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 peculiar 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 recently measured [3], roughly 4+ kilometer per second [4,5,6] discrepancy between two rigorous measurements of lambda known as “The Hubble Tension” is similarly unsatisfying.

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 [7] and the Extragalactic Distance Database (EDD) [8] as possible.

To challenge the simplifying assumptions, we segregated the SNe 1a from the SAI Supernova Catalog and observations from the EDD into directional cohorts by right ascension (RA) and compared the uncorrected redshift (RS)-magnitude relations. To obtain sufficient samples of Sne 1a in each cohort required a cone spanning 60 degrees of right ascension and +-90 degrees of declination. The larger EDD allowed narrower samples of 15 degrees of right ascension and 60 degrees of declination.

Measurements of Sne 1a magnitude are acknowledged to be affected by multiple independent random variables. [9] The comparison of cohort samples assumes that the various random variables that affect the measurements of Sne 1a magnitude apply to the entire population of observations in SAI Catalog. The population is sufficiently large that the magnitudes of cohort Sne 1a samples are candidates for a comparative statistical analysis. Applying the Central Limit Theorem [10], we posit that the sampling distribution of the cohort magnitudes may be compared with function-fit mean trend lines of magnitude-redshift plots. This allowed direct comparison of supernova magnitude-redshift values at regular intervals by applying the mean functions to a scale of redshift values.

The local Earth, solar system, Milky Way [11] reference frame is known to be moving roughly toward the Shapley Galactic Supercluster [12], and the more distant Great Attractor [13] with a velocity of approximately 630km/s [14]. 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.

The EDD analysis made use of distance, heliocentric radial velocity, and peculiar motion measurements to obtain a function-fit mean of distance-velocity plots of the RA2, RA14, RA8 and RA20 cohorts. This allowed direct comparison of the narrower cohort’s distance-velocity and distance-redshift values at regular intervals by applying the mean functions to a scale of velocities and corresponding redshifts.

3. Research

3.1. Type 1A Supernova Data Analysis

The Perlmutter et al and Reiss et al papers used Type 1A supernova astrometry to propose a dark energy induced acceleration of the expansion of spacetime. While the astrometry is exceptionally well grounded in theory and the analysis is thorough, the work may have been misled by observational bias and limited data. A broader statistical analysis of a larger, more recent Sn3 1a data set is presented here.

We plotted the Sne 1a redshift-magnitude relation for the RA 2.0000 through 5.9999 (RA 2-5) and RA 14.0000 through RA 17.9999 (RA14-17) cohorts of the test group. The current 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-magnitude (RSM) 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 RSM plots in the recent past that converges in the distant past. Conversely, a negative acceleration would produce a smaller difference in the RSM plots in the recent past.

Plots 1 through 6 Illustrating Cohort sn_mag Redshift Relations

The first pass through the data yielded similar plots from data found here and here:

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:

Since lambda is common mode in these two plots, any difference must be due to motion of the populations relative to 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 compare individual stars, we used the mean trend lines to filter the data in both sets to the exemplars of each trend from data found here and here:

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.

Individual stars still cannot be compared directly, but we can use the mean 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

Plots 7 and 8 Illustrating the Distance Modulus and ∆V vs Redshift

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

Plot 8 shows the computed change in velocity of the Milky Way on the RA 2-5 ->14-17 vector during the last 480 million years. After correction for doubling of the separation distance, this plot shows that during the most recent 3.26-million-year period that we can measure the velocity of the Milky Way in the direction of RA2..5 increased by an average of 2.46 km/s every 3.26 million years, and increasingly less in the past.

Plots 8 and 9 Displaying RA8..11 and RA20..23 sn_mag Redshift Relations

Plots of the control cohorts RA 8..11 and RA 20..23 are distinctly different in appearance than test plots 1 and 2 which illustrates potential recent anisotropic changes in velocity. Data for plots 8 and 9 is found here and here:

The 20..23 cohort contains no observations below redshift = 0.003, so to achieve representative plots for comparison we removed extreme outliers 2014J, 1999by, and 2011by from cohort ra8..11 cohort for the control comparison. The plots converge near redshift 0.025 as shown in Table 3 and Table 4, and plot 11.

