Resolving Quiescent Galaxies at z ≳ 2. II. Direct Measures of Rotational Support

The NIR spectroscopic data and their reduction were described in Paper I. Briefly, we used the FIRE echellette spectrograph (Simcoe et al. 2013) on the Magellan Baade telescope to observe MRG-M0150, MRG-P0918, and MRG-M2129 over the full NIR wavelength range. We also used MOSFIRE (McLean et al. 2012) at the Keck I telescope to observe MRG-M0138 Image 1 in the J and H bands and MRG-M0150 Image 1 in the H band. The mean seeing for each target ranged from 042 to 079, with a median of 057 (see Paper I, Table 3). We note that FIRE observations were also undertaken for MRG-M0138 Image 2, as mentioned Paper I, but since the slit orientation is close to the minor axis for this image, it is not well suited to measure rotation; we rely on the MOSFIRE observations of Image 1 instead.

For each lensed galaxy, we extracted spectra in a series of bins along the slit. The orientations of the slits are shown in Figure 1. (The slit was slightly misaligned for MRG-P0918, since the orientation of the image was initially estimated from a shallower ground-based image; we will discuss the small effects of this misalignment below.) The total spatial extent of the bins was determined by examining the intensity profile along the slit and selecting the region that exceeded ≃15% of the peak. Within this region, the size of the bins was approximately matched to the mean seeing during the observation, although in some cases the outer bins were enlarged to increase the signal-to-noise ratio (S/N). This procedure produced spectra with S/N ≈ 10–80 per 300 km s⁻¹ (approximately one velocity dispersion element), which is adequate to measure velocities and, in most of the bins, velocity dispersions.

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We used the spectral range from λ_rest = 3600 Å to the end of the H band for the kinematic measurements. We omitted the K-band spectra, since they do not contain strong absorption features besides Hα, which is contaminated by emission. A mask was created for each spectrum to exclude spectral regions that (1) contain strong telluric absorption bands, (2) contain emission lines, or (3) deviate from the stellar population model in ways that are likely to arise from data reduction problems (e.g., a small region blueward of Hβ for MRG-M2129 and the 4300 Å region of MRG-M0138 where residual telluric absorption is evident). A small number of outlier pixels were identified via a σ-clipping algorithm and masked.

In the case of MRG-M0138, the MOSFIRE J-band spectrum begins around the Ca K line, so the continuum blueward of 4000 Å is not well constrained. For this galaxy, we omitted the short spectral region blueward of 4000 Å. We also masked Mg b and Na D, since these lines show clearly nonsolar abundances.

In each bin, we used ppxf (Cappellari & Emsellem 2004) to measure the velocity V and velocity dispersion σ. By default, the template spectrum was constructed by ppxf as a linear combination of Vazdekis et al. (2015) simple stellar populations with ages ranging up to the age of the universe at the observed epoch and solar-scaled abundances with metallicities ranging from approximately solar to twice solar. This template was broadened by a Gaussian line-of-sight velocity distribution, redshifted, multiplied by a linear polynomial, and added to a polynomial of degree N to best fit the observed spectrum. By default, we set N ≈ Δλ/(200 Å), where Δλ is the rest-frame length of the portion of the spectrum used in the fit. Uncertainties were derived by shuffling the residuals in chunks to maintain correlations, adding these to the best-fit model, refitting a large number of such realizations, and measuring the scatter in V and σ. The resulting uncertainty estimates were moderately larger than the formal uncertainties derived from the χ² surface. Figure 2 shows the spatially resolved spectra and the model fits used to extract kinematics (see Newman et al. 2015 for MRG-M0150). Table 1 lists the stellar kinematic measurements.

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We then varied our procedure in several ways in order to estimate the systematic uncertainties and assess the robustness of the measurements. First, we used solar-metallicity Bruzual & Charlot (2003, hereafter BC03) models instead of the Vazdekis et al. (2015) grid. Second, we fit Flexible Stellar Population Synthesis (FSPS; Conroy et al. 2009; Conroy & Gunn 2010) models using the pyspecfit (Newman et al. 2014) code instead of ppxf, as described in Paper I. Third, we varied the additive polynomial order by ±2. Fourth, we masked the Balmer lines, since these are sensitive to the star formation history and to the rotational velocities of the library stars.

