Probing the size of extra dimensions with gravitational wave astronomy

In the Randall-Sundrum II braneworld model, it has been conjectured, according to the AdS/CFT correspondence, that a brane-localized black hole (BH) larger than the bulk AdS curvature scale $\ell$ cannot be static, and it is dual to a four-dimensional BH emitting Hawking radiation through some quantum fields. In this scenario, the number of the quantum field species is so large that this radiation changes the orbital evolution of a BH binary. We derived the correction to the gravitational waveform phase due to this effect and estimated the upper bounds on $\ell$ by performing Fisher analyses. We found that the Deci-Hertz Interferometer Gravitational Wave Observatory and the Big Bang Observatory (DECIGO/BBO) can give a stronger constraint than the current tabletop result by detecting gravitational waves from small mass BH/BH and BH/neutron star (NS) binaries. Furthermore, DECIGO/BBO is expected to detect ${10}^5$ BH/NS binaries per year. Taking this advantage, we find that DECIGO/BBO can actually measure $\ell$ down to $\ell =0.33\mu \mathrm{m}$ for a 5 yr observation if we know that binaries are circular a priori. This is about 40 times smaller than the upper bound obtained from the tabletop experiment. On the other hand, when we take eccentricities into binary parameters, the detection limit weakens to $\ell =1.5\mu \mathrm{m}$ due to strong degeneracies between $\ell$ and eccentricities. We also derived the upper bound on $\ell$ from the expected detection number of extreme mass ratio inspirals with LISA and BH/NS binaries with DECIGO/BBO, extending the discussion made recently by McWilliams [Phys. Rev. Lett. 104, 141601 (2010)]. We found that these less robust constraints are weaker than the ones from phase differences.

Figure 1

The non-sky-averaged noise spectral density for BBO (thick solid curve) and LISA (thin solid curve). We also show the amplitudes of GWs from a $(10+{10}^6)M_{\odot }$ BH/BH binary at $D_L=500\mathrm{Mpc}$ (thin dotted line) and a $(1.4+10)M_{\odot }$ BH/NS binary at $D_L=3\mathrm{Gpc}$ (thick dotted line). Each dot labeled “1 yr” represents the frequency at 1 yr before the binary reaches ISCO.

Figure 2

The configuration of DECIGO and BBO. There are eight effective interferometers in total.

Figure 3

(Left panel) The upper bounds on $\ell$ [Eq. (57)] by detecting GWs from BH/BH binaries at $D_L=500\mathrm{Mpc}$ with LISA. We set $m_0=10M_{\odot }$ and vary the values of $M_0$. The (red) solid, (green) dashed, and (blue) dotted curves represent the bounds obtained from 1 yr, 3 yr, and 5 yr observations, respectively. The horizontal black dotted-dashed line shows the upper bound $\ell \le 14\mu \mathrm{m}$ obtained from the tabletop experiment [8]. The SNR for various mass binaries with each observation period is also shown at the bottom. (Right panel) We show the 5 yr results for $m_0=3M_{\odot }$, $m_0=5M_{\odot }$, and $m_0=10M_{\odot }$ with the (red) solid, (green) dashed, and (blue) dotted curves, respectively. Again, the SNR for each binary is shown at the bottom.

Figure 4

The upper bounds on $\ell$ by detecting GWs from equal mass BH/BH binaries at $D_L=3\mathrm{Gpc}$ with DECIGO/BBO. The meaning of each line is the same as in the left panels of Fig. 3. Here only one curve is shown for the SNR since 1 yr, 3 yr, and 5 yr results are almost indistinguishable.

Figure 5

The upper bounds on $\ell$ by detecting GWs from BH/BH binaries at $D_L=3\mathrm{Gpc}$ with DECIGO/BBO. The meaning for each line is the same as in Fig. 3. We set $m_0=10M_{\odot }$ for the left panels and $T_{\mathrm{obs}}=5\mathrm{yr}$ for the right panels.

Figure 6

(Top panel) The upper bounds on $\ell$ by detecting GWs from BH/NS binaries at $D_L=3\mathrm{Gpc}$ with DECIGO/BBO. The meaning for each curve is the same as in the left panel of Fig. 3. We set the NS mass to $m_0=1.4M_{\odot }$.

Figure 7

Probability distribution of NS/BH merger time $t_{\mathrm{merg}}$ [36].

Figure 8

The detection errors of $\ell$ [Eq. (50)] with statistical analysis for 1 yr (red solid line), 3 yr (green dashed line), and 5 yr (blue dotted line) observations of BH/NS binaries with DECIGO/BBO. The (black) dotted-dashed line represents $\Delta {\ell }^{(\mathrm{stat})}=\ell$. If $\Delta {\ell }^{(\mathrm{stat})}$ comes below this line, $\ell$ can be measured.

Figure 9

The error contour showing a 95% confidence level on the $I_e\mathrm{\text{−}}L$ plane for a $(1.4+10)M_{\odot }$ BH/NS binary at $D_L=3\mathrm{Gpc}$, such that the fiducial values lie at the center of the ellipse. It is assumed that the observation period is 1 yr with DECIGO/BBO, and we take $I_e^{(\mathrm{fid})}=L^{(\mathrm{fid})}=0$ for our fiducial values.

Figure 10

Same as in Fig. 8 but, here, the eccentricity is included in binary parameters with the fiducial value $I_e=0$.