Friedmann Cosmology: Exploring the Fascinating Aspects of the Standard Model

[jonmtkisco]

The purpose of this thread is to describe certain interesting aspects that are explicit or implicit in the “standard model” of cosmology for a flat universe, by reference to the Friedmann equations. I hope to provoke more interest and constructive dialogue about the substance of this topic.

In this thread, let’s refer to the two Friedmann equations as:

[F1] expansion equation:

$$\right)\frac{\dot{a}}{a} = \sqrt{\frac{8G\pi\rho+\Lambda}{3}}$$

[F2] acceleration equation:

3$$\frac{\ddot{a}}{a} = \Lambda-4\pi G \left(\rho +3p \right)$$

I have deleted the curvature parameter from the expansion equation in order to simplify the discussion of our universe, which is observed to be approximately flat.

1. As discussed previously in this Forum, the expansion equation can be further simplified by substituting M/V (mass/volume) in place of $$\rho$$ (rho), the density parameter. In this form, the equation speaks to “total mass/energy” rather than “mass/energy density.” For the purposes of this topic, we need not be concerned with the frequent statement that the “total mass/energy” of the observable universe does not have a reliable meaning in general relativity. Here, we use the term because it is convenient and entirely mathematically substitutable for density, regardless of any deeper meaning it may or may not have. And, in any event Birkhoff’s Theorem says that mass/energy outside the expanding sphere of our observable universe cannot have any gravitational effect on our observable universe. Peebles, Principles of Physical Cosmology, at 75.

Substituting the equation to derive the radius of a sphere from its volume:

r = $$\sqrt[3]{\frac{3V}{4\pi}}$$

the resulting form of the expansion equation is the familiar equation for “escape velocity”:

$$\dot{r} = \sqrt{\frac{2GM}{r}}$$

2. We can then deduce from the expansion equation that a flat universe must always expand at exactly the “escape velocity” of its total mass/energy contents. This must be exactly true at all times: when the expansion rate is dominated by free radiation, matter, or the cosmological constant, or during each transition between an era dominated by one form of mass/energy to another. There is no discontinuity in the equation.

3. The acceleration equation tells us that during the radiation-dominated era, the active gravitational density is doubled, because radiation’s pressure is 1/3 of its density (rho). Peebles, infra at 63. The equation of state of radiation is:

$$\omega = \frac{\rho}{p}= \frac{1}{3}$$

This doubled gravity does not cause the universe to depart from flatness, because total mass/energy simultaneously decreases due to redshift. Thus the first 1x of deceleration accommodates the volume dilution of gravity, and the second 1x of deceleration accommodates the next incremental decrease in mass/energy. The expansion equation holds, and the universe at all times continues to expand at current escape velocity. Quite conveniently, if the universe was flat before the radiation-dominated era, then the equation of state of radiation will conserve that flatness. Is this a coincidence, or an inherent preference for flatness?

4. The acceleration equation also tells us that during the cosmological constant-dominated era, the active gravitational density is -1x, because the cosmological constant’s “negative pressure” is equal to its mass/energy density (rho). The equation of state of the cosmological constant is:

$$\omega = \frac{\rho}{p}= -1$$

(Note that the equation of state alone would drive a net -2x gravitational density (1+ -3), but half of this is eliminated because the acceleration equation subtracts the effect of the equation of state from Lambda.) This net 1x anti-gravity does not cause the universe to depart from flatness, because total mass/energy simultaneously increases due to the cosmological constant. Thus the first 1x part of the increase in acceleration (starting from the 1x deceleration of matter-domination) neutralizes the volume dilution caused by matter; the second 1x part of the increase in acceleration accommodates the increased expansion rate needed to account for the existing mass/energy of the cosmological constant, and the third 1x part of the increase of acceleration accommodates the next incremental increase in mass/energy (due to the creation of more vacuum space containing more cosmological constant mass/energy). The expansion equation holds, and the universe at all times continues to expand at current escape velocity. Again conveniently, if the universe was flat before the cosmological-constant dominated era, then the equation of state of the cosmological constant will conserve that flatness. Is this a coincidence, or an inherent preference for flatness?

5. The expansion equation tells us that if flat Universe “A” contains more total mass/energy than flat Universe “B”, then Universe A will always expand faster, and be larger at any point in time, than Universe B. The acceleration equation tells us that Universe A will always have a higher deceleration rate at any scale size than Universe B, but never enough for Universe B to overtake A’s size or acceleration rate. It's a bit counterintuitive that heavy universes expand faster than light universes.

6. Although the deceleration rate has an important effect on expansion rate and size, the most important determinate factor is the initial expansion rate, sometimes referred to as "initial conditions". (That is, the absolute expansion rate at which inflation ends). A flat universe that starts out with a higher initial expansion rate will forever “outrun” the expansion of a universe that starts with a lower initial expansion rate. And, as inflation theories implicitly explain, the initial expansion rate is determined solely by the total initial mass/energy!

7. It is implicit in the various inflation theories that the inflationary expansion rate reaches its maximum at the time when it is equal to the total mass/energy that will be released during the reheating phase. This phenomenon is referred to as “the potential energy falling below the kinetic energy.” At the reheating phase, all remaining potential energy of inflation is dumped into the universe in the form of radiation at extremely high temperature. With expansion continuing rapidly thereafter, adiabatic cooling causes a determinate portion of elementary particles (the quark-gluon plasma) to become matter.

8. In net, inflation theories are designed to deliver a flat initial universe which is expanding at exactly the escape velocity of its mass/energy contents. It is flat because inflation “dumps” precisely the initial amount of mass/energy into the universe, the escape velocity of which exactly matches the maximum inflationary expansion rate. The universe is flat because it expands exactly at escape velocity, which is both the necessary and sufficient cause of flatness. If it is exactly flat at the start, the Friedmann equations tell us it will remain exactly flat for all eternity. Is this a coincidence, or an inherent preference for flatness?

9. It is implicit in the Friedmann equations that spatial expansion (at escape velocity) depends on, and derives from, the mass/energy of the contents. In other words, mass/energy must be the repository of the ongoing expansionary “momentum”. If vacuum space could exist with zero mass/energy (i.e., zero cosmological constant), it would have no mass and therefore could not possesses momentum in the Newtonian sense. Moreover, it is difficult to conceive how each individual quantum of vacuum space could possesses a unique "scalar value of expansion" which changes over time and is independent of all the surrounding quanta of vacuum. So the “momentum” of expansion is unlikely to reside in vacuum space itself.

10. The expansion of space involves no physical movement of mass/energy or vacuum space. Whatever “momentum” of expansion is stored in mass/energy, it is not a Newtonian momentum. The mass/energy is not “moving through space” in the sense of peculiar motion. Instead, each nugget of mass/energy is causally connected directly to the expansion of the vacuum space around it, at the escape velocity of that nugget. By process of elimination, expansionary momentum must be a scalar value intrinsic to mass/energy itself. When we examine matter or radiation, we can’t see or directly detect its expansionary scalar value.

11. How can we determine whether the expansion of space is an expression of expansionary “momentum” accrued during inflation, or instead is an ongoing “real-time” result of mass/energy’s gravity field? In other words, is the expansion scalar imparted to mass/energy by a prior expansionary event, or is it an intrinsic (self-powered) characteristic of mass/energy itself? In our flat universe, all mass/energy is associated with spatial expansion at its own escape velocity. Therefore we are unable to ascertain (so far) whether it is even possible for any mass/energy to possesses an expansionary scalar value that differs from escape velocity. Maybe yes, maybe no. (If the answer is no, then expansionary momentum is more properly considered a "constant" attribute of mass/energy, rather than a "scalar" value.

13. It is interesting to consider a hypothetical scenario in which, at an arbitrary point in time, the cosmological constant drops to zero for new vacuum space created in the future, but any then-existing vacuum space retains its existing equation of state of $$\omega = -1$$. According to the Friedmann equations, the universe could not remain flat in such a scenario. Supposedly, the “negative pressure” anti-gravity of the cosmological constant of the pre-existing vacuum would continue to drive a "residual" accelerating expansion rate. But a flat universe cannot accelerate if its total mass/energy does not increase, because the expansion equation demands expansion at escape velocity. Although this scenario is hypothetical, it is logically uncomfortable. It seems unnatural to constrain the cosmological constant to one and only one equation of state. This discontinuity would not arise if the cosmological constant were viewed solely as an ongoing expansion force arising directly from the expansionary scalar value of the mass/energy of the cosmological constant, and not generated separately by negative pressure.

Constructive comments are welcome.

Jon

 

[Wallace]

Hmmm, this is a bit of a minefield. Much of this isn't really wrong as such, but awkwardly worded! I would advise against calling these 'facts', then again the alternative is that they are conjectures on your part and then would be against PF guidelines.

In any case, let's leave that to the mods to decide (fortunately I'm not one and don't have to make the call).

2. We can then deduce from the expansion equation that a flat universe must always expand at exactly the “escape velocity” of its total mass/energy contents. This must be exactly true at all times: when the expansion rate is dominated by free radiation, matter, or the cosmological constant, or during each transition between an era dominated by one form of mass/energy to another. There is no discontinuity in the equation.

