Impact of model parameter degeneracy on leptonic radiation models. The case of blazar multi-wavelength spectra

TL;DR

The paper demonstrates that one-zone leptonic blazar SED fits suffer strong parameter degeneracy, driven by non-convex, non-smooth χ^2 surfaces and incomplete data. By comparing grid scans, Minuit, GA, CMA-ES, and MCMC across simulated datasets and applying them to PKS 0735+178 and Mrk 501, it shows that best-fit parameters depend on the optimization method and data coverage, with multiple distinct parameter regions yielding similar goodness-of-fit. The study highlights the necessity of complete, quasi-simultaneous multi-wavelength data (and potentially polarization information) to constrain models, and suggests that future MeV–TeV observations could substantially reduce degeneracy, improving physical interpretation for blazars and other high-energy sources.

Abstract

Leptonic one-zone radiation models are commonly used to describe multi-wavelength data and explore the physical properties of high-energy sources, such as active galactic nuclei. However, these models often require a large number of free parameters. In the context of possible parameter degeneracy and the complex landscape of the parameter space, we study how the choice of the fitting procedure impacts the characterization of the source properties. Furthermore, we examine how the data coverage and the uncertainties associated to the data influence the model parameter degeneracy. We generated simulated spectral energy distribution datasets with different properties, which we then fit with a numerical model. The model describes the relevant radiation processes with seven free parameters. We compare different optimization algorithms and study the parameter degeneracy using t-distributed stochastic neighbor embedding. Additionally, we applied the same fitting procedures to the observational data of two sources, Mrk 501 and PKS 0735+178. We demonstrate significant degeneracies in the seven-dimensional parameter space of the one-zone leptonic models caused by the incomplete wavelength coverage of the data. Given the same goodness-of-fit function, the best-fit result depends on the choice of the minimization algorithm. Source properties extracted from the best-fit solution to realistic datasets cannot be interpreted as the only solution due to significant degeneracies of the model parameters. Adding new energy ranges (e.g. MeV) and regular source monitoring would allow to reduce gaps in the data and significantly decrease the parameter degeneracy.

Figures (15)

• Figure 1: Example of the seven-dimensional parameter space mapping with t-SNE (grid scan results for simulated dataset 2, see Sec \ref{['sec:simdata']}). Each point represents one model consisting of seven parameters. Darker colors correspond to lower $\chi^{2}/$n.d.f. values. The plot shows the first 50000 points with the smallest $\chi^2$/n.d.f. value, and the perplexity is set to 30.
• Figure 2: Simulated datasets. The black dots show the data points in the simulated datasets. The dashed curve shows the original model that was used to generate the data. The grey triangles in the last two plots are upper limits.
• Figure 3: Results of the SED fitting for dataset 1. Left panel, upper plot: the best-fit results from all selected optimization algorithms. Left panel, lower plot: relative deviation between the true values of $\nu F_\nu$ and those of the best-fit solutions. Right panel: the location of the best-fit solutions in the global parameter space shown in a t-SNE map.
• Figure 4: Same as Fig. \ref{['fig:results_simdata1']} but for dataset 2.
• Figure 5: Same as Fig. \ref{['fig:results_simdata1']} but for dataset 3.