This study has placed the most stringent limit to date on the intensity of gravitational waves that are produced by pairs of supermassive black holes dancing around each other in distant galaxies, as well as those that may be produced by vibrating cosmic strings. Rutger van Haasteren has won the GWIC Thesis Prize of the Gravitational Wave International Community for his innovative work in various directions of the search for gravitational waves by pulsar timing.

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Front Matter Pages i-xi. In the absence of radio frequency interference RFI , the noise in most pulse profiles is radiometer-noise-dominated, which is highly Gaussian. We index each of the above frequency-dependent quantities with the letter n e. In other words, we seek to minimize the statistic.

The fundamental difference in this approach, besides allowing for an arbitrary evolution of the pulse profile with frequency encoded in p nk , is that we perform a global fit for both an achromatic phase and a dispersion measure DM by implementing the constraint. The first two definitions are functions solely of the data and the model portraits.

This definition highlights the fact that both FFTFIT and our extension of it across a discretized bandwidth can be thought of as cross-correlation techniques. Solving for these maximum-likelihood a n as a function of the other parameters leads to the constraint. It is easy to see that, for negligible profile evolution, if the dispersion measure is zero or, equivalently, the data have been correctly dedispersed for that observation's epoch, then this algorithm is akin to averaging TOAs obtained in the usual way using individually aligned templates.

However, if the pulsar's DM needs to be measured at every epoch, and profile evolution should be accounted for—which are both likely true for most observations of MSPs with wideband receiver systems—we claim that this is a natural extension to how TOAs are currently procured.

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At the very least, this algorithm should perform no worse than traditional techniques. We have implemented our wideband timing algorithm in publicly available python code, 11 which also includes a Gaussian-component-based portrait modeling routine, which is described below. The initial phase parameter value is estimated by using a one-dimensional brute-force routine in scipy , which is performed on the frequency-averaged data and template. The nominal DM from the pulsar's ephemeris is used in this dedispersion and also as the initial DM parameter value in the global fit.

It is also possible to exclude fitting for a DM and only determine a phase. With respect to the demonstration in the next section, our source is close to the ecliptic plane and has a large DM, so this correction was essential. It is obvious that there is freedom in the choice of model portrait to use and we stress that any arbitrary model can be used in the above algorithm for phase and DM measurements.

We have experimented almost exclusively with analytic Gaussian-component models, but it is also feasible to find an interpolation scheme based on an average of all the data portraits or, for example, a principal component analysis approach. We model pulse profile evolution with independently changing Gaussian components g i of the form.

## Severine Perus

The modeling code allows flexibility for any of the parameters to be fixed; for example, a "fiducial component" with no positional change as a function of frequency can be selected. Most MSP portraits we have experimented on seem to be sufficiently characterized by a few to roughly a dozen or so Gaussian components. We also include an option to include scattering in the fit for the model via a convolution with a one-sided exponential,.

One could imagine extending our algorithm to include a variable scattering parameter in the fit to the data instead of fixing it in the model.

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The benefits, applicability, and practical limitations of doing this are currently being investigated by the authors. The details of pulse portrait modeling and its physical interpretations are beyond the scope of this paper, but we demonstrate one application of Gaussian modeling in the next section.

Finally, one subtlety that we did not address is the averaging of the model within each frequency channel to match the channel bandwidth of the data. Insofar that the aim is to have a greater number of channels, this should have a negligible effect.

## Gravitational Wave Detection and Data Analysis for Pulsar Timing Arrays

Similarly, we assumed that a sufficient number of phase bins are used so that all of the harmonic content of the profile is retained and an averaging of the model into phase bins is well-approximated by the Gaussian model evaluated at the phase bin centers. However, a more rigorous representation for P nj would be one that multiplies the model by a sampling function that has been convolved with both the channel width and bin size of the data.

The M28A data set presented here consists of 25 epochs of multi-frequency observations spanning more than three years from the Green Bank Telescope. The data were obtained with GUPPI beginning in February soon after the implementation of its real-time coherent dedispersion capability, which was utilized for the taking of these observations in search-mode i.

The native resolution of the data is The DMs had the nominal unweighted average value of The fractional bandwidths are approximately 0. The complexity of the portrait is evinced by its asymmetries, its non-Gaussian features, the exchanging dominance of components from differing spectral indices, and the presence of an obvious scattering tail at the lower frequencies. In effect, the nuisance parameters attempted to "align" the data so that the Gaussian parameters can be optimized.

