Structural and Star-forming Relations since z ∼ 3: Connecting Compact Star-forming and Quiescent Galaxies

This paper is based on a sample of massive galaxies built from the HST/WFC3 F160W selected catalog for the CANDELS GOODS-S field (Guo et al. 2013). The multiwavelength catalog includes photometry in 14 passbands ranging from U to 8 μm, with seven high-resolution bands from HST/ACS and WFC3 (B₄₃₅, V₆₀₆, i₇₇₅, z₈₅₀, YJH) and the deepest Spitzer/IRAC data from SEDS (Ashby et al. 2013). The merging with lower resolution data (ground-based and Spitzer/IRAC) was computed using TFIT (Laidler et al. 2006). A comprehensive overview of this catalog can be found in Guo et al. (2013; see also Galametz et al. 2013 for more details). We use the official CANDELS photometric redshifts in the GOODS-S field presented in Dahlen et al. (2013). Briefly, these photometric redshifts are based on a hierarchical Bayesian approach that combines the full probability distribution functions (PDF(z)) derived by six CANDELS photo-z investigators using a variety of codes and modeling assumptions. The average accuracy of the merged PDF(z) yielded errors of ${\rm{\Delta }}z/(1+z)=3$%.

The stellar masses for all galaxies are drawn from the catalog of Santini et al. (2015). Similarly to the team effort aimed at computing accurate photometric redshifts, Santini et al. (2015) presented an analysis of the stellar masses of the CANDELS GOODS-S galaxies based on the estimates of 10 different investigators, who computed the stellar masses using the same photometry and redshifts described above, but with different codes, priors, and parameter grids. Overall, the results from the various teams are in good agreement despite these differences (see also Mobasher et al. 2015 for additional tests). Only a small fraction of the lowest-mass ($\mathrm{log}({M}_{\star }/{M}_{\odot })\lesssim 9$) galaxies, which are not the focus of this work, exhibit significant differences ($\gt 1$ dex) when emission lines are included in the stellar population fitting templates. Santini et al. (2015) combined all mass estimates using the same stellar templates and IMF by means of the Hodges–Lehmann estimator and computed the standard deviation of the methods as a first-order estimate of the systematic uncertainties. Here we choose to use the results of one of the methods to have a uniform set of modeling assumptions based on one of the most commonly used values in the literature. We adopt the stellar masses from method-2τ using FAST (Kriek et al. 2009b) and based on a grid of Bruzual & Charlot (2003) models that assume a Chabrier (2003) IMF, solar metallicity, exponentially declining star formation histories, and the Calzetti et al. (2000) dust extinction law.

In addition to the optical/NIR SEDs, we also include complementary mid-IR photometry in Spitzer/MIPS 24 and 70 μm (30 μJy and 1 mJy, $5\sigma$) from Pérez-González et al. (2008b) and far-IR photometry from the GOODS-Herschel (Elbaz et al. 2011) and PEP (Magnelli et al. 2013) surveys, including PACS—100 and 160 μm, and SPIRE—250, 350, and 500 μm. A description of the method used to derive consistent mid-to-far-IR SEDs is presented in Pérez-González et al. (2008a, 2010; see also Domínguez Sánchez et al. 2016). Based on these IR fluxes, we compute SFRs on a galaxy-by-galaxy basis using a ladder of SFR indicators as described in Wuyts et al. (2011a). The method essentially relies on IR-based SFR estimates for galaxies detected at mid- to far-IR wavelengths, and SED-modeled SFRs for the rest. As shown in Wuyts et al. (2011a), the agreement between the two estimates for galaxies with a moderate extinction (faint IR fluxes) ensures continuity between the different SFR estimates. For IR-detected galaxies, the total SFRs, ${\mathrm{SFR}}_{\mathrm{IR}+\mathrm{UV}}$, were then computed from a combination of IR and rest-frame UV luminosities (uncorrected for extinction; Kennicutt 1998 and Bell et al. 2005) and adopting a Chabrier (2003) IMF (see Barro et al. 2011 for more details), ${\mathrm{SFR}}_{\mathrm{UV}+\mathrm{IR}}=1.09\times {10}^{-10}({L}_{\mathrm{IR}}+3.3{L}_{2800})$. Total IR luminosities (${L}_{\mathrm{IR}}\equiv L$[8–1000 μm]) were derived from Chary & Elbaz (2001) templates fitting MIPS 24 μm fluxes, applying a Herschel-based recalibration (Elbaz et al. 2011), and L₂₈₀₀ is estimated from the best-fitting SED template. For galaxies undetected by MIPS below a 2σ level (20 μJy), SFRs come from rest-frame UV luminosities that are corrected for extinction as derived from SED fits (Wuyts et al. 2011a).

The half-light radii (${r}_{{\rm{e}}}$), measured along the major axis, and Sèrsic indices (n) were determined from HST/WFC3 H images using GALFIT (Peng et al. 2002), with PSFs created and processed to replicate the conditions of the observed data (van der Wel et al. 2012). Using a full battery of tests and simulations, van der Wel et al. (2012, 2014) showed that these fits are stable even for the smallest galaxies with ${r}_{{\rm{e}}}\sim 1\,\mathrm{kpc}$ at $z\sim 2$. Generally, the total uncertainties in n and ${r}_{{\rm{e}}}$ are smaller than $\lt 10$% for galaxies brighter that $H\lt 24.5$ mag.

The stellar mass profiles were computed by fitting multiband SEDs derived from surface brightness profiles in nine HST bands measured with IRAF/ellipse (see Liu et al. 2013 and F. Liu et al. 2017, in preparation for more details). Following Wuyts et al. (2012), we impose an additional constraint on the spatially resolved SED fit by requiring that the integrated profile match the observed flux in IRAC ch1 and ch2. We apply this constraint by assuming that the integrated IRAC-F160W color is the same at all radii.

The multiband HST mosaics were PSF-matched to the resolution of F160W, which has a half width at half maximum of HWHM = 009. The profiles have an intrinsic spatial resolution ranging from $r=0.6\mbox{--}0.7\,\mathrm{kpc}$ within the redshift range of the sample and thus resolve the inner 1 kpc of the galaxies. However, part of the light can be smeared to larger radius. To correct for this effect, we estimate a Sérsic and size-dependent correction to the mass profile within 1 kpc (see Appendix C). The correction is computed from a grid of Sérsic profiles with $n=0\mbox{--}4$ and ${r}_{{\rm{e}}}=0.1\mbox{--}10\,\mathrm{kpc}$ degraded to F160W resolution at a different redshift. The correction ranges from ∼0.4 dex at $n\gtrsim 2$ to $\sim 0.2-0$ dex at $n=1-0$. We note that we do not measure the (deprojected) mass in the central regions directly, but use the best-fitting Sérsic profiles as a proxy to estimate the mass at r < 1 kpc. We also note that we use integrated SFR measurements, which do not allow us to account for the spatial distribution of star formation. Therefore, the galaxy classifications of either star forming or quiescent galaxy are based on their overall SFR properties, i.e., we do not explicitly distinguish galaxies with quiescent centers and star-forming disks.

Our goal is to analyze the distribution of SFGs and quiescent galaxies at $0.5\lt z\lt 3$ in the structural scaling relations in order to understand their differential evolution. Owing to the depth of the CANDELS imaging data in GOODS-S ($H\sim 27$ mag; 5σ), the galaxy sample is remarkably complete up the highest redshifts ($z=4\mbox{--}7$; e.g., Guo et al. 2013; Grazian et al. 2015), and the quality of the HST-based SEDs ensures very accurate SED modeling to effectively distinguish between SFGs and quiescent galaxies even beyond a redshift of z = 3 (e.g., Nayyeri et al. 2014; Straatman et al. 2014). Nonetheless, our analysis requires not only robust multiband detections, but also precise morphological measurements. Therefore, we apply additional constraints to ensure that the galaxy sizes and surface brightness profiles are reliably measured. As discussed in van der Wel et al. (2012, 2014), a redshift-dependent mass threshold of $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 8.5\mbox{--}9.75$ and $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 9\mbox{--}10.3$ for SFGs and quiescent galaxies at $z=0.5\mbox{--}3$ ensures that these galaxies are brighter than H = 24.5 mag, and thus their morphological properties can be recovered with uncertainties smaller than $\lt 10$%.

In the following section, we will adopt fixed thresholds of $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 9$ and $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 10$ for SFGs and quiescent galaxies, respectively. These limits provide a compromise between having a wide enough dynamic range in stellar mass to allow a representative fitting of the structural scaling relations of both populations, while having a high signal-to-noise ratio in the morphological measurements for the majority of galaxies. At our highest redshift bin, $z\sim 2.75$, some of the lowest-mass quiescent galaxies are slightly fainter than the limit of van der Wel et al. (2014), and thus exhibit larger uncertainties. Nonetheless, we note that they are still clearly detected as they are 1.5 mag brighter than the 5σ detection limit. Furthermore, as shown in the following section, the quiescent sample at this redshift is relatively small, and therefore we fit the quiescent scaling relation by fixing the slope to the value of the previous redshift bin. In Section 5, we adopt a more conservative mass limit of $\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 10.3$ for both populations as the focus of the discussion is centered on the transition of massive galaxies from star forming to quiescent. This limit is consistent with the mass completeness estimates for red quiescent galaxies of previous works based on a similar photometry (e.g., Tal et al. 2014).

Following van der Wel et al. (2014), we draw our primary sample from the parent CANDELS catalog including all galaxies with $H\lt 25.5$ mag and a minimum stellar mass of $\mathrm{log}({M}_{\star }/{M}_{\odot })=9$. Then, we apply the redshift- and SFR-dependent mass limits described above in the analyses in the following sections. We exclude from the sample those galaxies with bad photometric flags (e.g., galaxies near defects, bright spikes, or sources classified as stars), or with catastrophic failures in the surface brightness profile fits. Hence, we have a final sample of 4958 galaxies at $0.5\lt z\lt 3.0$.

