Friiedman Fun Facts to know and tell

[jonmtkisco]

Hi Pervect,

I know basic calculus but I have to admit I'm rusty at it. I'd rather minimize the amount of it that I have to do. Yes I agree that's somewhat limiting for me, but I also find that the simpler the terms in which the math is described, the more it's likely to be answering the question I actually asked and not some different question.

It worries me that you feel like you need to throw a lot of calculus into answer my simple questions. I'd rather get as close as is reasonably possible to "yes" and "no".

I understand 100% that your 3 equations can be converted to each other. I am not confused about that, and I don't need it explained to me again. My two basic questions are different -- (1) does the Friedmann Initial Value equation alone, without modification or conversion to another formula, yield an exactly correct solution for expansion in the radiation-dominated era? (2) Does the rho symbol state exactly the same value (and mean exactly the same thing) in the Friedmann Initial Value and Dynamic Friedmann equations?

If you can answer those two questions, then I believe I can generate the quantities you requested.

Thanks for the help,

Jon

 

[Wallace]

Hi Jorrie & Wallace,

I explained my two scenarios clearly several notes ago, but I don't feel like you take the time to understand precisely what I'm saying and what point I'm trying to get at, before you jump on the opportunity to put your obviously excellent math and physics skills on display. You've already convinced me that you're smarter than me, but you have to pay closer attention to my point before you can know whether it is wrong.

No one is trying to convince you that they are smarter than you, it's just that you have to be careful when claiming that a simple equation that is used everyday by an army of Cosmologists contains a hitherto unseen paradox. I'm sure you would agree that the more likely explanation is that there is some error in your application or interpretation of these equations.

I'd encourage you to follow the steps Pervect suggested to let us all know what it is that you are actually calculating and how you are doing it. In addition I'd make some more comments.

I'll address each of your points:

1. Please stick to my terminology in this post. I'm going to call the Friedmann "Initial Value" or "Initial Conditions" equation (Pervect’s #2) the "Friedmann Initial Value" equation because that is the mentor’s terminology. I will refer to Pervect’s #1 equation as the "Dynamic Friedmann" equation.

done

2. In this post, please do not say that a factor is “in” an equation unless the actual symbol for that factor is explicitly in the written expression of the equation. You can say that a factor is “indirectly in” an equation if it is directly substitutable for, or implicit in, one or more factors that expressly appear in the equation. Using the terminology of this post, “pressure” is in the Dynamic Friedmann equation, and it is not in the Friedmann Initial Value equation. However, we all agree that pressure is indirectly in the Friedmann initial Value equation. Enough said.

Wrong. Pressure can appear as an explicit symbol in the IV equation if you haven't first used the energy density equation to calculate functional form of the density $$\rho (a)$$. If you have done this then pressure will not appear as an explicit symbol.

3. In the Dynamic Friedmann equation, both pressure and “rho” appear as separate, mutually exclusive symbols. The rho symbol states energy density and does not state the quantity of pressure; the pressure symbol states pressure and does not state the quantity of energy density. That’s why they use two symbols instead of one! They are related but separate quantities.

On the other hand, only the rho symbol appears in the Friedmann Initial Value equation. Hmmm, something is different here. Now I would appreciate a simple answer to a simple question: Does the rho symbol in the Friedmann Initial Value equation have exactly the same meaning it has in the Dynamic Friedmann equation? Please answer YES if the rho symbol in the Friedmann Initial Value equation states simply the energy density value. Please answer NO if (for example) the rho symbol in the Friedmann Initial Value equation states the SUM of energy density + pressure. You do not need to explain (again) that energy density and pressure are related, we all know that. I just want to know which specific quantity the rho symbol is stating in this equation.

The simple answer is yes. Density has the same meaning in all equations.

4. If the Friedmann Initial Value equation alone is applied exactly as written (by Pervect) to the actual, historical expansion during the radiation-dominated era, does it calculate the correct expansion rate? Yes or No, please. If the answer it yields is correct, then it must reflect the summed effects of both energy density and pressure. In that case, please reconsider your answer to question #3 above.

