Can environmental effects spoil precision gravitational-wave astrophysics?

No, within a broad class of scenarios. Gravitational-wave (GW) astronomy will open a new window on compact objects such as neutron stars and black holes (BHs). It is often stated that large signal-to-noise detections of ringdown or inspiral waveforms can provide estimates of the masses and spins of compact objects to within fractions of a percent, as well as tests of general relativity. These expectations usually neglect the realistic astrophysical environments in which compact objects live. With the advent of GW astronomy, environmental effects on the GW signal will eventually have to be quantified. Here we present a wide survey of the corrections due to these effects in two situations of great interest for GW astronomy: the BH ringdown emission and the inspiral of two compact objects (especially BH binaries). We mainly focus on future space-based detectors such as eLISA, but many of our results are also valid for ground-based detectors such as aLIGO, aVirgo, and KAGRA. We take into account various effects such as electric charges, magnetic fields, cosmological evolution, possible deviations from general relativity, firewalls, and the effects related to various forms of matter such as accretion disks and dark matter halos. Our analysis predicts the existence of resonances dictated by the external mass distribution, which dominate the very late-time behavior of merger and ringdown waveforms. The mode structure can drastically differ from the vacuum case, yet the BH response to external perturbations is unchanged at the time scales relevant for detectors. This is because, although the vacuum Schwarzschild resonances are no longer quasinormal modes of the system, they still dominate the response at intermediate times. Our results strongly suggest that both parametrized and ringdown searches should use at least two-mode templates. Our analysis of compact binaries shows that environmental effects are typically negligible for most eLISA sources, with the exception of very few special extreme-mass-ratio inspirals. We show, in particular, that accretion and hydrodynamic drag generically dominate over self-force effects for geometrically thin disks, whereas they can be safely neglected for geometrically thick disk environments, which are the most relevant for eLISA. Finally, we discuss how our ignorance of the matter surrounding compact objects implies intrinsic limits on the ability to constrain strong-field deviations from general relativity.

Figure 1

Toy model of two rectangular barriers of height $V_0$ and $V_1\ll V_0$. For $V_1=0$, one recovers the original toy model considered by Chandrasekhar and Detweiler [109].

Figure 2

QNMs of the double-barrier system. Top left: Some modes of the toy model of two rectangular barriers of height $V_0$ and $V_1\ll V_0$ as functions of the second barrier position $b$. For definiteness, we focus on $V_0=16/a^2$, $V_1={10}^{-3}/a^2$, and $c-b=0.1a$. Top right: The same but for the fundamental mode only as a function of $b$ with $V_1={10}^{-3}/(ab)$ and for different values of $c-b\equiv L$. Qualitatively similar results hold for different choices of the parameters. Bottom left: Eigenfunctions of the double-barrier potential. We show the first three QNMs, the fundamental mode in the vacuum case (${\omega }_{v}a=0.466-0.710i$), and two intermediate modes whose real part is close to that of ${\omega }_v$. Bottom right: QNM spectrum (cf. also Table 2). In both panels we set $V_0=16/a^2$, $V_1={10}^{-3}/a^2$, $b=10a$, and $c=11a$, but different choices of the parameters would give similar results.

Figure 3

Left: Waveforms for a Gaussian packet in the double-barrier potential. Top panel: The initial packet is located on the left of the first barrier, $x_0=-2a$ and $\sigma =a$. Bottom panel: The initial packet is located between the two barriers, $x_0=5a$ and $\sigma =a$. For both panels, $V_0=16/a^2$ and $c=b+a$. Other choices give qualitatively similar results.

Figure 4

Left panels: Percentage deviations of the real and imaginary parts of the QNMs of a Schwarzschild BH surrounded by a thin shell with respect to the case of an isolated BH (with the same horizon mass) as a function of the shell mass $\delta M$ and for different values of the shell radius, $r_0$, and $l=2$. Note that isospectrality is mildly broken and that polar modes refer to $v_s=0$ (the dependence on $v_s$ is shown in Fig. 5). Right: The same as a function of $r_0$ for different values of $\delta M$. Top panels refer to the case $r_0\lesssim 10M$, whereas bottom panels refers to $r_0\gtrsim 10M$.

Figure 5

Left panels: Linear coefficients of ${\delta }_R$ and ${\delta }_I$ in the small $\delta M/M$ limit as functions of the shell radius $r_0$ for $l=2$ and for the real and imaginary parts. In the polar case we have considered $v_s=0$. Right panels: Real and imaginary parts of the QN fundamental frequency for $l=2$ polar modes as a function of $v_s$ for $\delta M={10}^{-2}M$ and different values of $r_0$.

Figure 6

Fractional difference between axial and polar modes (normalized by the polar modes) of a thin-shell model as a function of the shell location $r_0$ for different values of $\delta M$ and $v_s$.

Figure 7

Percentage deviations of the real and imaginary parts of the fundamental $l=2$ mode of a matter-bumpy BH as a function of $r_0$ for different values of $\delta M$ and $L$. Left and right panels refer to models I and II in Eqs. (56) and (57), respectively.

