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Subject headings: galaxies: clusters: general—galaxies: elliptical and lenticular, cD—galaxies: evolution—galaxies: formation—galaxies: fundamental parameters—galaxies: general—galaxies: photometry |
1. INTRODUCTION Hierarchical galaxy formation models embedded in a [MATH] CDM cosmology predict strong size evolution for massive galaxies. A higher gas fraction in high-redshift galaxies leads to more dissipation and hence compact galaxies (e.g., Robertson et al., 2006 ; Khochfar & Silk, 2006b , and subsequent evoluti... |
(e.g. Loeb & Peebles, 2003 ; Naab et al., 2007 . The models predict the strongest sample-averaged size evolution for the most massive galaxies |
(Khochfar & Silk, 2006a because of large differences in the gas fraction at different redshifts and because the assembly of massive galaxies continues until very late epochs in a hierarchical framework (e.g., De Lucia et al., 2006 |
Evidence for significant size evolution between [MATH] and the present has been building up quickly over the past few years (e.g. Trujillo et al., 2004 2006b ; Franx et al., 2008 . In particular, galaxies with low star formation rates and high stellar masses [MATH] ) appear to be extremely compact from [MATH] |
(Daddi et al., 2005 ; Trujillo et al., 2006a 2007 ; Longhetti et al., 2007 ; Cimatti et al., 2008 ; Rettura et al., 2008 to [MATH] |
(Zirm et al., 2007 ; Toft et al., 2007 ; van Dokkum et al., 2008 ; Buitrago et al., 2008 . Given the similarity between many of their observed properties there is likely to be an evolutionary connection between these distant compact galaxies and the present-day early-type galaxies despite the measured large difference ... |
The measurements of sizes and densities of high-redshift galaxies are hampered by many systematic uncertainties (e.g., morphological |
-corrections, surface brightness dimming, errors in photometric redshifts, and mass measurements). Most of these errors, however, are unlikely to fully account for the observed strong size evolution. The uncertainty in the mass estimates may be the exception. For the work in the literature, these mass estimates are alw... |
[MATH] galaxies must have physical central densities that are 3 orders of magnitude higher than those of local galaxies (van Dokkum et al., 2008 , further verification of those apparently reasonable assumptions is warranted. For example, a stellar initial mass function (IMF) that is radically different |
(e.g., Larson, 2005 ; Fardal et al., 2007 ; van Dokkum, 2008 ; Davé, 2008 from a Salpeter-like IMF (Salpeter, 1955 ; Scalo, 1986 ; Kroupa, 2001 ; Chabrier, 2003 ; Hoversten & Glazebrook, 2008 could reduce the stellar mass estimates by an order of magnitude, producing perfectly normal galaxies by today’s standards. |
The spectacular nature of these compact galaxies at [MATH] could be confirmed by direct, kinematical mass measurements. However, the quiescent nature of these objects and their consequent lack of emission lines (Kriek et al., 2006 require absorption-line measurements of their stellar velocity dispersions, which should ... |
[MATH] (Toft et al., 2007 ; van Dokkum et al., 2008 . Unfortunately, with the currently available instrumentation this is not feasible. These [MATH] galaxies are prohibitively faint at observed optical wavelengths (see, e.g., Cimatti et al., 2008 , and near-infrared spectroscopy is still maturing as a technique. Contin... |
At lower redshifts ( [MATH] ) absorption-line spectroscopy has for years been a powerful tool to study the evolution of distant early-type galaxies (van Dokkum et al., 1998 ; van Dokkum & Stanford, 2003 ; van Dokkum & Ellis, 2003 ; Wuyts et al., 2004 ; van der Wel et al., 2004 ; Treu et al., 2005a ; Holden et al., 2005... |
In this paper we compile a sample of galaxies at redshifts [MATH] with measured absorption line velocity dispersions from the literature and that are visually classified as early-type galaxies with the aid of Hubble Space Telescope HST ) imaging from the Advanced Camera for Surveys (ACS; Ford et al., 1998 . We measure ... |
The advantage of this approach is that the size and density measurements are independent of the photometric properties of the galaxies apart from the surface brightness profile. The absence of luminosity and other photometric properties from our analysis assures us that our study does not suffer from the strong possibl... |
(e.g., Trujillo et al., 2004 2006b ; Cimatti et al., 2008 but do not extend this analysis to verify the consistency with size measurements of nearby galaxies. |
