Parameter Estimation Limits in Blazars

Abstract

Parameter degeneracy in blazar spectral energy distributions (SEDs) is known but rarely quantified. This paper introduces a Fisher Information approach to determine theoretical limits to information extraction in the context of one-zone models. By evaluating the total Fisher Information by varying $δ$, $B$, $p$, $γ_{\rm min}$ and $γ_{\rm max}$, we find that EC models encode Fisher information $\gtrsim10^4$ times less than that in SSC models, establishing differences in limits of physical information extraction even in the case of perfect sampling. Moreover, the Fisher information in both SSC and EC models exhibit strong fluctuations across the parameter space, but since the magnitudes are orders of magnitude lower in EC, limits of parameter inference are expected to be worse in FSRQ SEDs than BL Lacs. We also find that the Doppler factor $δ$ carries at least $10^{2-3}$ more Fisher information than that for $p$ and $B$ in both EC and SSC, making $δ$ the most constrained SED parameter. Applying our Fisher Information motivated framework to real flaring SEDs of Flat Spectrum Radio Quasars (FSRQs) CTA 102 and 3C 279, we show that mild variations in $δ$ and $p$ can appreciably produce the flaring SEDs starting from the quiescent model, while two other flares in 3C 279 simple geometric and spectral considerations cannot reproduce the flares, reducing the efficacy of one-zone models. We propose that time-resolved SED models are indispensable to constraining physical parameters in EC-dominated blazars.

Figures (7)

• Figure 1: Response functions $\mathcal{R}$ plotted for all relevant physical parameters for the SSC (left) and EC models (right panel). The behaviour of $\delta$ and $p$ are similar in both cases, where they are very sensitive to sharp changes in the spectra. For $\delta$, it is more dominant at EC due to stronger beaming. The magnetic field and electronic energies depict the synchrotron-SSC peak and the cut-off frequencies respectively and their response functions hence show sharp behaviour around those frequencies. $N$ and $R$ control SED normalization and hence do not show sudden changes due to change in model flux across frequency.
• Figure 2: Figure shows the total Fisher information, encoded in $\det\mathcal{F}$ for different parameter sets, marginalized over the maximum values for all the other parameters not displayed in the figure. The Total Fisher Information for $\delta$ v/s $B$ and $p$ (upper and lower panels respectively) for SSC and EC are shown in left and right panels respectively. The Fisher information is orders of magnitude higher $\gtrsim 10^{3-4}$ in SSC models than EC models.
• Figure 3: Figure shows the total Fisher information, encoded in $\det\mathcal{F}$ for $\gamma_{\rm min}$ v/s $\gamma_{\rm max}$ for SSC and EC (left and right panels respectively). The Total Fisher Information for $\delta$ v/s $B$ and $p$ (upper and lower panels respectively) for SSC and EC are shown in left and right panels respectively. The average Fisher information in SSC and EC models are very similar, but the lowest values of SSC are orders of magnitude smaller than that in EC. One notices a drop in the information at larger values of $\gamma_{\rm min}$ perhaps expected from smoothening out of the frequency gap between the low and high-energy components.
• Figure 4: Fisher Information Maps for $\delta$ for the SSC (left) and EC models (right). $\mathcal{F}_{\delta\delta}$ is remarkably high $\sim10^{4-8}$ throughout the two models, suggesting it is the most well-constrained parameter in one-zone SED fits.
• Figure 5: Left Panel : Fisher Information Maps for $B$ for the SSC (left) and EC models (right). $\mathcal{F}_{BB}$ is remarkably stable varying only by $10^{0.1}$ throughout the parameter space in SSC. The EC case suggests more fragility/sensitivity to the underlying parameter space, while the values span three orders of magnitude, are larger than that in SSC. Right Panel : The off-diagonal elements of the FIM between $\delta$ and $B$ for each of SSC and EC models. The magnitudes are lower, as expected, than $\sqrt{\mathcal{F}_{\delta B}\mathcal{F}_{\delta\delta}}$. In the SSC case the FIM map shows little variation $\sim 10^1$ and the magnitudes are $\sim10^4$ on average, while in the EC case, the magnitudes are few orders of magnitude larger than SSC, suggesting degeneracies between $\delta$ and $B$ in numerous parts of the parameter space are higher than in the SSC case.