Why would there be 'discontinuity'? I think the escape velocity analogy isn't very useful, and only really makes sense if the Universe is matter dominated.

3. The acceleration equation tells us that during the radiation-dominated era, the active gravitational density is doubled, because radiation’s pressure is 1/3 of its density (rho). Peebles, infra at 63. The equation of state of radiation is:

$$\omega = \frac{\rho}{p}= \frac{1}{3}$$

This doubled gravity does not cause the universe to depart from flatness, because total mass/energy simultaneously decreases due to redshift. Thus the first 1x of deceleration accommodates the volume dilution of gravity, and the second 1x of deceleration accommodates the next incremental decrease in mass/energy. The expansion equation holds, and the universe at all times continues to expand at current escape velocity. Quite conveniently, if the universe was flat before the radiation-dominated era, then the equation of state of radiation will conserve that flatness. Is this a coincidence, or an inherent preference for flatness?

You've really missed the point here. You've restricted the Universe to be flat and then marveled at the fact that you find it is flat and ask about a deeper meaning?

4. The acceleration equation also tells us that during the cosmological constant-dominated era, the active gravitational density is -1x, because the cosmological constant’s “negative pressure” is equal to its mass/energy density (rho). The equation of state of the cosmological constant is:

$$\omega = \frac{\rho}{p}= -1$$

(Note that the equation of state alone would drive a -2x gravitational density, but half of this is eliminated because the acceleration equation subtracts the effect of the equation of state from Lambda.) This net 1x anti-gravity does not cause the universe to depart from flatness, because total mass/energy simultaneously increases due to the cosmological constant. Thus the first 1x increase in acceleration (starting from the 1x deceleration of matter-domination) accommodates the increased expansion rate needed to account for the existing mass/energy of the cosmological constant, and the second 1x of acceleration (from zero acceleration) accommodates the next incremental increase in mass/energy (due to the creation of more vacuum space containing more cosmological constant mass/energy). The expansion equation holds, and the universe at all times continues to expand at current escape velocity. Again conveniently, if the universe was flat before the cosmological-constant dominated era, then the equation of state of the cosmological constant will conserve that flatness. Is this a coincidence, or an inherent preference for flatness?

Again your wording here is confusing, for instance 'Note that the equation of state alone would drive a -2x gravitational density, but half of this is eliminated because the acceleration equation subtracts the effect of the equation of state from Lambda' really makes no sense. I know what you are referring to but your interpretation is a little off.

Again you force the Universe to be flat and then wonder why it is flat? I cannot fathom what leads you to think that this is a co-incidence?

5. The expansion equation tells us that if flat Universe “A” contains more total mass/energy than flat Universe “B”, then Universe A will always expand faster, and be larger at any point in time, than Universe B. The acceleration equation tells us that Universe A will always have a higher deceleration rate at any scale size than Universe B, but never enough for Universe B to overtake A’s size or acceleration rate. It's a bit counterintuitive that heavy universes expand faster than light universes.

No no no! Again, you've set the Universe to be flat, this means you restrict the total energy content of the Universe to be

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

Clearly then, the higher the expansion rate H, the greater the critical density. What you've done is pick two points in time as t=t0, call these two different Universes (instead of realizing they are the same Universe at different times) then been astounded than a monotonic function behaves as a monotonic function.

6. Although the deceleration rate has an important effect on expansion rate and size, the most important determinate factor is the initial expansion rate, sometimes referred to as "initial conditions". (That is, the absolute expansion rate at which inflation ends). A flat universe that starts out with a higher initial expansion rate will forever “outrun” the expansion of a universe that starts with a lower initial expansion rate.

Again you've misunderstood the meaning of the dimensions and units you are using. A flat matter+radiation Universe uniquely defines the cosmology. The only free parameter you have is to specify what the relative fraction of the critical density is matter and what is radiation at some t=t0. We normally set t0 to today, rather than 'at the end of inflation' though this is not essentially. However, you cannot set the amount of radiation and the amount of matter in the Universe at the time just after inflation, in a flat Universe, and then treat the initial expansion rate as a free parameter. As explained above the expansion rate and critical density are an equality. I have no idea what you are actually doing in your calculations, but it is clear something is amiss

8. In net, inflation theories are designed to deliver a flat initial universe which is expanding at exactly the escape velocity of its mass/energy contents. It is flat because inflation “dumps” precisely the initial amount of mass/energy into the universe, the escape velocity of which exactly matches the maximum inflationary expansion rate. The universe is flat because it expands exactly at escape velocity, which is both the necessary and sufficient cause of flatness. If it is exactly flat at the start, the Friedmann equations tell us it will remain exactly flat for all eternity. Is this a coincidence, or an inherent preference for flatness?

No, inflation causes flatness because curvature goes as 1/a^2 whereas a the density of a significantly negative pressure component remains constant. Therefore if the universe starts of with curvature, inflation (the expansion of the universe to many times the initial size a very quickly) curvature quickly becomes minimal. The real question is why does the inflaton field 'turn off' at the end of inflation? Your escape velocity thing is a red herring.

You really need to read up on some more of the physics!

13. It is interesting to consider a hypothetical scenario in which, at an arbitrary point in time, the cosmological constant drops to zero for new vacuum space created in the future, but any then-existing vacuum space retains its existing equation of state of $$\omega = -1$$. According to the Friedmann equations, the universe could not remain flat in such a scenario. Supposedly, the “negative pressure” anti-gravity of the cosmological constant of the pre-existing vacuum would continue to drive a "residual" accelerating expansion rate. But a flat universe cannot accelerate if its total mass/energy does not increase, because the expansion equation demands expansion at escape velocity. Although this scenario is hypothetical, it is logically uncomfortable. It seems unnatural to constrain the cosmological constant to one and only one equation of state. This discontinuity would not arise if the cosmological constant were viewed solely as an ongoing expansion force arising directly from the expansionary scalar value of the mass/energy of the cosmological constant, and not generated separately by negative pressure.

Dark energy is a hypothetical energy component that has an equation of state of w~-1. There are many different proposed forms of w(a) for dark energy, none of which cause the Freidmann equations to break down (i.e. if it starts flat is stays flat!).

Much of the rest of your comments about momentum and expansion of space are problematic and make little sense I'm afraid.

You may not feel my comments are 'constructive' and may seem a little harsh, but you are attempting to effectively derive new physical results based solely upon one set of equations that you keep wanting to take out of context without understanding the physics that they represent.

I implore you to please, take the advice that has been given to you many times here and read some good cosmology textbooks. I won't bother giving you links as they've been given to you before. You are clearly interested in this stuff, which is good, and are thinking hard about it all, which is also good. This is why you would really benefit from learning some more about it.

 

[jonmtkisco]

Hi Wallace,

Why would there be 'discontinuity'? I think the escape velocity analogy isn't very useful, and only really makes sense if the Universe is matter dominated.

I don't think there would be a discontinuity, but Pervect referred to one in his final post on my prior Friedmann thread. Also, I am quite correct when I say that the Friedmann expansion equation works equally well in a radiation-dominated, matter-dominated, or cosmological constant-dominated universe. I respectfully request that you refrain from adding confusion to this topic.

You've really missed the point here. You've restricted the Universe to be flat and then marveled at the fact that you find it is flat and ask about a deeper meaning?

You misunderstand. What I think is interesting is that every aspect of the Friedmann equations implicitly constrains the universe to remain forever flat if it starts out flat.

Again your wording here is confusing, for instance 'Note that the equation of state alone would drive a -2x gravitational density, but half of this is eliminated because the acceleration equation subtracts the effect of the equation of state from Lambda' really makes no sense. I know what you are referring to but your interpretation is a little off.

Please re-read this section, I edited it. And if you are going to say my statement is wrong, I respectfully ask that you explain why.

No no no! Again, you've set the Universe to be flat, this means you restrict the total energy content of the Universe to be

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

Clearly then, the higher the expansion rate H, the greater the critical density. What you've done is pick two points in time as t=t0, call these two different Universes (instead of realizing they are the same Universe at different times) then been astounded than a monotonic function behaves as a monotonic function.

Wallace, why are you exaggerating and misrepresenting my post? I did not say I was "astounded" or "marvelled" at any point. Those are YOUR words. I just said that some people may not have considered that heavy universes must expand faster than light ones. I have no idea what you mean when you refer to picking two points in time, so unless you care to explain that comment, I think readers should ignore it. For example, if Universe "A" starts out with twice as much radiation and twice as much matter as Universe "B", they cannot possibly be the "same" universe at different points in time. Please think before you critisize.

Again you've misunderstood the meaning of the dimensions and units you are using. A flat matter+radiation Universe uniquely defines the cosmology. The only free parameter you have is to specify what the relative fraction of the critical density is matter and what is radiation at some t=t0. We normally set t0 to today, rather than 'at the end of inflation' though this is not essentially. However, you cannot set the amount of radiation and the amount of matter in the Universe at the time just after inflation, in a flat Universe, and then treat the initial expansion rate as a free parameter. As explained above the expansion rate and critical density are an equality. I have no idea what you are actually doing in your calculations, but it is clear something is amiss

I don't understand your comment. I agree with you that expansion rate and critical density are "an equality", and are set at that equality at the end of inflation. So nothing is "amiss". I ask readers to disregard this comment.

