ADDING THE 1/ (1+z) FACTOR TO THE RIESS ET AL (1998) AND PERLMUTTER ET AL (1999) REST-FRAME DATA REMOVES ANY EVIDENCE OF DARK ENERGY Richard K. Love, 2245 Peavine Valley Rd., Reno, NV 89523 Sean R. Love, 191 E. El Camino Real #124, Mountain View, CA 94040 ABSTRACT Aims: Demonstrate that the 0.22 increase in slope of high z SNe1a currently attributed to an increase of 10-15% distance believed to be caused by dark energy is actually caused by the time-dilated template subtraction of the factor 1/ (1+z) to create a rest-frame. Methods: Simple addition and subtraction of the 1/ (1+z) factor to rest-frame data from Riess et al. (1998) and Perlmutter et al. (1999). Because the 1/ (1+z) factor is non-linear, subtracting it reduces the slope of the low SNe1a much more than it reduces the slope of the high SNe1a so that it appears that the slope of the high SNe1a has increased. Adding the factor reverses the slope change which is linearly cumulative for multiple additions. Results: When we added back the 1/ (1+z) factor to the SNeIa data, it undid the error by recreating the observer-frame, so that the high z slope of 2.40 became 2.18 and the low z slope of 2.22 became 2.17. This addition made the two slopes essentially the same and thus eliminated the observation of an increase of 0.22 in the slope of the high z SNE1a which required the theory of dark energy. We then subtracted multiples of the 1/ (1+z) factor. Each subtraction increased the SNeIa high z slope by the same 0.22 amount. We developed a factor-to-slope correlation (R2=0.9998) by comparing the number of 1/ (1+z) correction factors to the 0.22 increase in the resulting high z SNe1a slopes. THIS PAPER IS BASED ON ADDITION AND SUBTRACTION, NOT COSMOLOGY. 1. INTRODUCTION Riess et al. (1998) and Perlmutter et al. (1999) interpreted the physical observation of a 0.22 increase in the slope of high z SNe1a peak magnitudes as a 10 - 15% luminosity distance increase in the metric expansion of the universe. This distance acceleration was attributed to an unknown (dark) energy which still has no proven source, and according to NASA (2014) would have to comprise 71.4 % of all energy. We propose that this 0.22 increase in peak magnitude slope of high z SNeIa, observed by Riess et al. (1998) and Perlmutter et al. (1999), is simply caused by subtracting the 1/(1+z) cosmic time dilation correction factor in order to create a rest-frame. Because the 1/ (1+z) factor is non-linear, it subtracts less from the high z SNe1a than from the low z SNe1a, making it appear that the peak magnitudes of the high z SNe1a have increased. . We will show proof that adding back the 1/ (1+z) correction factor, which was subtracted by the time-dilated template, eliminates the slope increase observed in the high z SNeIa peak magnitude. Our addition of 1/ (1+z) returns the data to its observer-frame slope by eliminating the observed 0.22 increase (2.18 to 2.40) of the high redshift SNeIa peak magnitude slope, which was the observation that led to the theory of dark energy. 1 Our theory can be falsified: the change in high z slope caused by adding or subtracting any number of 1/ (1+z) factors from the rest-frame of the observed high z SNe1a must be 0.22. We developed a factor-to-slope correlation by comparing varying the number of 1/ (1+z) correction factors to their corresponding high z slopes with R 2  0.9998 . 2. Reiss et al (1998) and Perlmutter et al (1999) data 2.1 Before we added 1/ (1+z) The increase in slope in the peak magnitudes of the high redshift SNe1a in the top graph in FIG.1 was proposed by Riess et al (1998) and Perlmutter et al (1999 to be an increase in distance caused by an unknown (dark) energy This graph, displaying 70 rest-frame data points, is the baseline graph to which we will add 1/ (1+z). Remember that the restframe data displayed in this graph was created by subtracting the 1/ (1+z) factor (curve seen on the bottom) so that adding 1/ (1+z) will return the data to the observer-frame. The graph on the bottom of FIG. 1 is the non-linear curve that results from plotting the 1/ (1+z) curve using the redshift values from the top graph. Note that the shape of the bottom curve is an inverse of the top curve. Therefore, subtracting the non-linear 1/ (1+z) curve from a straight line from the observer-frame data will subtract more from the low z SNe1a which will cause the high z SNe1a slope to appear to increase in the rest frame. 