In § we describe the sample of nearby early-type galaxies and derive the dynamical mass-size relation. In we construct the sample of [MATH] early-type galaxies, determining their masses and sizes in a manner that is consistent with the nearby sample. In § we quantify systematic effects in our size measurements through ... |
[MATH] 2. NEARBY EARLY-TYPE GALAXIES 2.1. Velocity Dispersions and Sizes We have extracted a large sample of early-type galaxies at redshifts |
[MATH] from the SDSS database (DR6; Adelman-McCarthy et al., 2008 based on the criteria as outlined by Graves et al. ( 2007 Briefly, galaxies on the red sequence and either without emission lines or with high |
[MATH] to [MATH] ratios are included in the sample. These criteria effectively exclude star-forming galaxies, but include genuine early-type galaxies with nuclear activity |
(see Yan et al., 2006 The dispersion as measured within the spectroscopic aperture [MATH] ) is corrected to match the average dispersion within the effective radius [MATH] (measured as described below) following Jørgensen et al. ( 1995 |
[EQUATION] where [MATH] is the radius of the SDSS spectroscopic fiber [MATH] ) in kpc at the distance of the galaxy. We use the correction from Jørgensen et al. ( 1995 for consistency with previously published results. We note that Cappellari et al. ( 2006 used better data to improve the aperture correction, but the re... |
[MATH] is only a few percent. We use GALFIT (Peng et al., 2002 to determine effective radii from the SDSS [MATH] -band imaging assuming an [MATH] profile, leaving the effective radius, the integrated magnitude, the position angle, the axial ratio, and the position of the center as free parameters. The point-spread func... |
[MATH] profile (Sérsic, 1968 may provide a more realistic description of the surface brightness distribution of individual early-type galaxies, especially in the presence of a significant disk. However, [MATH] provides a good description of the average profile of early-type galaxies both nearby (de Vaucouleurs, 1948 an... |
[MATH] as an additional free parameter results in unnecessarily large, redshift-dependent systematic uncertainties in the size measurements (see § ). |
The size parameter that we use in this paper is the circularized effective radius [MATH] , where [MATH] is the effective radius along the major axis (the output parameter of GALFIT, |
[MATH] is the effective radius along the minor axis, and [MATH] the axis ratio (as calculated by GALFIT); [MATH] is a good approximation for optically thin luminosity distributions such as the generally dust-poor early-type galaxies in our samples. The systematic and random errors of our size determinations are inferre... |
The SDSS spectroscopic catalog suffers from several biases that may mitigate size evolution measurements. First, compact sources are not targeted for spectroscopy as they may be mistaken for stars or because their central surface brightnesses, i.e., their fiber magnitudes, are too bright. Second, almost all galaxies th... |
(see the HyperLEDA database compiled by Paturel et al., 2003 have been claimed to have high, [MATH] velocity dispersions have dispersions of [MATH] in the SDSS (see Bernardi, 2007 , Appendix A) . The source of this discrepancy is unknown. While Bernardi convincingly argues that the SDSS dispersions are more reliable, t... |
2.2. The Mass-Radius Relation and the Mass-Density Relation From [MATH] and [MATH] we derive the total dynamical mass and the corresponding average surface mass density within [MATH] |
[EQUATION] [EQUATION] with [MATH] , which has been shown to hold for local galaxies (Cappellari et al., 2006 . Following Shen et al. ( 2003 we adopt the following characterization of the [MATH] [MATH] relation: |
[EQUATION] With a least-squares linear fit to all galaxies with mass [MATH] we find that the slope is [MATH] and the zero point normalized to a characteristic mass [MATH] |
is [MATH] kpc. We find statistically the same relation if we perform a linear fit to the values of the median [MATH] in 0.1 dex wide mass bins in the range [MATH] . The scatter around the best-fit relation is 0.14 dex. |
Using stellar masses, [MATH] Shen et al. ( 2003 find the same slope [MATH] for the [MATH] [MATH] relation. Their zeropoint, however, is larger ( [MATH] kpc). This is likely due to the difference between |
[MATH] and [MATH] as Cappellari et al. ( 2006 show for a Kroupa ( 2001 IMF (which is also used by Shen et al. 2003) that [MATH] This translates into a difference in [MATH] of [MATH] 20%, close to the observed difference. Furthermore, Shen et al. ( 2003 analyze SDSS |
