A census of star formation histories of massive galaxies at 0.6 < z < 1 from spectro-photometric modeling using Bagpipes and Prospector

We present individual star-formation histories of ∼ 3000 massive galaxies (log( M ∗ / M (cid:12) ) > 10.5) from 1 the Large Early Galaxy Astrophysics Census (LEGA-C) spectroscopic survey at a lookback time of 2 ∼ 7 billion years and quantify the population trends leveraging 20hr-deep integrated spectra of these 3 ∼ 1800 star-forming and ∼ 1200 quiescent galaxies at 0.6 < z < 1.0. Essentially all galaxies at this 4 epoch contain stars of age < 3 Gyr, in contrast with older massive galaxies today, facilitating better 5 recovery of previous generations of star formation at cosmic noon and earlier. We conduct spectro-6 photometric analysis using parametric and non-parametric Bayesian SPS modeling tools - Bagpipes 7 and Prospector to constrain the median star-formation histories of this mass-complete sample and 8 characterize population trends. A consistent picture arises for the late-time stellar mass growth when 9 quantiﬁed as t 50 and t 90 , corresponding to the age of the universe when galaxies formed 50% and 90% 10 of their total stellar mass, although the two sets of models disagree at the earliest formation times 11 (e.g. t 10 ). Our results reveal trends in both stellar mass and stellar velocity dispersion as in the local 12 universe - low-mass galaxies with shallower potential wells grow their stellar masses later in cosmic 13 history compared to high-mass galaxies. Unlike local quiescent galaxies, the median duration of late-14 time star-formation ( τ SF,late = t 90 - t 50 ) does not consistently depend on the stellar mass. This census 15 sets a benchmark for future deep spectro-photometric studies of the more distant universe. 16


INTRODUCTION
Galaxies are complex systems of multi-age stellar populations.Observational constraints on the timescales of star-formation and hence stellar mass growth and assembly enable us to understand the role of various physical mechanisms and their environment in guiding the formation and cosmic evolution of galaxies (Panter et al. 2007;Leitner 2012).Understanding these processes on various physical scales (radial and spatially resolved) and temporal scales also helps shed light on some of the longstanding puzzles, such as when and how galaxies stop forming stars (quenching) (Schawinski et al. 2014;Wild et al. 2016;Schreiber et al. 2016;Maltby et al. 2018;Carnall et al. 2018) and the transition from star-forming to quiescent that leaves a bimodal population.(Bell et al. 2004;Muzzin et al. 2013a;Vulcani et al. 2014;Smethurst et al. 2015;Taylor et al. 2015).
Our current knowledge of the formation and cosmic evolution of galaxies is fundamentally limited by tradeoffs between the quality and quantity of observations.Building a coherent picture of galaxy evolution demands (a) looking back in time for galaxies in the younger universe without compromising the quality of observations, and (b) a statistically large representative sample size to connect the dots between similar populations of galaxies at multiple epochs.This is difficult owing to technological and observational challenges as light from galaxies at increasing distances dims and shifts to less accessible infrared regime.
Broadband multi-wavelength spectral energy distributions (SEDs) contain information about total stellar mass in old evolved stars, short-lived stars and dust content.Stellar Population Synthesis (SPS) modeling of SEDs has been widely used owing to the speed and simplicity.However, broadband photometry itself may not be sufficient to completely break the dust-agemetallicity degeneracy or to constrain higher order moments of star formation histories (SFHs) (Leja et al. (2017(Leja et al. ( , 2019)); Tacchella et al. (2022), A. Nersesian et al. submitted).The numerous, old low mass stars have long lifetimes and slow spectral evolution, yet leave weak imprints on observed data owing to their faint intrinsic luminosities.Therefore, inferring SFHs from photometry only is strongly susceptible to even small perturbations in data, leading to different recovery of SFHs (see e.g.Ocvirk et al. (2006)).The recovered SFHs from mocks show biases up to 0.2 dex in the mass-weighted formation times.This is even worse with real-world broadband photometric data having more systematic uncertainties involved (Leja et al. 2019;Carnall et al. 2019a).Photometry alone is not always sufficient to differentiate amongst modeling assumptions (Belli et al. 2018;Carnall et al. 2019a;Tacchella et al. 2022), thus properties measured from SED-only fits will always suffer from systematic uncertainties e.g.flux calibrations and different physical models obtained from stellar templates that vary with uncertainties on the data.
To robustly recover and analyze the timescales of a galaxy's stellar mass growth, we need to decipher spectral signatures containing strong imprints of stellar evolution.Continuum spectroscopy in the rest-frame optical regime contains information about the nature of past star-formation (bursty, uniform, rising, declining etc.) and metal enrichment within a galaxy in features like G4300, Fe4383, Fe4531, M g 2 , Balmer lines and 4000-Å break.These signatures evolve most rapidly in young stellar populations, since more massive galaxies today are generally older by several Gyr, studies of their SFHs should be more accurate and precise at earlier times.To obtain better signal-to-noise ratios, many analyses of high-redshift galaxy surveys involve mass matched stacking of spectra, vanishing any information about any individual galaxy's evolution (Schiavon et al. 2006;Choi et al. 2014;Siudek et al. 2017;Cullen et al. 2019).High signal-to-noise continuum spectroscopic data when combined with deep broadband photometric data has the potential to produce much stronger constraints on SFHs of galaxies (Gallazzi et al. 2008;Pacifici et al. 2012;Thomas et al. 2017;Carnall et al. 2019b;Iyer 2019;Tacchella et al. 2022;Webb et al. 2020) and provide clues on the number of major star formation episodes and the timescales of rejuvenation, starbursts, and quiescence.Spectroscopy alone also suffers from systematics like instrumental noise and outlier pixels and emphasises the importance of spectro-photometric modeling.
It is worth noting that even spectro-photometric modeling can be insufficiently sensitive to more slowly evolving old stellar populations, more so at later epochs.However, it has been argued that photometry can only probe the last ∼1 Gyr of a SFH, whereas adding spectroscopic data can help to probe the SFH further back in time (Chaves-Montero & Hearin 2020).Many studies have been conducted to recover SFHs of local galaxies (Thomas et al. 2005;Cid Fernandes 2007;Panter et al. 2007;Tojeiro et al. 2009;McDermid et al. 2015;Citro et al. 2016;Ibarra-Medel et al. 2016).However, this approach of using fossil records at z = 0 may not lead to reliable results, especially for more massive galaxies for which the major episode of their star-formation happened very early in the universe and these galaxies are now left with predominantly old stellar populations that suffer from strong outshining effects.This underscores the importance deep spectroscopic studies with high signal-to-noise at large lookback times.
Recent advances in computational and sampling techniques (Skilling 2006;Feroz & Hobson 2008;Feroz & Skilling 2013;Feroz et al. 2019) have led to the development of a variety of tools that provide fast full-Bayesian fitting of models to spectro-photometric data, like MCSED (Bowman et al. 2020), BEAGLE (Chevallard & Charlot 2016), Prospector (Johnson et al. 2021;Johnson & Leja 2017), and Bagpipes (Carnall et al. 2018).These Bayesian methods give a better handle on the priors assumed and are robust to classical problem of over-fitting the data with complex models e.g.(Leja et al. 2019).However, different SSP libraries and modeling assumptions in these tools can influence the derived star-formation histories (Martins 2021;Pacifici et al. 2023).
Though real SFHs of galaxies are complex, a simplified way to model them is by parametrizing the SFHs using a functional form.The most commonly used forms are exponentially declining (Mortlock et al. 2017;McLure et al. 2018;Wu et al. 2018a), delayed exponentially declining (Ciesla et al. 2017;Chevallard et al. 2019), lognormal (Gladders et al. 2013;Abramson et al. 2015;Diemer et al. 2017;Cohn 2018) and double-power law (Ciesla et al. 2017;Carnall et al. 2018).These analytic prescriptions have been shown to match well with many SFHs from simulations (Simha et al. 2014;Diemer et al. 2017) and are widely used as they minimize computational requirements.Increasing complexity by adding bursts of star-formation can bring parametric SFHs even closer to realistic scenarios.However, capturing events like rejuvenation and sudden quenching can still be challenging for parametric models.A more flexible method is non-parametric SFHs -which adopts a series of periods of constant star-formation in fixed or flexible time bins (Chauke et al. 2018;Cappellari 2017;Leja et al. 2017;Iyer & Gawiser 2017;Ocvirk et al. 2006;Cid Fernandes et al. 2005).Non-parametric approaches have higher flexibility with a wider range of possible solutions and therefore can allow a broader range of priors on SFHs and decrease biases on recovered results with more realistic episodes of star-formation in predefined time bins (Iyer & Gawiser 2017;Leja et al. 2019;Suess et al. 2022a,b).
Until recently, SPS modeling of statistically representative populations has been limited to the analysis of photometry-only data sets.Numerous efforts have been put toward recovering the SFHs of both star-forming and quiescent populations from deep broadband photometric surveys spanning wide redshift ranges (Dye 2008;Wuyts et al. 2009;Pforr et al. 2012;Pacifici et al. 2016b,a;Iyer & Gawiser 2017;Iyer 2019;Aufort et al. 2020;Olsen et al. 2021) due to greater data availability and lower computational demands.In contrast, fewer studies have tested modeling spectrophotometric data for high redshift galaxies (Carnall et al. 2019b;Estrada-Carpenter et al. 2020;Forrest et al. 2020;Johnson et al. 2021;Tacchella et al. 2022;Khullar et al. 2022;Hamadouche et al. 2023) owing to the dearth of high signal-to-noise continuum spectroscopy at significant lookback times.Large spectroscopic surveys like DEEP2 (Newman et al. 2013), MOSDEF (Kriek et al. 2015) and KBSS (Rudie et al. 2012;Steidel et al. 2014;Strom et al. 2017) have been primarily sufficient to char-acterize emission line properties of thousands of starforming galaxies.On the other hand, smaller spectroscopic studies of hundreds of quiescent galaxies have provided windows into the stellar populations of quiescent systems out to z∼2, but with a significant bias towards the brightest, most massive subset.
The Large Early Galaxy Astrophysics Census (LEGA-C) (van der Wel et al. 2021;Straatman et al. 2018;van der Wel et al. 2016) provides a novel opportunity to characterize the full population of Milky Waymass and larger progenitors at significant lookback time.This deep spectroscopic survey of ∼3000 star-forming and quiescent galaxies looking 6-8 billion years back in time (0.6 < z < 1) includes deep imaging available for each galaxy from the UltraVISTA survey (Muzzin et al. 2013b).At this redshift, most stars in LEGA-C survey galaxies are <3 billion years in age, enabling more robust characterization of their SFHs.
The primary goal of this paper is to measure the SFHs of full LEGA-C DR3 sample using two complementary modeling tools.We use these SFHs to quantify their median formation times and population scatter of these properties for massive star-forming, quiescent and all galaxies separately.
The structure of the paper is as follows.Section 2 describes the LEGA-C dataset and our approach to spectrophotometric modeling using Bagpipes and Prospector, and some example demonstration of modeling results.Section 3 describes the SFHs of starforming and quiescent galaxies and of the full population as recovered from spectro-photometric fits.We show cumulative median mass growth trends in stellar mass bins and further quantify population trends with two late-time formation metricst 50 and t 90 corresponding to time stamps when galaxies formed 50% and 90% of their total stellar mass.In Section 4 we expand on the interpretation of our results with respect to previous low and high redshift studies and how this impacts our understanding of the formation of both star-formation and quiescent systems.In Section 5 we conclude our study and highlight some major takeaways from this analysis.We also speculate on potential future works that could be a successor of this study to help us better constrain the evolution of massive galaxies and answer more broader questions.Throughout this paper we assume Ω M = 0.3, Ω Λ = 0.7, H 0 = 70 km s −1 Mpc −1 and all magnitudes are in quoted in the AB system.for all 3005 primary LEGA-C Galaxies from the DR3 catalog with density distributions of star-forming (blue) and quiescent (red) populations based on rest-frame UVJ colors.Our effective sample selections are shown with dashdot lines (mass-selected) and solid lines (velocity dispersion-selected).

