The density of universe at the time of the Big-Bang was infinitely high. Does that mean that the mass was also infinitely high? ( the universe was extremely small at that time)
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Firstly, it is important to note that the old Big Bang cosmology is no longer the most widely accepted theory. We include inflation into the mix in current theories. That said, there is an ambiguity in the definition of the Big Bang (you can find information on that in my question here).
If we take the definition of the Big Bang as coming before inflation, then we are probably referring to a curvature singularity. During this time, our best theories vary by a great deal in what we would expect to find. Many of them theorize the existence of one or more massive inflaton fields that eventually drive inflation. While we say they have mass, it is not the same as how a brick of iron has mass. The inflaton has a mass in much the same way that radiation or dark energy has a mass; there is a gravitational/inertial mass but not necessarily a rest mass. The mass density of inflatons is not usually volumetric, which means it does not scale with the size of the container. Furthermore, the mass in this case is used more as a coupling constant for the field. However, this is all besides the point, at a curvature singularity, the energy scales would be well above the range of most of our theories. Standard general relativity is not expected to accurately describe the universe at a point like that; we need a GUT and quantum gravity to accurately describe the physics of an initial curvature singularity. Short answer: No, the mass is not infinitely high. It is finite with a non-zero probability of being zero.
If we use the second definition of the Big Bang, where it occurs after inflation, then there is no curvature singularity. Under this interpretation, it is also common to treat the Big Bang as an era rather than a single moment, however we can look at the start of this era to answer the question. After inflation, the universe enters into its more regular routine. There is matter and radiation and it is very hot; everything one expects when they imagine the beginning of the universe (it's only $10^{-30}s$ old after all). The matter and radiation present comes from a few sources, mostly from the decay of the inflatons and quantum fluctuations during inflation. Since the universe is not confined to a curvature singularity and since the amount of matter and radiation is finite and generated by mostly decaying inflatons, the total mass of the universe is certainly finite. Furthermore, at the initial moment of the Big Bang era, not all inflatons would have decayed; the full compliment of matter in the universe is not yet present.
Now, there is one important thing left to consider. What do I mean when I say "the universe"? The possibility of the universe being infinite in extent means that the total mass of the entire universe is possibly infinite. So to clarify, when I refer to "the universe" I mean the observable universe. More specifically, because I'm a cosmologist and we like this value, I am referring to the region bounded within the comoving Hubble radius at the time of interest. This means we can further answer your question. Even though the mass is finite, I cannot deny that the universe was significantly smaller and denser than it is now. However, not only was the universe unexpanded (a small scale factor), the Hubble radius was much smaller back then. The energy density of matter is volumetric; it is inversely proportional to the size of the container. This means that, if no matter is created or destroyed, the mass of the universe now would have to be greater than or equal to the mass of the universe at the beginning of the Big Bang era (the comoving Hubble radius is larger now, which means the present universe can encompass more matter that was not inside it in the past).
This brings us to the end, for which I have saved the simplest (if not the most accurate) answer to your question. Your premise that the density is infinitely high is based on the following logic:
1) The universe has a mass $M$ now and a large volume, therefore a low density
2) At the time of the Big Bang, the universe was a singularity with zero volume
3) Therefore, the density is $\rho=M/V$, $V=0$, so $\rho=M/0\to\infty$
4) If the density is infinite, mustn't the mass be infinite as well?
Do you see the problem with point (4)? You used $M$ and $V$ to solve for the density, then you are using that to solve again for $M$. Logically, you can only retrieve the value you initially used. In other words, it's the same as the mass of the universe now (according to your reasoning): $\rho=M/V=M_{now}/0\to\infty$ but $M=\rho V=(\infty)(0)=M_{now}$