[MATH] -band imaging whereas we use [MATH] -band imaging. 3. DISTANT EARLY-TYPE GALAXIES 3.1. Velocity Dispersions and Sizes Several authors have published velocity dispersion measurements of early-type galaxies at [MATH] . We compile the data from three different data sets for which the selection criteria are well und... |
[MATH] (Wuyts et al., 2004 . The seven cluster galaxies at [MATH] for which dispersions have been measured (van Dokkum & Stanford, 2003 ; Holden et al., 2005 are not included because of the paucity of this sample, which prevents us to accurately model selection effects. The final sample only contains galaxies with [MAT... |
For all galaxies ACS imaging is available. In order to produce an internally consistent data set, we re-measure the sizes of all galaxies. GOODS |
provides deep, publicly available ACS imaging of the CDF-S and the HDF-N (Giavalisco et al., 2004 in four filters. We use the F850LP [MATH] -band) images in order to match the rest-frame wavelength at which the sizes of the nearby comparison sample are measured (the SDSS |
[MATH] -band; see § ). For the MS 1054-0321 cluster ACS imaging has been taken as part of the guaranteed time observation program |
(Blakeslee et al., 2006 . We use the F775W ( [MATH] -band) imaging as the available [MATH] -band imaging is of lesser quality. At this redshift rest frame [MATH] falls in between [MATH] and [MATH] such that the morphological -correction is not a problem; the [MATH] |
galaxies in the sample of Treu et al. ( 2005b are only [MATH] smaller in the [MATH] band than in the [MATH] band. With GALFIT we determine effective radii in the same manner as for the nearby galaxy sample (see § 2.1 ). The PSF is constructed with Tiny Tim (Krist, 1995 , even though using stars results in negligible di... |
We use Eqs. and to compute masses and surface densities. Again, we adopt [MATH] , which has been shown to hold for distant nonrotating elliptical galaxies |
(van der Marel & van Dokkum, 2007 ; van der Wel & van der Marel, 2008 . For rotating early-type galaxies the situation appears to be more complex (van der Wel & van der Marel, 2008 |
in the sense that [MATH] is possibly [MATH] 20% larger than 5. We comment on the impact of this possible complication on our size evolution measurement in § 5.2 . A low-mass cutoff of [MATH] is applied since below this limit no useful samples are available due to severe incompleteness of the surveys |
(see, e.g., van der Wel et al., 2005 3.2. The Average Surface Brightness Profile In determining the sizes of the nearby and distant galaxies in the previous sections we assumed that an [MATH] profile provides an accurate description of early-type galaxies. We know this to be true for nearby galaxies, but not for [MATH]... |
(see, e.g., Blakeslee et al., 2006 ; Rettura et al., 2006 However, there is a possibility that the true values of [MATH] are different; at large radii the “wings” of the profile become quickly overwhelmed by background noise, even in the deepest HST |
imaging. Because [MATH] and the measured [MATH] are correlated, assuming [MATH] for all redshifts introduces systematic errors in case |
[MATH] evolves with redshift. In order to examine the profiles of the [MATH] galaxies at large radii, we median-stack the [MATH] -band images of all elliptical galaxies (S0s are excluded) without bright neighbors in our CDF-S and HDF-N samples (see Fig. ). The images of the individual galaxies are drizzled onto a commo... |
With GALFIT we subtract [MATH] profiles with integer values [MATH] (see Fig. ). The negative residuals outside [MATH] for models with large [MATH] and the positive residuals for models with small |
[MATH] indicate that these limiting cases provide poor fits of the outer regions of elliptical galaxies at [MATH] . The [MATH] and |
[MATH] profiles provide the best description of their average surface brightness distributions. This visual impression is confirmed by the [MATH] -values of the respective fits: [MATH] for both |
[MATH] and for [MATH] , whereas [MATH] for other values of [MATH] Interestingly, [MATH] does not evolve significantly with redshift, and we conclude that it is safe to assume that choosing [MATH] for both nearby and distant early-type galaxies does not introduce significant systematic errors. |
4. SIMULATIONS OF SIZE MEASUREMENTS To test the robustness of our size determinations of local and distant early-type galaxies in §§ and we simulate size measurements by using SDSS [MATH] -band imaging of 45 early-type galaxies in the Virgo Cluster (Mei et al., 2007 . The pixels of the mosaics of the Virgo Cluster gala... |
