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# A useful guide for gravitational wave observers to test modified gravity models 

E. O. Kahya 1, ∗ 

$^{1}$Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universit¨ at Jena, Max-Wien-Platz 1, D-07743 Jena, Germany 

(Dated: October 30, 2018) 

We present an extension of a previously suggested test of all modified theories of gravity that would reproduce MOND at low accelerations. In a class of models, called “dark matter emulators,” gravitational waves and other particles couple to different metrics. This leads to a detectable time lag between their detection at Earth from the same source. We calculate this time lag numerically for any event that occurs in our galaxy up to 400 kpc, and present a graph of this possible time lag. This suggests that, gravitational wave observers might have to consider the possibility of extending their analysis to non-coincident gravitational and electromagnetic signals, and the graph that we present might be a useful guideline for this effort. 

The last forty years have been a very active and an exciting period for cosmologists. The breakthrough was the discovery of cosmic microwave background radiation (CMBR), an open window to the very early stages of the universe [1]. We realized that the universe is homo- geneous and isotropic on large scales [2, 3]. And very recently we learned that the universe is accelerating con- trary to the common belief [4, 5]. 

Another very exciting observational discovery was the peculiar shape of the so called “galaxy rotation curve”. Instead of the velocity of stars orbiting spiral galaxies be- ing inversely proportional to the distance away from the center of the galaxy, it was observed to stay almost con- stant [6–8] after certain point. Then even more suprising observational fact was the excessive amount of distortion in the background galaxies near a foreground mass [9–14]. Assuming that there is only baryonic matter in the uni- verse, it is not possible to explain these two observations with Einstein’s equations, 

Two of the following can be done to avoid this discrep- ancy. We can modify the right hand side by assuming additional dark matter, that would interact only gravi- tationally. Or we can modify the left hand side of the equation, namely the Einstein’s tensor. The latter ap- proach suggests to modify general relativity on cosmo- logical scales. And it was partially achieved in 1983 by Milgrom [15] with the so called “MOND” for Modified Newtonian Dynamics. Partially, because it is not a mod- ification of general relativity but Newtonian dynamics on cosmological scales. The modification of GR would then be the underlying fundamental theory that would give MOND in the non-relativistic limit. 

PACS numbers: 98.80.Cq, 98.80.Hw, 04.62.+v 

## I. INTRODUCTION 

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In order to understand Eq.(1), one needs to be able to handle both sides of the equation. That means iden- tifying dark matter directly, by a non-gravitational ex- periment. Therefore direct detection is crucial for dark matter. For more than 30 years, experimentalists have been trying to detect dark matter without any success. Very recently some excitement arose about the detection, but turned out to be almost no signal with excluding new parameter space[16]. On the other hand, the early relativistic [17], and the Scalar-Tensor [18] formulations of MOND led to, so called TeVeS for “Tensor-Vector- Scalar”, which was recently proposed by Bekenstein [19]. It reproduces the MOND in the non-relativistic limit and does a good job on cosmological evolution, strong grav- itational lensing as well as post-Newtonian parameters [20, 21]. But weak lensing is problematic in the case of colliding clusters, such as the Bullet Cluster [22–25]. The observed third peak of new 5-year WMAP data[26] is also problematic in the context of MOND-like models. But still one can not claim that these models are ruled out [27, 28]. Thefore it is very important to find a way of testing these two explanations by means of observation. 

The possibility for comparing dark matter with MOND-like models by means of a model-independent test was suggested before[30–32]. It was proposed that one would see a time lag between arrival times of gravita- tional waves and other massless particles. And the time lag was calculated for three different sources; a particu- lar gamma-ray burst(GRB) GRB070201, the low mass x- ray binary(LMXB) SCo-X1, and the supernova SN1987a. Other proposed tests of TeVes with gravity waves are dis- cussed in [29] and references therein. In Sec. 2 we sum- marize the formulation of this suggested test. In Sec. 3 we extend this calculation numerically to any source in Milky Way galaxy. In Sec. 4 we discuss the conse- quences of this effect to the externally triggered searches for GWs. In Sec. 5 we investigate the ambiguities in this calculation, and see how sensitive the time lag is to the choice of using different data sets. Our conclusions are summarized in Sec. 6. 

