# Is the Friedmann expansion the ultimate 'free lunch'?

1. Oct 14, 2007

### jonmtkisco

Hi folks.

Is there any laboratory demonstration or theoretical model for the proposition that a fundamental attribute of the “fabric of space” is its ability to expand forever just because it got a single expansionary “kick in the pants” from inflation? In other words, why didn’t space stop expanding the instant the cosmic foot came off of the inflaton accelerator?

I’ll use the term “original expansion” to refer to the expansion after inflation ended, and excluding the expansionary contribution of the cosmological constant. As I understand it, mainstream cosmology (e.g., the estimable Prof. Peebles) describes the ongoing original expansion as being the result of space possessing a “kinetic energy of expansion”. In a flat universe (which is what we observe), this kinetic energy must forever exceed the mass/energy of the total matter and radiation contents of the universe. This kinetic energy is “momentum-like”, in that it is not depleted by expanding per se, but only by the drag of gravity tugging against its expansion over time (converting the kinetic energy to potential energy). This mainstream kinetic energy assumption gives rise to several problems. I would appreciate if anyone can explain why these concerns are misconceived, or direct me to literature where these issues are discussed in detail.

1. Why doesn’t the kinetic energy of expansion gravitate? Adding this kinetic energy to the right side of the Friedmann equation would result in a substantially different expansion rate calculation. It is widely held that in general, kinetic energy (heat, pressure stress, electromagnetic energy) gravitates. Kinetic energy of momentum is something of a question mark, because the relativistic mass of momentum depends on the reference frame of the observer. Although as I suggest that expansion energy is “momentum-like”, nothing is actually moving, and the expansion itself looks identical from every frame of reference available to us. (We don’t have the luxury of standing outside the universe looking in). Of course, the cosmological constant clearly is held to gravitate, and so has earned its own home in the right side of the Friedmann equation. I don’t see any impediment to assuming that kinetic energy of expansion has mass/energy and gravitates, and therefore also ought to be accounted for separately in the Friedmann equation.

2. Why doesn’t the expansionary contribution of the cosmological constant have momentum-like characteristics? If space retains the expansionary momentum-like velocity imparted to it by inflation, then one would expect the expansion resulting from the cosmological constant to retain a momentum-like contribution as well. The cosmological constant is considered to impart ongoing expansion energy to space by means of its constant negative pressure (dark energy). This is like an ion engine that provides an ongoing (but small) thrust to continue accelerating a rocket in deep space. The rocket’s velocity at every second builds on the velocity retained from the prior second.

However, the cosmological constant causes space to expand at exactly the escape velocity of its own mass/energy at each instant in time. If space retained that momentum from instant to instant, then in each subsequent instant the total expansion velocity should “build” upon the retained velocity of the prior instant, and so on faster and faster. The cosmological constant does cause the expansion rate to increase over time, but only because the expansion of space generates more cosmological constant, not because of any retained momentum. It is difficult to rationalize why the velocity vector of the cosmological constant is additive to the velocity of expansion at each instant in time, but is not added to the retained momentum-like vector.

3. If the expansion of the universe is thought of as being analogous to an expanding sphere, then ΔVolume / ΔTime (cubic meters/second) seems to me to be the most representative measure of a momentum-like expansion vector. Since volume expansion is proportional to r^3, it’s not terribly surprising to calculate that, at the standard ΔRadius / ΔTime Hubble expansion rate (converted to an absolute meters/second scale), ΔVolume / ΔTime would steadily increase over the Hubble Time, if the cosmological constant is omitted entirely from the Friedmann equation. I don’t see how the original expansion can be considered momentum-like when the expansion vector increases over time.

By comparison, it is easy to demonstrate that the expansion rate caused by the cosmological constant alone, divided by the total volume of the total universe, is constant over time. That rate is 5.80E-18 cubic meters of expansion / second / cubic meter of volume. Note that while the momentum-like vector of the original expansion becomes “diluted” over time as it is spread over an ever-larger volume, that is not true of the expansionary contribution of the cosmological constant. Additional cosmological constant is added to the equation with every passing instant, which exactly offsets the volumetric dilution. So it is all the more surprising that even the "diluted" momentum-like vector of the original expansion increases over time while that of any arbitrarily fixed quantity of cosmological constant does not.

