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What determines how far a telescope can see in the universe? How does recording data for a very long time (~10 years) help? If we could build a telescope which work at microwave region, will it be a better telescope (so that we could see even farther)?

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We do have space telescopes (COBE, WMAP, Planck) that operate in the microwave region, and indeed they do see much, much deeper into the universe than HST. The Hubble Extreme Deep Field renders images of a universe that is a few hundred millions of years old. In contrast, COBE, WMAP and Planck provide us with a view on a 0.37 million years old universe.

This doesn't mean that COBE, WMAP or Planck are 'better' telescopes than HST. It's a matter of different tools for different tasks: the microwave instruments observe the cosmic microwave background and in doing so tell us a lot about cosmology and the Big Bang. The Hubble observes galaxies, nebula and star systems that formed much later.

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has provided a good answer about how other types of telescopes using different wavelengths can see further than Hubble. Staying within the visual wavelengths though ... Think of a telescope as a bucket where we try to collect as many light photons as possible. The further away an object is, the fainter it will be. This means we need to use a large bucket to collect enough of its photons to light up the camera sensors. Hubble is a pretty reasonably sized bucket so it sees really well. –  Carl May 23 '13 at 20:37
Also, to stay with the metaphor, the longer time we hold the bucket out, the more photons we catch. –  Thriveth Jul 15 '13 at 22:17

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