Earth's orbit and the expansion of the Universe

[Randy Subers]

TL;DR Summary

Is the Earth's orbit getting larger due to the expansion of the universe?

Does the Earth's orbit get very slightly larger over long periods of time due to the expansion of the universe? If so does it stay at the slightly larger distance or somehow does the energy to go back to the earlier orbit go somewhere else and if so where? If the orbit does not get slightly larger then what is stopping it?

 

[Ibix]

No. The dynamics of bound systems like the solar system, our galaxy, and even our local group of galaxies, are not affected by the expansion of the universe.

Basically, the "expansion of the universe" is simply Newton's first law (that stuff that started off moving apart will continue to move apart), complicated by being in a general relativistic curved spacetime. Stuff that had a strong enough mutual gravitational attraction to stop that initial motion away isn't still fighting that motion, any more than a ball that you threw into the air ten seconds ago and is now at rest on the ground still secretly has its old upward velocity. Expansion isn't a force that pushes everything apart.

 

[timmdeeg]

Expansion isn't a force that pushes everything apart.

Yes, not everything. To my knowledge accelerated expansion is acting as a "tiny" tidal force which pulls apart very loosely bound systems e.g. galaxy supercluster.

 

[PeterDonis]

Isn't accelerated expansion acting as a "tiny" tidal force which pulls apart very loosely bound systems e.g. galaxy supercluster?

No. It's a tiny force that alters orbital parameters of bound systems by a tiny amount, but the tiny amount is constant in time (because the dark energy density that causes accelerated expansion is constant in time).

 

[timmdeeg]

Ah, I see. So to be a tidal force would require not to be constant in time. Is that correct?

 

[Ibix]

I deliberately didn't mention dark energy and thought I'd address it if anyone followed up, basically because it isn't needed for expansion to happen. So I'd say that dark energy is a thing that slightly modifies both the expansion of the universe and orbital parameters of bound systems. At least, we assume it modifies bound systems - the effect it is predicted to have is way too tiny to be measurable.

If you add dark energy to the mix, orbital periods do change very slightly, yes. It might change a bound system into an unbound system in the sense that if you choose some particular set of initial positions and velocities and run simulations with and without dark energy, the version with dark energy might show unbound behaviour while the version without shows bound behaviour. However, I don't believe it will take a bound system and, over time, convert it into an unbound one. In other words, Earth won't eventually escape the Sun's gravity due to the cumulative effect of the tiny dark energy density we see (not even if we stop the Sun going nova). But with a large enough dark energy density we would never be bound at all. (@timmdeeg - I think that's the point @PeterDonis was making. Not that a tidal force must be time varying, but that it would need to be time varying to convert a bound system into an unbound one.)

There is one exception, which is a slightly fringe theory called quintessence, wherein the dark energy density does grow over time. All systems are eventually torn apart, including atoms in the very final fractions of a second before a singularity at the end of the universe. That's called the Big Rip, and isn't quite ruled out by observation. I think most physicists consider it pretty implausible

 

[timmdeeg]

(@timmdeeg - I think that's the point @PeterDonis was making. Not that a tidal force must be time varying, but that it would need to be time varying to convert a bound system into an unbound one.)

I understood @PeterDonis such that in FLRW-spacetime with a Cosmological Constant we don't call said tiny force a "tidal force" (unless Dark Energy turns out to increase over time). A counter example would be Schwarzschild-spacetime where tidal forces increase with decreasing $r$.

 

[Ibix]

I doubt it, but we'll see what he has to say.

 

[PeterDonis]

So to be a tidal force would require not to be constant in time. Is that correct?

It depends on what you want the term "tidal force" to mean. If it just means any kind of spacetime curvature, then, well, dark energy aka a cosmological constant is a kind of spacetime curvature. But it's not the same kind as the kind that, for example, creates tides on the Earth due to the Moon.

My answer to your question had nothing to do with what you want the term "tidal force" to mean. It had to do with the actual physics, which is that the effect of dark energy on a gravitationally bound system is not to "pull the system apart" but only to change the orbital parameters of the bound system in a way that does not change with time. The change to the orbital parameters for a system of the size of the solar system, or even our galaxy, is too small to observe. The change for a large galaxy cluster might be observable, although there are so many other uncertainties in our observations of such systems that I don't think we'd be able to pick out a dark energy effect.

 

[PeterDonis]

Not that a tidal force must be time varying, but that it would need to be time varying to convert a bound system into an unbound one.

Not necessarily, no. Consider a system that, in the absence of dark energy, would be very weakly bound--say two galaxies moving at a relative speed just short of the escape speed for the distance between them. The presence of dark energy, even with an effect that is constant in time, might change the orbital parameters just enough to make this system unbound (heuristically, by decreasing the escape speed at that distance just enough to make it smaller than the relative speed of the galaxies).

