Orbit Dynamics in an Expanding Universe

[Buzz Bloom]

After learning a bit from the thread "The Dark Sky Ahead", I have tried to think through the dynamics of gravitational interactions in a simplified scenario. Unfortunately the GR math of the problem is over my head. My attempt below, using a simpler approach, leaves me quite confused.

If my analysis below is correct, this would seem to suggest that the approach that is commonly used, comparing escape velocity with Hubble velocity to determine if a system is bound with respect to the universe expansion, is wrong.

Consider an (almost) empty universe expanding exponentially with a single point mass and a test particle.

Let p be the test particle.
Let P be the point M.
Let M be the mass of P.
Let R be the distance between P and p.
Let h be the fixed constant value of the Hubble constant.
Let G be the Gravitational constant.

The questions I want to investigate are about stable configurations.

Scenario 1. p is stationary, that is, not moving with respect to P

Case (a): escape velocity = Hubble velocity
The square of the escape velocity

V_esc² = 2 G M / R​

The square of the radial velocity of p relative to P due to the expanding universe

V_exp² = h² R²​

The value of R for which these two velocities squared values are equal is

R_1a = (2 G M / h²)^1/3​

Question 1a:

If p is undisturbed by any outside force, is this static configuration with R = R_1a possible. (I understand that if it is, any disturbance would cause p to eventually move to infinity.)​

Case (b): gravitational acceleration = Hubble acceleration
The gravitational acceleration is

A_g = G M / R²​

The Hubble acceleration is what p would experience relative to P if M = 0. It is

A_h = h² R​

The value of R for which these two accelerations are equal is

R_1b = (G M / h²)^1/3
​

Question 1b:

Same as question (1a) except that R = R_1b.​

Note: Clearly Q1a and Q1b cannot both be answered "yes".Scenario 2. p is in an orbit about P.

Question 2a: Is a circular orbit possible?

Question 2b: If so, is the orbital velocity affected by the universe expansion?

Question 2c: If so, how?

The following are my guesses at answers.

Guess (2a): yes.

Guess (2b): yes.

Guess (2c):
The square of the orbital velocity with h = 0 is

V_orb² = G M / R​

The square of the radial velocity of p relative to P due to the expanding universe is

V_exp² = h² R²​

The total velocity squared is the sum

V_tot² = G M / R + h² R²​

The square of the escape velocity is

V_esc² = 2 G M / R​

V_tot = V_esc implies

G M / R = h² R²,​

and the value of R is

R_2c = (G M / h²)^1/3​

Note: R_2c = R_1b

My guess is that the inward gravitational acceleration must equal the sum of the outward accelerations: Hubble plus centrifugal.

G M / R² = h² R + V_orb² / R
V_orb² = G M / R - h² R²​

When V_orb = 0,

G M / R = h² R²​

This is the same result as in Scenario 1 Case (b).

The analysis in Scenario 2 seem to suggest the conclusion I included at the beginning of this post.

 

[mfb]

If my analysis below is correct, this would seem to suggest that the approach that is commonly used, comparing escape velocity with Hubble velocity to determine if a system is bound with respect to the universe expansion, is wrong.

That is not the approach commonly used. That is the approach you used in the other thread (why did you open a new one?).

Where does your "Hubble acceleration" come from?

A circular orbit is possible, sure. It is not influenced by the current rate of expansion, but the change of this rate has an incredibly tiny effect on orbits. I think someone calculated it a while ago, the influence on the orbit of Earth (at fixed orbital velocity) was below the size of a proton.

 

[Buzz Bloom]

Hi @mfb:

Thanks for your post.

That is not the approach commonly used.

I found the escape velocity compared with Hubble velocity approach in An Introduction to the Science of Cosmology by D J Raine & E G Thomas (2001). On page 13 this approach is used discussing material not escaping from a galaxy.

I have looked for, but not found in any references, using the acceleration approach I analyzed. Do you know of any?

why did you open a new one?

I felt the questions I was asking in the other thread were distinctly different from this thread. In the other thread I was assuming the "escape velocity" approach was correct, and asking about the effects of various false assumptions (e.g., sphericality) on t he calculations. Here I am asking whether the "escape velocity" approach is fundementally wrong, and whether the "acceleration" approach might be OK.

Where does your "Hubble acceleration" come from?

I consider a distance R at time t between two points. R increases wth the universe expanding expoentially with a constant Hubble constant.

dR/dt = h R​

Differentiating w/r/t t gives

d²R/dt² = h dR/dt = h² R​

It is not influenced by the current rate of expansion, but the change of this rate has an incredibly tiny effect on orbits.

I would very much appreciate your help in finding a reference for this calculation.

Regards,
Buzz

 

[mfb]

R increases wth the universe expanding expoentially with a constant Hubble constant.

If the objects are following the Hubble flow.
At distance where this is a good approximation, objects are not bound gravitationally (otherwise they would not follow the Hubble flow in the first place).

I would very much appreciate your help in finding a reference for this calculation.

I searched but didn't find it.

