What Are the Key Insights from Friedmann Cosmology's Standard Model?

[jonmtkisco]

Hi:

Jorrie, welcome to the club. I have learned that on forums of this type it's far more difficult to defend the literal wording of virtually anything one writes on this subject, than it is to have a reasonably accurate sense of how it all works.

In response to your supposition, I didn't ignore radiation density, and I don't think I've made any error. I can't figure out exactly what you think is wrong about what I said. I said:

"...if a flat universe could hypothetically start out with a higher total mass/energy after inflation, then the Friedmann equations would calculate that such universe would have a higher initial expansion velocity which would eternally stay ahead of the deceleration caused by its higher gravity, at any given point in elapsed time."

This is a theoretical calculation that I thought we all agreed in earlier threads was mathematically correct, regardless of whether it's physically "unrealistic."

In any universe that is even slightly similar to our own observed universe, at the end of inflation the radiation rho (density) will be enormously higher than the matter rho (it was by a factor of E+22 historically.) Radiation density remains dominant as the radius or scale factor a(t) increases by about E+23. So it would make no sense to try to model the very early universe without focusing primarily on radiation density, not matter density or the cosmological constant. We all know that.

Mathematically, it is obvious that if (hypothetically) the initial rho of radiation were were increased by some arbitrary amount as of the end of inflation (say by a factor of E+10), the Friedmann equations would require that the initial expansion rate "delivered by inflation" must be much higher than the historic figure, in order to preserve flatness. Then, until expansion causes the scale factor a(t) to reach a size of around a(t) = E-6, matter rho would have very little impact on the expansion rate, regardless of whether initial matter rho were: (1) held flat at the historical initial value or (2) increased by the same factor as radiation rho (e.g., E+10). Subsequently, when matter rho becomes dominant, then of course matter rho will make a big difference in the subsequent expansion rate. The resulting total expansion curve will look somewhat different depending on the initial radiation/matter ratio. (That is, depending on at what point in elapsed time and a(t) the radiation-dominated and matter-dominated curves intersect.)

As Wallace says, the simple matter-dominated expansion curve and the radiation-dominated expansion curves individually possess their fixed slopes at any given scale factor. For example, if we graph a(t) (y-axis) and t (elapsed time) on the x-axis for a simple radiation-dominated expansion curve. An increase in the initial radiation rho (compared to the historic case) causes a steeper upward initial expansion curve. After that, the slope of the curve decreases (becomes flatter) as time elapses. At every point of elapsed time on this simple radiation-dominated curve, a higher initial radiation rho results in (1) a higher absolute expansion rate (line slope), (2) a larger absolute scale factor a(t), and (3) a larger (more negative) deceleration rate.

All of this is simple mathematical application of the Friedmann expansion formula, and should not be controversial. So I assume that any "error" you perceive must be a misimpression about what I was trying to say.

Wallace, I agree with you that the simple formulas I gave don't work across era transitions, for example across the transition from radiation-dominated to matter-dominated eras. However, they ought to work within some restricted time range of an individual era. For example, the historic universe was radiation-dominated for the first 25,000 years or so. Therefore, the radiation-dominated formula ought to be pretty accurate (to 2 decimal places) for a restricted range starting at the end of inflation and ending at, say, 2.5 years after inflation. Even during that restricted time range, the historic scale factor increased by about a factor of about E-21, so a lot can be observed about the early expansion rate and scale growth.

The best way I know to do a calculation that accurately spans the era transitions is the method Jorrie suggested -- doing a backwards integration from now, or from some earlier known time (such as the CMB surface of last scattering). As I've mentioned, the problem with a backwards integration is that one must start with the present known value of Ho. Thus it's very limiting if you are trying to do a forward-in-time model of a hypothetical universe starting at the end of inflation. The other problem with a model based on integration of expansion rates is that it becomes very inaccurate over its total range, unless the spreadsheet uses relatively tiny increments of time and scale. Which results in a very large spreadsheet.

Also, even when I do calculations within a tightly restricted time range, I am finding somewhat different results of the calculations using the formulas I suggested, as compared to the results calculated by Jorrie's method. Differences on the order of magnitude or more. Given that all of these equations are exact solutions to the Einstein Field Equations, I find these discrepencies to be very annoying. I'm sure they don't point to any inaccuracy in the formulas; it's just that the practicality of integrating the calculations seems to introduce enough fudge factor that the precision of the results becomes somewhat suspect.

