The Evolution of Bulge-dominated Field Galaxies from z ≈ 1 to the Present

The field galaxies we analyze here were observed serendipitously with the galaxy clusters MS0451.6-0305 (z = 0.54), RXJ0152.7–1357 (z = 0.83), and RXJ1226.9+3332 (z = 0.89), which are part of the GCP. The data for cluster galaxies in MS0451.6-0305, RXJ0152.7–1357, and RXJ1226.9+3332 are analyzed and published in Jørgensen et al. (2005) and Jørgensen & Chiboucas (2013).

All spectra of the field galaxies in this paper were obtained with Gemini North using the Gemini Multi-Object Spectrograph (GMOS-N; Hook et al. 2004) in the multi-object spectroscopic mode. The spectroscopic sample selection for this paper is based on GMOS-N imaging using r', i', and z' filters. Galaxies were classified as expected cluster members based on their colors. For galaxies imaged in the field of cluster MS0451.6-305, (r'–z') and (i'–z') colors were used. For galaxies in the field of cluster RXJ0152.7–1357, (i'–z') was used. For galaxies in the field of cluster RXJ1226.9+3332, (r'–z'), (i'–z'), and (r'–i') colors were used. When mask space was not filled with assumed cluster members, expected non-members were added. Our sample of field galaxies was selected from the observed galaxies that were determined to be non-members. For a detailed description on the methods of the spectroscopic sample selection refer to Jørgensen et al. (2005) and Jørgensen & Chiboucas (2013).

This is the first analysis of the data for the 82 field galaxies and the first publication of their spectra. The data tables, grayscale images, and sample spectra of the field galaxies can be found in the Appendix. Since the field and cluster galaxy data were obtained simultaneously, they were reduced in the same manner, the details of which are published in Jørgensen et al. (2005) and Jørgensen & Chiboucas (2013). The median S/N per $\mathring{\rm A}$ in the rest-frame of the 82 field galaxies is S/N = 28, while the 30 passive bulge-dominated field galaxies have a median S/N = 35.

From the spectra of the galaxies, we measure redshifts, velocity dispersions, and line indices. In this paper we use a subset of the Lick/IDS absorption line indices (Worthey et al. 1994); see the data tables and sample spectra in the Appendix. Considering the redshift intervals spanned by our spectra, indices blueward of 4700 Å in the rest-frame are available across our whole sample of field galaxies ($0.3\lt z\lt 1.2$) and for all of our comparison sources from Perseus and Abell 194 ($z=0.02$) to RXJ1226.9+3332 ($z=0.89$).

Specifically, we use the higher-order Balmer line indices ${\rm{H}}{\delta }_{{\rm{A}}}$ and ${\rm{H}}{\gamma }_{{\rm{A}}}$ (Worthey & Ottaviani 1997) as well as the ${\rm{H}}{\zeta }_{{\rm{A}}}$ index (Nantais et al. 2013) as primary indicators of the ages of the stellar populations. In our analysis, we use the combined index $({\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}})^{\prime}$ as defined in Kuntschner (2000), namely:

$$\begin{eqnarray}&&({\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}})^{\prime} \equiv -2.5\mathrm{log}\left(1-\displaystyle \frac{{\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}}}{43.75+38.75}\right).\end{eqnarray} \tag{ 1 }$$

As metallicity indicators we use the indices CN3883 (Davidge & Clark 1994), Fe4383 (Worthey et al. 1994), and C4668 (Worthey et al. 1994; Tripicco & Bell 1995).

We adopt the typical uncertainties on the line indices from Table 19 in Jørgensen & Chiboucas (2013), which were estimated from internal comparisons of them. This is possible because the field sample was observed at the same time as the cluster sample, guaranteeing similar observing conditions and the same observational setup. For a detailed explanation on the calculation of the uncertainties, we refer the reader to Jørgensen & Chiboucas (2013).

The cluster and field galaxies in this paper were observed with the HST's Advanced Camera for Surveys (ACS) and the data were obtained from the HST archive. The photometric parameters for our sample of field galaxies are published in Chiboucas et al. (2009) and Jørgensen & Chiboucas (2013). They used the fitting program GALFIT (Peng et al. 2002) to derive the mean surface brightnesses, effective radii, and total magnitudes. The magnitudes were calibrated to rest-frame B using colors from the ground-based imaging and the same technique as described in Jørgensen & Chiboucas (2013); see also Blanton et al. (2003). The Sérsic profile (Sérsic 1968) was used to determine the Sérsic index, n_ser, to select bulge-dominated galaxies. In our investigation of the FP, we use parameters that were fit using a ${r}^{1/4}$ profile (de Vaucouleurs 1948) to be consistent with the analysis of our low-redshift comparison sample. In Section 4.2 we present a brief test on the possible bias of this choice.

From a total of 82 field galaxies with spectroscopy, we select our passive field galaxy sample with the same criteria as those in Jørgensen & Chiboucas (2013) to be consistent with the cluster galaxy comparison samples. The selection criteria are the following (also see Table 1):

• 1.  
  spectroscopy with ${\rm{S}}/{\rm{N}}\ \geqslant \ 20\ \mathrm{per}\,\mathring{\rm A}$ in the rest frame
• 2.  
  a redshift cutoff of $z\ \geqslant \ 0.3$ to exclude a few low-redshift galaxies from our intermediate-redshift sample
• 3.  
  a Sérsic index of ${n}_{\mathrm{ser}}\geqslant 1.5$ (Sérsic 1968) to select for bulge-dominated galaxies
• 4.  
  $\mathrm{EW}[{\rm{O}}\ {\rm{II}}]\leqslant 5\,\mathring{\rm A}$ to exclude emission line galaxies, which may be star-forming or contain an active galactic nucleus.

