Why is it assumed that the Big Bang produced equal parts matter and antimatter?

The strongest motivation for initially equal amounts of matter and anti-matter actually comes not from the standard model but from cosmology - there is a staggeringly large number of photons in the Universe, relative to baryons$^1$. It's possible to arrive at this conclusion a couple of different ways. One is to look at the cosmic microwave background, which is the single largest contributor to the overall photon number density in the Universe. It has an energy density of about $0.25\,{\rm eV}\,{\rm cm}^{-3}$, which works out to about $n_{\gamma}=500\,{\rm photon}\,{\rm cm}^{-3}$. Compare this to the baryon energy density of about $240\,{\rm eV}\,{\rm cm}^{-3}$, which works out to $n_{\rm bar}=2.6\times 10^{-7}\,{\rm proton}\,{\rm cm}^{-3}$. There are therefore about $n_\gamma/n_{\rm bar}\sim2\times 10^9\,{\rm photon}\,{\rm baryon}^{-1}$.

The argument is then that at early times there were near-equal amounts of matter and anti-matter, most of which (except for about one part per billion) annihilated into photons. This either leads to a fine tuning problem - why was there a $10^{-9}$ excess of matter over anti-matter in the initial conditions? - or you can instead try invoking a CP-asymmetric process to break the initial matter/anti-matter symmetry. Most physicists and cosmologists feel more comfortable with the latter option; we tend to be very fine-tuning averse.

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$^1$I'm using the astronomer's baryon here, which is a loose term encompassing more or less all non-relativistic matter (mainly atoms, leptons), but not (cosmological) dark matter, neutrinos, or photons.