You're not all that far from describing dark energy.
Gravitational attraction tends to cause the trajectories of freely-falling ("floating in empty space") masses to converge. Dark energy tends to cause the trajectories of freely-falling masses to diverge. We see this in the sky: distant galaxies are freely-falling but their trajectories are away from each other (and from us).
As far as we can tell though, the "source" of dark energy is the cosmological constant. Physical interpretations usually start by slicing spacetime into spaces aligned along the cosmological time (the scale factor). In successive spatial slices ordinary matter, radiation, dark matter, and so forth all get sparser towards the future. We treat these as a sort of fluid or gas that dilutes away with the expansion of the universe, therefore the density of each fluid at a typical point differs from spatial slice to spatial slice (it's lower in the future). The cosmological constant is constant in every spatial slice. So if we were to turn it into one of these fluids, it would not dilute or thin out over time: its density at a typical point is always the same. Representing a geometrical feature (the cosmological constant) as a space-filling fluid with certain properties makes it easier to study what happens if the cosmological constant isn't constant . (Does it fluctuate? Does it get weaker or stronger over time or in the presence of overdensities of matter?)
Cosmic inflation introduces another of these fluids that like dark energy serves to separate rather than draw together the others. It's very very strong in the early part of the universe (the inflationary epoch), but disappears very early too. In some flavours of inflation, the "inflaton" field decays into standard-model matter; in others it just decays into weaker and weaker versions of itself and by the time there are electrons, positrons and photons it's basically so weak that it's undetectable.
The inflationary epoch somewhat matches your intuition that a big source of energy pushed space apart. The most important difference (at this "explain like I'm..." level) as far as I can tell is that there was no point in space that someone in the early universe could point to as the centre or source of inflation. A gravitating object has a centre of mass, and you can point to and away from that object, i.e. there is an up and down with respect to it, because it doesn't surround you like a shell. The inflaton field was strong everywhere, there was nowhere in space "outside" it.
Gravitational attraction tends to cause the trajectories of freely-falling ("floating in empty space") masses to converge. Dark energy tends to cause the trajectories of freely-falling masses to diverge. We see this in the sky: distant galaxies are freely-falling but their trajectories are away from each other (and from us).
As far as we can tell though, the "source" of dark energy is the cosmological constant. Physical interpretations usually start by slicing spacetime into spaces aligned along the cosmological time (the scale factor). In successive spatial slices ordinary matter, radiation, dark matter, and so forth all get sparser towards the future. We treat these as a sort of fluid or gas that dilutes away with the expansion of the universe, therefore the density of each fluid at a typical point differs from spatial slice to spatial slice (it's lower in the future). The cosmological constant is constant in every spatial slice. So if we were to turn it into one of these fluids, it would not dilute or thin out over time: its density at a typical point is always the same. Representing a geometrical feature (the cosmological constant) as a space-filling fluid with certain properties makes it easier to study what happens if the cosmological constant isn't constant . (Does it fluctuate? Does it get weaker or stronger over time or in the presence of overdensities of matter?)
Cosmic inflation introduces another of these fluids that like dark energy serves to separate rather than draw together the others. It's very very strong in the early part of the universe (the inflationary epoch), but disappears very early too. In some flavours of inflation, the "inflaton" field decays into standard-model matter; in others it just decays into weaker and weaker versions of itself and by the time there are electrons, positrons and photons it's basically so weak that it's undetectable.
The inflationary epoch somewhat matches your intuition that a big source of energy pushed space apart. The most important difference (at this "explain like I'm..." level) as far as I can tell is that there was no point in space that someone in the early universe could point to as the centre or source of inflation. A gravitating object has a centre of mass, and you can point to and away from that object, i.e. there is an up and down with respect to it, because it doesn't surround you like a shell. The inflaton field was strong everywhere, there was nowhere in space "outside" it.
Here is a reasonably accessible "outreach" article on cosmic inflation: https://www.ctc.cam.ac.uk/outreach/origins/inflation_zero.ph...