Plot 11 Displaying Magnitude-Redshift Variations in Directional Cohorts

Plot 11 clearly displays the changes in the relative velocity of the means of the cohort samples. Note that the changes in slope of the mean trend lines differ, with the greatest differences from redshift 0.001 to 0.005. This is crucial to disambiguating apparent change in velocity due to an increase in the rate of expansion of spacetime from changes in velocity due to gravitational acceleration.

Table 4. displays the values of the changes in velocity displayed in plot 11. Note the differences in the rows of the ∆v columns. Plot 11 and Table 4 were derived from the samples of SAI Supernova Catalog data found here:.

Table 4. Cohort Mean Trendline Distance Modulus and Velocity Changes from Redshift 0.001 to 0.035.

Redshift	RA 2-5 ∆distmod	RA 8-11 ∆dstmod	RA 14-17 ∆distmod	RA 20-23 ∆distmod	RA 2-5 ∆v kps	RA 8-11 ∆v kps	RA 14-17 ∆v kps	RA 20-23 ∆v kps
0.001	0	0	0	0	0	0	0	0
0.002	0.856175397	1.012226938	1.166081502	0.809665222	3.431996899	3.687713685	3.958476834	3.359269537
0.003	0.500830502	0.631917371	0.682113951	0.473623793	2.913921901	3.095246787	3.167631095	2.87764069
0.004	0.355344896	0.44835251	0.48396755	0.336041429	2.725089601	2.844345243	2.891380964	2.700972069
0.005	0.275626915	0.347769225	0.375394396	0.260653982	2.62686159	2.715599076	2.750367241	2.608810921
0.006	0.225203587	0.284148146	0.306719555	0.21296981	2.56656652	2.637190099	2.664745382	2.552147493
0.007	0.19040692	0.240243835	0.259327689	0.180063409	2.525766456	2.58440501	2.607217995	2.513763924
0.008	0.164937976	0.208108675	0.224639862	0.15597802	2.496315093	2.546440555	2.565900313	2.486035988
0.009	0.145485606	0.183564861	0.198146401	0.137582364	2.474052598	2.517820583	2.534784761	2.465064459
0.01	0.130141309	0.164204364	0.177247995	0.123071618	2.456631799	2.495471878	2.510506825	2.448646717
0.011	0.117727134	0.148540915	0.160340315	0.111331821	2.442627454	2.477536095	2.491035234	2.435444132
0.012	0.107476453	0.135607231	0.14637924	0.101637989	2.431123928	2.462823299	2.475070963	2.424596116
0.013	0.098868753	0.12474656	0.134655847	0.093497887	2.421506046	2.450536182	2.461744502	2.415524155
0.014	0.091538167	0.115497275	0.124671842	0.086565522	2.413345166	2.440120436	2.450451863	2.407824958
0.015	0.085219995	0.10752539	0.116066708	0.080590573	2.40633344	2.431178718	2.440760402	2.401208776
0.016	0.079717981	0.100583285	0.108573154	0.075387447	2.40024406	2.423418754	2.432352077	2.395462059
0.017	0.074883533	0.094483473	0.101988801	0.070815621	2.394906231	2.41662076	2.424987853	2.39042395
0.018	0.070602073	0.089081388	0.0961576	0.066766743	2.390188882	2.410616278	2.418484605	2.385970973
0.019	0.066783832	0.084263764	0.090957286	0.063155921	2.38598975	2.405274019	2.412699669	2.38200677
0.02	0.063357477	0.079940599	0.086290709	0.059915697	2.382227878	2.400490144	2.407520252	2.378455042
0.021	0.060265611	0.076039471	0.082079693	0.056991791	2.37883834	2.396181448	2.402856006	2.375254587
0.022	0.057461523	0.072501444	0.078260622	0.05434003	2.375768457	2.392280478	2.398633704	2.372355742
0.023	0.054906817	0.069278072	0.0747812	0.051924104	2.372975043	2.388731968	2.394793371	2.369717786
0.024	0.052569636	0.066329159	0.071598039	0.049713886	2.370422356	2.385490213	2.391285416	2.367307012
0.025	0.050423328	0.063621078	0.068674841	0.047684172	2.368080561	2.382517081	2.388068475	2.365095282
0.026	0.048445425	0.061125481	0.065981006	0.045813715	2.365924558	2.379780511	2.385107777	2.36305892
0.027	0.046616853	0.058818301	0.063490554	0.044084477	2.363933079	2.377253347	2.382373875	2.361177862
0.028	0.044921314	0.056678973	0.061181288	0.042481045	2.362087983	2.374912438	2.379841672	2.359434994
0.029	0.043344798	0.054689822	0.059034127	0.040990171	2.3603737	2.372737924	2.377489638	2.357815625
0.03	0.041875197	0.052835568	0.05703258	0.039600403	2.358776795	2.370712671	2.375299205	2.356307078
0.031	0.040501989	0.051102939	0.055162319	0.038301792	2.35728561	2.368821821	2.373254271	2.354898352
0.032	0.039215992	0.049480346	0.053410835	0.037085655	2.355889984	2.367052424	2.371340804	2.353579855
0.033	0.038009153	0.04795763	0.051767161	0.035944374	2.354581014	2.365393142	2.369546521	2.352343189
0.034	0.03687438	0.046525843	0.05022164	0.034871246	2.353350874	2.363834005	2.367860623	2.351180962
0.035	0.035805406	0.045177076	0.048765733	0.033860342	2.352192649	2.362366213	2.366273576	2.350086651

