Measuring Cosmological Redshift Using Gravitational Waves from Compact Binaries with Mass Transfer

Abstract

The mass transfer process is prevalent during the inspiral phase of compact binary systems. Detection of gravitational waves from the inspiral phase of binaries with white dwarfs will allow us to measure the mass transfer rate. Mass transfer effects provide additional contributions to the phase of gravitational waves, which can break the degeneracy between binary masses and redshift. Based on the analytic mass transfer rate to the first order post-Newtonian evolution of orbital angular frequency, we use the Fisher matrix to forecast the ability of DECIGO to measure the redshift of compact binaries with mass transfer. We conclude that for compact binary systems containing white dwarfs, the redshift can be determined to an accuracy of $10\%$ for $z=0.01$ with a $SNR\thicksim 30$.

Figures (5)

• Figure 1: (a) Matching between analytic and numerical results of maximum frequency $f_{max}$. The red solid line is the analytic maximum frequency and the black dotted line is the numerical maximum frequency in 10.1093/mnras/stab626. (b) Matching between analytic and numerical results of MT rate $\chi_{fmax}$ at the maximum frequency. The red solid line is the analytic MT rate obtained by using Eq.\ref{['Chi*']} and the black dotted line is the numerical MT rate in 10.1093/mnras/stab626. When the WD mass ranges from $0.2 M_\odot$ to $0.9 M_\odot$, the analytical results and numerical results exhibit agreement. We have set the value $\mathcal{A}=5\times 10^{-8}$ and $m_p=1.4 M_\odot$.
• Figure 2: The relationship between MT rate $\chi$, GW frequency $f$, and WD mass $m_{WD}$ with the mass of primary star $m_p=1.4 M_\odot$. $\chi'=\chi/(1 M_\odot/\text{yr})$ is the dimensionless MT rate. The "forbidden zone" refers to the frequency range that cannot be reached for a given WD mass or the frequency higher than the maximum frequency. The "region without MT" denotes the area where MT has not yet occurred because the WD radius is smaller than the Roche lobe radius. The MT rate increases with the increase of frequency while with the decrease of WD mass.
• Figure 3: The evolution of MT rate and GW frequency for different initial WD masses. The dashed lines represent the evolution of GW frequency, the solid lines represent the evolution of MT rate, and the red stars denote the maximum points of each curve. The initial conditions are $f_0=0.01$ Hz, $m_p=0.2 M_\odot$, and $m_{WD}=0.3,0.5,0.8 M_\odot$.
• Figure 4: The relative uncertainty of redshift under different GW frequencies and WD masses. The primary star in the left figure is a $1.4 M_\odot$ neutron star, while the one in the right figure is a $5 M_\odot$ black hole. Within the parameter space of the blue region above the contour line of $\log(\Delta z/z)=-1$, the redshift $z$ can be detected with an accuracy higher than $10\%$. The luminosity distance of the binary systems is 40 Mpc and the redshift is 0.01.
• Figure 5: The relative uncertainty in redshift measurements of binaries across different redshifts. The redshift uncertainty under four sets of parameters $(\bar{f}_0=0.035\text{Hz}$ and $\bar{m}_{WD}=0.7M_\odot)$, $(\bar{f}_0=0.04\text{Hz}$ and $\bar{m}_{WD}=0.7M_\odot)$, $(\bar{f}_0=0.04\text{Hz}$ and $\bar{m}_{WD}=0.8M_\odot)$, and $(\bar{f}_0=0.045\text{Hz}$ and $\bar{m}_{WD}=0.8M_\odot)$ are plotted as functions of redshift $z$ ranging from $10^{-3}$ to $0.1$.