Surface density dependent evolution to z=3

Structure and star formation in galaxies out to : evidence for surface density dependent evolution and upsizing11affiliation: Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5–26555. 22affiliation: Based on observations collected at the European Southern Observatory, Chile (ESO Programme LP168.A-0485)

Marijn Franx33affiliation: Leiden Observatory, Leiden University, P.O. Box 9513, NL-2300 RA Leiden, Netherlands , Pieter G. van Dokkum44affiliation: Department of Astronomy, Yale University, New Haven, CT 06520-8101 , Natascha M. Förster Schreiber55affiliation: MPE, Giessenbackstrasse, D-85748, Garching, Germany , Stijn Wuyts66affiliation: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138 77affiliation: W. M. Keck postdoctoral fellow , Ivo Labbé88affiliation: Carnegie Observatories, 813 Santa Barbara Street, Pasadena, CA 91101 99affiliation: Hubble Fellow , Sune Toft1010affiliation: European Southern Observatory, D-85748 Garching, Germany
Abstract

We present an analysis of galaxies in the CDF-South. We find a tight relation to between color and size at a given mass, with red galaxies being small, and blue galaxies being large. We show that the relation is driven by stellar surface density or inferred velocity dispersion: galaxies with high surface density are red and have low specific star formation rates, and galaxies with low surface density are blue and have high specific star formation rates. Surface density and inferred velocity dispersion are better correlated with specific star formation rate and color than stellar mass. Hence stellar mass by itself is not a good predictor of the star formation history of galaxies. In general, galaxies at a given surface density have higher specific star formation rates at higher redshift. Specifically, galaxies with a surface density of are “red and dead” at low redshift, approximately 50% are forming stars at , and almost all are forming stars by . This provides direct additional evidence for the late evolution of galaxies onto the red sequence. The sizes of galaxies at a given mass evolve like . Hence galaxies undergo significant upsizing in their history. The size evolution is fastest for the highest mass galaxies, and quiescent galaxies. The persistence of the structural relations from to , and the upsizing of galaxies imply that a relation analogous to the Hubble sequence exists out to , and possibly beyond. The star forming galaxies at are quite different from star forming galaxies at , as they have likely very high gas fractions, and star formation time scales comparable to the orbital time.

Subject headings:
cosmology: observations — galaxies: evolution — galaxies: formation – galaxies: high redshift
slugcomment: accepted for publication, ApJ, 2008, 689

1. Introduction

The dramatic progress in observational capabilities of the last decade have enabled studies of galaxy evolution and formation to epochs which were completely inaccessible only 15 years ago. Nevertheless, despite this dramatic progress, many open questions remain. Some of these are due to very fundamental problems related to studies of galaxy formation. The most important problem is that it has become clear that galaxy evolution is a complex process. Galaxy mergers change the mass function continuously. Furthermore, star formation does probably not occur with nice, smooth exponentially declining star formation rates - starbursts can change the luminosities of galaxies on a short timescale. As a result, studies of galaxy evolution and formation are of a statistical nature: we cannot directly establish which galaxies at would evolve into what galaxies at , and we therefore have to analyze statistically the sample as a whole.

Modern analyses therefore employ large samples, and attempt to use stellar masses whereever possible, as these are less likely to evolve rapidly (e.g., Brinchman & Ellis 2001, Kauffmann et al. 2003b, Dickinson et al. 2003, Rudnick et al. 2003, Fontana et al. 2005, Borch et al. 2006, Faber et al. 2007). However, as stressed by Faber et al. (2007), the mass function does not show a strong mode or other feature. As a result, we have to characterize the mass function by the characteristic mass at which it turns over. The determination of this characteristic mass and the determination of the general shape of the mass function is hard.

A different type of information may come from studies of the structure of galaxies. By studying the sizes, colors, velocity dispersions, etc, one may be able to derive tight correlations between galaxy properties. These tight correlations can then be used to study galaxy evolution. For example, by measuring velocity dispersions, sizes, and luminosities of early-type galaxies one can establish correlation between mass-to-light ratio and mass and size of a galaxy (e.g., Djorgovski et al. 1988, Faber et al. 1987). The evolution of these correlations with redshift (e.g., Franx 1993, van Dokkum and Franx 1996, van der Wel et al. 2005, Treu et al. 2005, van Dokkum & van der Marel 2007) provide the evolution of the mass-to-light ratio with redshift, which is essential for a proper interpretation of the evolution of the luminosity function. In addition this evolution puts strong constraints on the evolution of the early-type galaxies per se, and especially the time at which they formed their stars. However, we have to add that this interpretation is not entirely insensitive to the statistical evolution of the class of early-types (e.g., van Dokkum and Franx 2001). Nevertheless, detailed studies of galaxy properties can provide important additional constraints on galaxy evolution, even when relatively small samples are studied compared to the full statistical analyses.

In this paper we follow the second approach, where we study a fairly limited sample of -band selected galaxies in the CDFS (1155 galaxies from Wuyts et al. 2008), augmented by a local comparison sample based on the Sloan Digital Sky Survey (Strauss et al. 2002). We use size measurements to explore the relations between color, masses, star formation rates and size. We show that a good correlation exists between the sizes, masses, colors and specific star formation rates of the galaxies from to , and we analyze this correlation in this paper. The paper is built up in the following way: section 2 presents the data used in the analysis, both the high redshift sample, and the low redshift sample. Section 3 presents the correlation between mass, size, color at redshifts , 1, 2 and 3. In addition, the relation between mass, size and specific star formation rate is presented. Section 4 presents the evolution of the specific star formation rate as a function of redshift, and section 5 presents the evolution of size. The results are discussed in section 6, and in section 7 we discuss the potential biases that may occur in current samples of high redshift galaxies and analyses. The results are summarized in section 8.

2. Observations and derived quantities

2.1. High redshift ()

Figure 1.— The redshift distribution of the final CDFS sample. The galaxies are selected in the observed band, which lies redward of the rest-frame Balmer-4000Å break for this sample.
Figure 2.— Comparison of the star formation rates estimated from the UV + 24m MIPS fluxes, and the star formation rates estimated by SED fits. There is generally a good correspondence at redshifts below 2.5, with mild offsets smaller than 30%. For the remainder of the paper, we use the UV + 24m MIPS star formation rates. The typical formal error in the estimated star formation rates is shown in the lower right corner. The systematic uncertainties are a factor of 2 or more.

The high redshift sample of galaxies is taken from the GOODS-CDF-South field as presented by the FIREWORKS study by Wuyts et al. (2008). Briefly, Wuyts et al. (2008) combined the available optical-near IR-mid IR observations on the GOODS-CDF-South field. The optical imaging consisted of ACS GOODS imaging (Giavalisco et al 2004), and deep imaging with the ESO/MPG 2.2-m telescope (Arnouts et al. 2001, Wolf et al. 2004). The Near-IR imaging was taken with the VLT and ISAAC and presented in a reduced form by Vandame et al. (2001) and Vandame (in preparation). The GOODS IRAC and MIPS 24 m imaging was taken from Dickinson et al. (in preparation). Wuyts et al. (2008) homogonized the PSFs of the GOODS ACS and VLT Near-IR imaging data, to derive optical-near-ir photometry, based on a K-selected sample. As the PSFs of the other data sets were significantly worse, the PSF convolution and fitting technique of Labbé et al. (2006) and Labbé, (in preparation) was used to derive colors in the remaining bands. The overall photometry derived by Wuyts et al. (2008) agrees well with that of Grazian et al. (2006), with the exception of an offset in the IRAC photometry in Grazian et al. (2006), which was removed in a later update of the Grazian et al. (2006) catalogue (Grazian, private communication). The total effective area of the survey is 138 arcmin.

Wuyts et al. (2008) derived photometric redshifts using EAZY (Brammer et al. 2008), and found good correspondence between the photometric redshifts and the available spectroscopic redshifts. The overall difference amounted to 0.03 for the full sample with spectroscopic redshifts, and was slightly larger for galaxies with : .

The Wuyts. et al. (2008) sample, and the sample used here, is selected in the -band. To assure high quality photometry in the near-IR, only galaxies with a total -magnitude below 22.5 were used, and with a signal-to-noise higher than 10 in the -band. Galaxies with redshifts between 0.5 and 3.5 were used, assuring that the detection band is always redward of the restframe Balmer/4000 Å break. The resulting redshift distribution of the sample used in the paper is shown in Fig. 1.