Since a major point of this paper is to compare the scaling relations for SFGs versus quiescent galaxies, we start by dividing the sample into these two categories using the distribution of SFR versus stellar mass for the galaxies in GOODS-S. The results are shown versus redshift in Figure 1, which shows SFR versus stellar mass. The majority of SFGs follow a relatively tight relation between SFR and stellar mass. The observed relation suggests that galaxy star formation histories are predominantly regular and smooth, i.e., galaxies grow in a secular mode, which is usually referred to as the SFR–MS (Elbaz et al. 2007, 2011; Noeske et al. 2007; Salim et al. 2007; Pannella et al. 2009, 2015; Magdis et al. 2010; Rodighiero et al. 2010; Wuyts et al. 2011a; Whitaker et al. 2012). Following previous works, we characterize the SFR–MS as a single power law at $\mathrm{log}({M}_{\star }/{M}_{\odot })\gtrsim 10$:

$$\begin{eqnarray}&&\mathrm{log}\,\mathrm{SFR}=\mu \left[\mathrm{log}\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)-10.5\right]+\mathrm{log}\,C.\end{eqnarray} \tag{ 1 }$$

The power law was fit iteratively while removing galaxies that are more than 0.7 dex below the line. Note that at lower masses, the SFR–MS exhibits a steeper slope (e.g., Whitaker et al. 2014; Schreiber et al. 2015). The resulting best-fit SFR–MS at every redshift is in excellent agreement with the results of Whitaker et al. (2014) and Schreiber et al. (2015) at the high-mass end. The slope and the zero point are consistent with their values within the errors (Table 1). We select SFGs and quiescent galaxies above and below a threshold of ${\rm{\Delta }}\mathrm{SFR}=-0.7$ dex (dashed red line), respectively, where ${\rm{\Delta }}\mathrm{SFR}\equiv \mathrm{log}\,\mathrm{SFR}-\mathrm{log}\,{\mathrm{SFR}}^{\mathrm{MS}}$. This classification line is $\sim 2\sigma$ below the SFR–MS, which has a typical observational scatter of 0.3 dex (Speagle et al. 2014; Whitaker et al. 2014; Schreiber et al. 2015). Appendix A discusses the impact of using an alternative quiescent selection criterion on the analysis of the structural scaling relations. The quiescent galaxies are shown with red points in Figure 1, while the SFGs use the blue and green color-coding as explained in Figure 2.

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Table 1.  Power-law Fits

	${\mathrm{SFR}}_{\mathrm{MS}}$–SFGs		${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$–Quiescent		${{\rm{\Sigma }}}_{{\rm{e}}}^{\mathrm{SF}}$–SFGs		${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$–Quiescent		${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}$–SFGs
Redshift Range	μ	$\mathrm{log}\,C$	${\alpha }^{{\rm{Q}}}$	log A${}^{{\rm{Q}}}$	${\alpha }^{\mathrm{SF}}$	log A${}^{\mathrm{SF}}$	${\beta }^{{\rm{Q}}}$	log B${}^{{\rm{Q}}}$	${\beta }^{\mathrm{SF}}$	log B${}^{\mathrm{SF}}$
$0.5\lt z\lt 1.0$	0.19 ± 0.08	1.21 ± 0.02	−0.42 ± 0.13	9.19 ± 0.06	0.60 ± 0.04	8.46 ± 0.04	0.65 ± 0.03	9.53 ± 0.05	0.89 ± 0.03	9.12 ± 0.03
$1.0\lt z\lt 1.4$	0.53 ± 0.07	1.44 ± 0.04	−0.45 ± 0.14	9.53 ± 0.05	0.60 ± 0.05	8.54 ± 0.05	0.65 ± 0.04	9.64 ± 0.04	0.88 ± 0.03	9.16 ± 0.04
$1.4\lt z\lt 2.2$	0.64 ± 0.06	1.75 ± 0.05	−0.52 ± 0.14	9.91 ± 0.07	0.64 ± 0.05	8.68 ± 0.04	0.64 ± 0.03	9.74 ± 0.05	0.86 ± 0.04	9.25 ± 0.03
$2.2\lt z\lt 3.0$	0.68 ± 0.06	1.92 ± 0.05	−0.52	10.28 ± 0.05	0.56 ± 0.06	8.80 ± 0.04	0.67 ± 0.04	9.80 ± 0.05	0.89 ± 0.04	9.33 ± 0.05

Note. Power-law coefficients parameterizing the evolution of the SFR (log SFR $=\,\mu \mathrm{log}\,({M}_{\star }-10.5)+\mathrm{log}\,C$), the effective mass surface density (log ${{\rm{\Sigma }}}_{{\rm{e}}}=\,\alpha \mathrm{log}\,({M}_{\star }-10.5)+\mathrm{log}\,A$), and the mass surface density within the central 1 kpc ($\mathrm{log}{{\rm{\Sigma }}}_{1}=\,\beta \mathrm{log}\,({M}_{\star }-10.5)+\mathrm{log}\,B$) relations with stellar mass as a function of redshift.

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We turn now to the density scaling relations that are the main focus of this paper. The top row (A panels) of Figure 2 shows the distribution in effective surface mass density, ${{\rm{\Sigma }}}_{{\rm{e}}}\,=0.5{M}_{\star }/\pi {r}_{{\rm{e}}}^{2}$, versus mass for the galaxies in Figure 1. The points are color-coded as in Figure 1. The error bars indicate the typical uncertainty for galaxies at the median stellar mass computed by bootstrapping on the observational errors in stellar mass and effective radii, and propagating those uncertainties to ${{\rm{\Sigma }}}_{{\rm{e}}}$. SFGs (blue and green) and quiescent galaxies (red) follow well-defined size–mass relations, which are characterized by log-linear relations, $\mathrm{log}{r}_{{\rm{e}}}\propto a\mathrm{log}M$ (e.g., Law et al. 2012; Mosleh et al. 2012; Newman et al. 2012; Szomoru et al. 2012; van der Wel et al. 2014). Both relations can be expressed in terms of ${{\rm{\Sigma }}}_{{\rm{e}}}$:

$$\begin{eqnarray}&&\mathrm{log}\,{{\rm{\Sigma }}}_{{\rm{e}}}=\alpha \left[\mathrm{log}\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)-10.5\right]+\mathrm{log}\,A(z),\end{eqnarray} \tag{ 2 }$$

where α is related to the slope of the size–mass relation a as $\alpha =1-2a$. The red and blue lines in Figure 2(A) show the best-fit ${{\rm{\Sigma }}}_{{\rm{e}}}$ relations for quiescent galaxies and SFGs, and Table 1 gives the slopes and zero points. The best-fit slopes are relatively constant with time, ${\alpha }^{\mathrm{SF}}\sim 0.6$, ${\alpha }^{{\rm{Q}}}\sim -0.5$, and agree well with the results of van der Wel et al. (2014) for the ${r}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relations of both populations (${a}^{\mathrm{SF}}\sim 0.2$ and ${a}^{{\rm{Q}}}\sim 0.8$). There are too few quiescent galaxies at $2.2\lt z\lt 3.0$ to accurately fit the slope, and so we fix the slope (but not the zero point) in this bin to match the value at $1.4\lt z\lt 2.2$. Note that quiescent galaxies have a steep slope in the ${r}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relation, which leads to an anti-correlation in ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$. This is consistent with the observed trend for the most massive quiescent galaxies at low z (e.g., Graham & Guzmán 2003; Ferrarese et al. 2006; Kormendy et al. 2009). Meanwhile, SFGs have a shallow slope in ${r}_{{\rm{e}}}\mbox{--}{M}_{\star }$ (${a}^{\mathrm{SF}}\lt 0.5$) and a positive correlation between ${{\rm{\Sigma }}}_{{\rm{e}}}^{\mathrm{SF}}$ and mass. The scatter in ${{\rm{\Sigma }}}_{{\rm{e}}}$ is consistent with ∼2× that of the size–mass relations, $\sigma (\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}})\sim 0.5$ dex and 0.3 dex, for SFGs and quiescent galaxies, as expected from ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}}\propto 2{\rm{\Delta }}\mathrm{log}{r}_{{\rm{e}}}$. The redshift-dependent zero points decline from z = 3 to z = 0.5. Such a decline is much steeper for quiescent galaxies than for SFGs (1 dex versus 0.3 dex), as noted in previous works (Buitrago et al. 2008; Newman et al. 2012; van der Wel et al. 2014).

The bottom row (B panels) of Figure 2 shows the redshift evolution of the central surface mass density within 1 kpc, ${{\rm{\Sigma }}}_{1\mathrm{kpc}}={M}_{\star }{(\lt 1\mathrm{kpc})/\pi (1\mathrm{kpc})}^{2}$, versus the stellar mass. The error bars indicate the typical uncertainty for galaxies at the median stellar mass computed by bootstrapping on the observational errors in stellar mass, effective radii, and Sérsic index, and propagating those uncertainties to ${{\rm{\Sigma }}}_{1}$. Similarly to ${{\rm{\Sigma }}}_{{\rm{e}}}$, we characterize the observed correlation in ${{\rm{\Sigma }}}_{1}$ as a log-linear relation:

$$\begin{eqnarray}&&\mathrm{log}\,{{\rm{\Sigma }}}_{1}=\beta \left[\mathrm{log}\left(\displaystyle \frac{{M}_{\star }}{{M}_{\odot }}\right)-10.5\right]+\mathrm{log}\,B(z).\end{eqnarray} \tag{ 3 }$$

Again, we find clear correlations for both SFGs and quiescent galaxies at every redshift since $z\sim 3$. The slopes of each of these relations (see Table 1) are positive and relatively constant with time, ${\beta }^{\mathrm{SF}}=0.9$ and ${\beta }^{{\rm{Q}}}=0.7$. By comparison with ${{\rm{\Sigma }}}_{{\rm{e}}}$, the dispersion is $\sim 2\times$ tighter, $\sigma (\mathrm{log}{{\rm{\Sigma }}}_{1})\sim 0.25$ dex and 0.14 dex for SFGs and quiescent galaxies, respectively. The zero point of the star-forming ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}$ relation declines by ∼0.3 dex from z = 3 to z = 0.5, similar to the evolution in ${{\rm{\Sigma }}}_{{\rm{e}}}^{\mathrm{SF}}$. Interestingly, for quiescent galaxies, ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ declines by a similar amount, in stark contrast to the strong decline of ∼1 dex in ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$.

A summary of the main conclusions from Figure 2 is as follows.

