Andromeda & Milky Way Merger: Gravitational Waves

The peak strain of GW150914 was about $10^{-21}$. Strain scales linearly with the total mass of the system, and inversely proportionate to the distance. A merger of the of two supermassive black holes at the center of the galaxy, would be about (give or take an order of magnitude) a million times more massive and a million times closer than GW150914, giving a strain of $O(10^{-9})$. Across the size of the Earth this would still only translate to a few millimeters. This might cause measurable seismic activity across the globe, but would hardly be catastrophic.

Add: The peak strain would be somewhere in the mHz regime, i.e. the correct regime for eigenmodes of the Earth's crust. Consequently, the gravitational waves can couple to seismic activity relatively effectively.

UPDATE: In the comments it was questioned whether any seismic activity would exceed the typical background activity on Earth. So lets be a bit more precise. Sagittarius A* has a mass of $4cdot 10^6 M_odot$. The black hole at the center of Andromeda is much more massive, weighing in at about $1.2cdot 10^8 M_odot$. This gives a mass-ratio of about 1/32, so we can use one of the simulations from https://arxiv.org/abs/2006.04818 as a model. Our distance to the merger would be highly uncertain in the newly merged galaxy. For now let us assume a round 1 kpc. This would lead to a peak strain of $2.52cdot 10^{-10}$ at a frequency of 0.434 mHz. Applied to the Earth, this translated to a peak power spectral density of $2.12cdot 10^{-16} (mathrm{m}/mathrm{s}^2)^2/mathrm{Hz}$.

The ambient background seismic activity on Earth is given by the NLNM (new low noise model). At 0.434 mHz, this gives $1.63cdot 10^{-17} (mathrm{m}/mathrm{s}^2)^2/mathrm{Hz}$. Consequently, the signal would come in just above this noise floor, meaning it might be just measurable by sensitive seismic monitoring stations at quiet locations.

Some caveats:

• As mentioned the distance to the merger would be highly uncertain. Increasing the distance by a factor of 10 (well possible) would reduce the power spectral density by a factor 100 and put it well below the ambient seismic background.
• The viewing angle of the merger can affect the observed strain by a factor of a few, which at the given margins could make the difference between being detectable, and not.
• The above assumed that the black hole were not spinning. A significant amount of spin on the heavier component could lead to a significantly higher peak strain at a higher frequency.

All that being said, the effect would be order of magnitude smaller than the type of seismic events that happen on a daily basis, and would not pose any sort of threat to anything on Earth.