for the distant sample. The physical sizes of the simulated galaxies are thus conserved. For each redshift ( [MATH] ) we run [MATH] 200 simulations with different values for the flux density of the simulated galaxies, which are chosen such that the simulated galaxies have the same range in apparent magnitude as the obs... |
The results of the simulations are shown in Fig. . Within the nearby sample we find systematic, redshift-dependent differences, of up to [MATH] 10%. Random errors, derived from the scatter in the sizes inferred from the simulated images, are typically less than 5%. Systematic differences between the nearby and distant ... |
Adopting a [MATH] law with [MATH] as a free parameter may improve the quality of the fits to individual galaxies. However, our simulations reveal that the random errors increase to [MATH] 20–25%, without much change in the systematic errors. Together with the analysis of stacked images (§ 3.2 ), this test justifies our... |
The sizes we derive in the §§ and , and the derived quantities [MATH] (Eq. ) and [MATH] (Eq. ), are corrected to account for systematic measurement errors. Those corrections depend on redshift and are interpolated between the values listed in Fig. . For simplicity the dependence on size is not taken into account, such ... |
5. STRUCTURAL EVOLUTION OF EARLY-TYPE GALAXIES 5.1. Evolution of the [MATH] -Radius Distribution In Figure we compare the [MATH] [MATH] distributions of the nearby and distant early-type galaxy samples. This unusual projection of the fundamental plane (FP; Dressler et al., 1987 ; Djorgovski & Davis, 1987 has a very lar... |
[MATH] kpc) are rare in the local universe. In the nearby sample, galaxies with [MATH] have much larger sizes, and galaxies with sizes [MATH] kpc have dispersions of [MATH] . These numbers are only intended to guide the eye. A quantitative analysis of the differences between nearby and distant galaxies is presented bel... |
The distant sample is not directly comparable with the nearby sample in its entirety ( left ), as the nearby sample reaches to much lower masses. In order to assess the question whether the true, underlying [MATH] [MATH] distribution of distant galaxies is different from the [MATH] [MATH] distribution of nearby galaxie... |
Treu et al. ( 2005b do not precisely scale linearly with luminosity [MATH] , where [MATH] is the surface brightness, but as [MATH] . This implies that, at fixed luminosity [MATH] [MATH] . Since a dispersion measurement requires a minimum |
[MATH] [MATH] ), a galaxy with luminosity [MATH] has a maximum radius [MATH] for which its dispersion can be determined. We use the luminosity limits of the surveys discussed above to normalize the dependence between luminosity and maximum size; we simply assume that for the smallest galaxies ( [MATH] kpc) the luminosi... |
[EQUATION] One would expect that for galaxies smaller than 3 kpc the signal-to-noise ratio of the spectra would not depend on size any longer since seeing generally dominates the apparent sizes of such small galaxies at [MATH] . Because of the variety of telescopes, weather conditions, and data reduction techniques, th... |
The difference between the [MATH] [MATH] distributions at low and high redshift is highly significant, even after taking selection effects into account (Fig. right ). The two-dimensional Kolgomorov-Smirnov statistic has a high value [MATH] ), which implies that it is extremely unlikely that the nearby and distant sampl... |
5.2. Evolution of the Mass-Radius Relation The structural difference between the nearby and distant samples described in the previous section implies that the [MATH] [MATH] and |
[MATH] [MATH] relations evolve with redshift. In Fig. we show the [MATH] [MATH] relation for the distant sample, and compare this with the equivalent relation for nearby galaxies derived in § 2.2 . Clearly, the relation shifts to smaller radii from low to high redshift. |
Parametrized as in Eq. we find [MATH] and [MATH] with a scatter of [MATH] dex (after subtracting the observational uncertainties in quadrature). The errors are estimated with a bootstrap/Monte-Carlo simulation in which the data points are randomly sampled and varied according to the (correlated) measurement errors whic... |
The treatment of the selection effects described in the previous section shows that the observed size evolution seen in Fig. is not an artifact. However, given the nature of the selection effects, which favor small galaxies over large galaxies, the intrinsic amount of size evolution and possible evolution in the slope ... |