 

[Bandersnatch]

Not necessarily, no. Consider a system that, in the absence of dark energy, would be very weakly bound--say two galaxies moving at a relative speed just short of the escape speed for the distance between them. The presence of dark energy, even with an effect that is constant in time, might change the orbital parameters just enough to make this system unbound (heuristically, by decreasing the escape speed at that distance just enough to make it smaller than the relative speed of the galaxies).

But that DE presence would already be included when determining whether the system is bound or not. You're not adding DE to a borderline bound system. It was there from the beginning, and the system is either bound or not from the beginning.

 

[Ibix]

Not necessarily, no. Consider a system that, in the absence of dark energy, would be very weakly bound--say two galaxies moving at a relative speed just short of the escape speed for the distance between them. The presence of dark energy, even with an effect that is constant in time, might change the orbital parameters just enough to make this system unbound (heuristically, by decreasing the escape speed at that distance just enough to make it smaller than the relative speed of the galaxies).

Yes - that was a point I was trying to make in the post you quoted. "Turning on" dark energy in a simulation may change a bound system to unbound. But given dark energy exists, its effects won't build up and build up until suddenly the Earth flies away from the Sun (even if we allow the Sun to live forever). For that, you need an increasing dark energy density, quintessence/"phantom energy".

 

[PeterDonis]

that DE presence would already be included when determining whether the system is bound or not.

In actual fact, yes, either DE would be there or it would not. I am comparing two different models into which we are plugging the same parameters (distance and relative speed) to see what they say about the orbit being bound or unbound.

 

[PeterDonis]

given dark energy exists, its effects won't build up and build up

Yes, exactly.

 

[timmdeeg]

It depends on what you want the term "tidal force" to mean. If it just means any kind of spacetime curvature, then, well, dark energy aka a cosmological constant is a kind of spacetime curvature.

I meant "To my knowledge accelerated expansion is acting as a "tiny" tidal force which pulls apart very loosely bound systems e.g. galaxy supercluster." as I said in post #4. And I understood your "No" meaning that either the tiny force that pulls such loosely bound systems apart isn't a tidal force or that such systems aren't pulled apart. I think its obvious that in contrast stronger bound systems aren't pulled apart.

By the way according to

https://arxiv.org/abs/astro-ph/9803097
Acceleration on the scale of the Solar System
...
... This is to be compared to the predominant gravitational acceleration of the Earth towards the sun g = GM⊙ r 2 0 = 6 × 10−3m/sec2 (3.2) which completely overwhelms the effect of the cosmological expansion by 44 orders of magnitude.

This article is dated 1998 before dark energy was detected which seems to make a big difference to what we know today, because at that time the cosmological expansion was understood to be decelerating.
So it seems that the tiny acceleration on the solar system scale calculated in (3.1) still acts as a tidal force even in the absence of dark energy. But I'm not sure. Could one generalize that tidal forces in curved spacetime are consequences of growing distances between comoving objects?

 

[kimbyd]

I meant "To my knowledge accelerated expansion is acting as a "tiny" tidal force which pulls apart very loosely bound systems e.g. galaxy supercluster." as I said in post #4. And I understood your "No" meaning that either the tiny force that pulls such loosely bound systems apart isn't a tidal force or that such systems aren't pulled apart. I think its obvious that in contrast stronger bound systems aren't pulled apart.

By the way according to

https://arxiv.org/abs/astro-ph/9803097
Acceleration on the scale of the Solar System
...
... This is to be compared to the predominant gravitational acceleration of the Earth towards the sun g = GM⊙ r 2 0 = 6 × 10−3m/sec2 (3.2) which completely overwhelms the effect of the cosmological expansion by 44 orders of magnitude.

This article is dated 1998 before dark energy was detected which seems to make a big difference to what we know today, because at that time the cosmological expansion was understood to be decelerating.
So it seems that the tiny acceleration on the solar system scale calculated in (3.1) still acts as a tidal force even in the absence of dark energy. But I'm not sure. Could one generalize that tidal forces in curved spacetime are consequences of growing distances between comoving objects?

I think it was incorrect for them to make that comparison.

One way to dig at this is to look at the equations for structure formation, which in their simplest form use linearized gravity. In that formulation, if you look at what happens to gravitational potentials in an expanding universe is that below a certain radius, those potentials become constant. Basically, this picture is one where the universe is expanding, but some regions are dense enough to stop expanding. Once they stop expanding, they remain constant, bound systems.

This isn't exact, of course. This is the linear approximation. It's valid only at scales much larger than even galaxy clusters. But what's interesting about it to me is how dark energy modifies the picture: the gravitational potentials of these overdense regions are no longer constant when dark energy is around. They decay over time. Indicating that there are large systems which appear, for a time, to be bound systems, which prove to not be bound systems after all once the effects of dark energy become more apparent at a later time.

