The Big Bang theory is the approved theory of the "birth" of Universe. Are there any other theories which can explain how our Universe came into existence?
This is not an answer but a comment which is too long to post as such.
There is no such thing as a scientifically approved theory, there are theories consistent with the evidence some of which are simpler in some sense than others, and philosophy of science tells us that we should prefer those that are simpler.
Also I doubt that the big-bang theory is what you think it is, which is simply that at a certain time in the past the Universe was in a hot dense state which then expanded, forming the lightest chemical elements... The first part of this is a simple extrapolation of the observation of the way that galactic Doppler shift increases with distance and the observed cosmic microwave background. The second part is the consequence of the application of well know nuclear properties of matter.
Now there are plenty of more-or-less widely accepted additions to this, mainly to explain the uniformity and fluctuations in the microwave background, and the emergence of first stars and galaxies etc. But all of these are to some extent speculative since we have no generally accepted unification of Quantum Mechanics and General Relativity or require physics that has no predicted consequence (so far) beyond the phenomena it has been proposed in order to explain.
To make this approximate an answer I will point you to the Wikipedia page on Non-Standard Cosmologies