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Kairos
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I don't understand the discovery of "accelerating expansion" in 1998 (hence the postulate of dark energy, etc..), because Hubble's old law already described exponentially accelerating expansion in 1929, right?
No. All Hubble’s law says is that velocity is proportional to distance. This is true in any expanding FLRW universe.Kairos said:Hubble's old law already described exponentially accelerating expansion in 1929, right?
Under Hubble's law the recession speed of objects varies with distance, but it may or may not vary with time. Whether it varies with time and how depends on the mix of matter, radiation and dark energy in the universe. You cannot match the observed history of the expansion rate without some dark energy, and the observed mix leads to a Hubble constant that increases with time.Kairos said:Hubble's old law already described exponentially accelerating expansion in 1929, right?
This is incorrect. The Hubble constant decreases with time towards a constant value in a universe dominated by a cosmological constant.Ibix said:the observed mix leads to a Hubble constant that increases with time.
Wrong, because ##k## is not constant in time--the proportionality between velocity and distance changes with time.Kairos said:Velocity dx(t)/dt is proportional to distance x(t), so dx(t)/dt=k*x(t), so x(t)=exp(k*t)
Ah - that makes sense. Thanks.Orodruin said:An accelerated universe refers to a universe where the second derivative of the scale factor, ##\ddot a##, is larger than zero.
Are you responding to my post #6? If so, you are wrong, the fact that ##k## is not a constant is highly relevant since it makes your claimed mathematical solution in terms of an exponential wrong. And that claimed mathematical solution is the basis of your claim in the OP.Kairos said:of course, but that's not relevant to my question.
This is certainly very important, but I pointed out that even if k had been constant, the expansion would still have accelerated.PeterDonis said:k is not a constant is highly relevant since it makes your claimed mathematical solution wrong
Actually, it does vary with time in every FRW solution except a pure de Sitter spacetime, i.e., pure dark energy, no matter or radiation at all.Ibix said:Under Hubble's law the recession speed of objects varies with distance, but it may or may not vary with time. Whether it varies with time and how depends on the mix of matter, radiation and dark energy in the universe.
You're looking at it backwards. The case with ##k## constant is a pure de Sitter spacetime, just dark energy (aka cosmological constant), no matter or radiation at all. This is the "pure" case of "accelerated expansion". Adding matter or radiation makes ##k## not constant, and may take away the acceleration, depending on the relative densities of matter/radiation as compared to dark energy. Until a few billion years ago, the density of matter in our universe was high enough that the expansion was decelerating, not accelerating.Kairos said:I pointed out that even if k had been constant, the expansion would still have accelerated.
Hubble's law certainly describes well de Sitter's universe, but beyond the details, I'm just saying that this law describes an acceleration of expansion. You seem to agree?PeterDonis said:You're looking at it backwards. The case with ##k## constant is a pure de Sitter spacetime, just dark energy (aka cosmological constant), no matter or radiation at all. This is the "pure" case of "accelerated expansion". Adding matter or radiation makes ##k## not constant, and may take away the acceleration, depending on the relative densities of matter/radiation as compared to dark energy. Until a few billion years ago, the density of matter in our universe was high enough that the expansion was decelerating, not accelerating.
Up until the late 1990s, cosmologists thought the expansion of our universe was decelerating now because they thought there was zero dark energy. That decelerating solution is the original one that Hubble's evidence was taken to support. What changed in the late 1990s was that evidence was discovered that the expansion of our universe is accelerating now, and had been until a few billion years ago. That required adding nonzero dark energy to our best current model in order to account for the data.
No. Hubble's law, as @Orodruin has already told you, is true in any expanding FRW universe, not just de Sitter spacetime; de Sitter spacetime is the particular solution where the proportionality between velocity and distance, what you call ##k##, is not a function of ##t##. If ##k## is a function of ##t##, you do not get exponential expansion. You might not even get accelerating expansion at all. But you will still get a solution in which Hubble's law is true.Kairos said:Hubble's law certainly describes well de Sitter's universe, but beyond the details, I'm just saying that this law describes an acceleration of expansion. You seem to agree?
Hubble’s original law said nothing about the time dependence of the proportionality constant. Just that observed velocity now is proportional to distance now. As such, it would allow any type of expansion. Accelerated, constant, or decelerated.Kairos said:Hubble's law certainly describes well de Sitter's universe, but beyond the details, I'm just saying that this law describes an acceleration of expansion. You seem to agree?
OK, if k is a constant of proportionality, then the expansion is exponential and therefore accelerated.PeterDonis said:particular solution where the proportionality between velocity and distance, what you call k, is not a function of t.
You are completely missing the point.Kairos said:OK, if k is a constant of proportionality, then the expansion is exponential and therefore accelerated.
You are not even responding to the point I made in post #15. Go back and read it again. Carefully.Kairos said:OK, if k is a constant of proportionality, then the expansion is exponential and therefore accelerated.
If k is not constant, this changes everything (but can we still talk about proportionality?).Orodruin said:You are completely missing the point.