4. As I suggested previously in this forum, the idea that the original expansion retains “momentum-like” kinetic energy seems to require that each of an almost infinite quanta of vacuum retains its own separate kinetic energy level "account balance" that can range anywhere from near the original post-inflation energy, to far below zero. The more intense and prolonged exposure to gravitation an individual quantum of space has experienced, the less kinetic energy it currently retains. There is no reason to imagine that this kinetic energy does not go negative in local regions, signifying a local contraction of space under the influence of strong gravity. (Is kinetic energy allowed to go negative?) This variation in local energy levels seems mandatory, if for no other reason than the event horizons which prevent any causal connection or normalization of kinetic energy across distant regions of the universe.

I recognize that there are many questions in cosmology which don’t currently have definite answers. However, my impression is that the major missing links underlying mainstream cosmology theory get a lot of attention. This expansion problem doesn’t seem to have gotten attention.

Jon

2. Oct 14, 2007

### SpaceTiger

Staff Emeritus
It sounds like you're still thinking within a classical physics framework. The expansion of the universe is described with general relativity, so it's not quite right to think of space as having kinetic energy. Perhaps one of the resident GR experts will take the time to explain where Newtonian reasoning can and can't be applied in this case, but if that doesn't happen, check out the GR description of the expansion of the universe. The Friedmann Eqs. are pretty simple (by GR standards), so I'd suggest reading through a derivation of them.

3. Oct 15, 2007

### jonmtkisco

Hi SpaceTiger,

Thanks for the suggestion. Can you point me to a reference to check out the GR description of expansion? I wasn't aware that GR attempts to examine the subject. Typically the expansion is described as "arising from initial conditions", which I think means it's taken as a "metric", a given value, without delving into its cause or working mechanism. The Friedmann equation itself was built as a model of GR, by Friedmann, Lemaitre, Robinson, Walker, and eventually was embraced personally by Einstein after he saw Hubble's observations demonstrating that expansion was a likely fact. But the Friedmann model does not describe the cause or mechanism of expansion. Lemaitre speculated that it might have been caused by the nuclear decay of a gigantic primordial atom; he also suggested that maybe cosmic rays were residual evidence of the explosion. Great idea, but it didn't pan out.

Prof. Peebles clearly is well versed in both GR and inflation, and he uses the term "kinetic energy" repeatedly (but without explanation) to describe the expansion, at least as recently as his 1993 textbook.

By the way, I need to point out an error in my #4. In a kinetic energy model, the energy of expansion doesn't actually need to go negative, and probably doesn't go negative. If the expansion force locally drops below the magnitude of the local gravitational force, even while remaining positive or at zero, contraction will occur. This raises the interesting question about whether an accumulated velocity of contraction has the same kind of kinetic energy as the accumulated velocity of expansion does...

Of course that's assuming, as you rightly question, that "kinetic energy" is the correct way to model the expansion.

SpaceTiger, are you aware of any literature that attempts to describe the original expansion in quantum mechanical terms? Maybe that can't be done unless/until a satisfactory theory of quantum gravity is developed.

Jon

4. Oct 15, 2007

### SpaceTiger

Staff Emeritus
No, but you already mentioned the favored theory for the origin of the expansion -- inflation. Your question was, "...why didn’t space stop expanding the instant the cosmic foot came off of the inflaton accelerator? " The classical physics analogy can help in answering this question, but it will only take you so far. To understand the universe's expansion before and after inflation, one just needs to understand the FRW metric. If you already understand the basic structure of it, try a search of astro-ph for a review article on inflation. It should explain how the scale factor evolves in the presence of an arbitrary inflaton potential.

I've not read the Peebles writings you're referring to, so I'm not in a position to explain his use of the term "kinetic energy", but I'm sure he had a good reason.

5. Oct 15, 2007

### jonmtkisco

Thanks SpaceTiger,

Alas, I have searched and searched, but I'll keep searching.

I found the following insights from the indubitable Prof. John Peacock (Cosmological Physics, 1999, p.86-88):

"Describing the origin of the expansion as an explosion is probably not a good idea in any case; it suggests some input of energy that moves matter from an initial state of rest. Classically, this is false: the expansion merely appears as an initial condition. This might reasonably seem to be evading the point, and it is one of the advantages of inflationary cosmology that it supplies an explicit mechanism for starting the expansion: the repulsive effect of vacuum energy....

"An inability to see that the expansion is locally just kinematical also lies at the root of perhaps the worst misconception about the big bang.... If we understand that objects separate now only because they have done so in the past, there need be no confusion."

So while it is marginally helpful that inflation could supply the initial "kick in the pants", this says nothing at all about whether or why space has this surprising capability to continue expanding forever after a single kick. I for one am confused by an explanation that objects separate now only because they have done so in the past. Does anyone else find that explanation perplexing?