Edit:
A related post
This thread has more mathematical details.

The main point: locally, and if we could ignore dark energy, you could work with Newtonian physics. In particular, if two galaxies don't move apart now, they won't start doing that just because space on large scale is expanding.
We cannot ignore dark energy, so there is actually some weak energy density driving expansion, but you don't get that contribution via today's Hubble constant.

 

[Buzz Bloom]

Hi mfb:

Thanks for your post, especially the links.

Unfortunately I still remain confused about the effect of universe expansion on orbits, which is the reason I started this thread. I understand that the proper way to analyze this is GR, but I don't understand GR well enough to set up the equations. If somewhat could setup the appropriate GR equations for me, I think I might be able to find a useful solution, perhaps numerically.

I did not find the "A related post" link helpful, but some if the discussion at the "This thead" did seem to relevant to the purpose of this thread, particularly post #8 of @pervect.

Here is a quote from that post, #8.

Here is (I think this is right) how to get an actual number for acceleration out of the Hubble constant.

We have v=Hr, where v is the relative velocity, H is the Hubble constant, and r is the distance.

Thus dv/dt = H dr/dt. But dr/dt is just v, which is H*r.

Thus dv/dt = H^2 r​

This is the same result got for A_h. Post #9 by @hellfire seemed to accept this approach also, but added a correction for a non-constant h which is not relevant to the specific simplying assumption of this thread that h is constant.

I will try to interest pervect or hellfire in this thread.

Regards,
Buzz

 

[Buzz Bloom]

Hi mfb:

Thanks for your post, especially the links.

The "A related post" link was interesting, but not specifically helpful with respect the reason I started this thread: to gain some understanding of the effect of the expansion of the universe on orbits. I hope an understanding of this simplified problem will give me some useful insights into the more general topic of the interaction of expansion with gravity.

The "This thread" link had two specific posts, #8 and #9, that seemed to be in agreement with my approach regarding acceleration. Here is a quote from post #8 by @pervect:

Here is (I think this is right) how to get an actual number for acceleration out of the Hubble constant.

We have v=Hr, where v is the relative velocity, H is the Hubble constant, and r is the distance.

Thus dv/dt = H dr/dt. But dr/dt is just v, which is H*r.

Thus dv/dt = H^2 r​

This result is the same as in my posts for A_h.

Post #9 by @hellfire made a correction based on H not having a constant value, but otherwise seems to accept this approach.

Regards,
Buzz

 

[mfb]

Well, H is not constant, and that is a crucial point. Actually, in an empty universe with negligible overall matter and dark energy density, H would fall with 1/T where T is the age of the universe. The two terms would exactly cancel and give zero acceleration.
In a universe which is dominated by matter, H falls of faster and the acceleration is actually a deceleration. If the matter density is high enough, everything is bound and the universe will collapse again.
In a universe which is dominated by dark energy, H is constant and you can indeed see an acceleration. There, comparing acceleration works for galaxies that are at rest relative to each other (and only for those). That's not the universe we live in (yet), matter effects are still relevant, and galaxies in general have some relative motion.

 

[Chronos]

This question resurfaces from time to time in various forms. The short answer is the effect of expansion at local scales is negligiible. For the brutal details see http://arxiv.org/abs/astro-ph/9803097v1, The influence of the cosmological expansion on local systems. You can naively derive a large value based solely on the change in the scale factor of the universe since the solar system formed, but, the flaw in this logic becomes apparent in the face of other evidence - like the paleoecologic record, which shows no evidence the Earth's climate was significantly warmer a few billion years ago than now, as would be expected if a significant increase in the earth-sun distance had occurred over that same time period. Does this imply a cutoff on the scale at which expansion becomes measurably significant? Cooperstock argues 'No' in the referenced paper.

 

[Buzz Bloom]

Well, H is not constant, and that is a crucial point.

Hi mfb:

Sorry. My bad. I intended that the universe for this thread's analysis to be empty of matter (except for P having mass M) and radiation, but also that it would be flat and having dark energy so that expansion is exponential and h has a constant value. That is:

everywhere: Ω_r = Ω_k = 0,
everywhere outside a sphere with center P and radius R_s:

Ω_Λ = 1, and Ω_m = 0, where
R_s = 2 G M / c² :​

Inside that sphere, there is a mass densitity

ρ = 3 M / Rs3 = (3/8) c⁶ / G³ M²​

The critical density is

ρ_c = (3/8) h² / π G.​

Therefore inside the sphere,

Ω_m = ρ / ρ_c = π c⁶ / M² G², and
Ω_Λ = 1 - Ω_m.​

The region of interest for orbits have values of R >> R_s, so the small sphere can be ignored, as well as the non-Newtonean space/time distorions near the sphere.

Regards,
Buzz

 

[Buzz Bloom]

The short answer is the effect of expansion at local scales is negligiible.

Hi @Chronos:

Thanks for your post.

