of the LRGs and their satellite galaxies are poorly fitted by a Schechter function due to a luminosity gap between the centrals and their most luminous satellites. We utilize a two-component fit of a Schechter function plus a log-normal distribution to demonstrate that LRGs are typically brighter than their most luminous satellite by roughly 1.3 magnitudes. This luminosity gap implies that interactions within LRG environments are typically restricted to minor mergers with mass ratios of 1:4 or lower. The luminosity functions further imply that roughly 35% of the mass in the environment is locked in the LRG itself, supporting the idea that mass growth through major mergers within the environment is unlikely. Lastly, we show that the luminosity gap may be at least partially explained by the selection of LRGs as the gap can be reproduced by sparsely sampling a Schechter function. In that case LRGs may represent only a small fraction of central galaxies in similar mass halos.

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Revealing velocity dispersion as the best indicator of a galaxy's color, compared to stellar mass, surface mass density or morphology

Using data of nearby galaxies from the Sloan Digital Sky Survey we investigate whether stellar mass, central velocity dispersion, surface mass density, or the Sersic n parameter is best correlated with a galaxy's rest-frame color. Specifically, we determine how the mean color of galaxies varies with one parameter when another is fixed. When the stellar mass is fixed we see that strong trends remain with all other parameters, whereas residual trends are weaker when surface mass density, n, or velocity dispersion are fixed. Overall velocity dispersion is the best indicator of a galaxy's typical color, showing the largest residual color dependence when any of the other three parameters are fixed, and stellar mass is the poorest. Other studies have indicated that both the halo and black hole properties are better correlated with velocity dispersion than with stellar mass, surface mass density or Sersic n. Therefore, our results are consistent with a picture where a galaxy's star formation history and present star formation rate are determined to some significant degree by the current properties and assembly history of its dark matter halo and/or the feedback from its central super massive black hole.

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Which galaxy property is the best indicator of its host dark matter halo properties?

In this work we investigate the link between galaxy velocity dispersion, mass and other properties (color, morphology) with the properties of dark matter halos by comparing the clustering of galaxies at both fixed mass and velocity dispersion. We use the Sloan Digital Sky Survey to define a volume limited sample of massive galaxies complete in both stellar mass (>6e10 Msun) and velocity dispersion (>75 km/s). Using this sample we show that at fixed velocity dispersion there is no dependence of the clustering amplitude on stellar or dynamical mass. Conversely when stellar or dynamical mass are fixed there is a clear dependence of the clustering amplitude on velocity dispersion with higher dispersion galaxies showing a higher clustering amplitude. We also show that whilst when stellar or dynamical mass are fixed there remains a dependence of clustering amplitude on morphology, there is no such dependency when dispersion is fixed. However, we do see a dependence of the clustering amplitude on color when both mass and dispersion are fixed. Despite this, even when we restrict our samples to only elliptical or red galaxies the relationship between dispersion and clustering amplitude at fixed mass remains. It seems likely that the residual correlation with color is driven by satellite galaxies in massive halos being redder at fixed dispersion. The lack of a similar morphology dependence implies that the mechanism turning satellites red is not changing their morphology. Our central result is that velocity dispersion is more closely related to the clustering amplitude of galaxies than either stellar or dynamical mass. This implies that velocity dispersion is more tightly correlated with the halo properties that determine clustering, either halo mass or age, and supports the notion that the star formation history of a galaxy is more closely related to its halo properties than its overall mass.

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