Table 1.  Field Galaxy Selection Criteria

	Ngal	nser	EW[O ii]
Total	82	⋯	⋯
Disk-dominated	30	<1.5	⋯
Bulge-dominated	⋯	⋯	⋯
Emission linea	13	≥1.5	>5 Å
Passive	30	≥1.5	≤5 Å
Excludedb	9	⋯	⋯

Notes.

^aWhen the spectra do not cover the [O ii] emission lines, we use ${\rm{H}}\beta$ and [O iii] as an indication of star formation. ^bField galaxies that did not meet one or more criteria described in the text.

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Throughout this paper, the plots contain both the passive field galaxies and emission line field galaxies, though we concentrate on the passive field galaxy sample to allow a more consistent comparison between the field galaxies and the cluster galaxies in Jørgensen & Chiboucas (2013). To provide approximately equally sized subsamples of passive field galaxies, we split our sample into two redshift bins, $0.3\lt z\lt 0.8$ and $0.8\lt z\lt 1.2$, with median redshifts of $z=0.68$ and $z=0.95$. Henceforth, we will refer to the galaxies in the lower-redshift range as the $z\approx 0.7$ sample, and those in the higher-redshift range as the $z\approx 1$ sample. The redshift distributions and velocity dispersion distributions for the passive and emission line field galaxies are shown in Figure 1. Throughout the paper, the number of field galaxies varies between plots because not all of the field galaxies have measurements for all line indices.

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The Coma, Perseus, and Abell 194 clusters make up our local comparison sample. Perseus and Abell 194 are used for the analysis of line indices, while Coma is used for the size–mass, size–velocity dispersion, M/L–mass, and M/L–velocity dispersion relations, as well as the derivation of the FP. Spectroscopic data for the Coma cluster (z = 0.024) are published and detailed in Jørgensen (1999), while the spectroscopic data for Perseus (z = 0.018) and Abell 194 (z = 0.018) are described in Jørgensen et al. (2005) and will be published in I. Jørgensen et al. (2017, in preparation). The same B-band photometry was used for the Coma cluster as in Jørgensen et al. (2005). All three samples were filtered to exclude spiral, irregular, emission line, and low-mass ($\lt {10}^{10.3}\,{M}_{\odot }$) galaxies in order to best compare with our field galaxy sample.

The interpretation of the line index values was carried out using the same models as in Jørgensen & Chiboucas (2013). This allows for a consistent comparison between their cluster galaxy data and our field galaxy data.

Our analysis is based on the single stellar population (SSP) models from Thomas et al. (2011) and Maraston & Strömbäck (2011). We use the models that assume a Salpeter (1955) initial mass function. The output of the Thomas et al. (2011) models are line indices for ages of 1–15 Gyr, a range in metallicity with specific values of $[{\rm{M}}/{\rm{H}}]=-0.33,\ 0.00,\ 0.35,\ 0.67$, and a range in α-element abundance with ratios $[\alpha /\mathrm{Fe}]\,=0.0,0.3,0.5$. The output of the Maraston & Strömbäck (2011) models are spectra with ages of 1–15 Gyr, a range in metallicity with [M/H] = −0.3, 0.0, 0.3 and solar abundance ratio $[\alpha /\mathrm{Fe}]=0$. As is customary, sets of age, [M/H], $[\alpha /\mathrm{Fe}]$ were used to create model grids for pairs of line indices that were then directly compared with the measured index values.

The CN3883 index has not been modeled by any of the commonly available models in the literature which also take into account non-solar $[\alpha /\mathrm{Fe}]$. Therefore, we follow the approach by Jørgensen & Chiboucas (2013) and adopt their Equation (4) (see also their Figure 5 and Table 9) that relates ${\mathrm{CN}}_{2}$ with CN3883. Then we used that equation and the ${\mathrm{CN}}_{2}$ values from Thomas et al. (2011) to obtain CN3883 model estimates. ${\rm{H}}{\zeta }_{{\rm{A}}}$ were derived from the Maraston & Strömbäck (2011) model spectra and are only available for $[\alpha /\mathrm{Fe}]=0$. Finally, we used M/L ratios modeled by Maraston (2005) with solar abundance ratios as the Thomas et al. (2011) models do not provide M/L predictions.