30 10 25 MB - [1 x 1/(1+z)] MB 20 15 10 5 High redshift slope= 2.40 Low redshift slope= 2.22 1 x 1/(1+z) factor curve 0 0.01 0.1 Redshift 1 10 FIG.1: Untouched rest-frame data from Riess et al (19980 and Perlmutter et al (1999) also showing the curve of the 1/ (1+z) factor caused by subtraction using time-dilated template. 2 The graph in FIG.1 at the top is the graph seen on p 568 in Perlmutter et al. (1999) plus data from Riess et al. (1998). It shows the increase in the high redshift SNeIa peak magnitudes slope that was attributed to distance acceleration due to dark energy. The graph at the bottom is the non-linear curve of a 1/ (1+z) factor using the z data from the top graph. Note: when the 1/ (1+z) factor (seen in the lower curve) is subtracted from the observer-frame data (seen in the higher curve) to form the rest-frame, it subtracts more from the low z SNe1a, making it appear that the high z SNe 1a have increased. 1.2 After we added 1/ (1+z) FIG 2 shows that the addition of the 1/ (1+z) factor makes the high and low z slopes essentially equal, returning the data to the straight line of the observer-frame. This elimination occurs because adding the non-linear 1/ (1+z) factor reverses the subtraction error. Adding 1/ (1+z) reduces the high z SNe1a slope more than the low z SNe1a slope, which is the exact opposite of what was done by subtracting 1/ (1+z). 30 25 MB - [0 x 1/(1+z)] High redshift slope=2.18 MB 20 15 Low redshift slope=2.17 10 5 Curve removed by addtion of 1/(1+z) factor. 0 0.01 0.1 Redshift 1 10 FIG. 2: Data from FIG.1 with our addition of 1/ (1+z) factor. Note that with our addition of 1/ (1+z), the slope of the high SNe1a now closely matches the slope of the low SNE1a.and that they are both lower. The apparent distance acceleration in high-z SNe 1a proposed by Reiss et al (1998) and Perlmutter et al (1999) to be due to dark energy is gone. The 1/ (1+z) non-linearity reduced the high z SNe1a slope much more than the low SNe1a slope making the two slopes essentially equal. The results went back to the observer-frame as they were before the 1/ (1+z) correction factor was subtracted by a time-dilated template to form the rest-frame. 3 FIGS 1 and 2 are Hubble log diagrams each with 70 SNeIa from Riess et al. (1998) and Perlmutter et al. (1999). Data for top graphs in FIG.1 and FIG.2 is found in Table 7 on P13. FIG.1 uses untouched data from columns 3 and 4. FIG.2 uses untouched data from column 3 and calculated data from column 8, which has the 1/ (1+z) factor added to the data in column 4. Note the outlier data in Table 4 on P.12. Data for bottom graph in FIG. 1 is also found in Table 7 on P.13. It uses the untouched data in column 4 and the calculated factor data in column 5. 2.3 Possible explanation for the slope change caused by adding 1/ (1+z) The presence of the cosmic time dilation factor is assumed to be in the observer-frame data, which implies that when the correction factor 1/ (1+z) is subtracted to remove the time dilation factor to form a rest-frame, it is also assumed that the two factors would cancel each other, leaving only the effective peak magnitudes of the SNeIa. But if the cosmic dilation factor is not present to cancel it, the uncancelled correction factor remains in the rest-frame data to cause the increase in high z SNe1a data Equations 1 -4 demonstrate how it’s possible that adding a 1/ (1+z) factor could cancel the negative 1/ (1+z) factor that was subtracted by the time-dilated template. Equation 1 shows the original observer-frame calculation with no native 1/ (1+z) factor. (1) mBeff  mB  K BR  AR Equation 2 shows the rest-frame calculation, unexpectedly containing the negative 1/ (1+z) factor, because there was no native 1/ (1+z) factor to cancel it. mBeff  mB  1  K BR  AR (1  z ) (2) Equations 3 and 4 show the effect when we added 1/ (1+z). mBeff  mB  1 1   K BR  AR (1  z ) (1  z ) (3) With the two factors cancelled, the calculation returns to the observer-frame (4) mBeff  mB  K BR  AR . For more detail, please go to Appendix A on page 18. 