Why haven't we seen the big bang?

The Andromeda galaxy is 2,538,000 light years away, so if we view Andromeda from a telescope, we see Andromeda how it was 2,538,000 years ago. Now the diameter of the visible universe is 92 billion light years so if we say that we are at the center so the radius is 46 billion light years. So, if we see the farthest we can from our point, that will be 46 billion light years, so it means that we have seen the universe how it was 46 billion years ago, so why can't we say that universe was born 46 billion years ago?

And if still someone argues that 13.8 billion years ago universe was born then we should have seen the universe of how it was even before the singularity.

And if my logic of 49 billion years seems right then the thing we say that universe was born 13.8 billion years ago should be wrong, shouldn't it?

Hoping for a excellent answer with excellent explanation! :)

There are two parts to your question. First, why can we see things "46 billion light years away" if the Universe is only about 13.8 billion years old? Because the Universe is expanding. How far does a photon travel in 13.8 billion years in an expanding Universe? It depends on the rate of expansion. I'll give a simplified example to illustrate the point:

Imagine a Universe that is 10 billion years old. Its expansion history is simple. It started out with size $R$, which remained constant for 5 billion years, then the size suddenly doubled to $2R$, and remained constant for another 5 billion years. So, after 10 billion years, how far is the photon from its starting point? Well, it travels 5 billion light years in the first 5 billion years. When the size of the Universe doubles, so does the space between it and its point of origin, so just after the doubling it is 10 billion light years from where it started. Then in the next 5 billion years it travels another 5 billion light years, ending 15 billion light years from where it started. Notice that the constant speed of light from special relativity comes through intact here - the photon ends 15 billion light years from where it started after 10 billion years, but always moves at speed $c$ and only "travelled" 10 billion light years. Obviously a sudden doubling of size is not realistic, but the same idea applies to a more gradual, smooth expansion. Note that the expansion does not necessarily need to be "faster than light" (whatever that means). Any expansion at all will mean that the photon will be further from its starting point than expected in a static Universe when it arrives.

Second part to your question: why can't we see the Big Bang? Because a photon cannot reach us directly from the Big Bang. On a sunny day, I can see the Sun clearly because most of its photons that I observe leave the solar surface and travel along unimpeded to reach me. If it's cloudy, there's still sunlight - solar photons are reaching me - but I can't see the Sun directly because the photons are scattered along the way by the clouds. The early Universe was "cloudy" - it was full of an ionized plasma that scattered photons very frequently. As the Universe expanded and the plasma cooled, it eventually became transparent to the photons and the "Big Bang photons" have been travelling more or less without scattering ever since. This "surface of last scattering" is visible as the cosmic microwave background, and is the closest we'll get to seeing the Big Bang (except perhaps we could observe neutrinos or gravity waves from beyond the CMB because they are not scattered as strongly in the early Universe).

• Kyle in your second paragraph you said that light travels 15 billion light years in 10 years, but i didn't get that and you said its because of relativity but there are many things in relativity and excatly what part of relativity are you talking about? Please edit your answer and explain me better of what you are talking about in second paragraph. Hope you give a better answer. :) – Bhavesh Apr 23 '15 at 17:18
• @Bhavesh hopefully that's clearer? – Kyle Oman Apr 23 '15 at 17:21
• @Bhavesh Simply put, imagine A and B are two points that are 5 billion light years apart. A photon travels from A to B in 5 billion years. But then the universe suddenly expands to twice it's size, the distance between A and B is now measured to be 10 billion light years. The light travelled from A to B (which we measure as 10 billion light years apart), taking only 5 billion years to do so. – Praneet Srivastava Apr 23 '15 at 18:03
• @Bhavesh The light only traveled 10 billion light years, but the point where the light originated is 15 billion light years away by the time the light from it has traveled that far. The intervening space expanded, making the distance larger. – Kyle Apr 23 '15 at 18:04
• @Bhavesh that's the idea, yes. The "sudden" doubling in size is not realistic, it is more gradual throughout the travel of the photon, but it's the same idea. The space already passed is growing behind the photon (of course so is the space ahead). – Kyle Oman Apr 24 '15 at 2:27