Probing modified gravity theories and cosmology using gravitational-waves and associated electromagnetic counterparts

The direct detection of gravitational waves by the LIGO/Virgo Collaboration has opened a new window with which to measure cosmological parameters such as the Hubble constant $H_0$, and also probe general relativity on large scales. In this paper we present a new phenomenological approach, together with its inferential implementation, for measuring deviations from general relativity (GR) on cosmological scales concurrently with a determination of $H_0$. We consider gravitational waves (GWs) propagating in an expanding homogeneous and isotropic background, but with a modified friction term and dispersion relation relative to that of GR. We find that a single binary neutron star GW detection will poorly constrain the GW friction term. However, a joint analysis including the GW phase and GW-GRB detection delay could improve constraints on some GW dispersion relations provided the delay is measured with millisecond precision. We also show that, for massive gravity, by combining 100 binary neutron stars detections with observed electromagnetic counterparts and host galaxy identification, we will be able to constrain the Hubble constant, the GW damping term and the GW dispersion relation with 2%, 15% and 2% accuracy, respectively. We emphasize that these three parameters should be measured together in order to avoid biases. Finally we apply the method to GW170817, and demonstrate that for all of the GW dispersions relations we consider, including massive gravity, the GW must be emitted $\sim 1.74\mathrm{s}$ before the gamma-ray burst. Furthermore, at the GW merger peak frequency, we show that the fractional difference between the GW group velocity and $c$ is $\lesssim {10}^{-17}$.

Figure 1

Schematic figure showing a binary neutron star merger at comoving distance $r_{\mathrm{com}}$ emitting a GW and a GRB. The blue dots represent the two binary neutron stars. In modified gravity, GWs may propagate with a frequency-dependent speed, and arrive with a relative time delay with respect to the electromagnetic counterpart. In this example, photons follow a null-like geodesic identified by an angle of 45 deg on the plot. Subluminal GWs follow a time-like geodesic identified by an angle lower than 45 deg (angle defined counterclockwise between the $r$ and $\eta$ axes).

Figure 2

The shaded area of the plot shows the allowed value for the parameter ${\alpha }_M$ with respect to the redshift. Any functional form of ${\alpha }_M$ in the shaded area, will result in a monotonically increasing GW luminosity distance.

Figure 3

Absolute value of the fractional change (color bar) on the GW luminosity distance introduced by ${\alpha }_M$ at different redshifts for a BNS detection during the O4 observing scenario. We assume a value for the Hubble constant of $H_0=70\mathrm{km}{\mathrm{Mpc}}^{-1}{\mathrm{s}}^{-1}$. The hatched region corresponds to BNSs that cannot be detected since they are above the GW horizon. The dotted region is excluded from Eq. (27).

Figure 4

Fractional deviation in the PN coefficients introduced by the GW dispersion parameter ${\widehat{\alpha }}_j$ with respect to the source redshift for a BNS O4 observing scenario. The Hubble constant has been fixed to $H_0=70\mathrm{km}{\mathrm{Mpc}}^{-1}{\mathrm{s}}^{-1}$. The hatched area corresponds to sources above the O4 GW horizon for detectability, and in each case the range of the $y$ axis satisfies Eq. (45).

Figure 5

Time delay in seconds (color bar) introduced by the GW dispersion parameter ${\widehat{\alpha }}_j$ with respect to the source redshift for a BNS O4 observing scenario. The Hubble constant has been fixed to $H_0=70\mathrm{km}{\mathrm{Mpc}}^{-1}{\mathrm{s}}^{-1}$. The hatched area corresponds to sources above the O4 GW horizon for detectability, and in each case the range of the $y$ axis satisfies Eq. (45).

Figure 6

Ratio of the fractional changes introduced on the GW phase and GW time delay with a GW dispersion relation.

Figure 7

The inferential model. The arrows between the nodes represent conditional probabilities between the variables, while the shaded nodes represent observed variables.

Figure 8

Marginal and joint posterior distributions for ${\alpha }_M$, ${\widehat{\alpha }}_{-2}$ for a software injection with SNR 27 in modified cosmology.

Figure 9

Marginal and joint posterior distributions from the combination of 100 BNS detection in a modified Universe with parameters $H_0=70\mathrm{km}{\mathrm{Mpc}}^{-1}{\mathrm{s}}^{-1}$, ${\alpha }_M=5$ and ${\widehat{\alpha }}_{-2}=9.23\times {10}^{-14}{\mathrm{Hz}}^2$.

Figure 10

Joint and marginalized probability density function for $H_0$ and ${\alpha }_M$ assuming ${\widehat{\alpha }}_j=0$ (no friction term).

Figure 11

Posterior distributions for $\mathrm{\Delta }{t}_d^{\mathrm{MG}}$ (GW-GRB observation delay) and $v_g/c-1$, both evaluated at $f_{R,d}$. Red/green: Scenario I. Blue/orange: Scenario II.

Figure 12

Posterior distributions for $H_0$, ${\alpha }_M$ and $m_g^2$. Black dashed line: $H_0$ only (namely GR). Green solid line: ($H_0$, ${\alpha }_M$). Blue dotted line: Scenario I ($H_0$, ${\widehat{\alpha }}_j$). Purple dash-dotted line: Scenario II ($H_0$, ${\alpha }_M$, ${\widehat{\alpha }}_j$). Red patches: cosmologically motivated constraints on $H_0$, ${\alpha }_M$ [77] (fixing $m_g^2=0$). Yellow line: upper bound on $m_g^2$ from GW170817 in Ref. [26] (fixing $H_0=69.3\mathrm{km}{\mathrm{Mpc}}^{-1}{\mathrm{s}}^{-1}$).

Figure 13

The color bar displays the value of the selection function normalized over its value for $H_0=70\mathrm{km}{\mathrm{Mpc}}^{-1}{\mathrm{s}}^{-1}$ and ${\alpha }_M=0$. Horizontal axis: Hubble constant. Vertical axis: GW friction parameter.