We found that the σ measurements in the outer bins of MRG-P0918 were unstable to these changes and so omitted them from our analysis. Newman et al. (2015) determined that only an integrated velocity dispersion could be robustly measured for MRG-M0150, and we adopt the σ measured in that paper within a ±06 aperture. The resolved velocity dispersions for MRG-M0138 and MRG-M2129 were stable in all of the spatial bins.

For the stable bins, we found that varying the measurement procedure has a minimal effect of ≲3% on the relative velocities, which are very robust. The main effect is to induce systematic shifts in σ. We quantified these as $\langle {\rm{\Delta }}\sigma /{\sigma }_{\mathrm{def}}\rangle ;$ here the mean is taken over all spatial bins within a galaxy, and σ_def is the measurement derived with the default procedure. The largest shifts and their origins were $\langle {\rm{\Delta }}\sigma /{\sigma }_{\mathrm{def}}\rangle =-5 \%$ for MRG-M0138 (BC03 templates), −11% for MRG-P0918 (BC03 templates), and +7% for MRG-M2129 (masking Balmer lines, or FSPS templates). For reference, Newman et al. (2015) found a systematic uncertainty of 14% for MRG-M0150. We will treat the absolute values of these shifts as perfectly correlated fractional systematic uncertainties within each galaxy when we perform dynamical modeling in Section 3. As expected, the spectra most dominated by Balmer absorption (MRG-M0150 and MRG-P0918) have the largest systematic uncertainties in σ, whereas the oldest system (MRG-M0138) is the most robust.

The resolved stellar kinematics are plotted in Figure 3. All of the galaxies are clearly rotating. Figure 4 shows the position of the slit in the source plane. For MRG-M0138, MRG-M0150, and MRG-M2129, the slit is approximately aligned along the major axis; this is by design for the multiply imaged cases (MRG-M0138 and MRG-M0150) and is fortuitous for MRG-M2129. (Recall that we lack a lens model for the fourth system, MRG-P0918.) Therefore, we expect the long-slit kinematics to capture the majority of the projected rotation, but because of seeing and the width of the slit in the source plane, these data cannot be directly interpreted as major-axis kinematics. That requires modeling of the velocity field (Section 3).

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From the raw velocity measurements, we can estimate only lower limits on the rotation speed, but we can do so in a model-independent way as half of the velocity difference between the outermost bins: ${V}_{\max }^{\mathrm{proj}}\gt 183\pm 16$ km s⁻¹ for MRG-M0138 Image 1, ${V}_{\max }^{\mathrm{proj}}\gt 155\pm 32$ km s⁻¹ for MRG-M0150, ${V}_{\max }^{\mathrm{proj}}\,\gt 72\pm 21$ km s⁻¹ for MRG-P0918, and ${V}_{\max }^{\mathrm{proj}}\gt 218\pm 15$ km s⁻¹ for MRG-M2129. The notation emphasizes that these are projected velocities, i.e., $V\sin i$, that are not corrected for inclination.

The position of the FIRE slit was slightly misaligned with the MRG-P0918 image. Likewise, MRG-M0138 Image 1 is curved and so is unavoidably miscentered in the slit for most of its length. This will lead to spurious shifts in the measured velocities. For each of these galaxies, we calculated the miscentering of the flux in the slit for each spatial bin using the HST image, and we converted these to velocity shifts based on the spectrograph parameters. The velocity shifts are ≲6 km s⁻¹ in all cases. Since these are much smaller than the measurement uncertainties, they will be neglected for the remainder of the paper.