Data and Sample
The Large Early Galaxies Astrophysics Census (LEGA-C) (van der Wel et al. 2021;Straatman et al. 2018;van der Wel et al. 2016) is a 130-night public spectroscopic survey of ∼3000 Ks-band selected galaxies targeting redshift range 0.6 < z < 1.0 in the COSMOS field, looking 6-8 billion years back in time.Each spectrum has an approximate observed spectral coverage of 6300Å -8800Å corresponding to 3600Å -5200Å rest-frame optical regime.The survey was conducted on Very Large Telescope (VLT) using ViMOS (VIsible Multi Object Spectrograph) (Le Fevre et al. 2000) and completed its third and final data release (DR3) in August 2021 (van der Wel et al. 2021).
The full spectroscopic sample consists of 4081 galaxies -3029 primary targets and 1052 fillers.Targets were Ks-band selected from the UltraVISTA catalog (Muzzin et al. 2013b) to be stellar mass-representative above ∼ 5 x 10 10 M .Targets were selected to have Ks-band magnitudes brighter than a redshift dependent limit of [20.7 -7.5log((1+z)/1.8)].This selection criterion is independent of any derived quantities from the spectra.Each galaxy was observed for ∼ 20 hours yielding S/N ≈ 20 Å −1 continuum with high fidelity absorption and emission line features for both dusty, blue as well as faint, red galaxies.
As the LEGA-C survey targets only a subset of the full photometric sample, we account for missing galaxies with appropriate weighting to individual galaxies to ensure spectroscopic completeness above the previously specified magnitude limit in a full census of the properties of massive galaxies at that epoch.Hence throughout this paper we apply a multiplicative factor Tcor -corresponding to volume and sample correction -from the DR3 catalog (van der Wel et al. 2021) to individual object counts to make it representative of that redshift.The photometric information is taken from UltraVISTA catalog (Muzzin et al. 2013b).We use BvrizYJ bands for our primary spectro-photometric analysis.
The stellar mass (log(M * /M )), spectroscopic redshift (z), signal-to-noise (S/N) and integrated stellar velocity dispersion (σ * ) density distributions for the primary sample are shown in Fig. 1.All values are taken from the LEGA-C DR3 catalog with (total) stellar mass estimates from Prospector photometry-only stellar populations fits (LOGM_MEDIAN), spectroscopic redshift from the LEGA-C spectra (Z_SPEC), the median S/N per pixel of the LEGA-C spectrum (S/N) and stellar velocity dispersion (SIGMA_STARS) estimated from Gaussian broadening of theoretical single stellar population models as described in Bezanson et al. (2018).We split the sample into star-forming and quiescent populations using Muzzin et al. (2013c) rest-frame U-V and V-J color-color cuts, identifying quiescent galaxies with U-V > (V-J) × 0.88 + 0.69.Note that throughout this study, we adopt the LEGA-C DR3 catalog value of σ * and M * /M to maintain a single set of labels for each individual galaxy and to facilitate consistent comparisons with other studies.There are 1774 star-forming (medians: log(M * /M ) ∼ 10.8 and σ * ∼ 140 km s −1 ) and 1231 quiescent galaxies (median: log(M * /M ) ∼ 11.2, σ * ∼ 200 km s −1 ) in the primary sample of 3005 objects shown in red and blue colors respectively (obtained using PRIMARY flag in LEGA-C DR3 catalog, more details in van der Wel et al. ( 2021)).Throughout this paper, we focus on objects above the approximate completeness limit of LEGA-C (10.5 < log(M * /M ) ≤ 12 and 100 < σ * ≤ 300 km s −1 ).For mass-limited sample we have an effective sample size of 2703 unique galaxies (1459 starforming and 1244 quiescents) and the velocity-dispersion sample includes 2823 unique galaxies (1575 star-forming and 1248 quiescents).This sample selection is shown with dash-dotted and solid lines in Fig 1.