No, inflation causes flatness because curvature goes as 1/a^2 whereas a the density of a significantly negative pressure component remains constant. Therefore if the universe starts of with curvature, inflation (the expansion of the universe to many times the initial size a very quickly) curvature quickly becomes minimal. The real question is why does the inflaton field 'turn off' at the end of inflation? Your escape velocity thing is a red herring.

Wallace, you are just saying the same thing I am in a different way. I'm saying it mathematically follows with 100% certainty that if the universe is flat at the end of inflation, it is expanding at exactly the escape velocity of its contents. The escape velocity "thing" is not a red herring. I ask readers to disregard this comment.

Dark energy is a hypothetical energy component that has an equation of state of w~-1. There are many different proposed forms of w(a) for dark energy, none of which cause the Freidmann equations to break down (i.e. if it starts flat is stays flat!).

Of course they don't cause the Friedmann equations to break down, because none of them have tackled the particular hypothetical scenario I proposed. I don't believe that you've spent any time at all considering my hypothetical scenario, so I am going to disregard this comment.

Much of the rest of your comments about momentum and expansion of space are problematic and make little sense I'm afraid.

Critisism without explanation is of no value. So I'll ignore this comment as well.

You may not feel my comments are 'constructive' and may seem a little harsh, but you are attempting to effectively derive new physical results based solely upon one set of equations that you keep wanting to take out of context without understanding the physics that they represent.

I implore you to please, take the advice that has been given to you many times here and read some good cosmology textbooks. I won't bother giving you links as they've been given to you before. You are clearly interested in this stuff, which is good, and are thinking hard about it all, which is also good. This is why you would really benefit from learning some more about it.

Wallace, my description of the Friedmann expansion theory is derived directly from Peebles' textbook and numerous other sources. I am willing to wager that I've spent far more time studying and analyzing this set of issues than you have.

All I've done is add the perspective of looking at the same standard equation from a fresh, but mathematically identical direction. It is unfair to characterize a fresh perspective on an old equation as a "new physics".

It is sad that so many of the replies on the Physics Forum are intended to discredit the original posters rather than to respond constructively to their ideas. Your response is slightly more constructive than some others I've seen, but overall I'd have to grade it an F due to the insinuations of my ignorance that you added. Personal attacks on an author are a cheap, lazy substitute for an open-minded, reasoned response.

I respectfully suggest that you use the "edit" function to transform your post into something constructive.

Jon

 

[Wallace]

[doubled posted somehow, post deleted]

 

[Wallace]

I don't think there would be a discontinuity, but Pervect referred to one in his final post on my prior Friedmann thread. Also, I am quite correct when I say that the Friedmann expansion equation works equally well in a radiation-dominated, matter-dominated, or cosmological constant-dominated universe. I respectfully request that you refrain from adding confusion to this topic.

It's your escape velocity analogy that isn't useful, not the equations!

You misunderstand. What I think is interesting is that every aspect of the Friedmann equations implicitly constrains the universe to remain forever flat if it starts out flat.

But this is exactly my point! You think it is interesting that if you set curvature to zero then it stays zero! This is as interesting as 1+1=2

Please re-read this section, I edited it. And if you are going to say my statement is wrong, I respectfully ask that you explain why.

It's still not quite right, again because you are not thinking about the physics. The acceleration equations is

$$\frac{\ddot(a)}{a} = -\rho + 3 P$$ (with some other constant factors)

so what matters is the density and the pressure. Your statement '(Note that the equation of state alone would drive a net -2x gravitational density (1+ -3), but half of this is eliminated because the acceleration equation subtracts the effect of the equation of state from Lambda.)' while being mathematically equivalent, does not describe the physics.

Wallace, why are you exaggerating and misrepresenting my post? I did not say I was "astounded" or "marvelled" at any point. Those are YOUR words. I just said that some people may not have considered that heavy universes must expand faster than light ones. I have no idea what you mean when you refer to picking two points in time, so unless you care to explain that comment, I think readers should ignore it.

In a nutshell! If you don't understand it you want it ignored. This lead to the last thread on this turning into a mess.

Lets simplify things for a second. Take a matter only FRW Universe. There is in fact only one valid solution for this Universe. You cannot have a 'heavier' or 'lighter' matter only Universe, nor can you arbitrarily decide the initial expansion rate. The only thing you can do is decide what you call t=t0. This sets H0 through which you find the current value of the mean density via

$$\rho = \frac{3H}{8 \pi G}$$

If you take two different initial times t0, they will give you different H0's and hence different initial densities. They are however, merely two different 'nows' of the same Universe. Mathematically, you are taking two different starting points in a monotonic function and pointing out that the starting point that has a higher value of the function remains higher than the other starting point at all times. Not wrong but not interesting or insightful either. Exactly the same argument holds for a flat Universe with radiation, except that you do have the additional free parameter of how much of the Universe is radiation and how much is matter at your chosen t0.

I don't understand your comment. I agree with you that expansion rate and critical density are "an equality", and are set at that equality at the end of inflation. So nothing is "amiss". I ask readers to disregard this comment.

What this means is that you can't hold the density constant while twiddling the initial expansion rate up and down, which is what your post implied by suggesting the Universes with the same energy content have different histories depending on their 'initial expansion rate'. Clearly this is not possible!

Wallace, you are just saying the same thing I am in a different way. I'm saying it mathematically follows with 100% certainty that if the universe is flat at the end of inflation, it is expanding at exactly the escape velocity of its contents. The escape velocity "thing" is not a red herring. I ask readers to disregard this comment.

Again you are absolutely right, but you don't see that what you are saying is obvious. If there is no curvature in the FRW equation there never will. This is obvious and doesn't require some strange escape velocity analogy to explain.

Of course they don't cause the Friedmann equations to break down, because none of them have tackled the particular hypothetical scenario I proposed. I don't believe that you've spent any time at all considering my hypothetical scenario, so I am going to disregard this comment.

They do, you just don't realize it. The scenario you propose leads to the equation of state altering from w=-1, since the energy density of your 'no longer quite like vacuum energy' stuff no longer remains constant but begins to diminish (since it is not being created at a rate equal to the additional space). This would lead to some value of w between -1/3 < w < -1, which is a class of models that have been studied thoroughly in the literature.

Wallace, my description of the Friedmann expansion theory is derived directly from Peebles' textbook and numerous other sources. I am willing to wager that I've spent far more time studying and analyzing this set of issues than you have.

I'll take you up on that!

It is sad that so many of the replies on the Physics Forum are intended to discredit the original posters rather than to respond constructively to their ideas. Your response is slightly more constructive than some others I've seen, but overall I'd have to grade it an F due to the insinuations of my ignorance that you added.

I'm not insinuating your ignorance, I'm trying to help you to see where you are going wrong, qhich you are, yet refuse to think about why. Instead of thinking about what I've said you simply suggest everything you don't understand be 'disregarded'. Fundamentally you don't like criticism, which is not an uncommon thing I suppose.

 

[Wallace]

Oh, another thing I forgot to mention. What would be useful is to have a look at where the Freidmann equations come from, i.e. how they are derived from the field equations and what that means. A lot of the confusion can be cleared up by a better general understanding of relativity. It is impossible to take the equations out of context of the physics that produced them and try and find fundamental new interpretations.

So I'd recommend focusing on understanding relativity more generally, then the specific example of the FRW Universe will be easier to understand. Can I recommend 'Gravity' by James Hartle. It's an excellent 'physics first' textbook on general relativity. You need to read it all though, not just grab some equations out of it!

 

[jonmtkisco]

It's your escape velocity analogy that isn't useful, not the equations!

Escape velocity is not an analogy. It is very basic and mathematically trival substitution of M/V for "rho" in the Friedmann expansion equation. Every attack you make on my references to escape velocity is an implicit attack on the Friedmann equation. I won't accuse you of being ignorant, but I respectfully request that you think this through before you critisize.

But this is exactly my point! You think it is interesting that if you set curvature to zero then it stays zero! This is as interesting as 1+1=2

I didn't set curvature to zero. As you and I agree, it started itself at zero at the end of inflation. I think it's interesting how the combination of the two Friedmann equations preserves flatness through transitions between various equations of state. And I haven't seen it explained at all clearly on this Forum before. If you think that's boring, why not keep that thought to yourself?

It's still not quite right, again because you are not thinking about the physics. The acceleration equations is

$$\frac{\ddot(a)}{a} = -\rho + 3 P$$ (with some other constant factors)

so what matters is the density and the pressure. Your statement '(Note that the equation of state alone would drive a net -2x gravitational density (1+ -3), but half of this is eliminated because the acceleration equation subtracts the effect of the equation of state from Lambda.)' while being mathematically equivalent, does not describe the physics.