No. The particular description of the equation I think you are referring to did not have a radiation term, so clearly isn't valid in the radiation era. In addition you need to make sure that that matter density term is correctly changing with scale factor 'a' as the dependence was not explicit in that formulation. Not that Pervect's post was wrong in any way, just trying to be clear.

To be clear, here is the correct IV equation that is valid at any epoch, with the explicit functional forms of the respective densities:

$$(\frac{\dot{a}}{a})^2 = \frac{8\pi G \rho_m(a_0)}{3 a^3} + \frac{8\pi G \rho_{rad}(a_0)}{3 a^4} + \frac{\Lambda}{3} - \frac{K}{a^2}$$

This is equivalent to every other instance of this equation that has been posted in this thread (although some ommitted radiation since it is negligible at most epochs)

5. The Friedmann Initial Value equation was invented before anyone knew whether the universe was expanding, contracting, or static. It was designed as a generic model whose math and logic should work consistently with a wide range of different values of energy density, matter/radiation mix, curvature, etc. So the Friedmann Initial Value equation cannot be intellectually defended by asserting that it works only with the specific values that actually occurred, or something very close to them. The math should work just as well in scenarios that are “unrealistic”. By definition, any test scenario run for comparison purposes is different than the universe we observe. Besides, the “paradox” I found applies at any difference in the matter/radiation initial mix, no matter how small. It’s just easier to see when the difference is larger.

At least we agree that maths is not to blame here. Yes the equations work for all kinds of wacky scenarios (with no paradoxes).

6. Since apparently no one is interested enough in the subject matter of this post to run the simple math for themselves (as opposed to lecturing from the peanut gallery), I can only explain the paradox to you in “intuitive” terms. If you boil the Friedmann Initial Value equation to its simplest conceptual mathematical form, it is this:

In a flat universe, at any given scale factor “a”: a’ $$\propto$$ $$\sqrt{total mass energy}$$

Therefore, any relative decrease in total mass/energy must result in a relative decrease in expansion rate at every scale factor.

You are confusing first and second derivatives "increase in the expansion rate" means that the second derivate of a is changing, not the first derivate that appears in the above equation.

In my opinion, the reason for this seeming "paradox" is that the authors of the equations expected any change in total mass/energy to result in a change in curvature, not in expansion rate. The paradox arises only when the universe is forced to be flat.

Whoa! I'm not sure where you are getting that statement from, but it is simply not true. I have no idea where that idea came from??

Jorrie, in response to your point about the initial expansion rate, I note that in a flat universe, a scenario with lower than historical initial mass/energy but with the historical matter/radiation mix requires a decreased initial expansion rate and results in a faster deceleration of the subsequent expansion. So the decreases are mutually reinforcing and don’t counteract one another. Again, the universe could not maintain flatness at every point in time if it worked any other way; in order to remain flat, any given universe must always expand at exactly the escape velocity of its contents.

What you are effectively doing here (I think) is in a round about way describing in words the definition of the critical density:

$$\rho_{crit} = \frac{3H}{8\pi G}$$

that relates the expansion rate H to the density required for spatial flatness at any time. This is good, you've realized independently an important step in understanding the equations.

I have to say though, I still don't see any paradox, just maths working as it is supposed to. Can you describe what part of this thing doesn't work as it should? You've posted number in previous posts and I think assumed that we would see what it is that you find disturbing about those values, but it is not clear. Please be explicit about what it is that is paradoxical.

 

[jonmtkisco]

Hi Wallace, et al,

You said: "The particular description of the equation I think you are referring to did not have a radiation term, so clearly isn't valid in the radiation era."

Yes! A straight answer to a simple question. Does ANYONE ELSE disagree with this statement, that the version of the Friedmann Initial Value equation quoted by Pervect, alone, DOES NOT accurately calculate the expansion rate during the radiation dominated era?

Jon

 

[Wallace]

How could it be valid in the radiation dominated era, there is no radiation term!? Is that really the source of all this cofuffle??

 

[Chris Hillman]

Sigh...

How could it be valid in the radiation dominated era, there is no radiation term!? Is that really the source of all this cofuffle??

Alas, I noticed numerous distinct misconceptions in this thread Having just spend some time trying to correct a similar morass of muddlement in another thread in this forum, perhaps I can leave it at that

 

[Jorrie]

Hi Wallace, et al,
You said: "The particular description of the equation I think you are referring to did not have a radiation term, so clearly isn't valid in the radiation era."