Spectro-photometric Modeling
In this work, we use two state-of-the-art Bayesian SPS modeling tools, Bagpipes (Carnall et al. 2018) and Prospector (Johnson & Leja 2017;Johnson et al. 2021), to perform a full spectro-photometric fitting of the primary LEGA-C galaxies.Details of the two methods used in this work are discussed in sections 2.2.1 and 2.2.2.We use BvrizYJ bands in our primary spectrophotometric analysis (iz overlapping the spectra) following the conclusions in van der Wel et al. ( 2021)'s Appendix B, where the authors analyzed systematic errors in the SED modeling propagating into derived parameters.We check the robustness of our results by performing comparisons with two more parametric set of fits altering the photometric data -(1) spectra +vriz bands, and (2) spectra + 20 bands (B to 24 micron, namely -B, g, IA484, IA527, V, IA624, r, IA679, IA738, i, z, y, J, H, Ks, ch1, ch2, ch3, ch4, mips24) covering 15 Subaru bands (B to Ks), 4 IRAC bands (ch1 to ch4) and 1 Spitzer MIPS24 band, ranging from UV to far-IR wavelengths.Results of this comparison are shown in Figure A1 in our Appendix A.

Bagpipes
Bagpipes (Bayesian Analysis of Galaxies for Physical Inference and Parameter EStimation) is a SPS modeling package built on the updated BC03 (Bruzual & Charlot 2003) spectral library 1 with the 2016 version of the MILES library of empirical spectra that includes 2.5 Å resolution in 3525Å -7500Å wavelength range (Falcón-Barroso et al. 2011).It is built on a Kroupa (2001) IMF assumption and utilizes a Multi-Nest nested sampling algorithm (Feroz et al. 2019) to produce posterior distributions of physical parameters.We perform parametric full spectral SPS modeling of LEGA-C spectra with BvrizYJ UltraVISTA broadband photometry using a double-power law SFH (Ciesla et al. 2017;Carnall et al. 2018), parameterized as: (1) This model has three free parameters describing the rising (β), falling (α) and peak (τ ) of star formation, whereas other widely used options (e.g.exponential, delayed-tau and log-normal) have two or less free parameters.Hence, by construction, the double-power law SFH has more flexibility.Tau models are shown to fail to recover mock SFHs (Carnall et al. 2019a) and simulation SFHs (Pacifici et al. 2012).A double-power law SFH is chosen owing to its ability to recover the redshift evolution of cosmic SFRD (Behroozi et al. 2013;Gladders et al. 2013;Madau & Dickinson 2014) and owing to the agreement with simulation results (Pacifici et al. 2016b;Diemer et al. 2017).The rising and falling slopes essentially do not change beyond the prior ranges chosen (become either flat or vertical) owing to the analytical functional form (Eq.1) and cover full variations of possible SFHs within this limit.The stellar metallicity is assumed to be the same for all stars born.This value is linearly interpolated on a grid of SSP models and is allowed as a free parameter varying from (0.02,2.5)Z uniformly in linear space.The Z value is assumed to be 0.02 in BC03 models.We test the impact of choosing linear and logarithmic priors on the stellar metallicity on the derived posterior values in our spectro-photometric fits for a subset of galaxies and find no strong deviations.The last free SFH parameter is the stellar mass formed in the entire lifetime of a galaxy until the point of observation (without mass return to the ISM); we allow log(M * /M ) to vary logarithmically from (0, 13).Two bursts on top of a double power law are included to account for any abrupt variation in star formation activity.Each burst is given flexibility of age, stellar mass and stellar metallicity and hence three free parameters.We use Charlot & Fall (2000) dust model with two free parameters -V-band attenuation and the slope of attenuation.We adopt a second order spectral calibration and white uncorrelated noise for spectral pixels, for which a detailed description can be found in Carnall et al. (2019b).Dust emission models from Draine & Li (2007) are implemented with fixed Q pah = 2, U min  = 1 and γ e = 0.01.Nebular emission and Balmer regions are masked in this modeling as the only excitation mechanism in the Bagpipes physical models is ionization by young stars and hence SFR estimates could be biased by emissions from other processes in play (Carnall et al. 2019b).Stellar velocity dispersion is another free parameter modeled with a variable Gaussian kernel in velocity space.Although the DR3 LEGA-C spectra used in this study are flux calibrated using the UltraV-ISTA photometry (van der Wel et al. 2021), we include an additional polynomial function of wavelength to address any higher order spectrophotometric calibration uncertainties.We use a second order Chebyshev polynomial function with Gaussian priors -0th order centered around unity and 1st and 2nd orders centered around zero (see §4.3.1 in Carnall et al. (2019b) for more details).This modeling requires an average of ∼70 CPU hours per galaxy.Further description of all modeling parameters and prior distributions is shown in Table 1.
Modeling results are included in Table 2.We notice a subset (total of ∼120 in the full sample) of quiescent galaxies in Bagpipes, clustering around mass-weighted ages of t M W ∼ 3 Gyr with similar SFHs, adding more to the overall population scatter in that regime.We inves- tigate these objects via the color-color diagram, best-fit models, dust-metallicity distributions and corner plots of the full parameter space and do not find anything unusual.Though these objects are captured in 16th -84th percentiles (more in Section §3), they are statistically insignificant for median trends.