I made what I think is a good attempt to add clarity to a topic that many people find to be confusing. If you think you can do a better job of it, be my guest. But critisism is not a substitute for doing the work to really explain things.

In a nutshell! If you don't understand it you want it ignored.

No, If you disagree with me then I expect a thoughful explanation, not just a knee-jerk dismissal or personal attack. But if you insist on the latter, then yes I intend to ignore it.

Lets simplify things for a second. Take a matter only FRW Universe. There is in fact only one valid solution for this Universe. You cannot have a 'heavier' or 'lighter' matter only Universe, nor can you arbitrarily decide the initial expansion rate. The only thing you can do is decide what you call t=t0. This sets H0 through which you find the current value of the mean density via

$$\rho = \frac{3H}{8 \pi G}$$

If you take two different initial times t0, they will give you different H0's and hence different initial densities. They are however, merely two different 'nows' of the same Universe. Mathematically, you are taking two different starting points in a monotonic function and pointing out that the starting point that has a higher value of the function remains higher than the other starting point at all times. Not wrong but not interesting or insightful either. Exactly the same argument holds for a flat Universe with radiation, except that you do have the additional free parameter of how much of the Universe is radiation and how much is matter at your chosen t0.

Wallace, you simply don't understand my premise, so your critisism is entirely off base. Our "historical" universe began with total matter at about 8.5E+53kg and total radiation at about 3.8E+76kg. Let's call that universe "B". Let's then define another universe "A" with twice the initial total matter (1.7E+54kg) and twice the total radiation (7.6E+77kg). Obviously, there is no sense in which universe "B" can be considered equal to universe "A" at a different point in time.

Your misunderstanding of my premise comes from thinking in terms of density rather than total mass/energy. You haven't taken the time to think this through clearly.

What this means is that you can't hold the density constant while twiddling the initial expansion rate up and down, which is what your post implied by suggesting the Universes with the same energy content have different histories depending on their 'initial expansion rate'. Clearly this is not possible!

My premise said nothing about holding density constant, nor was I talking about universes with the same energy content. As I said above, I was talking about universes with different initial total mass energy.

Again you are absolutely right, but you don't see that what you are saying is obvious. If there is no curvature in the FRW equation there never will. This is obvious and doesn't require some strange escape velocity analogy to explain.

The fact that flatness is conserved really follows more from the acceleration equation than from the expansion equation. I don't think it's necessary to convert the expansion equation to its escape velocity form in order to understand the acceleration equation.

They do, you just don't realize it. The scenario you propose leads to the equation of state altering from w=-1, since the energy density of your 'no longer quite like vacuum energy' stuff no longer remains constant but begins to diminish (since it is not being created at a rate equal to the additional space). This would lead to some value of w between -1/3 < w < -1, which is a class of models that have been studied thoroughly in the literature.

Well, maybe it was presumptuous of me to say that my particular hypothetical hasn't been studied. Nevertheless, the fact remains that a mid-course CHANGE to the equation of state of the cosmological constant is completely incompatible with retaining flatness. This is clear from the expansion equation. I find that to be unsatisfying.

I'm not insinuating your ignorance, I'm trying to help you to see where you are going wrong, qhich you are, yet refuse to think about why. Instead of thinking about what I've said you simply suggest everything you don't understand be 'disregarded'. Fundamentally you don't like criticism, which is not an uncommon thing I suppose.

If you look through my past posts on this Forum, I think you will find that I am more willing to fall on my sword and explicitly admit my mistakes than most people who make serious posts here. Maybe it is my willingness to do so that let's other posters feel it's fair game to take personal potshots at me.

I don't see that you've identified a single legitimate flaw in my reasoning on this thread, so far. Each point of discussion seems to end with your begrudging admission that I'm not only correct, but correct to the extent being mundane and boring.

Jon

 

[Wallace]

sigh, I feel we've been down this road before.

Wallace, you simply don't understand my premise, so your critisism is entirely off base. Our "historical" universe began with total matter at about 8.5E+53kg and total radiation at about 3.8E+76kg. Let's call that universe "B". Let's then define another universe "A" with twice the initial total matter (1.7E+54kg) and twice the total radiation (7.6E+77kg). Obviously, there is no sense in which universe "B" can be considered equal to universe "A" at a different point in time.

Your misunderstanding of my premise comes from thinking in terms of density rather than total mass/energy. You haven't taken the time to think this through clearly.

No, you still don't understand. If you specify the energy density of the Universe, and require it to be flat, then the expansion rate is no longer a free parameter. In the case of radiation and matter in the Universe, as in your above example, we have at out defined t0

$$H_0^2 = \frac{ 8 \pi G \rho_{m0}}{3} + \frac{ 8 \pi G \rho_{rad0}}{3}$$

See how by defining our initial matter and energy densities we have constrained the expansion rate H0, we cannot set it arbitrarily and be consistent with any initial energy densities. If we want a flat Universe, it is not possible to have any other expansion rate. Your two Universes in the above example must have different initial expansion rates H0, but the difference between these two is not due to one being intrinsically 'heavier' or 'lighter', but can only be because of the different matter/radiation ratios.

Do you agree that (for simplicity) a matter only Universe has only one solution to the FRW equation, not an infinite number of solutions with different 'initial expansion rates'?

My premise said nothing about holding density constant, nor was I talking about universes with the same energy content. As I said above, I was talking about universes with different initial total mass energy.

You have to be careful about what you call initial

Well, maybe it was presumptuous of me to say that my particular hypothetical hasn't been studied. Nevertheless, the fact remains that a mid-course CHANGE to the equation of state of matter is completely incompatible with retaining flatness. This is clear from the expansion equation. I find that to be unsatisfying.

No this is simply wrong. A change (no matter how abrupt) in the equation of state of an energy component is not incompatible with flatness. Type 'dynamical dark energy' into google scholar if you want to see oodles of papers on dark energy with w(a) equation of state (where w varies with a).

 

[jonmtkisco]

No, you still don't understand. If you specify the energy density of the Universe, and require it to be flat, then the expansion rate is no longer a free parameter. In the case of radiation and matter in the Universe, as in your above example, we have at out defined t0

$$H_0^2 = \frac{ 8 \pi G \rho_{m0}}{3} + \frac{ 8 \pi G \rho_{rad0}}{3}$$

See how by defining our initial matter and energy densities we have constrained the expansion rate H0, we cannot set it arbitrarily and be consistent with any initial energy densities. If we want a flat Universe, it is not possible to have any other expansion rate. Your two Universes in the above example must have different initial expansion rates H0, but the difference between these two is not due to one being intrinsically 'heavier' or 'lighter', but can only be because of the different matter/radiation ratios.

Do you agree that (for simplicity) a matter only Universe has only one solution to the FRW equation, not an infinite number of solutions with different 'initial expansion rates'?

Yes, I agree that you are now saying exactly the same thing I was saying all along. ANY universe resulting from inflation will be constrained to have an initial expansion rate (at the end of inflation) dictated by the Friedman expansion rate, which translates into the precise escape velocity of the total mass/energy of its contents. It can have no other value. So in that sense, the initial expansion rate is entirely determined by the total initial value of the mass/energy contents. And that is that answer to the question of why inflation ends at a particular expansion rate.

The only way that the initial expansion rate could be different is if the initial total mass/energy were higher or lower than our "historical" initial values. Which is my example of universes "A" and "B". If universe A starts out with more TOTAL mass/energy than universe B, it will forever expand faster, and will be larger at every point in time, than universe B. And I want to be very clear, for the purposes of this discussion I am assuming exactly the SAME proportional mix of matter and radiation in each of universes A and B.

No this is simply wrong. A change (no matter how abrupt) in the equation of state of an energy component is not incompatible with flatness. Type 'dynamical dark energy' into google scholar if you want to see oodles of papers on dark energy with w(a) equation of state (where w varies with a).

I'm sure there are oodles of such papers. I'll go check them out. But having apparently studied the matter, you could save us all some time by explaining how it is mathematically possible for the Friedmann expansion equation to preserve flatness in a scenario where expansion rate continues to accelerate after a point in time when total mass/energy of the universe becomes capped. Once the expansion rate and total mass/energy get out of synch, by definition the universe is no longer flat.

The only way I can see to make it work is if the "old" cosmological constant created before the cutoff time ceases to provide any further accelerative "impulse" after the cutoff time. In other words, the "old" cosmological constant starts acting like plain old matter. But that is "cheating" in the sense that the equation of state for the "old" cosmological constant changes from -1 to 0 at the cuttoff point. In my scenario, the equation of state of the "old" pre-existing cosmological constant does not change.

Jon

 

[Wallace]

Yes, I agree that you are now saying exactly the same thing I was saying all along. ANY universe resulting from inflation will be constrained to have an initial expansion rate (at the end of inflation) dictated by the Friedman expansion rate, which translates into the precise escape velocity of the total mass/energy of its contents. It can have no other value. So in that sense, the initial expansion rate is entirely determined by the total initial value of the mass/energy contents. And that is that answer to the question of why inflation ends at a particular expansion rate.