relation at [MATH] that reproduces the observed [MATH] [MATH] distribution after applying the selection criteria. We take an iterative approach due to the interdependence of the selection criteria and the amount of evolution in zeropoint, slope, and scatter of the [MATH] [MATH] relation. In the following we de-evolve t... |
[MATH] [MATH] relation. The simplest evolutionary scenario is a change in the zero point [MATH] (see Eq. ). For each object in the nearby sample the size is reduced by the same amount [MATH] , and those that do not satisfy the selection criteria described in the previous section are removed. From the remaining sub-samp... |
[MATH] [MATH] relation is determined. The different selection criteria and sample sizes for field and cluster galaxies are taken into account in this process, which is repeated for many different values of [MATH] . We find that an intrinsic value of |
[MATH] reproduces the observed value of [MATH] Hence, it appears that selection effects do not strongly affect the inferred size evolution. |
However, the scatter of the assumed intrinsic distribution (0.14 dex) is higher than the observed scatter ( [MATH] dex). This cannot be explained by selection effects in the simple scenario described above. It is therefore required that the scatter, as well as the zero point, is also treated as an evolving parameter. T... |
relation by a given fraction. Doing so, we find that the best-fitting zero point [MATH] is not different from the earlier estimate based on a nonevolving scatter. We also find that the evidence for evolution in the scatter is weak ( [MATH] ). This exercise mainly serves to show that our size-evolution result in not sen... |
A similar verification must be carried out for evolution in the slope of the [MATH] [MATH] relation. Allowing only the scatter and the size to evolve, as described above, the inferred slope of the “observed” [MATH] [MATH] relation is 0.59, marginally consistent with the true observed slope of [MATH] . If we treat the s... |
We conclude that, despite (weak) evidence for evolution in the slope and the scatter of the [MATH] [MATH] relation with redshift, there is no significant improvement in modeling the observations by adopting slope and scatter as free parameters. Modeling the evolution by a fractional change in size, regardless of mass a... |
[MATH] times smaller than at [MATH] The weak evidence for a change in slope of the [MATH] [MATH] relation may also be interpreted as a difference between field and cluster galaxies, as the more massive galaxies in our sample tend to be cluster galaxies. Assuming that slope and scatter remain constant but that the zero ... |
kpc for field galaxies and [MATH] kpc for cluster galaxies. The [MATH] error on this difference of 0.57 kpc is 0.32 kpc. The true error may be larger since in this estimate it is assumed that scatter and slope behave the same in the different environments and that there are no relative systematic errors in the size det... |
(see also Rettura et al., 2008 . However, we have to keep in mind that so far only a very small number of clusters is considered. Future studies will need to extend the existing analyses to a larger number of clusters to verify the general validity of the results. |
So far, we have assumed that the masses of the galaxies do not change. Our justification is that the scatter hardly depends on mass; the effect of a changing mass function on modeling selection effects is expected to be small. However, physically speaking, it is unnatural to propose size evolution without changes in th... |
We recall that the nearby sample is biased against compact early-type galaxies (§ 2.1 ). The potentially underestimated number of galaxies with dispersions [MATH] is unlikely to drastically affect the size-evolution determination for the sample as a whole as the average dispersion of the galaxies in the distant sample ... |
[MATH] [MATH] relation is possibly overestimated, which would lead to an underestimate of the slope evolution. More important is the problem that small galaxies are missed because of their photometric misclassification as stars in the SDSS. To fully address this issue a complete analysis of the SDSS photometric catalog... |
In § we noted that the dynamical mass estimate as adopted in this paper (Eq. , with [MATH] ) may be too low for rotating early-type galaxies. If this is the case, then size evolution for these galaxies will be underestimated by [MATH] 10%. Since this is within the uncertainties of our measurements we do not take this f... |