While k could technically be a constant in time - Hubble's original law says nothing about this. It only states that, for relatively nearby galaxies, velocity is proportional to distance. The data Hubble had available only contained relatively nearby objects and so he could have concluded nothing about the time dependence of the constant that would later bear his name ... and not be a constant. In fact, in modern cosmology, it is often referred to as the Hubble parameter or Hubble rate - not the Hubble constant.
More precisely, if k is not constant in time. Which it isn't in our universe, as you were already told in posts #3, #5, #6, #9, #13, #16, #18, and #19.Kairos said:If k is not constant, this changes everything
Proportionality with respect to distance at an instant of time (more precisely an instant of FRW coordinate time).Kairos said:(but can we still talk about proportionality?).
You were already told that in post #5. Go back and read it.Kairos said:So how does the Hubble constant evolve over time according to the 1998 discovery?
No, decreases with time. See @Orodruin's post #5. What increases with time is ##\dot{a}##. But ##H## is ##\dot{a} / a##, and that decreases with time (asymptotically towards the constant de Sitter value associated with the dark energy density).Ibix said:the observed mix leads to a Hubble constant that increases with time.
Orodruin, Reading other threads on the same topic, I read in one of your posts, "In fact, in a dark energy dominated universe, H is constant.Orodruin said:And to continue on that theme. People contemporary with Hubble would most likely have assumed it to be decreasing with time simply due to gravitational attraction and - in the absence of gravity - that the objects would end up further and further away, thereby decreasing the constant.
You need to quote the specific post so we can see the context. The term "dark energy dominated" is ambiguous: many cosmologists describe our current universe as "dark energy dominated" because the expansion is accelerating, even though ##H## is not constant with time in our current universe, it is decreasing, as @Orodruin said in post #5 of this thread.Kairos said:I read in one of your posts, "In fact, in a dark energy dominated universe, H is constant.
Yes, and I suspect that's what you were actually saying in whatever other thread the OP read and quoted from out of context in post #24.Orodruin said:... and given enough time to evolve, it will approach a constant value given by the cosmological constant.
... or it was not worth making the distinction of being dominated by a cosmological constant or just having a cosmological constant ... I have probably said things like that a lot of times over the years.PeterDonis said:Yes, and I suspect that's what you were actually saying in whatever other thread the OP read and quoted from out of context in post #24.
If H will approaches a constant value given by the cosmological constant, there will still be a continuous acceleration ?(exponential).Orodruin said:... and given enough time to evolve, it will approach a constant value given by the cosmological constant.
Kairos said:If H will approaches a constant value given by the cosmological constant, there will still be a continuous acceleration ?(exponential).
If ##dH/dt = 0## then clearly ##\ddot a = aH^2 = \dot a^2/a## and therefore the universe is accelerating.Orodruin said:An accelerated universe refers to a universe where the second derivative of the scale factor, ##\ddot a##, is larger than zero. This is not the same thing as ##dH/dt > 0## as ##H = \dot a/a## implies
$$
\frac{dH}{dt} = \frac{\ddot a}{a} - \frac{\dot a^2}{a^2} = \frac{\ddot a}{a} - H^2
$$
Thus, the Hubble constant can indeed be negative even if ##\ddot a > 0##.
If ##H## is asymptotically approaching a constant value, the expansion of the universe is asymptotically approaching exponential. At what point during that process the expansion becomes accelerating depends on the details, but it will become accelerating at some point, and once it does, it will stay accelerating.Kairos said:If H will approaches a constant value given by the cosmological constant, there will still be a continuous acceleration ?(exponential).
Thank you. I understand that the densities decrease and lambda is left alone at the end. What I don't understand is why rho and lambda affect H in the same direction. I would have thought - and + signs since lambda increases expansion while rho slows it down.DAH said:In a spatially flat universe the Hubble parameter is given by $$H^2=\frac{8\pi G(\rho_m+\rho_r)}{3}+\frac{\Lambda c^2}{3}$$
##\rho_m## and ##\rho_r## decrease with time and ##\Lambda## stays constant, so the Hubble parameter decreases with time. In the remote future ##\rho_m## and ##\rho_r## will tend to zero and the Hubble parameter will become a constant given by $$H=\sqrt{\frac{\Lambda c^2}{3}}$$
First, what you mean by "expansion" here is not ##H##, its ##\dot{a}##. They're not the same.Kairos said:What I don't understand is why rho and lambda affect H in the same direction. I would have thought - and + signs since lambda increases expansion while rho slows it down.
yes, that's why I spoke of expansion in this post, and not acceleration. dark energy favors expansion contrary to gravityPeterDonis said:First, what you mean by "expansion" here is not ##H##, its ##\dot{a}##. They're not the same.
Second, when you say "lambda increases expansion while rho slows it down", what you mean is that dark energy, by itself, causes expansion to accelerate, while rho, by itself, causes expansion to decelerate. But the equation for ##H^2##, by itself, tells you nothing about acceleration or deceleration. What you need to look at is the equation for ##\ddot{a}##.