Yes! A straight answer to a simple question. Does ANYONE ELSE disagree with this statement, that the version of the Friedmann Initial Value equation quoted by Pervect, alone, DOES NOT accurately calculate the expansion rate during the radiation dominated era?
Jon

Despite what Wallace said in his next reply, I understood pervect's original initial values equation

$$\left(\frac{a'}{a}\right)^2 = \frac{8 \pi G}{3} \rho + \frac{\Lambda}{3} - K \frac{c^2}{a^2}$$

to mean that $\rho = \rho_m+\rho_r$, as George has confirmed later in reply#25/26 before. I agree that it is better to state such things explicitly, but normally it is clear from the context what is meant, i.e. since $\Lambda$ is specifically shown, vacuum energy density is obviously not included in $\rho$.

 

[Jorrie]

Hi Jorrie & Wallace, ...

Jorrie, in response to your point about the initial expansion rate, I note that in a flat universe, a scenario with lower than historical initial mass/energy but with the historical matter/radiation mix requires a decreased initial expansion rate and results in a faster deceleration of the subsequent expansion.

(Emphasis mine)

Right on the first count, but wrong on the second, as clearly shown by the dynamic Friedmann equation:

$$3 \frac{a''}{a} = \Lambda - 4 \pi G \, \left(\rho+ \frac{3P}{c^2}\right)$$

Reduce rho and/or P and the early acceleration becomes less negative (smaller deceleration). I'm curious as to what gave you the opposite idea.

 

[Wallace]

Ah yeah sure, as long as the density is decomposed into matter and radiation and their respective densities are properly calculated as functions of $$a$$ then that equation is valid.

This really is a silly thread!

 

[jonmtkisco]

Hi Jorrie, Wallace, et al,

Despite the fact that we've taken a few u-turns (for which I bear at least my share of muddle), I think we're getting to a helpful consensus here. I really do appreciate your energetic engagement in the dialogue.

Let me attempt to recap the Friedmann discussion:

1. The symbol "rho" has precisely the same meaning in the Friedmann Initial Value and Dynamic Friedmann equations. Rho states the quantity of energy density only, and does not directly state the quantity of pressure. Rho is the sum of Rho$$_{M}$$ and Rho$$_{R}$$. The Rho$$_{R}$$ figure indirectly captures the relationship of energy density to pressure as a function of time.

2. Therefore the Friedmann Initial Value equation is 100% valid to use alone to describe the expansion rate during both the radiation-dominated and matter-dominated eras. Regardless of the particular mix of matter-to-radiation entered into the formula, the calculated expansion rate should be correct without requiring any additional step of adding or subtracting a separate pressure component.

3. Therefore, for example, when we run a "100% matter" scenario using the historical initial value for rho but setting the mix at 100% matter, 0% radiation, the expansion rate calculated at every point in time (post inflation) by the Friedman Initial Value equation should be correct, without any need to double or halve the calculated result to correct for a pressure component.

If we are now finished torturing the poor Friedmann formula (whew!) I hope we can turn back to consider the concern I raised about the specific result I calculated for the 100% matter scenario, as compared to the vanilla "historical" scenario. I'm not going to refer to it as a "paradox" anymore, because that term is too grandiose. In my next post I will try to walk through the specific numbers. By the way, I'm happy to share my spreadsheet with anyone who is in interested enough to actually examine it.

Jorrie, you correctly pointed out a mistake in the final paragraph of my post #34. Oops. What I should have said was as follows:

"A scenario with lower than historical initial mass/energy but with the historical matter/radiation mix must have (as compared to the pure "historical" scenario) a relatively decreased initial expansion rate and a relatively decreased expansion rate at every point of the subsequent expansion. So these decreases are mutually reinforcing and don't counteract one another. This supports the underlying message that in order to remain flat, any given universe must always expand at exactly the escape velocity of its contents. When the matter/radiation mix is kept constant, then at any given scale factor "a", a universe with lower total mass/energy will always have a slower expansion rate. And at any given time, that universe will always be smaller."