4 3. Effect of subtracting the non-linear 1/ (1+z) factor We have demonstrated that when the 1/ (1+z) factor is added to Riess et al. (1998) and Perlmutter et al. (1999) data , the increase in high z SNeIa peak magnitudes currently attributed to Dark Energy is eliminated. The next question must be: What happens when the 1/ (1+z) factor is subtracted? As we will prove, the obvious answer is the correct one: It increases the high z peak magnitudes. We will also confirm a 0.9998 correlation between the slope and the number of 1/ (1+z) factors in the data. Remember that the original data from Riess et al. (1998) and Perlmutter et al. (1999) already includes a 1/ (1+z) factor subtracted by the time-lengthened template, so it has a native high z SNeIa slope of 2.40. In Fig.3, to help you see the mirror image of the two curves three additional factors have been subtracted from that data, making it easier to see the non-linear effect of 1/ (1+z) factor on both the high and low z SNe1a. 25 20 MB - [4 x 1/(1+z)] High redshift slope =3.06 MB 15 Low redshift slope =2.39 10 5 4 x 1/(1+z) factor curve 0 0.01 0.1 1 10 Redshift FIG.3: Same data as FIG.1 with the subtraction of 3 additional 1/ (1+z) factors FIG.3 shows a total of four (one already subtracted, three we subtracted) 1/ (1+z) factors (seen on the bottom of the page) have been subtracted from each of the 70 Riess et al. (1998) and Perlmutter et al. (1999) SNeIa peak magnitude values (on top of the page). As more clearly seen with additional factors, this subtraction affects the high redshift SNeIa peak magnitude values much more than it does the low redshift peak magnitude values, making it appear that only the high redshift SNeIa peak magnitudes have increased. Note the inverse correlation between the top and bottom curves 5 By subtracting the same 1/ (1+z) correction factor from the baseline data, it increases the slope of the baseline data by that same factor of 0.22 to increase the slope of the peak magnitude from a slope of 2.40 to a slope of 2.62. Equation 3 shows both the first 1/ (1+z) factor subtracted by the time-dilated template and the second 1/ (1+z) factor subtracted mathematically. mBeff  mB  1 1   K BR  AR (1  z ) (1  z ) (5) For more detail to see how the subtraction of 1/ (1+z) reduces the absolute value of the data as well as the relative slopes, go to Appendix B on page 20. 3. Summary of proof To recap, a 0.22 increase in high z SNe1a slope was observed by Riess et al. (1998) and Perlmutter et al. (1999) in their baseline data but was reported as a 10-15% increase in distance. It was seen that adding the 1/ (1+z) factor to that baseline data removes the cosmic time dilation correction factor which decreases the slope of the high redshift SNeIa by a factor of 0.22, while subtracting another 1/ (1+z) correction factor from the baseline data augments the correction factor already subtracted which increases the slope by the same 0.22 factor. 1. In Fig. 1, we started with the untouched baseline data of 70 SNeIa as originally published by Riess et al. (1998) and Perlmutter et al. (1999). The data has one 1/ (1+z) correction factor subtracted by a time-dilated template and displays a 2.40 slope in the high redshift peak magnitude of the SNeIa. 2. Then in Fig.2, we added one 1/(1+z) correction factor to the baseline data to cancel the correction factor that had been subtracted by Riess et al. (1998) and Perlmutter et al. (1999). This addition of the correction factor reduced the slope of the high redshift peak magnitude from 2.40 to 2.18 for a change of 0.22 in the negative direction. 