Deltaage_n: SFH Duration Descriptor

• Deltaage_n is a dimensionless descriptor quantifying the relative spread between the 10% and 90% mass assembly ages normalized by the median, providing a non-parametric summary of galaxy SFHs.
• It is derived by comparing spectral indices and broadband colours against comprehensive composite stellar population libraries using Bayesian inference.
• The metric distinguishes between burst-like and extended star formation processes, offering insights into multi-episode SFHs with practical applications in large spectroscopic surveys.

$\Delta\mathrm{age}_n$ is a dimensionless descriptor of galaxy star formation history (SFH) duration defined from mass-fraction formation ages rather than from a parametric timescale. In the formulation developed for Bayesian stellar-population analysis, it is the normalized interval between the look-back ages at which $10\%$ and $90\%$ of the total formed stellar mass has been assembled, divided by the median formation age $\mathrm{age}_{50}$ (Rossi, 8 Jul 2025). It is introduced to characterize SFH extension beyond mean stellar age, and is inferred statistically by comparing observed spectral indices and broadband colours against large composite stellar population (CSP) libraries.

1. Definition from cumulative mass assembly

The construction starts from the SFH $SFR(t)$, written as a function of time since the onset of star formation. The cumulative formed stellar mass is

$M(t)=\int_0^t SFR(t')\,dt',$

with normalization $M(t_{\rm form})=M_{\rm tot}$, where $t_{\rm form}$ is the time of observation since the beginning of the SFH. For a fixed formed-mass fraction $f$, the formation time $t_f$ is defined by

$10\%$0

and the corresponding look-back age is

$10\%$1

The specific percentile ages used are $10\%$2, $10\%$3, and $10\%$4, corresponding to the epochs when $10\%$5, $10\%$6, and $10\%$7 of the total formed stellar mass has been produced. These are explicitly mass-fraction ages, not light-fraction ages (Rossi, 8 Jul 2025).

The absolute 10–90 duration is

$10\%$8

and the normalized duration is

$10\%$9

By construction, $90\%$0 is a post-processed descriptor of the SFH. It is not a directly specified free parameter of the population-synthesis generator, but is computed from the mass-assembly history of each model.

2. Interpretation as a relative SFH-duration measure

$90\%$1 measures how extended the interval between $90\%$2 and $90\%$3 mass assembly is relative to the median formation age. Large values, $90\%$4, correspond to very extended or multi-episode SFHs; values of order unity correspond to star formation persisting over a duration comparable to the median age; small values, $90\%$5, indicate short, burst-like SFHs (Rossi, 8 Jul 2025).

Its normalization by $90\%$6 makes it a relative timescale rather than an absolute one. This permits direct comparison of galaxies with very different absolute ages and redshifts on the same footing. The quantity is therefore intended as a shape measure of the SFH, not simply as another age estimator.

The paper contrasts $90\%$7 with moment-based age summaries. The mass-weighted age is

$90\%$8

and the light-weighted age is

$90\%$9

where $\mathrm{age}_{50}$0 is the luminosity of a $\mathrm{age}_{50}$1-$\mathrm{age}_{50}$2 SSP of age $\mathrm{age}_{50}$3. These are moments of the SFH, whereas $\mathrm{age}_{50}$4 probes the spread of formation times around the median. A key motivation is that a single age, even a mass-weighted one, does not distinguish a short early burst from a prolonged SFH that yields the same mean age.

A common misconception is to read $\mathrm{age}_{50}$5 as a detailed reconstruction of the SFH. The underlying result is narrower: it summarizes global duration between the $\mathrm{age}_{50}$6 and $\mathrm{age}_{50}$7 formation milestones. The paper states explicitly that it cannot distinguish detailed shapes such as two widely separated bursts versus a smoothly rising SFH if they share similar 10–50–90 percentiles.

3. Role in SFH parameterization and Bayesian inference

The CSP models are built from a smooth Sandage continuous component,

$\mathrm{age}_{50}$8

with parameters $\mathrm{age}_{50}$9 and $SFR(t)$0, plus up to 6 stochastic bursts, each with its own age, metallicity, and mass fraction. Metallicity evolves as a power-law function of cumulative formed mass, and dust attenuation follows the Charlot & Fall (2000) two-component law (Rossi, 8 Jul 2025).

