# Acceleration of the Universe

1. May 30, 2013

### falcon32

Hi,

I know generically that the expansion rate of the Universe is increasing, hence theoretical dark energy.

But is the acceleration rate itself constant, or does it change? And if it does change, do we have a formula for it yet?

Thanks!

2. May 30, 2013

### cepheid

Staff Emeritus
3. May 30, 2013

### falcon32

Alright thank you, didn't realize I could wiki it. I've got much to learn...exciting to see the equation that Einstein modified. Did a bit of reading...this equation was created before Hubble discovered the expanding universe, right? So do astronomers still use just this one, or have they made a special one from empirical redshift observations alone?

4. May 30, 2013

### cepheid

Staff Emeritus
Yes, the Friedmann equations are derived from the Einstein Field Equations of General Relativity, and Einstein finished formulating General Relativity in 1915, a few years before Hubble discovered that distant galaxies all appear to be moving away from us.

General Relativity is still the best theory of gravity we have. So, yes, astronomers still use the Friedmann equation in the form given there to this day. However, that is less restrictive than you would think. The Friedman equation doesn't describe a single mathematical model with a single expansion history for the universe, but rather a whole set of models called the "Friedman World Models." The reason for this is that the equations contain certain parameters, and a particular model with a particular expansion rate and history is given by one particular set of values for those parameters. Change the values of the parameters, and you change the outcome predicted by the model. A key task of modern cosmology has been using measurements and observations to figure out which set of parameter values (and hence which world model) accurately describes the universe we live in.

A concrete example may help. One of the main parameters appears in those equations in the form of the density, rho (ρ). A key result of General Relativity is that the geometry of spacetime is affected by its mass-energy content. In the context of the universe as a whole, what this means is that the dynamics of the expansion are affected by the total energy density, ρ, of the universe. This total density rho consists of contributions from all the major constituents: dark energy, dark matter, ordinary matter, and radiation (photons and relativistic particles). There is a separate density parameter for each of these, and we can measure them separately. The total density can be either larger than, equal to, or less than some critical density, ρcr. In the classic case (with no dark energy), these three cases lead to the following possibilities:

1. If ρ > ρcr, in other words, if there is enough mass in the universe, the universe's expansion slows, stops, and reverses itself, leading to a collapse (the so called "Big Crunch" scenario). You can think of it as there being enough matter in the universe that its mutual gravitation slows down the expansion enough to stop it and reverse it. In this case, there is also positive spatial curvature, meaning that the geometry of the universe is like the geometry on the surface of a sphere.

2. If ρ = ρcr, the universe continues to expand forever, albeit at a steadily decreasing rate (there is not enough mass to pull everything back together). In this case, the geometry of the universe is "flat" (meaning that it is Euclidean: it behaves like the geometry you learned in high school). So, at the critical density, there is just enough mass to have no spatial curvature.

3. If ρ < ρcr, the universe will also continue to expand forever at a steadily decreasing rate. There will also be negative spatial curvature, meaning that geometry will behave the way it does on the surface of a "saddle."

So, the Friedman equation gives you distinctly different results when you modify the parameters (in this case the total density). Notice that in all three of these scenarios (or "world models"), the universe expands, but that expansion slows down. It turns out that NONE of these three models is right, because in our inventory of the total "mass budget" of the universe, we were missing a major contributor: dark energy. When you throw dark energy into the mix, there is no longer such a straightforward relationship between the mass density of the universe, its geometry (curvature), and its ultimate fate. In the link I sent you, dark energy appears in the form of an additional constant Lambda ($\Lambda$) in the equations. This is the famous "Cosmological Constant." Einstein originally added the cosmological constant because his models predicted that the universe was not static: it could either expand or contract. At the time, this was inconceivable to him. He wanted to make the universe static, so he tried to throw in the extra $\Lambda$ term and tune it to achieve a delicate balance in which the universe neither expanded nor contracted. It turns out that this was futile, because the solutions are unstable. Even if you tune $\Lambda$ to make the universe static NOW, it will not remain so LATER. This was a rookie mistake. If Einstein had stuck to his convictions and followed his own theory to its logical conclusions (the way he did so well when he formulated Special Relativity ten years earlier), he could have predicted the expansion of the universe before Hubble observed it. But Einstein just wasn't thinking outside of the box in this case, and he didn't do that. That's why he later referred to the introduction of the cosmological constant as his "greatest blunder." For a long time, $\Lambda$ fell by the wayside.

