Explaination of the universe is generaly excepted

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wolram
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the expantion of the universe is generaly excepted but what
governes this expantion? and what would be the outcome if
if the "control" for expantion differed by x%
best wishes
 
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marcus
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Originally posted by wolram
the expantion of the universe is generaly excepted but what
governes this expantion? and what would be the outcome if
if the "control" for expantion differed by x%
best wishes
the expansion is governed by a simple formula called Friedmann
and the "control" for the expansion is energy density


for the greatest simplification assume space is flat (which they tell us is true to within a percent or so, good enough for government work) and pretend all the constants like G and c are equal to one.

Then Friedman says the square of the expansion rate H is equal to 8pi/3 times the density

John Baez recommends a trick that even makes the 8pi go away---theoreticians like the maximum mental economy and comfort.

So if you do things his way and sweep the 8pi under the rug (and why not get rid of the 3 as well while we're at it) then the square of the expansion rate H is just EQUAL to the density.

BTW negative expansion is contraction. So the equation

H2 = density

has two solutions a positive H and a negative contraction-rate H.

There is also a second Friedmann equation we can call Mrs. Friedmann which governs the rate H changes
It is also very simple, like this one, and the "control" term
is - (density + 3 X pressure)
that is, minus the quantity (density plus 3 times the pressure)

well if you look out at night you see that the average density in space is pretty sparse and you can imagine the pressure is incredibly skimpy and almost completely negligible

But the two equations have a heck of a lot of space to work with and a whole lot of time and they arent in a hurry, so these things
we think of as almost zero (what kind of average density and pressure do you get from a few stars scattered about?) nevertheless WORK in the long run and do control things

(in the slick tight-lipped way that differential equations control things, never quite revealing how they do it)

best wishes to you in return
 
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marcus
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Why not write the two Friedmann equations out? In case
someone likes symbols.

The all-important thing in cosmology is the spatial scale factor a(t) which is a scale-factor in the formula for distance, normalized so that a(present time) = 1
The way cosmologists show space expand/contract is to have a(t) get larger or smaller---so the distances you compute with the metric formula get larger or smaller accordingly

So this a(t) is a simple function that you can plot on graph paper that was at least approximately zero at time zero and which is now equal to one, and it is an increasing function that people draw as a convex-then-concave ramp. They can study the thing by observations of things in the past whose light is getting to us.
This a(t) is the basic function in cosmology

the time drivative or slope of a(t) is called da/dt or "a-dot"

the acceleration or rate of change of "a-dot" is d(da/dt)/dt or "a-doubledot"

You think this is gobbledegook? well how else are you going to talk about the expansion of the universe or the acceleration of the expansion. You have to have "a-dot" and "a-doubledot".


The famous Hubble parameter H(t) is simply a-dot/a
that is the percentage rate of change
that's all it is
by definition
you take a-dot at time t and divide by a at time t.
The two friedmann equations are just about a-dot and doubledot.

I'd better try to finish this later

YOU wolram, who pretend to be an intellectual lowbrow, asked about what controls the expansion rate, and it is these Friedmann equations.

Maybe I had better call the time rate of change of a(t) by the name a'(t) and the time rate of change of THAT by the name a''(t).

OK, then the Hubble parameter H(t) = a'(t)/a(t) by definition.

Notation is the shoelaces of mathematics and if you do not keep it tied neatly you trip on it.

I had better call the two equations Big Friedmann and Little Friedmann, instead of Mr. and Mrs. (so as not to raise gander issues)

Big Friedman says

H2 = (a'(t)/a(t))2 = (8pi/3)(G/c2)(average density of energy)
 
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marcus
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Big Friedmann also has a "k" term in it where k represents the overall spatial curvature (not of space time, merely of space) but they tell us k is zero. Overall, aside from little humps and pits that effect things locally, space is not curved, it is merely expanding. this is great because it simplifies the equation for us. We dont have to put in the "k" term.
-----------------------------

Now how about the other Friedmann equation, Little Friedmann, the auxilliary controlling a''(t)----the rate of change of the rate of change of the scalefactor.

Little Friedmann says

a''(t)/a(t) = - (4pi/3)(G/c2)(average density of energy + 3 X average pressure)

BTW these two are simplifications of the original Einstein equation of GR which were derived from the main GR equation by you guessed it a man living in St. Petersburg Russia around 1922 except it was Leningrad then. They are an integral part of all the
GR results, light-bending blackholes etc etc and Big and Little Friedmann provide the basic theory of the big bang. You get a bargain with them because they are simpler than a lot of other
influential equations---so at least there's that to be happy about.
 
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marcus
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the Little Friedmann equation which you see there makes it plain as day why Einstein dreamed up something with negative pressure.

think about it

the righthand side of Little Friedmann is threatening to be negative unless you put in something with negative pressure----because energy density is positive and most kinds of pressure are positive, and there is a minus sign

so because of the minus sign the a''(t) would normally be forced to be negative

that seemed to spell collapse to Einstein, the way he happened to be looking at it, and he did not enjoy the prospect of collapse, so he dreamed up something called Lambda that contributes a negative pressure
and threw that into the mix on the righthand side.

and then for 70 years people thought that was just a mistake and there wasnt any Lambda

Until, in 1998 they discovered that a-doubledot, or a'', is actually POSITIVE
(that was what the supernova data said) and they all suddenly
came back to the idea of this Lambda

All the noise we have been having for the past 5 years about "accelerating expansion" is just about a term in the Little Friedmann equation. But almost nobody except the cosmologists can understand the discussion because they are the only ones who know the equation.

Better say "second" Friedmann equation. If you say Little they wont know what you are talking about.
 
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  • #6
wolram
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wow, isnt it incedible that our universe is so simple
mathmaticaly ,such a perfect ballance.
but maybe perfection is in the eye of the beholder,
i can only hope that no pesky numeron tips the scale in the future
many thanks.
 
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well since the universe worked as it has long before we were around... and we invented math. as a way to explain things.. math is really just a language... the most universal of any... and of course since we created math it is logical that the universe fits into the equations we draw up based on it...
 
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marcus
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Originally posted by wolram
the expantion of the universe is generaly excepted but what
governes this expantion? and what would be the outcome if
if the "control" for expantion differed by x%
best wishes
compliments on the clarity of the question
 
  • #9
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the question is realy pretty clear. first we should know what governs the inflation and than model it.
 

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