• (1)  
  Since ${{\rm{\Sigma }}}_{{\rm{e}}}$ is based on r_e and M_⋆, the nature and evolution of the ${{\rm{\Sigma }}}_{{\rm{e}}}$ relations in panel A can be inferred from the corresponding ${r}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relations, which have been known for some time (e.g., Newman et al. 2012; van der Wel et al. 2014). In particular, the large scatter of both relations, the offset of quiescent galaxies to higher densities, the constancy of the slopes versus time, and the diminishing difference between quiescent galaxies and SFGs—all of these properties are implied by the ${r}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relations.
• (2)  
  The general distribution of points in the ${{\rm{\Sigma }}}_{1}$ relations in panel B is essentially identical to the distribution of points found by Fang et al. (2013) for SDSS galaxies at $z\sim 0.1$. Moreover, the ridge line of Fang's quiescent galaxies has the same slope here, and the Fang zero point is a smooth extrapolation from higher redshifts. This suggests that the ${{\rm{\Sigma }}}_{1}$ relations were already fully in place at z = 3 and, except for modest declines in zero points, have remained essentially constant down to the present day.
• (3)  
  If a scaling relation versus mass is narrow and if galaxies have increased their masses greatly during the time over which the relation is seen to exist, then one can make the case that the objects that populate the relation could evolve roughly along it. This could be the case for the progenitors of massive galaxies at $z=0(\mathrm{log}({M}_{\star }/{M}_{\odot })\sim 11$) that grew in total mass by a significant amount since $z\sim 3$.There are several methods to estimate such growth, which yield slightly different results but similar orders of magnitude (e.g., van Dokkum et al. 2010; Behroozi et al. 2013; Moster et al. 2013; Patel et al. 2013; Marchesini et al. 2014). For example, the predictions from abundance-matching techniques imply growth rates of 0.75 dex to 1.5 dex for descendants with masses ranging from $\mathrm{log}({M}_{\star }/{M}_{\odot })(z=0)\sim 11$ to 10.75, respectively (see e.g., Figure 4 of Papovich et al. 2015). Here we adopt  mass growth by a factor of 20 to follow the growth of a population of SFGs that probe a significant fraction of the parameter space in the structural relations.An evolutionary vector of length ×20 (1.3 dex) in mass is large compared to either the scatter about the star-forming ${{\rm{\Sigma }}}_{1}$ relation (∼0.35 dex), or the evolutionary change in the zero point of 0.3 dex from z = 3–0.5 (Table 1). Thus, we argue that the ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}$ relation could be an approximate evolutionary track. The scatter in ${{\rm{\Sigma }}}_{{\rm{e}}}^{\mathrm{SF}}$ is twice as large as that in ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}$ but is nevertheless still small compared to a ×20 growth in mass. Hence, it is reasonable to infer that the ${{\rm{\Sigma }}}_{{\rm{e}}}^{\mathrm{SF}}$ relation could also be an evolutionary track, with the average galaxy moving close to the slope of the best-fit line, ${{\rm{\Sigma }}}_{{\rm{e}}}\sim {M}_{\star }^{0.6}$ (also ${r}_{{\rm{e}}}\sim {M}_{\star }^{0.2}$). A similar evolutionary picture was suggested by van Dokkum et al. (2015) with ${r}_{{\rm{e}}}\sim {M}_{\star }^{0.3}$ (also see Rodriguez-Puebla et al. 2017). The possible evolutionary paths of SFGs are discussed further in Section 4.
• (4)  
  An implication of the possible evolutionary track in ${{\rm{\Sigma }}}_{1}$ is that galaxies build up their central stellar mass densities monotonically in a smooth and regular way (averaged over time) as long as they continue to form stars. This places a constraint on galaxy mass assembly models, including the popular "inside-out" growth scenario for disk formation (e.g., Nelson et al. 2016) and suggests that galaxies did not form their core completely first and then the remainder of the galaxy formed around it.
• (5)  
  The clustering of virtually all quiescent galaxies on the ${{\rm{\Sigma }}}_{1}$ quiescent relation (red points) implies that forming a dense core within 1 kpc is a prerequisite to quenching, as argued in previous works (Cheung et al. 2012; Fang et al. 2013; van Dokkum et al. 2014). At the same time, some SFGs are also found on the quiescent relation (green points), which indicates that a dense core is a necessary but not sufficient condition for quenching. We discuss these galaxies further in Section 5.2.
• (6)  
  The ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}\mbox{--}{M}_{\star }$ relation implies that the quiescent threshold in ${{\rm{\Sigma }}}_{1}$ is not fixed but increases with stellar mass. Previous works advanced a slightly different view in which quenching occurs when galaxies pass over a fixed threshold in effective (i.e., global) velocity dispersion that varies with redshift but not with mass (e.g., Franx et al. 2008; Wake et al. 2012; van Dokkum et al. 2015). Figure 2 indicates that the accuracy of a threshold in ${{\rm{\Sigma }}}_{{\rm{e}}}$ or ${{\rm{\Sigma }}}_{1}$ depends on the slope of the quiescent correlation with stellar mass, being more effective for weak correlations—when the slope is close to zero. Thus, a threshold in ${{\rm{\Sigma }}}_{{\rm{e}}}$ or σ (e.g., Belli et al. 2014) is more effective than a threshold in ${{\rm{\Sigma }}}_{1}$ due to their shallower slope (assuming ${{\rm{\Sigma }}}_{1}\propto 0.5\sigma$; Fang et al. 2013). Nonetheless, the most reliable indicator would be the distance to the quiescent relation itself. We discuss this topic further in Section 5.1.

The preceding section derived scaling relations for both ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$. This section takes a short detour to consider the relationship between these two scaling relations, specifically whether one can be derived from the other simply from the known shapes of galaxy surface brightness profiles. Galaxy surface brightness and stellar mass profiles are typically described by Sérsic (1963) profiles, in which case, the central and effective densities are related to one another by the profile parameters. Assuming the following characterization of the mass profile:

$$\begin{eqnarray}&&M(r)={M}_{{\rm{e}}}\,\exp (-{b}_{n}[{(r/{r}_{{\rm{e}}})}^{1/n}-1]),\end{eqnarray} \tag{ 4 }$$

where n is the Sérsic index and M_e is the effective mass at r_e, ${{\rm{\Sigma }}}_{1}$ and ${{\rm{\Sigma }}}_{{\rm{e}}}$ can be determined from one of them for a given value of n and r_e. Note that in doing so we assume that the mass and light profiles are the same, ${r}_{{\rm{e}},\mathrm{mass}}={r}_{{\rm{e}},\mathrm{light}}$, a reasonable assumption for quiescent galaxies, which exhibit only weak color gradients (e.g., Guo et al. 2012; Szomoru et al. 2012; Wuyts et al. 2012).

By integrating Equation (4) to r = 1 kpc, we obtain a relation for ${{\rm{\Sigma }}}_{1}$ as a function of r_e (or ${{\rm{\Sigma }}}_{{\rm{e}}}$) and M_⋆:

$$\begin{eqnarray}&&\mathrm{log}\,{{\rm{\Sigma }}}_{1\mathrm{kpc}}-\mathrm{log}\,{M}_{\star }=-\mathrm{log}\pi -\mathrm{log}\gamma (2n,{b}_{n}{r}_{{\rm{e}}}^{-1/n}).\end{eqnarray} \tag{ 5 }$$

This can be expanded in terms of the log-linear size–mass and ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relations to calculate the slope and scatter of the ${{\rm{\Sigma }}}_{1}\mbox{--}{M}_{\star }$ relation [β, $\sigma (\mathrm{log}{{\rm{\Sigma }}}_{1})]$ in terms of the slope and scatter of the other two ([a, $\sigma (\mathrm{log}{r}_{{\rm{e}}})]$ and [α, $\sigma \mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}}$]). We find:

$$\begin{eqnarray}\beta & \sim & 1+0.5{c}_{1}(\alpha -1);\sigma (\mathrm{log}{{\rm{\Sigma }}}_{1})\sim 0.5{c}_{1}\sigma (\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}})\\ \beta & \sim & 1-{c}_{1}a;\sigma (\mathrm{log}{{\rm{\Sigma }}}_{1})\sim {c}_{1}\sigma (\mathrm{log}{r}_{{\rm{e}}}),\end{eqnarray} \tag{ 6 }$$

where $\alpha =1-2a$, and c₁ is the linear coefficient of the r_e polynomial expansion of the incomplete $\gamma ({r}_{{\rm{e}}},n)$ function (see Appendix B for a detailed calculation). These equations indicate that if galaxy profiles are described by single Sérsic profiles, the ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$ and ${{\rm{\Sigma }}}_{1}\mbox{--}{M}_{\star }$ relations are not independent and can be derived from one another (see also Saracco et al. 2012). In other words, the mapping of ${{\rm{\Sigma }}}_{1}$ into ${{\rm{\Sigma }}}_{{\rm{e}}}$ (and vice versa) depends on the Sérsic profile.

Figure 3 illustrates this idea by showing the same panels as Figure 2 but overplotting the predicted ${{\rm{\Sigma }}}_{{\rm{e}}}$ relations for SFGs and quiescent galaxies (left panel) based on the best-fit ${{\rm{\Sigma }}}_{1}$ relations and vice versa (right panel). For example, the red line in the left panel represents the value of ${{\rm{\Sigma }}}_{{\rm{e}}}$ predicted from the mean ${{\rm{\Sigma }}}_{1}\mbox{--}{M}_{\star }$ relation for quiescent galaxies in Figure 2 using the transformation relations described in Appendix A with n = 4. This predicted trend is an excellent fit to actual n = 4 galaxies. The cyan line represents the same exercise but using the ${{\rm{\Sigma }}}_{1}\mbox{--}{M}_{\star }$ relation for SFGs with n = 1.5, which also exhibits a good agreement with n = 1.5 galaxies.

