Could the uncertainty principle create matter in the Universe? According to quantum mechanics we have the uncertainty principle:
$$\Delta E \ \Delta t \geq \frac{\hbar}{2}$$
Near the beginning of the Universe $\Delta t$ is very small and therefore $\Delta E$ is large enough to produce matter-antimatter particle pairs just by the uncertainty principle alone.
If the Universe is expanding fast enough then these pairs would be unable to recombine and annihilate.
Maybe this process is the source of particles in the Universe rather than a singularity of infinite energy density at the big bang?
Edit: I don't assume that the uncertainty principle itself creates the particles. Instead the particle-antiparticle pairs borrow energy from the vacuum when the Universe is young (and possibly naive!) and then 'go on the run' in such a way that their creditor can never catch up with them.
 A: 
Could the uncertainty principle create matter in the Universe?

The uncertainty principle does not stand alone in quantum mechanics. It is a consequence of the operator structure of the theory, and an  uncertainty relation can be derived for all the conjugate variables describing the system. A good analysis of how the energy/time uncertainty is derived from the theory is in this answer here. 

Near the beginning of the Universe Δt is very small and therefore ΔE 
  is large enough to produce matter-antimatter particle pairs just by the uncertainty principle alone.

The beginning of the universe is defined in the standard cosmological model as a locus in space time where the energy momentum tensor of general relativity has very high curvatures, and is represented in the Big Bang model which developed from having a singularity for t=0, to an undefined region,  

for times less than 10^-32 seconds.
In that region the model uses an effective quantum mechanical theory, because at the moment there is no definitive quantization of gravity. This effective theory involves inflatons, hypothetical particles that carry all of the energy processes at that time, and create an inflation as seen in the graph.
Between 10^-32 seconds and 1microsecond the universe is a soup of elementary particles and finaly protons are formed. 
In this region virtual loops of particle antiparticle can be imagined, whose "reality"  starts at the proton formation time.

If the Universe is expanding fast enough then these pairs would be unable to recombine and annihilate.

It is these pairs in the soup that start becoming real protons and antiprotons  and other particles , but this is not because of the heisenberg uncertainty, but because of the the universe cooling enough due to the expansion that the available energy can coalesce into masses and remain stable because of lack of interactions that would destroy/annihilate them.

Maybe this process is the source of particles in the Universe rather than a singularity at the big bang? 

From the moment one assumes an expansion of the universe, as you do, the model implies a beginning region much compressed if not a singularity. Which brings us to the current cosmological model.
So it is not the heisenberg uncertainty that generates particles , but the available energy and the thermodynamic interactions that allow the persistence of matter. In the  Big Bang model using the standard model of particle physics, it is not yet known how the asymmetry between particles and antiparticles ( our universe is composed out of particles) is generated,   this is an ongoing research question.