Clearly in that particular scenario, the lower expansion rate over time is primarily an artifact of the decreased initial expansion rate, and the deceleration rate at every point in time is lower (less negative) in the smaller universe than in the larger one.

Jon

 

[Jorrie]

3. Therefore, for example, when we run a "100% matter" scenario using the historical initial value for rho but setting the mix at 100% matter, 0% radiation, the expansion rate calculated at every point in time (post inflation) by the Friedman Initial Value equation should be correct, without any need to double or halve the calculated result to correct for a pressure component.

Jon, what you wrote is not far off, but be aware that setting the matter portion to 100% depends on when (in which epoch) you do it. If you set radiation energy to zero today, it obviously has negligible influence. If you do it at $t = 10^{-32}$ seconds, while maintaining the 'historical' energy density, the mass energy density must be $\Omega_r(t_0)/a(t) \approx 10^{22}$ times what it was in your historical case. This gives a completely different evolution of $a(t)$. Similarly, if you keep the historical matter density what it was, with zero radiation energy, the total energy density drops by $\approx 10^{22}$ times, again with a vastly different $a(t)$.

I have a suspicion that part of your "paradox" originates from treating these facts incorrectly. Finally, your corrected paragraph is still a little confusing, e.g.

... When the matter/radiation mix is kept constant, then at any given scale factor "a", a universe with lower total mass/energy will always have a slower expansion rate.

The matter/radiation energy density mix does not remain constant over time, it evolves with "a". Again, it is confusing to talk about "total mass/energy" - why not stick to densities, because that's what the Friedmann equations are designed to handle.

 

[Chris Hillman]

Suggest a do-over, after taking time off for further study

IMO, this thread has been rendered unreadable by the failure of several participants, particularly jonmtkisco, to simply write out equations, to clearly define nonstandard terms (or to employ standard terms with the standard meaning), and so on. It is also terribly mistitled, since the "fun facts" appear to be largely common misconceptions, as jon seems to have partially acknowledged.

I am requesting that this thread now be locked since it seems to be marching in circles. Jon, if you still wish to discuss the FRW models, I suggest that you

• take a few months/weeks to study the excellent advanced undergraduate textbook by D'Inverno, Understanding Einstein's Relativity (the last chapter offers an excellent overview of the FRW models, but I stress that it seems to me that you would benefit greatly from reading the entire book as closely as time permits),
• start a new thread with a post in which you write out what you think the relevant equations are and ask what if anything is wrong with your understanding.

In my opinion, these steps should result in much better informed, more efficient, less contentious, and more interesting discussion!

 

[jonmtkisco]

Here are the simplified versions of the Friedmann equations I have been using, which are of course directly derived from the forms Pervect originally provided:

Friedmann Initial Value equation:

H$$^{2}$$ = (a'/a)$$^{2}$$ = 8$$\pi$$G$$\rho$$/3

Dynamic Friedmann (or Acceleration) equation:

a"/a = -(4$$\pi$$G/3)($$\rho$$ + 3P)

As of today, I have resolved the "anomaly" to own satisfaction. I believe that (despite multiple assertions by others to the contrary), my math was completely correct and my scenarios were completely valid test cases for the Friedmann calculations. The only thing I did wrong was to misinterpret the meaning of the results I calculated.

I erred in concluding that the more rapid decrease in mass/energy (or mass/energy density, for those who insist) during the radiation-dominated era was directly causally connected to the more rapid decline in the expansion rate. But in reality the relationship is indirect and much more subtle.

It turns out that during the radiation-dominated era, the universe does expand at the "escape velocity" of its mass/energy contents at any given moment, but it decelerates twice as quickly (as compared to a 100% matter universe). In this scenario, the "gravity density" is about twice the mass/energy density. The higher gravity density (of course) is caused by the positive pressure of radiation, which as Pervect pointed out effectively doubles the gravity density ($$\rho$$+3P, where for radiation P=$$\rho$$/3).

This higher deceleration rate is what enables the expansion rate, over time, to drop in perfect lock-step with the decline in total radiation mass/energy. (For example, if the deceleration rate were driven only by mass/energy density, it would compensate only for volume dilution over time, not the extra decrease in radiation mass/energy caused by redshift.)

