Hubble's Law and Hubble's Constant

[tanvi]

Hi there! I'm having some trouble understanding Hubble's Law and Constant and would be grateful for some help.

So Hubble's Law shows that the most distant galaxies have the greatest red-shift, as they are moving away from us fastest. This shows that all galaxies are moving away from each other and therefore that the Universe is expanding. Would it be correct to say that the increased area of space to be stretched (ie expanding space) between us and the most distant galaxies results in their increased velocity from us?

As I understand, Hubble's Constant (at about 42 miles per second per megaparsec) shows that a galaxy one megaparsec away is receding from us at a velocity of 42 miles per second. Therefore, is Hubble's Constant increasing as the universe's expansion accelerates?

Thanks very much in advance. Sorry that my questions are so basic!

 

[RyanH42]

Yeah.$V=HD$ If you differantiate it respect to time find acceleration you get $dV/dt=(dH/dt)D+H(dD/dt)$.If you assume Hubble constant is not depend time then $A=HV$ this means speed is increasing then A will be increase.But If we assume H changes with time which tis is true.

 

[marcus]

... Therefore, is Hubble's Constant increasing as the universe's expansion accelerates?
...!

No, Hubble's "constant" has been decreasing since the start of expansion and it is expected to continue decreasing but to level out in the long term.

The Hubble growth rate is an instantaneous fractional growth rate, like an interest rate on a savings account at the bank. Distance growth is proportional to size, at any given time in history distances are increasing by some fraction or percentage of their size per some small time increment Δt.

A constant or even slightly declining interest rate will still cause your savings to "accelerate" (grow by an larger dollar amount each year) because the principal is growing. As long as the bank reduces the interest rate GRADUALLY ENOUGH your savings will "accelerate" in that sense.

So Hubble growth rate is DECLINING but with time it has declined more and more gradually and is tending to level out at a positive rate. As the decline has become more gradual the growth of distances has tended to accelerate. So far it is not very dramatic. Here is a picture, in this graph the present is at 0.8 on the x axis. The time unit is a "zeit" = 17.3 billion years. The gold curve is the Hubble rate. The blue curve is the growth curve of a particular distance---curves for other distances would be proportional. They all have an inflection point at time 0.44.

 

[tanvi]

Thank you so much, marcus. I think I understand now. I'm not too sure about the graph (don't yet have the maths I guess), but I'll do some further research to understand it.

So just to clarify, in the analogy of compound interest on a savings account, the principle can 'accelerate' with a declining interest rate, because the principle itself is becoming larger... For example, £200 is placed in a savings account at a compound interest of 5%. After four years, the amount in the savings account is £243.10. Even if the compound interest is decreased at this point, eg to 4.9%, the savings will still 'accelerate' because 4.9% of 243.1 is a larger amount than 4% of £200. Is this the way it works?

In the context of an expanding universe, the Hubble 'constant' can decrease gradually in an expanding universe, because as time goes on there is a greater distance between different galaxies. So even when the universe's expansion is accelerating, the recessional velocity of distant galaxies decreases because there is more space to be stretched between us and them?

And in an accelerating universe, Hubble's 'constant' decreases at a more gradual rate than in a universe whose expansion is constant?

Sorry if this doesn't make any sense!

 

[marcus]

...

So just to clarify, in the analogy of compound interest on a savings account, the principle can 'accelerate' with a declining interest rate, because the principle itself is becoming larger... For example, £200 is placed in a savings account at a compound interest of 5%. After four years, the amount in the savings account is £243.10. Even if the compound interest is decreased at this point, eg to 4.9%, the savings will still 'accelerate' because 4.9% of 243.1 is a larger amount than 4% of £200. Is this the way it works?
...
And in an accelerating universe, Hubble's 'constant' decreases at a more gradual rate than in a universe whose expansion is constant?

Sorry if this doesn't make any sense!

It makes sense to me. It sounds to me as if you understand.

In the picture the blue curve shows the way any given distance grows (like between two galaxies neither of which is moving relative to the space around them).
You can see by looking at the blue curve that there is no "acceleration" until around time 0.45 (actually the exact point is closer to 0.44 but it's hard to tell). At that point the curve changes from convex to concave upwards, the slope starts to steepen. this is "acceleration" of distance growth.

The reason for deceleration, at first, is that the gold curve (the percentage growth rate) is declining TOO STEEPLY
The reason for changing over to acceleration (between time 0.4 and 0.5) is the gold curve now declines LESS STEEPLY.

If the bank cuts your interest rate too rapidly, you savings will not accelerate.
If the bank cuts your interest rate more gradually, slowly enough, you will see "acceleration"---savings grow by more £££ each year.

So we need to look at how steep the gold curve (which tracks the Hubble parameter, the distance growth rate) is decreasing.

The cosmological constant Λ is related to the long term growth rate H_∞.

If Λ = 0 then H_∞ = 0. So naturally we will see the H decline more steeply because it is going to zero.

But observations indicate that Λ is positive and more to the point H_∞ is positive. so H is still declining as the gold curve shows, but it levels out at a positive level instead of going to zero. that is what makes acceleration in the blue curve kick in at a certain point.