Andromeda & Milky Way Merger: Gravitational Waves

Question

When the Andromeda galaxy and Milky Way merge in the future, the super-massive black holes at their respective galactic centers will likely eventually merge. Similarly to the gravitational waves detected by LIGO on 14th September 2015, I assume that the merger of these super-massive black holes will generate gravitational waves.

Whilst the gravitational waves detected by LIGO in 2015 were caused by a merger of two black holes approximately 1.3 billion light years away, the merger of black holes inside our own future galaxy (Milky Way & Andromeda merged together) will be much closer. Also, the size of the black holes merging will be bigger.

Question: The waves detected by LIGO caused a space-time distortion of the order of a fraction of an atomic radius. What is the size of the spacetime distortion likely to be caused by the merger of the super-massive black holes at the galactic centers of Milky Way & Andromeda? Will the gravitational wave generated be of any threat to humanity on Earth (if humanity still exists at that point in time)?

Answer 1

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 $4\cdot 10^6 M_\odot$. The black hole at the center of Andromeda is much more massive, weighing in at about $1.2\cdot 10^8 M_\odot$. 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 $2.52\cdot 10^{-10}$ at a frequency of 0.434 mHz. Applied to the Earth, this translated to a peak power spectral density of $2.12\cdot 10^{-16} (\mathrm{m}/\mathrm{s}^2)^2/\mathrm{Hz}$.

The ambient background seismic activity on Earth is given by the NLNM (new low noise model). At 0.434 mHz, this gives $1.63\cdot 10^{-17} (\mathrm{m}/\mathrm{s}^2)^2/\mathrm{Hz}$. 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.

Answer 2

Back of an envelope calculation:

The gravitational waves detected by LIGO cause strains (proportional changes in length) of the order of $10^{-22}$. They come from the merger of black holes with a mass typically $30$ times that of the sun at distances of the order of $10^9$ parsecs.

The black hole at the centre of the Milky Way is about $8 \times 10^3$ parsecs from earth and has a mass of about $4 \times 10^6$ solar masses.

The strength of gravitational waves is proportional to the mass of the merging black holes and inversely proportional to distance. So a merger of two mega-solar mass black holes at a distance of, say, $10^4$ parsecs would produce strains of the order of $10^{-12}$.

This strain is about a million time smaller than the change in length caused in everyday materials by a one degree change in temperature. It would certainly have a detectable effect on systems that depend on very accurate distance measurements (e.g. GPS) but would not be a threat to humanity.