The only way that the initial expansion rate could be different is if the initial total mass/energy were higher and lower than our "historical" initial values. Which is my example of universes "A" and "B". If universe A starts out with more total mass/energy than universe B, it will forever expand faster, and will be larger at every given scale, than universe B.

You have to be careful that you don't just move the goal posts! A universe with a higher 'initial' energy density will, as it evolves, have a lower energy density. In the case of matter only (for simplicity) it will, at some point, have the same energy density and expansion rate as the 'other' universe, that have a 'lower' initial energy density. They are the same Universe, we are just changing were we set t=t0, i.e. changing where 'initial' is.

For the case with matter and radiation it is not so simple, as the energy densities evolve differently. However in this case, in a realistic scenario, we are not free to choose the initial matter/radiation ratio as this is set by Nuclear Physics.

Let me make that clear, for a flat, matter + radiation universe that obeys the physics of the standard model there is only one valid solution. The 'initial' expansion rate at the end of inflation is not a free parameter.

To get around all this confusion, we normally set t=t0 to today, rather than a small fraction of a second, and then we ask 'what is H0?'. This tells us which of the possible whens of this solution we are at, since we know there is only one valid solution.

I'm sure there are oodles of such papers. I'll go check them out. But having apparently studied the matter, you could save us all some time by explaining how it is mathematically possible for the Friedmann expansion equation to preserve flatness in a scenario where expansion rate continues to accelerate after a point in time when total mass/energy of the universe becomes capped. Once the expansion rate and total mass/energy get out of synch, by definition the universe is no longer flat.

Whoa! back up a second. The issue is the unphysical nature of what you propose. If the equation of state of dark energy stays at w=-1, then it will continue to have the same energy density everywhere. If its equation of state is greater than -1 then it's energy density will diminish. However, it cannot both retain its equation of state of -1 and diminish its energy density, since the change of energy density with expansion defines the equation of state.

Your original question was:
[/quote]
It is interesting to consider a hypothetical scenario in which, at an arbitrary point in time, the cosmological constant drops to zero for new vacuum space created in the future, but any then-existing vacuum space retains its existing equation of state of w=-1.
[/quote]

This makes no sense? Do you mean that as the Universe expands outwards, not new energy is created at the expanding edge? (which is a horrible sentence that is full of flaws) If so then apart from this not being what the FRW equations describe, it would also mean that the vacuum energy was not longer homogenous which cannot be described by the FRW equations.

This may well not be what you mean though. Do you mean that something happens to the homogenous energy density of the vacuum? If so then any change in it will be described by the equation of state w for this change and everything is fine.

Please clarify what you meant though.

 

[jonmtkisco]

You have to be careful that you don't just move the goal posts! A universe with a higher 'initial' energy density will, as it evolves, have a lower energy density. In the case of matter only (for simplicity) it will, at some point, have the same energy density and expansion rate as the 'other' universe, that have a 'lower' initial energy density. They are the same Universe, we are just changing were we set t=t0, i.e. changing where 'initial' is.

For the case with matter and radiation it is not so simple, as the energy densities evolve differently. However in this case, in a realistic scenario, we are not free to choose the initial matter/radiation ratio as this is set by Nuclear Physics.

Let me make that clear, for a flat, matter + radiation universe that obeys the physics of the standard model there is only one valid solution. The 'initial' expansion rate at the end of inflation is not a free parameter.

To get around all this confusion, we normally set t=t0 to today, rather than a small fraction of a second, and then we ask 'what is H0?'. This tells us which of the possible whens of this solution we are at, since we know there is only one valid solution.

I agree with your 2nd, 3rd and and 4th paragraphs. We are not free to arbitrarily choose a different initial matter/radiation ratio, and I did not do so. I agree (and have always agreed) that the initial expansion rate is not a free parameter. And I agree that general practice is to approach expansion going backwards from the present. I just seek whatever enlightment may come from starting at the beginning and moving forward. It's definitely more work to look at old equations from a new perspective!

I do not agree with your 1st paragraph. While it may be difficult to calculate, our universe in fact had an initial total mass/energy that was specific and fixed at some figure. If you want to talk in terms of density, it had a specific initial density and radius. But you cannot extrapolate further backwards from there and say that at "t minus x" it had the same mass/energy density but a smaller radius. (Or a higher total mass/energy). According to inflation theory, all radiation and matter was released in the reheating phase at the end of inflation, and it was a specific amount, coupled with a specific initial expansion rate.

So if an alternative universe had a higher total/mass energy at the end of its inflation, it truly is a different universe from ours.

Whoa! back up a second. The issue is the unphysical nature of what you propose. If the equation of state of dark energy stays at w=-1, then it will continue to have the same energy density everywhere. If its equation of state is greater than -1 then it's energy density will diminish. However, it cannot both retain its equation of state of -1 and diminish its energy density, since the change of energy density with expansion defines the equation of state.

This makes no sense? Do you mean that as the Universe expands outwards, not new energy is created at the expanding edge? (which is a horrible sentence that is full of flaws) If so then apart from this not being what the FRW equations describe, it would also mean that the vacuum energy was not longer homogenous which cannot be described by the FRW equations.

This may well not be what you mean though. Do you mean that something happens to the homogenous energy density of the vacuum? If so then any change in it will be described by the equation of state w for this change and everything is fine.

Please clarify what you meant though.

Yes, you are correct that my scenario specifies that vacuum energy is no longer homogenous after the cutoff point. It does not mean that the "new" vacuum is all created at the "expanding edge" of the universe -- there is no such thing as an expanding edge. New vacuum of course would be interspersed regularly throughout the existing "old" vacuum, all over the universe.

You can dismiss my scenario as unrealistic, but I have to wonder what the criteria for "realism" is. It seems more realistic to me that "old" vacuum wouldn't magically lose the mass/energy of its pre-existing cosmological constant, just because "new" vacuum is created under newly-prevailing conditions. Wouldn't the elimination of the existing mass/energy of the cosmological constant violate the conservation of mass/energy?

In any event, it's just a hypothetical thought experiment to explore the characteristics of the Friedmann equations. I have no actual opinion about whether it could be real.

Jon

 

[Wallace]

If the energy is not homogeneous then you can't use the Freidmann equations (since they are derived from GR on the assumption of homogeneity and isotropy), so let's stop even talking about that, it makes no sense.

On the first point, you are still not understanding what you mean by 'initial'. Take again (for simplicity) a matter only flat Universe. If you have one Universe with a 'higher' mass density that you 'start' at some time, then everything that you derive from it (expansion rate etc) will be that same as what you get from a 'different' Universe that you 'start' at a later point in time.

Remember that we are here now, so what we measure is H0, and from this we can extrapolate back in time.

Try this calculation, for matter only flat universe, set two different 'initial' (in your examples) mass densities. Track say H as a function of density. What you will find is that you get the same answer for both, so for any H that someone would measure (at the time they exist in the Universe) they can't tell the difference between the 'two' universes, they are the same.

With radiation it is not so simple, however as I've pointed out there is additional physics that determines what matter/radiation ratio we have as a function of the total mean density of the Universe. When we calculate this (which is much harder) we find the same result, that for a given H that an observer might measure, they would find the same energy densities for both matter and radiation, and again the 'two' universes would be indistinguishable.

 

[pervect]

If you specify the energy density of the Universe, and require it to be flat, then the expansion rate is no longer a free parameter.

Yes, I agree that you are now saying exactly the same thing I was saying all along. ANY universe resulting from inflation will be constrained to have an initial expansion rate (at the end of inflation) dictated by the Friedman expansion rate,

These two statements are almost the same, but not quite, and I think the difference may be relevant to the issue that is being discussed. A non-flat universe will have a slightly different expansion rate at the end of inflation than a non-flat universe. The point is that the difference will be very small.

Suppose Kc/a^2 is large, but finite, before inflation. As a(t) increases as the universe evolves, the importance of the term Kc/a^2 is eventually going to diminish for a large enough value of a. Because inflation involves such a large increase in a, the end effect is to make the Kc/a^2 term very small. But it never actually vanishes totally, it just becomes small and unimportant to the dynamics of the expansion.

 

[jonmtkisco]

If the energy is not homogeneous then you can't use the Freidmann equations (since they are derived from GR on the assumption of homogeneity and isotropy), so let's stop even talking about that, it makes no sense.

The Friedmann equations do not require 100% homogeneity. Obviously the matter in the universe isn't perfectly homogeneous either. The Friedmann equations treat homogeneity as a reasonable approximation. I think that two different flavors of cosmological constant would constitute a reasonable approximation of homogeneity for some significant period after a hypothetical cutoff of the cosmological constant. As I said, the purpose of this scenario is just to highlight certain characteristics and limitations in the Friedmann equations. I think it serves that purpose well.

Wallace, thanks for your insights on that subject. However, I don't want others to be discouraged from offering thoughts about this scenario if they're interested.

On the first point, you are still not understanding what you mean by 'initial'. Take again (for simplicity) a matter only flat Universe. If you have one Universe with a 'higher' mass density that you 'start' at some time, then everything that you derive from it (expansion rate etc) will be that same as what you get from a 'different' Universe that you 'start' at a later point in time.