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As indicated by Equation (6), the relation between the scatter in ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ depends on the Sérsic index. Quiescent galaxies typically have high Sérsic values, $n=3\mbox{--}4$ (Williams et al. 2010; Bell et al. 2012). For a value of n = 4, the linear coefficient is ${c}_{1}(4)=0.55$ (see Figure 11), which gives $\sigma (\mathrm{log}{{\rm{\Sigma }}}_{1})\sim 0.3\sigma (\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}}$). This result indicates that the smaller scatter in ${{\rm{\Sigma }}}_{1}$ with respect to ${{\rm{\Sigma }}}_{{\rm{e}}}$ is a direct consequence of the smaller dynamical range in 1 kpc values for a given Sérsic profile (the same point was made by Saracco et al. 2012). Furthermore, the linear approximations in Equation (6) also show that, given the observed slope of the quiescent ${{\rm{\Sigma }}}_{{\rm{e}}}$ relation, ${\alpha }^{{\rm{Q}}}\sim -0.5$, the expected slope of the ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ relation is ${\beta }^{{\rm{Q}}}\sim 0.7$, which is broadly consistent with the best-fit value. Note that when using the second-order polynomial expansions of Equations (10) and (11) instead of the linear approximation, the mapping of the ${{\rm{\Sigma }}}_{{\rm{e}}}$ relations is not linear in ${{\rm{\Sigma }}}_{1}$ and vice versa (red lines in Figure 3). Nevertheless, this comparison shows that, to first order, the similarities and differences between the ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ relations for quiescent galaxies can be explained under the assumption of a single Sérsic profile with high values of $n\gtrsim 2\mbox{--}4$.

SFGs, on the other hand, exhibit a broad range in Sérsic values. This is illustrated in Figure 3, which, as mentioned above, repeats Figure 2 for galaxies in the redshift range $z=1.4\mbox{--}3.0$, but now for SFGs only and with pixels color-coded by median Sérsic values. Sérsic values of SFGs vary all the way from n = 0 to n = 4. At a given stellar mass, the Sérsic index increases with ${{\rm{\Sigma }}}_{1}$, i.e., the spread in the ${{\rm{\Sigma }}}_{1}$ distribution for SFGs reflects differences in their effective (i.e., global) structure and the same is also true of ${{\rm{\Sigma }}}_{{\rm{e}}}$. This trend has two interpretations. One is that as SFGs grow a dense core, their global mass profiles become more centrally concentrated. A second interpretation is that the trend is primarily driven by disk fading, whereby star formation in the disk fades, causing r_e to shrink and the Sérsic value to increase. Regardless of interpretation, we note that SFGs in the overlapping regions with the quiescent Σ relations (dashed gray lines) have Sérsic values that are very similar to quiescent galaxies, implying that SFGs in that region are structurally very close to being quiescent.

To summarize this section, we have shown that the two sets of scaling relations in ${{\rm{\Sigma }}}_{1}$ and ${{\rm{\Sigma }}}_{{\rm{e}}}$ are closely equivalent, given the known variation of the Sérsic index with the star formation level. We derived equations that relate the slope and the scatter of each relation with one another. In the next section, we will use these relations to compare the predictions of different evolutionary channels for quiescent galaxies.

As shown in Section 3.1, the zero points of the density–mass relations for SFGs and quiescent galaxies decline with time while their slopes remain approximately constant. These properties of the scaling relations constrain the possible structural evolutionary mechanisms for both populations. The present section focuses on quiescent galaxies, which evolve only as a result of non-star-forming mechanisms. SFGs have an evolutionary channel that is primarily fueled by in situ star formation; this is discussed with the help of models in Section 4. Here, we analyze some of the most common mechanisms proposed to explain the structural evolution of quiescent galaxies (e.g., mergers or adiabatic expansion) to study whether they can explain the observed decline in the zero point of the structural scaling relations at constant slope.

We quantify the change in the zero point of the ${{\rm{\Sigma }}}_{1}$ and ${{\rm{\Sigma }}}_{{\rm{e}}}$ relations by integrating Equation (4) and using the relation between ${{\rm{\Sigma }}}_{{\rm{e}}}$ and r_e (see Appendix B.1 for a detailed calculation). Furthermore, it is assumed that all processes (1) preserve the Sérsic profile, (2) affect all galaxies in the same structural relations equally, and (3) can be characterized as producing a size growth ${\rm{\Delta }}\mathrm{log}{r}_{{\rm{e}}}=\eta {\rm{\Delta }}\mathrm{log}\,{M}_{\star }$. Note that assumption (1) does not imply that the profiles of quiescent galaxies remain unchanged by the evolutionary mechanisms, but rather that the central part (which dominates the Sérsic profile) remains approximately constant. With these assumptions, the zero points of ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$ ($\mathrm{log}\,A$) and ${{\rm{\Sigma }}}_{1}\mbox{--}{M}_{\star }$ ($\mathrm{log}\,B$) change as ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}},1}-(\alpha ,\beta ){\rm{\Delta }}\mathrm{log}\,{M}_{\star }$, which leads to

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{log}\,A=(1-2\eta -\alpha ){\rm{\Delta }}\mathrm{log}\,{M}_{\star },\end{eqnarray} \tag{ 7 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{log}\,B\sim (1-{c}_{1}\eta -\beta ){\rm{\Delta }}\mathrm{log}\,{M}_{\star }.\end{eqnarray} \tag{ 8 }$$

Equations (7) and (8) can now be used to test whether the proposed evolutionary processes for quiescent galaxies are consistent with the observed evolution in the zero points of their structural relations (Table 1). Figure 4 summarizes visually this evolution and shows the best fits to power laws that characterize their redshift dependence (${B}^{{\rm{Q}}}\propto {(1+z)}^{0.8}$, ${A}^{{\rm{Q}}}\propto {(1+z)}^{3.5}$). For quiescent galaxies, where the relative evolution in the zero points of ${{\rm{\Sigma }}}_{{\rm{e}}}$ versus ${{\rm{\Sigma }}}_{1}$ is large, there are three main evolutionary channels that can explain this difference. (1) The leading explanation is that quiescent galaxies experience a large increase in effective radius compared to mass due to dry minor mergers (i.e., accretion of smaller satellite galaxies; Bezanson et al. 2009; Hopkins et al. 2009; Naab et al. 2009; van Dokkum et al. 2010, 2014; Oser et al. 2012). (2) Part of the increase in the overall radius can also be caused by additional processes, such as stellar mass loss associated with passive evolution (i.e., death of old stars), which causes adiabatic expansion (Damjanov et al. 2011; Poggianti et al. 2013; Porter et al. 2014; van Dokkum et al. 2014). (3) Lastly, the strong decline in ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ could also be caused by the arrival of new quiescent galaxies with progressively larger effective radii at lower redshifts (e.g., Carollo et al. 2013; Poggianti et al. 2013; Porter et al. 2014). Each of these scenarios leads to a different evolution in the zero points of ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ and ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$, and thus we can test whether any of them matches the observed evolution in the ratio of zero points. Based on the best fits to the redshift evolution of log A^Q and log B^Q shown in Figure 4, any evolutionary process must verify that the ratio of change is ${\rm{\Delta }}\mathrm{log}\,{A}^{{\rm{Q}}}$/${\rm{\Delta }}\mathrm{log}\,{B}^{{\rm{Q}}}\sim 3.49/0.80\sim 4$.

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In the first scenario, the evolution of quiescent galaxies is driven by minor mergers, and the expectation is that the core mass remains relatively unchanged (e.g., Naab et al. 2009; Oser et al. 2012). In such a case, instead of Equation (7), the zero point in ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ changes simply due to the total mass growth as ${\rm{\Delta }}\mathrm{log}\,B\sim -\beta {\rm{\Delta }}\mathrm{log}\,{M}_{\star }=-0.65{\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ (i.e., Δ${{\rm{\Sigma }}}_{1}$ = 0), while the zero point in ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ follows Equation (7) as ${\rm{\Delta }}\mathrm{log}\,A\,\sim -1.7{\rm{\Delta }}\mathrm{log}\,{M}_{\star }$, for a conservative value of $\eta \sim 1.6$ (e.g., Newman et al. 2012). The predicted relative change is then ${\rm{\Delta }}\mathrm{log}\,A$/${\rm{\Delta }}\mathrm{log}\,B\sim 2.6$. This is substantially smaller than the ratio of the observed evolution trends in Figure 4, which, as we have noted above, is closer to 4. This shortfall is illustrated by the gray dotted model curve labeled "minor mergers" in Figure 4, which shows the predicted trend in effective density due to mergers, assuming that the slope of the observed decline in the zero point of ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ is multiplied by 2.6. This curve falls far short of matching the steep slope of the observed effective density curve, shown as the heavy red line. This tension can be partially alleviated if mergers cause a larger size growth ($\eta \gg 1.6$) or if they also cause an increase in ${{\rm{\Sigma }}}_{1}$, e.g., due to projection effects in the 2D surface density. The latter, however, is inconsistent with the results of van Dokkum et al. (2014), who showed that the 3D mass density within 1 kpc tends to decline with time, at least for massive quiescent galaxies.

The second evolutionary scenario, which also preserves the shape (Sérsic index) of the mass profile, is adiabatic expansion. Quiescent galaxies must puff up due to a decline in the gravitational potential caused by the death of old stars that ejects gas outside the galaxy. For a value of $\eta =-1$ in this case (e.g., Damjanov et al. 2009), Equation (8) indicates that ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ decreases as ${\rm{\Delta }}\mathrm{log}\,A\sim 3.5{\rm{\Delta }}\mathrm{log}\,{M}_{\star }$, while ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ decreases as ${\rm{\Delta }}\mathrm{log}\,B\sim {\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ (note that ${\rm{\Delta }}\mathrm{log}\,{M}_{\star }\lt 0$ as it is a mass loss). Thus, the ratio between the two is ${\rm{\Delta }}\mathrm{log}\,{A}^{{\rm{Q}}}$/${\rm{\Delta }}\mathrm{log}\,{B}^{{\rm{Q}}}\,\sim 3.5$, which is larger than the prediction from minor mergers (black versus gray dotted lines in Figure 4), but still fails to match the observed steep decline in ${{\rm{\Sigma }}}_{{\rm{e}}}$ (heavy red line). Interestingly, the nearly unity relation in ${\rm{\Delta }}\mathrm{log}\,B\sim {\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ in this case implies that the mass loss required to reproduce the observed evolution in ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ from adiabatic expansion would have to vary with time as ${\rm{\Delta }}\mathrm{log}\,{M}_{\star }\sim {(1+z)}^{0.80}$, or 0.67 dex between z = 2 and z = 1 (Figure 4). However, as noted by van Dokkum et al. (2014), a simple formation model in which quiescent galaxies quenched as early as $z\sim 5$ implies a much slower mass decline at later times, ${\rm{\Delta }}\mathrm{log}\,{M}_{\star }\sim 0.03z$ or ${\rm{\Delta }}\mathrm{log}\,{M}_{\star }\sim 0.06z$ for Salpeter and bottom-heavy IMFs, respectively. These amounts are 10–20 times smaller than the required decrease in stellar mass above. The main reason is that most of the stellar mass is lost in the first few gigayears after quenching (e.g., Damjanov et al. 2009).

