Friiedman Fun Facts to know and tell

[jonmtkisco]

Hi SpaceTiger, Pervect & Hellfire,

Here are some follow-up thoughts about the Friedmann equation for expansion. Pervect, thank you for using Noether's Theorem to demonstrate that normal momentum (of movement) is conserved. The next challenge is to demonstrate that the "momentum-like" continuation of the original expansion of space is conserved.

1. By playing around with a spreadsheet, I have determined that if mass is held constant, the "momentum-like" continuation of expansion is conserved, by the formula:

PP = \Deltavolume^{2} /volume

where PP is the "momentum-like" continuation of expansion. So my earlier suggestion that \Deltavolume/ \Deltatime might be the metric turns out to be wrong. It's a relief to find that this "momentum-like" quantity remains constant in the Friedmann equation when mass/energy is constant.

2. However, that finding merely leads to the next question, which I find to be of great concern. That is, that the Friedmann equation calculates that when the mass/energy of the universe increases (e.g., due to the cosmological constant), the expansion rate increases. And most relevant, when mass/energy decreases, the expansion rate decreases. This is an important scenario, because under the traditional \LambdaCDM model, the total mass/energy of the universe is held to have decreased dramatically during the radiation dominated era, as radiation gave up energy to expansionary redshift. Sure enough, the Friedmann equation dutifully calculates that the "momentum-like" continuation of expansion declines throughout the radiation-dominated era. Yet if mass/energy is held constant at its pre-decline value, Friedmann calculates that the expansion rate remains higher (and higher than we observe). You can convince yourself of this just by noting that mass is in the top line of the Friedmann equation, and R is not.

Now I really need a clear explanation as to how, in a GR-based model, a large decline in mass/energy can cause a large decline in the expansion rate. That certainly violates the expected behavior of a "momentum-like" quantity. Less gravity ought to cause faster expansion than more gravity. That's pretty basic.

The Friedmann equation was created in the 1920's when there was no observational evidence that the universe was expanding, and therefore the idea that the universe might not be purely adabiatic, because its mass/energy might actually change over time, was not incorporated in the formula. If anyone is aware of this specific question having been addressed subsequently by mathematicians, I would very much appreciate a reference.

Jon

 

[pervect]

2. However, that finding merely leads to the next question, which I find to be of great concern. That is, that the Friedmann equation calculates that when the mass/energy of the universe increases (e.g., due to the cosmological constant), the expansion rate increases.

The detailed explanation is in terms of Einstein's field equations.

A short, terse, and somewhat popularized answer is that one has to consider not only energy, but pressure, as causing gravity.

Details can be found in Baez's paper:

http://www.math.ucr.edu/home/baez/einstein/einstein.html

which I'll quote from below. (I'll certainly encourage people to read the original paper in its entierty, though).

We promised to state Einstein's equation in plain English, but have not done so yet. Here it is:

Given a small ball of freely falling test particles initially at rest with respect to each other, the rate at which it begins to shrink is proportional to its volume times: the energy density at the center of the ball, plus the pressure in the 'x' direction at that point, plus the pressure in the 'y' direction, plus the pressure in the 'x' direction.

In the final section of this article, we will prove that this sentence is equivalent to Einstein's equation.

Note that "by the rate at which the ball shrinks", Baez means the second derivative of the volume divided by the volume, i.e (d^2 V / dt^2) / V. This is explained in the paper.

Here the volume V is measured in the co-moving frame of the particles (which are all at rest relative to each other).

So, while the cosmological constant causes empty space to have a positive energy, it also causes empty space to have a negative pressure. The gravitational effects of the positive energy are smaller than the gravitational effects of the negative pressure.

A small ball of particles will contain positive energy, but because gravity is driven by rho+Px+Py+Pz, which equals rho+3P when the pressure is isotropic, the net effect on a small ball of particles in a space-time with a cosmological constant will be that they will expand, i.e. that d^2V/ dt^2 will be positive, in spite of the fact that the ball of particles contains a positive amount of energy.

Note that there are no non-gravitational effects of the negative pressure because the pressure is the same outside the ball of particles as it is inside.

 

[jonmtkisco]

Hi Pervect,

Thanks for the explanation, although I am very familiar with the concept of negative pressure. You may have notice that I explained the same concept to Holocene in a recent post about why the expansion rate is expanding.

Because we are also familiar with negative pressure and the cosmological constant, that's why I focused in my post on the opposite situation, when mass/energy was decreasing during the radiation-dominated period. I said:

"And most relevant, when mass/energy decreases, the expansion rate decreases. This is an important scenario, because under the traditional CDM model, the total mass/energy of the universe is held to have decreased dramatically during the radiation dominated era, as radiation gave up energy to expansionary redshift. Sure enough, the Friedmann equation dutifully calculates that the "momentum-like" continuation of expansion declines throughout the radiation-dominated era. Yet if mass/energy is held constant at its pre-decline value, Friedmann calculates that the expansion rate remains higher (and higher than we observe). You can convince yourself of this just by noting that mass is in the top line of the Friedmann equation, and R is not.

Now I really need a clear explanation as to how, in a GR-based model, a large decline in mass/energy can cause a large decline in the expansion rate. That certainly violates the expected behavior of a "momentum-like" quantity. Less gravity ought to cause faster expansion than more gravity. That's pretty basic."

Subsequent to writing the post, it occurred to me that maybe the decline in mass/energy coincided with the decline in the expansion rate because gravity was much, much higher than the simple mass/energy of the radiation at the time. Since radiation is supposed to have positive pressure, and the temperature was supposed to be very high, maybe these factors increased the gravity dramatically above what matter and radiation alone would have caused. And then maybe the forced expansion of the universe, (due to the supposed original expansion driven by the 'initial conditions') happened to drive gravity down dramatically at a rate which coincided with the startlingly abrupt deceleration rate.

But then it occurred to me that this could not be a valid case, because it would have caused total gravity during that period to exceed the expansion rate. And that would have meant that the universe was not flat. But it is believed that it was flat.

So, I have no explanation still, as to why a decrease in mass/energy would cause a decrease in the expansion rate. Any suggestions or referrences would be appreciated.

 

[pervect]

Hi Pervect,"And most relevant, when mass/energy decreases, the expansion rate decreases.

OK, I'm going to have to get a bit "nitpicky" here, because there's some sort of communication issue here.

Einstein's equation doesn't say anything about "the expansion rate" being a function of mass energy.

Einstein's equation can be interpreted as saying that for a small volume, the second derivative of the volume of a sphere of comoving particles, is proportional to the mass-energy density at the center of the sphere plus three times the pressure (assuming isotropy of the pressure).

