# Does WMAP data imply spacetime is flat?

1. Aug 10, 2012

### johne1618

I understand that the Wilkinson Microwave Anisotropy Probe has made observations of the angular size of early fluctuations in the Cosmic microwave background.

As the observed angular size is the same as the expected angular size it seems that light travelling to us from those early fluctuations has not been distorted.

Thus it has been concluded that the intervening space must be nearly flat.

If light beams from the edges of an early fluctuation passed either side of an accelerating region of space then I assume that those beams would be deflected leading to a different observed angular size.

Thus it seems to me that the WMAP observations carry the stronger implication that space-time is flat as well (or equivalently that the Universal expansion has not been accelerating while the light from early CMB fluctuations have been travelling to us).

Last edited: Aug 10, 2012
2. Aug 10, 2012

### marcus

I think the direct answer to your title question (Does WMAP data imply spacetime flat?) is no, it does not.
COBE data already suggested spatial near flatness and WMAP confirmed this and narrowed the range on it. It increased confidence but didn't qualitatively change thinking as regards spatial flatness.
Here is a plot of the history of expansion, based on COBE results. The picture is still good.
http://ned.ipac.caltech.edu/level5/March03/Lineweaver/Figures/figure14.jpg
The dark solid curve is for the standard cosmic model with common estimates of matter and Lambda. This is from Lineweaver's 2003 article, which is still a good thing to read. Title is "Inflation and the Cosmic Microwave Background" so you can get the online preprint by googling "lineweaver inflation cosmic" or keywords to that effect.

You can see that for the first 6 or 7 billion years there has NOT been acceleration, according to the standard model. The growth in the scalefactor has been slowing. And then for the last 7 billion or so it has been picking up very slightly. There has been a slight acceleration during the past 7 billion years. And in the figure you can see that is projected to continue into the future. The slope of the curve gradually increases with time.

In the usual cosmic model it is NOT assumed that spacetime is flat, because there is expansion of distances. It appears however that there is near spatial flatness at large scale. Space is (overall largescale almost) flat, but not spacetime.

The effect of the cosmological constant Lambda is assumed to be uniform. So far the evidence supports the idea that it is a constant. So the effect is not confined to patches.

3. Aug 11, 2012

### cepheid

Staff Emeritus
Yeah, what marcus said. Spacetime cannot be flat, because the universe is expanding, which means there is curvature in the time direction.

However, the WMAP data show that the universe has very nearly flat spatial sections, or at least, that these sections appear flat over distances corresponding to our visible horizon.

4. Aug 12, 2012

### JonDE

It does beg the question, if it turns out that space is flat, is there any way that we will ever know for sure? Or will we be doomed to always discovering that the margin of error is even closer to flat then previously known.

5. Aug 13, 2012

### johne1618

The Universe could expand linearly which would be consistent with flat spacetime.

6. Aug 13, 2012

### cepheid

Staff Emeritus
Yeah, but it does not, for any non-empty universe. Even if you ignore the effects of dark energy and use one of the more standard/traditional cosmological models, the expansion of the universe always slows with time. The interpretation of this is also fairly simple: you don't even have to resort to general relativity to get an intuitive understanding of it. The basic idea is that all objects are moving away from each other, but they have mass, which means that they have a mutual gravitational attraction, which tends to want to bring them together. This has the effect of slowing the expansion. That's why when observations showed that the rate of expansion was increasing and not decreasing, it was so mind-blowing at the time.

But the discussion above is irrelevant, because even in an empty universe, for which the scale factor increases linearly with time, the geometry of spacetime still isn't flat (Euclidean) but rather it is hyperbolic. It's like how in special relativity (which ignores the effects of mass and therefore does not take gravity into consideration), the geometry of spacetime is described by a Minkowski metric, and not a Euclidean metric, leading to a hyperbolic geometry.

7. Aug 13, 2012

### George Jones

Staff Emeritus
Actually, in this case, spacetime is flat, but space is curved.

8. Aug 13, 2012

### Khashishi

It's a lot harder to prove that the universe is flat than that the universe is curved. At best, measurements put some constraints on the curvature of the universe.
From http://map.gsfc.nasa.gov/news/: [Broken]
"If the dark energy is a cosmological constant, then these data constrain the curvature parameter to be within -0.77% and +0.31%, consistent with a flat universe (value of 0)."

Last edited by a moderator: May 6, 2017
9. Aug 13, 2012

### cepheid

Staff Emeritus
Wow, I guess I got that wrong...I'll have to look into it some more.

10. Aug 14, 2012

### johne1618

There is Kolb's coasting cosmology model:

It is characterised by an equation of state

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

If one also assumes that $k=0$ and $\Lambda=0$ then one has an expanding FRW model in which both space and spacetime are flat.

I believe this equation of state describes a zero-energy Universe in which the positive matter energy density ($\rho$) is exactly balanced by the negative gravitational potential energy density ($-\rho$).

The negative gravitational potential energy density implies a negative pressure ($p=-\rho/3$) whose gravitational repulsion balances the gravitational attraction of matter.

Thus maybe "dark energy" is in fact (negative) gravitational potential energy.

Last edited: Aug 14, 2012
11. Aug 14, 2012

### George Jones

Staff Emeritus
No.

If $\Lambda = 0$, then a flat spacetime FRW model implies that $k \ne 0$. Equivalently, flat space $k = 0$ implies that an FRW spacetime is not flat.

In fact, If $\Lambda = 0$, then a flat spacetime FRW model implies that $k = -1$, i.e., space has non-zero negative spatial curvature.

12. Aug 14, 2012

### johne1618

But I am assuming the equation of state

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

and $\Lambda = k = 0$

These conditions together imply an expanding FRW model that is flat in space and spacetime.

13. Aug 14, 2012

### George Jones

Staff Emeritus
This is impossible.
If the stress-energy tensor is

$$T^{\mu \nu} = \begin{bmatrix} \rho & 0 & 0 & 0\\ 0 & -\frac{1}{3} \rho & 0 & 0\\ 0 & 0 & -\frac{1}{3} \rho & 0 \\ 0 & 0 & 0 & -\frac{1}{3} \rho \end{bmatrix},$$

then it has trace $T = 0$.

FRW universes are subject to relativity, and thus are subject to Einstein's equation

$$R_{\mu \nu} = 8\pi \left( T_{\mu \nu} - \frac{1}{2} T g_{\mu \nu} \right) .$$

For the above stress-energy tensor, clearly the right side of Einstein's equation is non-zero, thus the Ricci curvature tensor is non-zero, thus the Riemann curvature tensor is non-zero, and thus spacetime is not flat.

14. Aug 14, 2012

### Durandarte

Yes it does. But only for what we can observe, other parts might be severely curved

15. Aug 14, 2012

### phinds

No, it DOESN'T. Have you even read the posts in this thread?