Sunday, June 7, 2015

Constraints on Dark Matter & Dark Energy from the Hubble Expansion between z=0 and z~3

(Note: Updated on 6/19 to include Planck 2015 Best Fit for comparison)
Using data for the Hubble expansion as a function of time, this week I'm showing how data just on the Hubble Expansion between z=0 and z~3 alone can provide some tight constants on the amount of dark energy and dark matter in the universe.
First, I'd like to present the experimental data (without any fits to the data), and then I'll present the experimental data shown along with a "best fit equation" and with Planck's recent estimates. The z=0 to z=1.3 data shown below was found in Heavens et al. 2014 (which has references to where the original data was collected.) The data point at z=2.34 is from baryon acoustic oscillations (BAO) found in the Lyman-Alpha forest by the BOSS collaboration (Busca et al. 2013).


The figure above is a plot of the expansion of the Universe, H(z), as a function of time in the past. Here, I've plotted on the y-axis the Hubble expansion normalized by the Hubble Expansion Constant Today, and then squared. The x-axis is the inverse scale of the universe. A value of 4 means that linear dimensions in the universe would have been 4 times smaller. Also note that this is a log-log plot so that the data points near (1,1) are not scrunched together. One thing to note about the data is that there is a definite change in slope between the data near z=0 and the data at z>1.

Next, we'll discuss the theory behind why the Hubble expansion rate changes with time. We'll focus here on the case in which the total mass in the universe is equal to the critical density (=1). In that case, we can use Equation 2.18 of the Physical Cosmology Class Notes by Tom Theuns to determine how the Hubble expansion rate changes with time. This equation is listed below:

As seen above, if the only form of energy density in the universe were dark energy (Λ), then the Hubble expansion rate would be a constant. The data above is clearly not consistent with the case of only dark energy (i.e. a cosmological constant.) The other terms in the equations are: curvature (k), matter (m), and radiation (r). Radiation is defined as particles whose kinetic energy is greater than their rest mass energy. Matter is defined as particles whose kinetic energy is much less than their rest mass energy. Curvature is the the curvature of the universe.
If we were in a universe with only radiation, then the Hubble constant would decrease as (a0/a)4, which is the same function as how the radiation energy density decreases as the universe expands. If we were in a universe with only matter, then the Hubble constant would decrease as (a0/a)3, which is the same function as how the matter energy density decreases as the universe expands.

Next, I want to show that the best fit through the experimental data is a universe with approximately 30% matter and 70% dark energy (today.) I wanted to see what would be the best fit through this data (ignoring all other data...which also points to ~30%  matter and 70% dark energy.) So, in Excel, I created a quartic polynomial equation with 5 free variables (a+bx+cx2+dx3+ex4), and constrained the 5 free variables to sum to a value of 1 (i.e. to constrain the total mass to be equal to the critical density) and also constrained the free variables to be greater than zero. In this case, the best fit through the data was  (0.717, 0, 0, 0.283, 0). These values are pretty close to the values determined using Planck+BAO data. Interestingly, there is no sign of energy which would scale linearly, quadrically, or quarticly with (ao/a). This means that the best fit through the data is a world with only matter and dark energy.


The experimental data used in the figures in this post are listed below:

z a H H error a0/a (H/Ho)2
0.00 1.00 0.71 0.02 1.00 1.00
0.27 0.79 0.82 0.03 1.27 1.33
0.59 0.63 0.95 0.03 1.59 1.80
0.50 0.67 0.90 0.05 1.50 1.61
0.80 0.56 1.00 0.06 1.80 1.98
1.05 0.49 1.45 0.20 2.05 4.17
1.30 0.43 1.60 0.20 2.30 5.08
2.34 0.30 2.29 0.00 3.34 10.40


Updated 6/19/2015
I'm including a figure that shows the Planck Best Fits through its data sets:
Ho = 67.7
Lambda matter = 0.31
Lambda dark energy = 0.69
(This is the Planck TT,TE,EE+lowP+lensing+ext in the 2015 update of the Planck Cosmological Parameters.)


The Planck 2015 data values fit the data almost as well (except for the data point at z=0) as the Best Fit values listed above:
Ho = 71.0
Lambda matter = 0.283
Lambda dark energy = 0.717

The relative error per data point is 14.1% for the "Best Fit" and 14.8% for the Planck 2015 values.
All in all, the H(z) data by itself can help constrain some of the cosmological parameters, but without other data sets, the constraints are fairly loose (as seen above that Ho, Lambda matter, and Lambda dark energy can vary by +/-5% without larger changes in the overall error with the experimental data. 






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