Recently (in the mid to late 90's), observations of objects called Type Ia supernovae caused people to bring it back. Type Ia supernovae are thought to result from the thermonuclear explosion of a white dwarf star, and as a result, they are thought to be "standard candles" meaning that they all reach roughly the same peak brightness. So, if you look at how bright one of them appears to be, you can infer its distance. Different Friedman models (with different values for the density parameters) predict different variations of distance (as measured in this way) with redshift. After all, what use is a theoretical model unless if it makes testable predictions? Anyway, by measuring and plotting the Type Ia brightness vs redshift, and finding out which model fits the observed data the best, you can determine the values of the cosmological parameters. In the late 90's, observations of Type Ia supernovae *strongly* favoured a model with a NON-zero value for $\Lambda$. This result has since been corroborated by observations of the Cosmic Microwave Background (CMB). The presence of $\Lambda$ in the equations was attributed to the presence of some as yet unknown and mysterious substance that astronomers termed "dark energy." Dark energy has weird properties. Even though it adds mass to the universe, it also has a repulsive, anti-gravitational effect. So, instead of pulling things together, it forces them apart. So, not only do the observations show that the universe will continue to expand forever, but it also shows that that expansion, rather than slowing down as we thought, will instead get faster and faster and faster with time. This is a crazy result (it won the 2011 Nobel Prize in physics), and it's an example of what you wanted: empirical observations that improve our understanding of the universe. But they do so within the theoretical framework of General Relativity (for now). By the way, the observations also show that the geometry of the universe is very close to being flat (Euclidean): i.e. if you include dark energy, the total density is very close to the critical value.

Last edited: May 30, 2013
5. May 30, 2013

### marcus

Hi, I started this before supper--didn't see that Cepheid already answered. Then finished post and saw you already have an extensive reply. This is redundant, I guess, but I will let it stand rather than erase.

Falcon you mentioned the *empiricism* issue. My view is that the Friedman equation is good on both empirical AND theoretical counts. Only thing is it is classical---dates from around 1922 before quantum theory was well enough developed to be involved.

That's all right though because we only use Friedman at large scales where Nature behaves classically, so non-quantum equations can be successfully applied.
The Friedman equation has one or two adjustable parameters, the fit is remarkably good. So there is no compelling reason to abandon it. It has another thing going for it: GR as a law of geometry/gravity has been tested in Earth orbit and at solar system scale and at astrophysical scale, it has proven exquisitely accurate, we don't have anything better over a wide range of scale. And Friedman equation has been DERIVED from GR equation. So it agrees with past cosmological observation and has successfully predicted new observations AND it comes out of the best law of gravity we have so far (which itself has been tested every possible way people could think).

So Friedman has outstanding credentials on both empirical and theoretical sides.

The main place it DOESN'T work is right around the start of expansion where the energy density was extreme---and people think that quantum effects on the geometry take over. It is a "classical" (i.e. pre-quantum) equation just like its parent, the 1915 GR equation. So Friedman has not been tested at very small scale and very high energy density. (No more has GR).

That's why people are currently working on versions of GR and of the Friedman equation that incorporate quantum effects on the geometry. Maybe geometry is no longer smooth at very small scale. Maybe a quantum version of GR would not have a "singularity" that is a blow-up point. So they are working on a more rugged version of main GR equation and Friedman equation that will not blow up at the very start of expansion. but which REPRODUCES all the good stuff of the classical equations in the range of conditions where these have been proven to work so well.

That seems to be the next step.

Last edited: May 30, 2013
6. May 30, 2013

### cepheid

Staff Emeritus
Actually marcus, I think your answer complements mine nicely, hitting on some key points I missed, in spite of my verbosity.

7. May 30, 2013

### falcon32

Thanks cepheid and marcus, both of your answers were well-written and thoroughly thought out in my opinion, and I enjoyed reading them. What I love about science is there's always something new to learn: for me, that is about to be trying to understand Einstein's Field Equations. I do understand that GR is the best theory for gravity that we have at the moment, and I would probably shut up if I just had some redshift data that covered a few decades or so that I could play with lol.

8. May 31, 2013

### Naty1

So here we use INFLATION THEORY pioneered by Alan Guth....just after the Big Bang....something was needed to account for observations such as near flatness....

You might find this discussion worthwhile

How to prove the stretching of space

9. May 31, 2013

### Mordred

just to add another reference to the already excellent answers you have recieved, I recently wrote this article on geometry. I'm still trying to figure out how to simplify the FLRW metrics at the 3d and 4d stage. However it should provide some answers.

https://www.physicsforums.com/showpost.php?p=4400002&postcount=1

10. May 31, 2013

### falcon32

Thanks for that link, I did find it interesting! Note that my question has never been "is the universe expanding?" but was more along the lines of "did we work backwards from redshift data to arrive at a conclusion?" I appreciate the answers, which were very informative -- it does lead me to the next question though, since it seems the acceleration of the Universe is thought to follow, quite generically a(t)=Ct, rather than a(t)=C...that is, the acceleration increases with time.

