Erratum for: A&A, 660, A68 (2022), https://doi.org/10.1051/0004-6361/202141987

We have realized that there is an error in our code for the calculation of the gravitational wave background (GWB), specifically an incorrect source sampling and their contribution to the GWB. The corrected amplitudes of the GWB for the two binary lifetime scenarios (100 Myr and 1 Gyr) at the reference frequency of 1 yr⁻¹ are h_c, MYR = 8.41 × 10⁻¹⁶ and h_c, GYR = 7.88 × 10⁻¹⁶, whereas the original were h_c, MYR = 1.1 × 10⁻¹⁵ and h_c, GYR = 1.4 × 10⁻¹⁶, respectively. The conclusions of the original paper are therefore unchanged as, although the previously calculated amplitudes were overestimated, they are still lower than the signal recently reported by the Pulsar Timing Array experiments (EPTA Collaboration et al. 2023; Agazie et al. 2023; Reardon et al. 2023).

From a finite population of massive black hole binaries retrieved from simulations, we can obtain multiple realisations by drawing the true number of binaries in a given comoving volume from a Poisson distribution P centered at (Kelley et al. 2017)

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\mathit{ij}}=\frac{1}{V_{\mathrm{s}}}\frac{\mathrm{d}{V}_{\mathrm{c}}(z_{\mathit{ij}})}{\mathrm{d}{z}_{\mathit{ij}}}\mathrm{\Delta }{z}_{\mathit{ij}},\end{array}$$(1)

where V_s is the comoving volume of the simulation, dV_c(z) is the comoving volume element, and Δz_ij is the redshift step size with indices i and j iterating over each binary and time step, respectively.

The corrected GWB was calculated by multiplying Eq. (15) (original paper) by Λ_ij, which effectively can be written as the following (Kelley et al. 2017):

$$\begin{array}{c}\hfill h_{\mathrm{c}}^2(f)={\displaystyle \sum _{\mathit{ij}}}P({\mathrm{\Lambda }}_{\mathit{ij}}){\displaystyle \sum _k}\frac{f_{\mathrm{r}}}{\mathrm{\Delta }f}{h}_k^2(f_{\mathrm{r}}),\end{array}$$(2)

where f and f_r are observed and source rest-frame GW frequencies, respectively, and index k iterates over all binaries.

As various models presented in the original paper were indistinguishable in terms of their GWB amplitude, in Fig. 1 provided in this erratum, we present the results only for the default model (B4H10). The solid colored lines show the median characteristic strain averaged over 300 iterations, while the solid red line corresponds to the amplitude assuming a pure power law $h_{\mathrm{c}}=A_{{\mathrm{yr}}^{-1}}{\left(\frac{f}{f_0}\right)}^{-2/3}$, where f₀ = yr⁻¹. For reference, we also show the amplitude of the signal as measured by the PTA experiments, specifically the European and Indian PTA (EPTA Collaboration et al. 2023), the North American Nanohertz Observatory for Gravitational Waves (Agazie et al. 2023), and the Parkes PTA (Reardon et al. 2023).

Fig. 1. Characteristic strain of the GWB for the default model B4H10. Black lines show the amplitude of the signal constrained by the PTA experiments, while solid colored lines show median amplitudes for 100 Myr, 1 Gyr, and 5 Gyr scenarios.

References

© The Authors 2024

All Figures

	Fig. 1. Characteristic strain of the GWB for the default model B4H10. Black lines show the amplitude of the signal constrained by the PTA experiments, while solid colored lines show median amplitudes for 100 Myr, 1 Gyr, and 5 Gyr scenarios.
In the text