In the offline analysis for each record several parameters are evaluated. In the following an overview of the basic concept of the data analysis performed in CRESST-II is given.
The aspects important for the measurements done within this work are in particular discussed. More details on the data analysis techniques used in CRESST are given in [71, 76].
2.5.1 Pulse Height Evaluation
The height of a recorded pulse is a measure of the energy deposited during an interac-tion. Thus, the determination of the pulse height must be accurate. Using simply the maximum of the pulse leads to an overestimation of the pulse height as the maximum would be preferentially found where an upward noise fluctuations occurs.
Another aspect is, that the pulse shape changes for larger pulses, which is related to the shape of the superconducting transition of the TES. In general, close to the operation point the transition is linear, meaning ∆R/∆T is constant in this region and for small pulses the pulse shape stays the same. However, at the top of the transition
∆R/∆T becomes zero (and the transition curve becomes flat). This can be observed in the pulses in the same way, that they become flat on top. With atemplate fit this can be compensated for and the “real” pulse height, which is called amplitude, is determined.
Template Fit
For the template fit method first a template pulse, which describes the observed pulse shape, is required. To generate this template pulse, a set of some hundred pulses is averaged. These pulses are typically selected to have almost the same pulse height in the linear regime. Furthermore, strict selection criteria are applied, so that they provide a good description of the pulse shape. Typically, pulses induced by 122 keVγ’s from a57Co calibration source are used. Due to the averaging the template pulse has a reduced noise contribution. It has to be generated for each pulse type separately (e.g.
for particle and heater pulses).
In the template fit, the template pulse is fitted to each pulse separately and three free parameters are adjusted:
The baseline offset describes the absolute level of the baseline. The baseline can be described by a constant or a baseline model (e.g. an exponential that describes pile-up).
Thepulse amplitude is determined by scaling the template pulse.
Thepulse onset is determined by shifting the template in time.
The result of such a template fit is shown in figure 2.13. The deviation of the recorded pulse from the template is given by the RMS (root-mean-square) value which is min-imized during the fit. It is calculated for each pulse and quantifies the quality of the fit.
baseline offset
pulse onset
pulseamplitude
Figure 2.13:Data of a particle pulse (red) shown together with the result from a template fit (black). The three parameters, that can be varied to best match the template to the pulse, are indicated with arrows.
Truncated Fit
The determination of the pulse amplitude only works reliable in the linear regime of the transition. Higher energetic recoils heat the TES in its nonlinear region, which changes the pulse shape. This can be observed in a rise of the RMS of the fit for higher pulses, due to the reduced fit quality for pulses with a different shape (see figure 2.14).
The region where the pulse shape starts to change is usually far above the energies interesting for dark matter search. However, for the characterization of background sources and the performance of the detectors, these pulses are also reconstructed in the analysis.
To increase the fit quality, in the so-calledtruncated fit parts of the pulse, that are in the nonlinear regime of the transition are excluded from the fit. By doing so, only the linear part of the pulse is considered for the fit with the low-energy template and the resulting amplitude is an extrapolation of the low-energetic part of the pulse. The limit above which samples are not considered for the fit anymore is chosen from the RMS distribution (see figure 2.14) and is called truncation limit. With this method pulses with energies up to some MeV can be reconstructed reliably [77].
Correlated Fit
In CRESST phonon and light detector pulses are measured simultaneously. As more energy is deposited in the phonon detector, usually phonon pulses have a larger signal-to-noise ratio than light detector pulses. For this reason, for small energies phonon pulses can still be fitted well, whereas light detector pulses are often more difficult to fit. The fact that the pulse onset must be the same for both pulses is exploited in the correlated fit, where both pulses are fitted with a common pulse onset.
Figure 2.14:Pulses exceeding the linear range of the transition cannot be fitted properly with a low energy template, since the pulse shape changes in the non-linear region. The different pulse shape results in a decrease of the fit quality and is indicated by an increase of the RMS value for these pulses (left). With the truncated fit (right) pulses with a higher energy are reliably reconstructed as only the linear part is used for the fit. The truncation limit is set where the RMS distribution starts to increase (dashed red line, here: 1V pulse height). Image taken from [63].