My understanding is that this is all compatible with what other people have been saying in this thread so far. When systems are small enough, they no longer decay over time due to dark energy. And the systems that appear bound but later prove not to be could be described as never having been bound systems in the first place.

But a way to verify this is to use the Newtonian approximation of the effect of dark energy and see how it modifies the parameters of the two-body problem. What you'd probably find is that there's a radius below which the system remains bound forever, a radius beyond which the system can never be in a bound state, and an intermediate zone where it can appear to be bound for a time but proves to be unbound.

 

[PeterDonis]

I meant "To my knowledge accelerated expansion is acting as a "tiny" tidal force which pulls apart very loosely bound systems e.g. galaxy supercluster." as I said in post #4. And I understood your "No" meaning that either the tiny force that pulls such loosely bound systems apart isn't a tidal force or that such systems aren't pulled apart.

Perhaps the problem is the "pull apart" language. As the series of posts in this subthread has made clear (for example see @Bandersnatch's post #11), a system is either bound or it's not bound. If it 's not bound, dark energy isn't "pulling it apart", because it wasn't "together" in the first place. And my point with my "No" was that dark energy is not "pulling", it's just changing the orbital parameters slightly from what they would have been if dark energy were not present. And for some systems, that change means the systems aren't bound with dark energy present, whereas, with the same separation between constituents, the system would have been bound (just barely) if dark energy were not present. But that doesn't mean dark energy "pulled the system apart"; as I said in post #13, I was not describing a process that actually takes place in our universe, I was comparing two different models of the same system, only one of which (the one with dark energy) is actually correct for our universe.

 

[PeterDonis]

it seems that the tiny acceleration on the solar system scale calculated in (3.1) still acts as a tidal force even in the absence of dark energy.

I don't think the analysis in that paper is correct. What they are doing is basically constructing a local inertial frame centered on the solar system and then calculating corrections to its metric coefficients based on the global FRW geometry of the universe. But that global geometry assumes a constant energy density everywhere, which is not actually the case. The local geometry of spacetime in the solar system is not the same as a small patch of the global FRW geometry (with a tiny global curvature due to the tiny average energy density of the universe) with the Sun and planets added on. So you can't calculate "corrections" to our local model of the solar system (which assumes that it's vacuum except for the Sun and planets) based on global FRW geometry and expect them to be valid.

The difference with dark energy is that dark energy is a constant energy density everywhere, even in the solar system, so it does make sense to calculate the (tiny) correction to the spacetime geometry everywhere due to the density of dark energy. But that only works for dark energy.

 

[timmdeeg]

Perhaps the problem is the "pull apart" language.

Yes it is.

As the series of posts in this subthread has made clear (for example see @Bandersnatch's post #11), a system is either bound or it's not bound. If it 's not bound, dark energy isn't "pulling it apart", because it wasn't "together" in the first place. And my point with my "No" was that dark energy is not "pulling", it's just changing the orbital parameters slightly from what they would have been if dark energy were not present. And for some systems, that change means the systems aren't bound with dark energy present, whereas, with the same separation between constituents, the system would have been bound (just barely) if dark energy were not present. But that doesn't mean dark energy "pulled the system apart"; as I said in post #13, I was not describing a process that actually takes place in our universe, I was comparing two different models of the same system, only one of which (the one with dark energy) is actually correct for our universe.

Thanks for clarifying this "pull apart" issue, I fully understand your "No" now . Indeed one can't pull apart what is apart already.
Just to be sure, do we call the tiny force which changes orbital parameters a bit tidal force?

 

[PeterDonis]

Just to be sure, do we call the tiny force which changes orbital parameters a bit tidal force?

Strictly speaking, tidal gravity isn't a force. It's geodesic deviation. Objects moving in a spacetime with geodesic deviation will in general experience stresses, but those stresses aren't due to "tidal force"; they're due to internal forces between the parts of the object that are induced to compensate for the geodesic deviation.

The general effect of dark energy by itself is geodesic deviation: geodesics that are initially parallel will diverge in the presence of dark energy. This is what causes the accelerated expansion of the universe. But in a situation like the solar system, where there are isolated massive bodies present as well as dark energy, the effect is not the same as dark energy by itself, because the isolated massive bodies do not have a constant energy density everywhere; there is a huge energy density inside the bodies and vacuum (except for the dark energy) everywhere else. The net effect of that is not well described by "geodesic deviation"; it's more like, as I said before, a tiny alteration in orbital parameters.

A good idealized example to study for this is Schwarzschild-de Sitter spacetime: this is a spacetime with a Schwarzschild black hole plus dark energy (aka a positive cosmological constant). Comparing bound orbits in Schwarzschild-de Sitter spacetime with a very small cosmological constant, with bound orbits in Schwarzschild spacetime with the same $M$, is the basis for the statements I've been making about tiny alterations in orbital parameters.