Remember that we are here now, so what we measure is H0, and from this we can extrapolate back in time.

Try this calculation, for matter only flat universe, set two different 'initial' (in your examples) mass densities. Track say H as a function of density. What you will find is that you get the same answer for both, so for any H that someone would measure (at the time they exist in the Universe) they can't tell the difference between the 'two' universes, they are the same.

With radiation it is not so simple, however as I've pointed out there is additional physics that determines what matter/radiation ratio we have as a function of the total mean density of the Universe. When we calculate this (which is much harder) we find the same result, that for a given H that an observer might measure, they would find the same energy densities for both matter and radiation, and again the 'two' universes would be indistinguishable.

Actually Wallace, I do understand what you're getting at. I just consider it to be a distraction that has little to do with the subjects my thread is aimed at.

I agree that the matter density of both "heavy" and "light" universes becomes diluted by volume over time, following the same Friedmann curve. Therefore, at a given time "Ta" universe A can have the same matter density as universe B has at time "Tb". That is inevitable. Again, I "get it", I agree.

However, that in no way demonstrates that universe A and B are, in effect, the same universe at different points in time. Clearly they are not identical, for several reasons:

1. Universe A at all times in its history contains more total kilograms of matter than universe B. At the separate points in time when the matter densities of the two universes match, the matter of Universe A is distributed over a larger radius than is the case for Universe B.

2. Universe A at any given point of time in its history will have a higher (more negative) deceleration rate than universe B. (Assuming that each universe has the same matter/radiation ratio).

3. Universe A has a faster expansion rate than universe B both initially, and at any given point in time thereafter.

4. Because of Universe A's faster expansion rate, and because radiation density declines faster over time than matter density, the shape of universe A's expansion curve will show a sharper downward dip during the radiation-dominated era, compared to universe B.

5. Since universe A's absolute expansion rate is faster than universe B's at any given point in time, universe A will reach any given scale size more quickly than universe B. So at any given point in time, universe A's event horizon must have a smaller radius than universe B's. There will have been less total time for light to traverse universe A, and that light will be "swimming upstream" against a faster average "current" of spatial expansion.

 

[cristo]

A non-flat universe will have a slightly different expansion rate at the end of inflation than a non-flat universe.

I think there's a typo here, pervect, and that one of these should read "flat."

 

[Wallace]

The Friedmann equations do not require 100% homogeneity. Obviously the matter in the universe isn't perfectly homogeneous either. The Friedmann equations treat homogeneity as a reasonable approximation. I think that two different flavors of cosmological constant would constitute a reasonable approximation of homogeneity for some significant period after a hypothetical cutoff of the cosmological constant. As I said, the purpose of this scenario is just to highlight certain characteristics and limitations in the Friedmann equations. I think it serves that purpose well.

No, the Friedmann equations require the energy to be precisely homogeneous and isotropic. It is in fact an open question whether our Universe, which is inhomogeneous, does follow closely enough the FRW solution. There is a lot of literature on this subject. There are other metrics, the Lematire (sp?) Tolman-Bondi model, the Szerkes (sp?) model, Buchart's model to name a few, that describe various types of inhomogeneous universes. Each of these define dynamical equations that are different from the FRW dynamical equations.

Even if your 'two flavours' of cosmological constant are evenly distributed, your proposal still makes no sense. Again there are plenty of examples in the literature of 'two fluid' (or more) dark energy models and they also don't make the FRW break down. You can't separately define what the energy density evolution of a component is as well as its equation of state, they depend on each other and this dependence ensures all the equations hold. To violate this dependence violates local conservation of energy, which would cause the equations to break down, but only because you violate the physical principles that give meaning to the maths.

1. Universe A at all times in its history contains more total kilograms of matter than universe B. At the separate points in time when the matter densities of the two universes match, the matter of Universe A is distributed over a larger radius than is the case for Universe B.

You are mistaking the interpretation of radius. Remember that the model you are dealing with is an infinite Universe, therefore there is not sensibly defined 'radius' of the Universe (we could talk about say the radius of the observable Universe, but that is not what you are referring to). Instead we use the dimensionless scale factor a. This defines the ratio of distances between co-moving observers at different times. You cannot simply take the absolute value of a from one of you examples and compare it with that from another.

Again, what we do normally is to define a=a0=1 today, and then define everything relative to that. IF you do this normalization (so at the same value of H0, find a and set to 1) you will see that the two Universes are in fact identical. You are not comparing like with like and don't realize what dimensions the quantities you are dealing with are in.

2. Universe A at any given point of time in its history will have a higher (more negative) deceleration rate than universe B. (Assuming that each universe has the same matter/radiation ratio).

Only because you start at different points. Re-read what I've written previously. All you are doing is asserting that a monotonic function behaves monotonically, since you are starting two integrations at different points on the same curve. The reason you think they are different is because you are setting t=0 at different places in the two models and seeing a numerical difference (i.e. at the same numerical value of t in the two models things are different). What you are not doing is realizing how the time needs to be interpreted. What you will find is if you ask how much time elapses between the times when H takes a particular value and when it takes a different value is the same in the two models. Hence they describe the same evolution.

3. Universe A has a faster expansion rate than universe B both initially, and at any given point in time thereafter.

As above, this is solely by when you arbitrarily set t=0. The Universe does not have a clock that tells you the time since the Big Bang, we have to evolve things backwards from today, and what you will find if you do this is that your models are identical.

4. Because of Universe A's faster expansion rate, and because radiation density declines faster over time than matter density, the shape of universe A's expansion curve will show a sharper downward dip during the radiation-dominated era, compared to universe B.

Again, we do not measure time in the way you obtain numerical values of it. You need to make the connection between the number you crunch and what they mean physically.

5. Since universe A's absolute expansion rate is faster than universe B's at any given point in time, universe A will reach any given scale size more quickly than universe B. So at any given point in time, universe A's event horizon must have a smaller radius than universe B's. There will have been less total time for light to traverse universe A, and that light will be "swimming upstream" against a faster average "current" of spatial expansion.

Again, the scale factor is dimensionless and only makes sense to a ratio with respect to something else. You have to make sure both models are getting their ratio with respect to the same thing, otherwise you cannot simply compare numbers.

 

[jonmtkisco]

Hi Pervect,

Suppose Kc/a^2 is large, but finite, before inflation. As a(t) increases as the universe evolves, the importance of the term Kc/a^2 is eventually going to diminish for a large enough value of a. Because inflation involves such a large increase in a, the end effect is to make the Kc/a^2 term very small. But it never actually vanishes totally, it just becomes small and unimportant to the dynamics of the expansion.

I agree. As I have come to understand it, during a radiation- or matter-dominated universe, with their decelerating expansion rates, any pre-existing curvature will decline more slowly than total density. Therefore, curvature becomes more noticeable as the scale size increases. So any pre-existing non-flatness increases over time in a decelerating universe.

On the other hand, during inflation or a cosmological constant-dominated universe, expansion accelerates at a geometrically increasing rate. This tends to make curvature less noticeable over time as the scale size increases. So a geometrically expanding universe becomes relatively more flat as it expands.

I think the above description of flatness is (and by definition must be) completely consistent with the concept of flatness mandated by the Friedmann expansion equation -- that a universe becomes less flat only if its expansion rate diverges more and more over time from the escape velocity of its mass/energy contents.

I do question, however, whether the current inflation theories "cheat" in reaching this outcome. As I understand it, inflation theories make no effort to maintain flatness at every instant during the inflationary period. Instead, regardless of what happens during inflation, they hypothesize that inflation ends by "dumping" a total mass/energy whose escape velocity exactly equals the maximum expansion rate reached during inflation. I would find these inflation theories to be more compelling if they required the universe to remain perfect or near-perfect flatness at all times during inflation. There seems to be a presumption that the universe did not need to be flat prior to the start of inflation. Some people may view that as desirable because it places fewer demands on the "initial conditions" before inflation. But I am not comfortable with any assumption that the universe was ever anything other than flat. It's fair to say that's a mere prejudice on my part...

 

[jonmtkisco]

Hi Wallace,

No, the Friedmann equations require the energy to be precisely homogeneous and isotropic. It is in fact an open question whether our Universe, which is inhomogeneous, does follow closely enough the FRW solution.

Now we're just wasting time on semantics. The Friedmann equations "assume" homogeneity, they don't "require" it. According to Peebles, the inhomogeneity of matter is on the order of 10E-4. So as long as any inhomogeneity of the cosmological constant is less than that, it introduces less error than is tolerated in the "standard model".

Even if your 'two flavours' of cosmological constant are evenly distributed, your proposal still makes no sense. Again there are plenty of examples in the literature of 'two fluid' (or more) dark energy models and they also don't make the FRW break down. You can't separately define what the energy density evolution of a component is as well as its equation of state, they depend on each other and this dependence ensures all the equations hold. To violate this dependence violates local conservation of energy, which would cause the equations to break down, but only because you violate the physical principles that give meaning to the maths.

