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jonmtkisco
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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:
[tex]\right)\frac{\dot{a}}{a} = \sqrt{\frac{8G\pi\rho+\Lambda}{3}}[/tex]
[F2] acceleration equation:
3[tex]\frac{\ddot{a}}{a} = \Lambda-4\pi G \left(\rho +3p \right)[/tex]
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 [tex]\rho[/tex] (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 = [tex]\sqrt[3]{\frac{3V}{4\pi}}[/tex]
the resulting form of the expansion equation is the familiar equation for “escape velocity”:
[tex]\dot{r} = \sqrt{\frac{2GM}{r}}[/tex]
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:
[tex]\omega = \frac{\rho}{p}= \frac{1}{3}[/tex]
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:
[tex]\omega = \frac{\rho}{p}= -1[/tex]
(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 [tex]\omega = -1[/tex]. 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
In this thread, let’s refer to the two Friedmann equations as:
[F1] expansion equation:
[tex]\right)\frac{\dot{a}}{a} = \sqrt{\frac{8G\pi\rho+\Lambda}{3}}[/tex]
[F2] acceleration equation:
3[tex]\frac{\ddot{a}}{a} = \Lambda-4\pi G \left(\rho +3p \right)[/tex]
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 [tex]\rho[/tex] (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 = [tex]\sqrt[3]{\frac{3V}{4\pi}}[/tex]
the resulting form of the expansion equation is the familiar equation for “escape velocity”:
[tex]\dot{r} = \sqrt{\frac{2GM}{r}}[/tex]
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:
[tex]\omega = \frac{\rho}{p}= \frac{1}{3}[/tex]
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:
[tex]\omega = \frac{\rho}{p}= -1[/tex]
(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 [tex]\omega = -1[/tex]. 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
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