 

[DaveC426913]

No. It's a tiny force that alters orbital parameters of bound systems by a tiny amount, but the tiny amount is constant in time (because the dark energy density that causes accelerated expansion is constant in time).

Can I ask for even more clarification?

So CE does affect the orbits of our SS planets, just not in a way that changes over time? i.e. it has caused our planets to orbit a little further from the sun than they would without it - but that it is not a cumulative effect over time?

 

[Orodruin]

Can I ask for even more clarification?

So CE does affect the orbits of our SS planets, just not in a way that changes over time? i.e. it has caused our planets to orbit a little further from the sun than they would without it - but that it is not a cumulative effect over time?

If we are talking about an actual cosmological constant yes, but its value is so tiny that the effect would be practically undistinguishable.

If dark energy is not a cosmological constant ... you would need to model it properly to find its behaviour.

 

[Ibix]

So CE does affect the orbits of our SS planets, just not in a way that changes over time?

Dark energy does (probably - as Orodruin notes it might not, since we don't know what it is and we can't measure its effects on this scale) but not expansion itself. Dark energy modifies the expansion rate too.

 

[PeterDonis]

So CE does affect the orbits of our SS planets, just not in a way that changes over time?

Assuming that "CE" is a cosmological constant, yes. (In practice the effect is much too small to measure.)

 

[timmdeeg]

Strictly speaking, tidal gravity isn't a force. It's geodesic deviation. Objects moving in a spacetime with geodesic deviation will in general experience stresses, but those stresses aren't due to "tidal force"; they're due to internal forces between the parts of the object that are induced to compensate for the geodesic deviation.

Thanks, understood.

The general effect of dark energy by itself is geodesic deviation: geodesics that are initially parallel will diverge in the presence of dark energy.
This is what causes the accelerated expansion of the universe.

And in case matter density is dominating dark energy then geodesics are still diverging ( as long as the universe expands) but decelerated, correct?

A good idealized example to study for this is Schwarzschild-de Sitter spacetime: this is a spacetime with a Schwarzschild black hole plus dark energy (aka a positive cosmological constant). Comparing bound orbits in Schwarzschild-de Sitter spacetime with a very small cosmological constant, with bound orbits in Schwarzschild spacetime with the same $M$, is the basis for the statements I've been making about tiny alterations in orbital parameters.

Great explanation. I appreciate your clear and enlightening answers.

 

[PeterDonis]

And in case matter density is dominating dark energy then geodesics are still diverging ( as long as the universe expands) but decelerated, correct?

No. If matter (or radiation, for that matter) dominates then geodesics converge. That's what expansion decelerating means. Geodesics only diverge (expansion accelerating) if dark energy dominates. "Diverging but decelerated" is an oxymoron.

Remember that in an expanding universe, comoving worldlines (the simplest geodesics to think of) aren't initially parallel. So "diverging" in the context of geodesic deviation and spacetime curvature doesn't mean "moving apart" for them; it means "moving apart faster and faster" (i.e., accelerating expansion). Whereas "converging" for them doesn't mean" moving closer together"; it means "moving apart slower and slower" (i.e., decelerating expansion). The intuitive correspondence between "diverging" and "moving apart", and "converging" and "moving closer together", for geodesic deviation and spacetime curvature only works if the geodesics start out parallel (i.e., neither moving apart nor closer together).

 

[timmdeeg]

No. If matter (or radiation, for that matter) dominates then geodesics converge. That's what expansion decelerating means. Geodesics only diverge (expansion accelerating) if dark energy dominates. "Diverging but decelerated" is an oxymoron.

Remember that in an expanding universe, comoving worldlines (the simplest geodesics to think of) aren't initially parallel. So "diverging" in the context of geodesic deviation and spacetime curvature doesn't mean "moving apart" for them; it means "moving apart faster and faster" (i.e., accelerating expansion). Whereas "converging" for them doesn't mean" moving closer together"; it means "moving apart slower and slower" (i.e., decelerating expansion). The intuitive correspondence between "diverging" and "moving apart", and "converging" and "moving closer together", for geodesic deviation and spacetime curvature only works if the geodesics start out parallel (i.e., neither moving apart nor closer together).

Another example for wrong intuition! Thanks for correcting. Your last sentence is very helpful.

 

[Orodruin]

Sometimes it is quite relevant to remember that the FLRW coordinates generally do not constitute local normal coordinates with everything that entails. Going to local normal coordinates, you will find that comoving observers do move apart exactly according to Hubble’s law in those coordinates. I think I wrote an Insight on that some time ago.

Edit: Ah yes, here it is https://www.physicsforums.com/insights/coordinate-dependent-statements-expanding-universe/