Another ingredient in the analysis presented here is the sizes of the galaxies. The sizes have been determined in the band redwards of the redshifted 4000 Å break and closest to the rest-frame band. Sersic models convolved with the PSF were fit to each galaxy. The sersic index was allowed to vary between 1 and 4. The procedure is identical to that used by Trujillo et al. (2006a), and Toft et al. (2007). The procedure was as follows: for each near-IR tile in the field, the PSF was determined by averaging the normalized PSFs from the stars in the tile. For the ACS imaging, the PSF was similarly determined from stars. Then the GALFIT program was used (Peng et al. 2002) to derive the best fitting position, flux, circularized half light radius , sersic index, ellipticity and position angle. Independently, the r4fit program written by the first author was used to verify the results. This program has been extensively used in the past (e.g., van Dokkum & Franx 1996, van Dokkum et al. 1998). The results agreed well, with a median systematic offset as small as 3%. We verified that all the results presented later do not change when the method is changed. Furthermore, the results change rather little when the sizes from just the -band imaging are used. Trujillo et al. (2006a) studied in great detail the possible systematic effects that can arise from low signal-to-noise, small input sizes, and other effects. In general, the systematic effects are small at high signal-to-noise. At sizes above 0.1 arcsec, the systematic effects in are very small - which is surprising, as the FWHM of the PSF is about 0.45 arcsec. As the instrumental setup of Trujillo et al. (2006a) was identical to the one used here, we expect similar errors, and hence we expect that the sizes are reliable for measured sizes above 1 kpc (corresponding to 0.12 arcsec at ).

Galaxy masses have been estimated from SED fits to the full photometric dataset shortward of 24m, and are presented in Förster Schreiber et al. (in preparation) following similar procedure as described by Förster Schreiber et al. (2004). The masses used here are based on fits with stellar population models by Bruzual & Charlot (2003). The models used in the study were based on star formation histories with exponentially declining star formation rates. The time scales were 0 (single stellar population), 300 Myr, and infinity (constant star formation). A Calzetti et al. (2000) extinction curve was used, and the extinction in the V band was allowed to vary between 0 and 4. The single stellar population models did not include dust extinction. The fits also produced estimates of the star formation rates, and these are briefly used below for comparison. The IMF used was the Salpeter (1955) IMF. The results were normalized to a Kroupa (2001) IMF, by multiplying the masses and star formation rates by . We verified that this approximation is accurate to a few percent. We notice that the masses presented here decrease on average by 1.4 when Maraston (2005) models are used, without a dependence on redshift (see also Wuyts et al. 2007b).

Figure 3.— Top row: the restframe color against stellar mass, for the low redshift sample (from the SDSS), and the redshifts 1, 2 and 3. A large spread in restframe colors exists for masses between and , for all redshifts. The absence of red, low mass galaxies at is due to selection effects. The thick curves in the higher redshift panels delineate the area where the samples are less than 75 % complete. The red galaxies to the upper left of the curve are missing as they are too faint in the observed near-IR. Bottom row: The relation between size and stellar mass for our redshift intervals. A large spread in sizes exists for masses between and , for all redshifts. The thin dashed line indicates the mass-size relation from Shen et al. (2003) for early-type galaxies at . It is clear from the figure that the scatter in color and size is very large for masses between 10 and 10 . As in subsequent plots, the typical formal error is shown in the lower right corner.

Star formation rates were also estimated from the UV and 24 m MIPS fluxes. First, mid-IR fluxes were estimated using the models by Dale & Helou (2001), and the prescription by Labbé et al. (in preparation) and Wuyts et al. (2008). In short, a large range of models was used to convert the MIPS flux to a bolometric flux, and the mean of the log of the bolometric fluxes was used as a best estimate. Typically, the top and bottom estimates varied by a factor of 3-5 from our best estimate. We note, however, that Wuyts et al. (in preparation) found that the conversion used here agreed within 10% with the conversion given by Papovich et al. (2007), based on observed 70 m and 160 m fluxes for galaxies in the E-CFDS. The star formation rate of the galaxies was estimated assuming the Kroupa IMF : . This is the relation by Bell et al. (2005), and it is based on the relations given by Kennicutt (1998). The Bell et al. (2005) relation has been adapted to the Kroupa IMF used in this study.

The star formation rates have been measured for all galaxies, and the 1-sigma errors on the measured star formation rates dependend on redshift. The typical random errors are less than 1 year at , less than 5 year at , increasing rapidly to 25 year at .

We compare the star formation rates derived from the UV+ 24 m MIPS fluxes with the star formation rates derived from the SED fits in Fig. 2. We compare the star formation rates at different redshift intervals, , , . We can see a good correlation in the first two redshift bins. There is a small offset at , with a median of SFR(UV+MIPS)/SFR(SED) = 0.77. At , the median ratio is 1.29. These deviations are much smaller than the uncertainties in the 24 m MIPS flux to total IR flux conversion, which are up to a factor of 3. At , the correspondence is less good. This is expected: observational errors on the 24m fluxes become more important, and the 8m PAH emission feature shifts out of the 24m band at those redshifts. Overall, the agreement between the star formation rates from UV+MIPS and SED-fits is surprisingly good at . This may come as a surprise, as other authors found that the MIPS based star formation rates are significantly too high for a substantial fraction of the galaxies (e.g., Daddi et al. 2007). We do not find such an effect. This may be due to the fact that we use a linear conversion from MIPS flux to full IR flux. This apparently works well. The assumption is further justified by the work of Papovich et al. (2007), who found that the earlier empirical conversions of MIPS flux to star formation rate for galaxies generally over-predict the star formation rate. As a matter of fact, the results of Papovich et al. (2007) are in very good agreement with a simple linear conversion. However, we have to note that our estimates of the star formation rates may still be wrong: we have not measured the full IR flux and depend on extrapolations. We remark, however, that Daddi et al. (2007) concluded that the star formation measured from the dustcorrected UV is a good representation of the true star formation rate. Our SED fits should be a good approximation of those values. But at the same time, we find that our UV+MIPS star formation rates correspond well to the star formation rates based on SED fits. Hence, in the rest of the paper, we use the UV+MIPS star formation rates. Galaxies with X-ray detections (Giacconi et al. 2002) were omitted from the sample, as these have likely AGN.

For consistency, we use the MIPS derived star formation rates also at . We note that at , the MIPS star formation rates are higher than the SED derived star formation rates by a factor than 2.8 (Fig. 2). We verified that the results obtained in the paper remain valid when using the SED derived star formation rates. The main difference is a systematic decrease in derived star formation rates and specific star formation rates at and higher.

2.2. Low redshift sample: SDSS

For comparison at low redshift, we use the SDSS sample, as used by Kauffmann et al. (2003a,b). These authors derived masses for galaxies from the DR2 release (Abazajian et al. 2004). The masses are based on mass-to-light ratios estimated from analysis of the nuclear spectra. We applied a small correction to the mass-to-light ratios: as the colors of the nuclei are generally redder than the color of the galaxy as a whole, the mass-to-light ratio is generally slightly overestimated. We corrected the mass-to-light ratios with a factor of , where . Furthermore, we corrected the masses by the fraction of light missed in the Petrosian aperture. We used the sersic fits by Blanton et al. (2003) to derive the total fluxes, and the ratio of the total flux to the Petrosian flux.

The rest-frame colors of the galaxies were derived using INTEREST (Taylor et al. 2008, in preparation), and are determined in a way consistent with the high redshift sample. INTEREST determines the rest-frame colors and fluxes from the observed magnitudes, using the algorithm defined by Rudnick et al. (2003). The star formation rates were taken from Brinchmann et al. (2004). Finally, the sizes (half-light radii) were taken from Blanton et al. (2003).

In the rest of the paper, we use the sample between redshifts of 0.05 and 0.07. The lower limit is used to avoid a strong bias against massive galaxies, resulting from the magnitude limit imposed on the sample with masses from Kauffmann et al. (2003a). The high redshift limit is used to avoid the worst selection effects against apparently small galaxies, and to avoid large uncertainties in the derived sizes. This sample contains 21722 galaxies. The sample includes both star-forming galaxies and galaxies without star formation.

3. Correlations between size, mass, color and star formation rates

3.1. Mass-color and Mass-size relations

We can now start to analyze the correlations between galaxy parameters out to . We start by showing the well known correlation between color and stellar mass (e.g., Kauffmann et al. 2003a, Borch et al. 2006). The top row in Fig. 3 shows the result for the galaxies in our sample. Here, and in following plots, we divide the galaxies in 4 bins: low redshift ( from SDSS), , , and , the latter all from the CDFS. Fig. 3 shows that there is a correlation between color and mass out to : low mass galaxies being generally blue, and high mass galaxies being red. At low redshift we see the well known red sequence extending to high masses. At high redshift we lack the accuracy to establish the red sequence, but a correlation between color and mass probably persists as massive blue galaxies are scarce. These results agree well with those of Kauffmann et al. (2003a), Borch et al. (2006) for galaxies at .