We compare the field galaxies to cluster galaxies at similar redshifts and to cluster galaxies at $z\approx \ 0$ by fitting scaling relations to the samples. We assume that the relations are linear in log-space, since previous studies of larger samples in clusters do not find any nonlinearities, except in the log $\sigma$ versus log mass relation for galaxies with masses above ${10}^{12}\,{M}_{\odot }$ (Jørgensen & Chiboucas 2013; Jørgensen et al. 2014). We use the same fitting technique as Jørgensen et al. (1996), which minimizes the sum of the absolute residuals perpendicular to the relation. This technique is robust to outliers and does not assume any particular distribution of the residuals. The uncertainties of the coefficients are determined using a bootstrap method. Fits to the field galaxy samples show that the slopes of the relations are consistent at the $1\sigma$ level with the better-determined cluster galaxy relations (Jørgensen & Chiboucas 2013; Jørgensen et al. 2014). We therefore adopt the scaling relations from Jørgensen & Chiboucas (2013), except for the scaling relation for ${\rm{H}}{\zeta }_{{\rm{A}}}$ which is derived in Jørgensen et al. (2014). These relations are listed in Table 2, Column 1. The median zero-points of the three samples are listed in Table 2, Columns 2, 5, and 8. The uncertainties on these values are calculated using bootstrap resampling.

Table 2.  Scaling Relations

Relation	Local Sample			$z\approx 0.7$ Passive Field			$z\approx 1$ Passive Field
	$\gamma$	Ngal	rms	$\gamma$	Ngal	rms	$\gamma$	Ngal	rms
(1)	(2)	(3)	(4)	(5)	(6)	(7)	(8)	(9)	(10)
$\mathrm{log}{r}_{{\rm{e}}}=(0.57\pm 0.06)\mathrm{log}\ \mathrm{Mass}+\gamma$	$-{5.73}_{-0.15}^{+0.16}$	105	0.16	$-{5.75}_{-0.07}^{+0.17}$	14	0.20	$-{5.77}_{-0.15}^{+0.17}$	16	0.14
$\mathrm{log}\ \sigma =(0.26\pm 0.03)\mathrm{log}\ \mathrm{Mass}+\gamma$	$-{0.67}_{-0.07}^{+0.07}$	105	0.08	$-{0.66}_{-0.07}^{+0.04}$	14	0.10	$-{0.64}_{-0.10}^{+0.06}$	16	0.07
$\mathrm{log}\,{\text{}}M/L=(0.24\pm 0.03)\mathrm{log}\ \mathrm{Mass}+\gamma$	$-{1.75}_{-0.10}^{+0.08}$	105	0.09	$-{2.29}_{-0.40}^{+0.16}$	14	0.24	$-{2.47}_{-0.22}^{+0.33}$	16	0.25
$\mathrm{log}\ {\text{}}M/L=(1.07\pm 0.12)\mathrm{log}\,\sigma +\gamma$	$-{1.56}_{-0.08}^{+0.14}$	105	0.11	$-{2.10}_{-0.35}^{+0.26}$	14	0.25	$-{2.23}_{-0.23}^{+0.27}$	16	0.23
$({\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}})^{\prime} =(-0.085\pm 0.015)\mathrm{log}\,\sigma +\gamma$	${0.10}_{-0.01}^{+0.01}$	65	0.02	${0.16}_{-0.02}^{+0.05}$	9	0.04	${0.19}_{-0.06}^{+0.06}$	8	0.08
$\mathrm{CN}3883=(0.29\pm 0.06)\mathrm{log}\,\sigma +\gamma$	$-{0.41}_{-0.05}^{+0.05}$	65	0.05	$-{0.43}_{-0.05}^{+0.07}$	10	0.05	$-{0.45}_{-0.03}^{+0.06}$	14	0.05
$\mathrm{log}\,\mathrm{Fe}4383=(0.19\pm 0.07)\mathrm{log}\,\sigma +\gamma$	${0.26}_{-0.07}^{+0.05}$	65	0.06	${0.11}_{-0.05}^{+0.21}$	12	0.18	${0.07}_{-0.22}^{+0.27}$	10	0.27
$\mathrm{log}\,{\rm{C}}4668=(0.33\pm 0.08)\mathrm{log}\,\sigma +\gamma$	${0.11}_{-0.04}^{+0.06}$	65	0.06	${0.03}_{-0.20}^{+0.10}$	10	0.16	${0.01}_{-0.00}^{+0.05}$	3	0.03
$\mathrm{log}\,{\rm{H}}{\zeta }_{{\rm{A}}}=(-0.76\pm 0.29)\mathrm{log}\,\sigma +\gamma$	${1.76}_{-0.21}^{+0.21}$	45	0.25	${2.00}_{-0.17}^{+0.20}$	4	0.14	${1.96}_{-0.08}^{+0.27}$	13	0.18
$\mathrm{log}\,{r}_{{\rm{e}}}=(1.3\pm 0.08)\mathrm{log}\sigma -(0.82\pm 0.03)\mathrm{log}\,\langle I{\rangle }_{e}+\gamma$	$-{0.44}_{-0.08}^{+0.06}$	105	0.08	${0.00}_{-0.35}^{+0.12}$	14	0.20	${0.15}_{-0.19}^{+0.27}$	16	0.21
RXJ0152.7–1357, RXJ1226.9+3332a
$\mathrm{log}{\text{}}M/L=(0.55\pm 0.08)\mathrm{log}\,\mathrm{Mass}+\gamma$	$-{5.85}_{-0.10}^{+0.25}$	49	0.14	$-{5.70}_{-0.15}^{+0.25}$	14	0.21	$-{5.90}_{-0.12}^{+0.32}$	16	0.20

Note. Column 1: scaling relation for the local comparison samples unless otherwise noted. Column 2: zero-point for the low-redshift sample; uncertainties are calculated with bootstrap resampling. Column 3: number of galaxies included from the low-redshift sample. Column 4: rms in the Y-direction of the scaling relation for the low-redshift sample. Columns 5–7: zero-point number of galaxies, and rms in the Y-direction for the $z\sim 0.7$ passive field galaxies. Columns 8–10: zero-point, number of galaxies, and rms in the Y-direction for the $z\approx 1$ passive field galaxies.