The "expansion rate" will be some function of the first derivative of the rate of change of the volume of the comoving sphere.

So if we define the scale factor of the universe as a(t) (the a(t) in the FRW metric), and the expansion rate as da/dt, then the volume of a sphere of particles will be V0*a(t)^3, the rate of change of the volume will be 3 v0 a(t)^2 da/dt, and the second derivative of the rate of change of the volume will be

d^2V/dt^2 = 6 v0 a(t) (da/dt)^2 + 3 v0 a(t)^2 d^2 a/ dt^2

This is what's proportional to energy density + 3*pressure (rho+3P for short).

So if by "expansion rate" we mean da/dt, then you can see that dV/dt is proportional to the expansion rate, and that d^2V/dt^2 depends on both the expansion rate and it's first derivative (the first derivative of da/dt, the second derivative d^2a/dt^2).

Thus, we can see that we cannot solve for da/dt knowing rho+3P, because it could have literally any value.

This is an important scenario, because under the traditional CDM model, the total mass/energy of the universe

You've totally lost me here.

The "total mass-energy of the universe" is a phrase which is rather ill-defined. For example, see MTW's "Gravitation" which says

There is no such thing as the energy (or angular momentum, or charge) of a closed universe, according to general relativity, and this for a simple reason. To weigh something one needs a platform on which to stand to do the weighing.

One might guess that you mean by "total mass energy of the universe" that you multiply the density, rho, by the comoving volume.

But we don't need to know the "total-mass energy of the universe", and I don't see why it's relevant, anyway.

Einstein's field equation, which gives us the Friedman equation, is a purely local equation, and the question of what the "mass-energy of the universe" is is irrelevant.

Furthermore, while there are tricky issues with regards to the global defintion of energy conservation, there aren't any issues with regards to local energy conservation, which is already built into Einstein's field equations.

So in short, you only need Einstein's field equations to get Friedman's equations, and the question of what "the total mass-energy of the universe" might be isn't relevant to the problem - i.e. we only need to know the density of mass energy, and the pressure.

 

[jonmtkisco]

Hi Pervect,

Well you can feel free to analyze the problem using the Einstein Field Equations. I haven't gotten there yet, I'm still working with the 2nd Friedmann equation.

You must be kidding if you are suggesting that "total mass/energy" isn't relevant to the Friedmann equation. As you yourself pointed out, and Hellfire documented, and I subsequently accepted, the reference in the 2nd Friedmann equation to "mass/energy density" is nothing more than "total mass/energy" divided by volume. It couldn't be more straightforward.

Don't give me a bunch of mumbo-jumbo about how "total mass/energy" is a meaningless quantity. If that's so, then the 2nd Friedmann equation by your definition is invalid. Again, all the 2nd Friedmann equation does is to divide this supposedly fictitious "total mass/energy" by volume to calculate energy density, which is the cornerstone of the equation.

 

[jonmtkisco]

Hi Pervect,

Given that, for purposes of the 2nd Friedmann equation "total mass/energy" and "energy density" are essentially interchangeable (the former being the latter multiplied by volume), my original question is equally valid if asked using energy density terminology:

"When energy density decreases, the expansion rate decreases. This is an important scenario, because under the traditional LCDM model, the energy density of the universe is held to have decreased dramatically during the radiation dominated era, as radiation gave up energy to expansionary redshift. Sure enough, the Friedmann equation dutifully calculates that the "momentum-like" continuation of expansion declines throughout the radiation-dominated era. Yet if energy density is held constant at its pre-decline value, Friedmann calculates that the expansion rate remains higher (and higher than we observe). You can convince yourself of this just by noting that energy density is in the top line of the Friedmann equation, and R is not.

Now I really need a clear explanation as to how, in a GR-based model, a large decline in energy density can cause a large decline in the expansion rate. That certainly violates the expected behavior of a "momentum-like" quantity. Less gravity ought to cause faster expansion than more gravity. That's pretty basic."

 

[hellfire]

Yet if energy density is held constant at its pre-decline value, Friedmann calculates that the expansion rate remains higher (and higher than we observe). You can convince yourself of this just by noting that energy density is in the top line of the Friedmann equation, and R is not.

Although you seem to be able to understand the math, you rely on some wrong heuristic arguments to understand the physics. Such a scenario we had already in this thread where I was tring to put in math terms your words and questions, and at the end I think we got the right answer. So humbly I suggest to get a good reference about cosmology or general relativity and rethink carefully in math terms what you have written above; what are the conditions for this to hold and how it relates to the apparent (incorrect) "paradox" in your argument.

 

[pervect]

Less gravity ought to cause a slower deacceleraton than more gravity. And it does. Less gravity doesn't have anything to do directly with the rate of expansion at all.

Gravity causes acceleration - it doesn't cause velocity.

The dynamic Friedman equation

<br /> 3 \frac{a&#039;&#039;}{a} = \Lambda - 4 \pi G \, \left(\rho+ \frac{3P}{c^2}\right)<br />

(FromThe Wikipedia

should be a consequence of the initial value equation

<br /> <br /> \left(\frac{a&#039;}{a}\right)^2 = \frac{8 \pi G}{3} \rho + \frac{\Lambda}{3} - K \frac{c^2}{a^2}<br />

along with the local energy conservation principle which says that density and pressure must satisfy

<br /> \frac{d}{dt} \left(\rho V \right) = -P \frac{d V}{dt} <br />

where V, the volume of a fluid element, can be taken as V0 * a(t)^3 for some constant V0.

This is discussed on MTW, pg 728-729, where it is also pointed out that the full version of Einstien's equation automatically gives you the the above conservation principle without having to assume it separately. (It would take some work to get this conservation principle out of Baez's approach, though Baez says it's possible).

(I was originally having some problem confirming this, but I think I'm now in a position to say that's what I get, too. My textbook uses geometric units, the Wiki uses standard units, just to make life interesting).

Note that with the re-defintion of variables suggested by the Wikipedia, and the assumption that a'=0 this dynamic equation is essentially what Baez derives - the main point is that

a'' / a is proportional to (rho+3P) when rho and P include contributions from Lambda rather than Lambda being a separate variable. Don't sweat the units overmuch, Baez is using some funky units, the proportionality is what matters.

I assume it should also be possible to work backwards from the dynamic equation, and the above relation between rho and P to get the initial value equation, one should be able to regard this as a "solution" to the second order differential equation given by the dynamic equation along with the energy conservation equation, since it's possible to work forwards from the initial value equation + the local energy conservation equation to the dynamic equation.