The third scenario to explain the fast decline in the zero point of ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ is that newly quenched galaxies at lower redshifts may arrive on the quiescent relation with significantly larger r_e, and hence smaller ${{\rm{\Sigma }}}_{{\rm{e}}}$, at a given mass than did earlier galaxies (e.g., Carollo et al. 2013; Poggianti et al. 2013). This is an example of the "progenitor bias" effect pointed out by van Dokkum & Franx (2001), in which galaxy samples are polluted by a mixture of galaxies forming at different times with different properties. At face value, the effect would appear to be small since the zero points of ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}$ and ${{\rm{\Sigma }}}_{{\rm{e}}}^{\mathrm{SF}}$ evolve together, and so there is no obvious differential effect to make ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ fall more steeply than ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$. However, this would change if quenching SFGs were drawn from a different region of the ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relation at different times. Say, for example, that late-quenching SFGs have systematically larger r_e and smaller ${{\rm{\Sigma }}}_{{\rm{e}}}$ (at fixed stellar mass) than early-quenching SFGs. Due to the large scatter in the ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relation, these late arrivals could pull down the zero point of ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ substantially. Hydrodynamic models of SFGs to be discussed in Section 4.2 show some evidence that late-evolving SFGs actually do follow lower tracks in ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$ than early-evolving SFGs, giving some weight to this possibility. The effect is larger in ${{\rm{\Sigma }}}_{{\rm{e}}}$ but is also significant in ${{\rm{\Sigma }}}_{1}$, helping to explain the fall in zero point of that relation as well.

To summarize, we study four density scaling relations versus mass: for SFGs versus quiescent galaxies, and for ${{\rm{\Sigma }}}_{{\rm{e}}}$ versus ${{\rm{\Sigma }}}_{1}$. The zero points of three of the four relations evolve almost identically, with projected density declining by ∼0.3 dex at fixed mass from $z\sim 2.6$ to z = 0.75. The fourth, ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}\mbox{--}{M}_{\star }$, shows a much larger fall of ∼1 dex. The important conclusion is that the "central" properties of galaxies as a function of mass are quite stable with time and they evolve in parallel with the effective properties as long as galaxies are star forming. In quenched galaxies, this link is broken (i.e., ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ declines much faster), most likely because extra mass is added to the outer parts from minor mergers without affecting the inner parts. However, under the typical assumptions for the effects of minor mergers, such growth alone seems unable to account for the observed evolution of the ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ zero point from $z\sim 2.6$ to z = 0.75—although minor mergers may play a more dominant role at low z (e.g., Newman et al. 2012; Belli et al. 2014). Two other factors—adiabatic expansion due to stellar mass loss and the late arrival of SFGs with systematically larger effective radii—may also be substantial contributors to the effect. Minor mergers change both mass and Sérsic value and are thus in principle distinguishable from the other two mechanisms, which have smaller effects on these quantities.

As pointed out in previous works, the most likely evolutionary scenario, particularly at $z\gtrsim 1.5$ where the number density of quiescent galaxies is still small, is that all these evolutionary channels play a role in the evolution of the zero points (e.g., Belli et al. 2014; van der Wel et al. 2014; van Dokkum et al. 2014). Nonetheless, quantifying their relative contribution as a function of time requires precise estimates of the ages and quenching times of quiescent galaxies, as well as large enough samples to characterize the number densities at the extremes of the distributions (i.e., the smallest or densest galaxies) whose disappearance indicates the need for both size growth and stellar mass loss.

Here we address the topic of whether the population loci in the various scaling relations shown in Figure 2 can be interpreted as approximate evolutionary tracks, and more generally how individual galaxies are moving in these diagrams. This topic was touched on in Section 3.1, where we noted that the ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ relation for SFGs is both relatively narrow in mass (σ(${M}_{\star })\sim 0.3$ dex)) and that its zero point changes by only a factor of two over the redshifts probed. Since galaxies are believed to evolve in mass by much more than either of these numbers, i.e., by $\sim \times 20$ over the same redshift range (van Dokkum et al. 2010; Behroozi et al. 2013; Moster et al. 2013; Papovich et al. 2015), the excursions and zero point changes are therefore both small compared to the overall mass trend. This suggests that ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ can be interpreted as an approximate evolutionary track and that galaxies evolve along it while still star forming (neglecting the small decline in zero point). A relevant conclusion based on this assumption is that galaxies would build their core densities continuously over time as long as they are in the star-forming phase, and do so in a way that is closely correlated with their instantaneous stellar mass. A similar evolutionary trend was proposed by Fang et al. (2013) at z = 0, and by Tacchella et al. (2015) at $z\sim 2$. Tacchella et al. (2015), however, suggest that all massive SFGs exhibit fully quenched cores at that redshift, and therefore the slope of their evolutionary tracks flattens $\beta \sim 0$ at $\mathrm{log}({M}_{\star }/{M}_{\odot })\gtrsim 10.5$.

The scatter in the ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relation is larger (σ(M_⋆) ∼ 0.6 dex), but it is still smaller than the estimated mass growth. Furthermore, as shown in Figure 3, the ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ relations are tightly related to one another for a given Sérsic profile. Thus, on average, galaxies would evolve along this relation as well. A similar statement can also be made about the ${r}_{{\rm{e}}}\mbox{--}{M}_{\star }$ relation, as shown in van Dokkum et al. (2015), which is equivalent in information to ${{\rm{\Sigma }}}_{{\rm{e}}}\mbox{--}{M}_{\star }$. In that paper, the authors advocate for a slightly steeper slope (a = 0.3) compared to that of the observed size–mass relation (a = 0.22; van der Wel et al. 2014). This slope is meant to be a time and population average of the overall size growth trend rather than the evolutionary track of individual galaxies.

In the following, we assume that SFGs approximately follow evolutionary tracks in density–mass space characterized by the instantaneous slopes from Table 1, ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{({\rm{e}},1)}\,=(\alpha ,\beta ){\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ with $\alpha \sim 0.7$ and $\beta \sim 0.9$, and we discuss the physical processes that could be involved in creating such pathways. These evolutionary vectors are shown in Figure 5 with blue arrows. SFGs growing along the ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}$ relation progressively increase their surface density with time. However, since $\beta \lt 1$ the mass in the core grows more slowly than the total mass of galaxy, i.e., the core-to-total mass ratio decreases with time. This is consistent with the notion of inside-out growth of an exponential profile (disk) due to galactic-scale star formation and accretion of higher angular momentum material, which causes both r_e and ${{\rm{\Sigma }}}_{1}$ to increase along with stellar mass (e.g., Nelson et al. 2012, 2013). This scenario also agrees with Figure 3, which shows that the ${{\rm{\Sigma }}}_{{\rm{e}}}^{\mathrm{SF}}$ and ${{\rm{\Sigma }}}_{1}^{\mathrm{SF}}$ relations (cyan lines) approximately match the distribution of galaxies with a constant, low Sérsic index.

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Next, we focus on the evolutionary paths that take SFGs to the higher-density structural relations of quiescent galaxies. To first order, SFGs growing strictly along the ${{\rm{\Sigma }}}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ relations would intersect the quiescent structural relations at $\mathrm{log}({M}_{\star }/{M}_{\odot })\gtrsim 11.5$. However, confining quenching to that track would likely overproduce the number of massive quiescent galaxies (e.g., Figure 27 of van Dokkum et al. 2015). An alternative scenario to explain the emergence of denser quiescent galaxies is that at least some SFGs follow a steeper path upwards from the ${{\rm{\Sigma }}}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ relation as a result of some period(s) of fast core growth relative to the smooth growth along the structural ${\rm{\Sigma }}\mbox{--}{M}_{\star }$ relations.

At high z, these periods are probably caused by strongly dissipational, wet "compaction" events, e.g., major mergers (Hopkins et al. 2008) or disk instabilities (Elmegreen et al. 2008; Dekel et al. 2009b) which bring existing stars and large amounts of new gas to the center of the galaxy. These events enhance star formation in the core, increasing the central density and, in the most extreme cases, collapsing the whole galaxy to a much smaller r_e. At low z, the frequency of rapid wet compactions declines with the observed gas fractions in galaxies and other slower events, such as minor mergers and/or secular evolution (e.g., Kormendy & Kennicutt 2004), can play a more important role.

The wet compaction scenario was discussed in Barro et al. (2013, 2014a) on the basis that some dense massive SFGs at $z=2\mbox{--}3$ were found to lie on the size–mass relation of quiescent galaxies. Such galaxies are shown by the green points in Figure 2. These compact SFGs exhibit smaller sizes and higher Sérsic indices than typical SFGs, suggesting that they have indeed experienced a substantial structural transformation.

In the following, we refer to any evolutionary paths that move upwards from the ${{\rm{\Sigma }}}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ relation due to (wet or dry) phase(s) of high-efficiency core growth as compaction tracks. We emphasize that compaction is, primarily, a core building process, and thus it is more efficient at increasing ${{\rm{\Sigma }}}_{1}$ than ${{\rm{\Sigma }}}_{{\rm{e}}}$, as the latter also depends on the overall galaxy size. This is illustrated by the green points in Figure 2 which have uniformly dense cores but vary greatly in ${{\rm{\Sigma }}}_{{\rm{e}}}$. We further discuss this difference in Section 5.2.

In order to provide further support for the structural evolutionary paths discussed above and to obtain some insight on the physical mechanism(s) that could motivate them, we study the evolutionary paths of a sample of high-resolution hydrodynamical galaxy simulations in the same parameter space. Figure 5 shows the evolutionary tracks in ${{\rm{\Sigma }}}_{1}$ and ${{\rm{\Sigma }}}_{{\rm{e}}}$ versus mass for the Vela set of 35 simulations described in Ceverino et al. (2014) and Zolotov et al. (2015). In the latter, the authors analyzed the SFR, structure, and kinematics of the simulated SFGs, showing that the driving force behind the most significant changes in all these properties is a phase (or phases) of gas-rich (wet) compaction caused by, e.g., mergers, disk instabilities, interactions, counter-rotating streams, or tidal compressions, which trigger intense episodes of gas inflow and SFR, as mentioned in the previous section.