^aThe slope derived from a fit treating RXJ0152.7–1357 and RXJ1226.9+3332 as one sample.

References. Columns 1–4 adopted from Jørgensen & Chiboucas (2013), except the relation for $\mathrm{log}\ {\rm{H}}{\zeta }_{{\rm{A}}}$ which is from Jørgensen et al. (2014).

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We chose RXJ1226.9+3332 as the high-redshift comparison cluster because the X-ray structure of this cluster is the most relaxed of the high-redshift clusters in Jørgensen & Chiboucas (2013). The smooth, symmetric X-ray emission implies that RXJ1226.9+3332 has not undergone a recent cluster–cluster merger interaction, as opposed to the RXJ0152.7–1357 cluster at $z=0.84$, which shows an elongated X-ray morphology, indicative of an ongoing merger (Demarco et al. 2005, 2010; Girardi et al. 2005; Jørgensen et al. 2005).

We then derive the median zero-point differences, ${\rm{\Delta }}\gamma$, between the field galaxy samples and the local comparison sample. The random uncertainties on these, ${\sigma }_{{\rm{\Delta }}\gamma }$, are derived using two methods: (1) paired bootstrap resampling and (2) using the following equation:

$$\begin{eqnarray}&&{\sigma }_{{\rm{\Delta }}\gamma }={({\mathrm{rms}}_{\mathrm{local}}^{2}/{N}_{\mathrm{local}}+{\mathrm{rms}}_{\mathrm{FG}}^{2}/{N}_{\mathrm{FG}})}^{0.5},\end{eqnarray} \tag{ 2 }$$

where subscripts "FG" refer to the field galaxy sample and "local" refer to the $z\approx \ 0$ cluster sample. These methods produce similar results and we therefore use the values derived from Equation (2) throughout the text and figures.

In the following, we compare the scaling relation zero-points for the two passive field galaxy samples to the local comparison sample. For the M/L–mass relation, we also compare the passive field galaxies to the cluster galaxies at high redshift. Jørgensen & Chiboucas (2013) fit RXJ0152.7–1357 and RXJ1226.9+3332 separately and determined that the slopes were consistent with one another. We therefore use the relation of these two clusters treated as one sample to compare to the passive field galaxies. The zero-points and scatter are summarized in Table 2.

Figure 2(a) shows the effective radius versus dynamical galaxy mass and Figure 2(b) shows the velocity dispersion versus dynamical galaxy mass. The galaxy masses are derived using the approximation $\mathrm{Mass}=5{r}_{e}{\sigma }^{2}{{\rm{G}}}^{-1}$ (Bender et al. 1992). We assume that field and cluster galaxies at z ≈ 0 have identical size–mass relations (Huertas-Company et al. 2013). We then compare the slopes of the size–mass and velocity dispersion–mass relations between the local comparison sample and the passive field galaxies and find no significant difference. We therefore adopt the slope of the local relation for the field galaxy sample and find the median offsets in the zero-points for the passive field galaxies, see Table 2. The zero-point differences with respect to Coma in $\mathrm{log}\ {r}_{{\rm{e}}}$ for the $z\approx 0.7$ and $z\approx 1$ passive field galaxy samples are ${\rm{\Delta }}\mathrm{log}\ {r}_{{\rm{e}}}\,=-0.02\pm 0.06$ and −0.04 ± 0.04, respectively, and therefore not significant. The zero-point differences with respect to Coma in $\mathrm{log}\sigma$ for the $z\approx 0.7$ and $z\approx 1$ passive field galaxies are ${\rm{\Delta }}\mathrm{log}\ \sigma =0.01\pm 0.03\ \mathrm{and}\ 0.02\pm 0.02$, respectively, and therefore also not significant. The random uncertainties were calculated using Equation (2). The passive field galaxies follow the same relations as found for the Coma cluster sample and so we conclude that at a given dynamical mass, the passive field galaxies show no significant evolution of size or velocity dispersion between $z\ \approx \ 1$ and the present.

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The zero-point differences with respect to RXJ1226.9+3332 in log ${r}_{{\rm{e}}}$ for the $z\approx 0.7$ and $z\approx 1$ passive field galaxy samples are ${\rm{\Delta }}\mathrm{log}\ {r}_{{\rm{e}}}=-0.02\pm 0.06$ and $-0.05\pm 0.05$, respectively, and therefore not significant. The zero-point differences with respect to RXJ1226.9+3332 in $\mathrm{log}\sigma$ for the $z\approx 0.7$ and $z\approx 1$ passive field galaxies are ${\rm{\Delta }}\mathrm{log}\ \sigma =0.02\pm 0.03\ \mathrm{and}\ 0.04\pm 0.02$, respectively, and therefore also not significant. We conclude that the passive field galaxies follow the same size–mass and velocity dispersion–mass relations as cluster galaxies at similar redshifts.

