Deltaage_n,min: SFH Duration Resolution

• Deltaage_n,min is the minimum normalized SFH duration above which extended star formation can be detected, marking a time-resolution limit in integrated-light data.
• It is determined using a Bayesian analysis of composite stellar population models that leverage spectral indices like D4000n and Balmer lines across varying metallicities and SNRs.
• The study finds a nearly constant log value (~ -0.3 dex) over a wide age range, implying that distinguishable SFH durations must be at least about one-third of the median age.

$\Delta \mathrm{age}_{n,\min}$ is the limiting value of the normalized star-formation-history duration parameter $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$ above which an extended star formation history can be distinguished from one of negligible duration in integrated-light data. In the formulation introduced for Bayesian characterization of galaxy star formation histories, $\Delta \mathrm{age}_{n,\min}$ is explicitly the value of $\Delta \mathrm{age}_n$ from which the adopted set of spectral features starts depending on the duration of the SFH, and therefore acts as a time-resolution limit for SFH duration measurements (Rossi, 8 Jul 2025).

1. Formal definition and percentile-age framework

The construction begins from mass–formation percentiles. If $SFR(t)$ is the star formation rate as a function of time since the beginning of star formation, the cumulative formed stellar mass up to time $\bar t$ is

$$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$

For a mass fraction $f$, the formation time $t_f$ is defined by

$f=A\int_0^{t_f} SFR(t')\,dt',$

and the corresponding look-back age is

$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$0

The specific percentile ages used are $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$1, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$2, and $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$3, corresponding to the look-back times when $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$4, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$5, and $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$6 of the stellar mass has formed. The absolute duration of the central $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$7 of stellar mass assembly is

$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$8

Because $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$9 scales with the absolute age of the system, the analysis introduces the dimensionless relative duration parameter

$\Delta \mathrm{age}_{n,\min}$0

Within this framework, $\Delta \mathrm{age}_{n,\min}$1 is the minimum relative SFH duration for which the adopted indices and colours carry detectable information about duration. For $\Delta \mathrm{age}_{n,\min}$2, the models are treated as observationally indistinguishable from an instantaneous burst. The interpretation encoded in the same framework is qualitative but direct: $\Delta \mathrm{age}_{n,\min}$3 corresponds to an SFH that is essentially a short burst around the median age, $\Delta \mathrm{age}_{n,\min}$4 corresponds to a moderately extended SFH, and $\Delta \mathrm{age}_{n,\min}$5 corresponds to a very extended SFH (Rossi, 8 Jul 2025).

2. Idealized CSP experiment used to determine $\Delta \mathrm{age}_{n,\min}$6

The time-resolution limit is derived first in an idealized composite stellar population library designed to isolate the information content of the observables from additional astrophysical degeneracies. The library contains $\Delta \mathrm{age}_{n,\min}$7 million CSP models, organized as $\Delta \mathrm{age}_{n,\min}$8 subsets of $\Delta \mathrm{age}_{n,\min}$9 million models each, with fixed metallicities

$\Delta \mathrm{age}_n$0

Each model uses a pure Sandage (1986) continuous SFH with no random bursts,

$\Delta \mathrm{age}_n$1

no dust attenuation, and randomly drawn $\Delta \mathrm{age}_n$2 and $\Delta \mathrm{age}_n$3, producing a wide range of ages from $\Delta \mathrm{age}_n$4 to $\Delta \mathrm{age}_n$5 yr (Rossi, 8 Jul 2025).

The observables used for the time-resolution experiment are the same as those later used in the Bayesian analysis. The spectral set contains five indices: $\Delta \mathrm{age}_n$6, $\Delta \mathrm{age}_n$7, $\Delta \mathrm{age}_n$8, $\Delta \mathrm{age}_n$9, and $SFR(t)$0. The photometric set begins from SDSS ugriz fluxes, but because the absolute normalization is a free parameter in the time-resolution test, only the four independent colours $SFR(t)$1, $SFR(t)$2, $SFR(t)$3, and $SFR(t)$4 are used. In this setup, $SFR(t)$5 is defined as

$SFR(t)$6

while $SFR(t)$7 and $SFR(t)$8 are defined by

$SFR(t)$9

$\bar t$0

The noise model assumes constant spectral SNR per rest-frame $\bar t$1, with tested values $\bar t$2, $\bar t$3, $\bar t$4, $\bar t$5, $\bar t$6, $\bar t$7, and $\bar t$8. Photometric errors are fixed at $\bar t$9 for $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$0 and $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$1 for $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$2. For $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$3, an additional $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$4 flux-calibration term is added: $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$5 This idealized construction is explicitly intended to provide a theoretical limit free from the extra degeneracies introduced by dust, bursts, and metallicity evolution (Rossi, 8 Jul 2025).