Table 4. Cohort Mean Trendline Distance Modulus and Velocity Changes from Redshift 0.001 to 0.035.

Redshift	RA 2-5 ∆distmod	RA 8-11 ∆dstmod	RA 14-17 ∆distmod	RA 20-23 ∆distmod	RA 2-5 ∆v kps	RA 8-11 ∆v kps	RA 14-17 ∆v kps	RA 20-23 ∆v kps
0.001	0	0	0	0	0	0	0	0
0.002	0.856175397	1.012226938	1.166081502	0.809665222	3.431996899	3.687713685	3.958476834	3.359269537
0.003	0.500830502	0.631917371	0.682113951	0.473623793	2.913921901	3.095246787	3.167631095	2.87764069
0.004	0.355344896	0.44835251	0.48396755	0.336041429	2.725089601	2.844345243	2.891380964	2.700972069
0.005	0.275626915	0.347769225	0.375394396	0.260653982	2.62686159	2.715599076	2.750367241	2.608810921
0.006	0.225203587	0.284148146	0.306719555	0.21296981	2.56656652	2.637190099	2.664745382	2.552147493
0.007	0.19040692	0.240243835	0.259327689	0.180063409	2.525766456	2.58440501	2.607217995	2.513763924
0.008	0.164937976	0.208108675	0.224639862	0.15597802	2.496315093	2.546440555	2.565900313	2.486035988
0.009	0.145485606	0.183564861	0.198146401	0.137582364	2.474052598	2.517820583	2.534784761	2.465064459
0.01	0.130141309	0.164204364	0.177247995	0.123071618	2.456631799	2.495471878	2.510506825	2.448646717
0.011	0.117727134	0.148540915	0.160340315	0.111331821	2.442627454	2.477536095	2.491035234	2.435444132
0.012	0.107476453	0.135607231	0.14637924	0.101637989	2.431123928	2.462823299	2.475070963	2.424596116
0.013	0.098868753	0.12474656	0.134655847	0.093497887	2.421506046	2.450536182	2.461744502	2.415524155
0.014	0.091538167	0.115497275	0.124671842	0.086565522	2.413345166	2.440120436	2.450451863	2.407824958
0.015	0.085219995	0.10752539	0.116066708	0.080590573	2.40633344	2.431178718	2.440760402	2.401208776
0.016	0.079717981	0.100583285	0.108573154	0.075387447	2.40024406	2.423418754	2.432352077	2.395462059
0.017	0.074883533	0.094483473	0.101988801	0.070815621	2.394906231	2.41662076	2.424987853	2.39042395
0.018	0.070602073	0.089081388	0.0961576	0.066766743	2.390188882	2.410616278	2.418484605	2.385970973
0.019	0.066783832	0.084263764	0.090957286	0.063155921	2.38598975	2.405274019	2.412699669	2.38200677
0.02	0.063357477	0.079940599	0.086290709	0.059915697	2.382227878	2.400490144	2.407520252	2.378455042
0.021	0.060265611	0.076039471	0.082079693	0.056991791	2.37883834	2.396181448	2.402856006	2.375254587
0.022	0.057461523	0.072501444	0.078260622	0.05434003	2.375768457	2.392280478	2.398633704	2.372355742
0.023	0.054906817	0.069278072	0.0747812	0.051924104	2.372975043	2.388731968	2.394793371	2.369717786
0.024	0.052569636	0.066329159	0.071598039	0.049713886	2.370422356	2.385490213	2.391285416	2.367307012
0.025	0.050423328	0.063621078	0.068674841	0.047684172	2.368080561	2.382517081	2.388068475	2.365095282
0.026	0.048445425	0.061125481	0.065981006	0.045813715	2.365924558	2.379780511	2.385107777	2.36305892
0.027	0.046616853	0.058818301	0.063490554	0.044084477	2.363933079	2.377253347	2.382373875	2.361177862
0.028	0.044921314	0.056678973	0.061181288	0.042481045	2.362087983	2.374912438	2.379841672	2.359434994
0.029	0.043344798	0.054689822	0.059034127	0.040990171	2.3603737	2.372737924	2.377489638	2.357815625
0.03	0.041875197	0.052835568	0.05703258	0.039600403	2.358776795	2.370712671	2.375299205	2.356307078
0.031	0.040501989	0.051102939	0.055162319	0.038301792	2.35728561	2.368821821	2.373254271	2.354898352
0.032	0.039215992	0.049480346	0.053410835	0.037085655	2.355889984	2.367052424	2.371340804	2.353579855
0.033	0.038009153	0.04795763	0.051767161	0.035944374	2.354581014	2.365393142	2.369546521	2.352343189
0.034	0.03687438	0.046525843	0.05022164	0.034871246	2.353350874	2.363834005	2.367860623	2.351180962
0.035	0.035805406	0.045177076	0.048765733	0.033860342	2.352192649	2.362366213	2.366273576	2.350086651