The reason I didn't recognize this subtlety earlier in this post is that I was concerned that if the expansion rate declined faster than mass/energy density, the universe could not preserve its flatness. I had assumed that if gravity density was different from mass/energy density, geometric flatness would be determined by the gravity density. But apparently this is not so, which I find to be a bit puzzling.

In that respect, the Friedmann Initial Value equation is misleading if taken at face value. Although it is completely accurate at calculating the expansion rate at any given scale factor, what it doesn't alert you to is that the initial expansion rate (at t=0 when inflation ends) must start out twice as high (for any given mass/energy density) if the mass/energy mix is radiation-dominated. (I don't think I ever would have noticed that interesting phenomenon if I had done my calculations using only density terms rather than total mass/energy).

Thanks again for your helpful insights and patience,

Jon

 

[Chris Hillman]

Judges 16:12

Here are the simplified versions of the Friedmann equations I have been using, which are of course directly derived from the forms Pervect originally provided:

You are not using the latex markup correctly; your equations are unreadable. To learn how to use latex markup, find a post with properly formatted equations and hit the "reply" button but don't submit the "reply"; rather, in the pane you should see the post quoted with the markup visible. You can also ask for help at "Forum Feedback"

Now let us break off this thread.

 

[Wallace]

Inflation really must be a sophisticated concept in order to generate exactly the right expansion rate for a flat universe. The expansion rate at which it stops inflating must not only match the density of the mass/energy that's about to be created, it also must match the exact matter/radiation mix that it's going to create.

Inflation is actually much more ham fisted than that. In order to get inflation to occur there must be a component of the energy density with a significantly negative pressure. As should be clear to you now, components with a lower (more negative) equation of state will tend to dominate over components with a higher (more positive) equation of state as time goes on. For instance radiation begins as the dominant term, but it's high w=1/3 means that its energy density drops more rapidly than matter with w=0, leading to the matter dominated era.

We can describe curvature by analogy to energy and see that it has an equation of state of w=-1/3. Therefore if there is a period in the early Universe in which some 'inflaton' field with a very negative equation of state (w~-1) then this field will dominated and reduce curvature to a negligible level. The real trick is to find out why the inflaton field turned off at some point. The take home message though is that inflation is not as or finely tuned as you suggest, it will ensure flatness regardless of what is in the Universe.

I have to agree with Chris in that you should try and read some good cosmology textbooks. Clearly you would enjoy them as you seem interested, but they would give you a much broader base for your knowledge. Your interpretation of the results is still somewhat backwards and a view of the big picture would help you greatly.

 

[pervect]

This thread has been dragging on. I've had one call to lock it, and another comment that the thread is silly, but one other comment by email that it has been educational.

Personally, I'm rather tired of it too, however.

I've decided that I'm going to give the thread a rather short fuse - 24 hours - and that after that, it gets locked.

This will give people a chance to "wrap up" any technical points they want to make, but prevent it from being a distraction, drain, and irritant to the forum.

 

[jonmtkisco]

Hi Pervect,

I ran a "radiation only" scenario, and happily, the universe remains geometrically flat because its instantaneous expansion rate is always exactly equal to the escape velocity of the total radiation mass/energy. As with all of these flat-universe expansion curves, the expansion continues forever at an ever decreasing rate.

In that scenario of course, the number of photons doesn't decrease, but the energy-per-photon continues decreasing forever. Which raises a question, is there an absolute minimum energy-per-photon threshold, or can a photon possesses an energy that is infinitely close to zero, if its wavelength is stretched long enough?

Also, although I understand that the Friedmann equations mathematically treat the gravity added by radiation pressure as having no direct effect on geometric curvature, is there a straightforward "physical" explanation for why, in the abstract, one "flavor" of gravity directly affects geometric curvature while another concurrent "flavor" of gravity does not?

I guess that's a dumb question because it isn't really a case where there are two different "flavors" of gravity. The reality is simply that twice as much gravity (of any flavor) is needed to offset the simultaneous decline in total mass/energy, and thereby maintain flatness.

It strikes me that in a flat universe like ours, "there is no such thing" as free radiation that isn't constantly degenerating (losing energy). Free radiation is fundamentally unstable.

