DETERMINING THE DETECTABILITY OF CHRISTODOULOU MEMORY WAVES BY LISA
G.M. Stange1,2, D. Kennefick2,3, 1Department of Physics, University of Florida, Gainesville, Florida, 32612, 2Arkansas Center for Space and Planetary Sciences, University of Arkansas, Fayetteville, Arkansas, 72701, 3Department of Physics, University of Arkansas, Fayetteville, Arkansas, 72701
Introduction: According to the theory of general relativity, masses cause warps in space-time. The acceleration of masses causes a warping of space-time that propagates outward at the speed of light. Such propagations are called gravitational waves [2]. Gravitational waves cause space and its contents to grow and shrink in alternating perpendicular directions as they pass by.
A gravitational wave has energy. General relativity theory states that all energy has mass. Therefore, a gravitational wave constitutes a flux of mass which emits its own gravitational wave. This wave of the wave causes a permanent relative displacement in any system of free masses that it passes through [1], [3]. Christodoulou [4] showed mathematically that this kind of wave with memory exists. For the above reasons, these byproduct waves are known as Christodoulou memory waves.
LISA, the laser interferometer space antenna, is a joint mission of NASA and ESA designed to detect gravitational waves through interferometry. It is hoped that LISA will also be capable of detecting Christodoulou memory waves emitted by two inspiraling bodies. The purpose of this project is to determine the feasibility of such detections.
To analyze the capability of LISA to reliably detect memory waves, a program was constructed to produce the signal-to-noise ratio of a wave detected by LISA. If this ratio is equal to 3 or above, the detection would be considered 99.75% reliable.
The Model: The first part of the code was designed to produce the amplitude of a memory wave based on several variables dependent on the system emitting the wave. Because the purpose of this research was to determine if any Christodoulou memory waves would be detectable by LISA, a system was chosen that would be the easiest to detect.
It is theorized that a super massive black hole, or SMBH, of millions or billions of solar masses resides at the center of every galaxy. These black holes should capture many of the stars and stellar mass black holes that inhabit the galactic core, creating an accelerating binary system that will emit gravitational waves and memory waves. Due to the large mass ratio involved, such a system would produce a memory wave of relatively high amplitude. In addition, it is thought that such systems would occur fairly often among all the galactic cores within the detection range of LISA. For these reasons, a super massive black hole-stellar mass black hole binary system was chosen as the model.
Programming: The first step in programming was to input, using FORTRAN 90, the following function [1] for the amplitude of a memory wave (all equations except (5), (6), and (7) are presented in geometrized units for which c=G=1).
In this equation μ is the reduced mass of the system, M is the total mass of the system, r is the distance from the system to LISA, t is the time until the coalescence of the two objects, and ι is the orientation of the system (0 degrees corresponds to a face-on orientation to Earth and 90 degrees to an edge-on orientation). To produce a larger amplitude, and therefore better chance of detection, the orientation of the system is assumed to be edge-on, meaning ι is equal to 90 degrees. This simplifies the amplitude equation to:
(1)
At a certain point, the two objects in question would touch, causing this equation to no longer be accurate. After some unknown fluctuations in the amplitude, the function should simply become horizontal as the two objects finally merge into one. This horizontal line would be present at the peak amplitude because the displacement from the memory function is permanent. To determine the point at which the function would become horizontal, the time at which the two objects first touch must be determined. This cutoff time would be the time at which the orbital radius of the stellar mass black hole was equal to the radius of the super massive black hole. The orbital radius is given by [1]:
(2)
The radius of a black hole (assumed to be non-spinning) is given by the Schwarzschild equation:
(3)
The mass of the larger black hole, m, is so much larger than the mass of the smaller black hole that it can be replaced by the total mass of the system, M. Setting the two equations equal and simplifying, the cutoff time can be determined:
(4)
Equation (1) was put into the code with a stopping point at tkand a horizontal line with amplitude equal to h(tk) was coded in with a starting point at tk. To remove any discontinuities, a smooth curve was interpolated at the meeting point of the two functions.
In order to disambiguate the units involved, the following set of geometrized units, with c=G=1, is used. One mass unit is equal to one solar mass, or 1.989x1030 kg. One distance unit is equal to the Schwarzschild radius of a one solar mass black hole, or 2948.14 m. One time unit is equal to the time it would take light to travel one distance unit, or 9.827x10-6 s.
The following graph was produced from the memory amplitude code, where M = 1x106, μ= 1, and r = 2.093x1021 each with the corresponding units stated above.
Fig. 1. The Christodoulou memory wave amplitude (h) as a function of time (t). Time decreases to the left. h has no units and the units of t are explained above.
The remainder of the programming process was taken on by Olga Petrova.
Checking the Code: The next step was to confirm the accuracy of the program. To accomplish this, the values put out by the code were compared to analytically calculated signal-to-noise ratios. This calculation required the following approximation [1]:
(5)
In this approximation Mc,, the “chirp mass,” is defined by the following equation [1]:
(6)
Mo is equal to one solar mass. Assuming optimum conditions, F+ is set equal to 1 and ι is set equal to 90 degrees. For this analytic approximation to be accurate, two conditions must be met [1]:
- tk< 0.16 seconds
- Mc< 300 Mo
Meeting the Conditions. Several possible systems were found that would meet these requirements. The calculations for a binary system composed of two 1 solar mass white dwarfs will be shown here. To confirm that this system meets the requirements, equations (2) and (5) will be referred to.
For these white dwarfs, M = 2 Mo and μ = 0.5 Mo. A new function for the cutoff time must be produced because the mass-radius relation of a white dwarf is different from that of a black hole. A white dwarf of mass m has a radiusapproximated by:
(7)
Setting this equal to equation (2) and solving for t gives:
(8)
Using equation (8) for two 1 solar mass white dwarfs, tk = 0.00977 time units, or 9.6x10-8 seconds < 0.16 seconds. Using equation (6) Mc = 0.87055 Mo < 300 Mo.
The conditions being met, equation (5) can now be used to approximate the signal-to-noise ratio for LISA detecting the Christodoulou memory waves emitted by a pair of inspiraling 1 solar mass white dwarfs at a distance of 200 Mpc from the detector.
Using the values stated above, equation (5) yields a signal-to-noise ratio of 0.077696. Such a system would not be reliably detectable by LISA, but this calculation is only to see whether or not the code will produce the same number.
The White Dwarf Case. The distance that has been used thus far, 200 Mpc, is a distance presumed to contain a number of neutron star-neutron star coalescences per year. White dwarf-white dwarf binaries should be significantly more common. Assuming that such systems are common enough to coalesce frequently at 2 Mpc, the signal-to-noise ratio of a detection would increase to 7.77, making a very reliable detection.
Conclusions: Several runs have already been conducted with the numbers for this and other binary systems. So far the calculations and the code have not produced the same number. Once this problem is solved, the number output by the program will tell how sensitive LISA will be to Christodoulou memory waves.
References: [1]Kennefick, D. (1994) Physical Review D, 50, 3587-3595. [2]Thorne, K.S. (1995) Gravitational Waves. [3]Thorne, K.S. (1992) Physical Review D, 45, 520-524. [4]Christodoulou, D. (1991)Physical Review Letters, 67, 1486-1489.