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We then again run the genetic algorithm, exploring the remaining parameter space i. The end result is a set of independent local maxima in likelihood, among which we choose the largest highest in value.
We repeat the entire procedure three times to verify convergence. This algorithm has proven to give very robust results, even in the case of pathological likelihood surfaces with multiple maxima. In the Bayesian approach, the parameters describing the model are treated as random variables. The principles of Bayesian statistics provide a robust framework to obtain probability distributions of those parameters for a given set of observations, while also folding in our prior knowledge of them.
We use non-informative and conservative priors in our analysis: they are always uniform in all parameters. Both techniques also permit a recovery of the aforementioned model-comparison statistic known as the Bayesian evidence. We describe the different searches in detail below. For a pulsar data set, that amounts to sampling a dimensional parameter space. By doing so, we not only rapidly recover the posterior PDFs, but also achieve quick and accurate Bayesian evidence values. The sampling of the posterior is performed by multinest.
We also employed the Bayesian formalism to construct a search for evolving signals. In this case, we can either i use the full signal of equation 6 , or ii restrict ourselves to the Earth term only. Note that our parametrization of the full signal does not rely on any knowledge about the distance to the pulsar. We name this search Bayes-EP-Ev. We now turn to the description of the main results of our analysis.
As in the previous section, we present the outcome for the frequentist and Bayesian analyses separately, discussing first the detection results and then the upper limit computation for the various techniques. The two are consistent with a p-value of 0. A standard way to do this in frequentist analysis is through signal injections. We also want to obtain an upper limit that takes into account the uncertainty of the single-pulsar noise parameters.
In doing this, the procedure outlined above is modified as follows:. The motivation behind criterion iii is that it resembles the criterion for the ML upper limit and it is much more conservative than a Kolmogorov—Smirnov test. It is also possible to choose even more conservative criteria, susceptible to possible non-Gaussian tails in either distribution, but we do not explore this possibility here. Upper limits as a function of f are shown in Fig.
This is consistent with Fig. The blue line corresponds to noise parameters fixed to the ML values obtained in the time-domain SPA, while the dashed lines take into account the full uncertainty in the noise estimation by sampling from the PDF distributions of either the pure time domain green or the time-frequency red SPA. At 1 per year the peak , the limit is poor because the GW signal at this frequency is absorbed in the fitting of the pulsar positions. The points are the trials of the search. The horizontal lines are the detection thresholds for different FAPs.
Results are shown in Fig. As mentioned in Section 3 , we also ran a full dimensional search over noise and signal parameters on the restricted set of the six best pulsars in our PTA c. By analysing Fig. We also observe that methods based on fixed noise ML parameters slightly underestimate the upper limit, which could be because the ML values are not always representative of the posterior distribution of the noise parameters.
Most of the searches outlined above are also sensitive to the source location on the sky. As already mentioned, the upper limit is evaluated through massive signal injections according to the procedure outlined in Section 4. This is much more computationally expensive than the evaluation of the sky-averaged limit; we therefore generate the sky map at 6. This is shown in Fig. As expected we are more sensitive in the region of the best pulsars.
Black dots indicate the location of the Virgo and the Coma clusters. We can also produce targeted upper limits as a function of sky location by means of Bayesian techniques. The colour scale and points are the same as for Fig. The qualitative agreement between the two maps is quite good. As expected, the most constraining i. The closest galaxy clusters in the Universe, i. A number of potentially interesting sources have been proposed in the literature. Should their binary nature be confirmed, these two systems would likely be among the thousands of contributors to the stochastic GW background.
In a nutshell, galaxy merger rates are obtained using a selection of galaxy mass functions and close galaxy pair fractions from the literature; merging galaxies are populated with SMBHs following empirical black hole—galaxy host relations; finally, each binary is assumed to emit GWs while inspiralling in a quasi-circular orbit. Given the broad range of different models taken into account and their uncertainties, numerous simulations are created in order to cover all possible configurations consistent with the observations. The SMBHB populations obtained in this way are consistent with the results of semi-analytic halo merger trees and cosmological N -body and hydrodynamical simulations.
More details on the simulations and the models employed to produce them can be found in Sesana The detection probability at any frequency obtained for each of the upper limits is given in the legend of Fig. Therefore, we can safely conclude that a non-detection is consistent with the theoretical expectations. GW strain amplitude versus GW observed frequency. The coloured lines represent the different upper limits presented in this work.
The shading gives the probability of detecting an SMBHB in a particular interval of strain and frequency. That detection probability increases towards lower frequencies and smaller values of strain on the lower-left corner. In the legend, the percentage of detection probability is given for each of the upper limits. In our searches, we distinguished between evolving and non-evolving GW signals, presenting distinct search methods for each of them. One may therefore ask, whether one type of signal is more likely than the other, in order to better focus development efforts on specific analysis pipelines.
