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If we travel with speed close to the speed of light, the light waves becomes shorter and their energy increases (because we flight through much more light for the same time).

So, even background radiation can turn into x-ray and gamma on some speed. So, what is the maximum speed with which we still can travel safely without a lot of radiation produced by relic microwave radiation?

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You can answer the question yourself if you look up three things, namely the representative frequency of the background radiation (remember it is black body radiation so it has a spectrum rather than a single frequency), the frequency at which EM radiation gets classified as gamma rays, and the formula for the relativistic Doppler effect, which tells you how the change in frequency is dependent on velocity. On principle I am not going to do that work for you, as I think you should have tried to work it out for yourself. However, I seem to remember that for visible light to appear as gamma radiation, the relative speed of the source and observer would need to be something like 0.999999999c, so that will give you an idea.

You might also like to look up how much energy you would need to accelerate a spaceship to such a speed relative to the CMB, as that will tell you how unlikely it is that the gamma radiation would prove to be the biggest obstacle in the short term. You might also like to consider how much energy would be required to steer your space ship- a u-turn for example, would require double the energy you used to accelerate in the first place, as would a right hand turn now I think about it.

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You can look up the peak frequency of the cosmic microwave background and the frequency where X-rays begin to become dangerous. Then you can use the formula for the relativistic Doppler effect $$\nu_\text{observed} = \nu_\text{source} \sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$$ where $v$ is the velocity between observer and source, $c$ is the speed of light, $\nu_\text{observed}$ is the observed X-ray frequency, and $\nu_\text{source}$ is the cosmic microwave frequency.

Solve this equation for $v$, plug in the numbers, and you will get a velocity very close to speed the of light.

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