Search for Sources of Systematic Error in Astrometric Measurements of the Cosmological Constant

1. Introduction

The results of the 1998 study published by Dr. Saul Perlmutter et al. [1], and the 2004 study from Dr. Adam Reis et al. [2] presented a view of Big Bang cosmology that was, and remains unsatisfying. Though both papers presented theoretically rigorous and well reasoned arguments for a “dark energy” acceleration of the universe, their conclusions published in their respective papers rest upon data from fewer than 80 type 1a Supernova (SNe 1a).

In statistical analysis, a result is only considered reliable if the sample data is representative of the population under study. Considering the literally universal size of the stellar population, 80 is a miniscule number of samples upon which to base such an important conclusion. To do so must rest on at least two simplifying assumptions:

• The expansion of the universe is isotropic. Thus, a sample of any size must be representative of the full population spanning the same distance and age.
• The velocity of the local reference frame is known and effectively constant throughout the span of history covered by the observations, or any change would be too small to affect the results and may be ignored.

The estimated 3 kilometer per second discrepancy between two rigorous measurements of lambda known as “The Hubble Tension” is similarly unsatisfying. If lambda is increasing over time, then the current measurement is just a present-day snapshot, not necessarily a reflection of lambda through history.

2. Methods

These widely acknowledged theoretical problems motivated a deeper look into astrometry concerning dark energy. To address the small dataset problem, we used as many of the stars listed in the SAI Supernova Catalog [3] as possible. To challenge the simplifying assumptions, we segregated the SNe 1a into directional cohorts by right ascension (RA) and compared the uncorrected red shift (RS)-luminosity relations.

The local Earth, solar system, Milky Way [4] reference frame is known to be moving roughly toward the Shapley Galactic Supercluster [5], and the more distant Great Attractor [6] with a velocity of approximately 630km/s [7]. We selected only confirmed SNe 1a from the catalog, then chose the RAs toward the Great Attractor (RAs 14-17), and those diametrically opposed (RAs 2-5) for the test comparison. For the control comparison we used two cohorts at right angle RAs 8-11 and 20-23. The data used to create the plots are found in the Data section.

3. Research

We plotted the redshift-luminosity relation for the low-RA and high RA cohorts of the test group. The positive velocity of the Milky Way of 630kps is additive to the distance from the low-RA cohort and subtractive to the distance from the high-RA stars. If the expansion of the universe is isotropic, and if the velocity of the Milky Way has been constant through the period under study, that would introduce a fixed offset to the redshift-luminosity (RSL) plots when compared. The plots should not converge.

However, if the Milky Way has positively accelerated on its current heading it would produce a plot with a larger difference in the RSL plots in the recent past that converges in the distant past. Conversely, a negative acceleration would produce a smaller difference in the RSL plots in the recent past.

The first pass through the data yielded similar, too-dense plots from data found here and here :

Figure 1.

Figure 1.

In plots 1 and 2 note that the low RA plot trends 2 magnitudes higher (lower luminosity) than the high RA plot at redshift (RS) near 0, but the two plots converge, possibly intersecting beyond redshift=1.6 as shown in Table 1. This pair of plots hypothetically illustrates a potentially consequential acceleration of our galaxy during the last 480 million years or more.

A sharper focus on low-redshift supernovae yielded these plots from data here and here:

Figure 2.

Figure 2.

Since lambda is common mode in these two plots, any difference must be due to motion of the local reference frame. We know that the Milky Way is moving away from the low RA stars toward the high RA stars which exaggerates the change over time by a factor of two. Since there is much more data in the low RA set, and it is difficult to line up individual stars for direct comparison, we used the trend lines to filter the data in both sets to the exemplars of each trend from data found here and here:

Figure 3.

Figure 3.

Note that though the trends appear to be quite similar, the high RA trend begins 0.002 to the right on the X or redshift axis, yet the traces nearly converge at RS = 0.035 as shown in Table 2.

Stars still cannot be compared directly, but we can use the trend line equations to make straightforward comparisons at each redshift in the interval of interest. Each 0.001 of RS is equivalent to approximately 13.8 million years. By using the high RA magnitude as the absolute magnitude, and the low RA magnitude as the apparent magnitude at each 0.001 interval in the trendline equations we can obtain regular changes in distance measurements in approximately 13.8-million-year intervals over the last 480 million years with this formula:

m – M + 1

d = 10 ⁵

Here are the plots of the distance modulus and computed acceleration from this data:

Figure 4.

Figure 4.

Plot 8 shows the computed change in velocity of the Milky Way during the last 480 million years. After correction for doubling of the separation distance, this plot shows that over the most recent 13.8-million-year period the velocity of the Milky Way increased by an average of 2.43 km/s every 3.3 million years, and increasingly less in the past.

Plots of the control cohorts RA 8..11 and RA 20..23 are distinctly different in appearance than test plots 1 and 2 using date found here and here:

Figure 5.

Figure 5.

We used the SNe 1a observations in the control cohorts without filtering outliers. The 20..23 cohort contains no observations below redshift = 0.03. As a result, the trendlines are substantially different below redshift 0.1 but rapidly converge as expected of observations roughly 90 degrees from our galaxy’s heading as shown in Table 3.

4. Discussion

Our search of papers concerning dark energy and the Hubble tension provided no evidence that astrometric measurements of them did not implicitly assume that the local reference frame is inertial. While a purely statistical analysis that treats SNe 1a observations as a random population could not disprove that assumption, we believe that it does support a hypothesis to that effect.

If the local reference frame is currently accelerating at nearly 2.5kps/MPC, it could it be consequential to the dark energy results if the data contains a selection bias in favor of SNe 1a with RAs in the range 2..5. In fact, that is the case in the results from Perlmutter et al. [1] and Reis et al. [2]. Of the 60 SNe 1a in tables 1 & 2 in Perlmutter, 14 have right ascensions in the range 2..5, only 1 in the range 14..17. Of the 16 SNe 1a in Reis Table 1, fully half are in range 2..5, and none are from 14..17. Hypothetically, data from RA range 2..5 used in isolation under an assumption that the local reference frame is inertial would clearly present as an increase in lambda. As a substantial fraction of the data in the Perlmutter and Reis studies they would bias the results more subtly. Recent measurements of lambda closer to 3 kilometers per second renders the measurements here somewhat more compelling.

However, this investigation is a relatively unsophisticated statistical analysis of raw, uncorrected SAI SNe 1a measurements. Though there is plenty of data, it is of unknown quality. The finding of recent acceleration could be due to noise in the data. Despite this, an acceleration of the local reference frame coupled with a very small data set and selection bias is a simpler explanation for an apparent acceleration of the expansion of the universe than dark energy. We believe that the hypothesis presented is the simplest explanation for the finding of a relatively recent increase in lambda and is sufficiently credible to merit further investigation.