1. From your link, we know that the Sachs-Wolf effect slightly blue-shifts photons traveling through gravity wells.
2. We also assume that the Universe is undergoing an accelerating acceleration.
3. Now this takes energy, but this is o.k. since Dark Energy is theorized to perform this work.
4. But the 1st Law of Thermodynamics says that energy cannot be created or destroyed, so we know that we must have a finite amount of Dark Energy to perform the work observed.
5. Therefore, shouldn't we run out of Dark Energy at some point, which would then decelerate a(t) to 0? And if not, does that mean that Dark Matter is converting to Dark Energy along the lines of E=mc2? But that is only delaying the inevitable, since there is a finite amount of Dark Matter available as well.

11. May 31, 2013

### Mordred

Thats the unusual property of dark energy. Its better described as vacuum energy. For the reason you supplied.
the vacuum energy or cosmologucal constant stays consistent in value per m3. As expansion occurs the total mass energy of the cosmological constant which is positive increases. However the density which is negative pressure stays constant.
The source of energy in regards to the cosmological constant is not fully understood. There are some theories such as virtual particle production. However nothing conclusive.

Last edited: May 31, 2013
12. May 31, 2013

### marcus

falcon in the most common notation a(t) is the size of a generic distance normalized to a(present)=1 and is called the scalefactor. So watch out there is a potential for confusion when you use a(t) to stand for the acceleration (of what???) there is no definite velocity with which the U expands. One has to pick some distance but there is no agreed on choice.

I guess one quantity that would be proportional to the acceleration of the growth speed of a generic distance would, in conventional cosmo notation, be a"(t) the second derivative of the scalefactor a(t). Just a heads-up about notation that you may be reading sometimes without realizing what is meant. Just in case.

a(t) is a dimensionless quantity, a pure number. So a'(t) is a reciprocal time. The Hubble rate H(t) is defined to be a'(t)/a(t), so it is a reciprocal time. E.g. like 1/144 of one percent per million years.
A number per unit time.

13. May 31, 2013

### Mordred

Good to note Marcus.

14. May 31, 2013

### falcon32

Alright marcus, thanks for that information, I'll be using better terminology from here on out.

15. May 31, 2013

### falcon32

Ah ok, so I'm not the only one wondering where the energy is coming from, good to know.
It has been established that the energy, no matter what the ultimate source, is finite, correct?

16. May 31, 2013

### Mordred

Personally in regards to the term "source" in this case I prefer "property".

I would have to answer no in the case total lamnda. For one we don't know if the universe is finite. In terms of density then yes its a finite amount. Also the total amount of DE will continue for as long as the universe continues to expand. Pressure due to DE will stay the same.

If your not aware the energy density and pressure per volume
are equivelent terms in units. You probably are, however saying so supplies that info for others reading this good info thread

17. May 31, 2013

### falcon32

The missing wide-angle correlations in the CMB are good support for a finite Universe, so say many.

18. May 31, 2013

### Mordred

Yes I agree good support however thats not the same as conclusive.

19. Jun 1, 2013

### Naty1

not sure what you have in mind here...since about 7B years the acceleration has been increasing...as we leave a matter dominated era [mass closer together] and enter a radiation dominated era [mass, like galaxies, more widely separated].

For an introduction to expansion concepts, try Leonard Susskind Cosmology lecture #3, Youtube, the first 20 minutes or so....He derives, using simply concepts and math, expansion measures. It's a recap. If you want the full very slow and deliberate version watch Lecture #2.

yes, based on observations, we seem to have evidence...like Hubble's observations

That idea has come into come recent question. When the cosmological constant is associatred with dark energy, whether expansion requires energy, I don't believe is resolved one way or the other.

Conservation of energy does not apply to cosmology...that is, it is undefined for curved spacetime....

So far the consensus is that expansion will continue. It's been going for 13.8 B years and is NOT 'running out of steam'...no evidence that energy for expansion, if that is in fact present, is going to be in short supply. As Susskind's lecture shows, one expansion gets started it seems to keep going....and, yes, that sure seems strange.

Good. There is much to be puzzled about!!Much we have yet to understand.

20. Jun 1, 2013

### falcon32

Thanks for the link Naty1, I am definitely watching the lecture, its probable I'm confused because I don't have the basics down for expansion yet. By the way, you seemed to contradict yourself in that last sentence. If expansion does not require energy, then why did astrophysicists introduce it in the first place upon noticing the acceleration? Why add something frivolous?

This is a bit much, I am expected to understand that the 1st Law holds true in regions like our galaxy, and yet, does not hold true on a Universe scale?

Heartily agree.