 

[pervect]

Add: if you like Newtonian analogies, the dynamic equation is a bit like

m d^2 x / dt^2 = force

the initial value equation is like

(m/2) (dx/dt)^2 = E^2 - V(x)

V(x) being a potential function.

note the formal similarities. I'm not sure how far this analogy can be pushed, though - that's why I say "a bit like".

Specifically, the dynamic friedman equation and the dynamic Newtonian equation both involve a second derivative with respect to time, while the IV versions involve only the square of the first derivative.

You also go from the initial value equation to the dynamic equation in much the same way - differentiate with respect to time. In the Newtonian example, we get

m (dx/dt) = -dV/dx = force.

This analogy may make it clearer as to what I mean when I say gravity causes acceleration (d^2 x/dt^2), and not velocity (dx/dt).

 

[jonmtkisco]

Hi Pervect,

I agree with you that gravity causes acceleration, and that any subsequent change in expansion velocity is therefore a second order effect. If you interpreted me to say anything different, that's sloppy wording on my part. Sorry about that.

Thanks also for the reference to Baez's GR website. I was very encouraged that he describes Vol"/Vol to be a critical metric in the Einstein Equations, as I found it to be when fiddling with the 2nd Friedmann equation. As I said, I found that Vol"/Vol is a constant during the Friedmann expansion, if total mass/energy (density*volume) is held constant.

In any event, the "paradox" (as Hellfire describes it) that I'm trying understand here is so simpleminded that the communication gap obviously exists only because of my inability to explain myself clearly.

Maybe it will help to just walk through the steps I took. I have a spreadsheet that calculates radius, volume, mass/energy, radius', vol', and vol" at various intervals between the end of inflation and 1400GY in the future. I've run various "what if" scenarios.

Here is the "what if" scenario that confuses me. I asked, "how would the Friedmann expansion rate calculation change, if the total mass/energy (and density) had its historical value at the end of inflation, but if that mass/energy were comprised 100% of matter and 0% of radiation." To simplify the question, I omitted the cosmological constant from the calculation. My reason for using 100% matter is to create a scenario where mass/energy remains a constant through time. No other significance.

The result is that in my test case, at any given absolute volume, the calculated radius' (meters/second) is more than 1000 times faster than what is calculated using the historical original mix of matter and radiation.

I'm not suggesting that this difference in calculated expansion rates between the "what if" and "historical" scenarios is a surprise. I'm just trying to interpret what it means. The following interpretation seems reasonable to me:

"For any given initial expansion rate and energy density, the 2nd Friedmann equation will calculate lower total expansion over any given time interval, if total mass/energy declines during that time interval (due to redshift of free radiation), than if total mass/energy had remained constant during that time interval." (Again, "total mass/energy" = Friedmann energy density*volume).

It indeed seems paradoxical if, starting from the same initial energy density value, a more rapid decrease in density over time results in a smaller universe over time. What gives?

(As one possible "fix" to this problem, I understand that adding positive pressure (of radiation) to the "historical" calculation might eliminate the discrepency, but as I mentioned earlier, adding a pressure factor would increase gravity during the early universe. I think that a higher gravity might cause the universe to be "closed" rather than "flat".)

 

[pervect]

Hi Pervect,

Maybe it will help to just walk through the steps I took. I have a spreadsheet that calculates radius, volume, mass/energy, radius', vol', and vol" at various intervals between the end of inflation and 1400GY in the future. I've run various "what if" scenarios.

Here is the "what if" scenario that confuses me. I asked, "how would the Friedmann expansion rate calculation change, if the total mass/energy (and density) had its historical value at the end of inflation, but if that mass/energy were comprised 100% of matter and 0% of radiation." To simplify the question, I omitted the cosmological constant from the calculation. My reason for using 100% matter is to create a scenario where mass/energy remains a constant through time. No other significance.

The result is that in my test case, at any given absolute volume, the calculated radius' (meters/second) is more than 1000 times faster than what is calculated using the historical original mix of matter and radiation.

Let me see if I understand this. You assumed that the denisity, rho, remains the same, and when you say that it "was composed 100% of matter", you've kept rho constant, but you've reduced P to zero. You have not kept K, the Gaussian curvature, constant.

Assuming that this is actually what you did, it's easiest to explain this from the dynamic equation.

Numerically, using the dynamic form of the equation, rho remains consant, and P goes to zero. Because P = 1/3 rho for radiation, rho+3P should drop to half its value when you convert from a radiation dominated universe to a matter dominated universe, meaning that the deaccleraton also drops to 1/2.

The consequence of this is that you reduce the deacceleration of the universe, meaning that if you project the universe forward to the current age, it will be a lot bigger.

I'm assuming that that's when you talk about the calculated radius, you talk about the calculated radius "now".

I'm not suggesting that this difference in calculated expansion rates between the "what if" and "historical" scenarios is a surprise. I'm just trying to interpret what it means. The following interpretation seems reasonable to me:

"For any given initial expansion rate and energy density, the 2nd Friedmann equation will calculate lower total expansion over any given time interval, if total mass/energy declines during that time interval (due to redshift of free radiation), than if total mass/energy had remained constant during that time interval." (Again, "total mass/energy" = Friedmann energy density*volume).

I think this is wrong, or perhaps you just lost me again. I suspect, however, that you are under the illusion that total mass energy causes gravity. This is not correct. Pressure also causes gravity, i.e. what is important is not rho, but rho+3P.

An modified version of an example I worked out and put in the wikipedia might help clarify what this means, though it's not cosmogical, it's related to the Komar mass.

The original article is at http://en.wikipedia.org/wiki/Mass_in_general_relativity with a different slant.

Suppose you have a heavy hollow spherical pressure vessel, and in the center you have an matter-antimatter bomb. You put accelerometers at various locations to measure the force of Newtonian gravity - the interesting points are on the outside and inside surface of the pressure vessel, and at some distant location.

You explode the bomb, converting matter into energy, and generating relativistically significant pressures in the interior of the pressure vessel in the process. What happens to the gravity at the various accelerometers (assuming that you wait until the system reaches some sort of equilibrium to take your readings?).

For the accelerometers far away from the pressure vessel, and even for the accelerometers on the outside surface of the pressure vessel, there is no change in the gravity readings on the accelerometers. Not so on the accelrometer mounted on the inside of the pressure vessel. It's reading essentially doubles when you set off the bomb.

What happened? Pressure causes gravity (not just total mass energy), and the extra pressure due to the explosion causes the extra gravity on the inside accelerometers. The gravity does not change in the exterior region, because there is a tension in the walls of the pressure vessel. This reduces the gravitational contribution of the walls of the pressure vessel, so that there is no change in the accelerometers measuring the gravity outside the pressure vessel.