3. It is important to note that the 2.18 slope of the high redshift peak magnitude essentially equals the 2.17 slope of the low redshift peak magnitude. The fact that these two slopes are equal proves that the non-linearity has been exactly (not more and not less) removed from the data by the addition of the 1/ (1+z) correction factor. This puts the non-linear 1/ (1+z) correction factor as the source of the peak magnitude increase and the time-dilated template as the source of the non-linear correction factor. 4. Then in Fig.3, we subtracted three 1/ (1+z) correction factors from the baseline data. This subtraction increased the slope of the high redshift peak magnitude from 2.40 to 3.06, which confirmed the same change of 0.22 per one 1/ (1+z) correction factor, but in a positive direction. 6 4. STATISTICAL BACKUP SUPPORT FOR THE EXPLANATION The previous explanation that we provided for the observed high redshift SNeIa increase in peak magnitude may appear to use only three data points to determine the confirm the plus or minus effect of the slope by the 1/(1+z) correction factor. In actuality, we used ten data points and performed several statistics to confirm our findings. . 4.1 Data points The first step in the confirmation is to determine the peak magnitude slope values of both the high and low redshift SNeIa. The starting slope values are from the untouched baseline data from Riess et al. (1998) and Perlmutter et al. (1999) (These slopes are 2.40 on the high z slope and 2.22 on the low z slope). Note that this point (point 1*) has one 1/ (1+z) factor that was subtracted by the time-dilated template. The other slope values were created by the adding and subtracting of multiple 1/ (1+z) factors to the 1* point as seen in Table 1 to form the data basis for the “Standard Addition” technique. Table 1 shows all the addition and subtraction done to the baseline point (1*) (with ½ increments of the 1/ (1+z) correction factor). As can be seen, the change in slope between full factors, whether added or subtracted, averages very close to 0.22 over the entire range. This consistent, quantitative change per action (0.22) is both bidirectional and solidly based on a zero value that is the complete absence of the non-linear 1/ (1+z) factor. A quantitative change per action is strong proof of causation, for example when applied to the medical field where it is referred to as “dose dependence.” In case of the SNeIa, the action is the addition or subtraction of one 1/ (1+z) factor and the change is a corresponding 0.22 increase or decrease in the peak magnitude slope. 1/(1+z) Factors 0 ½ 1* 1½ 2 2½ 3 3½ 4 4½ 5 Avg. Added or subtracted 1 Added ½ Added Subtracted* Subtracted Subtracted Subtracted Subtracted Subtracted Subtracted Subtracted Subtracted High z Slope 2.18 2.29 2.40 2.51 2.62 2.74 2.84 2.96 3.06 3.19 3.30 Low z Slope 2.17 2.20 2.22 2.25 2.28 2.31 2.34 2.36 2.39 2.42 2.45 Diff n/a 0.11 0.11 0.11 0.11 0.12 0.10 0.12 0.10 0.13 0.11 0.112 7 Diff n/a 0.03 0.02 0.03 0.03 0.03 0.03 0.02 0.03 0.03 0.03 0.028 Table 1: Slopes of 70 high and low redshift SNeIa with multiple added and subtracted 1/ (1+z) factors in the baseline data. The 1* data point was in the data and untouched by us. 4.2 Regression analysis with prediction equations and R 2 The curves presented in FIG. 4 include the data point 1* with one 1/ (1+z) factor. This is untouched data where the one 1/ (1+z) correction factor was subtracted by Riess et al. (1998) and Perlmutter et al. (1999) using a template, not by us. Note that this point was not included when developing either the high or low prediction equations. The “standard addition technique” is used to derive the equations to predict the slope values by adding or subtracting ½ of the 1/ (1+z) factor to or from the data in point 1*. The high redshift SNeIa points generated the equation: y  0.02236 x  2.1764 R 2  0.9998 (6) The low redshift SNeIa points generated the equation: y  0.0555 x  2.1699 R 2  0.9992 (7) 3.4 High redshift slope equation slope vakue 3.2 y = 