The second data release from the European Pulsar Timing Array - IV. Implications for massive black holes, dark matter, and the early Universe

\footnotesize{Properties of the common correlated signal detected in \texttt{DR2new}. Left panel: free spectrum of the RMS induced by the excess correlated signal in each frequency resolution bin (with width defined by the inverse of the data span, $\Delta{f}=T^{-1}$). The straight line is the best power-law fit to the data. Right panel: joint posterior distribution in the $A-\gamma$ plane. \it{Note that we normalise $A$ to a pivotal frequency $f_0=10{\rm yr}^{-1}$.}}

\footnotesize{Properties of the common correlated signal detected in \texttt{DR2new}. Left panel: free spectrum of the RMS induced by the excess correlated signal in each frequency resolution bin (with width defined by the inverse of the data span, $\Delta{f}=T^{-1}$). The straight line is the best power-law fit to the data. Right panel: joint posterior distribution in the $A-\gamma$ plane. \it{Note that we normalise $A$ to a pivotal frequency $f_0=10{\rm yr}^{-1}$.}}

\footnotesize{GWB amplitude distributions predicted by the RSG15 models. The thin-dashed yellow line is for the full set of models in RSG15, whereas the thick-dashed orange line is for the subset considered here. The solid blue line is the distribution predicted by the 108 down-selected sample used in the analysis. The vertical line marks the median value of $A$ at $f_0=1{\rm yr}^{-1}$ reported in PaperIII when fixing $\gamma=13/3$ in the search. Note that the lower $x$-axis scale is for A at $f_0=1{\rm yr}^{-1}$, whereas the upper $x$-axis is for A at $f_0=10{\rm yr}^{-1}$ (the normalization used in this paper). Since $\alpha=-2/3$ for circular GW-driven binaries, there is a shift of 0.666 dex between the two.}

\footnotesize{Free spectrum violin plot comparing measured (orange) and expected (green) signals. Overlaid to the violins are the 100 Monte Carlo realizations of one specific model; among those, the thick one represents an example of a SMBHB signal consistent with the excess power measured in the data at all frequencies.}

\footnotesize{Expected properties of CGWs as a function of frequency. Top panel: free spectrum violin plot comparing the measured signal (orange) to the power distribution of CGWs (green). Empty violins show the full GWB produced by the models for comparison. Bottom panel: the probability of detecting a CGW with S/N$>3$ as a function of frequency (green circles, left $y-$axis scale). The average S/N of CGWs is also shown as red crosses (right $y-$axis scale).}

\footnotesize{$A-\gamma$ distribution of the measured signal (orange) compared to model predictions (green). 1$\sigma$ and 2$\sigma$ contours are displayed. Shown are also the marginalised $A$ (left) and $\gamma$ (right) distributions, with their 1$\sigma$ credible intervals highlighted as shaded areas.}

\footnotesize{ Marginalised posterior distributions for $\ndot$ using two SMBHB population models. The orange and green open histograms show marginalised posteriors for the agnostic and astrophysically-informed models, respectively. The filled-green histogram shows the prior for the astrophysically-informed model (the prior for the agnostic model is uniform in the range $[-20,3]$). The vertical dotted lines show the $5{\rm th}$ and $95{\rm th}$ percentiles of the posteriors. }

\footnotesize{Posterior distribution of the chirp mass function of merging SMBHBs for both the agnostic (orange) and astrophysically informed (green) models. For both models, shaded areas are the central 50\% and 90\% credible regions and the dashed lines show the medias. The black-dotted lines show the central 99\% region for the astrophysical prior.}

\footnotesize{ Posterior distribution of selected parameters for the astrophysically-informed model using nine frequency bins of the free spectrum for the inference. Parameters are described in Sec.~\ref{sec:smbhb_informed}.}

\footnotesize{Predictions for the GWB characteristic strain amplitude at $f=1/10$yr from a range of SAMs published in the literature, assuming quasicircular orbits and no environmental interactions (i.e. $\gamma=13/3$), but different physical prescriptions for the delays (increasing from left to right) between galaxy mergers and black hole mergers. The ``no delays'', ``medium delays'' and ``long delays'' models correspond respectively to the classes of models \textit{(i), (ii)} and \textit{(iii)} defined in the text. The ranges account for the finite resolution of the models. The shaded area is the DR2new 95\% confidence bound. More details about the models are provided in the text.}

