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Can we not escape the expansion of space to get outside the local group? I was doing some math, and the closest Group to us besides the Local Group is the M81 Group which is 11.4 Million light years away.

Math:

Space expands at roughly 68 km/s/Mpc

Local Group is 11.4 Million light years away from the M81 Group

11.4 Million Lightyears = 3.5 Mpc 

68 km/s/3.5

3.5*68 km/s = 238 km/s

The fastest space probe is the New Horizons probe which is currently flying at 15.73 km/s

Is it true that it is impossible to escape from the Local group? Please correct any math that is wrong.

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    $\begingroup$ I don't know why this matters; if we can't get something faster than 238 km/s, we can't possibly get a probe even a megaparsec away before the sun goes out. There's no way to design for multi-billion-year missions, nor will there be for a very long time, if indeed ever. $\endgroup$ – Nathan Tuggy May 30 '16 at 23:13
  • $\begingroup$ @NathanTuggy Those are the calculations that I came up with, I could be wrong, but I didn't include the exponential rate that technology is advancing. It is unlikely that we will get out of the local group at this rate. $\endgroup$ – Kryometric Gaming May 30 '16 at 23:24
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    $\begingroup$ My point is that the expansion of space is a red herring: by the time we can even try anything like that in the first place it's already a trivially-surpassed obstacle. (To be honest, even designing a 20-million-year mission — i.e., more than half the speed of light — is fairly ridiculous and unlikely to ever be possible. Leaving galaxy, never mind the Local Group, is FTL or nothing.) $\endgroup$ – Nathan Tuggy May 30 '16 at 23:30
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    $\begingroup$ Problem is that you're neglecting the role of Dark Energy, so you can't just naively use Hubble's law, because H_0 = 67 km/s/Mpc only represents the current expansion of the universe. The rate of expansion is set to increase with time as the universe expands. It may still be possible to escape the Local Group, but there will be some limit to the distance we could travel and be able to reach another galaxy. I'm not sure if we could reach a cluster that isn't gravitationaly bound to us. Does anyone know how to calculate this? $\endgroup$ – user16483 Aug 2 '16 at 2:02
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Your math is fine, but the speed of the fastest probe we have now is no indication of what is possible. 238 km/s is less than a thousandth of the speed of light.

Laser- or maser-powered light sails have been modeled as attaining perhaps 30% of the speed of light on interstellar missions. The Breakthrough Starshot project is hoping to use this technology to get to Alpha Centauri sometime later in this century. Many technologies would need to be developed to make this possible, but the physics of it are well understood and sails propelled by sunlight have already been flown.

An earlier project, Project Longshot, looked at doing the same thing using pulsed fusion propulsion. Controlled fusion would need to be developed in order for this to work, so it was more speculative, but again it relied on physical phenomena we know a lot about, though we can't reproduce them yet.

And there are several other options for such things. The issue is that it would take tremendous infrastructure and technology development to launch them. But we have good ideas as to how it might be done.

Of course, if it is going to take 30 million years for your craft to get outside the Local Group, you have other things to worry about besides how fast it can go.

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Nothing wrong with your maths. If the fastest you can travel is given by v, then the furthest galaxy you can eventually get to, is one that is presently at distance (just slightly less than) $(v/c)L$, where $c$ is the speed of light, and $L=c/H_0$ is the Hubble length (https://en.wikipedia.org/wiki/Hubble%27s_law#Hubble_length ~ 14.4 Billion light years).

So if we build a spaceship to reach speeds of 0.1 c, the furthest galaxy we can eventually reach is one that is currently 1.44 billion light years away. If we can go 0.5c, we can reach a galaxy that's presently 7.2 billion light years away.

Despite what some may think, most conventional theories of dark energy do not predict that Hubble's constant $(H_0)$ is increasing. For that to happen, you'd need to have phantom energy, which is ruled out by most. According to the standard Lambda CDM model, H_0 will slowly decline towards 57 (km/s)/Mpc (https://en.wikipedia.org/wiki/Hubble%27s_law#Time-dependence_of_Hubble_parameter). Thus, the Hubble length $L$ would not be decreasing, but very slightly increasing.

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