Figure 3 shows the FP edge-on. The parameters for the FP are fit using a ${r}^{1/4}$ profile. To test the validity of this choice, we investigate the residuals of the FP for the passive field galaxies determined from the parameters fit with a ${r}^{1/4}$ profile versus those fit from a Sérsic profile, see Figure 4. Our results do not significantly depend on our choice of using the parameters fit with the ${r}^{1/4}$ profile. We investigate the correlation between n_ser and the residuals of the FP determined from the ${r}^{1/4}$ profile and find no correlation. We therefore consistently use ${r}^{1/4}$ profiles to compare between samples.

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Figure 5 shows a projection of the FP using M/L ratios versus the dynamical masses and the velocity dispersions of the galaxies. It shows the predictions based on models from Thomas et al. (2005) for the relations of the median redshifts of the two passive field galaxy samples and of the RXJ1226.9+3332 cluster. These models assume that the low-mass galaxies formed their stars more recently while the high-mass galaxies formed theirs at higher redshifts. The passive field galaxies roughly follow the models; however, they show more evolution in the M/L versus velocity dispersion relation than the models. The slope of the FP for the passive field galaxies is steeper than the Coma relation and consistent with the slope found for the cluster galaxies in Jørgensen & Chiboucas (2013) at $z\approx 0.86$.

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We use the same method as in Jørgensen & Chiboucas (2013) to evaluate the evolution of the field galaxies as a function of mass and redshift. For this purpose, we divide the $z\approx 1$ passive field galaxies into two subsamples, one with velocity dispersions of $\mathrm{log}\sigma \lt 2.24$ and one with velocity dispersions of $\mathrm{log}\sigma \geqslant 2.24$. These two subsamples have average velocity dispersions of $\mathrm{log}\sigma =2.12\ \mathrm{and}\ 2.30$, respectively. The $z\approx 0.7$ field galaxy sample contains only three galaxies with $\mathrm{log}\sigma \geqslant 2.24$ (Figure 1), and therefore was not analyzed in this way.

We use the equation ${\rm{\Delta }}\mathrm{log}\ M/L=0.935{\rm{\Delta }}\mathrm{log}\ \mathrm{age}$ (Jørgensen & Chiboucas 2013) based on the SSP models in Maraston (2005) to find the predicted formation redshift of the passive field galaxies under the assumption of passive evolution. We determine formation redshifts with ${\chi }^{2}$ fits to the zero-point differences in the M/L versus mass relation for the sample of passive field galaxies, which implies ${z}_{\mathrm{form}}={1.35}_{-0.07}^{+0.10}$. We also determine formation redshifts for the median zero-point differences of the $z\approx \ 0.7$ and $z\approx \ 1$ samples. These results are listed in Table 3. For the $z\approx \ 1$ sample, the zero-point differences of the low-velocity dispersion subsample imply ${z}_{\mathrm{form}}={1.24}_{-0.07}^{+0.10}$ and for the high-velocity dispersion subsample ${z}_{\mathrm{form}}={2.16}_{-0.39}^{+0.81}$.

Table 3.  Formation Redshift Results from SSP Models

Relation	Field Galaxies			Galaxy Clusters
	${\chi }^{2}$ Fit	$z\approx 0.7$	$z\approx 1$	z = 0.86
(1)	(2)	(3)	(4)	(5)
${\rm{\Delta }}\mathrm{log}\,{\text{}}M/L=0.935{\rm{\Delta }}\mathrm{log}\,\mathrm{age}$	${1.35}_{-0.07}^{+0.10}$	${1.16}_{-0.11}^{+0.17}$	${1.39}_{-0.09}^{+0.12}$	⋯
$\mathrm{log}\,{\text{}}M/L$ (low-mass)	⋯	⋯	${1.24}_{-0.07}^{+0.10}$	1.24 ± 0.05
$\mathrm{log}\,{\text{}}M/L$ (high-mass)	⋯	⋯	${2.16}_{-0.39}^{+0.81}$	${1.95}_{-0.2}^{+0.3}$
${\rm{\Delta }}({\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}})^{\prime} =-0.126{\rm{\Delta }}\mathrm{log}\ \mathrm{age}$	${1.40}_{-0.18}^{+0.60}$	${1.32}_{-0.22}^{+0.47}$	${1.49}_{-0.29}^{+1.22}$	$\gt 2.8$
${\rm{\Delta }}\mathrm{log}\ {\rm{H}}{\zeta }_{{\rm{A}}}=-0.456{\rm{\Delta }}\mathrm{log}\ \mathrm{age}$	${2.60}_{-0.62}^{+\infty }$	${0.94}_{-0.19}^{+0.62}$	${3.50}_{-1.57}^{+\infty }$	⋯
${\rm{\Delta }}\mathrm{log}\,{\rm{C}}4668=0.121{\rm{\Delta }}\mathrm{log}\ \mathrm{age}$	${1.29}_{-\infty }^{+\infty }$	${0.96}_{-\infty }^{+\infty }$	${1.29}_{-0.13}^{+0.25}$	⋯
${\rm{\Delta }}\mathrm{log}\,\mathrm{Fe}4383=0.272{\rm{\Delta }}\mathrm{log}\ \mathrm{age}$	${1.38}_{-0.24}^{+7.75}$	${1.19}_{-0.29}^{+1.81}$	${1.53}_{-0.39}^{+20.1}$	⋯

Note. Column 1: scaling relation. Column 2: formation redshift determined with ${\chi }^{2}$ fits to the zero-point offsets in the scaling relation for the passive field galaxies. The formal uncertainties derived from the fits are unrealistically low, so the uncertainties listed were derived as the random uncertainties on the median zero-points. Columns 3, 4: formation redshift determined from the median zero-point offsets in the scaling relation for the $z\approx 0.7$ and $z\approx 1$ passive field galaxy subsamples, respectively. Column 5: formation redshift determined from the zero-point offsets in the scaling relation for the galaxy clusters at z = 0.86 (Jørgensen & Chiboucas 2013).

