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Outline

The broad question is how well can we eliminate different alternative theories to GR with current and upcoming gravitational wave observations. What this translates to is how well can we measure those parameters that distinguish GW solutions in alternative theories from those in GR. If we can measure them precisely enough, then we go and measure them for existing signals and see whether our measurements deviate from null (!!) or not. If we cannot measure them precisely enough with one signal, combining information from multiple signals would be useful, if possible. If we cannot measure them at all, then we have shown that those theories will remain unprobed, perhaps till we have LISA.

Now, in the PPE framework, the gravitational waveform h_{nGR} emitted by binary black hole sources as per a GR-alternative theory has the following relation:

h_{nGR} = h_{GR} (1 + \alpha_{PPE} v^{a_{PPE}}) exp( i \beta_{PPE} v^{b_{PPE}})

with the waveform h_{GR} predicted by GR. Different theories have different mappings from their internal parameters to the coefficients \alpha and \beta and the exponents a and b (dropping subscripts now). Therefore to estimate the measurability of theory-intrinsic parameters, it suffices to estimate the measurability of PPE parameters, provided the mapping between the two are invertible. We will focus on those cases here where it is.

Further, when trying to measure extra-GR parameters from realistic extra-GR GW signals, one could find that:

• the extra-GR effect is at very high order, i.e. is very weak, and needs more sensitive instruments to measure.
• the extra-GR effect is around the same order to which we know our templates, i.e. their measurements are contaminated with modeling errors.
• the extra-GR effects are low order, and are strong enough to be well constrained. In this case the only thing that may still lower the quality of extra-GR parameters' measurement is their degeneracy with GR terms.

Therefore in our studies, we will probe both a range of magnitude of extra-GR terms, and for the borderline-measurable cases we will probe both degeneracies (with GR parameters, and with model-NR differences).

This task consists of two parts. First is to create suitable simulated GW signals for injection, and second to estimate posterior distribution for the PPE parameters using PPE+SEOB waveform templates. Once we have those posteriors we can invert relations between theory and PPE parameters and use those to obtain posteriors for theory dependent parameters.

Overall, we have {\beta, b, BBH intrinsic parameters, BBH location & orientation parameters} to sample injections over. As a first cut, we plan to study two cases:

• For injections we use SXS catalog to provide the GR solution with PPE corrections applied to them. The catalog will fix {source intrinsic parameters} and we will only have to sample injections over {\beta, b, source location/orientation parameters}. We can fix the orientation to be optimal, so as to maximize the measurability of the signal, without loss of generality (?). Finally, we sample injections over {\beta, b, source distance}. When analyzed with SEOB+PPE templates, our posteriors will pick up both PPE/GR-parameter degeneracy and SEOB-errors/PPE degeneracy.
• Replace injections to also be SEOB+PPE, rest the same as above. Now our posteriors will pick up only the PPE/GR-parameter degeneracy.

Status

See project progress here

References

[1] Introduction to ppE

[2] ppE on GW150914 and GW151226

[3] Tests of GR with GWTC-1 (O1+O2 catalog)

[4] Use and abuse of Fisher matrix analyses in GW PE