Following the approach in Zolotov et al. (2015), we divide the sample into two groups according to their mass (in Figure 5, high mass is in black, low mass is in gray), and we show their median evolutionary tracks for the two groups at early (dashed; $z\geqslant 2.5$) and late (solid; $z\lt 2.5$) times. The late evolution roughly matches the redshift range of the observed galaxy sample, shown as the blue and red contours. Qualitatively, the tracks have a similar behavior in both panels and match the observed distributions rather well. To first order, the evolutionary paths can be broken down into two phases: an early phase of steady structural growth approximately following the slope of the ${{\rm{\Sigma }}}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ relations ($\alpha \sim 0.6$, $\beta \sim 0.9$) and a compaction phase, which causes a steep increase in ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}(\alpha \sim 1.5$, $\beta \sim 1.3$). The steep increase in ${{\rm{\Sigma }}}_{{\rm{e}}}$ implies an actual size shrinkage, particularly in the high-mass simulations (see also Figure 9 in Zolotov et al. 2015). Overall, the dependence of ${{\rm{\Sigma }}}_{{\rm{e}}}$ on r_e, which can change rapidly due to gas accretion and/or interactions, implies a larger spread in ${{\rm{\Sigma }}}_{{\rm{e}}}$ with respect to ${{\rm{\Sigma }}}_{1}$ for a given stellar mass.

The main differences between the black and gray tracks are as follows: (1) a downsizing effect, i.e., the most massive galaxies evolve earlier and faster, (2) the evolutionary tracks of massive galaxies have a higher zero point, i.e., are denser at fixed stellar mass, and (3) the main compaction event is stronger and happens earlier for the most massive galaxies. The most natural explanation of the higher densities of the more massive galaxies is that they evolve faster and reach a given mass when the universe was denser. The fast growth (i.e., higher SFRs) is also a natural consequence of the higher (gas) densities (Dekel et al. 2013; Zolotov et al. 2015). Hence, at the risk of some oversimplification, the manifold of model galaxies is a two-parameter family given by the mass and time of formation.

The star-forming locus (blue contour) consists mostly of lower-mass galaxies, which arrive on the high-mass region only at lower redshifts ($z\lt 2.5$, gray lines). These galaxies experience weaker compaction events, after which the core becomes quiescent due to gas exhaustion. However, the quenching is only temporary and, when the gas inflow reaches the center of the galaxy, the core becomes star forming again, and the galaxy continues to grow along the ${\rm{\Sigma }}\mbox{--}{M}_{\star }$ relations. The combination of a compaction phase followed by a quiescent-core phase can increase or decrease the zero point of the ${\rm{\Sigma }}\mbox{--}{M}_{\star }$ relation on which the galaxy evolves. The low-mass simulations can experience further compaction events at a later time, which would cause them to arrive on the quiescent ${{\rm{\Sigma }}}_{1}$ relation from below. This is illustrated by the green track of Vela13. This galaxy has the latest compaction event and thus spends most of its life along the star-forming ${\rm{\Sigma }}\mbox{--}{M}_{\star }$ relations. Interestingly, during this phase, the gas mass in the central 1 kpc remains almost constant, as expected in the case of a simple "bath-tub" model evolution (e.g., Bouché et al. 2010; Krumholz & Dekel 2012; Dekel et al. 2013). This "bath-tub" phase strengthens the notion that the growth along the ${\rm{\Sigma }}\mbox{--}{M}_{\star }$ relations describes a phase of smooth, steady-state evolution that coincides with the SFR main sequence.

The quiescent locus (red contour) consists, almost exclusively, of massive galaxies that experienced a strong compaction event at $z\gtrsim 3$, reaching the (high density) quiescent region in just a few hundred megayears ($z\sim 2.5;$ black lines in Figure 5). The evolution of these tracks at lower redshift, however, differs substantially from that of fully quenched galaxies, which are expected to grow in mass only by merging. These massive simulations have quenched cores, but they still exhibit star formation (and therefore mass growth) in a re-grown disk (Zolotov et al. 2015; Tacchella et al. 2016). The newly formed disk causes a sudden increase in size, which leads to decreasing tracks with mass in ${{\rm{\Sigma }}}_{{\rm{e}}}$, but nearly flat tracks in ${{\rm{\Sigma }}}_{1}$ (see also Figure 19 of Zolotov et al. 2015). Thus, the low-z tracks (solid black lines) do not strictly apply to fully quenched galaxies, but they may reflect the evolution of some compact SFGs (green points in Figure 2), which rather than quenching right away sustain low levels of star formation in an extended disk.

Based on the encouraging agreement between the models and data, we suggest that the overall long-term structural evolution of SFGs can be expressed in terms of two phases that follow approximately linear tracks in $\mathrm{log}{{\rm{\Sigma }}}_{1,{\rm{e}}}\propto [\alpha ,\beta ]\mathrm{log}\,{M}_{\star }$. First, the evolution along the ${{\rm{\Sigma }}}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ relations ($\alpha \sim 0.6$, $\beta \sim 0.9$), which is a relatively smooth phase of structural growth, is consistent with a period of continuous gas accretion and galactic-scale star formation. As a result, both central and effective densities increase with time, as does the overall galaxy radius, i.e., similar to the growth of an exponential disk. Compaction is a phase (or phases) of enhanced core growth ($\alpha \gt 1$, $\beta \gt 1$) relative to the ${\rm{\Sigma }}\mbox{--}{M}_{\star }$ relations. At high z, compaction is a wet process, fueled by strong gas infall to the galaxy center, as a result of gravitational instabilities and other processes. As a result, the central and effective densities increase steeply, and the galaxy's effective radius can decrease due to the large increase in stellar mass close to the center, or due to a structural collapse.

Overall, the intensity and duration of compaction events decreases with time due to the similarly decreasing gas fractions. This leads to the conclusion that at late times, when galaxies are less gas-rich, compaction processes must also include minor mergers and/or secular evolution that operate on longer dynamical timescales (e.g., Kormendy & Kennicutt 2004). The main idea is that even a largely dry compaction process can put some gas at the centers at any time until full quenching (e.g., Dutton et al. 2011; Wuyts et al. 2012; Forbes et al. 2014) so there is always star formation in the core of SFGs. Note also that there can always be a relative increase in core mass relative to total mass (compaction) from a global decline of the total mass growth (due to lower star formation in the disk) rather than an acceleration of the core growth.

As discussed in the previous section, the higher normalization of the quiescent structural relations with respect to those of SFGs suggests that quenching is preceded by an increase in the surface density above a certain threshold. However, as shown in Figure 2, such characteristic density scales with stellar mass, challenging the simple notion of a fixed quenching threshold at a given surface density or stellar mass (e.g., Kauffmann et al. 2003; Franx et al. 2008). In turn, the most effective quiescent criterion is a selection with respect to the ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ or ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ sequences with stellar mass, i.e., a relative offset from the structural relations. Such relative selection includes fewer SFGs. However, there is always overlap with quiescent galaxies at every redshift. A possible interpretation, outlined in the previous section, is that those overlapping SFGs acquire quiescent morphologies while they are still star-forming as a result of a compaction process, i.e., a structural transformation that increases the central density and Sérsic index, and possibly reduces the size before quenching star formation. Such an evolutionary sequence was confirmed in Barro et al. (2013, hereafter B13; Barro et al. 2014a) for SFGs at $z\sim 2$. In B13, the authors used a selection on specific SFR and relative distance to the quiescent size–mass relation to identify compact SFGs at $z\sim 2$, finding that those galaxies have similar sizes, Sérsic indices, and spheroidal morphologies to the quiescent population.

Building on this idea, in this work we define compact SFGs as those SFGs (${\rm{\Delta }}{\mathrm{SFR}}_{\mathrm{MS}}\gt -0.7$ dex) found within the $\sim 2\times$ the scatter of the quiescent structural scaling relations at a given redshift, ${\rm{\Delta }}{{\rm{\Sigma }}}_{{\rm{e}},1}^{{\rm{Q}}}\equiv \mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}},1}-\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}},1}^{{\rm{Q}}}(z)\gt -2\sigma (\mathrm{log}{{\rm{\Sigma }}}^{{\rm{Q}}})$, where ${{\rm{\Sigma }}}_{{\rm{e}},1}$ is either the central or effective mass density, and we use $2\sigma (\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}})\sim 0.6$ dex and $2\sigma (\mathrm{log}{{\rm{\Sigma }}}_{1}^{{\rm{Q}}})\sim 0.3$ dex, respectively. This definition differs from previous works where compact is an absolute term to identify the smallest galaxies at high z(${r}_{{\rm{e}}}\lesssim 1\,\mathrm{kpc};$ e.g., Cassata et al. 2011, 2013). Here, compact is a relative term referring to the densest/smallest SFGs at every redshift. Panels A and B of Figure 6 illustrate the selection of compact SFGs in ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ (dashed line). In the x-axis, we add the zero point at each redshift to illustrate the different time evolutions in the normalization of these relations. Panel A also shows the compactness threshold of B13 (purple line), which, by definition, matches ${\rm{\Delta }}{{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}\lt 0.5$ dex at $1.4\lt z\lt 2.2$. However, as the normalization of ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ declines with time, a fixed selection threshold obtains progressively fewer galaxies. In contrast, ${\rm{\Delta }}{\mathrm{SFR}}_{\mathrm{MS}}\mbox{--}{\rm{\Delta }}{{\rm{\Sigma }}}^{{\rm{Q}}}$ is essentially a redshift-independent extension of the method to identify star-forming progenitors of quiescent galaxies (see the next section).

