Ever heard that the universe is expanding? We may just be floating into the abyss of space until the universe just tears apart. As terrifying as that sounds, scientists have been busy trying to figure out the rate at which the universe expands: the Hubble Constant [1]. This number has been debated since the 1920s which was when Edwin Hubble, along with Georges Lemaitre [2] discovered one of the most crucial aspects of cosmology - that the universe is expanding and accelerating. It opened up a whole world of theories and mysteries like dark energy [3] and the ultimate fate of the universe [4-5]. Georges Lemaitre theorized the expansion whereas Hubble confirmed this by discovering that all galaxies are red-shifted [6], which meant that they were moving away. He also discovered that there is a linear relationship between the distance to a galaxy and its velocity (the speed at which it is receding). This relationship is described by the Hubble Constant (velocity(km/s) divided by distance (megaparsecs) ), whose value is still being disputed to this date.
We have come a long way since the number Hubble calculated, which turned out to be around ten times too high due to inaccurate data, and have two main methods of calculating it now. One is through studying the early universe [7], which helps us calculate the age of the universe and how it has evolved and expanded since the Big Bang. A detailed map of the cosmic microwave background radiation(CMB) [8], which is a remnant heat from the Big Bang, helps us calculate the constant. This is done by finding temperature fluctuations and patterns and using the standard cosmological model, which assumes a flat, homogeneous, and isotropic universe. Using its estimates for matter and energy in the universe as well as the temperature patterns from the CMB, the Hubble Constant is calculated at around 67.4 km/s/Mpc. The same number is calculated through “Baryonic Acoustic Oscillations”(BAO) [9], which uses the clustering and distribution of galaxies to study the history of the universe. These patterns work as “standard rulers” (similar to standard candles) [10] for measuring the expansion and age of the universe. Since this method uses the standard model of cosmology [11] as a reference for how the universe began, the value is very much subject to change if the model changes.
On the other hand, the second method uses standard candles like supernova Ia [12-13] and Cepheid variables [14] to calculate distance and velocity. Cepheid variables are incredibly bright stars that can be used as reference points due to their high luminosities to measure large distances. Their luminosities have a relation to their pulsation (period-luminosity relation), which can be used to measure magnitudes, and therefore, their distances. Supernovae Ia are a type of an explosion of stars that happen at the end of a star’s lifetime (when they run out of gas in their core), and they usually occur in white dwarf stars that are part of a binary system. A supernova’s luminosity is measured in optical light in the form of light curves [15]. The magnitude is plotted against the time and most supernovae have the same or similar basic light curve in the B band. Although we have tried to standardize other bands of light, the B band works best as it usually has only one peak luminosity whereas other bands may not have a clearly defined peak. But since there is a lot of scattering within this band of light, other methods need to be applied to make the distance more accurate. The estimate for the Hubble Constant using standard candles comes to around 74.0km/s/Mpc. New methods like using gravitational lensing [16-17] and pulsars to measure distances also come up with an estimate closer to the standard candles method, thus sparking the controversy [18] in cosmology for what the true rate of expansion is. Although these numbers may not seem far, they will make a huge impact in figuring out the ultimate fate and beginning of the universe. We may even have to change the standard model of cosmology [19], based on what the true rate is.
Method
For my estimate of the Hubble Constant, I used the standard candles method using the B band light curves of supernova Ia. From the Supernova Catalog [20], I picked 50 Type 1a supernovae and used their data to find the distances to their host galaxies. Using the Supernova Catalog, I was also able to find the peak apparent magnitude for all the supernovae (through the B band light curves) that I needed for my distance calculation. For the velocities of the host galaxies, I obtained the data from Simbad [21] along with their error bars. Simbad also helped me compare the distances I found with my methodology against the actual, verified distances to the host galaxies of these supernovae. Within the standard candles method, I tried three ways of finding the constant [22]: using Phillips Law (the equation), finding an average absolute magnitude, and using the luminosity decline rate relation. All these methods help reduce the little variation there is among supernova Ia light curves in the B band and helped me find the peak absolute magnitude, which was what I needed for the distance calculation [23].