\footnotesize{Binned spectrum of the predicted GWB amplitude for models ``HS-nod-SN-high-accr (B+20)'' and ``HS-nod-noSN (B+20)''. The distribution of the predictions represents the scatter among different realizations of the SMBHB population (``cosmic variance''). Also shown are power-law fits to the predictions.}

\footnotesize{Predictions for $A(f=1/10{\rm yr})$ in various SAMs, obtained by fitting the spectrum in the first 9 frequency bins with $\gamma=13/3$ for multiple realizations of the SMBHB population. The error bars represent the 95\% confidence interval for the predictions, and account for the scatter due to cosmic variance. For each model (except for the boosted accretion model HS-nod-SN-high-accr (B+20)), the higher prediction is the extrapolation to infinite SAM resolution, while the lower one is the finite-resolution prediction. The shaded area is the 95\% confidence interval for the measurement of $A(f=1/10{\rm yr})$, fixing $\gamma=13/3$. For HS-nod-SN-high-accr (B+20) we only show the result uncorrected for resolution.}

\footnotesize{Predictions for the GWB characteristic strain amplitude at $f=10/$yr$^{-1}$ from a range of \texttt{L-Galaxies} semi-analytical model versions, assuming that $h_c(f)\,{\propto}\, f^{-2/3}$. The error bars are computed taking into account the cosmic variance. To this end, we divided the \texttt{Millennium} box into 125 sub-boxes and we compute the GWB in each one. The standard deviation provided by the 125 GWBs corresponds to the extension of our error bars.}

\footnotesize{Orbital parameters (distance between the SMBHs, semi-major axis and eccentricity) of a SMBHB formed in a representative $N$-body simulation of a galactic merger with parameters drawn from progenitors of likely PTA sources in the {\tt IllustrisTNG100-1} cosmological simulation. Mergers are selected from the merger trees of the 100 most massive galaxies at $z=0$, based on galaxy mass ratio (major mergers with mass ratio $1:4$ or higher) and redshift ($z\leq2$). The dashed lines indicate the critical separation $a_f$ and the corresponding eccentricity $e_f$ at the time in the evolution marking approximately the end of the SMBH inspiral due to DF and the beginning of the hardening phase. Dots represent $a$ and $e$ computed from the apoastron-periastron separation of the two SMBHs {\it before} pairing in a bound binary.}

\footnotesize{Posterior distributions of the recovered GWB from injections on synthetic data mimicking \texttt{DR2new}. Top panel: statistical offset in an ideal dataset. The square marks the injected value and the blue contours are 1$\sigma$ and 2$\sigma$ of the recovered posterior. Bottom panel: effect of high-frequency noise mismodeling on the recovery. The orange, blue and green contours are respectively obtained when EFAC$=0.8, 1, 1.2$ are used for the recovery (injected EFAC$=1$).}

\footnotesize{Posterior distributions of the recovered GWB from injections on synthetic data mimicking \texttt{DR2new}. Top panel: statistical offset in an ideal dataset. The square marks the injected value and the blue contours are 1$\sigma$ and 2$\sigma$ of the recovered posterior. Bottom panel: effect of high-frequency noise mismodeling on the recovery. The orange, blue and green contours are respectively obtained when EFAC$=0.8, 1, 1.2$ are used for the recovery (injected EFAC$=1$).}

\footnotesize{2D posteriors of the tensor-to-scalar ratio (in $\log_{10}$) and the fractional energy density spectral index $n_T$ in the PTA frequency range. The 68\% and 95\% credible regions are displayed. The black dashed line represents the tensor-to-scalar ratio upper bound found in \cite{2022PhRvD.105h3524T} assuming single-field slow-roll inflation.}

