&= \frac{1}{\int_{0}^{\infty} \frac{d\omega}{2\pi}  \frac{4 |\frac{\partial P_{t}}{\partial E_{a}}(\omega)|^{2}}{S_{P_{t}-total}    (\omega)}}\\
                                        &\sim \frac{S_{P_{t}\,G_{tb}}(\omega=0)}{\left(\frac{G_{ta,a}}{G_{ta,a} \!+ \!G_{ab}}\right)^{2}\omega_{ta}}\\
                                        &\sim \sigma_{Et}^2 
                                                        \left(\frac{G_{ta,a} \!+ \!G_{ab}}{G_{ta,a}}\right)^{2}
                                                        \frac{\omega_{S/N \, \delta \! P_{t}}}{\omega_{ta}}                                     
\end{split}
\label{eq:bandwidth_mismatch}$$

Basically, any signal whose bandwidth is smaller than $\omega_{S/N \, \delta \! P_{t}}$ will suboptimally use the sensor bandwidth, leading to poorer resolution as discussed in [@Sadoulet_96_BandwidthMismatch]. To see this in another way, notice that both $\omega_{S/N \, \delta \! P_{t}}$ (Eq. \[eq:w\_OF\_Pt\]) and $S_{p\,total}$ (which is roughly $S_{pt\,G_{tb}}$ as in Eq.\[eq:Spt-Gtb\]) both scale linearly with $G_{tb}$. Thus, as long as the bandwidth of the signal that is being measured is larger than $\omega_{S/N \, \delta \! P_{t}}$, the baseline energy sensitivity will be independent of the size of $G_{tb}$; increasing $G_{tb}$ increases the noise floor, but this effect is balanced by an increase in sensor bandwidth. By contrast, when measuring signals with bandwidth below $\omega_{S/N \, \delta \! P_{t}}$ one has accepted a noise penalty but the gained bandwidth is useless. In summary, because $\omega_{ta} \ll \omega_{S/N \, \delta \! P_{t}}$, the CRESST phonon detector has relatively poor sensitivity to phonon energy that is thermalized within the absorber ($\delta_{Pa}$).

![[]{data-label="fig:Traces_CRESST_Composite"}](figs/Gen/CRESST_Traces_0912-0170.pdf){width="3.3in"}

The athermal phonon signal that is thermalized in the TES bypasses the $G_{ta}$ restriction and 