To do so, the templates for both pulses must be summed from the same set of pulses so that the relative onset time is fixed. In the correlated fit the templates are fitted to the pulses and, thereby, even very small light detector pulses can be fitted correctly due to the known relative timing of both pulses. The best fit is found by minimizing the sum of the RMS values of both signals.
2.5.2 Energy Calibration
For the absolute energy calibration of CRESST-II detectors an external γ-source is used. Usually the 122 keV-line of a57Co source is used to fix the energy scale4. In case of the light detector, the detected scintillation light from the 122 keV γ’s hitting the target crystal, is fixed as 122 keVee. Thereby the energy scale of the light detector is set with the electron-equivalent energy5. Due to this definition the light yield is 1 at 122 keV.
With the calibration source the energy scale is fixed only at one point and at the time when the calibration takes place. As already explained, during a measurement test pulses are sent to every detector heater regularly. They are used to calibrate the detector over an energy range from threshold up to MeV energies and to compensate for variations in the detector response over time.
Over the whole measurement period every ∼30 s one test pulse is injected in the heater. The test pulses can have up to twelve different amplitudes in order to obtain pulses with energies corresponding to energies which are relevant for the measurement.
Similar to particle pulses the amplitude of the test pulses is obtained via a correlated
4A lower energetic calibration source would be closer to the energies that are aimed to be measured.
However, such a source would not be able to penetrate the in total 12 mm thick copper shields around the detectors.
5In general it is possible to calibrate the light detector absolutely with a low-energeticγ-source directly hitting the light detector
truncated standard event fit. The test pulses serve two purposes:
Variations of the detector response in time can be corrected by fitting splines to the measured test pulse amplitude. With this the detector response can be evaluated at any point in time.
As the energy deposited by a test pulse is proportional to the injected voltage, the test pulse amplitude can be used to linearize the detector response. To obtain a continuous response function (i.e. the relation between the injected test pulse am-plitude and the corresponding measured pulse amam-plitude) over the whole energy range of interest, pulse heights resulting from discrete injected test pulse ampli-tudes are fitted with a low-order polynomial, which then serves to determine the energy from the pulse height.
From the calibration with the57Co source a relation between the injected voltage and the energy deposited in the target can be evaluated. Thereby, the detector can be calibrated for the whole measurement period and for all energies.
2.5.3 Cuts
For the analysis of CRESST detectors invalid pulses that are not induced by particle events (such as electronic disturbances) or particle events that are not properly energy reconstructed due to artifacts in the record, are removed from the data set. This is done by several data cuts identifying invalid pulses. In the following only a brief overview of the most important cuts, used for the measurements of this work, is given. A more detailed description of the cuts used in the dark matter analysis of CRESST can be found in [71, 76]. The following cuts are applied on the data set:
Stability Cut Time periods in which the detector was not running stably in its oper-ating point (e.g. due to external disturbances) are rejected by the stability cut.
This is done by evaluating the height of the control pulses, which is a measure for the detectors operating point in the transition (see section 2.4). For time periods, where the control pulse height deviates significantly from the set value, all records are completely removed from the data set.
Coincidence Cut Due to the low expected cross-section of dark-matter particles it is nearly impossible that the latter interact twice in the CRESST setup. Thus, all cryogenic events which are coincident in time with either a signal from the muon veto or another detector module are rejected by the coincidence cut.
Data Quality Cuts Several cuts are designed to identify invalid pulses where the energy reconstruction does not work properly. For example records that include SQUID resets or that triggered on noise are removed by these cuts. Also pulses with a different pulse shape (such as events in the TES carrier of a phonon detector) are removed by dedicated cuts. The most important and most generic data quality cut is a cut on the RMS value of the template fit, where pulses with measured shapes that deviate from the expected signal shape, are discarded. This includes strongly distorted pulses, pulses with a strong baseline tilt or pile-up events, i.e.
samples where more than one pulse appears in the time window recorded.
After applying all cuts,the collected exposure needs to be calculated and afterwards either a limit on the dark matter particle-nucleon cross-section can be given or, if events in the ROI are observed, a hint for a possible dark matter signal can be calculated.
These steps are explained e.g. in [71, 76]. In the following section only the results of the previous CRESST-II dark matter data-taking phases are explained.