Figure 4.— The correlation between color and size, at fixed stellar mass, for our sample. The top row shows the relation for the SDSS sample, the row below for galaxies with , the 3rd row for galaxies with , the bottom row for galaxies with . The top rows show a strong correlation between color and size in the mass bins, with large galaxies being blue, and small galaxies being red. This demonstrates that the scatter in color and size at a given mass is driven by a third parameter. A correlation is still present at , especially in the mass bin . At high redshift () the sample is too small to establish whether or not a relation exists.
Figure 5.— Left panels: the correlation between color and surface density, at redshifts 0, 1, 2 and 3. At all redshifts, a correlation is found, with high surface density galaxies being red, and low surface density galaxies being blue. This indicates that surface density is one of the main driving parameters for galaxy evolution. The color is more strongly correlated with surface density, than with mass; and this holds out to the highest redshift. Right panels: the correlation between color and velocity dispersion, at the same redshifts. Again, a good correlation is found, with high velocity dispersion galaxies being red, and low velocity dispersion galaxies being blue. The velocity dispersions have been estimated from .

As the redshifts used for this figure are mostly photometric redshifts, the errors in the colors are significant at higher redshift. Hence the absence of a tight red sequence at can be caused entirely by the observational errors. We note that Kriek et al. (2008) found a red sequence at for a sample with spectroscopy. It is striking that at and above, there appears to be a lack of low mass, red galaxies. We have to realize, however that red, low mass galaxies are progressively missed from the samples, as they have low mass-to-light ratios, and drop out of the -band selected samples first. We have indicated in Fig. 3 the lines of 75% completeness. We determined this in the following way. We divided the galaxies in a redshift bin into separate color bins. In each color bin, the galaxy fluxes and masses were scaled downward so that the resulting signal-to-noise in the K band was at our selection limit. For each galaxy, the resulting mass is the limiting mass at which that particular galaxy could have been observed. Finally, we determined the mass limit at which 75 % of the galaxies were detectable, and show it in Fig. 3 with the dashed curve. It is clear that the absence of red, low mass galaxies at and is mostly due to the incompleteness of the sample.

Whereas a general relation exists between mass and color, it is noticeable that at stellar masses between and , the spread in color is very large, up to 0.8 magnitude in . This is not only the case at low redshift, but also at higher redshifts. As the color is a function of stellar age, it implies a large range in star formation histories of galaxies at a given mass. Other indicators (D4000, specific star formation rate 111The specific star formation rate is the star formation rate divided by the stellar mass. derived from emission lines) show the same, large spread at these intermediate masses in the local universe (e.g., Kauffman et al. 2003b, Brinchman et al. 2004). Clearly, mass is not the only factor determining the star formation history, and galaxies are not uniform in this mass interval.

Next we show the relation between mass and effective radius in the bottom row of Fig. 3. There is a well defined trend at low redshift, with high mass galaxies being larger than low mass galaxies. The trend is weaker at high redshift, but this might be caused by selection effects, as small galaxies are preferentially missed first (see Appendix A). Just as for the mass-color relation, we find a significant scatter at a given mass at all redshifts. This was found earlier by Shen et al. (2003) for the SDSS sample, and it persists to high redshift. Shen et al. (2003) found that the rms in log radius is 0.3-0.5 dex at a given mass, and the distribution is log-normal at low redshift.

3.2. Mass-size-color relation

If we take these two results together, we conclude that galaxies at masses between and have a large spread in effective radius, and a large range in colors, from to ! This raises the simple question whether a different, underlying parameter might cause this variation in color, and size. This might be expected if, for example, the galaxies consist of large, blue disks, and small, red bulges, and if the bulge-to-disk ratio varies, at a given mass.

To investigate whether an additional parameter may cause the scatter, we present the relation between color and effective radius in narrow mass bins. Figure 4 shows these relations in our 4 redshift bins. Interestingly, we find a tight relation between color and effective radius, showing that the combination of mass and size can predict the color very well. The relation persists to , with too few galaxies at to establish it at that redshift. The average scatter in color around a simple fit to color-size relation at a given mass is very small at 0.16 mag in the mass bins between and . This is a significant reduction of the scatter, which is 0.33 mag in color in the same mass range if we do not correct for the color-size relation. The scatter listed here, and in the remainder of the paper is measured with the normalized Median Absolute Deviation (nMAD). This is the median absolute deviation multiplied by 1.48 so that the nMAD of a gaussian is equal to its dispersion.

This result suggests that indeed a single underlying parameter might drive the variations between galaxies. Hence we test whether the relation between color, mass, and size can be written in a simple form. We assume a relation of the form

and we derive the values of , , and which minimize the scatter. The selection effects discussed earlier make it hard to do this test unambiguously at high redshift. At low redshift ( and ), we find that the scatter is minimized for =1.05 and respectively, with a value of giving a similar, but slightly higher scatter. At , selection effects are already quite important, but we still find a similar value of . At , we find that produces the lowest scatter, likely caused by the selection effects against low mass, red galaxies, making it impossible to establish the mass-color-size relation at that redshift.

We note that the coefficient has no special physical meaning, as it is dependent on the color used on the left side of the equation.

However, the coefficient has a special meaning, as it can indicate the physical parameter underlying the relation. Our main result is that is close to 1-1.5 for most redshifts ( to ). This suggests that the relevant parameter is , related to velocity dispersion, or , related stellar surface density. In the following, we use the term “inferred velocity dispersion” for the quantity derived from , to make clear that it is not the true stellar velocity dispersion for the galaxies. The two agree well at low redshift, see, e.g., Drory, Bender, & Hopp (2004), but at high redshift this remains to be demonstrated, especially for gas rich galaxies.

Figure 6.— Left panels: the correlation between specific star formation rate and surface density, at redshifts 0, 1, 2, and 3. The SDSS specific star formation rate is based on an analysis of emission lines, whereas the high redshift points are based on the 24 m emission. The arrows denote upper limits. As can be seen, the highest surface density galaxies have low specific star formation rates out to redshift of 2.5, showing that specific star formation rate is causing to a large degree the relation between color and surface density. The limiting specific star formation rates for the redshift of 3 galaxies are too high to allow the detection of a correlation. Right panels: The correlation between velocity dispersion and specific star formation rate. Again, a good correlation persists to , and above that the specific star formation rates are not well enough determined.

This correlations obtained here are similar to the results of Kauffmann et al. (2003b, 2006) for the SDSS sample, who noted that the specific star formation rate correlates better with stellar surface density than with mass. The scatter in color is low, for both the color-inferred velocity dispersion and color-surface density relations: 0.21 and 0.22 mag respectively, for our bins at .

We emphasize that the single correlation with surface density or inferred velocity dispersion is unlikely to be complete: as can be seen in Fig. 4, at , at each mass interval, a “narrow red sequence” exists over a range in radii. The relation above does not represent this sequence properly, and the relation has less predictive power than, for example, the Fundamental Plane relation for early-type galaxies alone.

Given this result, we now show the relation between color and surface density in the left panels of Fig. 5. It is striking that a well defined relation exists at all redshifts, with red galaxies having high surface density, and blue galaxies having low surface density. The relation exists at all redshifts, but we have to note that at the number of points is low, and the relation is weak. More, and deeper data at high resolution would be needed to establish the relation better at that redshift.

The relation between color and inferred velocity dispersion is shown in the right panels of Fig. 5. We calculated the inferred velocity dispersion from , where the constant has been chosen so that the inferred velocity dispersions of the SDSS galaxies match the measured dispersions well. Again, we find a well defined relation between the inferred velocity dispersion and the color, with somewhat smaller scatter in color than for the relation between surface density and color.

Figure 7.— The evolution of the specific star formation rate in mass bins. In all mass bins, the specific star formation rates increase with redshift, and the spread remains large. The dashed line indicates the specific star formation rate equal to .

3.3. Specific star formation rates as a function of surface density

The color variations found above are interesting by themselves, but the interpretation is complex, as color is a function of both star formation history and dust. A more direct diagnostic of the star formation history of the galaxies is the specific star formation rate. The specific star formation rate is the star formation rate divided by the mass, and it is the inverse of the time it would take to form the galaxy if the star formation rate were constant. In general, the specific star formation rate and color of galaxies are well correlated, and the correlation between surface density and color suggests therefore that a similar correlation may exist between surface density and specific star formation rate.

Figure 6 presents the specific star formation rates plotted against the surface density and inferred velocity dispersion of the galaxies. As can be seen, good correlations exist. The high surface density galaxies have generally lower specific star formation rates, and the low surface density galaxies have generally high specific star formation rates. We note, however, that the scatter is larger for the SDSS sample than when the color is plotted against surface density, and there is no clear linear relation. Either aperture correction effects play a role (Brinchmann et al. 2004), or a correlation between dust and specific star formation rates produces lower scatter in color than in specific star formation rate by itself. At higher redshift, this difference is not strong.

We conclude that the relation between color and surface density is directly related to the relation between specific star formation rate and surface density. This result extends the earlier analysis of Kauffmann et al. (2003b, 2006) to significantly higher redshift.