In Paper I, we estimated the uncertainties in the source structural parameters via two methods for the multiply imaged galaxies MRG-M0138 and MRG-M0150: the image-to-image scatter (which addresses the internal consistency of the lens model) and a magnification uncertainty (which addresses systematic uncertainties in the lens model). So far, we have propagated the former component via the Gaussian priors described above, but we have not propagated the latter. This is more difficult because uncertainties in the lens model affect both the source structure and the mapping from the image-plane kinematics to the source plane. As a simple first-order test of the effect of changing the magnification by a factor x, we scaled the source luminosity and R_e by x⁻¹ and x^−1/2, respectively, and isotropically dilated the source-plane coordinates by a factor x^−1/2. The only effect of changing the magnification by the uncertainties estimated in Paper I was to shift $\mathrm{log}{M}_{\mathrm{dyn}}$ by 0.06 dex for MRG-M0138 and MRG-M0150. This is smaller than the fractional magnification uncertainty, which is expected since M_dyn ∝ σ² R_e ∝ μ^−1/2 to a first approximation. We have therefore added 0.06 dex in quadrature to the uncertainty in $\mathrm{log}{M}_{\mathrm{dyn}}$ for these systems and report these enlarged errors in Table 2. For MRG-M2129, the magnification uncertainty is sufficiently small (10%) that it does not contribute significantly to the error budget.

The velocity and velocity dispersion fields are completely specified by the parameters described above. However, it is also useful to express the results in terms of other derived parameters. We calculated three of these.

• 1.  
  The maximum projected velocity on the major axis is ${V}_{\max }^{\mathrm{proj}}$. The deprojected rotation velocity is ${V}_{\max }={V}_{\max }^{\mathrm{proj}}/\sin i$.
• 2.  
  To compare to local early-type galaxies, we calculated ${\lambda }_{{R}_{e}}=\langle {RV}\rangle /\langle R\sqrt{{\sigma }^{2}+{V}^{2}}\rangle$, where the averaging is weighted by flux and restricted within R_e. This parameter is a proxy for the projected specific angular momentum; it was defined by Emsellem et al. (2007, 2011) and applied to the SAURON and ATLAS^3D surveys.
• 3.  
  To compare to disk galaxies, we define ${(V/\sigma )}_{{R}_{e}}$. Disk galaxy kinematics, particularly at high redshifts, are often quantified by the ratio V/σ, where σ is assumed to be uniform and isotropic throughout the disk. Since σ varies with radius in our JAM models, we define a characteristic value ${(V/\sigma )}_{{R}_{e}}$, where $V={V}^{\mathrm{proj}}({R}_{e})/\sin i$ is deprojected, $\sigma ={\sigma }^{\mathrm{proj}}({R}_{e})$, and both quantities are evaluated on the major axis. We find that V is nearly maximal at R_e in our sample, so the main effect of this choice is to pin down a fiducial radius at which to measure σ. We note that this definition is intended to compare to disk kinematics and differs from the conventional use of the central ${\sigma }_{0}$ in local early-type galaxies and from the metric used by Newman et al. (2015).

Constraints on these parameters are listed in Table 2. The deprojected quantities ${V}_{\max }$ and ${(V/\sigma )}_{{R}_{e}}$ rely on further assumptions that we will describe in Section 4.

Toft et al. (2017) analyzed an X-Shooter spectrum of MRG-M2129 and inferred an extremely rapid rotation speed of ${532}_{-49}^{+67}$ km s⁻¹, which is 65% higher than our less extreme value of 323 ± 28 km s⁻¹. To investigate the origin of this difference, in Figure 6, we compare the observed kinematics. The Toft et al. velocities are ≃30% higher in the outer bins, although they are still consistent within the uncertainties. Our dynamical models differ from those used by Toft et al. in several respects, but the most important is the inclination angle. As discussed in the Appendix of Paper I, Toft et al. found a rounder source with an intermediate inclination of i = 54°, whereas our reconstructed source has a significantly higher projected ellipticity and so must have i ≈ 90°.⁶ After a discrepancy in their treatment of the PSF was corrected, Toft et al. found a higher ellipticity that now is lower than ours by only Δe = 0.1 (S. Toft et al. 2018, private communication). The difference between our inclination and the value published by Toft et al. (2017) translates to a factor of $\sin 54^\circ /\sin 90^\circ =1.24$ in the deprojected rotation speed. Thus, about half of the total difference in deprojected rotation speed is attributable to differences in the kinematic measurements, while the other half arises from the inclination.