3. Operational criterion for the minimum resolvable duration

The extraction of $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$6 is based on the empirical transition from a flat-response regime to a duration-sensitive regime. At fixed $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$7, the spectral features display a flat plateau for small $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$8, meaning that they are essentially independent of SFH duration. Above a critical $$M(\bar t)=A\int_0^{\bar t} SFR(t')\,dt', \qquad A=\frac{1}{\int_0^{t_{\rm form}} SFR(t')\,dt'}.$$9, the indices and colours begin to show systematic trends with duration (Rossi, 8 Jul 2025).

To quantify this transition, models with

$f$0

are defined as the reference regime, treated as indistinguishable from an instantaneous burst. For each model $f$1, a multi-feature distance statistic is then defined: $f$2 Here, $f$3 is the perturbed value of observable $f$4, $f$5 is the mean of that observable across the reference regime, and $f$6 is the total scatter within the reference regime, including both intrinsic scatter and observational noise.

The analysis is performed in narrow $f$7 bins of width $f$8 dex from $f$9 to $t_f$0, and separately for each metallicity and SNR. For each such combination, the running median, $t_f$1th percentile, and $t_f$2th percentile of $t_f$3 are measured as functions of $t_f$4. The minimum resolvable duration is then defined by the intersection

$t_f$5

Equivalently, $t_f$6 is the smallest $t_f$7 at which the running $t_f$8th percentile of $t_f$9 crosses the threshold given by the mean plus one standard deviation of the reference level. If no such intersection occurs, no time resolution is claimed in that age bin (Rossi, 8 Jul 2025).

An explicit methodological caveat is attached to this definition. The text states that dependence begins when the running median is away two times the standard deviation from the mean reference level, but the implementation described uses the running $f=A\int_0^{t_f} SFR(t')\,dt',$0th percentile and the threshold $f=A\int_0^{t_f} SFR(t')\,dt',$1. This makes the adopted criterion a conservative start-of-systematic-deviation threshold rather than a direct two-$f=A\int_0^{t_f} SFR(t')\,dt',$2 crossing.

4. Numerical behaviour across age, metallicity, and SNR

The principal numerical result is that $f=A\int_0^{t_f} SFR(t')\,dt',$3 is approximately constant over a very large age range. The study reports a roughly flat trend of $f=A\int_0^{t_f} SFR(t')\,dt',$4 around $f=A\int_0^{t_f} SFR(t')\,dt',$5 dex over $f=A\int_0^{t_f} SFR(t')\,dt',$6 orders of magnitude in age, from $f=A\int_0^{t_f} SFR(t')\,dt',$7 to $f=A\int_0^{t_f} SFR(t')\,dt',$8 yr (Rossi, 8 Jul 2025). Quantitatively, depending on age and SNR, the typical range is

$f=A\int_0^{t_f} SFR(t')\,dt',$9

This means that the SFH duration must generally be at least $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$00 to $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$01 of the median age in order to be distinguishable from a burst.

The age dependence is structured rather than monotonic. For very young populations, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$02 yr, the behaviour is irregular, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$03 is tiny, Balmer lines vary rapidly, and the time resolution fluctuates. For $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$04 yr, the paper reports $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$05, which corresponds to worse resolution; in this regime the spectra are dominated by strong Balmer lines and fewer metal lines, and SFH degeneracies are severe. For $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$06 yr, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$07, sometimes approaching $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$08, and the improvement is attributed to the emergence of multiple metal indices and strong $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$09 (Rossi, 8 Jul 2025).

A hard floor is also identified. The analysis emphasizes that time-resolution values lower than $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$10 dex are not achieved in any case, corresponding to

$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$11

In the authors’ interpretation, the approximately constant time resolution relative to $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$12 means that the available diagnostics recover information only on a minimum time interval of order $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$13.