4. Evolution of specific star formation rate with redshift

The tight correlation between color and specific star formation rate with surface density and inferred velocity dispersion suggests that evolutionary studies should focus on using these parameters for studying the evolution with redshift - in contrast to using mass (e.g., Cowie et al. 1996, Brichmann & Ellis, 2000, Juneau et al. 2005) Below, we explore the evolution of specific star formation rate as a function of mass and of surface density.

4.1. Specific star formation rates in bins of mass

It is now well established that galaxies generally had higher star formation rates at higher redshift (e.g. Lilly et al. 1996, Cowie et al., 1996, Bell et al. 2005). The evolution of the star formation rate is generally found to be dependent on mass: the most massive galaxies stopped forming stars the earliest (also known as down-sizing, e.g., Cowie et al. 1996, Brichmann & Ellis, 2000, Juneau et al. 2005, Zheng et al. 2007)

Fig. 7 shows the evolution of the specific star formation rate in narrow mass bins. The specific star formation rate is the star formation rate divided by the stellar mass. It is the inverse of the time in which the galaxy would have formed, given the observed mass and star formation rate. We can see in Fig. 7 that the specific star formation rates increase with increasing redshift, consistent with previous results (e.g., Juneau et al. 2005, Zheng et al. 2007, and references therein) The increase can be seen in all mass bins.

The specific star formation rates increase by a large factor: whereas at the highest specific star formation rates range around yr, by the maximum specific star formation rates increase to yr, and by to even higher values ( yr). In this discussion, we have to keep in mind that the specific star formation rates suffer from systematic uncertainties. Nevertheless, it is striking how strongly the specific star formation rates increase with redshift, and we notice that the trends at extrapolate at to values consistent with the local SDSS values.

Figure 8.— The evolution of the specific star formation rate in bins of surface density. The specific star formation rates increase in all bins. In the lowest surface density bin, the specific star formation rates are always high, and very few quiescent galaxies exist. We define quiescent galaxies as those having specific star formation rates below . The dashed line indicates a specific star formation rate of . In the intermediate surface density bin with the specific star formation rates are very low at , increase rapidly to high values at , to have only high values at . The highest surface density bins always have the lowest specific star formation rates.

We also find a trend with mass: in the lowest mass bins, the majority of the galaxies have already very high specific star formation rates by , whereas at the highest masses, the galaxies with high star formation rates start to dominate at significantly higher redshift. These results are very similar to those obtained before by many other authors (e.g., Cowie et al. 1996, Brinchmann & Ellis 2000, Zheng et al. 2007 and references therein), and are generally characterized as “down-sizing”.

However, we notice that all mass bins contain galaxies with very low specific star formation rates to very high redshifts. We have to keep in mind, that the upper limits on the specific star formation rates at are quite high, especially for the lower mass galaxies. This is simply caused by the limited depth of the MIPS 24 m imaging: Even though the MIPS exposure is extremely deep, the limiting depth produces rather high limits on the specific star formation rate for low mass galaxies.

Additionally, we note that the highest specific star formation rates occur for the lowest mass galaxies. These do not have the highest star formation rates in an absolute sense: a specific star formation rate of yr at corresponds to a star formation rate of 200 yr, whereas a specific star formation rate of yr at corresponds to a star formation rate of 240 yr. Obviously, characterizing galaxies by their absolute star formation rate over-emphasizes massive galaxies, which are actually not as extreme as the lower mass galaxies.

4.2. Specific star formation rate in bins of surface density

The wide range in specific star formation rate in the mass bins is a striking feature in Fig. 7. In Fig. 8, we show the specific star formation rate in bins of surface density. It is striking that at low surface density, the specific star formation rates are always high, and at high surface density, the specific star formation rates are generally low; but at intermediate surface densities, ( - kpc) we see a transition from low specific star formation rates at low redshift. to high specific star formation rates at high redshift. At a surface density of kpc this transition takes place in a fairly narrow interval between and , and at the surface density interval of kpc, it takes place around .

By comparing Figs. 7 and 8, we conclude that the evolution is better defined by the surface density of the galaxies, than by their mass. At all masses and nearly all redshift, we find galaxies with very high specific star formation rates, and galaxies with very low specific star formation rates. When galaxies are sorted by surface density, we see that low surface density galaxies have high specific star formation rates at all redshifts, high surface density galaxies have low specific star formation rates at nearly all redshifts, and we find a clear transition zone for intermediate surface density galaxies.

4.3. The evolution of threshold surface density with redshift

Kauffmann et al. (2006) analyzed a sample of local SDSS galaxies, and introduced a threshold surface density, below which galaxies have nearly constant specific star formation rates, and above which the specific star formation rate declines rapidly.

Our results imply that similar threshold surface densities can be defined at higher redshifts, and that the evolution of the galaxies can be well described by an increase in the threshold surface density with increasing redshift.

Figure 9.— The median specific star formation rate as a function of stellar surface density, at 3 redshift intervals: local galaxies from SDSS, and galaxies at , and . The median specific star formation rate is fairly constant at low surface densities, to decrease rapidly at higher surface densities. We have defined the threshold density, where the specific star formation rate is a factor of 3 lower than the specific star formation rate of the plateau.
Figure 10.— a) The evolution of surface density threshold, above which the specific star formation rate is low. At higher redshifts, the threshold has been measured in bins of width . The threshold evolves fast, close to , as shown by the curve. b). The evolution of the specific star formation rate below the surface density threshold. The specific star formation rate evolves very rapidly like , as shown by the curve. The dotted line curve the specific star formation rate equal to .

We define the threshold surface densities in the following way: we determine the median specific star formation rates in surface density bins of width 0.5 dex, sampled at 0.1 dex. The results are shown for in Fig. 9. We find that the median specific star formation rates are fairly constant at low surface densities, and decline rapidly at higher surface densities. The threshold surface density is defined by requiring that the specific star formation rate is 3 times lower than the median at the low surface density end.

We determine the thresholds in redshift bins of width for our CDFS sample, and we sample them at redshift steps of 0.5. The results are shown in Fig. 10. The left panel shows the threshold surface density versus redshift. As can be seen, the threshold evolves quite strongly with redshift - proportional to .

This evolution by itself would obviously cause a strongly increasing star formation rate density with redshift, as more and more galaxies fall below the threshold. However, it is not the full story, as we can also see that the specific star formation rate below the surface density threshold increases rapidly with redshift (Fig. 10b). The evolution is fast at low redshift, proportional to .

5. Evolution of mass-size relation with redshift

We saw above that the specific star formation rate is strongly correlated with the surface density of the galaxies, and evolves strongly at a given surface density. This is not the only evolution taking place: the surface densities and sizes of galaxies at a fixed mass are also expected to evolve. This evolution has been studied before (e.g., Trujillo et al. 2006a, and references therein). We revisit this issue here as our sample size is significantly larger at high redshifts. To be consistent with existing literature, we study the mass-size relation. The relation between surface density and mass follows directly from using .

We note that the evolution of the mass-size relation is a valuable diagnostic of the evolution of galaxies: in simple models of disc formation, the sizes of discs are assumed to evolve at the same rate as the halo size (e.g., Mo, Mao, & White 1998). This has been used to predict the evolution of the mass-size relation for such galaxies (e.g., Mo et al. 1998). More complex models find generally weaker evolution (e.g., Somerville et al. 2008). The evolution of the mass-size relation for spheroidal galaxies is expected to evolve even faster (e.g., Kochfar & Silk 2006, Hopkins et al. 2007,2008)

Figure 11.— The evolution of half-light radius with redshift, in narrow mass bins. The sizes steadily decreases with redshift, both for high mass and low mass galaxies. The curves show fits using the average coefficient . The dashed curves show the fit using , which is predicted from simple formation models. The dotted curves show the fits of to the individual bins. The index varies from 0.140.07 to 0.81 0.1, from the low mass to the high mass bin. The average index is 0.59 0.10. The triangles are quiescent galaxies with low specific star formation rates (as defined in the text). They evolve faster, like . Overall, the sizes evolve significantly with redshift, the strongest for the high mass galaxies, and the quiescent galaxies.

The first studies of the evolution of the size-mass relation to were done by Trujillo et al. (2006), based on ISAAC imaging data on the smaller but deeper fields of the FIRES survey (Franx et al. 2003, Labbe et al. 2003a, Forster Schreiber et al. 2006). The sample studied here is based on imaging with the same instrument, but on the much larger CDF-South field. Fig. 11 shows the evolution of the radii of galaxies with redshift in narrow mass bins. As we can see, significant evolution is present. On average, the sample produces an evolution of , or . This is an average of the evolution seen in the mass bins with , where the sample is more than 70 % complete at the bin. The evolution may still be an underestimate, as the lowest mass bins are still deficient in small, red galaxies at high redshift ().