\footnotesize{Comparison of the string tension posteriors for the two string models (BOS and LRS) in case \textit{(i)}, $N_c=2$ and $N_k =0$ ($\Gamma = 57$). Solid lines assume only a cosmic string background, dashed lines assume both a population of GW-driven circular SMBHBs and cosmic strings.}

\footnotesize{SGWB spectra (in terms of $\log_{10} h^2\Omega_{gw}$) for four different early Universe SGWB models considered in this paper. BOS/LRS correspond to a cosmic string background with $N_c=2$ and $N_k =0$ ($\Gamma = 57$), and $\logGmu = -10.1$/$-10.6$. The GWB from turbulence is plotted in solid line for $\lambda_* \mathcal{H}_* = 1$, $\Omega_*=0.3$, and $T_* = 140$ MeV. The inflationary spectra is shown for $\log_{10} r = -13.1$ and $n_T = 2.4$ (maximum a posteriori value). Power spectrum of the 2nd-order GWB from the scalar curvature perturbations described by the powerlaw model with $A_\zeta^{\text{10yr}}=-2.9$ and $n_s=2.1$ is shown with brown puncture-dot line. The nine first Fourier bins posteriors of the common signal are represented by the grey violin areas.}

\footnotesize{2D posteriors for the parameters of the background from turbulence around the QCD energy scale obtained using a free spectrum fit on \texttt{DR2new} data. The 68\% and 95\% credible regions are displayed.}

\footnotesize{Results for the monochromatic curvature perturbations described by Eq.~\ref{eq:monochromatic_sp}. Left panel: recovered slopes $\gamma$ of a simple power-law model as a function of characteristic scale $k^{*}$ of the injected GWB generated by the monochromatic curvature perturbations. The horizontal lines show the theoretical value of $\gamma$ from a population of circular, GW-driven SMBHBs (grey) and the one obtained in PaperIII (orange). Right panel: 1$\sigma$ and 2$\sigma$ contours of the posterior distributions on the amplitude $A_{\zeta}$ and characteristic scale of fluctuations $k^{*}$ for \texttt{DR2new} (orange colour). The posterior distribution is overlaid with the current constraints on the primordial power spectrum using Planck data (CMB). The grey colour depicts the 2-$\sigma$-confidence intervals. The purple shaded area represents the bounds from spectral distortions \citep{2012ApJ...758...76C}. For comparison in green we place the prediction of the primordial spectrum of scalar perturbations in the two-field model of inflation described in \cite{2020JCAP...08..001B} for a range of the model parameters. All three models result in PBH mass functions peaked at $\sim35$~$M_\sun$ with the brightest line corresponding to the dark matter fraction of PBHs of $\sim0.01$.}

\footnotesize{Results for the monochromatic curvature perturbations described by Eq.~\ref{eq:monochromatic_sp}. Left panel: recovered slopes $\gamma$ of a simple power-law model as a function of characteristic scale $k^{*}$ of the injected GWB generated by the monochromatic curvature perturbations. The horizontal lines show the theoretical value of $\gamma$ from a population of circular, GW-driven SMBHBs (grey) and the one obtained in PaperIII (orange). Right panel: 1$\sigma$ and 2$\sigma$ contours of the posterior distributions on the amplitude $A_{\zeta}$ and characteristic scale of fluctuations $k^{*}$ for \texttt{DR2new} (orange colour). The posterior distribution is overlaid with the current constraints on the primordial power spectrum using Planck data (CMB). The grey colour depicts the 2-$\sigma$-confidence intervals. The purple shaded area represents the bounds from spectral distortions \citep{2012ApJ...758...76C}. For comparison in green we place the prediction of the primordial spectrum of scalar perturbations in the two-field model of inflation described in \cite{2020JCAP...08..001B} for a range of the model parameters. All three models result in PBH mass functions peaked at $\sim35$~$M_\sun$ with the brightest line corresponding to the dark matter fraction of PBHs of $\sim0.01$.}

