# Flat Universe Question

1. Nov 15, 2005

### Iscariot

I realise this may be a really obvious question, but wikipedia and google haven't help settle my curiousity.

If the universe is in fact a flat universe, as I believe has been proven with experimentation, this, if I am not mistaken, means the universe will continue to expand to a limit without ever reaching this limit, continously expanding slower and slower.

However, according to many books I've read, by studying the Doppler effect and the Hubble redshift, cosmologists have calculated that the rate of expansion in the universe is in fact speeding up, and the end result will be masses of dark space between stars in the distant future.

To me, these two seem to contradict each other, and my impression of what happened during the big bang. If the universe will never reach it's limit according to flat universe theory, then surely it has been forever slowing down since the big bang. However, if the universe is speeding up in it's expansion, it would suggest that at one point it will slow down, but where did this acceleration come from post-big bang? If stars, galaxies and planets are moving further apart at a greater speed than before, then surely the universe as a whole is expanding faster.

Maybe I have mis-interpretated what the books have said, or made another basic mistake in my thinking, but I'd really appreciate someone putting my mind to rest on this one.

Thanks a lot.

2. Nov 15, 2005

### mccrone

Yes, there are three alternatives. You can have flat space (which coasts to a halt at infinity), closed space (that will collapse) or openly expanding space (which has hyperbolic curvature and accelerates away).

The recent fuss about dark energy, quintessence or cosmological constant is due to observations that suggest Universe is in fact accelerating and so not flat but hyperbolic.

Everything still looks flat locally as deviation is so small. But "big rip" scenarios suggest that acceleration might one day pick up to inflation levels and we could once again see the open curvature locally (although briefly before it rips us apart)....

3. Nov 15, 2005

### hellfire

This is true as long as there is no dark energy (something that exerts negative pressure) in the universe. If dark energy is absent there is a relation between geometry of space and fate of the universe, but as soon as there is dark energy this relation does not necessarily hold. A universe might be geometrically closed due to a high energy density of dark energy (greater than the critical density), it may, however, expand forever and its expansion will accelerate due to the negative pressure of dark energy. Note that it is only energy density which determines the geometry but pressure counts three times more than energy density for the acceleration of expansion $\ddot a \sim a (\rho + 3p)$

Last edited: Nov 15, 2005
4. Nov 15, 2005

### matt.o

this is incorrect. The acceleration is implied based on the Universe being flat, or very close to it (WMAP) and the extra redshift of supernovae at cosmological distances. Ie. the supernovae are receeding faster than we think they should be in a flat universe.

5. Nov 15, 2005

### mccrone

Whoops! You are quite right. The visible Universe is flat (which implies it is at the critical density) and the rate of expansion is a separate issue to the geometry.

The old story was that the rate was decelerating due to gravity acting on mass - the coasting to a halt scenario. The new story is that it could be accelerating - expanding still flatly, but with increasing speed.

But is the difference between the flat and hyperbolic situation described by imagining two particles (with no force interactions between them)?

In a flatly expanding universe, two particles at rest would see the distance between them grow. The rate of this increase could be accelerating or decelerating.

But would things look any different to the two particles if they happened to be in an expanding hyperbolic space? Would the rate look just the same or would there be some increased divergence?

Then consider the same two particles moving together on parallel paths or geodesics for a long time. In a flat space they would remain forever the same distance apart. In a hyperbolic space, they would observe a divergence in which they seemed to grow apart.

But then even two comoving parallel particles in a flatly expanding world should see themselves diverging? They would still see an expansion of the flat space between them?

I'm still confused as to what would look different about an expanding flat space and an expanding hyperbolic space apart from perhaps the rate of divergence.

Help me out here!

6. Nov 18, 2005

### pervect

Staff Emeritus
This is not an obviou question - a good answer to it becomes rather tehcnical. Hopefully you can not be "thrown" by the technical parts which you would need to study GR in depth to understand, and attempt to follow the less technical parts of the argument.

Assuming the universe is flat - and also homogeneous and isotropic - the universe can be represented by the FRW metric

ds^2 = dt^2 - a(t)^2*(dx^2+dy^2+dz^2)

a(t) is the expansion factor, which defines the "size" of the universe at any instant of cosmological time 't'.

da/dt is the "velocity" of the expansion, and d^2a/dt^2 is the "acceleration" of the expansion.

The above metric uniquely defines the Einstein tensor. The motivation for calculating the Einstein tensor, a property of the curved space-time, is to find the stres-energy tensor, a property of matter and it's density. The calculations are very technical, as I mentioned above, but given the metric we can find the Einstein tensor.

$G_{00}=\rho$ $G_{ii}=P$ ,i=1..3

with

$$\rho = 3 \frac{\left( \frac{da}{dt} \right)^2 }{a^2(t)}$$
$$P = -2 a \frac{d^2 a}{dt^2}-\left(\frac{da}{dt}\right)^2$$

Very technical note: the above parameters come from computing G in an orthonormal basis, not from the computation of G in a coordinate basis.

This determines the density and pressure of the matter that makes up the universe, _assuming_ that Einstein's field equations are correct. Einstein's field equations relate the Einstien tensor, which we just calculated from the metric, to the properties of the matter in that universe (technically, the stress energy tensor). There are only two properties that matter - the density of the universe (just mass/unit volume, but it's "relativistic" mass in that it also includes energy), and the pressure of the universe.

We can see that given $\rho$ and P, we uniquely determine the velocity of expansion of the universe da/dt, and the acceleration of this expansion d^2 a / dt^2.

This is two variables in two unknowns, not enough to draw any conclusion. However, on physical grounds, we can relate density and pressure. This is the "equation of state" of the matter that makes up the universe.

The previous assumption was made that the equation of state of the universe was such that the pressure term, P would go to zero as the density term $\rho$ went to zero. (The density term will go to zero if the universe expands, because it is _assumed_ that you have the same amount of matter, and more volume, when the universe expands.)

[revision]
Note to self: is this really an extra assumption, or does it follow from the differential energy conservation law of GR?

However, the observation of accelerated expansion implies that P is not only non-zero, but is in fact negative. (Refer to the equation for P - the term in da/dt is always negative, and the term due to d^2 a / dt^2 will also be negative if the expansion of the universe is accelerating.

This negative pressure is the "dark energy" that people are scratching their heads about.

Note that the above analysis was ONLY for a spatially flat universe, as per the original posters request. The most general metric has a different form, depending on the spatial geometry of the universe, which gives different expressions for $\rho$ and P

Last edited: Nov 18, 2005