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Phillips Law [24] is described as a relationship between the peak luminosity of a supernova and the rate at which the luminosity declines after the peak (typically 15 days). Pskovskii was the first to propose the rate “β”, which was the mean rate of decline within a hundred day period after the peak, and this rate was used to find the peak absolute magnitude of a supernova. But once the availability of data increased within supernova light curves, Mark Phillips was able to derive a different, easier rate and equation to calculate the peak absolute magnitude (Mmax (B)):
- where “m15(B)” is the peak apparent magnitude subtracted from the apparent magnitude 15 days after the peak. To calculate the constant, I used the peak absolute magnitudes from the Phillips equation and plugged that into the distance modulus equation:
- where “m” is apparent magnitude, “M” is absolute magnitude, and “d” is distance. The equation helped me find the distance in parsecs since I already had the peak apparent magnitudes from the Supernova Catalog’s data. I then converted this to megaparsecs to fit the standard units for the Hubble constant. As shown in Figure 1, the Hubble Constant using this method came out to be 6.82 ± 6.39 km/s/Mpc, which is off by a factor of 10. The primary reason for it being significantly off is that the Phillips Law equation is outdated after the availability of new and more accurate data.Figure 1: Hubble Diagram using the Phillips equation with the equation of the line. As shown, there is a lot of scatter in the plot.
For my second method, which was finding an average peak for the absolute magnitude, I picked 10 out of my 50 Type Ia supernovae to solve for the peak absolute magnitude using already verified distances and the distance modulus equation. I took out the average of these 10 peaks and used that average absolute magnitude for all 50 of my supernovae. I then used the distance modulus equation again to find the distances in parsecs (with one peak absolute magnitude) and converted them to megaparsecs. As shown in Figure 2, the estimate of the Hubble Constant using this method came out to around 73.53 ± 23.36 km/s/Mpc, which proves that most supernovae Ia have pretty similar magnitudes and luminosities. I did have to do a little bit of a point clipping (12 galaxies in total) to keep my dataset consistent with the independently calibrated distances of those host galaxies and only removed points that were off by a factor of more than 1.5 but less than 0.75. Figure 2: Hubble diagram found using one average peak absolute magnitude as well as the equation of the line. It shows a fairly linear relationship between distance and velocity.
For the third and final method, I used the luminosity decline rate relationship [25]. It describes a relationship between brighter and dimmer supernovae, stating that brighter supernovae decline more slowly after the peak than supernovae with dimmer peaks. This relationship is described by a graph, as shown in Figure 3 [26], with the rate (peak apparent magnitude-apparent magnitude 15 days after the peak) on the x-axis and the peak absolute magnitude on the y-axis. I used this graph to find the peak absolute magnitudes based on my rates and the rates that were present in the plot. This plot is also the more accurate version of the equation Mark Phillips found. Next, like the other methods, I used the distance modulus equation to calculate the distance in parsecs (using the peak absolute and apparent magnitudes) and then converted it to megaparsecs. Using this method, my estimate for the Hubble Constant came to approximately 52.75 ± 10.05 km/s/Mpc, as depicted in Figure 4. As with the second method, I had to do some point clipping (12 galaxies in total) to keep my dataset consistent with the independently calibrated distances. I also set a condition which removed any points that had rates over two, since it wouldn’t fit the interpolation range of the graph I used to find the peak absolute magnitudes. Figure 3: Graph showing the updated version of the Phillips Law. To find the peak absolute magnitudes, I interpolated over the data to find the magnitudes from the rates, instead of using an equation.
Conclusion
Overall, the supernova light curves that I used had similar luminosities, which was proved with the second method (finding an average peak absolute magnitude) but the fact that I had to do some point clipping to get a more reasonable data set (and distances) shows that supernovae Ia that are measured within the B band of light still do contain a lot of scatter. If we compare the Phillips Law equation method and the luminosity decline rate relation method, we can see that even though they describe the same relationship, the luminosity decline rate relation graph was more accurate. Phillips Law described a linear relationship between the rate and peak absolute magnitude, when in reality, the graph should have been a curve, describing that dimmer supernovae decline faster from the peak than brighter supernovae. Because of the excessive scatter within this standard candles method, other standard candles like Cepheid variables and TRGB (tip of the red giant) [27] need to be used to verify the distances found with supernova Ia. Since there is a lot more variation among light curves within the B band of light, researchers are trying to use different optical bands of light, like the J band [28]. These bands may not have one clearly defined peak, but they will have less variation and scatter between supernovae. With more accurate data, we can hope to standardize supernova Ia better or completely remove them from our list of standard candles if the similarity between light curves decreases. It is still pretty impressive that two completely different methods of measuring the Hubble Constant (not to mention complete opposites) come up with numbers that are fairly close to each other. But if we want to solve this conundrum and figure out the history and future of the universe, we’ll need to come to a consensus on what the true rate of expansion is. Hopefully, the new methods being discovered will bridge the gap between the two numbers.
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