6. Oct 15, 2007

### SpaceTiger

Staff Emeritus
I'm a little bit confused about why you're confused. You seem to understand and accept the applicability of the FRW metric if an origin for the initial kick can be found. Furthermore, you seem to accept inflation as a possible origin for the initial kick. In classical physics, after that kick has been provided, we might think of the material in the universe as having a "momentum" that continues to carry it after it's no longer being kicked. In GR, classical momentum is not conserved, but we can use the FRW metric (which literally gives the expansion as a function of time after the initial kick of inflation) in lieu of the classical treatment, assuming we have accounted for all of the dominant constituents of the universe. If GR is correct and the FRW assumptions (e.g. homogeneity), are reasonable, then there's nothing left to explain. The FRW metric covers both time and space.

Obviously, there are a lot of potential complications (e.g. dark energy, inhomogeneities) to the above simple picture, but those don't seem to be what's confusing you. My impression is that it's the basic dynamics and energy balance of a general relativistic universe that are causing your hangup. You've given a number of quotes from famous people, but are you sure you understand what the FRW metric is actually describing? How is your math background?

7. Oct 16, 2007

### jonmtkisco

Hi SpaceTiger,

My math background is just fine. If you parse the FLRW metric, for a flat universe you will find that it does nothing more than calculate the escape velocity of the total mass/energy in the universe at any given scale factor. The inputs are derivatives of mass and radius. Anyone can easily calculate escape velocity without resort to FLRW, but FLRW also "automates" the process of adjusting the mix of densities between matter, radiation, and CC change over time. FLRW would also help to model geometric curvature, but we don't use that aspect because we know that the universe has no significant curvature.

In other words, for our purposes FLRW models the expansion based on a simple mathematical formula which appears to coincide with observations, but it makes no effort to explain or justify why. It was developed by Friedmann and Lemaitre by assuming the universe to be adabiatic and then simply applying the first law of thermodynamics to GR. That yields the formula which describes the shape of the cannonball-at-escape-velocity curve. That was the major insight Friedmann and Lemaitre both brought to the table.

But FLRW is merely a mathematically model, so it would be entirely backwards to suggest that the universe must expand at escape velocity just because FLRW says it must! Applying circular logic lets us evade questions about causation and mechanism. But I want to evade the evasions!

Last edited: Oct 16, 2007
8. Oct 16, 2007

### SpaceTiger

Staff Emeritus
No, the FRW metric is a general relativistic model of the universe, it is not just a calculation of the classical "escape velocity". If you believe GR, the cosmological principle, and inflation, the FRW metric tells you how the universe evolves with a particular composition. No further assumptions are needed. The composition of the universe can be determined from observations.

You seem pretty hung up on the Newtonian approximation, but the argument is essentially the same there. If a clump of matter explodes in free space, the future expansion can be
determined from the magnitude of the initial kick and the amount of matter that was expelled. If its self-gravity is stronger than the energy imparted by the kick, it will fall back in on itself. If not, it will expand indefinitely.

9. Oct 16, 2007

### jonmtkisco

Hi SpaceTiger,

I agree that the FLRW metric has provided a very good model for the observed expansion of the universe. I take no issue with those observations, or with their adherence to what FLRW predicts. Bravo. Friedmann and Lemaitre (the "L" in FLRW) hit a home run with their abstract model, especially considering that at the time, there was no observational evidence that the universe was expanding. Meanwhile, Einstein was convinced that it was static.

I don't want to be 'hung up' on Newton, so I appreciate your help in overcoming any lingering addiction there. But your closing statement that 'a clump of matter (e.g., cannonball) launched with insufficient energy will fall back' is as much a Newtonian statement as a GR statement. Can I suggest we share a therapist on this subject(?) (Just an attempt to lighten the mood...)

Here's the thing: When Friedmann and Lemaitre applied the First Law of Thermodynamics, they imagined the universe as a 'perfect fluid', like a pressurized gas. The Law says, "energy in = energy out, minus work done". In this case, the 'work done' is the adabiatic expansion of the universe against the resistance of its gravitational binding energy. And 'energy out' means any residual expansion rate after the 'work' was done (which could be negative if the universe were contracting). It's a perfectly reasonable, even brilliant, approach to the problem.

