Welcome to Our C161 Project Home Page!

Created by: Brianna Peck, Eli Gendreau-Distler and Dex Bhadra.

For our Astronomy C161 Final Project, we calculate the cosmological parameters \(\Omega_{m,0}\), \(\Omega_{\Lambda,0}\), and \(H_0\) from observations of Type 1a supernovae. We begin with datasets describing supernovae redshifts and luminosity distances from several important studies in the field. Using equations from the course, we derive a relationship between redshift and luminosity distance in terms of the cosmological parameters. Then we fit our data to this model in order to solve for the best fit values of \(\Omega_{m,0}\), \(\Omega_{\Lambda,0}\), and \(H_0\). We separately determine the best fit values of these parameters by sampling the phase space and calculating the probability associated with each point in phase space. After calculating \(\Omega_{m,0}\), \(\Omega_{\Lambda,0}\), and \(H_0\), we integrate the Friedmann equation to illustrate the evolution of the universe under the predicted model and in matter-dominated, dark energy-dominated, and empty universes. Lastly, we compare our parameters to the accepted results and discuss the implications of these results for the history and fate of our universe.

Background

In 1917, Albert Einstein published his paper describing the general theory of relativity. However, in this paper, Einstein assumed a static universe (\(\omega = 1 \)). Just over ten years later, Hubble published an article in 1929 where he presented empirical evidence that we instead live in an expanding universe. By observing the redshift of distant galaxies, Hubble derived his famous equation: \[v = H_0 D \]

Known as Hubble’s Law, this equation states galaxies are moving farther away at a speed proportional to their distances. If we examine Hubble’s data explicitly:

Hubble's Galactic Data.

From his infamous formula, his data gives a Hubble constant \(H_0 \approx 450 \). Despite applying the correct equation, Hubble incorrectly derived his own constant because he did not look far enough away. Supplied with more sensitive instruments, scientists can derive much more accurate values for Hubble’s constant and cosmological parameters. For further information, refer to the theory page for further information on how we derived our values.

Data

For our data, we choose to work with three different Type 1a supernova surveys: SCP, Pantheon and SNLS. The plot below contains all of the raw data for redshift and luminosity distances. To display the improvements in the astronomical observation, we included Hubble’s initial data from his 1929 paper.

All Data.

Going forward, we will be comparing the results of these three data sets. We will not be calculating cosmological parameters from Hubble’s data as there is not enough information to produce accurate phase space maps. Here are the three raw data sets plotted next to each other.

SCP Data Pantheon Data SNLS Data

Results from Phase Space Sampling

We fit the supernova luminosity distance versus redshift data to the model described on our Theory page in two different ways: first using the SciPy curve_fit function, and second by sampling the \((\Omega_{m,0},\Omega_{\Lambda,0},H_0)\) phase space. The luminosity distance versus redshift plots below demonstrate that our cosmological parameters obtained via curve fitting provide a good fit to the data.

SCP Curvefit Pantheon Curvefit SNLS Curvefit

The cosmological parameters obtained using phase space sampling are very close to those obtained using curve fitting (see table below). Since the corresponding luminosity distance versus redshift plots are practically identical, we do not reproduce them here. However, we will zoom in on the high redshift regions of the plots from above to demonstrate that our models approximate the data well even in the most difficult regions to fit. This time we will use the cosmological parameters obtained by sampling the phase space instead of by curve fitting (but, as noted above, the two sets of parameters are equivalent for all practical purposes). In addition, we will plot the luminosity distance as a function of redshift for a few alternative values of the cosmological parameters to demonstrate that our model performs better than these alternatives.

SCP Phase Space Pantheon Phase Space SNLS Phase Space

To perform the phase space sampling, we selected many different combinations of cosmological parameters and computed the reduced chi-squared values that result from using those cosmological parameters to fit our data. The plots below demonstrate how the reduced chi-squared value varies across the \((\Omega_{m,0},\Omega_{\Lambda,0})\) phase space (keeping \(H_0\) fixed at its best fit value). We selected the cosmological parameters with the lowest reduced chi-squared as the best fit values.