So if you look only at the exterior of the pressure vessel, you might think that "total mass-energy causes gravity". But when you look at the accelerometers on the interior of the pressure vessel, (which are quite well engineered so that they don't melt :-)), you realize that pressure also causes gravity.

 

[jonmtkisco]

Pervect,

Thanks for the answer. I agree that pressure causes gravity.

What befuddles me is why there are two separate Friedmann equations, and how to apply them both at the same time. Can one derive a correct expansion solution by using only one of the two equations?

In the 2nd Friedmann equation, why is there no pressure component? How can the expansion rate be modeled accurately without including pressure?

Is rho the same thing in both Friedmann equations? That is, is it always equal to mass/energy "without" the pressure component? Is it always "total" mass/energy, including both "matter energy" and "radiation energy"? If so, then why are the two equations not precisely interchangeable?

How can a"/a require a pressure component, when a'/a does not? Mathematically, the two figures should be directly related.

Jon

 

[Jorrie]

2nd Friedmann Equation

In the 2nd Friedmann equation, why is there no pressure component? How can the expansion rate be modeled accurately without including pressure?

Jon, just for clarification: I know the first Friedmann equation as the 'energy equation' or the 'expansion rate equation' (which is often just called the 'Friedmann equation') and the second one as the 'acceleration equation', which includes a pressure term. It seems that you refer to them the other way round, perhaps causing some confusion.

[Edit: I see Wallace and Pervect have clarified the issue and the standard names of the Friedmann equations below.]

 

[Wallace]

What befuddles me is why there are two separate Friedmann equations, and how to apply them both at the same time. Can one derive a correct expansion solution by using only one of the two equations?

In the 2nd Friedmann equation, why is there no pressure component? How can the expansion rate be modeled accurately without including pressure?

To throw a spanner in the works there are actually three equations that are at times referred to as the Friedmann equations. They are not independent however, any two can be combined to give the third, so you only ever need two of them.

In any case all three contain a pressure term, I'm not sure what has lead you to believe otherwise?

Here are the three equations in the most general way I can think to express them:

The energy conservation equation for each individual energy component (radiation, matter, dark energy, curavture etc etc):

\frac{d\rho_i}{dt} = -3H\rho_i(1+w_i)

where H is the Hubble parameter and w_i is the equation of state of the energy component i, defined as w_i = \frac{p_i}{\rho_i}

The expansion equation

\frac{H}{H_0} = \sqrt{\frac{8 \pi G }{3}\Sigma_i [\rho_i]}

The acceleration equation

\frac{\ddot{a}}{a} = -\frac{4 \pi G}{3}\Sigma_i [ \rho_i(1+3w_i)]

As an example of how you can reduce this to only two equations, if the equation of state, w, of a component is constant for all time the energy equation for that component can be solved to give:

\rho_i(a) = \rho_i(a_0)a^{3(1+w_i)}

and therefore the expansion equation becomes

\frac{H}{H_0} = \sqrt{\frac{\pi G 8}{3}\Sigma_i [\rho_i(a_0)a^{3(1+w_i)}]}

As you can see, pressure plays a part in all three equations. You may have been reading something that was a simplified equation assuming there is only matter in the universe (which is pressure less).

 

[pervect]

Pervect,

Thanks for the answer. I agree that pressure causes gravity.

What befuddles me is why there are two separate Friedmann equations, and how to apply them both at the same time. Can one derive a correct expansion solution by using only one of the two equations?

I thought I wrote something about this before, but I've been a bit distracted recently.

Let me quote from my textbook, MTW's "Gravitation", pg 728, with a few modifications in notation to clarify this:

Note that one only has to differentiate (F1) (the Friedmann equation which involves only the first derivative a') and combine it with the relation satisfied by the pressure

\frac{d}{dt} \left(\rho a^3\right) = -P \frac{d}{dt} \left(a^3 \right) \hspace{1 in} (3)

("law of conservation of energy") to get the acceleraton equation (F2), the Friedmann equation which involves the second derivative a''). Without any loss of information, one can therefore ignore the "accelration equation" or "dynamic equation" (27.39b) henceforth, and work with the analog of an energy expression or what is more properly known as "initial-value euation".

While MTW choses to ignore the acceleration equation, I utilized it in my reply, because I like it from an intuitive standpoint. The point is that you should get the same answer either way, because both equations are equivalent when combined with the "energy conservation equation" which I have numbered (3) for future reference.

So we've got three equations (F1), (F2), and (3).

(F1) is the so-called initial-value equation:
<br /> \left(\frac{a&#039;}{a}\right)^2 = \frac{8 \pi G}{3} \rho + \frac{\Lambda}{3} - K \frac{c^2}{a^2}<br />

(F2) is the so-called dynamic equation
<br /> 3 \frac{a&#039;&#039;}{a} = \Lambda - 4 \pi G \, \left(\rho+ \frac{3P}{c^2}\right)<br />

and we have a third equation, which is the "energy conservation equation:
\frac{d}{dt} \left(\rho a^3\right) = -P \frac{d}{dt} \left(a^3 \right) \hspace{1 in}

Any two of the above equations implies the third, i.e.:

(F1) + (3) -> (F2)
(F2) + (3) -> (F1)
(F1) + (F2) ->(3)

though I have personally only verified that (F1)+3 -> (F2)

I don't know if telling you that (3) for the relationship between pressure,density and the scale factor comes from

\nabla \cdot T = 0

where T is the stress-energy tensor of the cosmological fluid, will give you any insight into why it gets called the "law of conservation of energy", because I'm not quite sure of your background. But I'll add it in case it helps. So there are good reasons to assume that the energy equation is true on its own, though it turns out to be automatically generated by Einstein's field equations. (The full field equation route generates (F1) and (F2), which implies (3).)

Now, given all of these equations, how do we solve them?

I'm still very hazy on exactly what *you* are doing, other than you've got some sort of spreadsheet, which doesn't help much. I guess the solution is to describe what *I* am doing a bit more clearly.

Basically, the equations of state based on your matter model are going to imply some relation between rho and P. You have to combine these with the Friedmann equations to get the final answer, which is an expression for a(t).

If you consider a universe of pure matter, things are simple because P=0, always. (Actually if you have hot matter this might not be strictly true, a more exact statement is that P is negligible rather than zero, because the "hot" matter is not hot enough to have a relativistically significant pressure).

You can use P=0 plus (3) to find that rho*a^3 = constant for a universe of pure matter, because (d/dt) rho*a^3 = 0 via the energy equation.