You're marching around in circles. First you insist that an inhomogeneous cosmological constant automatically causes the Friedmann equations to break down, then you say that it doesn't, then you say it does. Please pick a position and stick to it !

I reiterate that it obviously makes no sense for the expansion rate to continue accelerating if total mass/energy is capped. The Friedmann equations cannot generate a "flat" result from that scenario. And of course my underlying assumption is that lambda has a mass/energy, the two go hand in hand. That's my point: if you hypothesize a cutoff time, you should still have both the lambda and the mass/energy of the cosmological constant from before that time. After that time, "new" vacuum possesses neither.

When exactly is "today" in universe A? In other words, what is the "lookback time" to t=0in universe A?

The reason you think they are different is because you are setting t=0 at different places in the two models and seeing a numerical difference (i.e. at the same numerical value of t in the two models things are different).

Earth to Wallace: I am setting t=0 to exactly the same time in universes A and B: the time when inflation ends.

The Universe does not have a clock that tells you the time since the Big Bang, we have to evolve things backwards from today, and what you will find if you do this is that your models are identical.

Whaaaaa? I just cited 5 reasons why the two models are not identical.

Wallace, unless you have a new point to make, can we please return to the substance of my thread? I will be happy to engage with you at great length in a separate thread on the subject of "Why a universe that weighs twice as much is, or is not, identical to a universe that weighs half as much." Do you want me to initiate that thread?

Jon

 

[Wallace]

Hi Wallace,

Now we're just wasting time on semantics. The Friedmann equations "assume" homogeneity, they don't "require" it. According to Peebles, the inhomogeneity of matter is on the order of 10E-4. So as long as any inhomogeneity of the cosmological constant is less than that, it introduces less error than is tolerated in the "standard model".

Sure, but be aware that the assumption that it doesn't matter (the inhomogeneities) is not universally accepted, there are plenty of papers around discussing this issue.

You're marching around in circles. First you insist that an inhomogeneous cosmological constant automatically causes the Friedmann equations to break down, then you say that they don't, then you say they do. Please pick a position and stick to it !

No, I'm not. Please calm down and stop over-reacting. The Friedmann equations are derived assuming homogeneity and isotropy. We think we can still use them as long as the deviations from this are 'small enough'. If they are not, then the Friedmann equations are not valid for our Universe. It's not that they would 'break down' it's just that they would no longer be based on physical conditions that match our Universe.

Earth to Wallace: I am setting t=0 to exactly the same time in universes A and B: the time when inflation ends.

Please don't be childish, I am trying to patient explain the misconception you have in interpreting your results. I'm sure your numbers are okay, if they are interpreted correctly. I can see exactly what you are doing and where you are going wrong in your interpretation. If you choose not to listen to me, that's fine. But try and be civil.

Now to the substance. Your point is mine, that you are setting t=0 to 'the end of inflation' in both models. This is your error. There is no universal signpost in the Universe that we can pin an absolute time to, so doing this is not physically meaningful. It's perfectly fine to do this and run your calculations, however, once they have been completed, you need to determine what the physical meaning of your time co-ordinate and the values it takes is. The only time we can sensibly do this is 'today', where we specify today by measuring the Hubble Constant at the present epoch. If you specify today in both models to be the same (as in, they have the same Hubbles constant) and normalise the times you get (by dividing all times by the time today for each model) you will then find that both of your models are identical.

Note that this only applies to a matter only Universe, with radiation the ratio that you start with makes a difference, but as I've said, this ratio is set by other physics, so you can't do it arbitrarily.

Whaaaaa? I just cited 5 reasons why the two models are not identical.

and I've just explained, once again, the procedure you need to go through to find the way in which they are identical.

What I mean by 'identical' is that someone looking at the Universe now would not distinguish between them, they would be exactly the same. Remember that the only way we can determine 'when' we sit in the solution of the Freidmann equations is by measuring H today. That is why it is such an important parameter in cosmology. It is true that your two models would have a very slight difference in the look-back time to inflation, however this will be negligible and in any case unmeasurable since it occurs well before re-combination. All this means is that you would be tracking back in time on identical curves and one stops at a slightly earlier time than the other. At all times after this early period, the evolution of the Universe is identical, when properly normalized to physically meaningful quantities.

Wallace, unless you have a new point to make, can we please return to the substance of my thread? I will be happy to engage with you at great length in a separate thread on the subject of "Why a universe that weighs twice as much is, or is not, identical to a universe that weighs half as much." Do you want me to initiate that thread?

This issue strikes at the heart of the interpretations you've made in this thread. Therefore it needs to be sorted out to proceed sensibly. The shift in your thinking that you need to make is very slight really, but crucial. You need to realize that numbers, in order to have meaning, need to be tied to physical concepts and things that we can observe. This is especially true in Relativity when we often have a lot of co-ordinate freedom. It is very important to understand what the co-ordinates we are using actually mean.

Please try and play around with the procedure I've explained for sensibly comparing the results from your two models and consider carefully what I'm trying to explain. I'm not a muppet when it comes to Relativity and I can see exactly what you are doing. It is very close to correct but you need to make a final important step.

If you cannot continue this conversation in a civil way then I've nothing more to add.

 

[jonmtkisco]

Hi Wallace,

I appologize for being snippy. You have a tendency to be condescending in your notes, referring to every difference of interpretation as a misconception or misunderstanding on my part. I will avoid that and I hope you will too.

I will work through your idea about "identical universes" before I respond again.

Jon

 

[Wallace]

Let me clarify something. In my last post I suggested you need to take the ratio of all your times with respect to a common t0 specified by the same H0 in both models. This isn't right, you need to take the difference between the times and t0. I did explain earlier than you need to take the difference, I just confused myself in the last post.

The essential point remain the same though, and that is that you need to move beyond the numerical values you get and work out what they mean physically.

I hope the following makes it even clearer. Let's start take a matter only Universe, then we have

$$H^2 = \frac{8 \pi G}{3} \rho$$

Now, this is a differential equation so we need to specify boundary conditions. Initially we have

$$H_0^2 = \frac{8 \pi G}{3} \rho_0$$

Now, if we define the dimensionless scale factor to be unity at the initial time then we know that

$$\rho = \rho_0 a^{-3}$$

Combining the three above equations then gives us

$$\frac{H^2}{H_0^2} = a^{-3}$$

But

$$H \equiv \frac{da}{dt} \frac{1}{a}$$

and therefore

$$\frac{da}{dt} \frac{1}{a} \frac{1}{H_0} = a^{\frac{-3}{2}}$$

solving this differential equation we get

$$\frac{2}{3} a^{\frac{3}{2}} = H_0 t$$

Now, the above line is crucially important. It describes the interrelated nature of the scale factor, time and expansion rate. It is the solution to the Friedmann equations (for matter only) so your numerical calculations should agree with it if they are correct.

The really important thing to note though is that of the three quantities, there is only one that we can measure, the expansion rate today H0. This is why we use today as the boundary condition and not as you are doing and specifying it at the unobservable 'end of inflation'.

You can do the integration 'forwards' as you are doing (which is the reverse direction from the way it is normally performed) and there is nothing wrong with this, providing that when you finish, you correctly adjust your numerical answer by reference to physically observable quantities. What you will see when you do this is that your 'two' Universes are the same, in that observers within them, who cannot measure t but can only measure H, will see them as indistinguishable.

 

[jonmtkisco]

I have tried several ways construct a spreadsheet with Ho set at the end of inflation (around t=E-33 seconds), but integrating the Friedmann formulas that way introduces so much error over time that I have not been able to generate any useful result. That's equally true whether I work from density or absolute mass and radius values.

Studying the problem conceptually, I conclude the following:

1. Inflation models, nuclear physics, and the Friedmann equations together establish fairly firm initial rho (density) values for all of the key parameters at the end of inflation: total rho, matter rho, radiation rho, and cosmological constant rho. Total rho is fixed as a function of the temperature necessary for nucleosynthesis of the light elements. The matter/radiation density ratio is set by nuclear physics. The cosmological constant is inherently a fixed value which has been estimated by observations. Therefore, any "thought experiment" which assumes more than a de minimus variation to any of these values must automatically be labelled "unrealistic."

2. If rho is a fixed value at a given point in elapsed time, then the ratio of mass to radius as a function of elapsed time is completely constrained as well. Any observable universe with the same rho (for matter, radiation, and CC) must have the same historic expansion curve, radius, and total mass/energy. Positing a larger radius without changing the lookback time is meaningless, because then the radius won't correspond to the actual observable horizon at whatever point in time one chooses to label as "now".

3. As Wallace points out, matter density follows a fixed curve over time, so any difference in matter density alone can be corrolated simply to a different point in elapsed time.

4. However, if the initial matter/radiation ratio were changed significantly (without changing total rho), the resulting expansion curve and elapsed time to the present can be significantly changed. Changes in matter/radiation ratio during the radiation-dominated era can have a very large effect on the total expansion curve, which can transcend the subsequent matter-dominated era. The same is true if radiation rho is kept at its historical value while initial matter rho is treated as a free parameter.