Figure 1. The horizontal gaps are where radio frequency interference was excised, but the widest one is a coverage gap between receivers. Note that the UBB has an instantaneous bandwidth of almost 1. MSPs like M28A exemplify how the choice of a "fiducial point" is not simple e. Figure 2. We used a 10 component model that captures both the finer structure seen at the higher frequencies and the scattering at the lower frequencies.

Although the model and residuals show that the Gaussian modeling is not perfect, the model still proved sensible for timing and DM measurements. The second component listed is the "fiducial component. The precision of all the parameters is arbitrary, since we offer no interpretation of the model in this paper.

The point estimate of the scattering timescale is marginally consistent with that found independently from a separate analysis of giant pulses in this M28A data A. Bilous , private communication. This profile marginalizes over all of the frequency structure and scattering tails, and so it would be imprudent to use such a profile as a template for obtaining phase measurements.

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The phase-averaged spectral flux density profile in the left panel was fit with a power law yellow dashed line. Details of M28A's spectra from this data set will also be presented in a forthcoming paper A. If this frequency-dependent bias were constant, in would be absorbed into the timing model, but varying scintillation patterns can change which segments of the bias are weighted more significantly or, similarly, what the frequency-averaged profile looks like , thereby introducing random systematic noise into the timing residuals. Figure 3. Each point represents The top panel "TEMPO" does not account for any profile evolution, but only uses a single template profile per receiver band, producing the observed scatter.

A comprehensive, global model for the evolution would ideally show flat residuals. Next generation wideband receivers will simultaneously cover more than this entire spectrum. The fitted a n values provide information about diffractive scintillation from the ISM, and they effectively act as weights for individual multi-channel TOAs that have been fitted for a DM and averaged together to obtain.

In principle, the a n values could also be used to determine if there is residual RFI in the data, although we have not yet investigated how the presence of RFI will affect the fitting. Figure 4. Simple demonstration of how the algorithm automatically manages diffractive scintillation. The fitted model is shown in the middle panel, which encodes the scintillation in the a n scale parameters, and the fit residuals are to the right.

The residual statistics match the off-pulse noise and mean from the input data. Figure 5. The calendar range of the data spans February 11 to May In what follows, we have compared our measurements labeled "PP" with those obtained from the same data using a more traditional procedure which we collectively label "TEMPO". For the latter, multi-channel TOAs were obtained via standard techniques: each time-averaged epoch's band was divided into 64 channels, and each channel's profile was cross-correlated with a smoothed template profile that was obtained by averaging all the data from a given receiver.

No consideration of profile evolution is taken into account besides the usage of three separate template profiles for the three bands. A somewhat less arbitrary approach is to characterize the trend with a simple function that can be included in the timing model. Ignoring profile evolution altogether and using frequency-averaged profiles may still be a sufficient practice for particular pulsars.

However, this strategy will become untenable with the next generation of receivers. It seems more appropriate and simple to directly model the profile evolution based on the folded profiles, as we have done, and then simultaneously measure a TOA and a DM. These "residuals" were calculated by independently cross-correlating each channel in the fitted two-dimensional model with the corresponding frequency channel in the data portrait using our own FFTFIT routine. The scatter of both sets of points is about the same as the corresponding average residual uncertainty. Alternatively, our attempt to concatenate disparate bands from epochs that may have significantly different scattering timescales may have negatively affected the lowest frequencies.

Note that these residuals are specific examples from the whole data set, and that the goodness of the model's fit will vary from observation to observation. The absolute DM is not a useful measure for comparison because its values depend on how the profile and its evolution are modeled, and the DM can even vary based on its inclusion in a timing model fit not applicable here. Consequently, the average values of differently measured DMs will differ by a constant. Figure 6. Comparison of mean-subtracted DM trends as measured in the M28A data by our technique and a more usual approach.

The solid line traces equality. The calculation of the error on our DM measurements is provided in the Appendix. See the text for further discussion. These significant differences may be relevant when interpolating the behavior of the ISM between observations. We address the measurement uncertainties in the next section. The offsets seen in the figure between the DMs measured in each receiver band come from the different modeling of profile evolution in each band. Kuniyoshi, A. Alexov, R.