Physically-Motivated Spectral Library for MS SFGs

• Physically-motivated spectral libraries are catalogs of galaxy spectra that encode realistic variations in star formation history, dust properties, and emission mechanisms using physics-driven models.
• The approach employs the Extended Regulator model and Gaussian Process sampling to generate synthetic spectra, linking stochastic SFH variations directly to observable spectral diagnostics.
• This framework enhances SED fitting and parameter inference in galaxy surveys, providing a robust tool for bridging theoretical models with observational data.

A physically-motivated spectral library of main-sequence star-forming galaxies (MS SFGs) is a structured catalog of galaxy spectral energy distributions (SEDs) and spectral templates that encode realistic, physics-informed variations in star formation history (SFH), dust properties, and emission mechanisms. Such libraries aim to reflect the underlying stochastic and regulatory processes in galaxy formation and evolution, providing a foundation for more accurate SED fitting, k-corrections, diagnostics, and parameter estimation in galaxy surveys. This approach utilizes analytic models (e.g., the Extended Regulator formalism), Gaussian Process kernels based on physical covariance functions, and population synthesis calculations to map physically-motivated priors onto observable spectral features, with explicit dependence on timescales for gas inflow, equilibrium, and dynamical processes.

1. Stochastic Regulation of Star Formation in Galaxies

Star formation in MS SFGs exhibits variability across multiple timescales, arising from stochastic processes governing gas inflow, reservoir equilibrium, and giant molecular cloud (GMC) formation/destruction. The Extended Regulator model formalizes this paradigm by expressing the SFR as a product of damped random walks, each characterized by distinct physical timescales: τ_in for inflow, τ_eq for equilibrium, and τ_dyn for GMC lifecycles. The combined impact of these processes is captured in the power spectral density (PSD) of SFR variability:

$$S_{\rm ExReg}(f) = \frac{s_{\rm gas}^2}{1 + [(2\pi \tau_{\rm eq})^2 + (2\pi \tau_{\rm in})^2] f^2 + (2\pi \tau_{\rm eq})^2 (2\pi \tau_{\rm in})^2 f^4} + \frac{s_{\rm dyn}^2}{1 + (2\pi \tau_{\rm dyn})^2 f^2}$$

The corresponding auto-covariance function (ACF), determined via the Wiener–Khinchin theorem, is used to encode the time-domain correlations in SFR:

$$C(\Delta t) = \int_{-\infty}^\infty S(f) e^{+2\pi i f \Delta t} df$$

This formalism provides a direct link between the physical burstiness in SFH and its observable imprint on the SED.

2. Gaussian Process Modeling of Star Formation Histories

Physically-motivated spectral libraries incorporate Gaussian Process (GP) models in log(SFR) space, enabling generation of ensemble SFHs consistent with analytic covariance structures. Each SFH realization is sampled as:

$$\ln\,{\rm SFR}(t) \sim {\rm GP}(\ln\,{\rm SFR}_{\rm base}(t), C_{\rm phys}(\Delta t))$$

where $C_{\rm phys}$ is the ACF determined from the physical model (e.g., Extended Regulator kernel), and $\ln\,{\rm SFR}_{\rm base}$ is a deterministic baseline. This GP formulation is advantageous because it provides fast, non-parametric sampling with physically meaningful correlations, facilitating efficient construction of synthetic spectra for a range of underlying physical scenarios. The spectral library can therefore encode variations in stochastic parameters $$(\tau_{\rm in},\ \tau_{\rm eq},\ \tau_{\rm dyn},\ s_{\rm gas},\ s_{\rm dyn})$$ that systematically sample the plausible diversity of MS SFG SFHs.

3. Spectral Predictions from Stochastic SFH Realizations

The variability in SFH, as modeled above, is propagated to synthetic spectra using stellar population synthesis codes (e.g., FSPS). Perturbations to the underlying stochasticity model—such as adjustments to timescales or scatter parameters—produce differentiable changes in key spectral diagnostics:

• H$\alpha$ and UV flux: traces star formation on $\sim$5–10 Myr timescales.
• H$\delta$ absorption: sensitive to bursts and quenching events on 0.1–1 Gyr scales.
• Ca–H,K absorption lines: react to the age distribution and duration/smoothness of SFH.
• $D_n$(4000) break: shifts with changes in median stellar age and star formation continuity.
• Broadband colors: reflect the integrated effects of burstiness, quenching, and age distributions.