If you consdier a universe of pure radiation, then P = 1/3 rho. In a similar manner, you can find that rho*a^4 = constant for a universe of pure radiation.

You can then use whichever of the Friedmann equations you like to find the evolution of a(t). (F1) is computationally more convenient, but I think that it doesn't give you as good an insight as to what's going on as (F2) does because (F1) is rather involved, and (F2) is much simpler.

 

[jonmtkisco]

Hi Pervect, Wallace and Jorrie,

First, thank you for your insights. I'll use Pervect's terminology to refer to the equation I've been using as the "Friedmann Initial Value" or "Friedmann IV" equation. I'm glad that the results work out the same with any of the 3 equations.

Second, Wallace, the version of the Friedmann IV equation you posted was the first time I've ever seen it containing an explicit pressure component. I'm confused, can you explain?

Third, Pervect, you said:

"Let me see if I understand this. You assumed that the denisity, rho, remains the same, and when you say that it "was composed 100% of matter", you've kept rho constant, but you've reduced P to zero. You have not kept K, the Gaussian curvature, constant.

"Assuming that this is actually what you did, it's easiest to explain this from the dynamic equation.

"Numerically, using the dynamic form of the equation, rho remains consant, and P goes to zero. Because P = 1/3 rho for radiation, rho+3P should drop to half its value when you convert from a radiation dominated universe to a matter dominated universe, meaning that the deaccleraton also drops to 1/2.

"The consequence of this is that you reduce the deacceleration of the universe, meaning that if you project the universe forward to the current age, it will be a lot bigger."


I want to be clear that I deleted both \Lambda and gaussian curvature elements in order to simplify the interpretation of my calculations. Shouldn't curvature always be zero, since the universe is believed to have been flat throughout the original expansion (post-inflation)? As a purely mathematical perspective, I think that by forcing the curvature=0 over time, I forced the equation to generate a faster deceleration in the 100% matter scenario. Still, I don't see why doing so is in any way invalid.

I'm a bit unclear as to whether the effect of positive radiation pressure is most properly incorporated by doubling gravity in the historical case of the Friedmann IV equation, or on the contrary by halving the gravity in the "what if 100% matter" case of the same equation. Since doubling gravity in the historical case seems...er...non-historical, I will follow your suggested route of halving gravity in the 100% matter case.

Initially, as you say, this must cause the expansion rate to decelerate more slowly in the 100% matter case. However, the critical problem here is that in the "historic" case, both total mass/energy density of radiation and radiation pressure decline over time (due to redshift), at the rate of a^{2}. At that rate, the halving of gravity (in the 100% matter case) caused by the inclusion of pressure in the formula quickly becomes a tiny rounding error, with virtually no impact on the long-term deceleration of the expansion rate, as compared to the "historic" case. In fact, its effect is entirely washed out within the first tiny fraction of a second after inflation ends. In the "historic case", by 3.6 seconds into the expansion, total mass has declined by a factor of 10^{16}, which obviously has a far more significant impact on the calculation. By comparison, that enormous reduction does not occur at all in the 100% matter case.

So I continue to find that dR/R remains a healthy 10^{3} slower in the "historical" case than in the 100% matter case, at every scale factor "a" higher than a*10^{-5}.

The seeming paradox remains: a faster decline in mass/energy density results in a slower expansion and smaller universe.

 

[pervect]

I want to be clear that I deleted both \Lambda and gaussian curvature elements in order to simplify the interpretation of my calculations.

What you need to think about (I have to think about it too) is whether or not this assumption about K satisfies the energy-conservation relationship that I described as (3).

 

[Jorrie]

Hi Jon. My 2 cents for what it's worth.

I'll use Pervect's terminology to refer to the equation I've been using as the "Friedmann Initial Value" or "Friedmann IV" equation.

If you refer to Wallace's

\frac{H}{H_0} = \sqrt{\frac{\pi G 8}{3}\Sigma_i [\rho_i(a_0)a^{3(1+w_i)}]}

as "Friedmann IV", I suggest you rather don't - it may just confuse the issue further. It's the same as the expansion (or initial values) equation, just expressed differently for the purpose of showing the pressure influence clearly.

Shouldn't curvature always be zero, since the universe is believed to have been flat throughout the original expansion (post-inflation)? As a purely mathematical perspective, I think that by forcing the curvature=0 over time, I forced the equation to generate a faster deceleration in the 100% matter scenario. Still, I don't see why doing so is in any way invalid.

If you force the curvature to zero for a matter only case, then \Omega_m = 1 and for a radiation only case, then \Omega_r = 1. I think you cannot just delete the other density components and leave the rest at the same value. If you put \Omega_m = 1 and \Omega_r = 1 separately into the initial value (F1) equation (leaving Ho the same), then yes, the initial expansion rate (a') is faster for a radiation only case, but the deceleration (a'' from F2) is also larger when compared to a matter only case.

However, the critical problem here is that in the "historic" case, both total mass/energy density of radiation and radiation pressure decline over time (due to redshift), at the rate of a^{2}. At that rate, the halving of gravity (in the 100% matter case) caused by the inclusion of pressure in the formula quickly becomes a tiny rounding error, with virtually no impact on the long-term deceleration of the expansion rate, as compared to the "historic" case.

I don't know what you mean by "historic" case, but doesn't radiation energy density decline by a^{4}? The effect of initial radiation pressure may quickly become insignificant in terms of deceleration, but its effect on the future expansion rate is significant for a very long time afterwards.

The seeming paradox remains: a faster decline in mass/energy density results in a slower expansion and smaller universe.

Not necessarily if the initial expansion rate is larger, as it must be for a flat universe with more radiation energy density (and the same Ho, one must add).

 

[jonmtkisco]

Hi Jorrie,

I'm not sure I understand all of your points. The "historic" case is not "radiation only", it's the actual mix of radiation and matter that the historic universe is believed to have had. The only modified case is the "100% matter" case -- in which radiation = 0, so I haven't departed from "omega"=1.

In the "historic" case, radiation density declines at a^4, but I was referring to total mass/energy of radiation, which declines at a^2.

As far as the possibility that in the "historic" case the universe may have expanded so much in the first tiny fraction of a second that it outweighs all of the dramatically faster expansion after that point in the "100% matter" case, I guess my calculations haven't ruled that out, although I'm skeptical. I would be grateful if you can help me figure that out. I know the expansion rate at every absolute volume, but in the 100% matter case I don't know what "t" corresponds with each "r" or "v, since I can't start with a Hubble value at the present time.

Jon

 

[Wallace]

Second, Wallace, the version of the Friedmann IV equation you posted was the first time I've ever seen it containing an explicit pressure component. I'm confused, can you explain?

