The Big Bang theory is a cosmological model describing the early development of the Universe. According to this theory, the Universe expanded from a very high-density and high-temperature state, and continues to expand to this day. Mathematically, this is often described using the field of physics known as general relativity.

The mathematical foundation of the Big Bang theory is rooted in the field equations of general relativity, formulated by Albert Einstein. These equations relate the geometry of spacetime with the distribution of matter within it. The specific solution to the equations that describes an expanding universe is known as the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. This solution derives from the assumption of a homogeneous and isotropic universe, which appears to be the same in all directions (isotropic) and has the same distribution of matter throughout (homogeneous).

The FLRW metric leads to the Friedmann equations when applied to a cosmological scale, which describe how the universe expands over time. The equations include terms for the curvature of the universe, the density of different forms of matter and energy, including dark energy and dark matter, and the cosmological constant, which is related to the accelerated expansion of the universe.

One of the intriguing aspects of the Big Bang theory is the concept of the initial singularity. When we extrapolate the equations backward in time, they predict a moment when the universe was infinitely dense and hot. This is known as the singularity. However, the term “nothing” isn’t accurate here; the singularity isn’t “nothing,” it’s a state where our current understanding of physics breaks down, and the densities and temperatures are beyond what we can currently comprehend.

The equations do not actually imply that “nothing” existed before the Big Bang because the concept of “before” breaks down when discussing a singularity. Time and space as we know them are part of the universe, and talking about a time “before” the universe is outside the scope of our current physical models.

To address this singularity and the origin of the universe, one needs a theory that combines general relativity with quantum mechanics, the other great pillar of modern physics. This is the realm of quantum cosmology and is an area of active research, with theories such as the Hartle-Hawking state suggesting that the universe could have arisen from a quantum fluctuation in a pre-existing state, thereby avoiding the need for a beginning in time as we understand it.

In summary, the Big Bang theory does not try to prove that “nothing exists” at some point. Rather, it describes how the universe evolved from an extremely dense and hot state to its current state. The question of what came “before” the Big Bang, or indeed if “before” is even a meaningful concept at that extreme, is still an open question in physics.