As can be seen in Fig. 11, the evolution is fastest for galaxies with masses above 6.3 . These give . Again this difference may be partly or fully due to incompleteness at the lower masses: the smallest galaxies are typically red, and they drop out of the selected samples earlier than the larger, blue galaxies. Deeper data will be required to verify this trend. Furthermore, we note that the smallest galaxies are small with regards to the PSF, and hence the size measurements around 1kpc and below should be considered to be uncertain (see also section 2). Hence higher resolution imaging is needed to verify those sizes. We note that the SDSS size measurements agree well with the trend found for the full dataset. There is no indication that the SDSS measurements are biased compared to the low redshift CDFS measurements.

The evolution found here agrees well with that found by Trujillo et al. (2006a), who found an average evolution for the full sample of a factor of 0.48 0.05 out to z=2.5, compared to the evolution of a factor of 0.480.08 found here (derived from our average redshift evolution).

Given the strong evolution, one may wonder to what extent these results could be affected by serious systematic errors in establishing the stellar masses of galaxies. We note that the evolution is so strong, that it cannot be caused by an error of a factor of 3 in the stellar mass at . The typical size of galaxies with mass of 6.3-10 in our sample is 1.6 kpc at . If we had overestimated the mass by a factor of 3, we should compare their size with galaxies at z=0 of a mass of 2.1-3.3 . As we can see in Fig. 11, galaxies with that mass have a size of 2.5 kpc at , implying an evolution with a factor of 0.64. Hence it seems rather unlikely that systematic errors in the mass determination cause the observed evolution. Incompleteness could be another factor, as we may miss the larger galaxies more easily by surface brightness effects. This is straightforward to test by simulating the effects of such selection on the observed low redshift galaxies, by putting them at higher and higher redshift, and simulation the selection effects. In appendix A we show that this effect is almost negligible, which is due to the fact that the PSF is significant compared to the sizes of the galaxies. Hence we miss rather few large galaxies, and we cannot easily explain the absence of massive, small red galaxies at low redshift. The most significant bias is to miss low mass, red galaxies at high redshift, because they are fainter in , and this biases the sample towards larger galaxies at the same mass. Hence the true evolution of the mass-size relation may even be faster than the evolution observed here.

The scatter in the mass-size relation at is approximately 0.3 dex in log radius. This is comparable to the scatter in the relation for the SDSS sample for galaxies with masses lower than (Shen et al. 2003). The scatter is comparable to the evolution found here, showing that it is possible that some individual galaxies have no evolution in either parameter. The largest galaxies at , for example, are consistent with galaxies at . The small, massive galaxies at are not consistent with galaxies, a point made also by many other authors (e.g., Cimatti et al. 2008, van Dokkum et al. 2008, and references therein). Therefore, the population as a whole must evolve.

6. Implications for galaxy evolution

6.1. General evolution and transformation onto the red sequence

We have found that at all redshifts, color and specific star formation rates correlate well with surface density and inferred velocity dispersion (which is ). The lower the surface density and inferred velocity dispersion, the bluer the colors, and the higher the specific star formation rates. The implication of this result is simply that the high surface density galaxies are older, have low star formation, and must have formed their stars earlier than the low surface density galaxies. This had been found earlier by Kauffman et al. (2003b, 2006) at low redshift, and these new result show that the relations persist to at least .

The evolution in the relations is as expected: at higher redshifts, high specific star formation rates are found at higher surface densities/ inferred velocity dispersions than at lower redshift. This is consistent with a simple picture in which the high surface density galaxies which are dead at low redshift were forming stars at some higher redshift. The apparently smooth increase of the characteristic surface density at which high star formation occurs suggests that the star formation history is a simple function of the surface density of the galaxy. This is quite striking as the star formation rate of a galaxy is driven by complex processes like mergers, gas accretion, and other processes which will vary with time. Hence these processes affect both the star formation rates and the surface densities to result in a fairly simple relation between surface density and star formation history.

Interestingly, the results provide independent evidence that many galaxies which are on the red sequence at were not ”red and dead” at : their typical surface densities are around kpc at , and approximately half of the galaxies with such surface densities at are forming stars (see Fig. 8). We note that this evidence is completely independent of the evidence based on the mass density evolution of the red sequence galaxies, which has shown that the mass density has increased significantly between and (e.g., Bell et al. 2004, Faber et al. 2007). The evidence presented here shows that the galaxies with the structure of red-sequence galaxies at are generally forming stars at . Of course, as galaxy evolution is a complex process which may include both merging and star formation, we cannot uniquely identify what the progenitors are of red galaxies once we allow for structural evolution. However, we can confidently exclude the possibility that all red sequence galaxies at have passively evolved from red galaxies: many of the galaxies with the corresponding structural parameters at are not dead. The evolution of the galaxies onto the ”dead” zone of surface density kpc likely involves both star formation and merging: the first from the direct observational evidence presented here that the galaxies with surface density kpc were forming stars at higher redshift, the latter from theoretical predictions, and determinations of the merger rate (e.g., Bell et al. 2006, van Dokkum et al. 2005), which are still quite uncertain, however.

The steady increase with redshift of the threshold density (above which the specific star formation rates drop) suggests that similar processes were at play between and , but then at higher surface densities/inferred velocity dispersions.

6.2. Properties of quiescent galaxies out to

One of the striking features is that quiescent galaxies exist at all redshifts. At all redshifts, they are the galaxies with the highest surface densities, and the highest inferred velocity dispersions. However, their sizes at and above are much smaller than their sizes at , and their surface densities much higher. This had been noticed before by many authors (e.g., Trujillo et al. 2003, 2006a,b, Daddi et al. 2005, Toft et al. 2007, Cimatti et al. 2008, van Dokkum et al. 2008). The results here indicate that their small sizes are related to the general evolution of the mass-size relation. When we select only quiescent galaxies by requiring that the specific star formation rate is smaller than , we find a size evolution of , significantly faster than the evolution for the full galaxy sample. This is measured for masses larger than , where good limits on the specific star formation rates are achieved. The error is the formal error from comparisons between the mass bins. In reality, the error is larger as many of the galaxies have very uncertain sizes below 1 kpc. However, we note that van Dokkum et al. (2008) used high resolution NIC2 imaging and found a similar evolution for massive quiescent galaxies: a factor of 5 between and . Hence the evolution of the quiescent galaxies is faster than the full galaxy sample, by a factor of two or more. Given our selection effects at low masses, and our limited resolution, deeper studies with better resolution are needed to measure the evolution more accurately.

The fact that quiescent galaxies exist out to the highest redshift is very significant, as it shows that the mechanism which shuts off star formation in galaxies was already present at high redshift ( and above). In many ways, it does not come as a surprise that the quiescent galaxies are small: once star formation shuts off, they apparently do not continue to accrete cold gas in their outer parts which would make them grow in size. Hence their size remains fixed, whereas the star forming galaxies grow in size. However, it is surprising that the quiescent galaxies are much smaller than any quiescent galaxies in the nearby universe with reasonable number densities. Hence, in one way they must have ”disappeared”, or surveys of the nearby universe are incomplete.

Incompleteness in the SDSS survey certainly does play a role: due to the star-galaxy separation criteria used, several known compact galaxies from the 7 Samurai survey (Faber et al, 1989) are missing in the SDSS survey. These are galaxies like NGC 4342 and NGC 5845, which have velocity dispersions above 200 km sec, and sizes below 0.5 kpc. Such galaxies are extremely rare in the SDSS (less than 1 in 10). The volume in which the compact, massive galaxies can be found is rather small in the SDSS, as they are excluded when at small distance because they are too bright in the fiber aperture, and they are excluded at large distance because they are too small (Strauss et al. 2002). Despite these potential problems, it is clear that evolution also takes place between and , where such incompleteness should play no role. The simplest explanation is that the small galaxies grow by merging with larger galaxies, whether star forming or not. As all other galaxies are larger, any merger is expected to increase the size. Furthermore, additional accretion of gas and star formation in the outer parts would also scale up these galaxies.

6.3. A simple accretion model to explain why blue galaxies are large

The fact that blue galaxies are larger than red galaxies of the same mass may find a very simple explanation. In the nearby universe, we know that star forming galaxies with masses between and are generally multi-component: a red bulge, with a blue disk around it. As a matter of fact, the Hubble sequence is correlated with bulge-to-disk ratio, and is correlated with color, and the sequence satisfies the same general trend as the trend observed here. Hence a simple explanation for the trend between specific star formation rate and surface density is that the galaxies with significant star formation have accreted gas, which forms stars preferentially in the outer parts, and this why they are both larger, and bluer. This gas accretion is assumed to be fairly regular and possibly caused by minor mergers which do not stir up the galaxy. For some reason, gas accretion and star formation are shut off for high density galaxies, either due to AGN (e.g., Croton et al. 2006, Bower et al. 2006), or heating of the gas due to shocks occurring naturally in massive, forming galaxies (Dekel & Birnboim 2006, Birnboim et al. 2007, Naab et al. 2007).