However, I disagree with your statement that "No further assumptions are required." Consider the case of a pressurized gas. Individual molecules have kinetic energy (related to physical temperature) which causes them to move about, rebound off each other, and physically push outwards against any pressure gradient (e.g., a container wall). On the contrary, space is not viewed as having significant positive pressure (other than the limited pressure of radiation), its matter contents do not move about with great energy, rebounding from each other, and the universe is not thought to be pressing outwards against a pressure gradient. Most important, the expansion of space does not involve any movement at all, either by the vacuum, or by matter. (Thus your allusion to matter exploding in free space and being expelled is inapropose, as my earlier quote of the illustrious Prof. Peacock lectured us). Space simply expands everywhere and in all directions. There is no "momentum" in the normal Newtonian or GR sense. So it is a significant stretch to 'assume' that space itself observes the same physical laws applicable to the momentum of relativistic particles moving 'through space'. For example, the vacuum itself has no mass (other than the tiny mass of the cosmological constant), so how can it possess classical 'momentum'? As we know, Momentum = Mass*Velocity. No mass, no momentum. I think it's quite reasonable to use the term 'assumption' to refer to the hypothesis that the vacuum can possess momentum. If you step outside your ship in deep space and give the vacuum a big kick, how much 'momentum' do you think you'll impart to it? Not much, I think you would agree.

Moreover, the concept that space can retain a "momentum-like" capability to expand forever must be an assumption. Surely you wouldn't describe it as a "fact", or as even as a clearly established probability. The evidence is rather skimpy for that.

As it turns out, the FLRW metric doesn't depend on the particular expansion mechanism. The metric works equally well for a 'one-time kick with momentum', or for an 'ongoing' incremental negative-pressure-like force. So without attacking FLRW itself, we are free to explore the possibility that the ongoing original expansion is the result of an ongoing incremental application of force, not a one-time kick. If doubtful, please examine the math for yourself.

I'm not interested in tilting at windmills, but I have a pretty good feeling that I can demonstrate objectively why the best implementation mechanism for FLRW is an 'ongoing' application of force. I'll be back soon with yet another (can you believe it?!) reinforcing train of logic on that point.

Thanks, Jon

Last edited: Oct 16, 2007
10. Oct 16, 2007

### SpaceTiger

Staff Emeritus
Of course, I said at the beginning of the paragraph that it was going to be. I was trying to explain in terms that made sense to you.

Your thinking on this is entirely backwards. The FRW is describing spacetime in the context of general relativity. The fact that the equations are similar to expressions for escape velocity and conservation of energy for particles exploding outwards (as in the analogy I gave at the end of my last post) is merely a curiosity, it is not the origin of the model itself.

This "momentum-like" capability to expand is a prediction of general relativity theory, which we've already agreed to apply to the problem.

Obviously FRW can be used for an ongoing kick, there is one going on right now (dark energy/CC)! An ongoing kick is not necessary for expansion, however.

Personal theories are not allowed on this forum. My sense is that you're looking to have your misconceptions reinforced rather than cleared up, so I'm going to lock the thread.

Last edited: Oct 17, 2007
11. Oct 17, 2007

### pervect

Staff Emeritus
I'm not sure if this will help, but because a FRW universe has a space translation symmetry, a FRW space-time has a conserved momentum.

See for instance Noether's theorem

The FRW metric has this property of space translation symmetry. Noether's theorem applies to GR because GR can be formulated in terms of an action principle, and thus the FRW metric has a conserved momentum.

If we consider the flat FRW universe with the metric

ds^2 = -dt^2 + a(t)^2 (dx^2 + dy^2 + dz^2)

(Note that I've chosen to use geometric units where c=1)

then none of the metric coefficients is a function of x, so it has the required space translation symmetry. This should not be a surprise, the reason the metric was picked in the first place was because of the cosmological principle, which states that space-time should be homogeneous on a large scale.

Then if x(tau), t(tau) is a particle following a geodesic, we wind up with the fact that

$$a^2(t(tau)) * \frac{d \, x(tau)}{d \, tau} = constant$$

we can consider the quantity on the left hand side to be P_x / m the (subscripted, i.e. contravariant) component of momentum in the x direction per unit mass.

If you're familiar with the geodesic equation, you can verify that differentiating this expression with respect to tau gives you one of the two geodesic equations for the FRW metric.

If you look at a(t)=constant, you'll have a non-expanding universe, and that we have

$$p/m = \frac{dx}{d \, tau} = \frac{dx}{dt} \, \frac{dt}{d \, tau} = \frac{\frac{dx}{dt}}{\sqrt{1 - \frac{dx}{dt}^2}}$$

which is the expected expression for relativistic momentum (given that we've assumed geometric units, with c=1).

Last edited: Oct 17, 2007