SCP Chi-Squared Pantheon Chi-Squared SNLS Chi-Squared

From the reduced chi-squared plots, we can already begin to see that the region of phase space in which the true cosmological parameters are most likely to lie forms a diagonal stripe centered on the best fit value. To gain further insight into the region most likely to contain the true cosmological parameters, we also computed the survival function, which is defined as one minus the cumulative distribution function for the chi-squared distribution. We rescaled the probability distribution so as to measure probabilities relative to the best fit value. The results are shown in the plots below.

SCP Propability Pantheon Probability SNLS Probability

Lastly, we integrated the Friedmann equation using the cosmological parameters we calculated from each dataset. This allows us to determine the scale factor of the universe at all past and future times under each model. The results are labeled “Matter and dark energy” and appear in blue in the plots below. For comparison, we also included the scale factor for a matter-only universe (using the \(\Omega_{m,0}\) and \(H_0\) values we calculated but assuming \(\Omega_{\Lambda,0}=0\)) and for an empty universe (using the \(H_0\) value we calculated but assuming \(\Omega_{m,0} = \Omega_{\Lambda,0} = 0\)).

SCP Scale Factor Pantheon Scale Factor SNLS Scale Factor

Discussion

In summary, we have calculated the best fit values of \(\Omega_{m,0}\), \(\Omega_{\Lambda,0}\), and \(H_0\) in two ways: directly fitting a model to the data and sampling the phase space. Using the best fit values of the cosmological parameters, we integrated the Friedmann equation to find the scale factor as a function of time. Our values of the cosmological parameters, along with the accepted values from the Benchmark Model, are summarized in the table below.

Source Method \(\Omega_{m,0}\) \(\Omega_{\Lambda,0}\) \(H_0 (\text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1})\)
SCP Curve Fit
Phase Space		 \(0.323 \pm 0.069\)
\(0.32\)		 \(0.75 \pm 0.11\)
\(0.75\)		 \(70.49 \pm 0.43\)
\(70.53\)
Pantheon Curve Fit
Phase Space		 \(0.354 \pm 0.039 \)
\(0.37\)		 \( 0.83 \pm 0.07 \)
\(0.87\)		 \( 66.15 \pm 0.28 \)
\(66.32\)
SNLS Curve Fit
Phase Space		 \(0.280 \pm 0.076 \)
\(0.25\)		 \( 0.85 \pm 0.11 \)
\(0.80\)		 \(77.11 \pm 0.39\)
\(76.84\)
Benchmark Model Ryden :) \(.30\) \(.70\) \(70\)

Our results are close to those of the Benchmark Model, and in particular demonstrate that the universe contains a significant amount of dark energy—a monumental result first discovered by Riess et al. (1998) and Perlmutter et al. (1999) using an analysis similar to our own. The discrepancies that do arise between the energy densities predicted by our three datasets and the Benchmark Model are of comparable magnitude to the error bars. (The phase space sampling method did not provide exact error bars, but we anticipate that the errors would be comparable to those from the direct fit method.) For all three data sets, the energy densities we calculated suggest \(\Omega_0 = \Omega_{m,0} + \Omega_{\Lambda,0} > 1\), which indicates the universe is positively curved.

On the other hand, the Hubble constants we calculated using the Pantheon and SNLS data differ from the Benchmark value by \(4-7\text{ km s}^{-1}\text{ Mpc}^{-1}\), yet have relatively small error bars (on the order of \(0.3-0.4\text{ km s}^{-1}\text{ Mpc}^{-1}\)). It is interesting to compare the Hubble constants we calculated to state-of-the-art measurements from the CMB and supernovae (Kamionkowski and Riess 2023):

\[H_{0,\text{CMB}} = 67.4\pm 0.5\text{ km/s/Mpc}, \quad H_{0,\text{SNe}} = 73.0\pm1.0\text{ km/s/Mpc}\]

At first it might seem surprising that our results do not align more closely with the accepted \(H_{0,\text{SNe}}\) value given that our calculations were also based on supernovae. However, measurements of the Hubble constant have changed significantly over time and as observational techniques improve. The Hubble Tension is an active area of current research in cosmology and we are excited to see how the accepted values continue to evolve in the years to come.

Next Steps:

If we had unlimited time to complete this project, we would have liked to have a feature on the website that allowed users to import their values for the cosmological parameters and a scale factor plot would be generated. The feature is technically available in our jupyter notebooks (linked on our github, see about page) but we did not have enough time to make this part of our website.

Galaxy