The persistence of the relation at higher redshift, but then shifted to higher surface density, suggests that similar processes occur out to and beyond. The fact that the threshold surface density is higher may be a simple result from the fact that the halos of galaxies are smaller at high redshift, and hence the galaxies are denser. Galaxy size is correlated with the star formation history, as the galaxies which have accreted material recently are expected to be larger, and have higher specific star formation rates than those who have not accreted. In short, a very simple picture is one in which pre-existing star forming galaxies accrete material preferentially in the outer parts, where stars are formed. For some reason, high surface density galaxies do not accrete such material. In this picture, star forming galaxies with at are very analogous to : old centers, younger outer parts. This specific prediction of gradients can be tested with higher resolution imaging data, as would be provided, for example with WF3 on HST. The imaging study by Labbé et al (2003b) showed evidence for substructure and gradients for a sample of large galaxies to . As the angular momentum of the accreted material is thought to set the scale of the galaxies, it is also natural to expect that the blue galaxies may have disks of star forming gas, analogous to the low redshift galaxies. There is at least some evidence for this from kinematic studies of high redshift galaxies (e.g., Föster Schreiber et al. 2006, Genzel et al. 2006, Wright et al. 2007, Law et al. 2007) As we see below, and as noted in these kinematical studies, the nature of the star forming galaxies at may be significantly different from what we call ”disk galaxies” in the nearby universe.

6.4. The enigmatic nature of strongly star forming galaxies at .

As we have seen, at the strongly star forming galaxies have very high specific star formation rates, well above . For example, at the typical specific star formation rate is 6 yr, well above yr. At the typical specific star formation rate is 2 yr, compared to yr, off by almost a factor of 10. This phenomenon has been noted by many authors (e.g., Daddi et al. 2007, Davé, 2007). Taken at face value, it suggests that these galaxies may have formed the bulk of their stars in a very short time. We note that some authors have argued that the star formation rates are systematically over-estimated by a factor of 2-3, either because too many stars are produced (Wilkins et al. 2008), or because the specific star formation rates are much higher than theoretical models (Davé 2007).

If we accept the high star formation rates, we have to conclude that the time scale of star formation is getting very close to the orbital times of these galaxies. We estimate the orbital time simply from the inferred velocity dispersion and the size. We emphasize that they are uncertain at high redshift, as the velocity dispersions or circular velocities are not directly measured. We show the ratio of the star formation time over the orbital time in Fig. 12. As can be seen, at redshifts below 1, the star formation time is times the orbital time, but at the ratio gets close to 3. This is so short, that it is unlikely that the gas is settled in a cold disk. Simulations would obviously be needed to understand better the exact dynamical state.

Figure 12.— Left panel: The ratio of star formation time scale to dynamical time scale. Whereas at low redshift, the dynamical timescale is generally more than 30 times longer than the star formation time scale , it decreases to values of 3-10 at a redshift of 1.5 and above for the strongly star forming galaxies. For these galaxies, the gas has barely time to settle in a disk. Right panel: The evolution of the ratio of gas mass to stellar mass with redshift. The gas mass has been estimated from the star formation rate, and the kennicutt star formation law. Galaxies with significant star formation are indicated with a large symbol (specific star formation 1/3 threshold specific star formation rate). Whereas at low redshift the typical gas fraction is around 10%, it increases steadily with redshift, to reach values of 0.3-1 at and above. The most strongly star forming galaxies at and above have gas masses comparable to their stellar masses.

Furthermore, we can roughly estimate the gas fractions of these galaxies, by assuming that the gas has the same lengthscale as the (blue) light, and by assuming the Kennicutt (1998) relation between gas surface density, and star formation rate per area. Even though the estimated gas masses must be considered to be very uncertain (as the sizes are uncertain, and the Kennicutt relation has not been established at ), the results can be used for at least consistency checks with models which assume the Kennicutt relation as a general recipe for star formation. We note that Bouché et al. (2007) confirmed the Kennicutt relation for sub-millimeter galaxies at . The resulting gas mass to stellar mass ratios for the star forming galaxies are shown in Fig. 12. As can be seen, the gas to star ratios for the star forming galaxies increase from 0.2 at to 1 at . Even if the star formation rates have been overestimated by a factor of 2 at , this result will not change much. As a result, we have to conclude that the general relations between star formation and gas density at imply that the typical star forming galaxies at and above have very significant gas fractions. A similar result had been found before by Erb et al. (2006), based on H spectroscopy.

Another issue related to these strongly star forming galaxies is how long the star formation episodes last. If the star formation rate would be constant, the specific star formation rate would decline very rapidly (on a timescale of 1/specific star formation rate). As the epoch in which the specific star formation rate is high would last only briefly, the universe would be expected to be dominated by galaxies with specific star formation rates comparable to the inverse of the Hubble time. This is obviously not what is observed. Either the specific star formation rates are over-estimated, or the galaxies undergo brief bursts punctuated by periods of low star formation, or the galaxies form with constant specific star formation rate. The first explanation requires an over-estimate by a factor of 3 - which is entirely possible, given the uncertainties with the conversion to bolometric luminosities, and uncertainties in the IMF. The second mechanism would imply that we would have to see a substantial number of ”dead” galaxies, with the same masses and sizes of the ”live” galaxies. However, one of the basic results of this paper is that we see a good correlation between the structure and specific star formation rate of the galaxies, and this would be washed out if galaxies were to undergo frequent bursts. The only way to explain this would be to assume that the sizes and possibly masses of the galaxies have been estimated wrongly for the star forming galaxies. If high redshift galaxies are multi-component, with high gas fractions, and large dust masses, it might be possible that their derived quantities from just optical light might be seriously wrong. Even in the nearby universe this plays a role: color gradients in spirals are significant, and this does imply that the optical half-light radius can deviate significantly from the half-mass radius. The derived gas surface densities for the galaxies imply typical absorption in the V-band of , suggesting that a significant amount of starlight is completely obscured. This result appears entirely reasonable, but is somewhat inconsistent with the result that SED fits to the restframe UV-optical-near-IR give good estimates of the total star formation rate, suggesting that the galaxies are semi-transparent (e.g., Daddi et al. 2007). Obviously, this should investigated further.

Alternatively, the high specific star formation rates are correct, and many galaxies form very rapidly with nearly constant, very high specific star formation rate. This implies rapid expontial growth on a short timescale (e.g., Daddi et al. 2007): in this case, they would increase by a factor of 10 in mass in a time of 2.3 * 1/SSFR, corresponding to 1 Gyr at . For the galaxies, they would be less massive by a factor of 10 at . This does not appear completely unreasonable, but we have to note that current models of galaxy formation do not predict this type of behaviour (e.g., Davé 2007).

6.5. Comparing the evolution of the mass-size relation with models

As we have seen, the mass-size relation of galaxies evolves smoothly from to , like at a given mass. There is evidence that the relation evolves faster for high masses, and quiescent galaxies also tend to evolve faster. The general evolution is fairly close to what is predicted for simple disk galaxy evolution, where (Mo et al. 1998). In the redshift interval from 0 to 3.5, we find approximately , fairly close to the observed size evolution. We note that in a detailed analysis, Somerville et al. (2008) found that the expect rate of evolution is somewhat slower than , more consistent with the result obtained here. The fact that the general trend holds can be taken as evidence that the simple scaling of dark matter halos also determines the scaling of the galaxies.

Hence the question remains open whether the very compact quiescent galaxies at high redshift have formed in intrinsically very different ways compared to their cousins at low redshift. Kochfar & Silk (2006) invoked strong dissipation at high redshift to explain the very small sizes of galaxies at high redshift. The results obtained here suggests that star forming galaxies at redshifts as low as also had very high gas fractions, potentially undercutting this explanation for the evolution.

If we assume that our trends persist to , we would be let to conclude that the compact galaxies at formed their stars around : Their average specific star formation rate is approximately yrat . If we assume that the specific star formation rates evolve like (as for the specific star formation rate at the threshold surface density), we find that the specific star formation rate would be when the universe was about 1.6 times smaller, i.e., at . Obviously, this is an uncertain extrapolation of the relations found at .

The stellar ages of the quiescent galaxies at have been estimated around 1Gyr (Kriek et al. 2008), this would imply a similar formation redshift of 4.

