Constraining alternative theories of gravity using GW$150914$ and GW$151226$

The recently reported gravitational wave events GW$150914$ and GW$151226$ caused by the mergers of binary black holes [arXiv:1602.03841],[arXiv:1606.04855],[arXiv:1606.04856] provide a formidable way to set constraints on alternative metric theories of gravity in the strong field regime. In this paper, we develop an approach where an arbitrary theory of gravity can be parametrised by an effective coupling $G_{eff}$ and an effective gravitational potential $\Phi(r)$. The standard Newtonian limit of General Relativity is recovered as soon as $G_{eff}\rightarrow G_N$ and $\Phi(r)\rightarrow \Phi_{N}$. The upper bound on the graviton mass and the gravitational interaction length, reported by the LIGO-VIRGO collaboration, can be directly recast in terms of the parameters of the theory which allows an analysis where the gravitational wave frequency modulation sets constraints on the range of possible alternative models of gravity. Numerical results based on published parameters for the binary black hole mergers are also reported. Comparison of the observed phase of the GW$150914$ and GW$151226$ with the modulated phase in alternative theories of gravity does not give reasonable constraints due the large uncertainties in the estimated parameters for the coalescing black holes. In addition to these general considerations, we obtain limits for the frequency dependence of the $\alpha$ parameter in scalar tensor theories of gravity.