Obviously, size evolution at fixed mass translates into density evolution. This is illustrated in Fig. where we compare the density distribution of [MATH] early-type galaxies with the |
[MATH] [MATH] relation for nearby galaxies. Because [MATH] does not strongly depend on [MATH] , evolutionary trends are readily visible; [MATH] early-type galaxies are [MATH] 4 times more dense than their local counterparts. The apparent change in slope can possibly be explained by selection effects, completely analogo... |
relation. Note that compared to the increase in projected density, the increase in physical density will be even larger. Up until recently, early-type galaxies were thought to evolve more or less passively. This appears to be an over-simplification and may apply more to their stellar populations than to their structura... |
6. DISCUSSION 6.1. Comparison with Photometric Size-Evolution Measurements The main goal of this paper is to use dynamical measurements to investigate whether early-type galaxies were smaller and denser in the past. Previous work has shown that the stellar mass surface density is higher, but there are a number of issue... |
In Fig. we compare the size-evolution results presented in the previous section with size-evolution results for early-type galaxies based on photometric mass estimates. For all the literature samples we take the mean redshift and the mean stellar mass (normalized to the Kroupa IMF), and compute the mean offset from the... |
[MATH] , MS 2053-04 at [MATH] , and MS 1054-0321 and RX J0152.7-1357 , both at [MATH] . Note that we also include MS 1054-0321 in the present study with dynamical mass measurements. The agreement between the independent measurements confirms that at least out to [MATH] dynamical and photometric mass estimates based on ... |
van der Wel et al. ( 2006 Rettura et al. ( 2006 , and Holden et al. ( 2006 The literature samples have all been selected in different ways, and so a direct comparison with our work may not be straightforward. Not all samples are morphologically selected; many are selected by their spectral or photometric properties. In... |
(e.g., van Dokkum et al., 2008 ; Kriek et al., 2008 , but these issues need to be further addressed in the future. Even with this cautionary proviso, the broad agreement between the results presented in this paper and the photometric results at higher redshifts is striking. All studies included in Fig. are consistent w... |
[MATH] and the present for galaxies with a given mass. A linear fit in log-log space to our two data points at [MATH] and [MATH] gives [MATH] . With a linear fit to the photometric data the inferred rate of evolution is |
[MATH] , where the error is obtained via a bootstrap/Monte Carlo simulation. The broad agreement of our measurement of the size evolution of early-type galaxies with the photometric studies is encouraging and alleviates concerns about serious systematic effects that potentially could have compromised previous work. Mos... |
Other systematic uncertainties cannot explain the observed evolution either. In our size measurements, systematic effects have been taken into account (see Secs. 3.2 and ). We are confident, for example, that we would detect low-surface brightness envelopes around distant galaxies. Furthermore, we know that only a mino... |
(e.g., Rodighiero et al., 2007 ; van der Wel et al., 2007 such that it is unlikely that central point sources affect our size measurements. This is also clear from the fact that the residuals of our [MATH] profile fits generally do not show central point sources and that none of the deep spectra used to measure dispers... |
Despite the broad consistency between our results and those previously published, the agreement is not perfect. There is a marginal inconsistency at the [MATH] level between the size-evolution measurement from kinematic data and the size-evolution measurement from photometric data shown in Fig. . This could point to th... |
Our robust results strengthen the results from previous studies. We conclude that early-type galaxies at [MATH] are [MATH] times smaller than local early types with the same mass, and that at [MATH] this size difference is likely increased to a factor of [MATH] 4, as previously observed by Zirm et al. ( 2007 Toft et al... |
van Dokkum et al. ( 2008 , and Buitrago et al. ( 2008 6.2. Comparison with Model Predictions The fact that we see considerable evolution in galaxy size with redshift is not surprising from a theoretical perspective. Most semianalytic models of galaxy formation in a [MATH] CDM universe predict substantial size evolution... |
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