Monday, May 19, 2014

What is the curvature of spacetime?

This post was updated on June 30 2014. (And see news update added to the bottom of the post on May 2015 regarding BICEP2+Planck2015 estimates of gravitational waves.)
The astrophysicists community is currently in a heated debate about the implications of the BICEP2 measurements of B-mode waves in the cosmic microwave background (CMB.)
[For non-experts, the CMB is the nearly spatially-uniform (isotropic) radiation that we receive in whichever direction we look. The radiation matches with the radiation of a blackbody at a temperature of 2.726 /- 0.0013 Kelvin. This radiation is nearly uniform, with only small fluctuations. This near uniformity can be contrasted with the extreme density fluctuations we see in matter. Before the temperature of the universe cooled to below 3000 K (~0.3 eV), the density of hydrogen and helium in the universe was rather uniform because the hydrogen and helium were ionized (i.e. plasma), and in constant contact with the photons that today make up the CMB. Only after the temperature dropped below 3000 K, could the helium and hydrogen decouple from the radiation, and clump together to form local dense spots (which eventually turned into galaxies, stars, and planets.) It appears that the dark matter was already much more lumpy than photons and non-dark-matter at this point in time, so when the non-dark-matter decoupled from the photons, it started to fall into the local gravitational wells caused by the lumpy dark matter. (Note that my guess of why dark matter did not become extremely clumpy is that dark matter has a rest mass of ~2-10 keV and is prevented from being clumpy due to Fermi quantum degeneracy as neutrons are in neutron stars.)]

Returning back to the discussion of BICEP2:
There has been a lot of debate about whether BICEP2's measurements of B-mode waves in the CMB is indirect evidence for gravitational waves and whether BICEP2 has confirmed theories of inflation. From the data I've seen so far, it's pretty clear in my mind that the signal BICEP2 has measured is due to gravitational waves (with a tensor-to-scalar ratio of at least 0.1.) Though, I think that it's important to state that there is still a lot of uncertainty in the value of the tensor-to-scalar ratio because there are fairly large error bars on the data and because it's unclear exactly how much dust there is in the region where the BICEP2 and KECK telescopes were looking into the sky. I'm skeptical that the data can be entirely caused by dust because the slope of the curve at low values of mutilpole moment is in the opposite trend as what would be predicted by dust models. The preliminary data from the KECK Satellite (see figure below) seems to be confirming BICEP2 that the value of r is greater than 0.01, and likely on the order of magnitude of 0.1. However, as can be seen from the image below made by Raphael Flauger, there is the chance that a lot of the signal could be due to dust. The curves in orange and green below are models that include B-mode waves from dust and lensing, but not gravitational waves. However, if the results were purely due to dust and lensing, then it's hard to explain the drop in the signal at low values of multipole, l, to values below that which are expected in the dust models. While I'm really excited about the implications of these results, I think that the prudent thing to do is to wait until we get some confirmation from other research groups in order to make statements about discovery or confirmation.

(SideNote: The recent results from the POLARBEAR telescope can't help to confirm the BICEP2 results because POLARBEAR does not have a wide enough angle of view in order to collect data at multipole moments less than 700. In order to confirm gravitational waves, one needs to measure B-poliarization at multipole moments between roughly 10 and 200. BICEP2+KECK collected data between 50-300. So, BICEP2+KECK's data is a good start, but data points at lower multipole moment, i.e. wider angular searches such as currently being analyzed by the Planck satellite, are crucial for confirming the tensor-to-scalar ratio, and indirectly confirming gravitational waves as the cause of the B-mode waves in the CMB.)

One thing that interests me about these results is knowing what the results suggest about the curvature of spacetime. Knowing the value of  r and Ns could help constrain the density of baryons, the dark matter, and the dark energy in the universe. However, as shown in the plots below, (which can be found on page 36 of the Planck results in the Cosmological Parameters report), a value of r on the order of 0.1 to 0.2 will have virtually no change on the parameters from which we can estimate whether spacetime is flat, open or closed. For example, as shown below, the Planck best fit for the cold dark matter content goes through the part of the surface in which the value of r is flexible. This means that the BICEP2 results, unfortunately, will do very little to help constrain whether spacetime is flat.