Prospector
Prospector is a Bayesian SPS modeling tool that allows a non-parametric modeling of the SFH of a galaxy in piece-wise constant SFR time bins.(Johnson et al. 2021).It deploys the Flexible Stellar Population Synthesis (FSPS) package with the MILES stellar library and MIST isochrones to model stellar properties (Conroy et al. 2009).We use a Chabrier (2003) IMF and the Kriek & Conroy (2013) dust law with nebular continuum and line emissions modeled with CLOUDY (Ferland et al. 2013).We use dynesty (Speagle 2020) nested sampling option for posterior sampling, similar to Bagpipes.This non-parametric approach is capable of recovering complex SFHs and capturing abrupt star formation processes like sudden quenching and rejuvenation events.On the flip side, fitting both the galaxy spectra and SED is highly computationally expensive and requires about ∼100 CPU hours per galaxy to fit an 8 fixed bin SFH model with 14-free parameters.
In this work, we adopt a continuity prior piece-wise constant SFH with Student's-t distribution that fits the change in log(SFR(t)) in adjacent time bins while weighing against abrupt changes in SFR(t).This prior has also been shown to robustly reproduce mock SFHs.The pioneering work of Ocvirk et al. (2006) show that a max-  3 (Prospector).Thick lines show the medians (solid) and dashed 1-σ (16th -84th percentile) scatter.The median values follow closely for both the codes, except at the earliest times the analytic function requires SFR(t=0) = 0 and non-parametric models assign non-zero star-formation in the earliest bin.
imum of 8 episodes of SFH can be independently recovered from an optical spectra of resolution R=10,000, S/N=100 and wavelength coverage λ = 4000Å -6800Å, with the distinguishability of simple stellar populations proportional to the separation in logarithmic time.Hence we use an 8 time bin SFH model (5 logarithmically spaced) in our analysis.The 8 bins of constant SFRs are distributed as follows (in lookback time) -0 < t < 30 Myr 30 Myr < t < 100 Myr 100 Myr < t log1,2,3,4 < 0.85 t univ (5 log bins) 0.85 t univ < t < t univ The two most recent fixed bins capture signatures of any recent abrupt star formation, one earliest fixed bin corresponding to oldest stellar populations spans the first 15% of cosmic time, with 5 logarithmically-spaced bins in between.Redshift is set to the LEGA-C spectroscopic redshift with allowed ± 0.005 variation and stellar velocity dispersion is fixed to the LEGA-C DR3 catalog values.A full description of the free and fixed parameters of the Prospector model and their adopted priors are included in Table 3 and modeling results are reported in Table 4.

Modeling Results and Examples
Fig. 3 and Fig. 4 show representative examples of spectro-photometric modeling and recovered SFHs of four quiescent and four star-forming galaxies respectively.In the left column, each panel shows an observed LEGA-C spectrum and UltraVISTA photometry (grey), along with the best-fitting models from Bagpipes (orange) and Prospector (green).The insets show the full BvrizYJ photometric SEDs and models.We note that the models fit both the spectroscopic and photometric data very well.The fractional residuals are plotted in the bottom panel (∆ f rac = (observed flux -model flux)/observed flux).Gray bands indicate regions that are masked to avoid emission lines in the Bagpipes modeling.The right large panels show the corresponding SFHs.Note that we chose to logarithmically scale the lookback time (horizontal axis) to highlight the most robustly measured epochs at late times.
For quiescent galaxies (Fig. 3), we note that both models provide excellent fits to the observed data and agree reasonably well on the overall shape of the SFHs.One subtle difference emerges at the earliest times when parametric Bagpipes SFHs must rise from zero star formation at t univ = 0, where as Prospector consistently assigns finite star formation in the first time bin leading to older stellar populations.For star-forming galaxies (Fig. 4), we see better agreement in SFRs at both early and late times.
One might be concerned that priors on the SFHs might be driving some of these clear differences in the implied SFHs that are derived by the two software packages.We test this effect by drawing from the prior distributions in each set of models.Figure 2 depicts the prior probability density from Bagpipes (orange) and Prospector (green) when 1000 random samples are drawn from the SFH parametrization described in Table 1 and Table 3 (more details in §2.2.1 and §2.2.2).For ease of comparison, we assign a floor SFR value of 0.001 M yr −1 to arbitrarily small SFR values here as well.The median distributions of prior SFHs are shown in solid lines whereas 16th and 84th percentiles are shown with dotted lines.The median values follow closely for both the codes, except at the earliest times the analytic function requires SFR(t=0) = 0 whereas non-parametric models assign non-zero star-formation in the earliest bin.Note that for both prescriptions, 50th and 84th percentile values are in reasonable agreement, with some differences in 16th percentile values.This figure suggests no strong biases in SFRs with lookback time from the priors adopted in the two codes. .Four representative star-forming galaxies with modeling outputs in the same format as described in Figure 3.In general, parametric and non-parametric SFHs for individual star-forming galaxies agree reasonably well at all times.