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To study the intrinsic scatter relative to the FP, we adopt the slope for the log M/L versus log mass relation from the RXJ1226.9+3332 and RXJ0152.7–1357 cluster galaxies treated as one sample and then calculate the median offsets in the zero-points for the $z\approx 0.7$ and $z\approx 1$ passive field galaxies, see Table 2. We then calculate the scatter with respect to this offset relation for the passive field galaxies. To calculate the intrinsic scatter in the relation we subtract off in quadrature the measurement uncertainty. The intrinsic scatter in the M/L ratio versus log mass for the $z\approx 0.7$ and $z\approx 1$ passive field galaxies is $0.18\pm 0.05$ and $0.17\pm 0.04$, respectively. The intrinsic scatter in the M/L ratio versus log mass for the RXJ1226.9+3332 and RXJ0152.7–1357 cluster galaxies treated as one sample is $0.08\pm 0.01$. This may indicate that the passive field galaxies have less homogeneous stellar populations than the high-redshift cluster galaxies.

Figure 6 shows the line indices versus the velocity dispersions of the galaxies.

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The $({\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}})^{\prime}$ lines are stronger for the passive field galaxies than for cluster galaxies at similar redshifts. This indicates that the passive field galaxies have younger stellar populations than the cluster galaxies.

To find the predicted formation redshift of the passive field galaxies, we use the equation ${\rm{\Delta }}({\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}})^{\prime} \,=-0.126{\rm{\Delta }}\mathrm{log}\ \mathrm{age}$ (Jørgensen & Chiboucas 2013) based on the SSP models in Thomas et al. (2011). The ${\chi }^{2}$ fits to the zero-point offsets of the $({\rm{H}}{\delta }_{{\rm{A}}}+{\rm{H}}{\gamma }_{{\rm{A}}})^{\prime}$ index imply a formation redshift of ${z}_{\mathrm{form}}={1.40}_{-0.18}^{+0.60}$ for the passive field galaxies. This formation redshift is consistent with that found from the M/L ratios. The formation redshift was also determined for the median zero-point differences in the $z\approx \ 0.7$ and $z\approx \ 1$ passive field galaxy subsamples, see Table 3. The ${\rm{H}}{\zeta }_{{\rm{A}}}$ lines for the passive field galaxies indicate a formation redshift consistent with those found from the higher-order Balmer lines but, due to the large uncertainties on this index for the Perseus and A194 galaxies, more accurate measurements need to be taken to provide additional constraints. The zero-point differences for the C4668 and Fe4383 lines imply a formation redshift of ${z}_{\mathrm{form}}\approx 1.3\mbox{--}1.4$, generally consistent with the higher-order Balmer lines.

To calculate the intrinsic scatter in the relation for Fe4383, we subtract off in quadrature the measurement uncertainty. We find an intrinsic scatter for Fe4383 of $0.16\pm 0.05$ and $0.26\pm 0.08$ for the $z\approx \ 0.7$ and $z\approx \ 1$ passive field galaxy samples, respectively. The intrinsic scatter for Fe4383 of the local comparison sample is $0.05\pm 0.01$ and for RXJ1226.9+3332 is $0.04\pm 0.01$. Therefore, the $z\approx \ 0.7$ and $z\approx \ 1$ passive field galaxies show 4 and 6.5 times more intrinsic scatter, respectively, than that found for the cluster galaxies at similar redshifts. This supports the results from the scatter in the M/L–mass relation, further indicating that the passive field galaxies have less homogeneous stellar populations than the cluster galaxies at similar redshifts.

Figure 7 shows the line indices versus each other for the field galaxies together with the local cluster sample and the RXJ1226.9+3332 cluster sample. Panel (a) shows SSP models from Thomas et al. (2011) with $[\alpha /\mathrm{Fe}]=0$, 0.3, 0.5 and [M/H] = −0.33, 0.00, 0.35, 0.67. Panel (b) shows SSP models for ${\rm{H}}{\zeta }_{{\rm{A}}}$ based on Maraston & Strömbäck (2011) spectral energy distributions for $[\alpha /\mathrm{Fe}]=0$, while CN3883 is based on ${\mathrm{CN}}_{2}$ from Thomas et al. (2011). The models cover [M/H] from −0.33 to 0.67 and ages of 1–15 Gyr. Panel (c) shows SSP models from Thomas et al. (2011) for $[\alpha /\mathrm{Fe}]=0.3$ with metallicities [M/H] between −0.33 to 0.67 and ages of 1–15 Gyr. The figure shows that the sample of field galaxies contains a range of abundance ratios $[\alpha /\mathrm{Fe}$], see panel (a), and young stellar populations (1–5 Gyr), see panels (b) and (c). Compared to the field galaxies, the cluster galaxies span a similar range in the line indices and therefore have a similar range in [M/H] and $[\alpha /\mathrm{Fe}$].

