Is Cosmos seriously using that exact number? Egads... if they are, don't take it too seriously, but otherwise they're probably conceptually correct.
How do we know it was dark energy?
In cosmology, the ΛCDM model fits numerous observations, most notably those observed by the WMAP satellite, but also others, to the Friedmann-Robertson-Walker family of solutions of general relativity, which describe a homogeneous and isotropic universe. The $\Lambda$ refers to the cosmological constant, which is the simplest model of dark energy, and is equivalent to an intrinsic energy density of the vacuum--hence, it must be Lorentz-invariant, and since time and space mix in a Lorentz transformation, the only way for that to be so is for its pressure to be equal to exactly the negative of its energy density. With $\Lambda>0$, the pressure is negative, and this is what's responsible for the outward gravitational acceleration of the universe.
Given we can't define or detect dark energy, why do we believe this is the cause?
The cosmological constant was defined eight decades before the discovery of cosmic acceleration, and we believe it to be the cause because it's theoretically very simple and it fits observations. There are other models for dark energy, though.
How do we even know this new acceleration happened?
Cosmological evolution is described by the Friedmann equations, which has the principal contribution of radiation, matter, and dark energy, in addition to spatial curvature of the universe. As the universe expands, both radiation and matter get diluted, but dark energy does not. Therefore, if the universe keeps expanding, the proportion of dark energy becomes dominant over both radiation and matter.
Most likely, the article is simply an exaggerated reference to the transition between the matter-dominated era and the dark-energy-dominated era. The particular number $6,771,500,000$ years seems comically precise for much of anything in cosmology, but since the nine-year best-fit WMAP+eCMB+BAO+H₀ (see above WMAP link) for the age of the universe is $13.772\pm0.059\,\mathrm{Ga}$, I propose the following explanation:
- The author hears that the age of the universe is $13.772$ billion years, and completely ignores any error bounds on that estimate.
- The author hears from somewhere else that the dark-energy-dominated era began when the universe was around $7$ billion years old, and again treats this number as exact.
- The author then subtracts the two.
This accounts for all but the last $0.5\,\mathrm{Ma}$, which about one-ten-thousandth. I conclude that my theory of the origin of the number agrees rather well with the empirical evidence of the author's words. Perhaps the $0.5$ as the next digit was rationalized as some sort of midway-point in the author's mind.
As to where $7$ billion years comes from, I have two guesses.
- In cosmology, the Hubble parameter is sometimes taken to be unitless, by scaling to $100\,\mathrm{km/s/Mpc}$. For example, we sometimes say that $h = 0.704$ when we mean that $H_0 = 70.4\,\mathrm{km/s/Mpc}$. Thus, $100\,\mathrm{km/s/Mpc}$ is kind of a de facto standard of comparison. It might be a coincidence, but it's rather suspicious that if one ignores the contribution of radiation (which is justifiable, since it's negligible except in the very early universe) and pretends that $H_0 = 100\,\mathrm{km/s/Mpc}$, then the age of the universe when the matter and dark energy densities are equal would be
$$\frac{2\ln(1+\sqrt{2})}{3\sqrt{1-\Omega_\text{M}}}\frac{1}{H_0}\bigg\vert_{\Omega_\text{M} = 0.2865} \approx 0.70H_0^{-1}\bigg\vert_{h = 1.0} \approx 6.8\,\mathrm{Ga}$$
I'm using the Ryden reference given in wiki for matter-dominated era above. Thus, I think someone did a guesstimate using a nice, round number for the Hubble parameter and rounded to $7$ billion years.
- The transition between matter-dominated and dark-energy-dominated is a somewhat arbitrary matter of convention. We could take it mean when matter and dark energy densities are equal, as above, but on the cosmological scale, pressure is three times as important as energy density (cf. Friedmann equations). Thus, we could take it instead as the time when the matter density was twice the dark energy density. This would be earlier than the $9.8\,\mathrm{Ga}$ given in the Ryden reference, but arguably more appropriate because the Friedmann equations would predict zero acceleration, i.e., it would be the transition between inward and outward acceleration. I haven't calculated when that would be, but this sort of conventional fuzziness is might be responsible for the discrepancy between the wiki's matter-dominated era and the dark-energy-dominated era pages. Although the latter claims $5\,\mathrm{Ga}$ instead, rather than $7\,\mathrm{Ga}$ (but it doesn't explain the source).
So Cosmos may or may not be right about dark-energy-dominated era beginning when the universe about $7$ billion years old, but concluding that it happened $6.7715\,\mathrm{Ga}$ ago is certainly taking some numbers as too exact to be justifiable.
Something in the red shift?
I think at this point you're better off making a separate question on specific issues with the ΛCDM model, if you have any. For the purposes of this question, I'm taking ΛCDM as a given, as almost certainly Cosmos is doing as well.
Yes that is a lot of questions, but the main question is how do we know an accelerated expansion happened at that time?
The standard cosmological model is that dark energy is a cosmological constant. Thus, its energy density does not change. It then becomes straightforward that at some point, cosmological expansion makes matter too dilute to compete with dark energy, which is when dark energy starts dominating. It was there all along, but before then, matter was dense enough to be dominant instead.
