We find the inferred properties of the putative gravitational wave background in the second data release of the European Pulsar Timing Array EPTA to be in better agreement with theoretical expectations under the improved noise model. In particular if the gravitational wave background is from supermassive black hole binaries the observed gravitational wave emission is the dominant source of the binary energy loss with no evidence of environmental effects eccentric orbits or an overabundance of sources. Our improved noise model addresses the misspecification of noise priors which impacts the inference of temporal correlations of the background. We also find evidence for epochcorrelated temporallyuncorrelated white noise which has a stronger impact on the measurement in the presence of HellingsDowns correlations of the background.
['Astrophysics - High Energy Astrophysical Phenomena', 'Astrophysics - Cosmology and Nongalactic Astrophysics', 'General Relativity and Quantum Cosmology']
Fewer supermassive binary black holes in pulsar timing array observations
{'Fewer supermassive binary black holes in pulsar timing array observations': "Boris Goncharov, 1, 2, ∗ Shubhit Sardana, 1, 3 A. Sesana, 4, 5, 6 S. M. Tomson, 1, 2 J. Antoniadis, 7, 8 A. Chalumeau, D. Champion, 8 S. Chen, 10 E. F. Keane, 11 K. Liu, 10 G. Shaifullah, 4, 5, 12 L. Speri, 13 and S. Valtolina 1, 2  \n1 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), 30167 Hannover, Germany  \n2 Leibniz Universitat Hannover, 30167 Hannover, Germany  \n3 Department of Physics, IISER Bhopal, Bhauri Bypass Road, Bhopal, 462066, India  \n4 Dipartimento di Fisica 'G. Occhialini', Universit'a degli Studi di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy  \n5 INFN, Sezione di Milano-Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy  \n6 INAF - Osservatorio Astronomico di Brera, via Brera 20, I-20121 Milano, Italy  \n7 FORTH Institute of Astrophysics, N. Plastira 100, 70013, Heraklion, Greece  \n8 Max-Planck-Institut fur Radioastronomie, Auf dem Hugel 69, 53121, Bonn, Germany  \n9 ASTRON, Netherlands Institute for Radio Astronomy,  \nOude Hoogeveensedijk 4, 7991 PD Dwingeloo, The Netherlands  \n10 Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, P. R. China  \n11 School of Physics, Trinity College Dublin, College Green, Dublin 2, D02 PN40, Ireland  \n12 INAF - Osservatorio Astronomico di Cagliari, via della Scienza 5, 09047 Selargius (CA), Italy  \n13 Max Planck Institute for Gravitational Physics (Albert Einstein Institute), Am Muhlenberg 1, 14476 Potsdam, Germany (Dated: November 8, 2024)  \nWe find the inferred properties of the putative gravitational wave background in the second data release of the European Pulsar Timing Array (EPTA) to be in better agreement with theoretical expectations under the improved noise model. In particular, if the gravitational wave background is from supermassive black hole binaries, the observed gravitational wave emission is the dominant source of the binary energy loss, with no evidence of environmental effects, eccentric orbits, or an overabundance of sources. Our improved noise model addresses the misspecification of noise priors which impacts the inference of temporal correlations of the background. We also find evidence for epoch-correlated temporally-uncorrelated (white) noise which has a stronger impact on the measurement in the presence of Hellings-Downs correlations of the background.  \nSince 2020, Pulsar Timing Arrays (PTAs) have reported growing evidence for the nanohertz-frequency gravitational wave background in their data. The first tentative evidence came from a temporally-correlated stochastic process in pulsar timing data [1-4]. The Fourier spectrum of delays and advances in pulsar pulse arrival times exhibited the expected spectral properties of the background. Most recently, PTAs - with varying levels of statistical significance [5-8] - showed that this stochastic process exhibits Hellings-Downs correlations [9] consistent with the isotropic unpolarised stochastic gravitational wave background.  \nSupermassive black hole binaries at subparsec separations are expected to be a dominant source of the stochastic gravitational wave background at nanohertz frequencies [10]. However, the expected amplitude of the background is lower than the observations suggest. Previous EPTA analyses found the best-fit strain amplitude to be at the edge of the values simulated from supermassive black hole binary population synthesis models. This is visible in Figure 7 from [11] and Figure 5 from [12]. The inferred strain amplitude lies at the theoretical upper limit of the predicted astrophysical range [10, 13, 14]. It might also be in tension with the observed black hole mass function [15, 16], although see [17] for a different view. Furthermore, the strain spectral index of the grav-  \nitational wave background is in ≈ 2 σ tension with the value corresponding to binary inspirals driven by gravitational wave emission alone. This is visible in Figure 5 in [12] and Figure 11 in [5]. Although previous PTA results were consistent with a very broad range of assumptions about supermassive binary black hole populations [18], they suggested deviations from purely gravitational wave-driven binary evolution.  \nIt was pointed out in several studies [2, 4, 19] that the standard PTA models of how noise parameters are distributed across pulsars are incorrect. These models manifest as prior probabilities in PTA data analysis. To be precise, the models are incorrect because they are 'static', i.e. , the shape of the distribution of noise parameters is not influenced by the data. Although imposing such priors is very unlikely to influence our conclusions about evidence for the gravitational background [20-23], it may introduce systematic errors in our measurement of the strain spectrum [19].  \nWe address the aforementioned problem. We employ hierarchical Bayesian inference to account for the uncertainties in our noise priors. In particular, we parametrise noise prior distributions which are our models of how parameters governing pulsar-specific noise are distributed across the pulsars [19]. These newly-introduced hierarchical parameters are called hyperparameters . We use a new procedure of marginalisation over hyperparameters, which is described in Section 2.2.2 of the companion paper [24]. Furthermore, we discuss other sources of noise that may impact data analysis. Based on the revised  \n9  \nnoise models, we revisit the measurement of properties of the gravitational wave background with the European Pulsar Timing Array (EPTA) [25].  \nCorrelations. -We assess the properties of the gravitational wave background using the power law model of its characteristic strain spectrum: h c ( f ) = A ( f yr -1 ) -α . Here, A is the strain amplitude, and -α is the power law spectral index 1 . The corresponding spectral index of the power spectral density [s 3 ] of delays(-advances) [s] induced by the background in the timing data is -γ . These stochastic timing delays resulting from the gravitationalwave redshift of pulsar spin frequency are referred to as temporal correlations. The value of γ = 13 / 3 ( α = 2 / 3) corresponds to purely gravitational wave driven inspirals of supermassive black hole binaries [26].  \nModelling of pulsar noise is important because it can affect the conclusions about the properties of the gravitational wave background such as A and γ . Pulsar noise introduces temporal correlations which impossible to distinguish from the effect of gravitational background based solely on single pulsar data. However, unlike for the gravitational background, noise-induced timing delays have different Fourier spectra across pulsars [2]. Our hierarchical model describes the ensemble properties of these noise spectra.  \nDespite similar delay time series induced by both pulsar noise and gravitational background in pulsar data, the presence of the background can still be established. The Hellings-Downs function [9] determines the mean inter-pulsar correlations of the stochastic timing delays induced by the isotropic stochastic unpolarised gravitational background. The statistical significance of the background is established based on the Bayesian odds between (1) the Hellings-Downs-correlated stochastic process with the same spectrum of delays across pulsars and (2) the same spectrum of delays across pulsars and no inter-pulsar correlations [27].  \nResults. -Posterior distributions for A and γ are shown as contours in Figure 1. Solid blue contours correspond to our improved model. For comparison, dashed red contours correspond to the standard 'static' noise priors used in earlier EPTA analyses [6]. The results are shown for both the 10-year subset of the EPTA data which showed evidence for the Hellings-Downs correlations, and the full 25-year EPTA data where the evidence is not visible 2 . The value of γ = 13 / 3 ( α = 2 / 3) is shown as a dashed straight line. Our improved model results in a lower median-aposteriori strain amplitude of the gravitational wave background, as well as in a steeper spectral index, as shown with solid blue contours in Figure 1. Maximumaposteriori (lg A,γ ) obtained with our improved model and the respective measurement uncertainties based on 1 σ credible levels are reported at the top of Figure 1.  \nIn addition to revising noise priors, we consider the presence of epoch-correlated white noise [28] in the data. Epoch-correlated noise has not been considered in previous EPTA analyses because of a low number (1-4) of pulse arrival time measurements per observation epochs compared to other PTAs [6, 29]. Nevertheless, we find evidence for this noise in the data from certain backend-received systems. The posterior for our improved model in Figure 1b is marginalised over epochcorrelated noise. The choice leads to the consistency of the fully marginalised posterior on γ with 13 / 3 at 1 σ level. Revising noise priors alone, without considering epoch-correlated noise, yields a weaker consistency of the joint posterior on (lg A,γ ) with γ = 13 / 3 at 1 σ level (not shown in Figure 1b).  \nImplications. -If the energy loss in binary inspirals is dominated by the adiabatic emission of gravitational waves and if binaries are circular, the characteristic strain spectrum of the gravitational wave background is [26]  \nh 2 c ( f ) = 4 G 5 / 3 3 π 1 / 3 c 2 f -4 / 3 ∫ d 2 N dV dz M 5 / 3 (1 + z ) 1 / 3 dz, (1)  \nwhere G is the Newton's constant, c is the speed of light, z is redshift, M is the binary chirp mass, and d 2 N/ ( dV dz ), a function of ( M , z ), is the number density of binaries per unit comoving volume per unit redshift. The integral does not depend on a gravitational wave frequency, thus h c ∝ f -2 / 3 , as stated earlier. The background amplitude A depends on the mass spectrum and the abundance of supermassive binary black holes in the universe. The derived value of α = 2 / 3 ( γ = 13 / 3) is confirmed by population synthesis simulations, e.g. , Figure 7 in [11], where the theoretical uncertainty is only δγ ∼ 0 . 1 at 1 σ due to cosmic variance [31, 32].  \nTensions of γ ≈ 3 inferred from the previous EPTA analysis with 13 / 3 reported during the announcement of evidence for the gravitational background [5-8] has led to discussions on whether the signal is influenced by certain effects of binary evolution that make strain spectrum to appear flatter. Mechanisms of flattening h c ( f ) typically involve the introduction of a more rapid physical mechanism of a reduction in binary separation compared to a gravitational wave emission at < 0 . 1 parsec separations. Such a mechanism could be an environmental effect [33] such as stellar scattering [34, 35], the torques of a circumbinary gas disc [36]. It could also be due to the abundance of binaries in eccentric orbits which lead to a more prominent gravitational wave emission [37] 3 . In contrast, the results of our improved analysis are consistent with binary evolution driven only by the emission of gravitational waves.  \nOur improved model also impacts the measurement of the strain amplitude which is directly related to the number density of supermassive black hole binaries. The new  \nFIG. 1: Posterior distributions for the power-law amplitude A and spectral index γ of the putative gravitational wave background in the European Pulsar Timing Array (EPTA) data. The results of a fit to Hellings-Downs correlations are shown in the left panel (full 25-year data) and the middle panel (the 10-year subset of the data described in ref. [6]). The results of a fit of only temporal correlations to the 10-year data are shown in the right panel . Dashed red contours correspond to the result using the standard pulsar noise priors a , and the blue contours correspond to our improved model bc . The horizontal dashed line corresponds to the background from supermassive binary black holes inspiralling entirely due to gravitational wave emission (subject to cosmic variance). Our improved model results in a lower median-aposteriori strain amplitude of the background and mitigates tensions with γ = 13 / 3.  \n<!-- image -->  \nresults suggest that the supermassive black hole binaries are not as (over-)abundant as the earlier measurements implied. This is shown in Figure 2, which is a replica of Figure A1 in [11]. There, green horizontal bands correspond to theoretical uncertainties on the strain amplitude at the 16th - 84th percentile level in 24 studies [10, 14, 39-60]. Our improved model reduces tensions of the gravitational wave background strain amplitude with theoretical and observationally-based predictions for supermassive black hole binaries. Furthermore, the revised background strain amplitude corresponds to longer delay times between galaxy mergers and supermassive binary black hole mergers [61]. The caveat is that the reported amplitude is referenced to f = yr -1 , so the covariance between lg A and γ in posterior changes following a rotation of a power law about this frequency. Therefore, visible consistency with theoretical predictions may improve based on a different reference frequency.  \nObservational impact. -When the model closely matches reality, one would expect a reduction of the measurement uncertainty when adding extra data. This is not visible in the original EPTA analysis where the 1 σ range for (lg A,γ ) is roughly the same between the 10-year and the 25-year data before our improved noise model. As shown in Figure 1c, our improved analysis  \nyields a larger measurement uncertainty in the 10-year data, in agreement with our expectation. A decrease in 1 σ uncertainty levels in Figure 1c compared to Figure 1b indicates that inter-pulsar correlations in the 10-year data provide additional constraints on the background amplitude and spectral index.  \nThe shift of the posteriors in Figure 1 towards larger spectral indices and smaller amplitudes with our improved analysis 4 suggests that the louder pulsar-intrinsic noise with flatter spectra leaks into our measurement of the background strain spectrum when ensemble pulsar noise properties are not modelled.  \nBecause the 10-year data and the 25-year data are not independent data sets, a high degree of consistency is expected. In the original EPTA analysis, maximumaposteriori (lg A,γ ) in the 25-year data differ from those of the 10-year data by around (0 . 3 , 0 . 8). It is visible in red contours across all three panels in Figure 1. A match of the best-fit (lg A,γ ) between the 25-year data that does not exhibit Hellings-Downs correlations (Figure 1a) and the 10-year data when modelling only temporal cor-  \nFIG. 2: The bottom panel shows the predicted log-10 strain amplitude lg A from 24 studies. The top panel shows posteriors on lg A marginalised over γ . Dashed posteriors are obtained with the original pulsar noise model. Solid posteriors with a band of 1 σ credibility correspond to our improved model a . The green colour corresponds to the 10-year subset of the EPTA data, the grey colour corresponds to the full 25-year data.  \n<!-- image -->  \n- As in Figure 1b, additional modelling of the epoch-correlated noise is performed for a solid green contour.  \nrelations and not the Hellings-Downs correlations (Figure 1c) is achieved with our improved analysis. However, our improved analysis does not strongly impact best-fit (lg A,γ ) in the 10-year data when modelling inter-pulsar correlations. Because (lg A,γ ) obtained with only temporal correlations is still expected to match those obtained with temporal and Hellings-Downs correlations, it is also possible that we have not removed all noise model misspecification from the analysis of the EPTA data.  \nLet us briefly hypothesise about the nature of any other mismodelled noise. We note that the North American Nanohertz Observatory for Gravitational Waves (NANOGrav) has mitigated a tension of the background spectral index with 13 / 3 by adopting the Gaussian process model of the dispersion variation noise [5], as in the EPTA analysis [29]. Therefore, one potential source of a systematic error could be the mismodelling of the pulsarspecific noise that depends on a radio frequency. A very nearby binary is another example [62-65]. Frequencywise comparison of the inferred strain spectrum against  \nblack hole population synthesis models performed earlier by the EPTA (Figure 3 in ref. [12]) suggests that the deviation from γ = 13 / 3 may occur due to excess noise in two frequency bins, ≈ 10 -8 Hz and ≈ 3 × 10 -8 Hz. The rest of the spectrum seems to be consistent with γ = 13 / 3. The aforementioned potential sources of systematic errors may require better temporal and interpulsar correlation models of the data as part of future work.  \nMethods. -PTAs perform precision measurements of pulse arrival times from millisecond radio pulsars. The likelihood of delays(-advances) δt for a vector of pulse arrival times t is modelled as a multivariate Gaussian distribution L ( δt | θ ), where θ is a vector of parameters of models that describe the data. From Bayes' theorem, it follows that the posterior distribution of model parameters is P ( θ | δt ) = Z -1 L ( δt | θ ) π ( θ ), where Z is Bayesian evidence, a fully-marginalised likelihood. The term π ( θ ) is called a prior probability distribution, a model of how likely it is to find a certain value of θ in Nature. Model selection is performed by computing the ratio of Z for pairs of models, which is referred to as the Bayes factor. The Bayes factor is equal to the Bayesian odds ratio if both models are assumed to have equal prior odds. A form of the likelihood and the description of the standard analysis methodology can be found in [28]. Because the total PTA noise prior is a product of noise priors for every pulsar, PTA data will inform on the distribution of θ in Nature. Our improved analysis introduces hyperparameters Λ to parametrise priors: π ( θ | Λ ) π ( Λ ). We then perform a numerical marginalisation over Λ . For more details, please refer to the companion paper [24].  \nData availability. -Second data release of the European Pulsar Timing Array [25] is available at zenodo.org and gitlab.in2p3.fr.  \nCode availability. -The code to marginalise over the uncertainties in pulsar noise priors is available at github.com/bvgoncharov/pta priors. The PTA likelihood is incorporated in enterprise [66], and posterior sampling is performed using ptmcmcsampler [67].  \nAcknowledgements. -We thank Bruce Allen for helpful comments on the results, contributions to the structuring of this manuscript, and valuable advice on scientific paper writing. We thank Rutger van Haasteren for insightful discussions about hierarchical Bayesian inference. Some of our calculations were carried out using the OzSTAR Australian national facility (highperformance computing) at Swinburne University of Technology. European Pulsar Timing Array is a member of the International Pulsar Timing Array [68]. J. 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