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The other important kinematic parameters are M_dyn and V/σ. We agree with Toft et al. on the total dynamical mass to 0.1 dex. Toft et al. found ${V}_{\max }/\sigma \gt 3.3$ at 97.5% confidence. This parameter is not uniquely defined in our JAM models, since σ is not constant. However, at R_e, we find a lower value of ${(V/\sigma )}_{{R}_{e}}=1.74\pm 0.50$ (see Table 1 and the radial variation in Figure 4). The cause of this difference can be seen in Figure 6: we measure $\sigma \approx 200$ km s⁻¹ across the image, in contrast to Toft et al., who found much lower σ in the outer bins. The Toft et al. spectrum is shallower, and the derived σ measurements have much larger uncertainties, particularly in the outer bins where Toft et al. only measured upper limits on σ. This might explain the difference. Based on their V/σ, Toft et al. claimed that MRG-M2129 is as rotationally supported as local late-type disk galaxies. Our lower value makes this less likely, although as we will show below, it is still much more rotationally supported than local early-type galaxies.

In general, determining the inclination of early-type galaxies is difficult because of their wide range of intrinsic shapes. For MRG-M0138 and MRG-M2129, the situation is unambiguous: since they have highly elliptical isophotes in projection, they must be intrinsically flat galaxies viewed nearly edge-on. MRG-M0150, however, is nearly round in projection (b/a = 0.87); based on photometry alone, we cannot tell whether it is nearly spherical or a face-on flattened system. Kinematic data can distinguish these possibilities.

Figure 7 shows the location of our lensed sample in the ellipticity versus ${\lambda }_{{R}_{e}}$ diagram. From integral field studies of local early-type galaxies, it has been found that the locus of "fast rotators" in this diagram can be described by simple models of single-component systems that span a range of intrinsic ellipticities and inclinations and have a velocity anisotropy β_z that is proportional to the intrinsic ellipticity (Emsellem et al. 2011).

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Under these assumptions, it is possible to estimate the shape and inclination of the galaxies in our sample by comparing them to the model grid in Figure 7. (We will compare our sample to the local early-type galaxies shown in the figure in Section 5.1.) We infer that MRG-M0150 is an intrinsically flat galaxy (${e}_{\mathrm{intr}}\approx 0.75$) viewed nearly face-on (i ≈ 30°). The factor $1/\sin i=2.0\pm 0.4$, leading to a deprojected rotation speed of ${V}_{\max }=352\pm 87$ km s⁻¹. For MRG-M0138 and MRG-M2129, we find i > 80° (see footnote 6) and can approximate $\sin i\approx 1$ so that ${V}_{\max }=290\pm 21$ and 323 ± 28 km s⁻¹, respectively. Although the relations describing β_z in local early-type galaxies might not hold in z > 2 systems, we found that the inferred inclination of MRG-M0150 does not change much even if we compute the model grid in Figure 7 under the very different assumption that β_z = 0. Alternatively, we can estimate i by selecting ATLAS^3D galaxies (Emsellem et al. 2011; Cappellari et al. 2013) whose position in Figure 7 is consistent with MRG-M0150. These have a median i = 38°, lower than the 57^◦ expected for randomly oriented disks and consistent with our model-based estimate.

