Is there any credible evidence for big bang? The big bang model is first discussed when the people witnesses that almost every galaxies are speeding away from each other suggesting that at one point in time or more accurately at $t=0$ everything including space-time have to be congregated together. Is there any truth to this model?
 A: The Big Bang is the prevailing cosmological model because there is plenty of evidence supporting it. There’s some confusion on this page regarding what scientific evidence is. A model is evidenced by the empirical success of its predictions, some of which would otherwise be unexpected. The model therefore explains these predictions’ observed truth. Wikipedia provides a very concise summary of these predictions before going into detail: “The model accounts for the fact that the universe expanded from a very high density and high temperature state, and offers a comprehensive explanation for a broad range of phenomena, including the abundance of light elements, the cosmic microwave background, large scale structure and Hubble’s law.” For more information, see here. A refinement, the inflationary Big Bang, explains some other features of the universe; see here.
Evidence does need to be “repeatable”, but this doesn’t mean a historical event needs to be made to happen on queue. (How would that even be relevant to whether it happened before?) Reproducibility means that the findings of a piece of research should occur again when others repeat the research. For example, Hubble’s law is observed no matter who looks at the relative velocities of distant galaxies.
Misunderstanding here is not limited to flippiefanus. The Big Bang model is mathematical, but attempts to shoehorn its concepts into English words that weren’t developed for such concepts leads to further misconceptions, viz. user104372. For example:
a) The Big Bang is not an “explosion”; it’s a time-dependent scale factor in an FLRW metric. The scale factor varies over time according to the Friedmann equations, which follow from the general relativity’s equivalent of Newton’s second law, the Einstein field equations. Since the Einstein-Hilbert Lagrangian depends on the scale factor, it is explicitly time-dependent. This is why energy conservation is… a bit of a complicate issue in general relativity.
b) The speed of light is an upper bound on local relative velocities (all velocities are relative to something, e.g. Earth), but not on global relative velocities. Galaxies too far apart have global relative velocities > c, but they cannot observe each other because not enough time has passed, so there is no local relative velocity > c. (The local-global distinction I also important in other contexts, such as Alcubierre metrics.)
c) I have no idea what provable history of ideas is the basis for this nonsense, but as I said the expansion is a time dependence in the FLRW scale factor, which happens because the Friedmann equations does not have stable static solutions.
d) Curvature has two components, extrinsic and intrinsic. The FLRW metric has extrinsic curvature. The open question regards the universe’s intrinsic curvature, i.e. whether the “Friedmann parameter” is positive, zero or negative. Physicists discussing whether the universe is “curved” are referring to that question.
e) The observable universe is finite because the time since the Big Bang is finite, but general relativity and all observations are compatible with both a finite and an infinite universe. The universe could be finite yet unbounded. (To think of an analogy for this geometry, Earth’s surface is “unbounded” because you can travel an arbitrary distance along it, but the surface area is in fact finite; you’re just moving on an oblate spheroid.)
f) “Everywhere” is only vague if you do it in English. In the FLRW metric, the line element’s time dependence due to the scale factor is the same at all choices of spatial coordinates.
A: Present day physics is defined as a discipline that studies data, i.e. numbers, whether from experiments or observations, proposes mathematical models and considers them valid as long as they are not falsified. A physics mathematical model is falsified if a prediction of the model is false. The historic reaction of physicists to falsification is to expand the mathematical model in a manner that keeps the former fits as successes of the model and corrects the wrong predictions.
The Big Bang model of the universe is a good example of this process.

In 1929, Hubble examined the relation between distance and redshift of galaxies, combining his own measurements of galaxy distances based on Henrietta Swan Leavitt's period-luminosity relationship for Cepheids together with earlier data from fellow astronomer Vesto Slipher plus Milton L. Humason's measurements. He found a rough proportionality of these objects' distances with their redshifts, nowadays termed Hubble's law.

This is the data that needed a theoretical model different that a simple Newtonian model, to be described .

Yet the reason for the redshift remained unclear. It was Georges Lemaître, a Belgian Catholic priest and physicist, who found that Hubble's observations supported the Friedmann model of an expanding universe based on Einstein's equations for General Relativity, which is now known as the Big Bang theory. 

So the Big Bang is a mathematical model, within the framework of General Relativity. The words convey the meaning of an initial singularity .  It caught the imagination of physicists and the astronomical and astrophysical observations were fitted with this model at hand.
Until technology brought us the cosmic microwave background radiation.
The great isotropy in this radiation could not fit with the general General Relativity singularity model , because close to the singularity the various regions could not interact due to the nature of General Relativity, and so no thermodynamic black body isotropy of the early universe could be derived from the BB model as it then stood.
The models expanded and morphed, the present BB model here :


Timeline of the metric expansion of space, where space (including hypothetical non-observable portions of the universe) is represented at each time by the circular sections. On the left the dramatic expansion occurs in the inflationary epoch, and at the center the expansion accelerates (artist's concept; not to scale).

Physicists started working and are still working to bring in Quantum Mechanics to General Relativity. In this plot, an effective quantization is assumed for the very early ages of the universe, which, due to the probabilistic nature of quantum mechanics, can explain the isotropy of the comsic microwave background radiation.
Thus the falsification of the original mathematical  BB model gave rise to the present one, which keeps the successes of the original model and expands it in a manner that removes the falsification of the original model.
This is where we are with the mathematical modeling of the cosmos at present. New observations or new data might change the picture in the future, and more patches may have to be devised. 
Physics is not about truth. It is about modeling nature successfully within the experimental and observational limits of the times. 
In this frame,  the BB model is a successful model.
A: "The big bang model is first discussed when the people witnesses that almost every galaxies are speeding away from each other"
People didn't witness that. Rather, they (Hubble) observed that the redshift increases with distance, which can have a different explanation. For instance, as the photon travels through space (in a STATIC universe), it may bump into vacuum particles and as a result lose speed (energy), similarly to a golf ball losing speed due to the resistance of the air:
http://www.sciencedirect.com/science/article/pii/S0262407911603059 
 New Scientist: "Vacuum has friction after all. A ball spinning in a vacuum should never slow down, right? Wrong. It turns out quantum effects can create a type of friction in the void." 
http://www.nature.com/news/superfluid-spacetime-points-to-unification-of-physics-1.15437 
 Nature: "As waves travel through a medium, they lose energy over time. This dampening effect would also happen to photons traveling through spacetime, the researchers found."
A: From a purely scientific point of view, one can never say whether it really happened this way or whether there could not be other explanations. The reason is the big bang is a historic event that happened only once, while science requires repeatability.
Addition:
Perhaps it is worth pointing out some misunderstanding. There is a difference between a scientific theory that directly describes a physical phenomenon (or set of phenomena) such as electromagnetism and scientifically feasible theory that describes the historical events, such as the big bang theory. In the former case one can directly test the theory based on experiments that accesses nature behaviour. In the latter can one need to distinguish between the feasibile theory and the historic event. What the feasible theory can be tested in a scientific manner one cannot make the connection between between the feasibile theory and the historic event, unless one has direct evidence (i.e. direct observation of the event) or one can rule out all possible alternatives. In the case of the big bang theory neither is possible, because no one was present to record the event, nor can we know that there would not in the future be some other theory that can also explain the observations that are currently being made.
Now, I know people are very passionate about their favorite theories, especially if it is the theory that one spends one's career on, but there is something to be said for scientific honesty. Without scientific honesty, one can get into the situation where people are easily mislead. The big bang theory is a beautiful feasible theory and I'd be happy to say that it could well be the actual way that it happened, but such a conviction would not be based on science.
