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1

You ignored the $k$ term, but it's crucial here. $k$ is the curvature not of spacetime but of constant-$t$ spatial slices, so it depends not only on spacetime curvature (represented by $ρ$ and $Λ$) but also on the extrinsic curvature of the spatial slice in the spacetime (represented by $\dot a/a$). You can think of this equation as showing the relationship ...

2

The theory you're referring to is called "eternal inflation", and in that context, the universe isn't inflating today. It is expanding at an accelerating pace, but it is not inflating. The time when the universe did inflate was immediately after the big bang. Given the above, your question is rather vague and hard to answer as a result. However, we ...

2

Your intuition that $\rho$ ought to slow the expansion is correct, but that intuition concerns $\ddot{a}$ not $\dot{a}$ (and indeed in the other Friedman equation you do see a minus sign). The $(\dot{a})^2$ term comes from the expression for the curvature when the metric is of FLRW form. The Einstein field equation says there has to be a relationship between ...

1

$\dot a$ is the rate of change of the scale factor i.e. it tells us how fast the universe is expanding. Shortly after the Big Bang the universe was very dense and expanding very rapidly so both $\rho$ and $\dot a$ were high. Then as time went by the universe became less dense as the matter was diluted by the expansion, and at the same time the expansion ...

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