Local early-type galaxies show a bimodal distribution of κ, with the "fast rotators" centered on 0.99 with a small rms of 0.07 (Cappellari 2016). MRG-M0138 and MRG-M2129 are consistent with κ = 1 within small uncertainties. They therefore have the simple dynamical structure that characterizes local fast rotators: the rotation speed follows from the galaxy shape and σ_ϕ = σ_R, i.e., the velocity ellipsoid is oblate, so the rotational motion is just sufficient to produce the flattened galaxy shape without excess azimuthal dispersion ${\sigma }_{\phi }\gt {\sigma }_{R}$. For MRG-M0150, we find κ = 1.59 ± 0.47 when marginalizing over inclination. Values of κ > 1 are physically possible but somewhat unnatural; they imply more rotation than is needed to produce the flattened galaxy shape, which must be compensated by low σ_ϕ. However, κ is covariant with inclination (Figure 5): if i ≈ 30°, as we argued above, then we find κ = 1.20 ± 0.30 for MRG-M0150, closer to unity. This provides additional evidence that the three lensed galaxies have a similar dynamical structure viewed at different angles. It also supports the validity of the lens models and source-plane reconstructions, since we would not expect κ = 1 if the ellipticity were seriously in error.

Although we cannot place MRG-P0918 in Figure 7, we can put a lower limit on ${\lambda }_{{R}_{e}}$. To so do, we derived a relation between ${\lambda }_{{R}_{e}}$ and ${V}_{\max }^{\mathrm{proj}}/{\sigma }_{e}$ within the ATLAS^3D sample (Emsellem et al. 2011), where σ_e is the integrated velocity dispersion within R_e. We then used the lower limit ${V}_{\max }^{\mathrm{proj}}\gt 72$ km s⁻¹ from Section 2.3 and approximated σ_e = 223 km s⁻¹ using the integrated velocity dispersion measured in Paper I. We find ${\lambda }_{{R}_{e}}\gtrsim 0.2$ for MRG-P0918. Although the division shown in Figure 7 formally allows for a galaxy in this ${\lambda }_{{R}_{e}}$ range to be classified as a "slow rotator" if its ellipticity is sufficiently high, there are no such galaxies in the entire ATLAS^3D sample. We therefore consider that MRG-P0918 can be classified as a "fast rotator."

In summary, MRG-M0138, MRG-M0150, and MRG-M2129 are likely to be intrinsically flat, disk-dominated galaxies (${e}_{\mathrm{intr}}\approx 0.75\mbox{--}0.85$) that are rotationally supported (${(V/\sigma )}_{{R}_{e}}=1.6\mbox{--}2.3$; Table 2) and have rapid rotation speeds of V_max = 290–352 km s⁻¹. Our lack of a lens model does not allow us to construct a dynamical model of MRG-P0918, but we can place a lower limit on its rotational support and classify this galaxy as a "fast rotator."

We find high dynamical masses for all three galaxies spanning the range $\mathrm{log}{M}_{\mathrm{dyn}}=11.26\mbox{--}11.64.$ The stellar-to-dynamical mass ratios are $\mathrm{log}{M}_{* }/{M}_{\mathrm{dyn}}=0.05\pm 0.20$ for MRG-M0138, 0.24 ± 0.21 for MRG-M0150, and −0.42 ± 0.13 for MRG-M2129. (These include the magnification uncertainty.) This range is similar to that obtained from unresolved kinematics of $z\sim 2$ quiescent galaxies (Belli et al. 2017).

MRG-M0138 and MRG-M0150 are consistent with equality between the dynamical and stellar masses, assuming a Chabrier (2003) initial mass function (IMF). The stellar mass of MRG-M2129 is the best constrained and is significantly lighter than the dynamical mass, leaving room for additional dark matter, gas, or a heavier IMF. Given the uncertainties, both MRG-M0138 and MRG-M2129 could also support a Salpeter IMF, which would be in tension with the dynamical mass of MRG-M0150. Future observations of high-z quiescent galaxies might be able to constrain the normalization of the IMF close to the epoch of star formation, when the stellar population is less polluted by mergers, but such comparisons will have to wait for larger samples with well-resolved kinematics.

