Cosmology with space-based gravitational-wave detectors: Dark energy and primordial gravitational waves

Proposed space-based gravitational-wave (GW) detectors such as DECIGO and BBO will detect $\sim {10}^6$ neutron-star (NS) binaries and determine the luminosity distances to the binaries with high precision. Combining the luminosity distances with cosmologically induced phase corrections on the GWs, cosmological expansion out to high redshift can be measured without the redshift determinations of host galaxies by electromagnetic observation and be a unique probe for dark energy. On the other hand, such a NS-binary foreground should be subtracted to detect primordial GWs produced during inflation. Thus, the constraining power on dark energy and the detectability of the primordial gravitational waves strongly depend on the detector sensitivity and are in close relation with one another. In this paper, we investigate the constraints on the equation of state of dark energy with future space-based GW detectors with/without identifying the redshifts of host galaxies. We also study the sensitivity to the primordial GWs, properly dealing with the residual of the NS-binary foreground. Based on the results, we discuss the detector sensitivity required to achieve the forementioned targeted study of cosmology.

Figure 1

Noise curves of default DECIGO (solid blue curve) and those scaled by $r_n=1/2$ (green), $1/3$ (orange), $1/5$ (red), which include the confusion noise from a number of WD binaries. The WD-binary foreground $\sqrt{S_h^{\mathrm{WD}}}$ is shown with solid, black curve. The three diagonal lines represent the NS-binary foreground $\sqrt{S_h^{\mathrm{NS}}}$ before subtraction (or ${\mathcal{R}}_{\mathrm{NS}}=1$) with ${\dot{n}}_0={10}^{-5}$, ${10}^{-6}$, ${10}^{-7}{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$ from the top to the bottom.

Figure 2

Fraction of NS residual ${\mathcal{R}}_{\mathrm{NS}}$ after the subtraction as a function of ${\rho }_{\mathrm{th}}/{\overline{\rho }}_e(5)$. The fitting formula is presented in Appendix .

Figure 3

$d_L$—$X$ plot as a function of redshift. Left: $w_0$ is varied, while $w_a$ and ${\Omega }_{\mathrm{m}}$ are fixed. Each curve corresponds to $w_0=-1.2$ (red, dotted curve), $w_0=-1$ (green, solid curve), $w_0=-0.8$ (blue, dashed curve). Right: $w_a$ is varied, while $w_0$ and ${\Omega }_{\mathrm{m}}$ are fixed. Each curve corresponds to $w_a=0$ (green, solid curve), $w_a=0.5$ (red, dotted curve), $w_a=1.0$ (blue, dashed curve). The green solid curves in both figures are for $\Lambda \mathrm{CDM}$ universe. The points corresponding to $z=1$, 2, 3, 4, 5 are shown on the curve.

Figure 4

$w_0$—$w_a$ error ellipses marginalized over ${\Omega }_m$. The merger rate is fixed to ${\dot{n}}_0={10}^{-6}{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$. From the left to the right panels, the observation time is 3 yr, 5 yr, and 10 yr. In each panel, the larger to smaller ellipses denote those with $r_n=1$, $1/3$, and $1/5$, respectively. The dot at the center is our fiducial value: $w_0=-1$ and $w_a=0$.

Figure 5

FoM achievable for each parameter set of $r_n$, $T_{\mathrm{obs}}$, and ${\dot{n}}_0$. From the left to the right panels, the observation time is 3 yr, 5 yr, and 10 yr. In each panel, the merger rate is represented by curve types: ${\dot{n}}_0={10}^{-5}$ (dashed), ${10}^{-6}$ (solid), ${10}^{-7}$ (dotted), in the unit of ${\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$.

Figure 6

Redshift dependence of the measurement accuracies in $d_L$—$X$ measurement: $\mathrm{FoM}/\mathrm{FoM}(z_{\max }=5)$ (blue, solid), $\Delta w_0(z_{\max }=5)/\Delta w_0$ (red, dotted), $\Delta w_a(z_{\max }=5)/\Delta w_a$ (green, dashed). Parameters are fixed to $r_n=1/3$, $T_{\mathrm{obs}}=5\mathrm{yr}$, and ${\dot{n}}_0={10}^{-6}{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$.

Figure 7

FoM as a function of $\alpha$ defined in Eq. (19) or $N_z$. The observation time and the merger rate are fixed to $T_{\mathrm{obs}}=5\mathrm{yr}$ and ${\dot{n}}_0={10}^{-6}{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$. The horizontal dashed line represents $\mathrm{FoM}=100$. The solid and dotted curves are the case with $r_n=1$ and $1/3$, respectively.

Figure 8

${\Omega }_{\mathrm{gw}}$ detectable with $\mathrm{SNR}=5$ for each parameter set of $r_n$, $T_{\mathrm{obs}}$, and ${\dot{n}}_0$. The observation time is denoted by colors: $T_{\mathrm{obs}}=3\mathrm{yr}$ (red, solid), 5 yr (green, dotted), 10 yr (blue, dashed). For a fixed observation time, the merger rate only affects the $r_n=1$ case: the upper and lower lines correspond to ${\dot{n}}_0={10}^{-6}$, and ${10}^{-7}{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$, respectively. For ${\dot{n}}_0={10}^{-5}$ case, no data is plotted due to the same reason in Table .

Figure 9

FoM and ${\Omega }_{\mathrm{gw}}$ achievable with GW observation alone ($\eta =0$) for each parameter set of $r_n$, $T_{\mathrm{obs}}$, and ${\dot{n}}_0$. From the left to the right panels, the observation time is 3 yr, 5 yr, and 10 yr. In each panel, the merger rate is represented by curve types: ${\dot{n}}_0={10}^{-5}$ (dashed), ${10}^{-6}$ (solid), ${10}^{-7}$ (dotted), in the unit of ${\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$. On each curve, points from the top to the bottom correspond to $r_n=1/2$, $1/3$, and $1/5$. The horizontal lines correspond to $r=0.1$ and 0.01 at CMB scale in the de-Sitter inflation.

Figure 10

Measurement accuracy of the luminosity distance with a single binary as a function of redshifts. The curves tagged ${\sigma }_{\mathrm{inst}}$ are those determined only by instrumental noise and with the observation time 3 yr (red, solid curve), 5 yr (green, dotted curve), and 10 yr (blue, short-dashed curve), respectively. The lensing error and the peculiar velocity error are represented by magenta (long-dashed) and light blue (dot-dashed) curves.

Figure 11

Errors of $X(z)/H_0$ as a function of a redshift. The Solid curve denote the error due to instrumental noise in a GW observation, and the dotted curve denote the error propagating from the lensing error of $d_L$. Assumed parameters are $r_n=1$, $T_{\mathrm{obs}}=3\mathrm{yr}$, and ${\dot{n}}_0={10}^{-6}{\mathrm{Mpc}}^{-3}{\mathrm{yr}}^{-1}$.