Let's compare the initial value equation I posted:

\frac{H}{H_0} = \sqrt{\frac{8\piG}{3}\Sigma_i[\rho_i]}

with the one Pervect posted which was (with c=1 and some slight extra notations to make things crystal clear) :

(\frac{H}{H_0})^2 = \frac{8\piG}{3}\rho_m + \frac{\Lambda}{3} - \frac{K}{a^2}

Now I'm sure you're familiar with Pervects version and indeed that is how it is commonly presented. Then you look at mine and are surprised that pressure seems to have appeared from nowhere! In fact the effect of pressure is included in both, it's just more explicit in the general form I posted. The reason it's not usually written this way is that it's more general than is usually needed and harder to 'get a feel for' but none the less they are equivalent. The energy component represented in Pervects version are matter, vacuum energy (or cosmological constant) and curvature. These have equations of state of w=0,-1, -1/3 respectively. Substituting those values into the equations I gave leads to the form of the equation that Pervect posted.

Note that the fact that the energy components have different equations of state is due to the different pressure they exert. Out of interest, radiation has an equation of state of w=1/3, but is usually not included since the energy density drops so rapidly as to be unimportant for the global dynamics except in the early universe.

 

[Jorrie]

The "historic" case is not "radiation only", it's the actual mix of radiation and matter that the historic universe is believed to have had. The only modified case is the "100% matter" case -- in which radiation = 0, so I haven't departed from "omega"=1.

I think I understand what you are trying to do (correct me if I'm wrong). It seems you want to keep the initial expansion rate a' (or H') after inflation the same, but want to omit the radiation energy density and then see what happens with the deceleration and expansion rate during the first moments after inflation. Correct?

I think the moment you do that, you deviate grossly from flatness when a is small, simply because there is then not enough energy density (gravity) to balance out the (now excessive) expansion rate and the cosmos would have been very open. Removing energy density at the beginning must be accompanied by an appropriate reduction in the initial expansion rate. Alternatively, you have to 'balance the books' by converting that (removed) radiation energy density into matter density and so keep the initial expansion rate the same.

I'm not sure how to calculate that and whether it's even valid to do that, but I'll think about it (and hope the mentors come up with an answer in the meantime).

I know the expansion rate at every absolute volume, but in the 100% matter case I don't know what "t" corresponds with each "r" or "v, since I can't start with a Hubble value at the present time.

Absolute volume is poorly defined, as Pervect said many times before. It's much better to use the normalized expansion factor a and its evolution over time, which is easily calculated from the initial values equation for K=0 (or any value of K, if you like). For your 100% matter scenario, all you need to decide on is the total matter-energy density for your starting a and K and then integrate expansion rate over time.

I may have taken a few 'engineering-liberties' here, but I do not think it's too far off the mark.

 

[Jorrie]

... the one Pervect posted which was (with c=1 and some slight extra notations to make things crystal clear) :

(\frac{H}{H_0})^2 = \frac{8\piG}{3}\rho_m + \frac{\Lambda}{3} - \frac{K}{a^2}

Jon is particularly interested in the scenario immediately after inflation. Is it correct to bring radiation energy density into your above equation as follows:

(\frac{H}{H_0})^2 = \frac{8\piG}{3}(\rho_m + \rho_r) + \frac{\Lambda}{3} - \frac{K}{a^2}

or must rho_r be divided by a?

 

[George Jones]

Jon is particularly interested in the scenario immediately after inflation. Is it correct to bring radiation energy density into your equation above as follows:

(\frac{H}{H_0})^2 = \frac{8\piG}{3}(\rho_m + \rho_r) + \frac{\Lambda}{3} - \frac{K}{a^2}

or must rho_r be divided by a?

Yes, another a is needed. It's interesting to think of this from point view of the particle/wave nature of light. As the universe expands, the number density of photons decreases at the same rate as the number density of massive particles. However, the wavelength of light also expands, thus decreasing its energy density by another factor of a.

 

[Jorrie]

Yes, another a is needed. It's interesting to think of this from point view of the particle/wave nature of light. As the universe expands, the number density of photons decreases at the same rate as the number density of massive particles. However, the wavelength of light also expands, thus decreasing its energy density by another factor of a.

Thanks George, but what still bugs me is this: although not explicitly stated, I assume that in the equation:

(\frac{H}{H_0})^2 = \frac{8\piG}{3}(\rho_m + \rho_r) + \frac{\Lambda}{3} - \frac{K}{a^2}

\rho_m and \rho_r represent the energy densities when the expansion factor was a, so the dependency of the radiation energy density on a has already been accounted for. So why divide it by a again?

To clarify my uncertainty further, consider the radiation energy density shortly after inflation, at say a = 10^{-25}, which is in the order of \rho_r=10^{-30}/10^{-100} = 10^{70} kg/m^3. It already includes the 1/a^4 factor, so 1/a must not be introduced another time.

Or am I messing something up?

 

[George Jones]

Or am I messing something up?

No, I messed up. And I'm still a little confused.

with the one Pervect posted which was (with c=1 and some slight extra notations to make things crystal clear) :

(\frac{H}{H_0})^2 = \frac{8\piG}{3}\rho_m + \frac{\Lambda}{3} - \frac{K}{a^2}

I think that pervect posted (c=G=1)

H^2 = \frac{8\pi}{3}\rho + \frac{\Lambda}{3} - \frac{K}{a^2}

I see why Wallace's \pi's disappeared (look at the latex), but I can't see why an H_0 appeared.

And, yes, \rho = \rho_m + \rho_r when radiation is also included.

 

[Jorrie]

I think that pervect posted (c=G=1)

H^2 = \frac{8\pi}{3}\rho + \frac{\Lambda}{3} - \frac{K}{a^2}

I see why Wallace's \pi's disappeared (look at the latex), but I can't see why an H_0 appeared.

And, yes, \rho = \rho_m + \rho_r when radiation is also included.

Yes, pervect posted:

\left(\frac{a&#039;}{a}\right)^2 = \frac{8 \pi G}{3} \rho + \frac{\Lambda}{3} - K \frac{c^2}{a^2} = H^2

which does not require an H_0. So can we conclude that to be explicitly clear, one should write the intitial values Friedmann equation as:

\left(\frac{a&#039;}{a}\right)^2 = \frac{8 \pi G}{3} (\rho_m + \rho_r) + \frac{\Lambda}{3} - K \frac{c^2}{a^2}.

I prefer the c and the G to be there, because it makes the units more intuitive.