Distributions (and joint distributions) of these features across synthetic galaxy populations constitute testable predictions for surveys (e.g., Hubble, JWST, Roman). Spectral libraries constructed via this approach naturally inherit and encode the imprint of physically-motivated SFH variability.

4. Construction and Parameterization of the Spectral Library

Physically-motivated spectral libraries assign each entry/spectrum explicit values for the regulatory model parameters. For example, with GP sampling of SFHs using Extended Regulator kernels, each synthetic spectrum is indexed by $$(\tau_{\rm in},\ \tau_{\rm eq},\ \tau_{\rm dyn},\ \sigma_{\rm gas},\ \sigma_{\rm dyn})$$, along with a baseline SFH and metallicity. Critical mathematical forms include:

• Extended Regulator PSD (see above)
• Auto-covariance functions $C_{\rm phys}(\Delta t)$ calculated via Wiener–Khinchin
• Special cases: For $\tau_{\rm in} = \tau_{\rm eq}$,

$$C_{\rm Reg}(\Delta t) = \sigma_{\rm gas}^2 (1 + |\Delta t|/\tau_{\rm eq})\, e^{-|\Delta t|/\tau_{\rm eq}}$$

Systematic variation of these parameters generates a library that densely samples the physically reasonable range of burstiness and temporal correlation as governed by galaxy-scale gas physics.

5. Implications for SED Fitting and Galaxy Survey Analysis

The physically-motivated spectral library for MS SFGs represents a dynamic, high-dimensional ensemble rather than a static template grid. Its integration into SED-fitting frameworks (e.g., Prospector, Dense Basis) as a prior enables direct propagation of physically-relevant SFH covariance into parameter inference. The GP-based approach allows for flexible incorporation of non-parametric SFH structure and full time covariance. Empirically predicted spectral features (H$\alpha$, UV, H$\delta$, Ca–H,K, $D_n$(4000)) can be used as diagnostics to constrain SFH stochastic parameters in observed samples, thereby informing models of galaxy formation and evolution.

Combining high S/N spectra from current (Hubble, JWST) and future (Roman) observatories provides the opportunity to leverage the spectral library’s predicted feature distributions to test and refine regulatory scenarios, paper the scatter in the star formation main sequence, and constrain burstiness as a function of physical environment.

6. Future Directions and Applications

This physically-motivated framework has several key applications:

• Parameter inference in surveys: Enables robust fitting of observed SFGs, capturing realistic SFH variation.
• Predictive modeling: Allows for the forward modeling of spectral feature distributions under physical assumptions, enhancing the interpretability of future survey data.
• Simulations and theory linkage: Provides a direct bridge between analytic regulatory models, hydrodynamical simulations, and observations by encoding physical process timescales and variabilities in the observable SEDs.
• Expansion to multi-wavelength diagnostics: Such libraries can be extended to incorporate radio, infrared, and UV diagnostics, calibrated to regulatory SFH models.

The computational efficiency and flexibility of the GP–Regulator approach further facilitate integration into large-scale SED modeling pipelines, supporting both rapid parameter estimation and detailed exploration of the impact of physical stochasticity on galaxy observables.

7. Summary of Conceptual and Mathematical Framework

By explicitly linking the regulatory timescales and amplitudes of stochastic processes in galaxy formation to both the frequency-domain variability (PSD) and time-domain correlation (ACF), and sampling SFHs via physically-motivated Gaussian Processes, these spectral libraries encapsulate realistic, physics-driven spectral diversity. This enables more accurate k-corrections, SFR measurements, understanding of main sequence scatter, and empirical constraints on star formation regulation physics in MS SFGs. The formalism articulated in the Extended Regulator model and its GP kernel forms the core of this approach, with direct applicability to survey analysis and model testing across the next generation of galaxy studies.