0.2236x + 2.1764 2 R = 0.9998 3 2.8 Low redshift slope equation 2.6 y=0.555x+2.1699 R2=0.0992 2.4 2.2 2 0 0.5 1* 1 1.5 2 2.5 3 3.5 4 4.5 5 # of 1/(1+z) factors in data FIG. 4: Plot of high and low z data with addition of partial 1/ (1-z) factors The “standard addition technique” is used to derive the equations to predict the slope values. In this technique, the other data points are created by adding or subtracting ½ of the 1/(1+z) factor to or from the data in point 1* 8 4.3 Predicted slope values The following Table 2 compares the actual and predicted slopes based on the derived equations. Note that the predicted slope for 1* factor (high redshift slopes) perfectly predicts the actual slope (2.40) observed in the Riess et al. (1998) and Perlmutter et al. (1999) data # factors Zero 1/2 1* 1½ 2 2½ 3 3 1/2 4 4 1/2 5 SDD High redshift slopes Actual 2.18 2.29 2.40 2.51 2.62 2.74 2.84 2.96 3.06 3.19 3.30 Predicted 2.18 2.29 2.40 2.51 2.62 2.74 2.85 2.96 3.07 3.18 3.29 Low redshift slopes Diff Actual 2.17 2.20 2.20 2.25 2.28 2.31 2.34 2.36 2.39 2.42 2.45 0.00 0.00 0.00 0.00 0.00 0.00 -0.01 0.00 -0.01 0.01 0.01 0.006 Predicted 2.18 2.20 2.23 2.26 2.29 2.32 2.34 2.37 2.40 2.43 2.45 Diff -0.01 0.00 -0.03 -0.01 -0.01 -0.01 0.00 -0.01 -0.01 -0.01 0.00 Table 2: Actual and predicted high and low redshift slopes. 4.4. Comparison between # of factors and total range slope and intercepts 1* factor predicts the slope (2.40) perfectly to match the slope of the high redshift increase in SNeIa peak magnitude observed in the Riess et al. (1998) and Perlmutter et al. (1999) data. Table 3 shows the increase in both the slope and the intercept for the total range (high redshift and low redshift) for all the additions and subtractions made with the 1/ (1+z) factors. This result illustrates that the observed effect was a subtraction of a non-linear factor. (See Fig.2) Factor Zero ½ One 1½ Two 2½ Three 3½ Four Total slope value 2.26 2.33 2.38 2.44 2.51 2.57 2.63 2.69 2.75 Intercept 25.30 25.00 24.75 24.46 24.17 23.87 23.58 23.29 22.99 9 4½ Five 2.81 2.87 22.70 22.41 Table 3: Decrease in total slope value and intercept as increasing number of 1/ (1+z) factors are subtracted 5. DISCUSSION 5.1.1 Historical evidence about problems with cosmic time dilation factor It is very logical to infer that the subtracted correction factor remains to cause the increase in the high z SNe1a because the cosmic time dilation factor was not in the observer-frame to cancel it, but it is impossible to prove a negative. Therefore, the truth of whether cosmic time dilatation exists or not rests on the balance of the evidence. It is therefore very important to critically evaluate all evidence that is used to support or deny the existence of the cosmic time dilation factor. Despite years of effort, except for Blondin et al. (2008) (which now has been shown to be a tautology by our supporting paper), as will be presented, there is no accurate quantitative evidence supporting the existence of cosmic time dilation and at least one other strong paper, plus our paper, denying its existence. There are three main areas (gamma ray bursts, quasar light curves and supernovae) where major investigations have been made thus far into the existence of cosmic time dilation. Technical problems prevented definitive results from gamma ray bursts. As examples, Deng & Shaefer (1998) couldn’t measure individual bursts and the results reported by Chang (2001) were limited by selection error. However, negative results from quasar light curves presented by Hawkins (2010), on the other hand, were clear and definitive: no evidence of cosmic time dilation was found in either low or high redshift light curve power spectra over a measurement period of 28 years. With supernovae, Blondin et al. (2008), a very highly regarded paper, was very critical of attempts using light-curve width technology to prove the existence of cosmic time dilation. Quotes from Blondin et al. (2008) make it clear that their new study was necessary because previous studies, using