On the uncertainty of the White Dwarf Astrophysical Gravitational Wave Background

Author(s)

Hofman, Sophie, Nelemans, Gijs

Abstract

The astrophysical gravitational wave background (AGWB) is a stochastic GW signal that is emitted by different populations of inspiralling binary systems containing compact objects. In the frequency range between 10^{-4} and 10^{-1} Hz it will be detected by future space-based gravitational wave detectors like the Laser Interferometer Space Antenna (LISA). Recently, Staelens & Nelemans 2024 (SN24) concluded that the white dwarf (WD) contribution to the AGWB dominates over that of black holes (BHs). We aim to investigate the uncertainties of the WD AGWB that arise from the use of different stellar metallicities, different star formation rate density (SFRD) models, and different binary evolution models. We use the code developed by SN24 to determine the WD component of the AGWB. We use a metallicity dependent SFRD based on Chruslinska et al (2019,2020,2021) to construct five different SFRD models. We use four different population models that use different common-envelope treatment and six different metallicities for each model. For all possible combinations, the WD component of the AGWB is dominant over other populations of compact objects. The effects of metallicity and population model are smaller than the effect of a (metallicity dependent) SFRD model. We find a range of about a factor of 5 in the level of the WD AGWB around a mid value of Omega_WD = 4 10^{-12} at 1 mHz and a shape that depends weakly on the model. We find an uncertainty for the WD component of the AGWB of about a factor 5. We note that there exist other uncertainties that have an effect on this signal as well. We discuss whether the turnover of the WD AGWB at 10 mHz will be detectable by LISA, and find that this is likely. We confirm the previous finding that the WD component of the AGWB dominates over other populations, in particular BHs.

Figures

Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six metallicity bins for the MZ19 SFRD model.

Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six metallicity bins for the MZ19 SFRD model.


Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six different SFRD models.

Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six different SFRD models.


Density plot of the initial properties of the WD population in the case of a $\gamma\alpha$, $\alpha$ = 4 population model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the Universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.

Density plot of the initial properties of the WD population in the case of a $\gamma\alpha$, $\alpha$ = 4 population model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the Universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.


WD components of the AGWB for six different metallicities, compared to the LVK results (upper limit to BH/NS AGWB \citep[dashed grey][]{2021PhRvD.104b2004A} and estimates for the BBH and BNS components in green and blue, \citealt{2023PhRvX..13a1048A}), the LISA Powerlaw Integrated sensitivity \cite[black][]{2013PhRvD..88l4032T,2020PhRvD.101l4048A}, and an estimate of the Galactic foreground (pink) based on \cite{2021PhRvD.104d3019K}. The population synthesis model used is $\gamma\alpha$, $\alpha$ = 4 and the SFRD model used is that of \cite{madau_cosmic_2014}.

WD components of the AGWB for six different metallicities, compared to the LVK results (upper limit to BH/NS AGWB \citep[dashed grey][]{2021PhRvD.104b2004A} and estimates for the BBH and BNS components in green and blue, \citealt{2023PhRvX..13a1048A}), the LISA Powerlaw Integrated sensitivity \cite[black][]{2013PhRvD..88l4032T,2020PhRvD.101l4048A}, and an estimate of the Galactic foreground (pink) based on \cite{2021PhRvD.104d3019K}. The population synthesis model used is $\gamma\alpha$, $\alpha$ = 4 and the SFRD model used is that of \cite{madau_cosmic_2014}.


As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. The MZ19 SFRD model is used. The dashed lines are the WD components of the AGWB for each of the six metallicity bins, the solid light green line is the sum of all six separate WD components.

As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. The MZ19 SFRD model is used. The dashed lines are the WD components of the AGWB for each of the six metallicity bins, the solid light green line is the sum of all six separate WD components.


As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. Each line represents the sum of the signal for a different SFRD model. The light green line is the same as the one in Fig.~\ref{fig:Omega ga4 MZ19}.

As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. Each line represents the sum of the signal for a different SFRD model. The light green line is the same as the one in Fig.~\ref{fig:Omega ga4 MZ19}.


As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. Each line represents a different choice for population synthesis model. The MZ19 SFRD model is used.

As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. Each line represents a different choice for population synthesis model. The MZ19 SFRD model is used.


