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If we currently can look into some of the furthest stars, actually seeing the past
Isn't it conceivable that given enough distance we should be able to see
Parts of the Big Bang? If the Universe is endless, it means we should be able to see into its beginning.
Has this theory ever been presented ? Is there anyone looking in to this possibility ?

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Related: physics.stackexchange.com/q/25205/2451 and links therein. –  Qmechanic Mar 27 '13 at 21:56
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Omar is correct that the furthest back we can "see" is the Cosmic Microwave Background because before this time photons, the carriers of light, were "coupled" with the protons and neutrons (and before that quarks). As the universe cooled the photons were able to be freed and create the Cosmic Microwave Background we see today. However we can theoretically look farther back with other techniques.

As Nic mentioned, the CMB is not entirely isotropic, meaning it is not the same in all directions. Slight variations (generally less than one part in $10^5$) can indicate structure from the time of decoupling (sometimes called the time last scattering). This is known as the Saches-Wolfe effect. However one has to be careful because there is also some redshifting (making the photons loose energy and appear more "red") cause along the path from the surface of last scattering to us, known as the "integrated Saches Wolfe effect".

There are also other backgrounds that can be theoretically seen if we get more precise instruments:

There is a Cosmic Neutrino Background, which could provide us information about further back in time. This is because, while photons decoupled about 400,000 years after the big bang, neutrinos decoupled significantly earlier, perhaps just a few seconds. However, because they were emitted earlier, they are even more redshifted and low energy (the CMB is about 2.7 K, a CNB would be ~1.9K) and neutrinoes are significantly harder to detect anyway. So far the only extraterrestrial neutrinoes detected have been from the sun and SN 1987a (a supernova).

There is also a gravitational wave background. It is very difficult to detect this, particularly because we haven't been able to really directly detect any gravitational wave (LIGO hasn't been very successful). There was a proposed space mission called the Big Bang Observer, which was never approved due to budget and technical concerns, which might have detected these early gravitational waves. They would have been emitted even earlier, probably just after inflation some $10^{-32}$ seconds after the big bang. EDIT: Note that the BICEP2/Keck paper of course shows great evidence for the existence of these gravitational waves, but they are still an ``indirect" in that we see their effects on the cosmic microwave background as opposed to directly seeing the space-time variations they cause.

As far as looking back to $t=0$, that seems unlikely. To say for certain we would need a good quantum gravity theory to understand the uber-early (before $10^{-42}$ seconds) universe. Perhaps some very strange exotic particle could exist that would give us a "view" of this period.

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Yeah....the CNB...speaking as a neutrino experimenter, right now we haven't got the first clue how to probe that. The energies are so low and the cross-sections so small (by our standards!) that nothing we use these days has a chance. –  dmckee Dec 13 '11 at 2:58
    
The inflation produced gravitational waves MIGHT be detectable in the "B-modes" of polarization of the CMB. Whether they are detectable depends on the details of the inflationary model. –  FrankH Dec 13 '11 at 8:13
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The BICEP experiment has now reported a positive result using the method suggested by @FrankH here. –  dmckee Apr 12 at 15:39
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The furthest back we can look is the Cosmic Microwave Background (CMB) radiation. It is not possible to observe any further back than that because prior to that the Universe was hot enough to only contain hydrogen plasma. This plasma (soup of electrons and protons) absorbed any photons therefore making the Universe opaque.

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There are inferences based on anisotropies in the CMB that allow us to picture early times. –  Nic Dec 12 '11 at 11:26
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