both light curve width and spectral analysis, had failed to provide sufficient quantitative evidence to prove the cosmic time dilation hypothesis. Blondin et al. (2008) first points out the critical errors found with the light curve width technique: “It is problematic to disentangle this intrinsic variation of light curve width and the effect of cosmic time dilation. To directly test the cosmic time dilation hypothesis, one needs to accurately know the distribution of the light-curve widths at z ≈0 and its potential evolution with redshift, whether due to a selection effect (not taken into account by Goldhaber et al. 2001) or an evolution of the 10 mean properties of the SN 1a sample with redshift, as possibly observed by Howell et al. (2007). Furthermore, one might argue that at high redshift we are preferentially finding the brighter events (akin to Malmquist bias). Such as selection effect would produce a spurious relation in which there would be broader light curves at higher redshifts, without any cosmic time dilation.” (Italics are ours) However, there is a newer technology: Spectral Analysis. The spectral analysis technique uses the change in the spectrum of the SNeIa to measure the time effects of cosmic time dilation. Blondin et al. (2008) asserts that the spectral analysis technique avoids the degeneracy between intrinsic light-curve width and the effects of cosmic time dilation. “The spectra of SNe 1a provide an alternative and a more reliable way to measure the apparent aging rate of distant objects. Spectral analysis is the technique used in Blondin et al. (2008). However, in discussing the two prior studies that also used spectral analysis, only their qualitative success was acknowledged by Blondin et al. (2008). This is due to the large error reported by both papers as follows. A quote from the paper using spectral analysis by Riess et al. (1997) shows a large error: “In the 10.05 days which elapsed between spectral observations, SN 1996bj aged 3.35 ± 3.2 days, consistent with the 6.38 days of aging expected in an expanding Universe and inconsistent with no cosmic time dilation at the 96.4% confidence level” Note: The difference between the expected value of 6.8 days and the predicted value of 3.35 days is a 49.8% error. Also note that the variability of the prediction is 3.2 days which is almost as large as the value of the prediction itself (3.35 days). In the second spectral analysis paper, from Foley et al. (2005), the age factor was observed to be 1.602, while the prediction was for 1.361, which is a 14.9% error. This prediction error is better than the 49.8% error found in Riess et al. (1997), but still very large. No claim was made by Blondin et al. (2008) that any of the previous studies (either light width measurement or spectral analysis) were ever able to determine the expected magnitude of cosmic time dilation. These serious problems with the prior studies with SNeIa made the publication of Blondin et al. (2008) stand alone as a definitive proof of the cosmic time dilation hypothesis. It was widely believed to present clear and quantitative evidence of the existence of cosmic time dilation. 5. New evidence proving Blondin et al. (2008) is a tautology 11 We have submitted to your publication a manuscript that proves that Blondin et al (2008) is a tautology by substituting the observer-frame ages published in the paper with random numbers, which couldn’t possibly contain the 1/ (1+z) factor, yet the procedure in Blondin et al (2008) still finds the factor. 6. CONCLUSIONS 1. The fact that adding a 1/ (1+z) factor to the rest-frame data removes the observed increase in slope of the high z SNe1a has been experimentally proven. However, this fact doesn’t prove in 2019 that dark energy doesn’t exist, any more than the fact of finding the increase in the slope of the high z SNE1a proved in 1998 that dark energy did exist. 2. This fact does, however, prove that if it weren’t for the subtraction of the 1/ (1+z) from a rest-frame, the good correlation between redshift and the peak magnitude of all z levels of SNe1a would have been observed as was expected. 