What is the evidence that dark energy exists? (as of 2020) Astrophysicists believe that dark energy exists and makes up ~70% of the universe's energy content. What is the evidence that dark energy exists? Wikipedia gives five lines of evidence:


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*Supernovae. A specific type of supernovae (type 1A) are supposed to be standard candles - that is, their luminosity is known. In turn this lets us measure distance to faraway galaxies, check how fast those galaxies are receding from us, and check if that recession speed is increasing over time.

*CMB. Data from the Cosmic Microwave Background indicates the universe is approximately flat. Visible matter + dark matter can account for ~30% of the energy content required to make the universe flat, leaving ~70% for something else - dark energy.

*BAO. Baryon acoustic oscillations act as a "standard ruler" that lets us measure how the Hubble constant varies with redshift (i.e. time), and see if the recession speed is increasing with time.

*Late-time ISW. ISW stands for integrated Sachs-Wolfe effect. The idea is that, usually, a photon that enters a potential well gains energy as it falls in and loses energy as it emerges, and gain/loss exactly cancel. If a universe is expanding in accelerated fashion, then this is untrue; the potential wells / hills are smoothed out and there is a permanent shift in the photon's energy and therefore temperature. If we see a correlation between hot and cold spots on the CMB and the locations of superclusters and voids, then it's a sign of accelerated expansion.

*Galaxy evolution. This uses (known) evolution of early-type galaxies as a standard clock. Once we know how long it takes for a galaxy to evolve from one state to another, as well as their redshifts, we can reconstruct how the Hubble constant varies over time, and see if the recession speed is increasing with time.
Two questions:


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*Did I understand any of these five methods wrong?

*Are there any other lines of evidence for the existence of dark energy?

 A: One should note that the evidence #1–5 of OP first and foremost counts as evidence of accelerated cosmological expansion. While one could also view these observations as evidence of dark energy, it is worth noting that there are other possible explanations for accelerated expansion that could not be classified as dark energy models.
A general name for such models is “modified gravity”. The distinction between dark energy and modified gravity comes from the way Einstein field equations are modified. Dark energy enters EFEs as a source term (either cosmological constant or slowly evolving field) and interacts with other matter gravitationally. Modified gravity, as  the name suggests, introduces new dynamic in the gravitational sector, possibly introducing new particles mediating “fifth force”. Examples of theories of modified gravity are massive gravity, brane world models, certain scalar-tensor theories. Note, there are also modifications of gravity that attempt to eliminate the need for dark matter, rather than dark energy, but we will not discuss these.
For more on the distinction between dark energy and modified gravity see review: 


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*Joyce, A., Lombriser, L., & Schmidt, F. (2016). Dark Energy versus Modified Gravity. Annual Review of Nuclear and Particle Science, 66, 95-122, doi:10.1146/annurev-nucl-102115-044553, arXiv:1601.06133
From the text:

… we  draw  a  distinction  between  Dark  Energy  and  Modified  Gravity  by  means  of  the strong equivalence principle (SEP). We classify any theory which obeys the SEP as Dark Energy, and any theory which violates it as Modified Gravity (see Sec. 2.3).  Heuristically, the strong equivalence principle forbids the presence of fifth forces.

The following image from this review illustrates the distinction:

At the level of Friedmann equations and even for the description of large-scale structure evolution the distinction between dark energy and modified gravity might appear nonexistent (and some authors do not make it), but such distinction offers a new class of evidence in support of dark energy: observations , not directly related to cosmic acceleration, but favoring dark energy over modified gravity models. So, if some astronomical observation or experiment eliminates (or constraints) modification of gravity that offers explanation for cosmic acceleration, we could consider this an evidence for the dark matter. 
For example, gravitational wave event GW170817 and corresponding gamma-ray burst GRB 170817A establishes with great precision that gravitational waves travel at the speed of light. This eliminates (or seriously constrain) many modifications of gravity and so could be seen as evidence of dark energy models (paper 1, paper 2, paper 3).
A: I think you have the main lines of evidence right. You didn't mention the overall age of the universe, but arguably this is part of your item 5. (The point being that if you don't include $\Lambda$ in a $\Lambda$CDM cosmology then even rough observations of the amount of matter and the amount of curvature cannot square with the age of the oldest stars.)
From these lines of evidence there is now a good overall case for a term like $\Lambda$ in the Einstein field equation. As I understand it, the case is not quite as strong as we would like, in that the data is not quite precise enough to make a "beyond reasonable doubt" or 5-sigma-like case, so it is useful to continue to accumulate evidence. There are complicating factors coming from the fact that the universe is not quite homogeneous and the local region is in motion compared to the average. There is also the well-known tension between different Hubble constant measurements, which suggests that there is some systematic effect somewhere which has not yet been properly understood. It could be something to do with calibration of standard candles and modelling of the effects of interstellar dust. But overall these look like things which will be ironed out as data gets more precise and it looks like $\Lambda$ will survive. Overall most physicists in this area would be prepared to bet that way.
Notice that I said $\Lambda$ rather than dark energy because the observations are consistent with a cosmological constant. But one should keep an open mind.