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The relative distributions of SFGs and quiescent galaxies in both panels of Figure 6 are qualitatively very similar. By definition, both ${\rm{\Delta }}{{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ and ${\rm{\Delta }}{{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ select the majority of the quiescent galaxies within the typical scatter of the average quiescent structural relations. Nonetheless, there are some noticeable differences between them. First, selecting quiescent galaxies using ${\rm{\Delta }}{{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ leads to a few more catastrophic outliers, most likely due to extreme values of the Sérsic index also affecting the r_e (see, e.g., van der Wel et al. 2012). Second, the selection of compact SFGs using ${\rm{\Delta }}{{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ yields ∼10%–15% more compact SFGs at each redshift. These SFGs characterized as compact by ${{\rm{\Sigma }}}_{1}$ but scattered outside the ${{\rm{\Sigma }}}_{{\rm{e}}}$ selection threshold have dense centers similar to those of quiescent galaxies, but more extended profiles with ongoing star formation. In terms of an evolutionary sequence, one could speculate that these galaxies will eventually become fully quiescent once the extended star-forming disk fades (e.g., Fang et al. 2013; Tacchella et al. 2015). Some of these galaxies exhibit radial profiles that can be characterized by two Sérsic components (bulge+disk; e.g., Bruce et al. 2012; Lang et al. 2014), where the "bulge" is already quiescent. Thus, ${\rm{\Delta }}{{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ appears to be a more efficient criterion to account for all the SFGs that are most likely to join the quiescent population based on structural considerations.

The difference between Δ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ and Δ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ is also a reflection that effective quantities are prone to larger fluctuations due to the continuous accretion and disruption processes in galaxies that make the evolution of r_e not monotonic with time. On the contrary, ${{\rm{\Sigma }}}_{1}$ is more stable and therefore closer to a clock (it only increases). It also exhibits a tighter scaling relation with mass and a slower evolution with redshift of the normalization. As shown in panel B of Figure 6, the compactness threshold in $\Delta {{\rm{\Sigma }}}_{1}$, ${{\rm{\Sigma }}}_{1}-0.65(\mathrm{log}\,M* -10.5)\gt \mathrm{log}$B(z)-0.2, exhibits a small range between 9.6 and 9.3 M_⊙ kpc⁻², and thus a threshold of ∼9.5 M${}_{\odot }$ kpc⁻² identifies the majority of compact SFGs and quiescent galaxies at redshift $z\gtrsim 1$. Hereafter, we will refer to compact SFGs as those selected in terms of ${\rm{\Delta }}{{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$.

The remarkable similarity in the galaxy distributions of Figure 6 as a function of time suggest that the relative distance from the SFR–MS and ${{\rm{\Sigma }}}_{{\rm{e}},1}^{{\rm{Q}}}$ relations describes a common evolutionary sequence toward quiescence for massive galaxies, which is independent of redshift. Figure 7 illustrates this sequence showing the stacked distribution in ${\rm{\Delta }}{\mathrm{SFR}}_{\mathrm{MS}}$ versus ${\rm{\Delta }}{{\rm{\Sigma }}}^{{\rm{Q}}}$ for massive galaxies ($\mathrm{log}({M}_{\star }/{M}_{\odot })\gt 10.3$) in the redshift range $0.5\lt z\lt 3.0$. The upside-down, L-shaped sequence summarizes the notion that forming a dense core is a pre-requisite for quenching star formation (cf. Bell et al. 2012; Cheung et al. 2012; Fang et al. 2013; van Dokkum et al. 2014). The color code of the Sérsic index emphasizes that the evolution requires both the growth of a dense core and a structural transformation from an exponential (disk) profile to more of a concentrated ($n\gtrsim 2$) Sérsic profile. Incidentally, Figure 7 also illustrates why the Sérsic index is a better quiescent indicator than a constant threshold in ${{\rm{\Sigma }}}_{{\rm{e}}},{{\rm{\Sigma }}}_{1}$ or velocity dispersion, σ (e.g., Bell et al. 2012).

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In terms of the galaxy evolutionary paths discussed in Section 4, the nearly horizontal branch, which consists of main sequence SFGs, can be qualitatively described as the inside-out growth of an exponential disk ($n\sim 1$) due to in situ star formation. This growth increases the overall size of the galaxy as well as ${{\rm{\Sigma }}}_{1}$ and ${{\rm{\Sigma }}}_{{\rm{e}}}$. The slight tilt of this branch, ${\rm{\Delta }}{\mathrm{SFR}}_{\mathrm{MS}}\sim -0.20{\rm{\Delta }}{{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$, ${\rm{\Delta }}{\mathrm{SFR}}_{\mathrm{MS}}\sim -0.25{\rm{\Delta }}{{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ (gray lines), which follows an increase in the Sérsic index, suggests that the onset of quenching goes hand in hand with a structural transformation that increases the concentration of the galaxy. Following the discussion in the previous section, we refer to this transition phase at the knee of the L-shaped distribution as the compaction phase, as it involves a stronger increase in ${{\rm{\Sigma }}}_{1}$ and ${{\rm{\Sigma }}}_{{\rm{e}}}$ relative to the average evolution during the inside-out growth phase, which makes the mass profile more concentrated ($n\gtrsim 2$) and possibly leads to smaller ${r}_{{\rm{e}}}$. The vertical branch shows that quenching, defined as a hard threshold in sSFR (or ${\rm{\Delta }}{\mathrm{SFR}}_{\mathrm{MS}}\gt -0.7$ dex), takes place at maximum central density in compact SFGs that have already acquired similar morphologies to the quiescent population, i.e., compaction precedes the full shutdown of star formation, although it may start approximately at the same time as the quenching process.

Following the interpretation of B13, we characterize the evolution in the L-shape diagram as declining from fast to slow as a function of time. The gradient in quenching speed arises naturally from the global decline in sSFR with time (e.g., Speagle et al. 2014). Since ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{1,{\rm{e}}}\propto {\rm{\Delta }}\mathrm{log}M\propto \mathrm{log}(\mathrm{sSFR}({\rm{z}}))$, the central densities increase more slowly with time. The larger SFR at high z is associated with higher gas fractions (Tacconi et al. 2010; Tacconi et al. 2013), which are likely related with larger accretion rates from the dark matter halos (Dekel et al. 2013; Dekel & Burkert 2014). Simple stability arguments predict that such gas-rich galaxies are also prone to stronger gravitational instabilities, and thus favor more rapid compaction, and hence quenching, processes. In such case, the downward tilt in Figure 7 could be due to gas starvation after an instability-induced starburst, i.e., a wet inflow that causes a peak in the SFR at the maximum gas density and declines progressively with the gas supply, while the core mass increases (see, e.g., Feldmann & Mayer 2015; Zolotov et al. 2015 in simulations and Ikarashi et al. 2015 and Barro et al. 2016 for recent observations of compact nuclear starburst at $z\sim 2$). The truncation of the gas supply to the galaxy center could be caused by a combination of feedback (e.g., outflows) and stabilization of the disk due to the increasing central density (Martig et al. 2009).

At lower redshift, the smaller gas fractions and longer dynamical timescales suggest that compaction mechanisms would likely become a mixture of weaker instabilities, which can still cause enhanced gas inflows or inward migration of stellar clumps (e.g., Dekel et al. 2009b; Bournaud et al. 2011; Genel et al. 2012), and secular processes (Kormendy & Kennicutt 2004) associated with torques and dynamical friction in the presence of bars and spiral arms. The latter play an important role in increasing the central density in SFGs that already exhibit relatively quiescent centers, i.e., lacking enough star formation to sustain the core growth required to reach the ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ relation (Wuyts et al. 2012, 2013; Bruce et al. 2014a, 2014b; Lang et al. 2014). In those SFGs, quenching is also expected to be a slow process, related with gas consumption in the star-forming disk (fading; e.g., Fang et al. 2013; Tacchella et al. 2015). The slow quenching process also depends on additional mechanisms to prevent further gas accretion onto the disk, e.g., virial shock heating in massive halos (Croton et al. 2006; Dekel & Birnboim 2006) or AGN feedback. As mentioned in the previous section, some of these SFGs are compact in ${{\rm{\Sigma }}}_{1}$ but not in ${{\rm{\Sigma }}}_{{\rm{e}}}$. Therefore, the tilt in ${\rm{\Delta }}{\mathrm{SFR}}_{\mathrm{MS}}$ versus Δ${{\rm{\Sigma }}}_{{\rm{e}}}^{{\rm{Q}}}$ can be caused by the size shrinkage (r_e) associated with fading rather than with an increase in their core mass (see, e.g., Feng et al. 2013).

The structural similarities between compact SFGs and quiescent galaxies and the increasing number density of the latter as a function of time supports the idea that the L-shaped diagram is indeed an evolutionary sequence, i.e., compact SFGs in the knee of the relation are immediate progenitors of the quiescent galaxies at lower redshifts. Figure 8 shows the evolution in the number density of compact SFGs, selected using Δ${{\rm{\Sigma }}}_{1}^{{\rm{Q}}}$ $\gt \,-0.2$, and all quiescent galaxies, regardless of their structural properties, since $z\sim 3$. The latter grows approximately as a power law, ${N}^{{\rm{Q}}}\sim {10}^{-0.5(1+z)}$, in good agreement with previous results (e.g., Muzzin et al. 2013). Assuming that quiescent galaxies are descendants of compact SFGs at higher redshift, the number of quiescent galaxies is the cumulative distribution of quenching compact SFGs as a function of time. To quantify the relative numbers of these two populations, we assume that the number density of compact SFGs also follows a power-law evolution as the result of two opposite processes that cause either an increase (due to compaction) or decrease (due to quenching) in their total number: ${{dN}}_{\mathrm{SFG}}/{dz}={\lambda }_{{\rm{C}}}{N}_{\mathrm{SFG}}+{\lambda }_{{\rm{Q}}}{N}_{\mathrm{SFG}}$. In this model, the evolution in the number of quiescent galaxies is proportional to the number of quenching SFGs and thus inherits the power-law dependence: ${{dN}}^{{\rm{Q}}}/{dz}=-{\lambda }_{d}{N}_{\mathrm{SFG}}\sim -{\lambda }_{{\rm{Q}}}{10}^{({\lambda }_{{\rm{C}}}+{\lambda }_{{\rm{Q}}})(1+z)}$. There are two additional parameters that determine the initial number of compact SFGs and quiescent galaxies, ${N}_{z0}^{\mathrm{SFG}}$ and ${N}_{z0}^{{\rm{Q}}}$. We set ${N}_{z0}^{{\rm{Q}}}=0$ at $z0=4$, and we fit the other three parameters to the observed number density of compact SFGs and quiescent galaxies, which yields ${\lambda }_{i}=1.27\pm 0.15$, ${\lambda }_{d}=-0.75\pm 0.12,$ and ${N}_{z0}^{\mathrm{SFG}}=5\cdot {10}^{-6}$ Mpc⁻³. Thus, the characteristic quenching timescale for a compact SFG in units of redshift is ${z}_{1/2,Q}=\mathrm{ln}(10)/\mathrm{ln}(2)/{\lambda }_{{\rm{Q}}}=0.38$, which implies that the quenching time increases from ${t}_{{\rm{Q}}}=700\,\mathrm{Myr}$ to 1.1 Gyr at z = 3 and z = 1. In spite of its simplicity, this model provides a more realistic approximation than previous assumptions that all compact SFGs form and quench in discrete intervals of time. In this case, the number densities vary continuously while preserving the evolutionary connection as a characteristic quenching time.

