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In the video linked in the Space.com article What Would It Mean for Astronomers If the WFIRST Space Telescope Is Killed?, (available in YouTube as WFIRST: The Best of Both Worlds) after 02:00 Caltech cosmologist Yun Wang says:

WFIRST will allow us to potentially make ground-breaking discoveries, finding out what dark energy is. So this will tell us if dark energy is an unknown form of energy, or if its a modification of general relativity.

Without going too deep in to GR, how might that happen?

Question: What kinds of WFIRST observations and interpretations would be used to potentially address the nature of dark energy?

Based on comments, I'll reclassify the following as an afterthought:

Is this in any way related to news about a recent observational non-exclusion of Erik Verlinde's theories (the guy who's "Trying to kill dark matter"), discussed in Margot Brouwer's 2016 video Verlindes new theory first tested with weak gravitational lensing about the VLT Survey Telescope kilo-degree survey results (see also Cosmos Magazine's New theory of ‘dark gravity’ passes first test, but Einstein’s still on top, or am I mixing apples and oranges?

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  • $\begingroup$ You aren't mixing apples and oranges. You are mixing blueberries and blue whales. Just because blueberries and blue whales both start with blue doesn't mean they have anything to do with one another. The same goes for dark energy and dark matter. WFIRST was proposed without a thought toward Verlinde's entropic gravity (which is a bit fringy). The goal is to get a better handle on what dark energy is. Dark energy is a placeholder name for the accelerating expansion of the universe, so that "better handle" means a better understanding of the accelerating expansion. $\endgroup$ – David Hammen Feb 14 '18 at 5:31
  • $\begingroup$ @DavidHammen thanks as always, and happy to see you here; but are you absolutely sure that WFIRST data can not be used to distinguish between theories attempting to explain the phenomenon? I'm not asking about the original proposal. If Verlinde's theory is a bit "fringy", wouldn't one want to try to rule it out decisively, and could what Yun Wang is talking about potentially include such an exclusion? $\endgroup$ – uhoh Feb 14 '18 at 5:40
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    $\begingroup$ Re If Verlinde's theory is a bit "fringy", wouldn't one want to try to rule it out decisively -- No, for several reasons. First off, there are two billion reasons, the number of US dollars WFIRST will cost. While NASA does spend small amounts on fringy ideas, it does not spend tens of millions, let alone two billion. Secondly, that's not how science works. Suppose I came with my own theory of gravity. The US government is not going to spend any money to rule out that concept. $\endgroup$ – David Hammen Feb 14 '18 at 6:01
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    $\begingroup$ @DavidHammen okay! Insert quip about Ponds and Fleischmann, $4.5 million, sate of Utah here ;-) $\endgroup$ – uhoh Feb 14 '18 at 6:08
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The WFIRST's study of dark energy is pretty straight forward. It's no different than the previous ways of studying dark energy. It's not studied directly, cause we can't see it or re-create it, it's studied by the effect it has on the relative motion of distant galaxies.

WFIRST will get much more accurate readings on the relative velocity of galaxies both relative to each other in their neighborhood and relative to us, this will get a significantly more accurate estimate for dark energy than was previously possible and questions can be answered such as, is it speeding up.

NASA's WFIRST Web page discusses this in more detail

The High Latitude Spectroscopic Survey will measure accurate distances and positions of a very large number of galaxies. By measuring the changes in the distribution of galaxies, the evolution of dark energy over time can be determined. The High Latitude Survey will measure the growth of large structure of the universe, testing theory of Einstein's General Relativity.

Type Ia Supernovae (SNe) Survey uses type Ia SNe as "standard candles" to measure absolute distances. Patches of the sky are monitored to discover new supernovae and measure their light curves and spectra. Measuring the distance to and redshift of the SNe provides another means of measuring the evolution of dark energy over time, providing a cross-check with the high latitude surveys.

High Latitude Imaging Survey will measure the shapes and distances of a very large number of galaxies and galaxy clusters. The shapes of very distant galaxies are distorted by the bending of light as it passes more nearby mass concentrations. These distortions are measured and used to infer the three-dimensional mass distribution in the Universe. This survey will determine both the evolution of dark energy over time as well as provide another independent measurement of the growth of large structure of the universe.

Dr. Yun Wang's comments appear to be written for a Science-TV show and/or youtube video, and video like those are designed for a general audience so I wouldn't analyze that particular sentences meaning too closely.

As for Erik Verlinde's . . . ideas. I'm not a fan of his proposals, but the answer is mostly the same. WFIRST will get more accurate readings and better readings should help get more specific answers.

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  • $\begingroup$ Thank you very much! I'm guessing the spectroscopic survey uses red shift to infer distances, but how would an imaging survey to this? (if it's not a quick/easy answer, I can ask later, separately) $\endgroup$ – uhoh Feb 14 '18 at 8:33

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