## 2 TES Physics and Models
### TES Noise  
The total noise of a TES detector is a combination of Nyquist-Johnson noise associated with the TES resistance itself and with the losses in the read-out circuit and bias circuit, and thermal fluctuation noise between the TES thermal elements. The noise sources can be classified in _external_ and _internal_ noise sources. Typical external Johnson noise sources are the additive white current noise from the SQUID amplifier and the white voltage noise \(e_{sh}=\sqrt{4k_{B}T_{bath}r_{sh}}\) from the Nyquist-Johnson noise of the shunt resistor \(r_{sh}\) in the ac or dc bias circuit. With a proper choice of the parameters of the bias circuit and the SQUID amplifiers, these noise terms can be made negligible. Other external noise sources like, for example, stray light or cosmic photons could generate power fluctuations (\(p_{ext,1}\) and \(p_{ext,2}\)) on the detector thermal elements. A careful thermal design of the focal plane assembly can reduce these noise sources to a negligible level. In Figure 6, we show, as an example, a typical current noise spectra for a Ti/Au TES developed at SRON, ac biased at 2.5 MHz in the transition at \(R/R_{n}\sim 0.2\). The different noise contributions are explained here below. There are three fundamental thermodynamic internal noise sources of a TES detector, described with a two-body model as in Figure 1. The first is the thermal fluctuation noise between the TES detector and the heat bath, already discussed in Section 2. It is prominent in the detector thermal bandwidth typically below \(\sim 200\) Hz. It has a white thermal noise given by \(p_{bath}=\sqrt{F(T_{0},T_{bath})4k_{B}T^{2}G_{bath}((T_{bath}/T)^{n+1}+1)(1+M _{ph}^{2})}\), where \(F(T_{0},T_{bath})\sim 0.5\), as discussed in Section 2, and \(M_{ph}\) is a parameter introduced ad-hoc to account for the excess noise observed in the experiments. In the literature, this noise term is typically referred to as _phonon_ or thermal fluctuation noise [14]. The noise contribution \(p_{2,b}\) to the thermal fluctuation noise from the conductance \(G_{2,b}\) is generally neglected since \(G_{2,b}\ll G_{bath}\). The second fundamental noise term is the so called internal fluctuation thermal noise (ITFN), which is generated by thermal fluctuations between distributed heat capacity (C and C2 of Figure 1b) internal to the TES-absorber system. It has a spectral density \(p_{lifn}\equiv p_{12}=\sqrt{4k_{B}T^{2}G_{12}}\), where \(G_{12}\) is the intrinsic thermal conductance. The presence of internal thermal fluctuation noise in TES microcalorimeter was first reported by [PERSON] et al. [71]. The origin of the thermal conductance \(G_{12}\) can be of different physical nature depending on the specific TES design. In TES bolometers, for example, the parasitic conductances and the distributed heat capacities is typically attributed to the electrical leads patterned onto the thermally isolating nitride legs [72] or to the SiN legs themselves [73]. In [PERSON] et al. [50], the observed dangling heat capacity arose likely from the thin SiN  
Figure 6: Current noise spectra of a Ti/Au \(80\times 40\) μm\({}^{2}\) TES X-ray calorimeter developed at SRON, biased at 2.5 MHz, at \(R/R_{n}\sim 0.2\) with \(\kappa\sim 370\) and \(\beta\sim 5\)[22]. membrane supporting the low \(C\) TES calorimeter. In [PERSON] et al. [53], the internal thermal fluctuation noise is assumed to be generated by a poor thermal conductance of the TES bilayer and \(G_{12}\) was estimated from the Wiedeman-Franz relationship \(G_{WF}=L_{o}T_{0}/R_{0}\), where \(L_{o}=24.5\times 10^{-9}\,\mathrm{W}\Omega\mathrm{K}^{-2}\) is the Lorenz number. [PERSON] et al. [54] investigated the contribution of the internal thermal fluctuation noise on TES microcalorimeters with different design and the origin of the finite \(G_{12}\) term was thoroughly discussed.