6.6. Relation to sub-mm galaxies

One of the striking results in this work is that the star forming galaxies at are large compared to the quiescent galaxies, in agreement with earlier studies (e.g., Zirm et al. 2007 and Toft et al. 2007). Furthermore, the star forming galaxies have very high star formation rates and short specific star formation times, and would be called starbursts if they occured in the nearby universe. One of the interesting questions is whether they are related to the ULIRGS at low redshift, and the sub-mm galaxies at high redshift. We note that ULIRGS in the nearby universe have very concentrated star formation in very small volumes, with typical sizes significantly smaller than 1 kpc (Tacconi et al. 2006, and references therein). Hence the high redshift star forming galaxies have comparable IR luminosities (typically ), but are inferred to have a very different structure than the local sub-mm galaxies. Unfortunately, catalogues of sub-mm sources in the CDF-South are not yet available. Size measurements of sub-mm galaxies at high redshift are rare. Tacconi et al. (2006, 2008) found typical sizes of the gas smaller than 2 kpc at , which is small for the star forming galaxies found here at the same redshift. A detailed comparison of the sizes measured for the stars, gas, and star formation region would be valuable. As the space density of the sub-mm galaxies at high redshift is low, it may be that they simply lie in the tail of the size distribution for star forming galaxies. Larger samples are needed to test this.

6.7. Caveats and further work

The results obtained here illustrate the power of structural studies of galaxies. It is appropriate to discuss the potential errors that may have occured in the derivation of the masses, sizes, and star formation rates.

1) masses: it is well known that the stellar masses derived from sed fitting are rather uncertain, even though they are the most stable outcome of such fits. Problems include the uncertainties in stellar population models themselves (without considering differences in star formation history, for example Maraston 2005 versus Bruzual & Charlot 2003), uncertain star formation histories, uncertain geometries of the dust, and potential correlations between absorption and stellar age, and the possibility that some components are entirely hidden throughout the rest-frame near-ir due to very high extinction. In addition, the IMF of the stars may vary (e.g., van Dokkum 2008, Davé 2008, Wilkins& Hopkins et al. 2008). Only more detailed observations can provide answers. High resolution observations with HST can determine whether galaxies have strong color gradients, complicating the SED fitting. Direct spectroscopy is urgently needed to establish the stellar velocity dispersions of the compact, quiescent galaxies, and the star forming galaxies (which will be even harder). High spatial resolution observations of molecular lines can provide detailed information on the mass distribution in the inner parts, and can maybe provide insight into hidden populations, whether old or young. Rest-frame optical spectroscopy can potentially probe the mass distributions in the outer parts.

2) Sizes: the sizes used here are sizes measured in the rest-frame optical. For the small galaxies, the sizes are just barely resolved. Obviously, higher resolution imaging is required for the rest-frame optical sizes at , and, in addition, the determination of color gradients is important to see whether the measured sizes could be affected significantly by color gradients. The color gradients in the nearby universe vary from 0.07 in B-R per dex radius for ellipticals to 0.2 in g-r per dex radius for late-type spirals, (e.g., Franx & Illingworth 1990, Peletier et al. 1990, de Jong 1996). If we use simple relations between mass-to-light ratio and color, we find that the half mass radii are smaller by about a factor of 0.87 and 0.56, respectively. This is insufficient to wash out the effects which we have seen, but larger effects could be present at high redshift. Simulations can also play a role here: Hopkins et al. (2008) have shown that the optical sizes of the merger remnants can be smaller by a factor of 2 than the true half-mass size. This is caused by the concentrated young population of stars formed at the end of the merger. Joung et al. (2008) find that the apparent sizes of simulated star forming galaxies at are a factor of 3 higher due to extinction by dust. Obviously, the interpretation of the apparent sizes may not be straightforward, and other diagnostics may be needed (e.g. the spatial distribution of the star forming regions). The models by Guo & White (2008) predict the correct qualitative rise in specific star formation rate, but unfortunately do not predict sizes.

3) Star formation rates: It is well known that the derivation of reliable star formation rates is still very hard. The extrapolation of the measured 24 m flux to a total bolometric IR flux is uncertain (although the results of Papovich et al. 2007 imply that a simple linear relation may suffice). Studies with Herschel may improve upon this situation. Furthermore, even if the bolometric flux is well determined, the star formation rate is not, as the IMF may vary with redshift. For example, the IMF may vary at very high masses (around the masses of O stars, 50 ,), at the full mass range between 1 and 50 , and at the low mass range (as suggested by Davé 2008 and van Dokkum 2008). There is direct evidence that the star formation rates estimated traditionally are too high: the mass in stars seems to be over-produced (Wilkins et al 2008, Davé 2008). van Dokkum (2008) emphasized that a change in the IMF results in both changes in the derived star formation rate, and the derived masses, and the changes depend on the exact form of the IMF evolution. Omitting or adding low mass stars ( ) essentially does not do very much, as all masses and star formation rates are changed by the same factor, and the specific star formation rates remain the same (and too high). Changes in the characteristic mass of Chabrier type IMFs will change the star formation rates more than the masses, which is the type of change that is desired; and changing the slope above will have the largest effect on the derived star formation rates, and smaller effects on the masses. Obviously, the exact investigation of these effects is beyond the scope of this paper. The calibration of the stellar masses of quiescent galaxies at high redshift can play an important role in these investigations, in addition to the high resolution dynamical studies of gas and stars to decompose galaxies. The comparison of many different star formation indicators will also provide further insight in this issue.

4) cosmic variance: the current field is fairly small, and does not have large numbers of high mass galaxies. It will obviously be important to study the same relations on larger fields, thereby making a fairer sample of the universe. Furthermore, such studies will allow the determination of the distribution of surface densities, and inferred velocity dispersions as a function of redshift.

7. Implications for high redshift galaxy studies: biases and consequences

Figure 13.— Simulated selection effects for our sample. The left panel shows the selection on observed flux. The flux has been estimated from the star formation rate, and extinction corrected, assuming the extinction from the SED fit. Galaxies with H flux ergs sec and are indicated with full squares, the galaxies with H flux ergs sec and are indicated with open triangles, and the other galaxies are indicated with open squares. The V band limit is used as most often Near-IR spectroscopy is done on galaxies with optical redshifts measured first. The right panel shows the selection on UV-slope and UV continuum flux (right), simular to those used for Ly-break and BM-BX galaxies (e.g., Steidel et al. 2006). Filled squares are galaxies with and . In both panels, we see that the selected galaxies are prefentially large compared to the sample as a whole. It is obviously difficult to study galaxies at the full range of sizes, given the mass. We note that our full sample is also likely to be incomplete at low masses and small sizes.

The correlations which we found above have important consequences for observational studies. First of all, many studies have flux limits or color selection criteria which may make them to pick up specific subsamples in the space of mass, size, and specific star formation rate. For example, studies have now begun of the H emission line kinematics and spatial distribution (e.g., Erb et al. 2003, 2006, Genzel et al. 2006, Förster Schreiber et al. 2006, Kriek et al. 2006, Law et al. 2007, Wright et al. 2007). Many of these studies impose a flux limit on the H emission line flux before the detailed observations are performed. The H fluxes have not been measured for this sample, but we can estimate them using the star formation rate, and the estimated extinction . We used the conversion by Kennicutt (1998) to transfer the star formation rate to unobscured H flux. We impose a flux limit of ergs sec, and we show the selected galaxies in the left hand panel of Fig. 13. As is clear, the galaxies with strong H are preferentially large, and have high specific star formation rates for galaxies at that mass, as might be expected. Their star formation timescales are generally a few times the dynamical time, and hence their dynamical state is not typical for the median galaxy at that redshift.

Another, often used selection technique is the Lyman Break selection technique, or BM-BX selection employed by Steidel et al. (1999, 2004), and many other authors. We show in the right hand panel those galaxies which satisfy the criterion , and , which are the criteria often used for these studies. Again, it is clear that the selection picks up preferentially large galaxies from the samples. Near-IR spectroscopic follow-up of these samples introduces an additional bias (through the flux limit on H).

Apart from the selection effects discussed above, the results imply that the interpretation of the SEDs of the galaxies may be over-simplified in the current models. As the relation between mass and radius evolves with time, and as galaxy masses increase with time, it is likely that galaxies have old subcomponents which are smaller than the new additions. If so, the subcomponents will have different extinction as well, and the modeling of the galaxies as simple populations with constant extinction may give biased results. Without more detailed information, it is hard to quantify these effects, and first of all, direct dynamical mass estimates are needed. However, simulations can also be used to estimate the sizes of such effects. We notice that Wuyts (2007a) found that the stellar mass can be underestimated by a factor of 2 in the phase of strong star formation in a gas rich merger. These effects are in addition to the known uncertainties in stellar populations models.

8. Summary and conclusions

We have shown that massive galaxies from out to have a strong correlation between size and color and size and specific star formation rate, at a given mass. Galaxies with high specific star formation rates are large, galaxies with low specific star formation rates are small. At increasing redshifts, the overall specific star formation rates go up.