THE STAR FORMATION HISTORIES OF MASSIVE GALAXIES
In this section, we combine the star-formation histories derived for each galaxy in the LEGA-C sample to characterize the overall growth histories of massive galaxies.We first interpolate all SFHs on a uniform age grid from (0.01, 8) Gyr and impose a minimum starformation rate (floor) of 0.001 M yr −1 to arbitrarily small SFR values.For each galaxy, we take the median SFH from the posterior distributions and combine these medians to calculate population medians in bins of -(1) stellar mass and (2) stellar velocity dispersions from LEGA-C DR3 catalog.For ease of presentation, we first focus on three coarse stellar mass bins representative of LEGA-C primary sample -a) 10.To demonstrate our approach to combining the posteriors from individual fits, we show the individual median and population trends in an example subset of LEGA-C data in Figure 5.The left panels show the individual SFHs for the 54 most massive quiescent galaxies, defined as the median SFHs from the posteriors of each fit (top: Bagpipes and bottom: Prospector).The middle and right panels show these SFHs in specific star-formation rate (sSFR) as a function of time.sSFR at an epoch is obtained by dividing SFR at that epoch by the total stellar mass formed up until that epoch excluding mass loss.The center panel includes median SFHs for individ-ual galaxies and the right panel collapses these to show only the population distributions.Note that Bagpipes has more variety in onset, duration and quenching of star-formation for individual galaxies, which are often imprinted on the population trends even though the individual parametric SFHs are smooth.
Figure 6 expands the right panel in Figure 5 to show the sSFR evolution in three stellar mass bins (rows) for: star-forming galaxies (left), quiescent galaxies (center), and the full population (right).Each panel shows the population medians (dashed lines/solid lines) and 16th -84th percentile population scatter (hashed/filled regions) for Bagpipes and Prospector respectively.The bottom row combines the median trends of all mass bins with thin shaded regions showing 1-σ error on the medians calculated from bootstrapping.We adopt the transition and quiescence boundaries as (1/3 t H ) and (1/20 t H ) from Tacchella et al. (2022), where t H is the Hubble time at z obs .Although individual galaxies show a variety of SFHs, the population median trends are largely independent of stellar mass, with slight trends at early times that are dominated by the demographics of the LEGA-C sample and modeling priors.In part, differences in the onset of star formation can be partially due to slight differences in the mass-dependent redshift distributions of the LEGA-C sample.However, as discussed in §2, the modeling priors also introduce subtle differences.Bagpipes consistently measures slightly higher median sSFRs, especially within the quiescent population.For star-forming galaxies, both codes infer that massive galaxies fall to lower sSFRs at the point of observation than the less massive ones.Some differences are seen in the median trends of the most massive galaxies (red) from the two codes that could be attributed to small sample size.For quiescent galaxies, we see a large diversity of Bagpipes sSFR tracks.We do not find that  the population of massive quiescent galaxies shut off before lower mass counterparts.Note that this is different from local universe findings (Gallazzi et al. 2006;Nelan et al. 2005).
Figure 7 focuses on the median trends in the cumulative stellar mass growth and adopts the same plotting conventions as Figure 6, except the horizontal axis now shows the age of the Universe, starting at the Big Bang.The hashed and filled regions show the 16th -84th percentile distributions of individual median SFHs from Bagpipes and Prospector respectively.Horizontal grey dotted lines correspond to 10%, 50% and 90% formation thresholds.These timescales, which we hereby define as t 10 , t 50 , t 90 , are widely used metrics to quantify the timescales of stellar mass growth in a galaxy including all progenitors up to the point of observation (Behroozi et al. 2019;Pacifici et al. 2016a;Weisz et al. 2011Weisz et al. , 2008)).These values are quantified for the full primary sample in Table 3 and Table 4.As we find in the previous two figures, parametric models in general show higher population scatter than non-parametric ones due to galaxy to galaxy variations.The bottom row compiles all median trends to compare the rate of stellar mass growth across different masses.This cumulative view highlights both mass trends within the populations and discrepancies at early times due to modeling degeneracies and prior assumptions.At any given time, low mass galaxies have grown less than massive galaxies.Significant discrepancies in t 10 , and to some degree in t 50 , suggest that the LEGA-C dataset is inadequate to characterize the earliest growth histories of massive galaxies.Perhaps similar deep spectroscopic observations of galaxies at even earlier times will help to break modeling degeneracies and resolve this issue.
Figures 8 and 9 show t 50 (teal) and t 90 (brown) the formation time scales for the populations of star-forming (left), quiescent (center) and all galaxies (right) versus stellar mass (6) and stellar velocity dispersion (8).We exclude t 10 because of the dramatic disagreement between the two sets of models at early times, suggesting that its measurements are completely prior dominated.Bagpipes median trends and population scatter are shown with dotted lines and error bars and Prospector as solid lines with shaded bands.The bottom row shows the difference between the median values derived from the two methods with (very small) error bars indicating the standard error in the medians.The measured late formation times (t 90 ) agree well across all stellar masses, with a slight offset in early time formation especially of quiescent systems (t 90 : ∆ median < 0.3 Gyr t 50 : ∆ median < 2 Gyr).We find clear trends in both t 50 and t 90 formation times with stellar mass amongst star-forming galaxies.In contrast, the median formation times are mostly independent of mass amongst the quiescent galaxies.The overall correlation with mass is reflected by the full population, in which the younger star-forming galaxies dominate at low masses and are rare at the massive end.We note that similar correlations with stellar mass remain within finer redshift bins (∆z = 0.1).Correlations of formation times with stellar velocity dispersion qualitatively similar to those with stellar mass (Figure 9).Again, the median t 50 and t 90 formation times are essentially uniform for the quiescent population.However, because the fraction of starforming galaxies plummets at high stellar velocity dispersion (especially above σ 200 km s −1 ) (Taylor et al. 2022), the formation timescales of the full population exhibits an even stronger correlation with stellar velocity dispersion (Fig. 9,right panel).
Finally, we investigate the duration of late time star formation (t 90t 50 ) from Prospector (green) and Bagpipes (orange) in Figure 10.Some studies have used (t 80t 20 ) and (t 90t 10 ) timescales (Pacifici et al. 2016a;Tacchella et al. 2022) to quantify full duration of starformation, but we choose to avoid large uncertainties at earlier times and quantify late-time star-formation as τ SF,late ≡ t 90t 50 .Figure 10 shows these timescales versus stellar mass (top row) and stellar velocity dispersion (bottom row) with error bars/shaded regions representing the population scatter, following the plotting conventions of Figures 8 and 9.Here we see that while agreement is reasonably good between e.g., median t 50 in the two models, systematic offsets in the duration of star-formation remain.Non-parametric models from Prospector exhibit consistently more extended star formation histories than those derived with double-power law models.This discrepancy at the earliest times has been well-documented (e.g., Carnall et al. 2019a;Leja et al. 2019).Notably, the median τ SF,late from Bagpipes is dramatically (∼2 Gyr) more rapid for quiescent galaxies, with an enormous population scatter, reflecting the larger variety of SFHs (see e.g., Fig. 5).In fact, the population scatter on Bagpipes star formation duration detracts from the informative nature of the median trends.However, significant offset between τ SF,late of star-forming and quiescent galaxies does imprint a trend in the full population median value with stellar velocity dispersion (bottom row, right-most panel), where demographics are more cleanly separated than in stellar mass (e.g., Franx et al. 2008;Belli et al. 2014;Taylor et al. 2022, Fig. 9).The median duration of star-formation derived by Prospector does not vary significantly within the star-forming population, but quiescent galaxies exhibit a slight decrease with stellar mass and stellar ve- Age of Universe (Gyr) Age of Universe (Gyr) .The cumulative stellar mass fractions versus time for star-forming (left), quiescent (center) and all (right) galaxies sorted by stellar mass.The color and plotting scheme is the same as in Fig 4 .This represents a complete census as individual galaxies are weighted by correction factors to reflect the LEGA-C targeting.Lower mass galaxies grow more slowly than their massive counterparts.Non-parametric SFHs (Prospector, solid) start earlier and are more extended, especially for quiescent galaxies.  .Similar to Fig. 8, now binned in stellar velocity dispersion.We find a striking correlation between the formation timescales and stellar velocity dispersion in full population (right).This trend is in part driven by demographics; the quiescent fraction depends more strongly on stellar velocity dispersion than stellar mass (e.g., Taylor et al. 2022).
locity dispersion ( 0.5 Gyr across the full range).This is consistent with weak correlations between alpha enhancement and stellar velocity dispersion for a subset of quiescent galaxies from the same LEGA-C dataset Beverage et al. (2023).