3. This fact also makes it improbable that there was a cosmic time dilation factor in the observer-frame data. 7. Data The four outliers removed from the data in this paper before graphing are detailed in Table 4. (For instructions on how to check our numbers, go to Appendix C on page 21) Outliers # in order Name 1 1992al 26 1997O 59 1997K 60 1997S Redshift 0.014 0.374 0.592 0.612 Table 4: Removed outliers . In Table 5, with the 70 SNeIa, arranged in order of ascending redshift, the first 20 are the low redshift SNeIa range and the second 50 are the high redshift SNeIa range Low redshift SNeIa range # in order Name Redshift 1 1992al 0.014 20 1997N 0.180 High redshift SNeIa range #in order Name Redshift 21 1996J 0.300 70 1997ck 0.970 Table 5: High and low redshift SNeIa ranges. 7.2 Table 6 contains explanation for Riess et al. (1998) and Perlmutter et al. (1999) SNeIa Data columns. 12 Column # 1 2 3 Designation [Source] [ Name] [ z] 4 [mb] 5 [1 (1  z )] 6 zero 7 1/2 8 one 9 1.5 10 two 11 2.5 12 three 13 3.5 14 four 15 4.5 16 five Explanation Literature reference IAU name assigned to supernova (from source Redshift of supernova (from source) Untouched Stretch peak magnitude-corrected B-band peak magnitude (from source) Original data. (also in column 8) ( Cosmic time dilation correction factor) (calculated) Stretch peak magnitude-corrected B-band peak magnitude plus one cosmic time dilation correction factor: mB  1 (1  z ) (calculated) Adding factor canceled subtraction. Has zero factors, Removed evidence of dark energy Stretch peak magnitude-corrected B-band peak magnitude plus ½ cosmic time eff dilation correction factor: mB  1 (1  z ) (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus one cosmic time dilation correction factor: mB  1x[1 (1  z )] Untouched Original data eff Stretch peak magnitude-corrected B-band peak magnitude minus 1 ½ cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus two cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus 2 ½ cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus three cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus 3 ½ cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus four cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus 4 ½ cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Stretch peak magnitude-corrected B-band peak magnitude minus five cosmic time dilation correction factor: mB  1x[1 (1  z )] (calculated) eff Table 6: Data legend for Riess et al. (1998) and Perlmutter et al. (1999) SNeIa data in Table 9 Table 7 has the 70 SNeIa from Riess et al. (1998) and Perlmutter et al. (1999) Source 1 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) Name 2 z 3 mb 4 1/(1+z) 5 Zero 6 1/2 7 One 8 1 1/2 9 Two 10 2 1/2 11 Three 12 3 1/2 13 Four 14 4 1/2 15 Five 16 1992al 0.014 14.47 0.986 15.46 14.96 14.47 13.98 13.48 12.99 12.50 12.00 11.51 11.02 10.53 1992bo 0.018 15.61 0.982 16.59 16.10 15.61 15.12 14.63 14.14 13.65 13.15 12.66 12.17 11.68 13 p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 1992bc 0.02 15.18 0.980 16.16 15.67 15.18 14.69 14.20 13.71 13.22 12.73 12.24 11.75 11.26 1992P 0.026 16.08 0.975 17.05 16.57 16.08 15.59 15.11 14.62 14.13 13.64 13.16 12.67 12.18 1992ag 0.026 16.28 0.975 17.25 16.77 16.28 15.79 15.31 14.82 14.33 13.84 13.36 12.87 12.38 1990O 0.03 16.26 0.971 17.23 16.75 16.26 15.77 15.29 14.80 14.32 13.83 13.35 12.86 12.38 1992bg 0.036 16.66 0.965 17.63 17.14 16.66 16.18 15.69 15.21 14.73 14.25 13.76 13.28 12.80 1992bl 0.043 17.19 0.959 18.15 17.67 17.19 16.71 16.23 15.75 15.27 14.79 14.31 13.83 13.35 1992bh 0.045 17.61 0.957 18.57 18.09 17.61 17.13 16.65 16.17 15.70 15.22 14.74 14.26 13.78 1990af 0.05 17.63 0.952 18.58 18.11 17.63 17.15 16.68 16.20 15.73 15.25 14.77 14.30 13.82 1993ag 0.05 17.69 0.952 18.64 18.17 17.69 17.21 16.74 16.26 15.79 15.31 14.83 14.36 13.88 1993O 0.052 17.54 0.951 18.49 18.02 17.54 17.06 16.59 16.11 15.64 15.16 14.69 14.21 