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Our sample of passive field galaxies follows the same size–mass and velocity dispersion–mass relations as found for the Coma cluster sample and we therefore conclude that, at a given dynamical mass, the passive field galaxies show no significant evolution of size or velocity dispersion between $z\approx 1$ and the present. Oldham et al. (2017) find an extremely small amount of structural evolution in their sample of z = 0.545 cluster galaxies. In contrast to this, several studies have found that galaxies at a given mass are smaller at higher redshifts (Trujillo et al. 2007; van Dokkum et al. 2010; Newman et al. 2012; Belli et al. 2014). However, these authors all use stellar masses while we use dynamical masses in our relations and models. It is unclear whether the different method of mass estimation is the reason for the disagreement between our results.

Jørgensen & Chiboucas (2013), Jørgensen et al. (2014), and Saglia et al. (2010) use dynamical masses for deriving the size–mass and velocity dispersion–mass relations. Jørgensen & Chiboucas (2013) and Jørgensen et al. (2014) found no significant evolution in galaxy sizes or velocity dispersions as a function of redshift at a given dynamical mass for their sample of $z=0.86$ and $z=1.27$ cluster galaxies, respectively. Saglia et al. (2010) analyze field galaxies and galaxies in less rich cluster environments (compared to Jørgensen & Chiboucas 2013), and found that both field and cluster galaxies at a given mass are smaller and increase in velocity dispersion at higher redshifts. Jørgensen & Chiboucas (2013) already established that the differences between our conclusions are not due to selection effects, as Saglia et al. (2010) use the same limit in ${r}_{{\rm{e}}}$ as ours, excluding galaxies with ${r}_{{\rm{e}}}\lt 1\ \mathrm{kpc}$ from the samples. For a better comparison with our results, more studies of size evolution using dynamical masses are needed.

Our sample of passive field galaxies follow the same size–mass and velocity dispersion–mass relations as the cluster galaxy comparison sample at similar redshift. The Saracco et al. (2017) sample of passive field galaxies at $z=1.3$ also follows the same size–mass scaling relations as their sample of cluster galaxies, meaning that at fixed stellar mass, they have the same structural parameters. They also find a significant lack of passive bulge-dominated field galaxies at ${M}_{\mathrm{dyn}}\,\gt 2\cdot {10}^{11}\,{M}_{\odot }$ and ${r}_{{\rm{e}}}\gt 5\ \mathrm{kpc}$. Our sample also displays these trends but this may be due to selection effects since our sample is by no means complete.

We find that the slope of the M/L ratio versus dynamical mass of our sample of $z\approx 0.7$ and $z\approx 1$ early-type field galaxies is steeper than that of Coma. Supporting this conclusion, steeper FP slopes at high redshift have also been reported by di Serego Alighieri et al. (2005), Treu et al. (2005a, 2005b), and van der Wel et al. (2005) for their samples of field galaxies. This effect is also seen in high-density environments as shown by Jørgensen et al. (2006, 2007) and Jørgensen & Chiboucas (2013). However, Gebhardt et al. (2003) find no significant difference between the slopes of the FP for their sample of field galaxies and the local comparison sample. The discrepancy between our results and those of Gebhardt et al. (2003) can possibly be attributed to their sample not containing galaxies of low enough masses, with only two galaxies in the redshift range $0.75\lt z\lt 1.0$ with masses $\lt \ {10}^{10.8}\,{M}_{\odot }$.

The steepening of the FP slope is consistent with "downsizing" (Cowie et al. 1996), where the high-mass galaxies formed their stars at higher redshift and in a shorter time period than low-mass galaxies. Figure 5 shows the predicted locations of the relations for passive evolution models from Thomas et al. (2005) for the median redshifts of the $z\approx 0.7$ and $z\approx 1$ passive field galaxies and the RXJ1226.9+3332 ($z=0.89$) cluster galaxies showing galaxies with high mass forming their stars at high redshift and low-mass galaxies forming theirs more recently. All three model lines use the high-density environment predictions. The data roughly follow the models, implying that the passive evolution model and "downsizing" can explain our results. However, the passive field galaxies appear to show more evolution in the M/L ratio versus log $\sigma$ relation than predicted by the models, meaning that age depends more strongly on the velocity dispersion for this sample than stated in Thomas et al. (2005).

For a deeper understanding of the evolution of field galaxies as a function of mass, we used the zero-point differences of the M/L ratios to determine formation redshifts for the low-mass and high-mass passive field galaxy subsamples, see Table 3 and Figure 8. These findings are consistent with those from other authors. For example, van Dokkum & van der Marel (2007) find ${z}_{\mathrm{form}}={1.95}_{-0.08}^{+0.10}$ for field galaxies with ${M}_{\mathrm{dyn}}\gt {10}^{11}\,{M}_{\odot }$ while for cluster galaxies within the same mass range they find ${z}_{\mathrm{form}}={2.01}_{-0.17}^{+0.22}$ (see also van der Wel et al. 2005; Oldham et al. 2017). Similarly, Treu et al. (2005a) found ${z}_{\mathrm{form}}\approx 1.2$ and ${z}_{\mathrm{form}}\gt 2$ for their low-mass and high-mass subsamples, respectively. They interpret the results to mean that the majority of the stellar mass in all systems was formed at $z\gt 2$, but that the low-mass systems had secondary episodes of activity that revived the older population.