As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. The red line represents the WD component of the AGWB for a choice of $\alpha\alpha$, $\alpha$ = 4, with Z = 0.02 and the SFRD of \cite{madau_cosmic_2014}, which is the result from \citetalias{2024A&A...683A.139S}. The light green line represents the WD component of the AGWB for a choice of $\gamma\alpha$, $\alpha$ = 4 with the MZ19 SFRD model. The light green band represents the uncertainty estimate.

As in Fig.~\ref{fig:Omega ga4 SFH madau&dickinson}. The red line represents the WD component of the AGWB for a choice of $\alpha\alpha$, $\alpha$ = 4, with Z = 0.02 and the SFRD of \cite{madau_cosmic_2014}, which is the result from \citetalias{2024A&A...683A.139S}. The light green line represents the WD component of the AGWB for a choice of $\gamma\alpha$, $\alpha$ = 4 with the MZ19 SFRD model. The light green band represents the uncertainty estimate.


Comparison of the WD AGWB (salmon) and the broken power law fit (dark red) to a purely $f^{2/3}$ signal (green dashed) by dividing the curves by $f^{2/3}$. The WD AGWB deviates from the expected  $2/3$  slope but the residuals (bottom) between the WD AGWB and the fit are below 2\%.

Comparison of the WD AGWB (salmon) and the broken power law fit (dark red) to a purely $f^{2/3}$ signal (green dashed) by dividing the curves by $f^{2/3}$. The WD AGWB deviates from the expected $2/3$ slope but the residuals (bottom) between the WD AGWB and the fit are below 2\%.


Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six metallicity bins for the LZ21 SFRD model.

Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six metallicity bins for the LZ21 SFRD model.


Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six metallicity bins for the HZ21 SFRD model.

Total star formation rate density (SFRD) in $M_\odot$ Mpc$^{-3}$ yr$^{-1}$ versus redshift (z), for the six metallicity bins for the HZ21 SFRD model.


As in Figure \ref{fig:Omega ga4 SFH madau&dickinson}. The population synthesis model used is $\alpha\alpha$, $\alpha$ = 1 and the SFRD used is that of \cite{madau_cosmic_2014}.

As in Figure \ref{fig:Omega ga4 SFH madau&dickinson}. The population synthesis model used is $\alpha\alpha$, $\alpha$ = 1 and the SFRD used is that of \cite{madau_cosmic_2014}.


As in Figure \ref{fig:Omega ga4 SFH madau&dickinson}. The population synthesis model used is $\alpha\alpha$, $\alpha$ = 4 and the SFRD used is that of \cite{madau_cosmic_2014}.

As in Figure \ref{fig:Omega ga4 SFH madau&dickinson}. The population synthesis model used is $\alpha\alpha$, $\alpha$ = 4 and the SFRD used is that of \cite{madau_cosmic_2014}.


As in Figure \ref{fig:Omega ga4 SFH madau&dickinson}. The population synthesis model used is $\gamma\alpha$, $\alpha$ = 1 and the SFRD used is that of \cite{madau_cosmic_2014}.

As in Figure \ref{fig:Omega ga4 SFH madau&dickinson}. The population synthesis model used is $\gamma\alpha$, $\alpha$ = 1 and the SFRD used is that of \cite{madau_cosmic_2014}.


Density plot of the initial properties of the WD population in the case of a $\alpha\alpha$, $\alpha$ = 1 population synthesis model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.

Density plot of the initial properties of the WD population in the case of a $\alpha\alpha$, $\alpha$ = 1 population synthesis model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.


Density plot of the initial properties of the WD population in the case of a $\alpha\alpha$, $\alpha$ = 4 population synthesis model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.

Density plot of the initial properties of the WD population in the case of a $\alpha\alpha$, $\alpha$ = 4 population synthesis model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.


Density plot of the initial properties of the WD population in the case of a $\gamma\alpha$, $\alpha$ = 1 population synthesis model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.

Density plot of the initial properties of the WD population in the case of a $\gamma\alpha$, $\alpha$ = 1 population synthesis model: chirp mass, $\mathcal{M}$, versus GW frequency at the time of formation. Each panel shows a different metallicity of the universe. The dashed lines indicate frequencies above which there is significant (10\%) frequency evolution in a Hubble time.


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