Within that setup, the fundamental SFH inputs are $SFR(t)$1, $SFR(t)$2, and burst parameters, but the proposed analysis emphasizes a smaller set of model-independent derived quantities: light-weighted age, mass-weighted age, median age $SFR(t)$3, and $SFR(t)$4. The rationale is that percentile ages and their ratios provide a non-parametric, physically transparent summary of when the bulk of the mass was formed.

Inference is Bayesian. For an observed galaxy, the data vector $SFR(t)$5 includes five spectral indices—D4000n, $SFR(t)$6, H$SFR(t)$7, $SFR(t)$8, and $SFR(t)$9—together with SDSS ugriz photometric fluxes, used as four independent colours $M(t)=\int_0^t SFR(t')\,dt',$0, $M(t)=\int_0^t SFR(t')\,dt',$1, $M(t)=\int_0^t SFR(t')\,dt',$2, and $M(t)=\int_0^t SFR(t')\,dt',$3. The posterior over model parameters $M(t)=\int_0^t SFR(t')\,dt',$4 is written as

$M(t)=\int_0^t SFR(t')\,dt',$5

with Gaussian likelihood

$M(t)=\int_0^t SFR(t')\,dt',$6

and

$M(t)=\int_0^t SFR(t')\,dt',$7

The posterior of a derived parameter $M(t)=\int_0^t SFR(t')\,dt',$8, such as $M(t)=\int_0^t SFR(t')\,dt',$9, is then obtained by marginalization over all models whose derived value falls in the relevant bin. The median of this PDF is used as the estimator, and the 16–84 percentile range as the uncertainty. Thus $M(t_{\rm form})=M_{\rm tot}$0 is not fitted directly; it is inferred statistically through model comparison.

4. CSP libraries, observables, and the time-resolution limit

The CSP libraries are generated with the SEDlibrary C code using BC03 (2016) SSPs with MILES-based spectra and a Chabrier IMF. Two principal libraries are used. The first is an idealized 5 million model library for time-resolution tests, with no dust, no bursts, fixed metallicity in each of five subsamples, and pure Sandage SFHs. The second is a realistic 500,000-model library with variable metallicity, dust, and up to 6 bursts (Rossi, 8 Jul 2025).

For each model, the SFH is discretized and integrated to obtain $M(t_{\rm form})=M_{\rm tot}$1, from which $M(t_{\rm form})=M_{\rm tot}$2, $M(t_{\rm form})=M_{\rm tot}$3, and $M(t_{\rm form})=M_{\rm tot}$4 are solved and converted into ages. The same model library also yields the spectral indices and colours used for Bayesian comparison. The observables have differentiated diagnostic roles: D4000n and the Balmer lines are primarily age-sensitive; $M(t_{\rm form})=M_{\rm tot}$5 and $M(t_{\rm form})=M_{\rm tot}$6 are mostly metallicity-sensitive and help break the age–metallicity degeneracy; colours encode broad SED shape, recent star formation, and dust effects.

The paper defines a minimum observable SFH duration, $M(t_{\rm form})=M_{\rm tot}$7, as the minimum $M(t_{\rm form})=M_{\rm tot}$8 above which the chosen spectral features start depending on SFH duration. Operationally, for narrow bins in $M(t_{\rm form})=M_{\rm tot}$9 and metallicity, a distance from effectively instantaneous SFHs is defined as

$t_{\rm form}$0

where the reference set uses models with $t_{\rm form}$1. $t_{\rm form}$2 is then identified where the running 16th percentile of $t_{\rm form}$3 crosses $t_{\rm form}$4, corresponding to deviations significant at approximately $t_{\rm form}$5.

Over 4 orders of magnitude in $t_{\rm form}$6, the relative time resolution is reported as remarkably flat, with $t_{\rm form}$7 to $t_{\rm form}$8 depending on SNR and metallicity. For a standard SNR$t_{\rm form}$9 case and $f$0 yr, the result is $f$1 dex within approximately $f$2 dex, implying $f$3. The abstract summarizes this as a roughly flat $f$4 around $f$5 dex over 4 magnitude orders in age.