3.2. Extragalactic Distance Database Analysis

The EDD provides numerous measurements including distance, heliocentric velocity, and peculiar motion of over 50 thousand objects. This provided sufficient data of cohort samples of 15 degrees of right ascension and +- 30 degrees of declination. An analysis of right ascension 2.0000 to 2.9999 (RA2), 14.0000 to 14.9999 (RA14), 8.0000 to 8.9999 (RA8) and 20.0000 to 20.9999 (RA20) is presented here. All observations are identified by their Principal Galaxies Catalog (PGC) [15] entries.

We plotted the distance- peculiar velocity-corrected heliocentric velocity relation for the EDD data for the RA 2 and RA 14 cohorts of the test group. The current positive velocity of the Milky Way of 630kps is incorporated into the velocity measurement. 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 distance-velocity (DV) 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 DV plots at low distance (recent past) that converge at high distance (distant past). Conversely, a negative acceleration would produce an increasing difference in the DV plots over increasing time and distance.

Plots 12 and 13 Displaying RA2 Observations by Redshift and Velocity

The plots 12 and 13 from EDD data displaying the RA2 distance-velocity relation on redshift and velocity scales respectively, are found here:

Plots 14 and 15 Displaying RA14 Observations by Redshift and Velocity

The plots 14 and 15 from EDD data displaying the distance-velocity relation on redshift and velocity scales respectively, are found here:

The curve in the RA2 and RA14 mean trendlines indicate a change in velocity over time in both cohorts, however, the plots do not clearly display any difference between them.

Plots 16 through 18 Illustrating Heliocentric ∆V Changes by Redshift and Velocity

Plots 14, 15 and 16 display the RA2-RA15 differences in the distance-redshift and distance-velocity relations change from data found here:

Plot 18 shows the computed change in velocity of the Milky Way relative to objects in RA2 and RA14 during the last 480 million years. From peculiar velocity-corrected heliocentric velocity measurements this plot shows that during the most recent 3.26-million-year period that we can measure, the velocity of the Milky Way in the direction of RA14 relative to RA2 increased by an average of 3.09 km/s every 3.26 million years, and increasingly less in the past.

Plots 19 through 22 Displaying Samples at Ninety degrees to RA2-RA14 by Redshift and Velocity

Plots of samples at 90 degrees from the RA2->RA14 heading, RA8 and RA20, are presented below for comparison with the analysis of the R2-R14 changes in velocity.

