## 2 TES Physics and Models  
In this section, we will focus on the theory of TESs. We will discuss primarily the case of X-ray microcalorimeters. However, the theory presented is general enough to be adapted to other TES-based energy or power detectors. A figure of merit that characterizes single photons detectors is the resolving power \(E/\Delta E\) for photons with energy \(E\). A fundamental limit for the minimum energy resolution \(\Delta E\) achievable with a calorimeter is given by the random exchange of energy between the detector and the thermal bath [13]. This thermodynamic limit is given by \(<\Delta E_{TD}^{2}>=k_{B}T^{2}C\). It depends quadratically on the temperature \(T\) of the calorimeter, linearly on the detector heat capacity \(C\), and it is independent on the thermal conductance \(G\) of the thermal link. This value however does not set a limit to how accurately the thermal fluctuations can be measured. This can be understood by looking at the signal and noise power spectrum of the thermodynamic fluctuations. Each frequency bin provides an uncorrelated estimation of the signal amplitude and the signal-to-noise ratio can be improved as the square root of the number of bins averaged [13]. The two most important parameters used to estimate the detector signal-to-noise ratio and the energy resolution are the unit-less logarithmic temperature and current sensitivity \(\alpha=\partial\log R/\partial\log T\) and \(\beta=\partial\log R/\partial\log I\), calculated, respectively, at a constant current \(I\) and temperature \(T\). They have been conveniently introduced [17] to parametrize the TES resistive transition  
\[R(T,I)\approxeq R_{0}+\alpha\frac{R_{0}}{T_{0}}\delta T+\beta\frac{R_{0}}{I_{ 0}}\delta I, \tag{1}\]  
and they can be estimated experimentally at the quiescent operating point (\(R_{0}\), \(T_{0}\), \(I_{0}\)) in the transition. A detailed calculation of the minimum energy resolution achievable with a TEScalorimeter is possible. After a careful analysis of the thermometer sensitivity, the detector noise and the signal bandwidth, gives  
\[\Delta E_{FWHM}\simeq 2\sqrt{2\ln 2}\sqrt{4k_{B}T_{0}^{2}C\frac{\sqrt{\zeta(I)}}{ \alpha}\sqrt{\frac{nF(T_{0},T_{bath})}{1-(T_{bath}/T_{0})^{n}}}}. \tag{2}\]  
The unit-less parameter \(F(T_{0},T_{bath})\) depends on the thermal conductance exponent \(n\), which is related to the physical nature (radiative or diffusive) of the thermal link between TES and the heat sink at \(T_{bath}\)[13]. For a TES \(F(T_{0},T_{bath})\simeq 0.5\)[14]. The parameter \(\zeta(I)\) takes into account the non-linear correction terms to the linear equilibrium Johnson noise, which conveniently helps to write the Johnson voltage spectral density as \(S_{V}=4k_{B}TR_{0}\zeta(I)\). For a linear resistance \(\zeta(I)=1\). For a TES, it is usually defined as \(\zeta(I)=(1+2\beta)(1+M^{2})\) where the factor \(1+2\beta\) is the first order, near equilibrium, non-linear correction term as discussed in [18] and the unit-less factor \(M^{2}\) parameterizes any _unexplained_ observed deviation from this approximation. More details on this important noise contribution are given further in Section 2.3. From Equation (2), it is clear that low heat capacity devices, operating at very low temperature \(T_{0}\), could achieve very high energy resolution. The \(C\) value, however, is not a free parameter and is typically defined by the dynamic range, DR \(\propto C/\alpha\), requirement of the specific application. By developing TES with a large temperature sensitivity \(\alpha\), the photon energy can be measured with a much higher resolution than the magnitude set by thermodynamic fluctuations. To achieve then the ultimate sensitivity, for given \(C\) and \(T_{0}\), the factor \(\sqrt{\zeta(I)}/\alpha\) has to be minimized. The details of the TES electro-thermal response and noise will be discussed here below. In Section 2.1, we will review the recent development in the understanding of the physics of the TES resistive transition.