DISCUSSION
Star-formation activity has been demonstrated to depend both on stellar mass (Calvi et al. 2018;Pacifici et al. 2016a;Kauffmann et al. 2004) and on stellar ve-locity dispersion (Franx et al. 2008).It has been long debated whether stellar velocity dispersion is a more fundamental property than stellar mass (Sharma et al. 2021;Zahid et al. 2018;Wake et al. 2012), as it is directly connected to the total gravitational potential well (including effects of dark matter halo and supermassive black hole at galactic center) in which a galaxy resides (Elahi et al. 2018;Dutton et al. 2010).
This census of star formation histories builds on extensive studies based on complete photometric (e.g., Olsen Prospector.Non-parametric models are consistently more extended than the double-power law SFHs.We find no correlation between τ SF,late and stellar mass or stellar velocity dispersion for star-forming galaxies.The weak trends amongst the quiescent galaxies are insignificant relative to the population scatter.This lack of correlation is in contrast with trends found amongst analogous massive elliptical galaxies today (e.g., Graves et al. 2009;Greene et al. 2022Greene et al. ). et al. 2021;;Aufort et al. 2020;Iyer 2019;Iyer & Gawiser 2017;Pacifici et al. 2016b,a;Dye 2008) and smaller and/or less complete, and more observationally expensive, spectroscopic dataset (e.g., Gallazzi et al. 2008;Sánchez-Blázquez et al. 2011;Carnall et al. 2019bCarnall et al. , 2022;;Tacchella et al. 2022).For the first time at significant lookback time, we present a comprehensive view of the star-formation histories of the full population of massive galaxies (not just quiescent or star forming galaxies separately).This builds on a preliminary study by Chauke et al. (2018) based on early LEGA-C data (607 galaxies) with simpler modeling assumptions: adopting fixed solar metallicity and piece-wise constant star-formation rates using CSP templates from FSPS package (Conroy et al. 2009).This current paper expands in both scope and detail to yield a comprehensive analysis of the full LEGA-C sample.
Our results are in qualitative agreement with previous analyses, and quantitative differences can gener-ally be attributed to modeling systematics.Previous studies have found similar correlations between formation times and stellar masses or stellar velocity dispersions (Tacchella et al. 2022;Ferreras et al. 2019;Chauke et al. 2018).Interestingly, both of our modeling methods recover significant population scatter in the late time formation timescales of star forming galaxies, however the median τ SF,late is almost independent of stellar mass.This result is in contrast with previous studies.One interesting comparison is with Estrada-Carpenter et al. (2020), in which τ SF,late is defined as quenching timescale (t Q ).Using Prospector to fit non-parametric SFHs to a much smaller sample of ∼30 quiescent systems at 0.7 < z < 1, finding similar median and population scatter in formation redshifts (z 50 ).In contrast, they find a broader range of quenching timescales, specifically spanning down to ∼0.4 Gyr for several galaxies.We can only speculate that this subtle difference could be driven by priors, sample size or differences in data characteristics (e.g., the use of low-resolution grism spectra).Beverage et al. (2023) study 135 LEGA-C massive quiescent galaxies based on elemental abundance patterns and similarly found only a weak, but qualitatively similar, correlation between average starformation timescale and stellar velocity dispersion such that the highest dispersion galaxies formed the earliest and are most metal rich.Interestingly, although the qualitative trends agree well with cluster galaxies at a similar epoch, there may be some hint of environmental effects on SFHs.Khullar et al. (2022) find t 50 of massive quiescent cluster galaxies to be inversely correlated with stellar mass at 0.3 < z < 1.4 with values within 2 Gyr.This is ∼ 1 Gyr/2 Gyr earlier than our Prospector/Bagpipes estimates.Again, this difference could be driven by modeling systematics (e.g. that study adopts a delayed tau SFH) or a physical manifestation of environmental processes that combine to quench galaxies earlier or the generally speed up galaxy evolution in clusters.Even within the LEGA-C dataset age sensitive indices (Dn4000 and Hδ) of quiescent galaxies depend on the environment, not just in cluster versus field; galaxies in over-dense regions are older and formed earlier than those in less extreme environments Sobral et al. (2022).We note that direct comparisons between different studies can be challenging; Webb et al. (2020) perform a cluster analysis at similar redshifts and find little difference between the ages of galaxies in the field versus cluster environments at fixed mass.
Next, we investigate whether the view from z ∼ 0.8 differs significantly from our understanding of this early time period from modeling local massive galaxies or not.We start by comparing to SPS modeling of local elliptical galaxies from McDermid et al. (2015).Figure 11 shows a comparison of the median formation times t 50 and t 90 as a function of the Virial mass for all galaxies in our sample at < z > = 0.8 from Bagpipes (dashed line) and Prospector (solid-line) relative to quiescent systems at z ∼ 0 (diamonds) from McDermid et al. (2015), after subtracting the minimal cumulative stellar mass growth between z ∼ 0 and z ∼ 0.6.The vertical axis is forward in time with t=0 corresponding to the Big Bang.Note that these elliptical galaxies at z ∼ 0 could either be quiescent or star-forming at our redshift, hence for a first order comparison with LEGA-C, we compare to the full (left) and quiescent only (right) population.Virial masses have been calibrated for the majority of the LEGA-C sample (van der Wel et al. 2022) using spatially resolved stellar kinematics (van Houdt et al. 2021).For this simplistic calculation we adopt a median offset of 0.33 dex between the stellar and dynamical mass for each LEGA-C individual object.McDermid et al. (2015) find that most massive galaxies (log(M * /M ) > 11) have already accumulated > 95% of their stellar mass by z ∼ 0.6, whereas low mass galaxies have formed ∼90% of their stellar mass.They also find ubiquitously earlier (t 50 = 2Gyr) stellar mass growth (diamonds with dotted line), than the view from LEGA-C using either modeling framework.This reveals a natural uncertainty of stellar population modeling for the oldest local stellar systems.The fact that we see a better agreement in the left panel emphasizes the importance of including star-forming progenitors in driving the strong mass trends observed in local elliptical galaxies, even at low redshifts.Thomas et al. (2005Thomas et al. ( , 2010) ) provide clear evidence from a z ∼ 0 sample of galaxies that the duration and timing of galaxy formation depends strongly on the stellar mass, arguing that this ultimately reflects halo assembly.However, even at z ∼ 0.8, we do not find these trends, even when star-forming progenitors are included.Some of these differences could be attributed to modeling systematics e.g.use of Gaussian SFH prescription to describe population average, chemical enrichment models/abundance patterns, various environmental dependence and most importantly aperture effects -probing the central few kpc (not the full galaxy) with a 3" diameter fiber that makes results more mass dependent for older ages and higher metallicities, whereas slit aperture effects are not that extreme.Furthermore, the view from maximally old galaxies in local Universe will always be complicated by the relative challenge of modeling slowly evolving old stellar populations.However, it is hard to escape the conclusion that suggests that the trends found in local ellipticals reflect the transformation and assembly/merger history of massive galaxies and not just their in-situ SFHs.
Finally, we compare our measured formation time trends for quiescent galaxies with similar studies at other epochs.Figure 12 shows the comparison of median formation times (t 50 ) of massive quiescent systems from our study (orange and green bands indicating populations scatter) with other empirical (spectroscopic) studies and simulations across redshifts.Choi et al. (2014) results (grey solid lines) are derived from full-spectral fitting based on alpha abundance measurements and SSPequivalent ages.The grey band indicates mean formation times recovered from the VANDELS survey (1 < z < 1.3) in Carnall et al. (2019b) using Bagpipes.The single JWST spectroscopic observation around z = 4.65 from Carnall et al. (2023b) is shown with a star-symbol.We note that the incredibly small uncertainty on formation time reflects the incredible data quality and limits imposed by the age of the Universe at that early time.