The plots 19 and 20 from EDD data displaying the RA8 distance-velocity relation on redshift and velocity scales respectively, are found here:

The plots 21 and 22 from EDD data displaying the RA20 distance-velocity relation on redshift and velocity scales respectively, are found here:

As with the RA2 and RA14 curves in the mean trendlines the RA8 and RA20 trendlines indicate a change in velocity over time in both cohorts, and the plots do not clearly display any difference between them as well. Plots 23, 24 and 25 display the RA8-RA20 differences in the distance-redshift and distance-velocity relations change from data found here:

Plots 23 through 25: Plots Illustrating Heliocentric ∆V Changes by Redshift and Velocity

Plot 25 displays the computed change in velocity of the milky way relative to objects in RA8 and RA20 during the last 480 million years. Unfortunately, neither of the cohorts contains any observations below redshift 0.004 which skews the trend to higher redshifts thus making direct comparison with the RA2-RA14 comparison difficult. However, these plots display changes in velocity quantitively different from the RA2-14 cohort analysis. From peculiar velocity-corrected heliocentric velocity measurements this plot shows that during the most recent 3.26-million-year period that we can measure, 58.9 million years ago, the velocity of the Milky Way in the direction of RA20 relative to RA8 increased by an average of 2.04 km/s every 3.26 million years, and increasingly less in the past.

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 or sufficiently close to inertial that it can be ignored. While the purely statistical analysis that uses SNe 1a and EDD observations as a random population could not disprove that assumption, we believe that it does support a hypothesis to that effect.

4.1. Type 1a Supernova Data

If the Sne 1a sampled cohort means analysis data here is taken at face value then the local reference frame is currently accelerating at nearly 2.5kps/MPC on the RA 2-5, 14-17, vector. If the changes in velocity as shown in Plot 11 and Table 4 were due to a change in the isotropic increase of the cosmological constant, then the velocity changes between the same redshift values in Table 4, and the curves in the cohort means in Plot 11 should be nearly identical. However, the differences could be due to the uncorrected random variables affecting the measurements of apparent magnitude and peculiar motion.

4.2. EDD Data

The narrower RA2 and RA14 cohort means from EDD data yields a higher calculated acceleration of nearly 3.1kps/MPC the direction of RA14 and crucially away from RA2, contributing to the redshift of nearby objects with right ascensions of 2.0.0 to 2.9.99. As with the Sne 1a redshift/velocity curves, the EDD curves show an anisotropic change in velocities. The changes are presented in Plot 17 and the EDD data table for Plots 15 through 17 here.

5. Conclusion

An acceleration of the local reference frame could be consequential to the finding of a recent acceleration of the expansion of spacetime if the data used in the Perlmutter and Reis papers 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 only a substantial fraction of the data in the Perlmutter, and Reis studies, they would bias the results more subtly.

The analysis of the narrower RA2 and RA14 cohorts using EDD data confirmed the Sne 1a analysis of an anisotropic acceleration and yielded a higher acceleration of 3.09kps/MPC away from RA2 alone toward RA14. Though an acceleration of the local reference frame could also contribute to the Hubble tension this value is insufficient to account or all of it. However, if, as the analysis of the calculated means suggests, nearby objects display recent anisotropic changes in velocities relative to the local reference frame.

The analysis of the Sne 1a data is a relatively unsophisticated statistical analysis of raw, uncorrected SAI SNe 1a measurements. Though there is plenty of data, it is of unknown quality. Therefore, finding of recent acceleration could from the Sne 1a data be due to noise in the data. However, a similar analysis using a completely different dataset which contains heliocentric data with peculiar motion correction eliminated two potential sources of noise and yielded a finding in qualitative agreement with the Sne 1a analysis.

Therefore, if these findings are valid, one or more of the following must be true, and possibly consequential to the finding of a recent increase in the rate of expansion of spacetime:

• The recent rate of expansion of spacetime near the local reference frame is not isotropic and/or
• the nearby stars and galaxies in the cohort samples are not inertial and/or
• the local reference frame is not inertial.

An acceleration of the local reference frame coupled with a very small data set and selection bias is a simpler explanation for the finding of acceleration of the expansion of spacetime than dark energy. We believe that acceleration of the local reference frame is most parsimonious explanation for the finding of a relatively recent increase in lambda and is sufficiently credible to merit further investigation.