> Dark energy, for maths reasons I don't really understand[0], acts like negative pressure even though it's positive energy

First, let's understand a bit about the cosmological frame: it lets us consider the universe as 3-dimension-of-space ordered by a "scale factor" dimension of time. The coordinates of each 3d space are Euclidean, but the coordinates don't line up exactly between different scale factors. Colloquially, the coordinates expand with the expansion of space, or equivalently the space between coordinates grows over time. You could think of it this way: if at some early point we label every point in space with an integer, in the future of that point we have to add new labels between the integers.

If we add in a pair low-mass freely-falling test matter at the time when everything is labelled with an integer, e.g. at point (1,1,1) and (2,1,1), then they stay at those coordinates even as more and more labelled (with non-integer) space appears between them.

To this we add a set of gasses, dusts, or fluids representing radiation, ordinary nonrelativistic matter, and maybe others (e.g. relativistic dark matter (e.g. neutrinos), non-relativistic dark matter ("cold dark matter"/"particle dark matter")). Again, any "mote" of the ordinary matter dust stays at the same coordinate forever. Rather than coping with the "motes" of radiation not staying put, we average every point in space and see there is some quantity of matter (a mote, or a fraction thereof), some quantity of radiation (a mote, or a fraction thereof), some quantity of dark matter, and so forth.

The dusts dilute away because more and more space appears between each original coordinate. In our averaging picture, we get a smaller and smaller fraction of a mote at eacn point on average as the universe expands.

This is the essence of the Friedmann-Lemaître-Robertson-Walker model that is the standard cosmology.

The view here is that of boring old observers freely floating in deep inter-galaxy-cluster space. That leads to (from that view point) concrete calculations of the various contributions to the averaged energy-density at each point in space at a particular scale factor ("at a given age of the universe"). In general, that figure is higher in the past and lower in the future, with different contributions to the total (average) energy density at a point dropping at different rates (this is the "equation of state" for each of radiation, baryons, neutrinos, dark matter, ...), but they all drop away towards nothing in the far future.

We can then think of where stress-energy goes at an average point. For freely falling baryons, almost nothing interferes with the whole of the stress-energy flowing from (microsecond-before,0,0,0) to (now,0,0,0) to (microsecond-after,0,0,0). There's some cosmic mircowaves and neutrinos that have a tiny ghost of a chance of transferring in some momentum via scattering at each point, but that falls away when we consider the average across each of these three 3d spaces.

Flipping things back around, while a "mote" of the baryon gas stays "at the integers" as we add more and more digits after the decimal point as space expands, each time we add digits we get more dark energy fluid "motes". If we change coordinates, the coordinate-distance between "motes" of the fluid grows with the expansion, e.g. as the distance between these proxies for galaxy clusters goes from 1 to 10 to 100 to ... the 9, 99, ... has "new" motes of dark energy.

Given this it is straightforward to treat some aspects of the expansion as another fluid with an energy density that is the same at every point in every space. It does not dilute away like the others. It imposes a tension ("negative pressure") on the other sources of energy-density.

If we think of a single point in one of these 3d spaces as being imprisoned within a tiny six-sided cubical cell, we can track the flow of momentum through each (pair of) face(s) of the cell. At very early times radiation flowing through the cell dominates. The inflow is tracked as the normal stress <https://external-content.duckduckgo.com/iu/?u=https%3A%2F%2F...> on each face. When the normal stress is identical on all six faces (or spherically symmetrical if we switch from a cube to a sphere), we call that pressure.

Negative pressure is just flipping the arrows around. We can call that "tension".

Colloquially we're interested in how much the pressure changes the energy level of the imprisoned matter. In general, large positive pressure leads to the imprisoned matter becoming more energetic. Large negative pressure would lead to the imprisoned matter becoming less energetic. "Large" here is relative to the energy-density within the cell, and varies by component ("equation of state" again).

In the comsological frame, freely-falling imprisoned matter (baryons, dark matter) in a cell at (t,1,1,1) will thus cool with the expansion across (t',1,1,1), (t'',1,1,1) and so forth, with the energy sucked out by the constant outward tension.

> 16 simultaneous partial differential equations

You can start with understanding the stress-energy tensor.

This is it laid out in a 4x4 matrix form <https://en.wikipedia.org/wiki/Stress%E2%80%93energy_tensor#/...>. The indices running 0,1,2,3 correspond to the timelike dimension and the three spacelike ones. Each element of the matrix has two indices i and j (e.g. for T^00 in the top left, we have i=0, j=0) indicating the "comesinfrom" and "goesoutto" directions.

Energy pouring in and staying in is the orange column. Energy already there and staying put is the top left.

Let's use spherical coordinates and call the 1 direction "in/out", i.e., described by the radial coordinate. We'll place our cell of interest microscopically displaced radially from the spherical coordinate origin.

If we were in a dense object like the core of a planet or star, T^11 would be dominated by the inwards flow of inwards-momentum and the reaction-pressure of outwards-momentum flowing outwards. In high-mass stars' cores, this number will dominate all the others in the stress-energy tensor. Also in general T^ij, i=j, i!=0 (the green bar) will not be completely identical.

However, if we're in a cosmological setting (deep inter-galaxy-cluster space), T^11 is [a] small, [b] it's the same as T^22 and T^33, and [c] they are interpretable as the flow of energy-momentum out of the cell. In the very very far future, T^00 drops to zero, and T^ii, i!=0 (despite being small) dominates.

Another difference here is that in the stellar core positive pressure case, T^11 is not a constant. However, if we got rid of all possible radiation pressure and the like, in the resulting cosmological vacuum T^11 would be a constant (and identical to T^22 and T^33).

This is an interpretation that depends on our choice of viewpoint (that of a freely floating low-mass observer who feels only the expansion and not attractive influences from dense matter, because all matter is completely evenly smeared out). The interpretation does not hold up well as we change our point of view (e.g. to a relativistic observer, to a different set of coordinates, to an observer who is close to or part of a self-gravitating mass overdensity).

Consequently it is probably better to start with the idea that dark energy is the Cosmological Constant (until this possibility is disproven, which has not happened yet). It is simply a constant of nature, like the speed of light or like the charge of an electron. Where does the constant come from? Who knows!

Sometimes it is convenient to think of this constant as if it were a substance with appropriate properties, or as if it were "the cost of empty space" (vacuum energy). However it's also possible to be misled by this. It's a scalar quantity, and it has an exact relationship to a tensor quantity (the metric). Scalars and tensors are generally covariant, so we can always make that pairing work for any possible observer, even e.g. an ultrarelativistic cosmic ray or a photon or a relativistic compact object like a black hole.

Finally, the reason for "sometimes" is: while the total tensor value of T is the same for everyone, the value of the individual components of the tensor depends on the frame of reference.

One of the sometimeses is the very early universe when radiation pressure is extremely important for smoothing out temperature and density differences; it is natural to want to do accounting of the expansion as an offset against that radiation pressure instead of the "truer picture" of the radiation pressure falling because the radiation is diluted and de-energized (redshifted) by the early expansion.