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Although the L-shape of the ΔSFR$-{\rm{\Delta }}{{\rm{\Sigma }}}^{{\rm{Q}}}$ sequence is nearly independent of redshift, the evolution in the zero points of the SFR–MS and the structural ${{\rm{\Sigma }}}^{{\rm{Q}}}\mbox{--}{M}_{\star }$ relations, from which the relative distances are computed, implies that both the star formation and the structural and morphological properties of the galaxies in a given point of the sequence vary with redshift. To illustrate this change in structural properties and visual appearances within the L-shaped evolutionary path, Figure 9 shows images of galaxies in the three regions of the L-sequence at different redshifts and with increasing stellar masses as a function of time following a typical mass growth track. For this particular example, we adopt a smaller mass growth factor of $\sim 6\times$ (compared to $\sim 20\times$ in Section 3.1) because the initial progenitors are more massive, $\mathrm{log}({M}_{\star }/{M}_{\odot })=10.5$ at z = 2.75 (see, e.g., the predicted mass tracks in Wellons et al. 2016).

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The upper panel of Figure 9 indicates the average values of the Sérsic index and effective radius for the galaxies in each of the three groups (blue, green, and red) at a given redshift snapshot. The bottom panel illustrates the possible evolutionary sequences within the L-shaped diagram following the mass-growth track. The blue arrow illustrates a possible track for "extended" SFGs that remain in the SFR–MS and the ${{\rm{\Sigma }}}^{\mathrm{SF}}\mbox{--}{M}_{\star }$ relations from redshift z = 3 to 0.5. These galaxies increase both their sizes and central densities with time and the change in their visual morphologies is consistent with the notion of an inside-out growing disk ($n\lesssim 2$), starting from smaller, more disturbed, and clumpier appearances at $z\gtrsim 2$ to progressively acquire more settled disk-like structures and to grow larger cores with redder colors at $z\lesssim 1$ (cf. Bruce et al. 2012; Wuyts et al. 2012; Lang et al. 2014; Tacchella et al. 2015).

The green arrows illustrate possible compaction tracks for galaxies around the knee of the L-shaped diagram. This evolutionary track implies a larger increase of the central densities, with respect to the evolution of "extended" SFGs in the blue line, for the same mass growth. The more massive cores in the compact SFGs are visually obvious. With time, the morphologies of galaxies in the compact phase change from pure spheroids to more disk-like appearances. The large central density dominates the Sérsic index, which is high at all redshifts ($n\gtrsim 3$), but the diffuse extended components visible at lower redshifts lead to an average size increase with time. This is consistent with the idea that compact SFGs are formed from extended SFGs, which are mostly disk dominated at low z, by increasing their central densities. The evolution toward disk-like morphologies with progressively bigger cores is consistent with the prominence of disks with bulges among massive SFGs at $z\gtrsim 1$, as reported in previous works (e.g., Bruce et al. 2012, 2014a; Buitrago et al. 2013; McLure et al. 2013; Mortlock et al. 2013; Lang et al. 2014; Huertas-Company et al. 2015). Note that rejuvenation and/or the re-growth of a star-forming disk in compact SFGs is also a possibility, although the increasing number of quiescent galaxies with time suggests that the net flow of galaxies is preferentially toward compaction and quenching.

Qualitatively, the morphologies of compact SFGs are almost identical to those of quiescent galaxies at the same redshift, which strengthens the notion that they are their immediate progenitors and that full quenching takes place at maximum surface density in SFGs that have acquired quiescent morphology (red arrows). Extended and compact SFGs also exhibit qualitatively similar morphologies at $z\lesssim 1.5$, showing increasingly larger cores (bulges) along the green compaction arrow. However, at higher redshifts, the formation of compact SFGs, with almost pure spheroidal morphologies, from irregular, clumpy, extended SFGs suggests the action of strongly dissipative processes that redistribute the gas reservoir toward the galaxy center, removing it from the extended irregular component and forming a dense core. An alternative formation scenario could be that compact SFGs follow a much faster mass-growth track (e.g., by a factor of $\sim 5\times$ from $z\sim 3$ to 2), evolving rapidly from small, low-mass SFGs (e.g., William et al. 2014; Wellons et al. 2016) that typically show smooth, spheroid-like morphologies, although their Sérsic indices are disk-like.

Lastly, we discuss the possible evolutionary path of the quiescent population (bottom row, red squares). The images in Figure 9 illustrate how quiescent galaxies are progressively increasing their size as a function of time. This increase becomes more evident at $z\lt 1.75$, which is the approximate epoch where the number density of "naked" compact SFGs experiences a steep decline (Barro et al. 2013; van Dokkum et al. 2015). Thus, an intuitive notion to explain the morphological evolution of the quiescent population is that the disappearance of "naked" star-forming cores leads to newly quenched galaxies coming from larger (more extended) progenitors (diagonal downward-pointing red arrow). Simultaneously, the already existing quenched galaxies may also grow in size (horizontal red arrow) either by minor merging¹³ or perhaps by accreting stars (and/or gas) that could rebuild a disk, which in some cases could host some star formation (diagonal up, not shown), and eventually lead to the formation of early-type (passive) disks (e.g., Bundy et al. 2010; Margalef-Bentabol et al. 2016), which nonetheless maintain the same stellar density within the central 1 kpc. The disk regrowth scenario could explain the apparent decline in the number of the most compact quiescent galaxies (defined as having small sizes of ${r}_{{\rm{e}}}\lt 1-2$ kpc) with time (e.g., Cassata et al. 2013; Damjanov et al. 2014, 2015; Trujillo et al. 2014; Ferré-Mateu et al. 2017) since they would be hidden as the bulges of today's massive early-type disks (Driver et al. 2013; Graham & Scott 2013; Graham et al. 2015).

In Section 3.3, we studied the change in the ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ relations as a result of evolutionary processes that cause a relative change in the effective radius and stellar mass of the galaxy, ${\rm{\Delta }}\mathrm{log}{r}_{{\rm{e}}}=\eta {\rm{\Delta }}\mathrm{log}\,M$. As shown above, the variations in ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ as a result of a change in stellar mass and r_e are

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{log}\,{{\rm{\Sigma }}}_{{\rm{e}}}={\rm{\Delta }}\mathrm{log}\,{M}_{\star }-2{\rm{\Delta }}\mathrm{log}{r}_{{\rm{e}}}=(1-2\eta ){\rm{\Delta }}\mathrm{log}\,{M}_{\star }\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{log}\,{{\rm{\Sigma }}}_{1}\sim {\rm{\Delta }}\mathrm{log}\,{M}_{\star }-{c}_{1}{\rm{\Delta }}\mathrm{log}{r}_{{\rm{e}}}=(1-{c}_{1}\eta ){\rm{\Delta }}\mathrm{log}\,{M}_{\star }.\end{eqnarray} \tag{ 21 }$$

The coefficients on the right-hand side terms can be interpreted as the slope of the evolutionary tracks in ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ versus mass as a result of the process η. In particular, for minor mergers ($\eta =1.6$), we would obtain ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}}=-2.2{\rm{\Delta }}\mathrm{log}\,M$ and ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{1}=0$ (assuming that the mergers do not change the central mass), while for adiabatic expansion we obtain ($\eta =-1$), ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{{\rm{e}}}=3{\rm{\Delta }}\mathrm{log}\,M$ and ${\rm{\Delta }}\mathrm{log}{{\rm{\Sigma }}}_{1}=1.6{\rm{\Delta }}\mathrm{log}\,M$ (assuming n = 4). Note that these relations apply to both SFGs and quiescent galaxies. However, SFGs also have ongoing star formation, which would also contribute to the change in size and mass.

Furthermore, if the evolutionary processes affect equally all galaxies lying in the ${{\rm{\Sigma }}}_{{\rm{e}}}$ and ${{\rm{\Sigma }}}_{1}$ scaling relations, the slope of those relations is preserved in the evolution, and the change in the zero points of the relations in terms of the stellar mass change is

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{log}\,A\\ &&\,={\rm{\Delta }}\mathrm{log}\,{{\rm{\Sigma }}}_{{\rm{e}}}-\alpha {\rm{\Delta }}\mathrm{log}\,{M}_{\star }=(1-2\eta -\alpha ){\rm{\Delta }}\mathrm{log}\,{M}_{\star }\end{eqnarray} \tag{ 22 }$$

$$\begin{eqnarray}&&{\rm{\Delta }}\mathrm{log}\,B\\ &&\,={\rm{\Delta }}\mathrm{log}\,{{\rm{\Sigma }}}_{1}-\beta {\rm{\Delta }}\mathrm{log}\,{M}_{\star }\sim (1-{c}_{1}\eta -\beta ){\rm{\Delta }}\mathrm{log}\,{M}_{\star }.\end{eqnarray} \tag{ 23 }$$

The equations above depend on the slope of the scaling relations. In Section 3.3, we discussed in particular the effect on the quiescent relations ($\alpha =-0.5$, $\beta =0.7$). For a minor merger scenario ($\eta =1.6$), we obtain ${\rm{\Delta }}\mathrm{log}\,A=-1.7{\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ and ${\rm{\Delta }}\mathrm{log}\,B=-0.7{\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ (assuming again that the mergers do not change the central mass). For adiabatic expansion ($\eta =-1$), we obtain ${\rm{\Delta }}\mathrm{log}\,A=3.5{\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ and ${\rm{\Delta }}\mathrm{log}\,B=0.9{\rm{\Delta }}\mathrm{log}\,{M}_{\star }$ (assuming n = 4).