Accelerated Hubble expansion -- Is the second derivative positive?

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Summary:

Which values takes the third and forth derivative of distances in universe? Acceleration needs third derivative at least positive. Dark energy as is most considered now leads to increasing third derivative.
Since distances increase, their first derivative which is velocity (Hubble constant) should be positive if not increasing too. Accelerated expansion needs the velocity to increase. What about the third derivative which is acceleration? An accelerated universe could have third derivative (called jerk) constant positive, increasing positive or even decreasing positive, even going to zero gradually. I made a quick graph table that shows what I am talking about.

Dark energy, by increasing with space (voids) as is most considered now, leads to increasing second derivative and possibly positive constant third derivative. Do we have any clue by observational data about these?

PeterDonis
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There is a key distinction here, between the second derivative of the scale factor (what you are calling "distances") and the derivative of the Hubble constant. They aren't the same.

The scale factor is ##a##, so the "velocity" related to it is ##da / dt##, or ##\dot{a}##, and the "acceleration" related to it is ##d^2 a / dt^2##, or ##\ddot{a}##. The fact that this acceleration is positive (greater than zero) is what we mean when we say that the expansion of the universe is accelerating.

The Hubble constant ##H##, in terms of the scale factor, is not ##\dot{a}##; it's ##\dot{a} / a##, i.e., the velocity divided by the scale factor. So the derivative of ##H## is ##\dot{H} = \ddot{a} / a - \left( \dot{a} / a \right)^2##. And this quantity is negative (the negative term on the RHS has larger magnitude than the positive term), even though ##\ddot{a}## is positive.

The third derivative of the scale factor, ##\dddot{a}##, is also positive now; but its value is decreasing (so the fourth derivative is negative). [Edit: the crossed out statement is not correct; see my exchange with kimbyd below.] In terms of ##H##, while ##H## is decreasing, its rate of decrease is slowing, and it will eventually approach, not zero, but a constant positive value (the value associated with the dark energy density). In a universe containing nothing but dark energy, ##H## would be constant and the third and all higher derivatives of ##a## would be zero.

Last edited:
The Count, Ibix and fresh_42
kimbyd
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Not quite. With H positive and constant in the far future, all derivatives of the scale factor will be positive.

This is because with H constant, the scale factor changes at the following rate:

$$a(t)=e^{H_0t}$$

All derivatives of an exponential function are positive (or alternating if the exponent is negative).

The Count
PeterDonis
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With H positive and constant in the far future, all derivatives of the scale factor will be positive.
Yes, I misstated it. First and higher derivatives of ##H## will be zero in a universe with pure dark energy; but all derivatives of ##a## will be positive.

The Count

You are right that Hubble constant is not exactly just velocity, but velocity over distance and the derivative is more complicated than just the third derivative of the scale factor.

You mentioned that the third derivative is also positive and decreasing eventually to approach a constant positive value. In my graphs I think it is the last line, No4. Can you give a reference on that?

If this is the case then it doesn't fit with dark energy models that have constant dark energy per space, and as space expands dark energy is getting bigger with it, probably exponentially. In this case we should have line No3, with increasing second derivative and constant positive third derivative. The universe should then be exponentially accelerating.

PeterDonis
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If this is the case then it doesn't fit with dark energy models that have constant dark energy per space, and as space expands dark energy is getting bigger with it, probably exponentially. In this case we should have line No3, with increasing second derivative and constant positive third derivative. The universe should then be exponentially accelerating.
The density of dark energy (and of matter and energy in general) is what drives the dynamics. The "total amount" does not. So your reasoning here is incorrect.

PeterDonis
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You mentioned that the third derivative is also positive and decreasing eventually to approach a constant positive value.
If you are referring to my statement in the last paragraph of post #2, please see my exchange with @kimbyd (and I have edited post #2 to reflect that exchange).

With H positive and constant in the far future, all derivatives of the scale factor will be positive.
H does not seem to be constant. It breaks down to approximately 1/(age-of-universe), or at least it would if scale factor was linear. Point is, it is only constant because it's known to 1 significant digit and humans exist for too short span of time to witness a change in it. It was far larger in the past and will continue to slow.

For example, a thousand years after the big bang, something a megaparsec away would be massively beyond the size of the visible universe at the time and receding at thousands times light speed. A constant H would have it receding at 70 km/sec.

So what do you mean by H being constant in the far future? It's going down even though expansion is accelerating.

Bandersnatch
So what do you mean by H being constant in the far future? It's going down even though expansion is accelerating.
What was meant is what was written. It's going down towards a constant positive value (around ~57 km/s/Mpc, if LCDM is correct):

Technically speaking, it's always going to be decreasing, but the downwards slope quickly (in cosmological terms) becomes negligible and the expansion approximates to a de Sitter universe.

The Count and Halc
The density of dark energy (and of matter and energy in general) is what drives the dynamics.
Yes, you are correct. But since the density of matter (dark or not) reduces with universe expansion, repulsive action of dark energy will be getting stronger and stronger in the comparison between them.