Figure 7 compares the angular momentum parameter ${\lambda }_{{R}_{e}}$ of the three galaxies in our sample for which we have a lens model to a sample of local early-type galaxies drawn from the ATLAS^3D (Cappellari et al. 2011; Emsellem et al. 2011) and MASSIVE (Ma et al. 2014; Veale et al. 2017a) surveys. Galaxies with $\mathrm{log}M/{M}_{\odot }\gt 11.2$ are plotted; all of the high-z galaxies have M_dyn in this range.⁷

It is immediately apparent that the high-z galaxies have much more rotation than typical early-type galaxies in the local universe that are comparably massive. The rarity of local analogs of the galaxies in our sample allows us to make a strong inference about their future evolution. Among galaxies in the ATLAS^3D and MASSIVE samples that are at least as massive as MRG-M0150, 16% have a higher ${\lambda }_{{R}_{e}}$.⁸ This implies that MRG-M0150 is atypical among possible descendants in this parameter, but not extremely so. On the other hand, among galaxies in the local sample that are at least as massive as MRG-M0138 (79 galaxies) or MRG-M2129 (92 galaxies), none has a higher ${\lambda }_{{R}_{e}}$ or projected ellipticity (e = 0.74 and 0.71 for MRG-M0138 and MRG-M2129, respectively; see Paper I). Although these galaxies are both viewed nearly edge-on, this rarity is not primarily an inclination effect: the model grid in Figure 7 implies that they are also intrinsically flatter than all but one of the local sample.

The comoving number density of ${M}_{* }={10}^{11\mbox{--}11.5}\,{M}_{\odot }$ quiescent galaxies at z = 2 is ≃10%–20% of the value at z ∼ 0 (Moustakas et al. 2013; Tomczak et al. 2014). Therefore, unless MRG-M0138 and MRG-M2129 were very atypical of z ∼ 2 quiescent galaxies, we would expect to find a significant number of local analogs if ${\lambda }_{{R}_{e}}$ did not decline after quenching. We conclude that a significant fraction of massive quiescent galaxies at z ∼ 2 (the majority of our small sample) must decrease their specific angular momentum and become rounder.

By how much has ${\lambda }_{{R}_{e}}$ declined in such galaxies? Since they likely follow a wide range of evolutionary paths, we can only estimate a rough number by assuming that our sample is representative both of z ∼ 2 quiescent galaxies and of the progenitors of some z ∼ 0 comparison population. Simulations and empirical arguments based on number densities suggest that galaxies in the mass range of our sample have grown in mass since z ∼ 2 by 0.3 dex, on average, with a wide dispersion (van Dokkum et al. 2010; Muzzin et al. 2013; Wellons et al. 2015). We therefore selected a comparison sample of local galaxies with masses that are 0.3 dex higher than each lensed quiescent galaxy. (In practice, in order to have adequate statistics, we chose galaxies within ±0.15 dex of that mass.) For MRG-M0138, MRG-M0150, and MRG-M2129, the median ${\lambda }_{{R}_{e}}$ of these candidate descendants is lower by a factor of 14, 6, and 10, respectively. This result is not very sensitive to the definition of the local comparison sample; for example, we find lower ${\lambda }_{{R}_{e}}$ by factors of 10, 5, and 8, respectively, if we simply select local galaxies that are at least as massive as each high-z galaxy. We therefore estimate that massive quiescent galaxies at z ∼ 2 have declined in ${\lambda }_{{R}_{e}}$ by a typical factor of 5–10.

