cosmology - Isn't Big Bang theory a violation of law of conservation of linear momentum?

What Big Bang theory assumed in the formation of universe?

Answer:-At its simplest, it says the universe as we know it started with a small singularity, then inflated over the next 13.8 billion years to the cosmos that we know today.

Now This nothing must be there outside this singularity as everything is contained in universe itself.So let's take the singularity as our system.Then,$$\vec{F_{ext}}=0$$

It means linear momentum of the system must be conserved.So,the centre of mass must not move even after 13.8 billion years as it was initially at rest.I am saying that it was initially at rest because it would never have acquired a velocity because nothing pushed it from outside because every force is an internal force for the universe.Initially,$$\vec{v_{com}}=0$$ $$\vec{p_{com}}=0$$

The velocity of centre of mass of universe must be zero for all time.Now after 13.8 billion years,the velocity of centre of mass by its definition must be:-

$$\vec{v_{com}}=\frac{\sum_{i=1}^{\infty}m_{i}\vec{v_{i}}}{\sum_{i=1}^{\infty}m_{i}}$$

$$\vec{v_{com}}=\frac{\sum_{i=1}^{\infty}\vec{p_{i}}}{\sum_{i=1}^{\infty}m_{i}}$$

$$\vec{v_{com}}=\frac{0}{unmeasurable}$$

It is an undetermined form in mathematics because dividing zero by a unmeasurable number is meaningless.So linear velocity of center of mass is not zero which is a complete violation of law of conservation of linear momentum.With due respect,it means our assumption of universe as a singularity may be wrong.

My question is,isn't Big Bang theory a violation of law of conservation of linear momentum?

EDIT

I made a major mistake in the above problem so I have corrected it now. The mistake was I quoted that $\sum_{i=1}^{\infty}m_{i}$ tends to be infinity which is a limiting process and it can't be a limiting process as universe is still expanding so it's mass must be undefined as we can't calculate the mass of this universe due its expansion.It is meaningless to say something about the mass of universe.

Answer

You’ve made two mistakes.

First, you can’t use Newtonian physics to understand the entire universe. If the universe had a center of mass (which it doesn’t), what would it be moving, or not moving, relative to? In mainstream cosmology, you have to use Riemannian geometry and General Relativity to understand the dynamics of the universe.

Second, zero divided by infinity isn’t undetermined. It’s well-determined and is just zero.