This means that dark energy's gradual stronger domination should cause exponential expanion with the third derivative "constant positive" or "positive and decreasing"? If decreasing it approaches a fixed value or zero?

What was meant is what was written. It's going down towards a constant positive value (around ~57 km/s/Mpc, if LCDM is correct):
View attachment 268724
Technically speaking, it's always going to be decreasing, but the downwards slope quickly (in cosmological terms) becomes negligible and the expansion approximates to a de Sitter universe.
So if I understood correct your answer with fixed Hubble parameter in the far future we should have (as scale factor is concerned) the forth line in my graph.
1. Space exponential expansion
2. The speed of that expansion first exponential and later linear (1st derivative),
3. Acceleration of expansion increasing in the begining, and reaching a fixed point in the future (2nd derivative),
4. As for the jerk (3rd derivative) positive in the begining reaching zero later (lineary or exponentially?).
After you correct any misconseptions of mine, can you please provide a reference?

Finally, my initial question was " Do we have any clue by observational data about these? "

PeterDonis
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the density of matter (dark or not) reduces with universe expansion, repulsive action of dark energy will be getting stronger and stronger in the comparison between them
Yes, that is why the expansion of the universe started accelerating a few billion years ago.

This means that dark energy's gradual stronger domination should cause exponential expanion
It means that exponential expansion should be the limit towards which the dynamics tend, yes.

with the third derivative "constant positive" or "positive and decreasing"?
Exponential expansion means the scale factor is an exponential function of time. All derivatives of an exponential function of time are also exponential functions of time (just with different constants in front).

The Hubble constant with exponential expansion is constant in time, so all derivatives of it are zero. That is what the diagram @Bandersnatch posted is showing the dynamics approaching as a limit.

PeterDonis
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my initial question was " Do we have any clue by observational data about these? "
Of course we do. Ned Wright's article on curvature in the Hubble diagram gives a good overview:

http://www.astro.ucla.edu/~wright/sne_cosmology.html

"Curvature in the Hubble diagram" is just another way of saying "data on the derivatives of the Hubble constant as well as the Hubble constant itself".

Exponential expansion means the scale factor is an exponential function of time. All derivatives of an exponential function of time are also exponential functions of time (just with different constants in front).
I am not sure I agree with that. Exponential expansion doesn't mean all the time e power that never goes away as you differentiate. It could be x^3 that goes to 3x^2, 6x, 6 and finally 0. That's why I keep asking which line of my graph charts you believe is best describing universe's expansion and you insist on not giving any answer. If you think that none of the lines are correct, please draw or describe a new one.

If exponential with e power is correct (according to ΛCDM) then we are concluding an expansion with accelerated acceleration (since all derivatives would be positive and exponential) which doesn't fit with your statement that density of dark matter is stable and this is what defines the dynamics. Because while the dynamics of expansion would indeed increase, from one point on (that all other energies are indifferent) the density will remain the same and the expansion rate should stop being accelerated (third derivative should be zero or approaching asymptoticaly zero).

No it couldn't because ##x^3## is not an exponential function: https://en.wikipedia.org/wiki/Exponential_function
ok, the correct word could be power function that has exponential growth. I think we still are in exponential expansion.

All the rest in my question I hope they are correct.

Bandersnatch
I think there's some misunderstanding here. The lambda-dominated expansion is exponential of the form shown in post #3. It comes from the Friedmann equations; it can't be a power function or anything else - there's no wiggle room. You can find how it is derived, step-by-step, e.g. in Ryden, chapter 5.
It's hard to relate this to your charts, since - as was already discussed - they purport to show derivatives of the Hubble parameter, whereas what's being talked about are derivatives of the scale factor. Which in the lambda-dominated universe are all exponential functions, just with different constants in front. So I guess those graphs showing the exponents would fit, but then again, you label them oddly (what's exp^2 supposed to indicate?), and they're all in the column for the scale factor.

PeterDonis
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2019 Award
Exponential expansion doesn't mean all the time e power that never goes away as you differentiate. It could be x^3 that goes to 3x^2, 6x, 6 and finally 0.
No, that's not exponential expansion, that's power law expansion.

Exponential expansion, as has already been said, means the scale factor goes like ##a = e^{H_0 t}##, where ##H_0## is a constant (and will be related to the cosmological constant aka dark energy density). Taking derivatives with respect to ##t## does not change the exponential function at all, it just puts more and more factors of ##H_0## out in front.

If exponential with e power is correct (according to ΛCDM) then we are concluding an expansion with accelerated acceleration (since all derivatives would be positive and exponential) which doesn't fit with your statement that density of dark matter is stable and this is what defines the dynamics. Because while the dynamics of expansion would indeed increase, from one point on (that all other energies are indifferent) the density will remain the same and the expansion rate should stop being accelerated (third derivative should be zero or approaching asymptoticaly zero).
I have no idea how you are coming up with any of this, but it's wrong. If dark energy is constant, then ##H_0## is constant in the exponential above. Which, as noted above, means the scale factor is an exponential function of time, forever, and so are all of its derivatives.

the correct word could be power function that has exponential growth
There is no such thing.

All the rest in my question I hope they are correct.
They aren't. See above.