 

[nrqed]

Thanks George, but what still bugs me is this: although not explicitly stated, I assume that in the equation:

(\frac{H}{H_0})^2 = \frac{8\piG}{3}(\rho_m + \rho_r) + \frac{\Lambda}{3} - \frac{K}{a^2}

\rho_m and \rho_r represent the energy densities when the expansion factor was a, so the dependency of the radiation energy density on a has already been accounted for. So why divide it by a again?

To clarify my uncertainty further, consider the radiation energy density shortly after inflation, at say a = 10^{-25}, which is in the order of \rho_r=10^{-30}/10^{-100} = 10^{70} kg/m^3. It already includes the 1/a^4 factor, so 1/a must not be introduced another time.

Or am I messing something up?

The densities there are functions of time themselves and they should not be divided by any factor of 'a" as far as I know.

 

[Wallace]

The H_0 is to do with how units are defined. Don't worry about it!

 

[jonmtkisco]

Hi folks,

I appreciate all the help, but now I'm more confused than ever. The "p" in Pervect's Friedmann IV equation is "energy density", not pressure. There is NO PRESSURE component in the equation. How can you just substitute pressure for energy density?

I'm back to my original question, why is there no pressure component at all in the Friedmann IV equation, when there is in the other two equations?

Jorrie, I'm sorry if I've been unclear about my "what if 100% matter" formula, but I don't think you've interpreted it correctly. Its EXACTLY THE SAME initial total energy density as the "historic" case, but I treated all of it as matter and none of it as radiation. So I haven't departed one iota from critical density or flatness.

To be as clear as possible, in the "historical" (actual) case, at t=3.6 seconds I have total mass/energy at 3.80E+76 kg (or 3.41E+93 joules), of which almost the entire amount is radiation. (Matter is only 8.53E+53kg). I have not made ANY adjustment to the "historical" case to reflect any pressure component. Total density is 2.53E+74kg/cubic meter. (I wonder if that's really the SUM of energy density and pressure? Well, whatever.)

In the "100% matter" case I started with total/mass energy the same as the other case. But I put it in as 100% matter, no radiation. Then, because everyone keeps suggesting I do so (I still don't understand why, if Friedmann IV is an exact formula), I divided this total mass/energy by 2. Yielding total mass/energy of 1.90E+76kg (or 1.17E+93 joules), and energy density of 1.71E+74kg / cubic meter.

I hope this is clear now. I took the simplest possible approach to maintain flat curvature at all times. The paradox is still there, as big and ugly as ever!

Jon

 

[Wallace]

Hi folks,

I appreciate all the help, but now I'm more confused than ever. The "p" in Pervect's Friedmann IV equation is "energy density", not pressure. There is NO PRESSURE component in the equation. How can you just substitute pressure for energy density?

I'm back to my original question, why is there no pressure component at all in the Friedmann IV equation, when there is in the other two equations?

There IS is pressure term in the initial value equation! There was no 'P' is Pervect formula, only \rho ('rho') the energy density.

BUT, that formulation somewhat hides the pressure term. Let's be more explicit. Here I will re-write Pervect's form of the equation with one modification, I will explicitly include the function form of the matter energy density:

(\frac{\dot{a}}{a})^2 = \frac{8\pi G \rho(a_0)}{3a^3} + \frac{\Lambda}{3} - \frac{K}{a^2}

So what we can see is that the energy density of matter goes with the inverse cube of a, curvatiure goes with the inverse square and the cosmological constant as its name suggest just stays constant.

Let's relate this to the energy conservation equation I posted, which contains Pressure through the equation of state w = \frac{P}{\rho}. Here is the equation again:

\frac{d\rho}{dt} = -3H\rho(1+w)

Matter is pressure less and hence w=0, solving that above with w=0 gives us

\rho = \rho(a_0) a^{-3}

similarly for the cosmological constant with negative pressure such that w=-1 we can trivially see that the density stays constant. Lastly for curvature for which w=-1/3 we can solve to get

\rho = \rho(a_0)a^{-2}

So, the IV equation shown at the top has used the energy conservation equation to find the functional form of the energy densities and then substituted these. The energy densities depend on pressure so pressure appears in this equation!

If you want to include matter in your calculations, you need to know that it has an equation of state of w=1/3 and this leads to

\rho = \rho(a_0)a^{-4}

If you go back and read what I wrote when I wrote out the general forms of the Freidmann equations you will see that I'm repeating myself!

I don't follow your description of how you are attempting to solve these equations for the early universe, but in any case it dosn't seem like your approaching the problem correctly. There is no paradox!

 

[jonmtkisco]

There IS is pressure term in the initial value equation! There was no 'P' is Pervect formula, only \rho ('rho') the energy density.

Let's relate this to the energy conservation equation I posted, which contains Pressure through the equation of state w = \frac{P}{\rho}. Here is the equation again:

\frac{d\rho}{dt} = -3H\rho(1+w)

Wallace, thanks again, but it would help if our communication could be literal, not figurative.

My literal understanding is that Pervect's Friedmann IV equation DOES NOT include an explicit Pressure term. It includes only energy density - "rho". You agree with that.

You encourage me to perform a substitution to bring pressure into Friedmann IV. But why should I? If Friedmann IV is accurate without adding in pressure, then I'd rather use it in the vanilla form Pervect quoted.

If I were to bring pressure into the equation, then I am convinced I should bring it into my "historical" (actual) scenario, NOT into my "100% matter" scenario. In the 100% matter scenario, pressure = 0, so there is no reason to introduce pressure.

My belief is that the "rho" I calculate in the "historical" scenario is NOT REALLY "rho" at all! It's really the SUM of "rho" + pressure of radiation. Which means that if I just use the number calculated by the simple, original Friedmann IV equation, it should be historically accurate and work just fine, since it includes both density and pressure.

It's a little presumptious of you to assert that the paradox I found isn't real, when you also claim that you don't understand the calculation I used!

The calculation is simple. [However, I now realize that the figures I gave in my last post were at 3.6E-32 seconds, not at 3.6 seconds. Oops.] In the "historical" scenario, the "initial value" of total "rho" at 3.6E-32 seconds = 2.53E+74 kg/cubic meter. That also is the "rho" of radiation at the same point in time. The "rho" of matter is 8.53E+53kg. The radius is 3.3 meters.

In the 100% matter case, total "rho" is the same as the "historical" case. The "rho" of radiation = 0, and the "rho" of matter = 2.53E+74 kg/cubic meter. Radius is still 3.3 meters.

Please perform the calculation yourself, using whichever version of the Friedmann equation you prefer, and convince us that there is no paradox.

