European Pulsar Timing Array Limits On An Isotropic Stochastic Gravitational-Wave Background

TL;DR

The paper presents a comprehensive Bayesian and frequentist search for an isotropic stochastic GWB using a six-pulsar, ~18-year EPTA dataset. By jointly modeling intrinsic pulsar noise, DM variations, and common noise processes (including clock and ephemeris errors), the authors derive a robust 95% upper limit $A<3.0\times10^{-15}$ for a SMBHB-like spectrum with $\\gamma=\\tfrac{13}{3}$, corresponding to $\\Omega_{ ext{gw}}(f) h^{2} < 1.1\\times10^{-9}$ at $f=2.8$ nHz. Allowing the spectral index to vary loosens the limit to $A<1.3\\times10^{-14}$, highlighting the degeneracy between amplitude and spectral slope. The study also places constraints on cosmic strings ($G\mu/c^{2} < 1.2$–$1.3\\times10^{-7}$) and relic GWB ($\\Omega^{\text{relic}}_{\text{gw}}(f) h^{2} < 1.2\\times10^{-9}$), demonstrating PTA power to probe beyond-SMBHB scenarios. Overall, the results underscore the importance of robust, simultaneous modeling of pulsar-specific and common noise terms in PTA analyses and set the stage for stronger future constraints as datasets grow.

Abstract

We present new limits on an isotropic stochastic gravitational-wave background (GWB) using a six pulsar dataset spanning 18 yr of observations from the 2015 European Pulsar Timing Array data release. Performing a Bayesian analysis, we fit simultaneously for the intrinsic noise parameters for each pulsar, along with common correlated signals including clock, and Solar System ephemeris errors, obtaining a robust 95$\%$ upper limit on the dimensionless strain amplitude $A$ of the background of $A<3.0\times 10^{-15}$ at a reference frequency of $1\mathrm{yr^{-1}}$ and a spectral index of $13/3$, corresponding to a background from inspiralling super-massive black hole binaries, constraining the GW energy density to $Ω_\mathrm{gw}(f)h^2 < 1.1\times10^{-9}$ at 2.8 nHz. We also present limits on the correlated power spectrum at a series of discrete frequencies, and show that our sensitivity to a fiducial isotropic GWB is highest at a frequency of $\sim 5\times10^{-9}$~Hz. Finally we discuss the implications of our analysis for the astrophysics of supermassive black hole binaries, and present 95$\%$ upper limits on the string tension, $Gμ/c^2$, characterising a background produced by a cosmic string network for a set of possible scenarios, and for a stochastic relic GWB. For a Nambu-Goto field theory cosmic string network, we set a limit $Gμ/c^2<1.3\times10^{-7}$, identical to that set by the {\it Planck} Collaboration, when combining {\it Planck} and high-$\ell$ Cosmic Microwave Background data from other experiments. For a stochastic relic background we set a limit of $Ω^\mathrm{relic}_\mathrm{gw}(f)h^2<1.2 \times10^{-9}$, a factor of 9 improvement over the most stringent limits previously set by a pulsar timing array.

Figures (17)

• Figure 1: Summary of key results from the analysis of a 6 pulsar dataset from the 2015 EPTA data release (D15). Results are presented in terms of $\Omega_{\mathrm{gw}}(f)$ as a function of GW frequency, with $H_0 = 70 \mathrm{km~s^{-1}~Mpc^{-1}}$. We indicate the 95$\%$ upper limits on the amplitude of a correlated GWB assuming a power law model with a spectral index of $\gamma=13/3$ (solid black line; Section \ref{['Section:Results']}) and for a more general analysis where the power is determined simultaneously at a set of discrete frequencies (dashed line) as discussed in Section \ref{['sec:bayesian']}. The red shaded areas represent the central 68%, 95%, and 99.7% confidence interval of the predicted GWB amplitude according to 2013MNRAS.433L...1S under the assumptions that a SMBHB evolves purely due to gravitational radiation reaction and binaries are circular (See Section \ref{['Section:implicationsSMBHB']} for more details). Only about 5$\%$ of the distribution is excluded, meaning that our limit does not place significant restrictions on the cosmic SMBHB population. We also indicate 95% upper limits obtained for a stochastic relic background (green dash-dotted line; Section \ref{['sec:relicGWs']}), and for cosmic string network backgrounds (blue triple-dashed line; Section \ref{['Section:implicationsStrings']}). The cosmic string limit plotted corresponds to a fiducial model for a population of cosmic strings, with the following parameters: string tension $G\mu/c^2=10^{-7}$, the birth-scale of loops relative to the horizon $\alpha_{\rm cs}=1.6\times10^{-6}$, spectral index $q=4/3$, cut-off on the number of emission harmonics $n_*=1$ and intercommutation probability $p=1$. Finally we indicate recent constraints placed by CMB 2012PhRvD..85l3002S, and BBN 1997rggr.conf..373A2000PhR...331..283M2015arXiv150201589P observations.
• Figure 2: Simulated clock error used used in our analysis (black line) after subtracting the maxmium likelihood timing models from the joint analysis, and the time averaged maximum likelihood clock signal with 1$\sigma$ uncertainties (red points with error bars). We find the recovered signal is consistent with the injected signal across the whole dataspan.
• Figure 3: Top: timing residuals as a function of Modified Julian Date (MJD) for the 6 pulsars included in the stochastic GWB analysis presented in this work, after the maximum likelihood DM variations signal realisation has been subtracted. From top to bottom these are PSRs: J0613$-$0200, J1012+5307, J1600$-$3053, J1713+0747, J1744$-$1134, and J1909$-$3744. While the overall timing baseline for this dataset is $\sim 18$ yr, only four of the 6 pulsars have data that extends across the majority of this timespan, and in particular, PSR J1909$-$3744 contributes only to the latter half of the dataset, significantly reducing our overall sensitivity to signals at the lowest frequencies supported by the dataset. Bottom: Frequency coverage for the 6 pulsars included in the stochastic GWB analysis presented in this work. The order of the pulsars is as in the top plot. Colours indicate observing frequencies $< 1000$MHz (red crosses), between 1000 and 2000 MHz (green circles) and $>$ 2000 MHz (blue squares). In addition to fewer pulsars extending across the full dataset, there is also less multi-frequency coverage in the early data. This further decreases our sensitivity to a stochastic GWB at the lowest sampled frequencies as the signal becomes highly covariant with the DM variations for the individual pulsars in the first half of the dataset.
• Figure 4: EFAC values obtained for all systems from the initial analysis performed for the 6 pulsars used in our analysis. All EFACs are consistent with values equal to or greater than 1 within uncertainties, with the exception of the Westerbork 1380 MHz data in PSR J1713+0747 which have values consistent with $\sim 0.5$. This could be the result of systematic effects that occur in the template forming phase, and is the subject of ongoing work.
• Figure 5: Comparison of the 1-dimensional marginalised posterior probability distributions for PSRs (left to right) J1909$-$3744, J1713+0747, and J1744$-$1134. The y-axis in all plots represents probability. In each case we show the spin-noise and DM variation power law parameters for the full noise analysis (red solid lines) and 5-dimensional analysis where the TOA error bars have been pre scaled by the mean value of the EFAC/EQUAD parameters for each pulsar backend (blue dashed lines). In both cases parameter estimates have been obtained using a uniform prior on the amplitude of the spin-noise and DM variations power law models. We also show the global EFAC parameters from the 5-dimensional analysis in each case. We find the posteriors are consistent between the two sets of analysis.