Hubble's Constant

[Alan Grayson]

It's measured about 70 km/sec/megaparsec. This is a direct measurement using red shift to measure recessional velocity, and different standard candles depending on the distance. So, at a distance of one megaparsec, the expansion rate is 70 km/sec; at two megaparsecs the expansion rate is 140 km/sec; and so on. This suggests the rate of expansion is greater as we go back in time; or conversely, that the rate of expansion is slower as we go forward in time. How is this reconciled with the 1998 measurements that the rate of expansion is actually speeding up? AG

[Lawrence Crowell]

Red shift data from supernovae SN1 give H = 74km/sec-Mpc. The velocity of a region at distance d is v = Hd. Using the CMB data H = 70km/sec-Mpc. The two data points appears distinct at a rather high sigma.

LC

[John Clark]

On Fri, Dec 2, 2022 at 5:47 AM Alan Grayson <agrays...@gmail.com> wrote:

> Regardless, both results contradict the slowing of the expansion. AG

Well of course it does, the expansion is accelerating.  

John K Clark    See what's on my new list at  Extropolis

ytq

 

[Alan Grayson]

The definition of the Hubble Constant implies the rate of expansion slows as time advances. I don't see that the two values contradict this conclusion. AG

[Lawrence Crowell]

On Friday, December 2, 2022 at 4:53:22 AM UTC-6 agrays...@gmail.com wrote:

The definition of the Hubble Constant implies the rate of expansion slows as time advances. I don't see that the two values contradict this conclusion. AG

The FLRW expansion factor is exp(Hd/c), where Taylor's theorem means exp(Hd/c) = 1 + Hd/c + 1/2(Hd/c)^2 + ... . For small distance d we have exp(Hd/c) =~ 1 + Hd/c, and for Hd = v this is exp(Hd/c) =~ 1 + v/c. To convert this to a time dependent expansion then d = ct.

[Jesse Mazer]

The wiki page on the Hubble parameter also says in the section at https://en.wikipedia.org/wiki/Hubble%27s_law#Time-dependence_of_Hubble_parameter that the expansion seems to be accelerating in such a way that the first derivative of the scale factor a(t) is increasing over time but the Hubble parameter H(t) is decreasing, and that this has the implication "The recession velocity of one chosen galaxy does increase, but different galaxies passing a sphere of fixed radius cross the sphere more slowly at later times". There's a more technical discussion of how these parameters are defined at https://en.wikipedia.org/wiki/Accelerating_expansion_of_the_universe#Technical_definition which mentions that physicists define "accelerating expansion" specifically in terms of the second derivative of the scale factor being positive, it doesn't require an increasing Hubble parameter.

On Fri, Dec 2, 2022 at 4:15 AM Alan Grayson <agrays...@gmail.com> wrote:

It's measured about 70 km/sec/megaparsec. This is a direct measurement using red shift to measure recessional velocity, and different standard candles depending on the distance. So, at a distance of one megaparsec, the expansion rate is 70 km/sec; at two megaparsecs the expansion rate is 140 km/sec; and so on. This suggests the rate of expansion is greater as we go back in time; or conversely, that the rate of expansion is slower as we go forward in time. How is this reconciled with the 1998 measurements that the rate of expansion is actually speeding up? AG

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[Alan Grayson]

TY. I think I get it, but I must devote more time to fully appreciate your comment. AG

[Lawrence Crowell]

On Friday, December 2, 2022 at 6:50:26 PM UTC-6 jessem wrote:

The wiki page on the Hubble parameter also says in the section at https://en.wikipedia.org/wiki/Hubble%27s_law#Time-dependence_of_Hubble_parameter that the expansion seems to be accelerating in such a way that the first derivative of the scale factor a(t) is increasing over time but the Hubble parameter H(t) is decreasing, and that this has the implication "The recession velocity of one chosen galaxy does increase, but different galaxies passing a sphere of fixed radius cross the sphere more slowly at later times". There's a more technical discussion of how these parameters are defined at https://en.wikipedia.org/wiki/Accelerating_expansion_of_the_universe#Technical_definition which mentions that physicists define "accelerating expansion" specifically in terms of the second derivative of the scale factor being positive, it doesn't require an increasing Hubble parameter.

That is the case. The Hubble constant is determined by the cosmological constant, and this gives an exponential law for the expansion of the universe. For a distance d the law for velocity is v = exp(Hd) - 1 = Hd + (Hd)^2/2 + ... , where for small enough d = distance v = Hd, which is the classic Hubble law. However, we may be faced with a variable Hubble constant, and data might suggest it is increasing. This means the accelerated expansion will asymptote to a divergence in a finite time in the future. The exponential acceleration will itself increase so that galaxies are shredded, then star systems, then stars, then planets, then atoms and hadrons as everything approaches a singularity with temperature T --> 0. It could be that in a few trillion years the entire universe will reach this singularity. The discrepancy between the CMB and SN1 data is beginning to suggest something odd about the expansion of the universe. It is not just dark energy, but phantom energy and the whole universe will reach a big rip.

LC