Jon

 

[Jorrie]

I appreciate all the help, but now I'm more confused than ever. The "p" in Pervect's Friedmann IV equation is "energy density", not pressure. There is NO PRESSURE component in the equation. How can you just substitute pressure for energy density?

Jon, you are still talking about "Pervect's Friedmann IV equation". There is only three Friedmann equations here and then there are numerous ways to represent each of them. Please, let's rather stick to the names rather than 'Friedmann N'. I guess you are referring to the initial conditions equation. Wallace explained it fully, so I think by now you realize that the energy densities include pressure, even if not stated explicitly. Even the \Omegas in this very common form of the initial conditions equation

\left(\frac{\dot a}{aH_0}\right)^2 = \frac{1-\Omega}{a^2}+\frac{\Omega_m}{a^3}+\frac{\Omega_r}{a^4}+ \Omega_v

includes pressure in every term, except for the first (curvature density) term.

In the "100% matter" case I started with total/mass energy the same as the other case. But I put it in as 100% matter, no radiation. Then, because everyone keeps suggesting I do so (I still don't understand why, if Friedmann IV is an exact formula), I divided this total mass/energy by 2. Yielding total mass/energy of 1.90E+76kg (or 1.17E+93 joules), and energy density of 1.71E+74kg / cubic meter.

I hope this is clear now. I took the simplest possible approach to maintain flat curvature at all times. The paradox is still there, as big and ugly as ever!

Jon

I also don't know where the 'divide by 2' comes from - perhaps just an artifact of the units used sometimes, but I think I understand now what you are struggling with. Look at pervect's (F2), the so-called dynamic or deceleration equation:

3 \frac{a&#039;&#039;}{a} = \Lambda - 4 \pi G \, \left(\rho+ \frac{3P}{c^2}\right)

where the pressure term has been separated. He wrote:

You can use P=0 plus (3) to find that rho*a^3 = constant for a universe of pure matter, because (d/dt) rho*a^3 = 0 via the energy equation.

If you consider a universe of pure radiation, then P = 1/3 rho. In a similar manner, you can find that rho*a^4 = constant for a universe of pure radiation.

You can then use whichever of the Friedmann equations you like to find the evolution of a(t). (F1) is computationally more convenient, but I think that it doesn't give you as good an insight as to what's going on as (F2) does because (F1) is rather involved, and (F2) is much simpler.

It is clear that the deceleration a&#039;&#039; must be larger as long as the radiation pressure is there and then it must settle into the \rho-only deceleration when the pressure is diluted away. If you start with the same \rho, matter only density and hence P=0, the deceleration magnitude must be lower from the start and hence the initial evolution of a(t) faster.

So your 'paradox' "a faster decline in mass/energy density results in a slower expansion and smaller universe" is no paradox. Perhaps the way you stated it is the problem: 'A faster decline in mass/energy' per se has little to do with expansion rate. It is only the initial expansion rate and the deceleration profile that determines the evolution a(t).

Edit: I just realized something: if you want to try a matter only initial condition with the same energy density as for the initial matter+radiation case, you have to increase the matter density by a factor \approx 10^{22}. This will surely not give the sort of universe that we observe and is perhaps not valid at all.

 

[Wallace]

Wallace, thanks again, but it would help if our communication could be literal, not figurative.

My literal understanding is that Pervect's Friedmann IV equation DOES NOT include an explicit Pressure term. It includes only energy density - "rho". You agree with that.

You encourage me to perform a substitution to bring pressure into Friedmann IV. But why should I? If Friedmann IV is accurate without adding in pressure, then I'd rather use it in the vanilla form Pervect quoted.

There is only one set of Friedmann equations! You can't prefer to use on person's 'version' over another. They are equivalent, as has been explained to you several times, by several people.

If I were to bring pressure into the equation, then I am convinced I should bring it into my "historical" (actual) scenario, NOT into my "100% matter" scenario. In the 100% matter scenario, pressure = 0, so there is no reason to introduce pressure.

My belief is that the "rho" I calculate in the "historical" scenario is NOT REALLY "rho" at all! It's really the SUM of "rho" + pressure of radiation. Which means that if I just use the number calculated by the simple, original Friedmann IV equation, it should be historically accurate and work just fine, since it includes both density and pressure.

Pressure is in the equation whether you choose to 'put it in' or not, unless of course you are not using the equations correctly. Density is density and pressure is pressure despite whatever you would like to believe!

It's a little presumptious of you to assert that the paradox I found isn't real, when you also claim that you don't understand the calculation I used!

What I am asserting is that you haven't discovered that mathematics doesn't work in the early universe. I don't need to fully comprehend how you are coming to this conclusion to be reasonably sure that this is the case!

The calculation is simple. [However, I now realize that the figures I gave in my last post were at 3.6E-32 seconds, not at 3.6 seconds. Oops.] In the "historical" scenario, the "initial value" of total "rho" at 3.6E-32 seconds = 2.53E+74 kg/cubic meter. That also is the "rho" of radiation at the same point in time. The "rho" of matter is 8.53E+53kg. The radius is 3.3 meters.

In the 100% matter case, total "rho" is the same as the "historical" case. The "rho" of radiation = 0, and the "rho" of matter = 2.53E+74 kg/cubic meter. Radius is still 3.3 meters.

Please perform the calculation yourself, using whichever version of the Friedmann equation you prefer, and convince us that there is no paradox.

Jon

What exactly is the paradox here?

 

[jonmtkisco]

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.

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.

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.

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.

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.

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.

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.

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.

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.

 

[pervect]

Hi Jorrie & Wallace,

I explained my two scenarios clearly several notes ago,

It seems like I'm not the only one having trouble following what you think you explained clearly several notes ago :-(.

I gather that you have two scenarios.

I would like you to give:

rho(t), P(t), and a(t) for both scenario #1 and #2, preferably in symbolic form (i.e. as equations written down as function of time).

Then we can see at least if we agree with these, whether there is some error in your basic scenarios, or whether the problem is solely in the interpretation of their significance of their existence.

Note that if you know rho(t) and a(t), it is possible to find P(t) by the relationship I call (3).

I don't know your mathematical background, but I'm almost getting the impression you're trying to tackle this problem without knowing how do do calculus symbolically, i.e. you use a spreadsheet of some sort to perform numerical integrations, and that's the limit of your mathematical abilities?

I'm not sure if I can explain what you need to know without at least basic, symbolic, calculus (I'd have to think about if this was even possible), but maybe I've misread the state of your knowledge and you do understand symbolic calculus...

 

[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&#039;}{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&#039;&#039;}{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\piG\rho/3

Dynamic Friedmann (or Acceleration) equation:

a"/a = -(4\piG/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.