Does the cosmological principle apply to the entire universe?

Some cosmological models assume that the universe is isotropic and homogeneous and that is also flat and infinite. If the universe is infinite now it was infinite immediately after the big bang. If universe was infinite immediately after the big bang, what explains isotropy and homogeneity?

• The Lambda CDM model does not assume that the universe is flat or infinite. Jun 28 '20 at 19:48
• The false presumption about ΛCDM assuming flatness and infinity aside, this is a brilliant question.
– pela
Jun 28 '20 at 20:29
• @JamesK you're right I'll correct the "at". About homogeneity, I read the it (and isotropy) is explained by inflation because the entire observable universe has been causally connected long enough to come into thermal equilibrium. But if the universe was infinite soon after the big bang time, there are regions that was never causally connected. So what explains isotropy and homogeneity of the entire universe?
– UFO
Jun 29 '20 at 7:27
• @RobJeffries in the [Lambda Nasa website][1] I read that there 6 independent parameters that completely specifies the cosmological model. 4 of them are related to matter content. Specifically: "The Ω parameters are defined as the ratio of the present day mean density of each component χ to the critical density: by definition ΣχΩχ = 1". If ΣχΩχ=1 than the universe is flat and infinite (to be precise it could be also finite like a three-torus) (1): lambda.gsfc.nasa.gov/education/graphic_history/parameters.cfm
– UFO
Jun 29 '20 at 7:41
• You can assume that $\Omega=1$ but there is no need to. Sometimes this is referred to as the "base $\Lambda$CDM model" or the "spatially flat $\Lambda$CDM model". to make it clear that this assumption has been made. Since $\Omega$ is consistent with 1, then the assumption is often made, since it is consistent with the predictions of inflation. Jun 29 '20 at 9:14