Redshift (z)
Figure 12.Formation times versus stellar mass for massive quiescent galaxies across redshifts.The grey lines and bands indicate relations at lower and higher redshifts respectively (Choi et al. 2014;Carnall et al. 2019b).The highest redshift measurement from JWST is shown with black star (Carnall et al. 2023a).Our results are shown in orange (parametric SFHs) and green (non-parametric SFHs).Scaling relations from cosmological simulations (dashed/dotted lines, Davé et al. 2019;Nelson et al. 2019) agree well with our Prospector models.The evolution in this relation for quiescent galaxies emphasizes the importance of including starforming progenitors and ex-situ contributions when comparing the stellar populations of galaxies across cosmic time.
details on the simulation selection, see Carnall et al. (2019b).Although modeling differences could introduce significant systematic offsets with respect to our fits (e.g.extended SFH versus SSPs, scaled solar metallicity versus alpha enrichment), it is interesting to speculate whether some of the observed offset in t 50 between epochs is physical, perhaps driven by late additions to the quiescent population.This progenitor bias could exaggerate the evolution of these scaling relations without changing the stellar populations of any individual galaxies.Superficially the older LEGA-C ages from Prospector are roughly consistent with the trend found for quiescent galaxies in VANDELS at higher z, however our Bagpipes modeling adopts much more similar priors and provide the more consistent comparison.Thus, the observed offset between the grey and orange bands suggests that the relation evolves even between z ∼ 1.15 and z ∼ 0.8.Although all of our modeling is significantly different from the Choi et al. (2014) analysis, the similar offset to later formation times in lower-redshift quiescent populations is consistent with new additions shifting the scaling relations, perhaps more efficiently at lower mass.Galaxy mergers are also major avenues of mass growth and therefore will also contribute to the spatially integrated ages of galaxies.With SPS modeling, we measure the formation times of stars from all progenitors of a galaxy at t obs , which can be quite different from assembly times (t a ).Hill et al. (2017b) demonstrate that this can be a significant effect, by z∼0.1 assembly times and light-weighted formation times can differ by up to ∼ 2.5 Gyr (see also e.g., Hill et al. 2017a;Muzzin 2017).
Spectroscopic studies can precisely pinpoint the emergence of massive quiescent systems and can help settle uncertainties at the earliest times.Recent results from JWST photometry-only SED analyses suggest that massive galaxies may indeed form efficiently and rapidly within 1 Gyr of the Big Bang (e.g.Labbé et al. (2023)).However, given uncertainties in the emission line contributions to photometry (e.g.McKinney et al. (2023)), spectroscopic data are needed more than ever.JWST/NIRSpec has been shown to provide the sensitivity required to constrain the SFHs and and quenching timescales of galaxies at higher redshifts and lower masses: e.g., the Carnall et al. (2023b)'s extreme massive quiescent galaxy at z = 4.65 included in Figure 12.Other recent studies include Looser et al. (2023) and Nanayakkara et al. (2023).Larger, more complete spectroscopic samples from JWST will be crucial to map the early evolution of today's massive quiescent galaxy population.These samples will hopefully shed more light on the modeling degeneracies at the earliest times.

CONCLUSIONS
In this paper, we analyze the median SFHs of massive (log(M * /M ) > 10.5) galaxies at 0.6 < z < 1.0 from the LEGA-C survey by quantifying their formation times (t 50 and t 90 ) and investigate the population trends in this census with stellar masses and stellar velocity dispersions.From our spectro-photometric analysis, we conclude that: 1.The two modeling methods (parametric and non-parametric) yield consistent late-time starformation histories for star-forming and quiescent galaxies.Non-parametric SFHs consistently prefer earlier stellar mass formation, especially for quiescent systems.Potential reasons could bestellar population outshining driving this inference into a systematics-limited regime emphasizing the dominance of modeling priors at the earliest times or a host of different modeling assumption e.g.SPS models, treatment of emission lines, dust, parametrization of SFHs, etc. (Figure 2,5).
2. Although individual galaxies show a variety of SFH pathways, the median sSFR evolution is sim-ilar for all mass bins.Bagpipes SFHs shows population scatter at any time.Both codes infer that massive star-forming galaxies exhibit falling sS-FRs with redshift.Neither model finds a massdependence in the median time at which quiescent galaxies cross a sSFR threshold (Figure 6).
3. We robustly find that quiescent galaxies form 90% of their stars well within ∼4 Gyr/∼5.5 Gyr for massive and less massive end respectively.We confirm the established trend that non-parametric fits infer older stellar populations, in this case by a median t 50 ∼1.5 Gyr).This is at least partially driven by the systematics introduced by priors on the SFH parametrization (Figures 7,8).
4. Lower mass galaxies are slightly younger than massive galaxies (Figure 7), however this trend is relatively weak.This is perhaps due to the fact that the full LEGA-C sample only probes a limited dynamic range in stellar mass.This picture is also consistent with simulations (Iyer et al. 2020;Torrey et al. 2018;Shamshiri et al. 2015).The formation times of star-forming galaxies correlate with stellar mass and stellar velocity dispersion, but similar trends do not exist not amongst the quiescent galaxies.For combined population, the formation times depend strongly on the M * and σ * as quiescent fraction increases with each property.
5. For the star-forming population, both codes recover consistent median formation times (|t Bagpipes − t P rospector | < 300 Myr); the most massive systems formed within t 50 /t 90 ∼3 Gyr/5.5 Gyr and the less massive ones within ∼5 Gyr/7 Gyr (Figure 8, or similarly for high/low σ , Fig. 9).Even for these galaxies, SFRs were on the decline by the time of observation, corroborating a number of previous studies (e.g., Feulner et al. 2005;Ilbert et al. 2015).
6.The late time duration of star formation (τ SF,late = t 90 − t 50 ) does not exhibit a significant correlation with M or σ for quiescent or star-forming galaxy populations.At face value, this is in contrast with expectations from the local universe (e.g.Thomas et al. (2005)).We posit that this suggests the importance of transformation, quenching and/or ex-situ evolution even at late times (z < 1) (Figure 10).
Remaining open questions include connecting quenching timescales (defined from recovered SFHs), environ-ment, and processes driving quenching in quiescent systems to stellar mass and stellar velocity dispersion.A number of studies have begun to answer those questions using photometric data (Mao et al. 2022) and age sensitive spectral indices (Sobral et al. 2022;Wu et al. 2018b), but not much effort has been put towards understanding this evolution from the SFH of individual galaxies.Another interesting comparison would be to compare our SFHs to those derived from simulation using PCA analysis (e.g., Sparre et al. 2015).
This SFH census of a representative sample of ∼ 3000 massive galaxies using two state-of-the-art Bayesian modeling tools will set the benchmark at z ∼ 1.It is a "cosmic mid-point" in time, providing a stepping stone to future spectroscopic studies of high redshift galaxies (e.g., from JWST/NIRSpec, VLT/MOONS (Maiolino et al. 2020) or Subaru/PFS (Greene et al. 2022)).By including two independent sets of models, we hope that this intermediate redshift study would be a link to other similar studies and help connect our understanding of evolution of stellar mass growth and assembly in galaxies from cosmic noon and beyond to present day universe.
represent the population scatter within the samples of that mass bin.It is evident that the derived formation times do not depend strongly on the photometric information provided to the modeling.
The population scatter is higher for earlier formation times compared to later ones with Prospector t 10 dominated by the priors on the very first time bin.For quiescent population, we do see highest variation in lowest mass bin 10.5 < log(M/M ) < 11, primarily driven by low number of samples (∼290) in that mass range.