In general, the specific star formation rates correlate better with surface density, and inferred velocity dispersion, than with mass. This suggest that surface density, or inferred velocity dispersion, is the driving parameter. We find that, as expected, specific star formation rates at a given surface density increase with redshift. We identified a threshold surface density at each redshift interval: below the threshold the specific star formation rates are high with little variation, above the threshold density galaxies have low specific star formation rates. As expected, the threshold increases with redshift: high specific star formation occurs at higher and higher surface density with increasing redshift. As a result, we find that many galaxies which are on the red sequence at are star forming at .

Furthermore, the sizes of galaxies at a given mass decrease with redshift steadily from to . This overall evolution shows that galaxies grow inside out. As this growth also occurs for ’red and dead galaxies’, it suggests that these galaxies keep evolving - and are never just ’passively’ evolving. In short, all galaxies show ’up-sizing’, and the most massive galaxies show the strongest evidence for it. There is no evidence in this sample that very massive galaxies do not evolve between and (Scarlata et al. 2007), but obviously larger area studies can address this issue better. The very small, red galaxies at have to evolve into larger galaxies by through merging, accretion, and star formation. Their small sizes may simply be due to the fact that they are on the tail of the distribution at , where all sizes are smaller. Two processes are likely responsible for their small sizes: the halos were smaller and denser, and additionally their gas fractions were higher, and dissipation during merging was stronger, as suggested by Khochfar & Silk (2006). The observed evolution is also consistent with merger simulations (e.g., Hopkins et al. 2008).

The similarity in structural relations for galaxies from to , and the fact that galaxies grow inside out suggests that one form or another of the Hubble sequence persists to , and possibly beyond. The older stars likely dominate in the centers, and younger stars are likely distributed over a larger radius - similar to bulges and disks in spiral galaxies. Although we don’t have the resolution to establish this directly for the galaxies, the evolution of the mass-size relation with redshift also strongly supports such inside-out growth of galaxies. The exact dynamical state of high redshift galaxies still needs to be determined.

The multicomponent nature of galaxies suggest that modeling these galaxies is much harder, as the different populations have different spatial distributions, and therefore different extinction, star formation history, etc. Analysis of simulations suggests that this can lead to under-estimates of the masses through SED fits, and over-estimates of the sizes (e.g., Wuyts et al. 2007a).

The galaxies with very high star formation rates at likely have very high ratios of gas mass to stellar mass (approximately 1 or above). Furthermore, they are large (), and therefore different from ULIRGs in the nearby universe, which have typical sizes of the star formation regions of . However, these high redshift galaxies may not be simple cold disks: their star formation time scale is only a few dynamical times, and therefore the gas had barely time to settle in discs, if accretion is causing the very high star formation. Alternatively, the star formation may have been overestimated, and this would allow the gas more time to settle.

Obviously, these results call for many follow-up observations. First and foremost, the mass estimates must be improved, hopefully through dynamical mass estimators, either through near-ir spectroscopy, or spectroscopy with ALMA. Second, higher resolution Near-IR imaging can determine the structure of nature of the high redshift galaxies better. High resolution imaging with ALMA will be able to establish the distribution of the star formation across the galaxies, and spectroscopy will allow the determination of the gas content and gas distribution.

Third, it will be important to extend this work to even higher redshifts, where samples selected in the rest-frame optical are very rare, and structural analyses absent. Fourth, the environment of the high redshift galaxies needs to be determined. This will allow a study of the the relation between structure, star formation history and environment at high redshift. At low redshifts, environment plays an important role in setting the star formation rate (e.g., Kauffmann et al. 2004), and a full understanding requires an extension to high redshift. Finally, a determination of evolution of the surface density function can help to confirm the simple picture in which the centers of massive galaxies formed first.

The results show that stellar surface density, or inferred velocity dispersion, is one of the main driving parameters of galaxy evolution. Studies of the correlation of other galaxy properties with these parameters would be extremely valuable: metallicities, observed circular velocities, but also correlation length, and environment. Such studies require surveys of much larger areas, and extensive spectroscopy. The velocity dispersions used here are estimated from , and it remains to be verified whether the two are well correlated at all redshifts, for all galaxies. If not, it is crucial to determine which is the driving parameter.

The comments of the referee helped to improve the paper. We thank the Leids Kerkhoven Bosscha foundation for providing travel support. We thank the Lorentz Center for hosting workshops during which this paper was written. We thank Joop Schaye, Phil Hopkins, Lars Hernquist, Rachel Somerville for discussions. Support from NASA grant HST-GO-10808.01-A is gratefully acknowledged. S. Wuyts acknowledges support from the W. M. Keck Foundation.

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Appendix A Appendix A

Here we analyze in more detail whether some of our evolutionary effects may be caused by selection effects. Trujillo et al. (2006a) presented the impact of surface brightness selection effects on the distribution of effective radii for a comparable set of galaxies in the field of MS1054-03, and found no strong effects. As our own dataset is very comparable in depth and uses the same instrument, we don’t expect strong selection effects here either.

We have to note, however, that selection effects in the mass-size and surface density specific star formation rate planes may be more complex, as these quantities are derived indirectly from the observables. To determine whether our evolutionary effects may be driven by selection biases, we perform an analysis where we use low redshift samples as a reference sample, and transpose the galaxies to higher redshift, while keeping their intrinsic properties the same. This allows us to determine directly whether the evolution observed here is caused by simple selection effects.

Figure 14.— Simulation of the selection effects. Top row: galaxies at have been shifted to higher redshift by , while keeping their intrinsic properties the same. The open symbols show the galaxies which would still be selected using the selection criteria in this paper. It is clear that they are not biased in the mass radius relation or in the relation between specific star formation rate and surface density. Bottom row: the same, but now shifting galaxies at by . Again, the resulting relations are the same. If anything, the selected galaxies are slightly larger than the original full distribution, caused by the fact that small galaxies are red, and therefor dropped earlier from the sample. The median radius increases by 15%. The simulations shows that the evolutionary effects observed in this paper are not caused by selection effects.

a.1. Moving galaxies from to

We first take the galaxies in the redshift bin as our reference sample. We increase the redshift of each galaxies by 1 unit in redshift, while maintaining constant intrinsic absolute magnitudes, and apparent size. For each galaxy, we then re-determine whether it has sufficient signal-to-noise ratio and band flux to be included in our sample. The resulting distributions in the mass-size plane, and the surface density-specific star formation plane are shown in the top row of Fig. A14. The open squares indicate galaxies which are still detected at , the small symbols are galaxies which are lost in the process. It is clear that in this simulation the incompleteness is very strong below a mass of 3  . Even above that mass significant numbers of galaxies are missing. However, it is also obvious from the plot that there is no bias towards losing large galaxies. We quantified this by determining the median radii of all galaxies, and the galaxies still selected when shifted at . This two radii are the same within 2 %.

The upper right figure shows the distribution in the plane of specific star formation against surface density. Again, it is clear that the galaxies which persist when shifted are not strongly biased.

a.2. Moving galaxies from to

In a similar analysis, we shifted the galaxies at to . The result is shown in the bottom row of Fig. A14. Again, a large fraction of galaxies is lost. This should not come as a surprise, as the simulation assumes no evolution, and therefore may very well overpredict the fraction of lost galaxies. We note a small bias towards keeping large galaxies in the sample - exactly the opposite from what might be expected from surface brightness selection effects. This is caused by the fact that red galaxies are lost the fastest, as they are faintest in (and all other bands), for the same mass. Hence the median radius of the galaxies still detected after shifting them is slightly higher (by 15%), than the median radius of all galaxies more massive than 6  . Overall, this is a small effect, and we ignore it in the analysis. We also see that the distribution in the plane of specific star formation against surface density is not affected by these selection effects.

In short, we conclude that the selection effects are not likely to cause the strong evolution in specific star formation rate and size. This is likely caused by the fact that the point spread function is rather large, compared to the size of the galaxies, and hence strong surface brightness selection effects are not likely to play a role. It is also clear that the main selection effect is caused by the band limiting depth. This causes quiescent galaxies of a given mass to be missed earlier than strongly star forming galaxies. As these quiescent galaxies are small, it causes us to miss small galaxies first below the mass limit at which we are complete.

TABLE 1: LIMITING MAGNITUDES FOR THE GOODS-SOUTH IMAGING

                          Camera Filter Magnitude
                          WFI 26.06
                          WFI 27.21
                          WFI 26.88
                          WFI 26.99
                          WFI 25.00
                          ACS 27.29
                          ACS 27.42
                          ACS 26.87
                          ACS 26.51
                          ISAAC 25.43 - 26.03
                          ISAAC 25.00 - 25.55
                          ISAAC 24.63 - 25.57
                          IRAC ch1 26.15
                          IRAC ch2 25.66
                          IRAC ch3 23.79
                          IRAC ch4 23.70
                          MIPS 24 m 21.30

The limiting magnitude is the total AB magnitude for point sources detected at 3-.

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