The second data release from the European Pulsar Timing Array: V. 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 normalize $A$ 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 normalize $A$ 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{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 marginalized $A$ (left) and $\gamma$ (right) distributions, with their 1$\sigma$ credible intervals highlighted as shaded areas.}

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.

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 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.}

\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 mis-modeling 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 mis-modeling 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$=1$).}

2-D 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.

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.

The 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). The nine first Fourier bins posteriors of the common signal are represented by the gray violin areas.

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.

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 $\sim0.01$.

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 $\sim0.01$.

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}.

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}.

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

Posterior probabilities for the ULDM amplitude $\Psi_c$ and mass $m_\phi$, from the correlated 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.

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

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) line represents the 95\% Bayesian upper limit on $\Psi_c$ obtained from the EPTA \texttt{DR2new} dataset with the correlated (uncorrelated) analysis. The purple line shows the expected ULDM abundance computed from Eq.~\eqref{eq:psi_c}.

Constraints on the ULDM density $\rho_{\phi}$ normalized to the DM background value $\rho_{\text{DM}} = 0.4 \text{GeV}/ \text{cm}^3$. The blue (orange) line represents the 95\% Bayesian upper limit on $\rho_{\phi}$, obtained from the EPTA \texttt{DR2new} dataset with the correlated (uncorrelated) analysis. 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.