The age dependence is structured. For $f$6 yr, the behaviour is irregular because very young populations have rapidly changing SEDs. For $f$7 yr, the resolution is worse, with $f$8, reflecting chaotic Balmer-line behaviour before their peak and the lack of strong metal lines. Around $f$9 yr, $t_f$0 reaches a minimum because Balmer lines evolve most rapidly there. For $t_f$1 yr, resolution improves again owing to the combined leverage of Balmer and metal-sensitive indices.

5. Recovery performance and degeneracy structure

The mock-analysis stage selects 12,500 models from the realistic 500,000-model library, perturbs their observables to simulate SNR$t_f$2, $t_f$3, and $t_f$4, and refits them with the Bayesian algorithm. The main reported result is that $t_f$5 can be constrained within $t_f$6 dex for most of the sample, while populations with strong Balmer absorption and mean stellar age $t_f$7 yr show uncertainties exceeding $t_f$8 dex because of SFH degeneracies (Rossi, 8 Jul 2025).

More detailed trends are consistent with that summary. At SNR$t_f$9 and $10\%$00, the Bayesian error on $10\%$01 is typically $10\%$02–0.4 dex for the bulk of models and $10\%$03 dex in regions with strong Balmer absorption. The problematic regime is the Balmer-peak region, where the light is dominated by A-type stars. In that region, short recent bursts, longer SFHs with an old underlying component, and multiple bursts can produce similar Balmer strengths and colours. The old component is overshined in light but still affects $10\%$04 and $10\%$05, which makes $10\%$06 strongly degenerate.

An important negative result is that increasing SNR from $10\%$07 to $10\%$08 does not dramatically improve $10\%$09 constraints. The limiting factor is SFH degeneracy rather than noise. The paper contrasts this with metallicity, for which higher SNR continues to help.

The favourable regime is more localized. For D4000n $10\%$10, $10\%$11, and typical $10\%$12 yr, the inferred distribution is

$10\%$13

corresponding to $10\%$14. In that locus, extended SFHs can therefore be measured with relatively high precision. By contrast, the paper notes that within many Balmer-plane bins the output scatter remains small even when the input library has broader $10\%$15 structure, which suggests that the posterior can be dominated by the library prior plus the location in the Balmer plane rather than by independent resolving power from the full observable set.

6. Applications, interpretation of values, and limitations

The intended application domain is large spectroscopic surveys such as WEAVE/StePS or LEGA-C, and more generally galaxy-archaeology analyses where one seeks a minimal but robust characterization of SFHs from spectra and photometry (Rossi, 8 Jul 2025). In that context, $10\%$16 complements median age and metallicity by encoding whether stellar mass assembly was rapid or extended relative to the observation-time age of the system.

The paper provides explicit interpretive anchors. Values around $10\%$17, or $10\%$18, correspond to a 10–90 duration about one third of $10\%$19, close to the minimum typically resolvable duration and therefore observationally consistent with a relatively short formation episode. Values near $10\%$20 indicate duration comparable to $10\%$21, characteristic of broader SFHs. Values $10\%$22, or $10\%$23, indicate very extended or re-juvenated SFHs and are found particularly in models with an old population plus a strong recent burst.

A concrete resolution example is given for an elliptical galaxy with $10\%$24 yr: if $10\%$25, then $10\%$26, so one cannot constrain SFH structure on scales $10\%$27 yr. This suggests that $10\%$28 should be read together with the time-resolution floor $10\%$29, not as an unrestricted chronometer.

The principal limitations are explicit. $10\%$30 is a global SFH descriptor; it does not resolve detailed temporal morphology if different histories share similar percentile ages. It is subject to strong degeneracies in blue, Balmer-dominated populations where old stars are weakly constrained in light. Practical constraints remain noise- and prior-dependent. Within those limits, the metric provides a compact and physically interpretable summary of SFH duration that is more robust across model families than raw parametric quantities such as $10\%$31 or $10\%$32.