Also, I think that it's important to remember that it's impossible to prove experimentally that spacetime is exactly flat! All we can do is to put bounds on likely values of the curvature, and say that curvature of spacetime is "nearly flat compared with the range of possible types of geometry." (Similarly, it's impossible for us to experimentally prove that the total angular momentum of the universe is exactly zero. There will always be error bars.)

In addition, if inflation is correct (and BICEP2/KECK seem to imply that some sort of GUT-scale inflation theory may be correct), then it's nearly impossible to know what was the value of OMEGA Total before inflation. Current versions of inflation require a seed sphere before inflation can start. For GUT-scale inflation theories, this seed sphere needs to be greater than ~100 grams. Since this size is still well above the Planck-scale, this seed sphere could have open, flat, or closed curvature. But we would probably never know because inflation drives curvature to flat/zero (i.e. drives OMEGA TOTAL to a value of 1.

One way of estimating the current (i.e post-inflation) curvature of spacetime is the OmegaM vs. OmegaL plot shown below. The y-axis is the ratio of the dark energy to the amount of energy required to create a flat universe, and the x-axis is  the ratio of the amount of matter to the amount of energy required to create a flat universe. Like in the figure above, values above the solid black line indicate a closed, convex universe and values below the line indicate an open, concave universe. The TT Power Spectrum wave data from the CMB provides the orange curve that does through the data. From the location of the first peak in the multipole distribution of the TT Power Spectrum of the CMB, one can estimate the curvature of the universe.

(Above: made in 2010. Below made in 2013 by Planck team. pg 41 of the Cosmological Parameters report. The right graph includes the constraint from the BAO, like in the plot above.)

What we can say from the graphs above is that spacetime is nearly flat, but as I mentioned before. There's no way to experimentally prove that spacetime is exactly flat (and especially no way to prove what was the curvature of spacetime in the seed sphere before inflation.) The opposite, however, is not true. It might be possible to prove that spacetime is open or closed with greater than 5 sigma certainty, if the values had fallen far enough away from the "Flat" line. (no pun intended)

Of course, plots like the ones above have a lot of hidden  assumptions that go into them, such as constant values of constant dark energy, cold dark matter, and (sometimes) isentropic expansion.
It will be interesting to see how these plots change when making other assumptions: such as (1) warm dark matter, (2) time varying  dark energy, and (3) non-isentropic expansion.

Also, it's important to remember that the universe can be "closed" in many different ways. One way for the universe to be closed is for the universe to start at a point, expand, and then contract back into a point. (Big Bang and Big Crunch) Another way to have a "closed" universe is for spacetime to be like the wrinkled surface of a 4D sphere, where the radius of the sphere is the time dimension. This is a closed, finite geometry, which could remain finite in size if the sphere stops growing after it reaches a certain radius. This would require that dark energy be non-constant (i.e. varying in time.) There would have to be a balance between matter and dark energy such that the 4D sphere kept from either contracting or expanding towards infinite.

What I'm suggesting here is just that we need to keep an open mind and not get too stuck in any of one of our models. For example, we have no viable theory about dark energy really is. We just have a number (Lambda) that physicists add to General Relativity in order to match with experimental data. (All we know for certain is that Lambda is not the vacuum energy predicted by quantum field theory. When integrateing up to the Planck scale,  the 1/2 hbar*omega of vacuum energy of quantum field theory completely overestimates the energy density in the universe.)

So to summarize this post, what I want to make clear are the following: (1) that the data from BICEP2+KECK is pretty good but not perfect evidence for gravitational waves, (2) that the exact curvature of the universe is unknown right now but it's fair to call it "near-flat", (3) that it's impossible to experimentally prove that the universe is exactly flat (both pre-inflation and post-inflation), and (4) that there are many different ways in which universe could be considered to be open or closed, and (5) that until we have a consistently theory that brings together cosmology and particle physics and that is confirmed by experimental data, I think that we need to avoid the trap of believing any one of our approximate laws of physics as the actual law of physics.

Update: May 2015
Planck released it 2015 data set, and there are now some tight constrains on the tensor-to-scalar ratio, r. Note also that in most Theories of Inflation, you can only generate large values of r if you also generate large values of the "running of the scalar pertubations" with k.  And Planck data sets some really tight constraints on d (n_s) / d ln(k). So, it's very likely that the BICEP2 results are not from primordial gravitational waves. However, I think that "it's just dust" is still not a sufficient answer at this point in time. We still need to quantify:  how much is due to dust?

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