Mergers are the likely mechanism to accomplish such a large reduction in spin. In recent years, the emergence of local early-type galaxies and their dynamical properties has been investigated by several groups using cosmological simulations (Naab et al. 2014; Penoyre et al. 2017; Lagos et al. 2018a, 2018b). Although these works disagree on some aspects (e.g., the relative important of mass ratio versus gas fraction in setting the remnant spin), many of the broad trends are common. Slow rotators were born as fast rotators, and few slow rotators are expected to be present at z ≳ 1. Mergers are the primary cause of the loss of angular momentum over time that today's slow rotators experienced. Major mergers tend to reduce angular momentum suddenly, although rare configurations can instead "spin up" the remnant. However, a gradual transformation by minor mergers is also possible, with the total merged mass emerging as the most important parameter in some studies. Bournaud et al. (2007) presented simulations of a disk galaxy undergoing a series of minor mergers that double its mass and reduce V/σ from 3 to $\lesssim 0.2$. Similarly, Naab et al. (2014) discussed a subset of galaxies in cosmological simulations whose assembly histories are dominated by gas-poor minor mergers and whose angular momentum declined by a factor of $\simeq 10$ since z = 2 (their Figure 3, Class F), consistent with our estimates. Therefore, major mergers are probably important but not required to produce the observed spin-down. Finally, although mergers are a key factor shaping the spin history of most systems, Lagos et al. (2018a) found that 30% of z = 0 slow rotators in the Illustris simulation have had no significant mergers. They suggested that such galaxies instead inherited their low spin from their dark matter halos.

A rare subset of galaxies may have remained virtually untouched since z ∼ 2. Searches have uncovered a few massive galaxies with old stellar populations and compact sizes (similar to high-z quiescent galaxies) in the local universe, including NGC 1277 (van den Bosch et al. 2012; Trujillo et al. 2014), PGC 032873, and Mrk 1216 (Ferré-Mateu et al. 2017).⁹ Interestingly, these galaxies have projected rotation speeds of V ≃ 200–300 km s⁻¹ and projected V/σ ≃ 1–2. The similarity of their kinematics to the high-z galaxies in our sample supports the idea that these are indeed "relic" galaxies. Such "relic" galaxies appear to be rare, as expected from cosmological merger histories (Quilis & Trujillo 2013).

Bezanson et al. (2018) recently measured the rotational support of 104 quiescent galaxies at z ∼ 0.8. By comparing to a homogeneously analyzed z ∼ 0 sample, they inferred a decrease in rotational support since z ∼ 0.8 by a factor of ∼2. Given that we estimated a factor of 5–10 decline since z ∼ 2 for massive galaxies (${M}_{\mathrm{dyn},z\sim 2}\gtrsim {10}^{11.3}\,{M}_{\odot }$), this suggests that much of the spin-down might have occurred at z > 1. That would be qualitatively consistent with the evolution of the mass–size relation, which is also thought to be driven by mergers and also evolves by a similar factor over z = 1–2 as over z = 0–1 (e.g., Newman et al. 2012). However, it is not consistent with the predictions of cosmological simulations, in which angular momentum loss mainly occurs at z < 1 (Penoyre et al. 2017; Lagos et al. 2018a). A robust test of that prediction will require larger samples of z > 1 stellar kinematics measured with the James Webb Space Telescope or 30 m-class ground-based telescopes.

Finally, the "spin-down" of quiescent galaxies may provide a new way to distinguish post-quenching evolution from progenitor effects. Since star-forming galaxies are continually growing in size, as they quench and join the quiescent population, they will increase the mean size of quiescent systems (e.g., Carollo et al. 2013). Disentangling this growth—which occurs in star-forming galaxies—from size growth in quiescent systems is challenging (e.g., Belli et al. 2015). The situation is different for V/σ, because it increases over time in star-forming galaxies (e.g., Wisnioski et al. 2015; Simons et al. 2017), whereas this paper and Bezanson et al. (2018) show that V/σ declines in quiescent galaxies. Processes that are coincident with or follow quenching are therefore needed to decrease V/σ in the quiescent population. The rarity of local analogs of our sample argues that post-quenching "spin-down" must occur. At the high masses sampled by our survey, the importance of mergers is fairly uncontroversial, but as future observations probe the kinematics of less massive quiescent galaxies at high redshifts, they will provide an interesting new constraint on evolutionary models.

