The question is in the title: are there more bosons or fermions in the universe? Or is there the same number of bosons and fermions?
I think there is the same number but I don't know why exactly.
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The answer is no.
Neither the number of bosons nor the number of fermions notr their difference is conserved in nature. This means that these number change all the time.
Should they be equal by coincidence for some time, this would last only for a moment. Even such a coincidence would be extremely unlikely, given the huge number of particles in the universe.
Edit: I had added in the discussion to other answers that due to the infrared problem, the number of each kind of massless particles (in particular that of photons, but possibly also that of one species of neutrinos) is infinite.
This makes the answer to the question somewhat trivial: If there is a massless species of neutrinos (still a possibilty - we now only know that not all neutrinos are massless), the answer would be yes, though not for an interesting reason, as both numbers are countably infinite and hence equal. On the other hand, if all neutrinos are massive, the answer would be no, as there are then infinitely many bosons but only finitely many Fermions.
The real answer to the question is is that counting particles makes no sense; particle number is not a physically relevant observable, and tells nothing interesting about the universe. (Except perhaps an entry in the Guinness Book of Records - but isn't the universe extremal in every respect?)
Arnold Neumaier mentioned "soft photons" and the "infrared problem"; this bears elaboration. Soft photons are photons with very very low energy, for example a photon can have a period of one year, a wavelength of one lightyear, and an energy of 1E-22 eV. These photons are unobservable by any means. In traditional quantum field theory, it is predicted that every time two electrons repel each other (for example), an infinite number of soft photons are created ("the infrared divergence problem"). This is not regarded as much of a "problem" because the total energy of the infinite number of photons is finite, as the energy of each photon can be arbitrarily small. The vast quantities of soft photons have no observable effect or consequence.
In reality, I imagine that the number of soft photons created by a scattering event is not (strictly speaking) infinite. For example maybe there's no such thing as a photon with wavelength larger than the visible universe. That's just a guess, I don't know. But even if there's some cutoff like that, I would certainly bet that the number of soft photons (and soft gravitons) is vastly more than the number of all other particles in the universe combined. Photons and gravitons are massless, so they can have arbitrarily low energy, which is not true of any other particle because of the mass rest energy.
Since photons and gravitons are both bosons, I would say "more bosons than fermions".
Broadly fermions are associated with matter, and bosons are associated with exchange (force mediation), at least if we are considering elementary particles. A given matter particle will produce and absorb many many exchange particles in its lifetime, so we can guess that there are more bosons than fermions.
It is not, however, fixed. A photon can be converted to an electron/positron pair, which would reduce the number of bosons by 1 and increase the number of fermions by 2. So at different times and places, the balance will be different, including at different stages of the evolution of the universe.
Immediately after the Big Bang, it is theorised that the universe was so small and dense that only photons existed; any matter particles created would be immediately obliterated. However, as the universe cooled, matter particles could survive long enough to form structures, until we have the cooler universe we see today. Perhaps, if the universe survived long enough and expanded far enough, particles would be so far apart that they would experience very little force and the balance would be in favour of fermions. But unless something changes significantly, the structures of matter - atoms, nuclei, crystals - are all stable and involve continual exchange of bosonic particles to mediate the forces that maintain them.
Finally, remember that at quantum scales everything is probabilistic; we cannot say that a photon exists or not in the classical sense until we measure its effect, so there can be many virtual particles with some probability that we do not measure for every particle that we do. For some fairly involved discussion of it, see Quantum Electro-Dynamics, and the electron self-interaction problem. It involves all the possible paths of electrons interacting with themselves, and held up fundamental quantum field theory for quite a while.
The question may be unanswerable because I'm pretty sure the number of photons is variable depending on your relativistic frame. At least that was an interesting observation that Feynman made, almost apologetically, in some of his early work; I'd have to look up the specific reference. (Hmm. Feynman's idea there would require variable-count photons to be generated in spin-cancelling pairs...)
I suppose that the question is ill-posed for at least three reasons.
Since Fermions and Bosons can be generated from each other, there is a certain dynamics involved. Therefore: When would you count those numbers? The problem I see is connected with defining the (global?) time in which to count - given a lack of simultaneity.
In QFT, there is a particle number operator. This operator depends on the coordinate frame. Which one would you chose? The "number of particles" is not independent of the coordinate system.
Finally, Unruh effect tells us that an accelerating observer sees the vacuum as a heat bath consisting of photons. So, we would need to know the acceleration state as well.