One can conclude that either: (a) any thought experiment which varies a physical parameter in the Friedmann equations is automatically invalid, or (b) it is interesting to examine how the output of the Friedmann equations changes when the inputs are varied, even if the resulting parameters are "unrealistic."

I continue to find it interesting that, 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. Monotonic, perhaps, but interesting.

Jon

 

[Jorrie]

I have tried several ways construct a spreadsheet with Ho set at the end of inflation (around t=E-33 seconds), but integrating the Friedmann formulas that way introduces so much error over time that I have not been able to generate any useful result. That's equally true whether I work from density or absolute mass and radius values.

Hi Jon.

I found it useful to start at some early known point, like decoupling (CMB) time, with known redshift and expansion factor a. Then I integrate backwards towards inflation and forward towards the present and future. I believe that the situation for t <~ 1 seconds may become tricky due to the fact the the equation of state of radiation energy are not known there. See https://www.physicsforums.com/showthread.php?t=195021".

I continue to find it interesting that, 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. Monotonic, perhaps, but interesting.

I guess you meant a "higher total mass/energy density after inflation..."? Or perhaps the mass-energy of the observable universe after inflation? The latter is a slippery concept, as has been pointed out many times in this forum...

The first one is a contradiction: the universe cannot then be flat. Also, if the energy density after inflation was significantly higher than the critical density, the universe would have first expanded and then contracted again before dark energy could have taken over as a dominant effect.

 

[Wallace]

I have tried several ways construct a spreadsheet with Ho set at the end of inflation (around t=E-33 seconds), but integrating the Friedmann formulas that way introduces so much error over time that I have not been able to generate any useful result. That's equally true whether I work from density or absolute mass and radius values.

As suggested by Jorrie and myself previously, don't set H0 at 'the end of inflation'. You should set it only where it can be measured, which is today, and integrate backwards from there. That will help keep you numerical errors in check as well.

2. If rho is a fixed value at a given point in elapsed time, then the ratio of mass to radius as a function of elapsed time is completely constrained as well. Any observable universe with the same rho (for matter, radiation, and CC) must have the same historic expansion curve, radius, and total mass/energy. Positing a larger radius without changing the lookback time is meaningless, because then the radius won't correspond to the actual observable horizon at whatever point in time one chooses to label as "now".

Be careful, your equations are not calculating any sensibly defined radius of the Universe, rather the dimensionless scale factor which is defined only as a ratio to its value at t0. You should avoid thinking of the scale factor as a radius! It does not have a one to one relationship with the radius of the observable Universe, so is in no sense a radius.

That doesn't make anything above wrong neccessarily (as long as you replace every reference you've made to radius with a sensibly defined scale factor), but sloppy use of terminology is not a minor issue but is crucially important (see any of the many threads on expanding space for instance).

I continue to find it interesting that, 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. Monotonic, perhaps, but interesting.

You're still missing a crucial point and that is that it is not physically meaningful to set 'the end of inflation' as a universal t=0 for different models and then compare what they are like after the same 'elapsed time'.

You can only set t=t0 at a time when you can measure H0 and then set a0=1 at that time. Once you do this you could then find the look-back time to some earlier epoch. What you would find is that there is a (very small) difference in look-back time to inflation, if you do define inflation as happening when the Universe has some particular density (i.e. set the density immediately after inflation as a free parameter).

I'm a little hazy on the physics of the very early Universe but I'm not sure that we know the precise density of the Universe immediately after inflation. I don't think that it would make much difference to observables. It certainly would make no difference to zeroth order observables such as Supernovae distances, and I can't see that it would make much difference to higher order observables such as the CMB and structure formation. We could constrain a combination of the density immediately after inflation and the amplitude of the power of the fluctuations at that time, but I can't see how we could measure them individually.

I'm getting a little beyond what I'm sure about here though, so does anyone else know much about this?

In any case, in the zeroth order (homogeneous) model you are dealing with, there is no difference in subsequent evolution for models where you set different mean densities immediately after inflation.

 

[jonmtkisco]

Hi Jorrie & Wallace,

I am quite familiar with how to start with Ho at the present and work backwards towards inflation. If that's all I wanted to accomplish then I would have no fresh perspective at all to test out in this thread. I'm trying to calculate things backwards or upside down (using equivalent versions of the same equations) and see if I get the same answers, and to see whether any insights emerge that are interesting to people with my level of knowledge. If nothing interesting emerges, then so be it. I would appreciate if you could help keep me honest but not try to discourage me from exploring.

Jorrie, in reference to the two variant mass/energy scenarios I briefly mentioned at the end of my last post, you commented:

"The first one is a contradiction: the universe cannot then be flat. Also, if the energy density after inflation was significantly higher than the critical density, the universe would have first expanded and then contracted again before dark energy could have taken over as a dominant effect."

As you know as well as anyone, a universe with higher than historical density can of course be flat, as long as the initial expansion rate (after inflation) is adjusted commensurately higher. This reflects simply that the expansion rate must always be equal to escape velocity, if the universe is going to be flat. To reinforce this point, I quote from your publication:

"...the inflationary epoch 'delivered' a universe with the expansion rate precisely balanced with the amount of matter (visible and dark) that caused the slowing down effect through mutual gravitational attraction. For the vacuum aided expansion curve, the amount of matter must today be less than what it would have been for a matter-only universe -- only 30% in the case plotted. So during early years, when there was negligible vacuum effect, the amount of matter to be balanced by expansion rate must have been less in the same proportion. This 'lighter' universe would initially have expanded slower than the 'heavier' one. It can be easily deduced from the expansion law equations..." p199.

Jorrie, I had forgetten that you wrote that, long before I "independently discovered" that a lighter universe would initially have expanded slower than a heavier one. So the credit (on this forum!) for that interesting insight properly belongs to you, not me.

Jon

 

[jonmtkisco]

Hi again Wallace and Jorrie,

Just thinking out loud here. One reason why it's difficult to calculate expansion rates going "forwards in time" from the end of inflation is that you don't have a known Ho value to work from. But can't I generate exact values for scale factor and time using a form of the equation that doesn't rely on Ho per se, such as the following equation for the matter-dominated era:

$$a = \left(\frac{t}{to} \right)^{\frac{2}{3}}$$

I can of course set "to" = E-33 seconds, and calculate everything else accurately from there. And the formula is different for the radiation-dominated period.

Jon

 

[Wallace]

Hi Jorrie & Wallace,

I am quite familiar with how to start with Ho at the present and work backwards towards inflation. If that's all I wanted to accomplish then I would have no fresh perspective at all to test out in this thread. I'm trying to calculate things backwards or upside down (using equivalent versions of the same equations) and see if I get the same answers, and to see whether any insights emerge that are interesting to people with my level of knowledge. If nothing interesting emerges, then so be it. I would appreciate if you could help keep me honest but not try to discourage me from exploring.

Sure, just remember that you are exploring physics, not maths, so you do need to think about the meaning of the numbers you generate and what that implies for physical conditions and, crucially, how they can be measured.

If you get different answers depending on which way you integrate the equation then you've made a mistake. Insight comes from realizing what the numbers mean, not what value they have, so integrating forwards or backwards (which are equivalent if done correctly) should be equally useful in gaining new insight.

 

[jonmtkisco]

Wallace, yes I'll not lose sight of the hypothetical nature of any variant numbers I produce.

As I understand it, the equation for the radiation-dominated era is:

$$a = \left(\frac{t}{to} \right)^{\frac{1}{2}}$$

 

[Wallace]

Yes that's true, however the transition between radiation dominated and matter dominated is not instantaneous, so you need to do the full calculation (as it sounds like you are) if you are integrating through the transition epoch.

I'm not sure exactly what you are proposing to do with that equation?

 

[Jorrie]

Jorrie, I had forgotten that you wrote that, long before I "independently discovered" that a lighter universe would initially have expanded slower than a heavier one. So the credit (on this forum!) for that interesting insight properly belongs to you, not me.

Jon, now you have exposed the "sins of my youth", forcing me to defend what I wrote!

I wrote that in order to explain the curves (attached, for Ho=64 Km/s/Mpc, Omega = 1, Omega_v = 0.7 for the bold curve, Omega_v = 0 for the 'dotted' curve). The mention of "the inflationary epoch 'delivered' a universe with the expansion rate precisely balanced with the amount of matter" together with "So during early years, when there was negligible vacuum effect, the amount of matter to be balanced by expansion rate must have been less in the same proportion" was perhaps confusing. It is certainly true for the "early years", but even the very coarse timescale of the attached curves indicate that the slopes converge at very early stages.

Given Ho, a detailed early epoch integration shows that at decoupling (z~1100), there was ~ 50% difference in expansion rate between a flat radiation + matter universe and a flat LCDM universe. At t ~ 1 year, there was only ~ 1% difference, diminishing to virtually no difference at t ~ 10 seconds. The reason for this is the dominance of radiation energy density at the very early epochs, which you cannot ignore. (I have a suspicion that is what you did?)

 

[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