Our analysis implies that rotational support generally declines in massive galaxies after quenching. It is interesting to consider whether there is also a more sudden decline in rotational support associated with quenching. This question is harder to address. The kinematics of similarly massive star-forming galaxies that are coeval with our sample have been measured. Tadaki et al. (2017a, 2017b) resolved the Hα and CO kinematics of 11 extended star-forming galaxies with masses ${M}_{* }\gtrsim {10}^{11}\,{M}_{\odot }$. They found that V/σ spans the range 3.5–7.1, significantly higher than the values of $\simeq 2$ we see in quiescent galaxies. Compact star-forming galaxies have been proposed as transitional objects that are the immediate progenitors of compact quiescent galaxies (Barro et al. 2013). Van Dokkum et al. (2015) inferred that such galaxies have rapidly rotating ionized gas disks based on marginally resolved Hα kinematics. Recently, Barro et al. (2017) measured the CO kinematics of such a galaxy and found $V/\sigma \simeq 2.5$. This is consistent with our quiescent sample and supports the idea that "blue nuggets" are immediate progenitors of some compact quiescent galaxies, assuming that the kinematics of the stars and CO are not too dissimilar.

Assessing a decline in V/σ associated with quenching requires comparing not to coeval systems but to massive star-forming galaxies at z = 3–4, the quenching epoch of the galaxies in our sample. Surveys of ionized gas kinematics in this redshift range do not sample galaxies that are sufficiently massive to be progenitors of galaxies in our sample (Gnerucci et al. 2011; Turner et al. 2017). The CO observations of a few massive starburst galaxies at z ≈ 4–5, which are probably progenitors of some massive quiescent galaxies (e.g., Toft et al. 2014), have shown a range of kinematic properties with both less and more rotational support than our sample (Hodge et al. 2012; Riechers et al. 2014; Oteo et al. 2016).

Given these few constraints, the lower V/σ seen in our quiescent galaxy sample compared to coeval star-forming galaxies could result from two effects. First, V/σ might decline when star formation is quenched, but our analysis in Section 5.1 indicates that it does not generally reach the lower levels seen in local early-type galaxies, which requires mergers after quenching. Second, the lower V/σ of $z\sim 2$ quiescent galaxies might be inherited from their star-forming progenitors at z ∼ 3–4, with less or even no change associated with quenching.

Theoretically, it has been shown that gas-rich major mergers can produce remnants with disks (e.g., Robertson et al. 2006; Governato et al. 2009; Wuyts et al. 2010; Sparre & Springel 2017). The persistence of disk-dominated kinematics therefore does not require a particularly "gentle" quenching process. The Wuyts et al. (2010) simulations of binary wet mergers are particularly relevant, since they compared the simulated remnants to the observed properties of z ∼ 2 quiescent galaxies. Provided that the gas fraction at coalescence is high ($\gtrsim 40 \%$), which is expected to occur more frequently at higher redshifts, the remnants can be quiescent galaxies with sizes as compact as observed. Wuyts et al. also made a prediction that these galaxies are much more rotationally supported than local early-type galaxies. Their simulated remnants lie near or above the isotropic rotator line in the V/σ versus ellipticity plot (their Figure 5), broadly consistent with the measurements in this paper. The sizes and kinematics of our sample may therefore be compatible with remnants of gas-rich major mergers.

The very flattened shapes of MRG-M0138 and MRG-M2129, however, present more of a puzzle, since the Wuyts et al. simulations do not produce such thin remnants. Furthermore, the simulated remnants have cuspy light profiles that are unlike the exponential disk observed in MRG-M2129. This galaxy is particularly interesting, since it demonstrates that it is possible for star formation to quench at z ∼ 3 without the formation of a significant bulge. It lacks a central enhancement of the stellar mass density (Paper I, Figure 7) that one might expect in major mergers or "compaction" scenarios where a central starburst precedes quenching (Dekel & Burkert 2014; Zolotov et al. 2015). Toft et al. (2017) suggested that MRG-M2129 was instead quenched as gas flowing onto the (presumably massive) halo of MRG-M2129 was shock-heated and prevented from accreting onto the galaxy. In this scenario, the recently quenched object could retain the structure and kinematics of its star-forming progenitor. Considering their range of bulge properties, it seems likely that massive quiescent galaxies at z ≈ 2 were formed through multiple channels.