Figure 8

Example of two modes of the bumpy BH configuration I (left panels) and II (right panels) for $\ell =2$, $\delta M=0.01M$, $L=10M$ as a function of $r_0$ in the large $r_0$ limit. The black curve represents the “vacuum-driven” mode, i.e., the one that resembles the fundamental mode of the vacuum BH at intermediate distance. The red curve represents the “matter-driven” mode, i.e., a mode which exists only in the presence of matter and that approaches $({\omega }_R,-{\omega }_I)\to \infty$ as $r_0$ decreases. In this limit, the mode becomes difficult to track further. Note that, at a large distance, this mode becomes lower lying with respect to the vacuum-driven mode.

Figure 9

Left: The linear in $\delta M/M$ correction to ${\delta }_{R,I}$ as a function of $r_0$ for the bumpy BH model I and for various values of $L$. Right: Example of a mode tracked as a function of $L$ for fixed $\delta M$ and $r_0\ll M$.

Figure 10

Scalar $l=2$ waveforms for a point particle falling into a Schwarzschild BH. The particle “suddenly” materializes at $r_p=6M$ with gamma factor $\gamma =2$ and subsequently falls into the BHs. The matter profile has extension $L=2,1.5$ in the top and bottom panels, respectively.

Figure 11

Scalar $l=1$ waveforms for a scattering in a bumpy BH background. Initial data are $\mathcal{S}={\psi }_0=0$ and ${\dot{\psi }}_0=f{e}^{-(r_{*}-r_0{)}^2/{\sigma }^2}$ with $\sigma =6M$ and $r_p=10M$. We used our bumpy profile II with $L=2M$ and $\delta M=0.1M$ (left panels) and $\delta M=2M$ (right panels), respectively. Each single panel corresponds to a different value of $r_S$ in the bumpy metric.

Figure 12

Percentage difference for the fundamental $l=2$ scalar QNM of the Brown-Husain solution (71) as a function of $Q_m$. The extremal limit corresponds to $Q_m/M=2^{5/3}/3\sim 1.06$ for $k=3/2$ and to $Q_m/M=3^{3/4}/2\sim 1.14$ for $k=2$.

Figure 13

Relative corrections to the fundamental axial QNM for a firewall model using a thin shell with mass $\delta M=0.01M$ located at $r_0$. Top and bottom panels, respectively, refer to the real and the imaginary part of the frequency as a function of the shell location in the limit where $r_0$ approaches the horizon (i.e., $r_0\to -\infty$ in tortoise coordinates). The oscillatory behavior is similar to that shown in the bottom right panel in Fig. 4. Finite numerical accuracy prevents us from tracking the modes when $r_0/(2M_0)-1\ll {10}^{-6}$, where much larger corrections are expected. The modes were computed by using an accurate continued-fraction method adapted from Ref. [112]. Polar modes and different values of $\delta M$ would give qualitatively similar results.

Figure 14

Percentage difference for the fundamental $l=2$ scalar and $l=1$ electromagnetic QNMs of the parametrized metric (12) and (13) as a function of various parameters. Each curve corresponds to a single parameter turned on, while the others are set to zero.

Figure 15

Coefficients appearing in the formulas (161), (163), (166), and (171) as defined in Table 6 as functions of $M$. We have set the observation time $T=1\mathrm{y}$, the final orbital radius $r_f=6M$, and the mass of the small particle reads $\nu M=1.4M_{\odot }$ and $\nu M=10M_{\odot }$ in the top and bottom panels, respectively. The coefficient $c_{\widehat{\alpha }}$ refers to $\widehat{\alpha }=3.5$; other values give qualitatively similar results.

Figure 16

Dephasing due to thin-disk self-gravity. The disk is modeled by superimposing the potentials in Eqs. (93) and (94) and by fitting the model parameters to match the Shakura-Sunyaev surface density (95). We have set the observation time $T=1\mathrm{y}$, the final orbital radius $r_f=6M$, and the mass of the small particle reads $\nu M=1.4M_{\odot }$ and $\nu M=10M_{\odot }$ for the solid and the dashed line, respectively. The dephasing is shown as a function of the central mass $M$ and for $f_{\text{Edd}}=1$ and $\alpha =0.1$. The fit parameters are $M_{\text{disk}}\approx 63M_{\odot }{f_{\text{Edd}}}^{7/10}[M/({10}^6{M}_{\odot }){]}^{11/5}(\alpha /0.1{)}^{-4/5}$, $R\approx 1016GM/c^2$ for the Kusmin-Toomre model and $M_{\text{disk}}\approx 1.5M_{\odot }{f_{\text{Edd}}}^{7/10}[M/({10}^6{M}_{\odot }){]}^{11/5}(\alpha /0.1{)}^{-4/5}$, $R\approx 74GM/c^2$ for the exponential model. The inset shows a comparison of the surface density (95) and of the fit.

Figure 17

Dephasing due to the parametrized metric element (12) and assuming GW dissipation through the lowest-order GR formula (141). Note that in this approximation the dephasing depends only on the metric coefficient $A$ and not on $B$. Higher-order corrections in $1/r$ correspond to smaller dephasing. We consider two different choices of the masses of the binary components and assume a $1-\mathrm{y}$ observation time terminating at $r=6M$.

Figure 18

Scattering of a scalar wave packet in the Schwarzschild-AdS background. The wave packet has a unit amplitude and width $\sigma =0.06$ and is centered at $r_0=0.1$, around a BH with radius $r_{+}=0.025$ (in units with $|\mathrm{\Lambda }|=3$).