Jon

 

[Jorrie]

Critical density

I said:

"...if a flat universe could hypothetically start out with a higher total mass/energy after inflation, then the Friedmann equations would calculate that such universe would have a higher initial expansion velocity which would eternally stay ahead of the deceleration caused by its higher gravity, at any given point in elapsed time."

I think Wallace and myself have issues with how you use the Friedmann equations, e.g. we want Ho to be the present measured value and then you cannot change the density at any epoch and still retain flatness. After all, critial density is a function of Ho, i.e. (with appropriate units):

\rho_{crit} = \frac{3H_0^2}{8\pi G}

Sure, if you assume Ho to be a free parameter and you increase the total energy density after inflation, the expansion rate will be higher for a flat universe and so will Ho. (Tip: don't talk about total mass/energy on this forum, ever... )

Wallace, I agree with you that the simple formulas I gave don't work across era transitions, for example across the transition from radiation-dominated to matter-dominated eras.

Jon, why do you want to use them separately and bother about transitions? Just use the compounded density parameter at any time and it automatically does the transitions for you, as per that most useful form (to me at least) of the Friedmann initial values equation:

\left (\frac{\dot a}{a}\right)^2 = H_0^2 \left (\frac{1-\Omega}{a^2}+\frac{\Omega_m}{a^3}+\frac{\Omega_r}{a^4}+\Omega_v \right)

The only time it may not work is for t < 1 seconds or so, where the radiation energy equation of state may be different.

Jorrie

 

[jonmtkisco]

Jorrie,

I agree with everything you say about the equations. I understood all of that when I wrote the first post in this thread, and I haven't made any error in my calculations. I used the "omega" formula you cited, and in fact I used the integration spreadsheet you sent me as the starting point to generate one set of values I used to compare to results I calculated with other methods.

I am frustrated that I have encountered nothing but critisism for trying to work out some version of the Friedmann equations that generates accurate results when starting at the end of inflation and calculating forward. No one has explained to me why that is such an unreasonable objective, and frankly I don't care anymore. My intuition tells me there is no more gold to be mined on this topic for now. So I guess I'll stop writing for this forum and try out "beyond the standard model".

Thanks, Jon

 

[Wallace]

I am frustrated that I have encountered nothing but critisism for trying to work out some version of the Friedmann equations that generates accurate results when starting at the end of inflation and calculating forward. No one has explained to me why that is such an unreasonable objective, and frankly I don't care anymore.

I'm gobsmacked mate, I've spent many a post explaining precisely why the 'version' of calculating the equation you are trying to achieve is no different to the 'standard' version. I was even getting the impression you were being to understand it all, but I guess not. You've completely mistaken repeated attempts to help you understand how it all works for criticism. Of course you've been integrating the equations in a numerically correct way, the leap you need to make is from the maths to the physics, which I've again tried to explain many times.

If you're not going to bother reading other people's posts, and take any input as merely criticism you can just dismiss, there's not much point posting in any forum now is there?

 

[jonmtkisco]

Wallace, thank you for the response.

I do feel that I have thoroughly read and considered all of the responses to this thread. I highlighted areas where I thought you made a good point, such as the fact that the matter-dominated expansion curve correlates each point of elapsed time (t) with a single point on the a(t) scale factor. I don't think this point was central to my thread, but it's interesting, and it's fine that you developed the point.

I agree that a forward-calculated expansion curve ought to be mathematically identical with a backward-calculated one. As I mentioned, the mechanical process of integrating the same curve in different directions seems to introduce enough error that the results don't match identically, and in some cases are very divergent. That is a frustrating fact of life in the real world. At the moment I don't have enough enthusiasm to try to manually reconcile all of the divergences. I am tempted to conclude that there is as much mechanical inaccuracy in the backward-calculated model as there is in the forward-calculated model. Which suggests that it could be worth the effort to reconcile them. Maybe I'll regain my enthusiasm later.

If I do write another thread on this forum, I would appreciate if I could get some support for the calculations I do correctly and the resulting perspectives. As opposed to monotonicaly repeated advice to stop trying to think independently and instead calculate everything in the pure orthodox form using pure orthodox terminology.

Despite all of the dialogue and suggestions, I still see no error in my original post. Items 1-8 and 13 are factual descriptions of basic attributes and limitations of the Friedmann equations, coupled with some minor editorializing on my part. Items 9, 10, and 11 are reasonable attempts to extrapolate from the 'standard model', which I suggested to address gaps that for some reason don't get a lot of focus in the current literature.

Thanks again.

Jon