Answer by Frank Heile, Physicist
The answer is complicated and is partly yes and partly no since it depends on what is meant by the terms "universe," "particles," and "stable"!
What is meant by the "Universe"?
The standard cosmological model is that the universe is infinite. This is because cosmologists assume the entire universe must be homogeneous and isotropic. It is not just an assumption, the universe really does look isotropic and homogeneous on very large scales. The only way that a homogeneous and isotropic universe could be finite if it has a constant positive curvature, but the current measurement of the curvature implies that the universe is approximately flat and therefore ("approximately") infinite - or at the very least many many times larger than we can see.
On the other hand, theis finite. The observable universe is that part of the universe that we can see - and since the universe is only 13.8 billion years old, we can therefore only see photons that reach us in less than 13.8 billion years. Therefore the observable universe is defined as only the parts of the universe that are within 13.8 billion light years of us.
If there were no dark energy, the observable would actually increase with time since as the universe gets older we would see out further and further. However, because of the dark energy, the expansion of the universe is actually accelerating with time, so structures that are at the edge or our current observable universe will eventually be receding at faster than the speed of light and will disappear from our universe observable forever.
However, we can define a comoving volume which takes the current observable universe and follows that volume forward and backward in time ignoring the issue of whether the whole volume would be observable at those other times. If we do this and define the "universe" as being this comoving volume of space, then we can try to understand what is happening to the particles in this co-moving volume. So that is the choice we will make for the meaning of the word "universe" for the rest of this answer - it will be the comoving universe of our current observable universe..
What is meant by "Particles"?
We most often think of particles as things like electrons, protons, and neutrons since these are the fundamental building blocks of atoms, our bodies, the earth, and the solar system. The term nucleon will be used to refer to either protons or neutrons. If we take this as the definition of particles, then the commonly accepted answer for the number of particles in the observable universe is . This number will not vary by large amounts in short periods of time so this number is "almost" stable (with some caveats that will be described later). I will refer to the combinations of protons, neutrons, and electrons as matter particles for the rest of this answer.
If by particles we mean atoms, then the number of atoms definitely decrease with time because stars fuse hydrogen atoms together to make helium atoms and all the other elements of the periodic table. For example, according to Wikipedia, thefuses 620 million metric tons of hydrogen each second. This means that in each second, atoms of hydrogen are fused to produce 1/4th that number of atoms of helium each second (see ). When this fusion takes place, the total number of nucleons remains constant. However the fusion of 4 hydrogen nuclei to make one helium nucleus also creates 2 positrons (anti-electrons) plus 2 neutrinos. These positrons will annihilate electrons in the sun so although the number of nucleons remains constant, the number of electrons will go down, and the number of neutrinos will increase by an equal amount so the total number of electrons plus neutrinos remain constant.
Are photons particles? They are relatively easily created and destroyed. For example, photons can be created on the surface of the sun, travel for 8 minutes, hit the surface of the earth and be destroyed when they are absorbed on the earth's surface. On the other hand, if they miss the earth the photons may continue to travel forever.
However, most of the photons in our universe are the photons from the cosmic microwave background (CMB) radiation which have been around for 13.8 billion years. It is estimated that there are of these CMB photons for every matter particle in the universe so that would make photons in the universe. However, it is clear that the average number of photons in the universe is slowly increasing because stars produce more photons than are absorbed by planets, dust clouds, and black holes.
Dark Matter & Dark Energy?
Until we know what the dark matter particle is, we cannot make an accurate estimate of the number of dark matter particles or how that count would change with time. We do know that the total mass of the dark matter is about 6 times the mass of the matter particles in the universe. Currently, the favored theoretical candidate for the dark matter particle is the WIMP - the weakly interacting massive particle. These particles are assumed to be much heavier (x100?) than a proton, so if this is the dark matter particle, then it would imply that the number of dark matter particles is significantly smaller than the number of matter particles in the universe. On the other hand, if the dark matter particle is the axion, it may be 1/1000th the mass of a proton (or less) so there could be 6000 or more dark matter particles for each matter particle.
In any event, we don't know much about dark matter particles so it is impossible to say if they are constant and stable or not. It is possible, that the dark matter particles are their own antiparticle so at some very low rate the dark matter particles could be decreasing if they slowly annihilate each other. However, when they annihilate, they may create other kinds of particles that could include electrons, positrons, photons, or even nucleons. We simply don't know at this time what happens with dark matter.
We know even less about the dark energy in the universe, but the leading estimate is that it is "just" a small constant vacuum energy density. If the dark energy is just vacuum energy, then dark energy is not a particle.
Finally, consider black holes. Is a black hole a single particle? Then the number of particles in the universe will definitely decrease since a huge number of both dark matter particles, matter particles and photons go into creating the black hole. In addition, particles of all types may continually fall into the black hole. However, current theory says that eventually even black holes will evaporate due to, so the count of black holes will eventually fall to zero as they evaporate. However black hole evaporation will result in the energy of the black hole being converted (mostly) into photons, so the photon count will go up significantly as the energy of the rest mass of all the matter particles that have fallen into the black hole gets converted into a huge number of mostly very low energy photons..
What do we mean by "Stable"?The claims that in a very short time after the initial big bang, the universe underwent a period of inflation which would have expanded space by such a huge factor () so that no particles would be left in the very small (at that time) co-moving volume that would eventually became our observable universe. So at the end of inflation, there were no particles in our universe but there was a large amount of vacuum energy that was stored in the inflation field (this the field that caused the inflation). At the end of inflation, a reheating process would have converted that vacuum energy into a huge number of particles and antiparticles - approximately equal numbers of each. So the particle count at that time would be much higher than the current particle counts in the universe.
There is a mechanism that is not understood during this reheating process that resulted in a slight excess of matter particles over antimatter particles. Thus eventually, after all the antimatter particles annihilated with most of the matter particles, there would have still been a small excess of matter particles left over. During this time of particle annihilation, the total particle count will decrease dramatically. In the end, there will be left over matter particles that constitute all the electrons, nucleons, and dark matter particles that we have today. The annihilation of all the antimatter would have produced the large number of photons that we see today in the CMB. Note that a large number of neutrinos would also have been produced, but we have not been able to measure these primordial neutrinos because they have very low energy today.
In any event, because we went from an empty universe to one that has all the matter, dark matter, and photons that we have today, we know the numbers of these particles was not constant in the process at the very beginning of the universe. So what about today: are the electrons and nucleons stable today?
Well, there are many particle physics theories, including the now currentthat predict that protons will be unstable and will slowly decay. Attempts to measure this decay rate have failed, but we can say that the decay rate half time must be larger than years. Now this is times older than the current age of the universe! So, I think it is safe to say that the number of nucleon particles are relatively stable over long periods of time - depending on your definition of "long." There is no theory (that I know of) that claims that the electron is potentially unstable - the problem is that there is no lighter charged particle that the electron could decay into. Thus electrons (and neutrinos) are forever!
However, if the theories are correct about the proton eventually decaying, then eventually all the protons will decay and would presumably create additional positrons, neutrinos, and photons. The positrons could annihilate with all the electrons in the universe to eventually leave only photons and neutrinos. However, this would be so far in the future and the universe would have expanded so much that the electrons and positrons may never be able to find each other. In any event, the universe will mostly be empty by the time we get to something like years from now - even black holes will have all evaporated by then. For more about the eventual fate of the universe, see my answer to The Universe: What would we see if we could watch as the universe is dying?