Figure 1 .
Figure1.Stellar mass vs redshift (left), median signal to noise of LEGA-C spectra (center) and stellar velocity dispersion (right) for all 3005 primary LEGA-C Galaxies from the DR3 catalog with density distributions of star-forming (blue) and quiescent (red) populations based on rest-frame UVJ colors.Our effective sample selections are shown with dashdot lines (mass-selected) and solid lines (velocity dispersion-selected).

Figure 2 .
Figure2.Prior density distribution of SFH obtained by drawing 1000 samples from parametrization described in Table 1 (Bagpipes) and Table3(Prospector).Thick lines show the medians (solid) and dashed 1-σ (16th -84th percentile) scatter.The median values follow closely for both the codes, except at the earliest times the analytic function requires SFR(t=0) = 0 and non-parametric models assign non-zero star-formation in the earliest bin.

Figure 3 .
Figure3.Spectro-photometric modeling of four representative quiescent galaxies using LEGA-C spectra and UltraVISTA photometry (grey) along with best fit models from Bagpipes (orange) and Prospector (green) (left).Lower sub-panels show χ values (data-model/error) with grey bands showing regions masked in parametric modeling.Reduced chi-squared values are quoted in inset boxes.The right panels show the median (lines) and 16th -84th percentile distributions of the SFH posteriors.The right column shows the posterior distributions of stellar mass (without mass loss), dust Av, and stellar metallicity.Both modeling techniques yield good fits to the data and overall the SFHs agree reasonably well, with some divergence at early times.

Figure 5 .
Figure5.Individual (left and center) and population (right) SFHs recovered from Bagpipes and Prospector for an example sample of massive (11.5 < log(M * /M ) < 12) quiescent galaxies.The middle panel shows individual SFHs in the sSFR and the right panel shows 16th, 50th and 84th percentile population distributions.Note that Bagpipes generates relatively higher population scatter due to variation in the onset and truncation of the double power law parametrization, in contrast with the early onset of star formation inferred from non-parametric models.
5 < log(M * /M ) < 11 b) 11 < log(M * /M ) < 11.5 c) 11.5 < log(M * /M ) < 12 and four coarse stellar velocity dispersion bins -a) 100 < σ * (km/s) < 150 b) 150 < σ * (km/s) < 200 c) 200 < σ * (km/s) < 250 and d) 250 < σ * (km/s) < 300 and later increase the resolution to finer bins to characterize population trends.Note that the mass and sigma values used to bin galaxies are taken from the LEGA-C DR3 catalog (LOGM_MEDIAN,SIGMA_STARS), described in detail in §2.In each mass/velocity dispersion bin, each individual SFH is weighted by T cor (which includes both a volume and sample correction factor, for details see van der Wel et al. (2021) Appendix A) in the calculations of population median and scatter to account for the LEGA-C survey targeting strategy.

Figure 6 .
Figure6.Median specific star-formation rates (sSFR) with lookback time for populations of star-forming (left), quiescent (center) and all (right) galaxies sorted in bins of their stellar mass.Colors indicate stellar mass bins -blue < green < red in increasing mass.Bagpipes/ Prospector results are shown in dotted-lines/solid-lines with hashed/shaded regions representing 16th-84th percentile population scatter.Bottom most row shows median trends of all mass bins with thin shaded regions representing standard error on the medians via bootstrapping.Transition region from star forming to quiescent is shown in grey band.Star-forming population shows overall better agreement between two methods than quiescent population.Note that Bagpipes has more bursty pathways to reach quiescence.

Figure 8 .
Figure 8. t90 (brown) and t50 (teal) formation times/redshifts of star-forming, quiescent and total galaxy populations versus stellar mass.Error bars/shaded regions indicate the 16th -84th percentile population scatter.The bottom row represents the difference in median values of the Bagpipes and Prospector formation times.To the far right, we show equivalent mass-weighted ages at a representative redshift of z ∼ 0.8.Late formation times (t90) are consistent between models, with greater disagreement at earlier times (t50), especially for quiescent galaxies.The strong mass trends within the full population (right panel) are primarily driven by star forming galaxies that dominate the low-mass end.

Figure 9
Figure 9. Similar to Fig.8, now binned in stellar velocity dispersion.We find a striking correlation between the formation timescales and stellar velocity dispersion in full population (right).This trend is in part driven by demographics; the quiescent fraction depends more strongly on stellar velocity dispersion than stellar mass (e.g.,Taylor et al. 2022).

Figure 10 .
Figure10.The duration of star-formation at late times (τ SF,late ≡ t90 -t50) versus stellar mass (top row) and stellar velocity dispersion (bottom row) for population of star-forming (left), quiescent (center) and all (right) as recovered from Bagpipes and Prospector.Non-parametric models are consistently more extended than the double-power law SFHs.We find no correlation between τ SF,late and stellar mass or stellar velocity dispersion for star-forming galaxies.The weak trends amongst the quiescent galaxies are insignificant relative to the population scatter.This lack of correlation is in contrast with trends found amongst analogous massive elliptical galaxies today (e.g.,Graves et al. 2009; Greene et al. 2022).

Figure 11 .
Figure11.Comparison of median formation times t50 (teal) and t90 (brown) derived for the progenitor populations from the LEGA-C sample compared to those inferred from local elliptical galaxies from ATLAS3D (after correcting for stellar mass growth between z ∼ 0.8 and z ∼ 0)(McDermid et al. 2015) in versus dynamical mass.The left panel shows the comparison with the full LEGA-C population (star-forming and quiescent progenitors) and the right panel compares only quiescent galaxies to the local ellipticals.SPS modeling of local ellipticals finds ubiquitously earlier (t50 = 2 Gyr) star formation (diamonds), which is even more extreme than even non-parametric SFHs for LEGA-C progenitors.The strong trends in later star formation (t90) suggest the importance of star forming progenitors joining the population at M dyn ∼ 10 11 M since z ∼ 0.8.

Table 1 .
Bagpipes modeling parameters and prior distribution functions

Table 2 .
Bagpipes Spectro-photometric modeling resultsIDMask LEGAC_ID zspec σ * ,cat log(M * ,cat/M ) log(M * ,f it /M ) * We display a table snippet for formatting; the full data table is available to download in the MRT format.

Table 3 .
Prospector modeling parameters and prior distribution functions

Table 4 .
Prospector Spectro-photometric modeling resultsIDMask LEGAC_ID zspec σ * ,cat log(M * ,cat/M ) log(M * ,f it /M ) * We display a table snippet for formatting; the full data table is available to download in the MRT format.