The two Friedman equations, combined with some (reasonable) simplifying assumptions about the possible types of matter --- 'hot' and 'cold' matter, 'quintessence' (dark energy/cosmological constant) and geometric curvature, and the independency of contributions from these types. The self-consistency of these assumptions can be verified. But it is always possible to invent alternative explanations, since the number of observations is small and the number of alternatives infinite.

https://en.wikipedia.org/wiki/Physical_cosmology

This relates to the expansion history of the universe. Additional constraints/equations come from primordial nucleosynthesis, and the growth of structures in the universe.

Thank you for your response. So could you please write the equation where "The two Friedman equations, combined with some (reasonable) simplifying assumptions about the possible types of matter --- 'hot' and 'cold' matter, 'quintessence' (dark energy/cosmological constant) and geometric curvature, and the independency of contributions from these types." will give the proportion of dark energy and matter at any given time in the universe? I am looking for an as simple an equation as possible.

Gentlemen,

The main problems of uasmology are solved relatively simply. The relativistic description of a homogeneous space surprisingly connects the density of the universe and the Hubble law. At this time, the distance from the observer changes with acceleration compared with the law. Hubble. At the same time, the dark energy, the concentration of which is dark meteoria, will be explained ...

Preprint Dark energy as zero energy of gravity field

Hi Ranku,

Unfortunately there isn't an analytic solution, you have to evaluate it numerically.

I dont think there is an equation to calculate.

The total density of the energy of space is related to the Hubble constant. This one was clear after the work of Friedman. From my work it follows that dark energy and dark matter is the energy of an empty homogeneous space.

"I am looking for an as simple an equation as possible."

It is not even an equation, but a metric of empty homogeneous space. The first version of the article contained one and a half pages of text.

There is no need to involve the cosmological constant hypothesis.

I think the Hubble law is linked not only with local energy density of an observer but also with an average energy density of the space in between an observer and an observed origin of a signal. That could be a reason why the Hubble parameter vary due to the method of measuring (cmb vs standard candles)...

That's right, the suggestion is that the local "KBC void" causes the locally higher values, it's about a 30% under-density.

It can be calculated very simply by noting that, with the expansion of the universe, the energy density of non-relativistic matter falls as 1/a^3 and energy density of the dark energy decreases as 1/a^{3(1+w)}, where w is the equation of state parameter (i.e. pressure = w. energy density). Here a is the scale factor describing the expansion. So, if one knows the ratio today, one can calculate the ratio at any epoch.

Ok, then how to connect the scale factor with any epoch of the universe?

Gentlemen,

The expansion of the universe is just a hypothesis.

Uniform gravitational field is described by the metric

ds2=exp(2kixi)(dx02-dx12-dx22-dx32)

If the gravitational field is zero

ds2=exp(2k0x0)(dx02-dx12-dx22-dx32)

This metric describes a homogeneous universe if we take the k0 Hubble constant. Then time in this universe will depend on the degree of removal from the observer following a law close to the Hubble law. This metric determines the density of the universe. Preprint Dark energy as zero energy of gravity field

Happy New Year to all

Happy New Year,

Ranku Kalita, there is a very useful program widely available for doing the calculation called "Cosmocalc". If you look up that term in your favourite search engine, you should be able to find open source code for it. To understand the maths, look for lectures on the Friedmann Equations and the "equation of state" for the components, radiation, matter and dark energy.

The amount of 'dark energy' is 1.0 minus (the mass density of the Universe divided by the 'critical' mass density at which the Universe would gradually slow down until it almost but doesn't quite stop expanding). The current mass density is thought to be around 0.27 or 0.28 of the critical mass density, so 'dark energy' is 0.72 or 0.73; but the actual mass density at any given time in the past can only be determined by difficult measurements and uncertain calculations. As a result, we don't have enough information to know whether there is any equation that would describe its evolution over time (there are various theories about this, but no one of them seems any better than any of the others, and they may all be wrong).

I should also note that the amount of 'dark energy' may well be zero. The term was invented at a time when it was thought that the mass density of the Universe must equal the critical density (this was foolish from the get-go and proven wrong in the 1990's, but that's the way most scientific theories end up on the trash heap of history). Although wrong, this idea was cherished by most cosmologists for most of their working lives, and when it was shown to be wrong, they clung to the idea that the 'total' value of whatever there was (or wasn't) should still add up to 1.0. So they invented the concept of 'dark energy' to make up for the 'missing' mass.

So you're wondering why 1.0 is so big a deal? The Universe is 'flat', as far as we can tell. That is, Euclidean geometry seems to work as well as any other. But for that to be true for the Universe as a whole, then the mass density (including whatever isn't mass) has to be the 'critical' mass density, which is defined as 1.0. In my opinion, the problem is the same as the one that leads flat-Earthers to cling to their foolish fancies. Namely, on the average, the Earth is as smooth and locally flat as a flagstone (or even flatter, if it's a rough flagstone). It's really 'round', but it's so much larger than our usual view of the Earth that it looks flat. Odds are that the Observable Universe (the part we can see, because light has had time to get from wherever it started to here) is much, much smaller than the real Universe. In that case, the real Universe could be as curved as the Earth or whatever other geometric shape you want to give it, and the Observable Universe could look flat just because it's too small to look any other way. But since we can never observe anything beyond the Observable Universe, there's no way to know; so cosmologists still cling to 1.0 as the correct determiner of shape for the Universe (which it certainly is, for the Observable part), without wondering whether that really has anything to do with the actual mass density of the (Observable or entire) Universe.

Thank you, that's how I present the current situation.

I assume that the metric of our universe is homogeneous. This is enough to connect the energy density of the universe with the Hubble constant. This density can be interpreted as vacuum energy. Like all energy, it must have weight, i.e. accumulate around massive objects (galaxies). Thus, the vacuum energy is considered as dark energy + dark matter.

Energy density + dark matter density is calculated from the metric and is numerically equal to the Friedmann critical density. But I do not use the lambda term hypothesis. Only a homogeneous metric and nothing but a metric.