Of course, Earth won't be here while that happens. The Sun will expand and swallow it up long before then.
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The expansion of the universe won't kill us off but at some point in the distant future thing will be so spread out that we won't be able to see any stars. The sky will be dark.
However the sun will blow up in a few billion years and that'll destroy planet earth and anybody living on it. If there's any humans left they'll be well advised to evacuate this solar system if they can.
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The 'heat death' of the universe is baked into the physics of reality. On the plus side, it will be a long time into the future.
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The Sun will become to hot for life to exist on Earth in about 1.2-1.3 billion years, and then decline into a white dwarf
All stars, including those not yet formed, will have burned out in about 500 trillion years
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The universe isn't actually expanding at all. The idea that it does comes from misinterpreting galactic redshifts as a doppler effect when they are actually a scattering effect. The galaxies are not generally receding from each other at all. Their light simply loses energy through it's interaction with the molecular hydrogen of the intergalactic medium:
http://www.newtonphysics.on.ca/universe/
If expanding space was responsible for cosmological redshifts it would cause a reduction in the surface brightness of distant galaxies but we find this is not the case:
http://www.sci-news.com/astronomy/scienc...
Also, as well as stretching out the wavelengths of light, expanding space would stretch out the light curves of quasars (their oscillation in luminance). No sign of this either:
http://phys.org/news190027752.html
This documentary may enlighten you:
There never was a Big Bang. The universe has most likely existed for eternity and always will.