I really don't intend to apply this analysis to planetary orbits. I would like to understand the effect of universe expansion on the motion of galaxies relative to each other. From the previous thread I started, "The Dark Sky Ahead", I learned a bit about the issues related to trying to decide if a system is sufficiently gravitationally bound to avoid disruption from universe expansion. I felt that I could gain a greater depth of understand by considering simpler scenarios. In particular, I felt that analysis based on comparing escape velocity to Hubble velocity was flawed, and that analysis based on the concept of Hubble acceleration might work better. In the simple scenarios above, I think it becomes clear that this is a plausible possibility.

I obviously failed to present these simplified scenrios with sufficient clarity to avoid misunderstanding. I would much appreciate your taking a look at all my earlier posts here which include clarifications, and letting me know what you think about the Hubble acceleration approach.

Regards,
Buzz

 

[Chronos]

it fails in the FRW model.

 

[Buzz Bloom]

it fails in the FRW model.

Hi @Chonos:

Thanks for your answer to my question. I would much appreciate some help in setting up the FRW equations for my scenario so I can understand why "it fails". Can you help me?

Regards,
Buzz

 

[Chronos]

You may find this reference helpful; http://arxiv.org/abs/0810.2712, Influence of global cosmological expansion on local dynamics and kinematics.

 

[Buzz Bloom]

You may find this reference helpful

Hi Chronos:

Thank you very much for your post. The reference you cited seems to be exactly what I was looking for. I expect to have a little difficulty with the math, but I am grateful for the opportunity to try my aging mind at it.

Regards,
Buzz

 

[Buzz Bloom]

Hi @Chronos:

I am just getting started with the Carrera & Giulini article you cited, but what I found so far seems to support the acceleration approach I take in this thread.

On page 7, equation 7 is found by differentiating the Hubble Law R' = H R, giving

R''_|cosm.acc. = H' R + H R' = (a''/a) R = -q H² R .​

Note: I use the apostrophe instead of the over dot because of format limitations. The subscript "|cosm.acc." reminds the reader that in this equation R refers to the radial coordinate between an observer and an observe object with respect to cosmological acceleration.

Since in my scenario H is assumed to be a constant, a''/a = -q = 1, and H' = 0, giving

R'' = H² R .​

This is the same result as I give for A_h in my post #1 case (b).

Here is another quote:

from the motions described by (7), which suggests that in Newton’s law, m¨~x = ~F, we should make the replacement R'' -> R'' - (a''/a) R .​

This is also the same as my approach in case(b).

Later the article says this approach is consistent with the FRW model.

In the light of this, can you help me find something in the article that supports your quote

it fails in the FRW model.

Regards,
Buzz

 

[Chronos]

I was thinking in terms of the appendix in the Cooperstock paper for which I concede the possibility I may have misconstrued.

 

[Buzz Bloom]

Hi @Chronos:

Thanks for your post. I have much enjoyed our dialog here.

Regards,
Buzz

 

[Buzz Bloom]

I AM STILL WORKING ON THIS POST.

DRATS!

I just accidentally hit a backspace (or at least I think that's what I hit) and I lost about an hour's work on this post. The EDIT window reverted to this much earlier state. I am not sure what I need to do to protect myself.

Any suggestions?​

I have read through the closed thread "Force to resist expansion", and I found the discussion there quite helpful, especially post #9 by @hellfire, and its equations

(1) r'' = a = v'= H' r + r' H, and
(2) r'' = a = H' r + H² r.​

Note that the apostrophe, ', represents d/dt.

​

That thread did not continue by calculating H', which I will do below.

I start with the LambdaCDM model assuming flatness.

(3) https://www.physicsforums.com/attachments/90771
See https://en.wikipedia.org/wiki/Lambda-CDM_model#Cosmic_expansion_history.​

Note that H₀, and the Ω's are constants corresponding the the present time, t = 0. If we assume radiation is negligible, we can ignore the Ω_rad term, and rewrite the equation as:

(4) H(a) = H₀ √f(a), where
(5) f(a) = Ω_Λ + Ω_m a^-3​

I will use ^ to represent d/da. Keep in mind that a = a(t), and a(0) = 1.

(6) H^(a) = H₀ [1/2√f(a)] f^(a)
(7) f^(a) = -3 Ω_m a^-4
(8) √f(a) = H(a)/H₀
(9) H^(a) = (-3/2) H₀ [H₀ / H(a)] Ω_m a^-4​

Now, I want the equation for H'(a) = a' H^(a):

(10) H'(a) = H^(a) a' = (-3/2) H₀ [H₀ / H(a)] Ω_m a^-4 a'

= (-3/2) H₀² Ω_m a^-3​

Substituting this equation for H' into (2) yields:

(11) r'' = a = H² r - (3/2) H₀² Ω_m a^-3 r​

The value of the function r''(r) for t = 0 is

(12) r₀'' = H₀² r - (3/2) H₀² Ω_m r = H₀² r [1 - (3/2) Ω_m]​

Now, going back to my original post, I was then assuming that a >> 1 and H was a constant. Now I can use (2) as assume a = 1.

Scenario 1, Case 1 becomes