Jon

 

[jonmtkisco]

Hi Mr or Ms Hillman,

I must protest your abusive and non-responsive posts.

Jon

 

[Chris Hillman]

Jon, click on "UserCP" at the top of the display shown (probably) by your browser when you are looking at (almost) any PF page. Click on buddy/ignore lists and add me to your ignore list. After I post this I will do the same for you. This will ensure that henceforth we don't see each other's posts, which I trust will be agreeable to you.

 

[jonmtkisco]

Well I guess there's no point in responding to Chris because he's ignoring me now.

I'm interested in robust dialogue, not cutting it off. Obviously Chris has tremendous expertise and I'm quite interested in reading what he has to say. I want him to direct some of that towards the substance of my questions...

I'm going to end this thread now and pursue questions in separate threads.

Jon

 

[pervect]

OK, here is my final $.02 on the topic. I'm going to lock the thread after this as I mentioned previously, because I think that everything that needs to be said will have been said and the natives here on the forum are getting restless (plus I'm getting tired of it, too).

This also implies that I don't want to see new threads on this same topic for at least a couple of weeks, if not longer.

I would like to go on record as to encourage people such as jonmtkisco to brush off their rusty calculus skills and actually do some work. Doing work does take work, but it pays off in understanding. That's another reason I'm going to lock the thread, I'm hoping it will encourage more thought and less typing, and this thread is getting way too long.

With all the factors of c and G put back in we get the following symbolic solutions for a(t), rho(t) and P(t) with K=0

case1: no pressure (matter dominated universe)$$a(t) = a_0 \, t^{\frac{2}{3}}$$
$$rho(t) = \frac{1}{6 \pi G t^2}$$
$$P(t) = 0$$

these match up with MTW's equations (pg 735) except for the unit conversion factors.

To verify that these are correct, put them into F1 and F2 (the dynamic and initial value Friedmann equations), and show that they satisfy both equations.

Satisfying both F1 and F2 (the dynamic and initial value equations) is necessary to satisfy Einstein's field equations. Furthermore, it is both necessary and sufficient for F1 and F2 to both be simultaneously satisfied to ensure local energy conservation.

case 2: radiation dominated
$$a(t) = a_0 \, t^{\frac{1}{2}}$$
$$rho(t) = \frac{3}{32 \pi G t^2}$$
$$P(t) = c^2 \frac{rho(t)}{3} = \frac{c^2}{32 \pi G t^2}$$

Again, these can be verified by showing that they satisfy F1 and F2.

In geometric units, for the radiation dominated case, P = rho/3. In non-geometric units, this becomes P = c^2 rho / 3. This is the proper relationship between density and pressure for a universe of pure radiation. P = c^2 rho/3 is an example of what is called an "equation of state" and is the "equation of state" of radiation. As implied previously, the "equation of state" for cold matter is just P=0.

If Jon's spreadsheet gives values that are compatible with these symbolic expressions, great. If not, there is probably a problem with the spreadsheet.

If Jon's spreadsheet is giving these (correct) results, then Jon's problem is with the interpretation of his results, not with the spreadsheet.

I believe that Jon may be attempting to splice these two solutions together, and that may be the source of his difficulty. A smooth splice must have the following properties at a minimum:

a(t) must be continuous (no sudden jumps) at the transition
da/dt must be continuous (no sudden jumps) at the transition, because d^2a/dt^2 must exist

Such a transition is *not* a part of standard cosmology. The standard Friedmann cosmology assumes that there is no significant coupling between radiation and matter.

[add]
Initially, I thought there wasn't any smooth way to splice together a radiation dominated cosmology to a matter dominated cosmology with K=0, but I've changed my mind.

I'll note that such a solution is not a standard textbook solution. Standard textbook solutions assume little coupling between matter and radiation.

The splice I think I've found is:

a(t) = sqrt(t), t<1
a(t) = (t+37/27)^(2/3) t>1

a(t) and a'(t) are continuous at t=1, there is however a step function change in a'' at t=1. (I don't think this is a problem).

One should be able to work forwards from this using both Friedmann equations and the assumption that K=0 and $\Lambda=0$ to get a valid solution for rho(t) and P(t). Interestingly enough, rho(t) does not appear to be continuous either.