# Grab a partner, it’s time for tonight’s dizzying doppler dance!

Authors: Aditya Vijaykumar, Avinash Tiwari, Shasvath J. Kapadia, K. G. Arun, and Parameswaran Ajith

First Author’s Institution: International Centre for Theoretical Sciences, Tata Institute of Fundamental Research, Bangalore, India; Department of Physics, The University of Chicago, Chicago, Illinois, USA

Status: Pre-print on arXiv (open access)

Around 100 detections of gravitational waves from merging binary black holes and neutron stars (collectively referred to as compact objects) have enabled us to understand more and more about the properties of these tiny spacetime ripples and their sources. Understanding where these binaries formed helps us understand how these objects got close enough together to merge in the first place. One possible environment to find merging compact binaries is within the orbit of supermassive black holes (SMBHs) in the centre of galaxies, either having formed in situ, or having migrated there from elsewhere, and today’s astrobite tackles how we can know if that’s where a gravitational-wave source originates from.

Just like light, gravitational waves can be stretched (redshifted) or squeezed (blueshifted) by the expansion of the universe. The further away a source is, the more of the universe’s expansion it will travel through, and consequently its wavelength will be stretched out longer while it will be shifted to lower frequencies. For gravitational waves from two merging compact objects, the profile of the waves is very well modelled, and we know that waves from less massive merging objects emit gravitational waves of higher frequencies. This means that redshifting a gravitational wave makes it look like it came from a more massive source!

However, it’s not just cosmological redshifts that affect gravitational waves. Doppler shift, the red or blueshifting of a source moving with some line of sight velocity, can also leave an imprint on the gravitational wave profile. If a compact object binary is emitting gravitational waves while the pair are also traversing round a larger orbit, as it approaches us on one side of the orbit, the gravitational wave will be blueshifted, and as it recedes from us on the other side it will be redshifted, as animated in Figure 1. Similar to how we use redshifting and blueshifting of spectra from stars to detect exoplanets, we should be able to use this doppler shifting of the gravitational waves to understand if the merging objects we detect are actually orbiting a much more massive central mass, such as a SMBH.

Today’s paper looks at the prospects of detecting doppler shifted gravitational waves due to line of sight acceleration, both using current gravitational wave detectors, and more sensitive future detectors. Through adding a term to the phase of the model gravitational waveforms used to hunt for gravitational waves, the authors account for doppler frequency shift for a given acceleration. If the acceleration is due to a circular orbit, this change in phase allows us to calculate the mass of the object at the centre of the orbit, this orbital radius, and the direction of the acceleration.

In order to pick up on this doppler shift in gravitational wave signals, we want to be able to hear the gravitational wave in our detectors for as long as possible, to maximise the change in phase imprinted on the signal. Because the current gravitational wave detectors have a lower limit on the frequencies of gravitational waves they can detect, higher frequency, lower mass compact objects can be heard for longer, and the prospects of picking up doppler shift signatures from these signals is larger.

Today’s authors look at two of the least massive signals detected so far, both from neutron star mergers, and find no evidence of line of sight acceleration when accounting for the additional doppler phase shift. However, assuming that these binaries were orbiting around an SMBH, we still can use the acceleration to learn about the properties of a possible SMBH orbit. Zero acceleration means that the orbit of the first signal has to be greater than 12.1 AU from the central SMBH, because if the orbital radius were any smaller than this then we would be able to confidently detect a non-zero acceleration in the gravitational wave signal. But, we also know constraints on this distance because we also observed this merger with photons. It turns out that the electromagnetic counterpart, having been traced back to a host galaxy, places the merger occuring at at least 10 million times further away from the centre of the galaxy, and so the current constraints from doppler shift are much less informative than the electromagnetic information.

While it seems that these two events didn’t have any noticeable doppler effects, we may still make future detections. Multiple ground-based future detectors are planned for the upcoming decades, each with increasing sensitivities – making for louder signals from further away sources – and lowering minimum frequencies, allowing us to hear more massive gravitational waves for longer. The uncertainties on the acceleration we could detect determines how confident we can be about uncovering doppler shift in a gravitational wave detection. Figure 2 shows us that for the current LIGO detectors, and two future detectors Cosmic Explorer and the Einstein Telescope, with minimum frequencies 15, 5, 2Hz respectively, this uncertainty in acceleration gets smaller and smaller for the same signal strength. The uncertainty is smaller for objects with lower mass, as these signals will be at higher frequencies, and therefore heard in the detectors for longer. At the same time, we expect that with less background noise, signals will be even louder in these future detectors, and if doppler shifts are present, they will get easier and easier to detect.

Furthermore, there are prospective space-based detectors that aim to detect gravitational waves at even lower frequencies, between 0.1 and 10 Hz, making it possible that we could hear gravitational waves from compact objects spiralling into each other for years instead of our current capacity for detections of just a few seconds. This would make the uncertainties on acceleration lower by several orders of magnitude, and make it possible to detect doppler shifts in black hole binaries with even larger masses. Uncovering doppler shifts to this precision would mean we could trace binary neutron stars not just up close and personal to a SMBH, but all the way out to the middle of the galactic disk of a Milky Way-like galaxy, and understand motions of binary compact objects in other dense stellar environments such as globular clusters, making for very exciting times in astronomical dynamics!

Astrobite edited by Lucas Brown

Featured image credit: modified by Storm Colloms from Event Horizon Telescope, Budgeron Bach