J. Ellis, F. Jenet, & M. McLaughlin Practical Methods for Continuous Gravitational Wave Detection Using Pulsar Timing Data J. Ellis, F. Jenet, & M. McLaughlin

Overview Aim: “Determine the most sensitive practical detection technique for continuous GW sources in PTA data” Methods compared: Matched filtering (various realisations) Power spectral summing Figures of merit: Minimum detectable amplitude (relative to other methods) Compute load Important simplifying assumptions: No frequency evolution Perfect sinusoids (no eccentricity) Data: white, regularly-sampled, “stationary” Builds from previous work:

Data used Up to 100 pulsars Uniform sky distribution, random distance distribution (0.5-3kpc) 10 years, 250 TOAs , regular (~bimonthly) sampling White residuals, 100-300ns RMS “In real pulsar timing data, the residuals will be unevenly sampled and the noise may have various red components. In addition, the pulsar timing residuals will not be stationary, as a quadratic must be fit out of the data to account for the spindown of the pulsar. Specifically, our definition of the inner product in Eq. 16 no longer holds as we will need to include the covariance matrix of the data and incorporate a linear operator that takes into account this fitting. This […] will be addressed in future papers. However, here we will deal with the simple case to illustrate the efficacy of the studied search techniques on a data set of optimal quality.” Usual caveats included which is to say “we’re simply comparing methods not doing any real data analysis”

Matched filtering process Define a vector of search parameters: Compute inner product of data (x) with template (r): Define a (log-)likelihood function: False alarm probability? Maximise this likelihood function using various l realisations. Binary orientation parameters, frequency, mass, distance, and a vector of N pulsar distances. Likelihood function includes a hypothesis test, so is some relative likelihood of probability of signal vs. probability of having no signal Determine threshold value over which this statistic must exceed to claim a detection

NGWtrials NDtrials^(Npulsars) 10400 “Coherence” N templates needed e.g. for NGWt = NDt = Np = 100 Notes FULL (earth+pulsar terms) Fully coherent NGWtrials NDtrials^(Npulsars) 10400 Pulsar distance added as search term. Basically impossible for low expected SNR signals EARTH-ONLY Pulsar term = noise source NGWtrials 100 Lower SNR detection than FULL method. D and W strongly correlated.

Full filter Earth term only

NGWtrials NDtrials^(Npulsars) 10400 “Coherence” N templates needed e.g. for NGWt = NDt = Np = 100 Notes FULL (earth+pulsar terms) Fully coherent NGWtrials NDtrials^(Npulsars) 10400 Pulsar distance added as search term. Basically impossible for low expected SNR signals EARTH-ONLY Pulsar term = noise source NGWtrials 100 Lower SNR detection than FULL method. D and W strongly correlated. PAIRWISE (earth + sum-of-likelihood-stat-for-all-pulsar-pairs) Semi-coherent NGWtrials  NDtrials2  Npulsars(Npulsars-1) ~1010 Positional inaccuracies remain unless large Npulsars

Power Spectral Summing Similar to Yardley et al. (2010) Claim detection when this value > threshold based on false alarm rate “Bad”: gives detection but no source parameters “Good”: no reliance on templates Npsr Calculate power spectrum, sum the power weighted by dataset variance, look for maximum value over all f bins

Result In the case that you already know about a SMBH and are just searching for *that one*. Lower false-alarm probability assuming only one template, matching the data

Result The “best we’ll ever do” might not be that good! (Note, longer Tobs not considered) In the case that you already know about a SMBH and are just searching for *that one*. Pairwise/Full: Similar results for current experiments!

lower false-alarm rate Summary Your analysis method choice will depend on: Your compute cluster size Your expected signal strength Whether you need to locate the position really well Whether you know anything about your target vs. blind sky search How many pulsars you have fewer templates = lower false-alarm rate