\footnotesize{Results for the power-law model of the curvature perturbations described by Eq.~\eqref{eq:powerlaw_sp}. Left panel: 1$\sigma$ and 2$\sigma$ contours of the posterior distributions on the amplitude $A_{\zeta}$ and the slope of the power spectrum $n_s$ obtained by the analysis of \texttt{DR2New}. Right panel: 1$\sigma$ and 2$\sigma$ contours of the power spectra inferred from the \texttt{DR2New} analysis by picking 1000 random samples from the posteriors overlaid with the current constraints on the primordial power spectrum using the latest Planck data. The grey colour depicts the 2$\sigma$-confidence intervals. The green lines and purple shaded areas are the same as in Fig.~\ref{fig:summ_cmb_delta}.}

\footnotesize{Results for the power-law model of the curvature perturbations described by Eq.~\eqref{eq:powerlaw_sp}. Left panel: 1$\sigma$ and 2$\sigma$ contours of the posterior distributions on the amplitude $A_{\zeta}$ and the slope of the power spectrum $n_s$ obtained by the analysis of \texttt{DR2New}. Right panel: 1$\sigma$ and 2$\sigma$ contours of the power spectra inferred from the \texttt{DR2New} analysis by picking 1000 random samples from the posteriors overlaid with the current constraints on the primordial power spectrum using the latest Planck data. The grey colour depicts the 2$\sigma$-confidence intervals. The green lines and purple shaded areas are the same as in Fig.~\ref{fig:summ_cmb_delta}.}

\footnotesize{Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the correlated (top row) and uncorrelated (bottom row) analysis of the \texttt{DR2new} dataset. The pulsar correlated analysis is not shown, but displays the same features.}Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the correlated analysis of the \texttt{DR2new} dataset.

\footnotesize{Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the correlated (top row) and uncorrelated (bottom row) analysis of the \texttt{DR2new} dataset. The pulsar correlated analysis is not shown, but displays the same features.}Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the correlated analysis of the \texttt{DR2new} dataset.

\footnotesize{Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the correlated (top row) and uncorrelated (bottom row) analysis of the \texttt{DR2new} dataset. The pulsar correlated analysis is not shown, but displays the same features.}

\footnotesize{Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the correlated (top row) and uncorrelated (bottom row) analysis of the \texttt{DR2new} dataset. The pulsar correlated analysis is not shown, but displays the same features.}Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the uncorrelated analysis of the \texttt{DR2new} dataset.

Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the uncorrelated analysis of the \texttt{DR2new} dataset.

\footnotesize{Constraints on $\Psi_c$ as a function of $m_\phi$ using the EPTA \texttt{DR2new} dataset from PaperIII. Previous analyses are shown for comparison, cf. \cite{Porayko_2014, Porayko_2018} for further details. The blue, orange and brown lines represent the 95\% Bayesian upper limit on $\Psi_c$ obtained from the EPTA \texttt{DR2new} dataset with the correlated, uncorrelated and pulsar correlated analysis, respectively. The purple line shows the expected ULDM abundance computed from Eq.~\eqref{eq:psi_c}.}

\footnotesize{Constraints on the ULDM density $\rho_{\phi}$ normalised to the DM background value $\rho_{\text{DM}} = 0.4 \text{GeV}/ \text{cm}^3$. The blue, orange and brown lines represent the 95\% Bayesian upper limit on $\rho_{\phi}$, obtained from the EPTA \texttt{DR2new} dataset with the correlated, uncorrelated and pulsar correlated analysis, respectively. The purple dotted line shows the fiducial local DM density value.}

Marginalised posterior distributions for all $18$ parameters of the astrophysically-informed model. The posterior and prior are shown in grey and green, respectively.

Marginalised posteriors for all five parameters of the agnostic SMBHB model.

Preliminary - (still running, this shows samples so far) Three versions of the astro-informed analysis showing the four parameters highlighted in M21. Left: using 3 frequency bins, middle: using 5 frequency bins, right: using 9 frequency bins.

Preliminary - (still running, this shows samples so far) Three versions of the astro-informed analysis showing the four parameters highlighted in M21. Left: using 3 frequency bins, middle: using 5 frequency bins, right: using 9 frequency bins.

Preliminary - (still running, this shows samples so far) Three versions of the astro-informed analysis showing the four parameters highlighted in M21. Left: using 3 frequency bins, middle: using 5 frequency bins, right: using 9 frequency bins.