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We investigate this further by showing the formation redshifts of individual field galaxies derived from the zero-point differences in the M/L ratios versus dynamical masses, see Figure 9. All of the field galaxies with log ${\mathrm{Mass}}_{\mathrm{dyn}}\lesssim 10.6\ {M}_{\odot }$ have experienced recent star formation. Above this mass, our sample contains a mix of galaxies, some that have experienced recent star formation and others that contain only older stellar populations. Since our sample is not complete, this analysis likely contains selection effects. A larger and more complete sample is required to determine whether this is a general feature of passively evolving field galaxies.

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In the case of cluster galaxies, Jørgensen & Chiboucas (2013) found ${z}_{\mathrm{form}}=1.24\pm 0.05$ for low-mass galaxies and ${z}_{\mathrm{form}}={1.95}_{-0.20}^{+0.30}$ for high-mass galaxies. The consistency shown between the field and cluster galaxy populations in terms of formation epoch strongly suggests that galaxy age depends more on mass than environment. Rettura et al. (2011) suggest that the timescale of the star formation histories of galaxies is determined by environment, but the timing of galaxy formation is regulated by the galaxy mass. This means that, at a given stellar mass, field and cluster galaxies have roughly the same age (or z_form) but galaxies in clusters tend to have a less extended phase of star formation. More extended star formation activity should lead to lower $[\alpha /\mathrm{Fe}]$ abundances as the products of SNe Ia are recycled with those from core-collapse supernovae. The $\alpha$-element abundances, as derived from line index fitting, may provide an independent assessment of the length of the star formation in galaxies as a function of environment. However, the typical star formation timescale difference between cluster and field reported in Rettura et al. (2011) is $\lt 1\,\mathrm{Gyr}$. This is too short a difference to be detected in $\alpha$-element abundances.

The intrinsic scatter in the M/L versus mass relation for Coma is consistent with that of the z = 0.86 clusters RXJ1226.9+3332 and RXJ0152.7–1357 treated as one sample. In contrast, Treu et al. (2005a) and van der Wel et al. (2005) find that the intrinsic scatter in the FP decreases with decreasing redshift, after allowing for evolution of the slopes.

The intrinsic scatter in the M/L versus mass relation for the passive field galaxies is $\sim 2.5$ higher than the intrinsic scatter of the cluster galaxies at similar redshifts. This may indicate that the scatter in the M/L versus mass relation is dependent on environment and not redshift. However, we recognize that this result depends on how precisely we understand and can quantify our measurement error. In contrast to this, Bernardi et al. (2006) show that the scatter in the FP at $0.05\lt z\lt 0.14$ does not depend on environment.

Line indices are used to determine characteristics of stellar populations such as ages and metallicities. For the cluster galaxies in Jørgensen & Chiboucas (2013), the offsets in the higher-order Balmer lines imply older ages than those found from the offsets in the M/L ratios. In contrast to this, for our sample of passive field galaxies, the ages determined from the strength of the higher-order Balmer lines agree with the formation redshifts determined from the zero-point differences in their M/L ratios. We confirmed there was no bias in our sample by calculating the formation redshift for only those field galaxies that had measured higher-order Balmer lines. We then compared that result to the formation redshift determined from the entire sample of passive field galaxies and found no significant difference. Consistent with our results, Schiavon et al. (2006) find, for red field galaxies at $z\sim 0.9$, a formation redshift of ${z}_{\mathrm{form}}\approx 1.1\mbox{--}1.3$ when modeled using SSP models with supersolar metallicity.

One line index we use in our analysis is CN3883. Our sample of passive field galaxies exhibits weaker CN3883 lines than the cluster galaxies at similar redshifts. However, because there are no models published for CN3883, the predictions for this index were based on ${\mathrm{CN}}_{2}$ models and the empirical relation between CN3883 and ${\mathrm{CN}}_{2}$. The passive evolution models predict a weakening of CN3883 that we do not detect in our data. We suggest that better or updated models are needed for this index.

Our analysis of the zero-point differences of the line indices and M/L ratios of the passive field galaxies is consistent with the passive evolution model. In agreement with this, Gallazzi et al. (2014) and Choi et al. (2014) find that their samples of $z\sim 0.7$ quiescent field galaxies are consistent with passive evolution. These authors also find no significant evolution in metallicities since $z\sim 0.7$.

An important line index in our analysis centers on the Fe4383 lines because we find that the $z\approx 0.7$ and $z\approx 1$ passive field galaxies show 4 and 6.5 times more intrinsic scatter, respectively, than the cluster galaxies at similar redshifts. This may indicate that the passive field galaxies are more diverse in Fe4383 than the cluster galaxies. One potential explanation for this is that, as compared to the cluster galaxies, the passive field galaxies have experienced a larger range in the duration of their star formation episodes, consistent with Rettura et al. (2011).