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I can understand how the percentage of dark matter compared to ordinary matter is calculated, because the amount of dark matter has a clear gravitational effect on the ordinary matter in a Galaxy.
However, calculating the percentage of dark energy in the universe seems less obvious. Is it something to do with the rate of expansion of space-time?
Please explain in lay-mans terms, I've never learnt any undergraduate level cosmology.
There are (at least) four ways in which the dark energy content of the universe influences things we can observe
The cosmic microwave background is formed in the early universe when atoms (of hydrogen) first formed and the universe became transparent to the radiation that was within it. There are small fluctuations in the CMB which reflect small differences in the density of regions within the universe at the time of this "decoupling" of radiation and matter. How big (in terms of an angle on the sky) these regions now look depends on the subsequent rate of expansion of space and this in turn depends on both the amount of gravitating matter (which slows the initial expansion) and the amount of dark energy (which accelerates the expansion). Hence, in broad terms, the angular size of fluctuations (about 1 degree) in the CMB allows one to infer the amount of dark energy, though it is mixed up in what the other cosmological parameters are (including the total amount of gravitating matter density).
The expansion of the universe can also be tracked using standard candles. That is we can measure the brightness of something, infer how far away it must be and then measure how fast it is receding away from us. In a decelerating universe, containing only gravitating matter, then the expansion rate would have been much larger in the past and would be witnessed in the recession velocities of more distant objects that are seen as they were in the distant past. Observations of type Ia supernovae - standard candles arising from the explosions of white dwarfs at nearly a fixed mass and with an extremely consistent peak luminosity - confound this expectation. Instead it appears that the expansion of the universe is accelerating and this is attributed to dark energy. Again, this interpretation is not wholly independent of the other cosmological parameters.
The large scale structure of the universe is dependent on both the amount of gravitating matter (especially dark matter, that was able to start clumping together even before normal matter and radiation became decoupled) and the amount of dark energy. Models of the how clusters of galaxies and filamentary structures in the universe form, suggest that the range and scale of structures we see in the universe today depends in detail on the nature of dark matter, but also on the nature of dark energy. Comparison of these models with observations leads to constraints on how much dark energy there must be.
In the 80s and early 90s there was something of a crisis in cosmology. If one measure the current expansion rate of the universe and extrapolates back in time, one can calculate how long ago the big bang occurred. Performing this exercise assuming that the expansion of the universe was only decelerating with time , due to the gravitating matter within it, resulted in an age of about 10 billion years. However, the oldest stars in the universe seemed to be at least 12 billion years old! Dark energy resolves this problem by allowing the expansion to accelerate with time after an initial period gravitational deceleration. An extrapolation based on the current expansion rate thus leads to an age that is too young. Instead, a revised extrapolated age for the universe, including dark energy, is 13.7 billion years old and comfortably older than the oldest stars.
Putting all these things together leads to the so-called "concordance" or Lambda-CDM model for the universe (see image, adapted from Kowalski et al. 2008). This posits that 68% of the total energy density in the universe is in the form of "dark energy", which causes the expansion of the universe to accelerate.
