Gravitational waves from quasinormal modes of a class of Lorentzian wormholes

S. Aneesh, Sukanta Bose, and Sayan Kar

Phys. Rev. D 97, 124004 – Published 5 June 2018

Abstract

Quasinormal modes of a two-parameter family of Lorentzian wormhole spacetimes, which arise as solutions in a specific scalar-tensor theory associated with braneworld gravity, are obtained using standard numerical methods. Being solutions in a scalar-tensor theory, these wormholes can exist with matter satisfying the weak energy condition. If one posits that the end-state of stellar-mass binary black hole mergers, of the type observed in GW150914, can be these wormholes, then we show how their properties can be measured from their distinct signatures in the gravitational waves emitted by them as they settle down in the postmerger phase from an initially perturbed state. We propose that their scalar quasinormal modes correspond to the so-called breathing modes, which normally arise in gravitational wave solutions in scalar-tensor theories. We show how the frequency and damping time of these modes depend on the wormhole parameters, including its mass. We derive the mode solutions and use them to determine how one can measure those parameters when these wormholes are the endstate of binary black hole mergers. Specifically, we find that if a breathing mode is observed in LIGO-like detectors with design sensitivity, and has a maximum amplitude equal to that of the tensor mode that was observed of GW150914, then for a range of values of the wormhole parameters, we will be able to discern it from a black hole. If in future observations we are able to confirm the existence of such wormholes, we would, at one go, have some indirect evidence of a modified theory of gravity as well as extra spatial dimensions.

Received 27 February 2018

DOI:https://doi.org/10.1103/PhysRevD.97.124004

Physics Subject Headings (PhySH)

Alternative gravity theories Gravitational waves

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

S. Aneesh^1,*, Sukanta Bose^2,3,†, and Sayan Kar^4,‡

¹Department of Physics, Indian Institute of Technology, Kharagpur 721 302, India
²Inter-University Centre for Astronomy and Astrophysics, Post Bag 4, Ganeshkhind, Pune 411007, India
³Department of Physics and Astronomy, Washington State University, 1245 Webster, Pullman, Washington 99164-2814, USA
⁴Department of Physics and Center for Theoretical Studies Indian Institute of Technology, Kharagpur 721 302, India

^*aneesh.s306@gmail.com
^†sukanta@iucaa.in
^‡sayan@phy.iitkgp.ernet.in

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Vol. 97, Iss. 12 — 15 June 2018

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Images

Figure 1
Variation of effective potential with the parameters $κ$ , $λ$ and $l$ : (a) $κ = λ = 0.5, M = 1$ , (b) $l = 0$ , $λ = 0.5$ , $M = 1$ .
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Figure 2
Time domain profile and quasinormal ringdown. Profiles have been calculated for $κ = 0.5 = λ$ , $M = 1$ and $l = 0$ , 1, 2 at $r_{*} = 7$ . Initial conditions are $ψ_{l m} (u, 0) = \exp [- \frac{(u - 10)^{2}}{100}]$ and $ψ_{l m} (0, v) = 1$ . The integration grid is $u, v \in (0, 200)$ with $h = 0.1$ : (a) $l = 0$ , (b) $l = 1$ , and (c) $l = 2$ .
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Figure 3
Real and imaginary parts of QNFs are plotted with respect to $λ$ . Here $M = 1$ and $κ = 1 - λ$ . Frequencies for all other values of $κ$ , $λ$ and $M$ can be obtained from this data. The smooth curve joining the data points is the analytical fit which is obtained in the next section.
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Figure 4
An example of Prony fit of time domain profile. The data is fitted with four complex frequencies.
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Figure 5
The estimated statistical error in $κ / λ$ (in percentage) is shown for various values of the wormhole mass $M$ (in the left figure, where $κ / λ$ is kept fixed at 0.01) and the wormhole parameter $κ / λ$ (in the right figure, where $M$ is kept fixed at $30 M_{⊙}$ ). These estimates were obtained for gravitational-wave observations of the breathing mode of the wormhole solution studied here with a detector like aLIGO. The maximum amplitude of the mode in all cases is set to be $10^{- 21}$ , which is approximately the peak amplitude of GW150914, whose final detector-frame mass had a median value of about $68 M_{⊙}$ . The wormhole QNM frequency for $l = 0$ is 64 Hz for $M = 68 M_{⊙}$ and $κ / λ = 0.1$ . For comparison, the frequency of the $l = 2$ , $m = 2$ QNM of GW150914 was at or above $≃ 243 Hz$ [3]. The error in $κ / λ$ increases with $M$ (left figure) primarily because the mode frequency decreases, thereby, placing the signal in less sensitive part of the detector band. The right figure shows that the error in $κ / λ$ initially reduces when the value of that parameter is increased. This happens because the mode frequency shifts to more sensitive parts of the detector band (from 64 Hz at $κ / λ = 3 \times 10^{- 3}$ to 85 Hz at $κ / λ = 10^{- 2}$ ). For higher values of $κ / λ$ , the error increases owing to decreasing time constant (and, therefore, the effective integration duration) of the mode.
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Figure 6
The two left-most plots are similar to the ones shown in Fig. 5, but for the Einstein Telescope [78]. The right-most plot is for $κ / λ = 10$ and the Einstein Telescope; it is interesting how the error decreases with increasing mass for the mass range shown here, before it rises again (not shown) owing to the signal frequency moving into the low-frequency seismic band). Moreover, whereas $κ / λ < 1$ in Fig. 5 here we raise that limit in the middle and right plots.
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