The SNR dependence shows an initial improvement followed by saturation. At low SNR, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$14–$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$15, time resolution is dominated by photometry alone and the spectral indices add almost no extra information. At intermediate SNR, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$16, the indices begin to contribute significantly. At high SNR, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$17–$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$18, the time resolution improves further, but beyond $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$19 the curves for $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$20, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$21, and $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$22 almost overlap, indicating that intrinsic model degeneracy, not statistical noise, has become the limiting factor. The metallicity dependence in the idealized fixed-$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$23 case is described as weak: the time-resolution maps are similar across the five metallicities, even though metallicity shifts the locus of stellar-population evolution in diagnostic planes (Rossi, 8 Jul 2025).

5. Relation to realistic Bayesian retrieval of SFH duration

The paper distinguishes sharply between the idealized time-resolution limit $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$24 and the practical recovery of $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$25 in a realistic CSP library (Rossi, 8 Jul 2025). The realistic library contains $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$26 models generated with SEDlibrary, combining a continuous Sandage SFH with up to $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$27 random bursts, variable metallicity following

$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$28

random initial and final metallicities between $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$29 and $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$30, and Charlot and Fall (2000) dust attenuation. Bayesian fitting is then performed with BaStA on $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$31 mock observations perturbed at $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$32, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$33, and $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$34.

The likelihood is written as

$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$35

and posterior PDFs are marginalized to derive $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$36, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$37, $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$38, and $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$39. In this realistic setting, the study reports that $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$40 can be constrained within $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$41 dex for most of the sample. For populations with strong Balmer absorption and mean age $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$42 yr, however, the uncertainty exceeds $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$43 dex because of SFH degeneracies on the resulting galaxy spectra (Rossi, 8 Jul 2025).

This contrast defines the role of $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$44. It is a best-case theoretical limit derived under fixed metallicity, zero dust, and smooth burst-free SFHs. The practical retrieval of $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$45 is broader because bursts, dust, and metallicity evolution introduce additional degeneracies. The study therefore states that, at odds with the limiting time resolution, for which a steady increase of performance is observed for increasing SNR up to $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$46, in the case of $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$47 for complex SFH the accuracy is limited by intrinsic degeneracies already at $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$48, and the gain in going at much larger SNR is marginal.

6. Interpretation, survey relevance, and limitations

$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$49 is not a direct posterior uncertainty and it is not an absolute duration scale; it is the minimum distinguishable relative SFH duration implied by the information content of the chosen integrated-light diagnostics. For a galaxy with known $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$50, the corresponding absolute best-case time resolution is set by

$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$51

Using the empirical floor $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$52, the analysis gives concrete examples. For $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$53 yr, the minimum distinguishable duration is $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$54 yr; for $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$55 yr, the theoretical limit is $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$56 yr, although realistic degeneracies make the usable precision weaker (Rossi, 8 Jul 2025).

The survey implications are correspondingly specific. For SDSS-like spectra with $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$57, the method is able to constrain $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$58 within $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$59 dex for most galaxies with $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$60 yr, while younger or Balmer-strong galaxies remain poorly constrained. For future deeper surveys such as StePS/WEAVE and LEGA-C, the study argues that intermediate-age and old populations can approach the theoretical time-resolution limit, but also emphasizes that claims of sub-Gyr time resolution at $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$61–$$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$62 for typical galaxies are physically unrealistic if they are based solely on integrated-light spectra and similar diagnostics (Rossi, 8 Jul 2025).

The principal limitations are tied to the modelling assumptions. The stellar population synthesis setup uses BC03 (2016) with the MILES stellar library and a Chabrier IMF. The idealized library assumes smooth Sandage SFHs, fixed metallicity per subset, and no dust, whereas the realistic library adds bursts, dust, and metallicity evolution. Age–metallicity degeneracy, dust–age degeneracy, and SFH degeneracies are explicitly identified as limiting factors, especially in the Balmer-peak regime where old components are overshined by A-type stars. A plausible implication is that $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$63 should be regarded as a model- and observable-dependent resolution limit rather than a universal constant, even though the reported value around $$\Delta \mathrm{age}_n \equiv (\mathrm{age}_{10}-\mathrm{age}_{90})/\mathrm{age}_{50}$$64 dex is remarkably stable within the adopted framework (Rossi, 8 Jul 2025).