As shown by Fig. Nevertheless, it is still meaningful to study the outcome of those realizations, since they would resemble the true ensemble of SMBHBs in the fortunate case of a detection in the near future. Their distribution is shown on the upper plot of Fig. Should EPTA detect an individual source in the near future, it could either be evolving or non-evolving with nearly equal probability. Decreasing the PTA sensitivity floor would make it sensitive to lower mass binaries which evolve faster , but would also improve the chance of detection at higher frequency, where evolution is more likely.
Detection strategy development for both classes of sources is therefore warranted. We adopted both frequentist and Bayesian techniques, searching for both frequency-evolving and strictly monochromatic signals.
In most of the cases, we fixed the value of the noise parameters in each pulsar to the ML estimated in a separate SPA. This choice was primarily dictated by computational feasibility, but is certainly non-optimal, since pulsar noise and GW signals might be degenerate and one should include both simultaneously in the search. Because of the high dimensionality of the search parameter space, the latter has been conducted on a restricted data set including only the six best pulsars in the array.
Limits on the strain amplitude can be converted to horizon distances as a function of source mass and frequency. Finally, we compared our limits to the predictions of state of the art models of the cosmic population of SMBHBs. The present analysis has also highlighted a few interesting technical issues related to the search methods and to the nature of the data set. Therefore, so long as the data do not support the presence of a signal, a computationally cheap analysis of this type can be carried out over an extensive data set of numerous pulsars, possibly yielding more interesting astrophysical constraints on the low-redshift SMBHB population in the near future.
Eventually, simultaneous searches over the signal and noise parameters will be required for a confident detection claim. The reason why the results of the full search on the restricted data set of six pulsars is consistent with those provided by fixed noise analysis on the full set of 41 pulsars, is that the current EPTA array is heavily dominated by a handful of ultrastable MSPs. As a result, the EPTA data set sensitivity has a strongly dipolar pattern across the sky, varying by almost a factor of four over the celestial sphere.
This work was supported in part by the National Science Foundation under grant no. SO is supported by the Alexander von Humboldt Foundation. EFAC is used to account for possible miscalibration of the radiometer noise and it acts as a multiplier for all the TOA error bars. A full use of the multimodal genetic algorithm for each injection is computationally expensive and not needed, in practice.
We therefore use a lighter and faster search for the injected signals. We construct a stochastic bank with minimal match 0. In each search we use only the portion of template bank covering the parameter space region around the injected signal. This is an arbitrary choice dictated by computational convenience only.
Results are, however, qualitatively unchanged if a different statistic e. F p is assumed. Anholm M. Ballmer S. Creighton J. Price L. Siemens X. D 79 Arzoumanian Z. Babak S. Quantum Gravity 25 Sesana A. D 85 Begelman M. Blandford R. Rees M. Burke-Spolaor S. Quantum Gravity 30 Chamberlin S. The Dams Raid was, like many British air raids , undertaken with a view to the need to keep drawing German defensive effort back into Germany and away from actual and potential theatres of ground war, a policy which culminated in the Berlin raids of the winter of — In May this meant keeping the Luftwaffe aircraft and anti-aircraft defences away from the Soviet Union; in early , it meant clearing the way for the aerial side of the forthcoming Operation Overlord.
The considerable amount of labour and strategic resources committed to repairing the dams, factories, mines and railways could not be used in other ways, on the construction of the Atlantic Wall , for example.
The pictures of the broken dams proved to be a propaganda and morale boost to the Allies, especially to the British, still suffering from the German bombing of the Baedeker Blitz that had peaked roughly a year earlier. Even within Germany, as evidenced by Gauleiters' reports to Berlin at the time, the German population regarded the raids as a legitimate attack on military targets and thought they were "an extraordinary success on the part of the English" [sic].
They were not regarded as a pure terror attack by the Germans, even in the Ruhr region, and in response the German authorities released relatively accurate not exaggerated estimates of the dead. An effect of the dam raids was that Barnes Wallis's ideas on earthquake bombing , which had previously been rejected, came to be accepted by 'Bomber' Harris. Prior to this raid, bombing had used the tactic of area bombardment with many light bombs, in the hope that one would hit the target.
Work on the earthquake bombs resulted in the Tallboy and Grand Slam weapons, which caused damage to German infrastructure in the later stages of the war. They rendered the V-2 rocket launch complex at Calais unusable, buried the V-3 guns, and destroyed bridges and other fortified installations, such as the Grand Slam attack on the railway viaduct at Bielefeld. Harris regarded the raid as a failure and a waste of resources. From Wikipedia, the free encyclopedia. Not to be confused with Operation Chastity.
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