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In theories with dark energy, an energy density is just a property of space — even “empty” space devoid of matter and radiation. It’s closely related to Einstein’s old idea of a “cosmological constant”. Dark energy isn’t something material that comes from somewhere. As space expands, you simply get more dark energy because there is more volume; a dark energy ...


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Velocities of galaxies When galaxies are gravitationally bound to each other in groups or clusters, they move on more or less elliptical orbits in the common gravitational potential from all the other galaxies (as well as all the dilute intracluster gas which is also a significant part of the total mass). I say "more or less" because galaxies do ...


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Assuming (as this chart does) that the universe contains only dust and dark energy, the scale factor satisfies $$\dot a^2 = H_0^2 \left( Ω_{Λ,0}\, a^2 + Ω_{k,0} + Ω_{m,0}\, a^{-1} \right)$$ where $Ω_k = 1 - Ω_m - Ω_Λ$. (The exponents are $-1{-}3w$ where $w$ is the equation of state parameter.) The boundary of the $κ=\pm1$ regions is just the line $Ω_{k,0}=0$....


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There is a close analogy: If you jump say 3 feet high, you can calculate your speed as a function of you position between 0 to 3 feet high. Then you wonder what is your speed at 6 feet high during your jump. The good old high school Newtonian mechanics offers an answer: your speed was imaginary at 6 feet high according to the total energy conservation ...


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One way to interpret Friedmann equation with an imaginary Hubble parameter is as arising from some solution with Euclidean metric signature. One class of such solution, termed “Euclidean wormholes” consists of two large asymptotic regions connected by a “throat”. Many FLRW cosmologies analytically continued into an imaginary time become such Euclidean ...


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RW assumes that matter is uniformly distributed over the spatial slices, and elliptic, flat, and hyperbolic geometries have very different distributions, in terms of the amount of matter within a given distance of any given point. There's no way you could move the matter around to be homogeneous in a different geometry without violating homogeneity in the ...


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Yes. "The curvature of the universe" is an imprecise term, and describing the curvature of a general four-dimensional spacetime takes 20 numbers at every point. But I'll assume that your phrase should mean the Ricci scalar curvature $R$, which is a single number at each point that is a kind of average curvature of spacetime (where the averaging is ...


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No The curvature parameter $k$ of the universe remains constant throughout its evolution. If the universe is open ($k < 0$) , it will stay open, and if it is closed ($k > 0$), it will stay closed. That's because the amount of matter-energy of the universe is conserved, so if the density is greater than critical now it will forever be greater than ...


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As far as we know, it's a coincidence. $1/H = a(t)/a'(t)$. In the early radiation-dominated era, $a(t) \propto t^{1/2}$, so $1/H$ is 2 times the actual age of the universe. In the later (but pre-modern) matter-dominated era, $a(t) \propto t^{2/3}$, so $1/H$ is 1.5 times the actual age. In the future dark-energy-dominated era, $a(t) \propto e^{t/t_0}$ where $...


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Yes. The answer is kind of obvious, but if you do not see this, just note that the FLRW metric is: $$g=-c^2dt^2+a(t)^2d\Sigma^2_t,$$ where $d\Sigma^2_t$ is metric induced up to the scale factor on the spacelike hypersurfaces of constant time $t$. This metric is independent of $t$, from which it follows, that if you have a curve living on one of this ...


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