13.74 1992bs 0.063 18.24 0.941 19.18 18.71 18.24 17.77 17.30 16.83 16.36 15.89 15.42 14.95 14.48 1993B 0.071 18.33 0.934 19.26 18.80 18.33 17.86 17.40 16.93 16.46 16.00 15.53 15.06 14.60 1992ae 0.075 18.43 0.930 19.36 18.90 18.43 17.96 17.50 17.03 16.57 16.10 15.64 15.17 14.71 1992bp 0.079 18.27 0.927 19.20 18.73 18.27 17.81 17.34 16.88 16.42 15.95 15.49 15.03 14.56 14 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Riess et al (1998) et al (1998) p. 1020 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Riess et al (1998) et al (1998) p. 1020 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et 1992br 0.088 19.28 0.919 20.20 19.74 19.28 18.82 18.36 17.90 17.44 16.98 16.52 16.06 15.60 1992aq 0.101 19.16 0.908 20.07 19.61 19.16 18.71 18.25 17.80 17.34 16.89 16.44 15.98 15.53 1997I 0.172 20.17 0.853 21.02 20.60 20.17 19.74 19.32 18.89 18.46 18.04 17.61 17.18 16.76 1997N 0.18 20.43 0.847 21.28 20.85 20.43 20.01 19.58 19.16 18.74 18.31 17.89 17.46 17.04 1996J 0.3 22.28 0.769 23.05 22.66 22.28 21.90 21.51 21.13 20.74 20.36 19.97 19.59 19.20 1997ac 0.32 21.86 0.758 22.62 22.24 21.86 21.48 21.10 20.72 20.34 19.97 19.59 19.21 18.83 1994F 0.354 22.38 0.739 23.12 22.75 22.38 22.01 21.64 21.27 20.90 20.53 20.16 19.80 19.43 1994am 0.372 22.26 0.729 22.99 22.62 22.26 21.90 21.53 21.17 20.80 20.44 20.07 19.71 19.34 1994H* 0.374 21.72 0.728 22.45 22.08 21.72 21.36 20.99 20.63 20.26 19.90 19.54 19.17 18.81 1997O* 0.374 23.52 0.728 24.25 23.88 23.52 23.16 22.79 22.43 22.06 21.70 21.34 20.97 20.61 1994an* 0.378 22.58 0.726 23.31 22.94 22.58 22.22 21.85 21.49 21.13 20.77 20.40 20.04 19.68 1996K 0.38 22.8 0.725 23.52 23.16 22.80 22.44 22.08 21.71 21.35 20.99 20.63 20.26 19.90 1995ba 0.388 22.65 0.720 23.37 23.01 22.65 22.29 21.93 21.57 21.21 20.85 20.49 20.13 19.77 1995aw 0.4 22.36 0.714 23.07 22.72 22.36 22.00 21.65 21.29 20.93 20.57 20.22 19.86 19.50 1997am 0.416 22.57 0.706 23.28 22.92 22.57 22.22 21.86 21.51 21.16 20.80 20.45 20.10 19.75 15 al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Riess et al (1998) et al (1998) p. 1020 Riess et al (1998) et al (1998) p. 1020 Riess et al (1998) et al (1998) p. 1020 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al 1994al 0.42 22.55 0.704 23.25 22.90 22.55 22.20 21.85 21.49 21.14 20.79 20.44 20.09 19.73 1994G 0.425 22.13 0.702 22.83 22.48 22.13 21.78 21.43 21.08 20.73 20.38 20.02 19.67 19.32 1996cn 0.43 23.13 0.699 23.83 23.48 23.13 22.78 22.43 22.08 21.73 21.38 21.03 20.68 20.33 1997Q 0.43 22.57 0.699 23.27 22.92 22.57 22.22 21.87 21.52 21.17 20.82 20.47 20.12 19.77 1996E 0.43 22.72 0.699 23.42 23.07 22.72 22.37 22.02 21.67 21.32 20.97 20.62 20.27 19.92 1996U 0.43 22.77 0.699 23.47 23.12 22.77 22.42 22.07 21.72 21.37 21.02 20.67 20.32 19.97 1997ce 0.44 22.83 0.694 23.52 23.18 22.83 22.48 22.14 21.79 21.44 21.09 20.75 20.40 20.05 1995az 0.45 22.51 0.690 23.20 22.85 22.51 22.17 21.82 21.48 21.13 20.79 20.44 20.10 19.75 1996cm 0.45 23.17 0.690 23.86 23.51 23.17 22.83 22.48 22.14 21.79 21.45 21.10 20.76 20.41 1997ai 0.45 22.83 0.690 23.52 23.17 22.83 22.49 22.14 21.80 21.45 21.11 20.76 20.42 20.07 1995aq 0.453 23.17 0.688 23.86 23.51 23.17 22.83 22.48 22.14 21.79 21.45 21.11 20.76 20.42 1992bi 0.458 23.11 0.686 23.80 23.45 23.11 22.77 22.42 22.08 21.74 21.40 21.05 20.71 20.37 1995ar 0.465 23.33 0.683 24.01 23.67 23.33 22.99 22.65 22.31 21.96 21.62 21.28 20.94 20.60 1997P 0.472 23.11 0.679 23.79 23.45 23.11 22.77 22.43 22.09 21.75 21.41 21.07 20.73 20.39 1995ay 0.48 22.96 0.676 23.64 23.30 22.96 22.62 22.28 21.95 21.61 21.27 20.93 20.60 20.26 16 (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 571 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Riess et al (1998) et al (1998) p. 1020 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Riess et al (1998) et al (1998) p. 1020 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 1995K 0.48 22.92 0.676 23.60 23.26 22.92 22.58 22.24 21.91 21.57 21.23 20.89 20.56 20.22 1996cg 0.49 23.1 0.671 23.77 23.44 23.10 22.76 22.43 22.09 21.76 21.42 21.09 20.75 20.42 1996ci 0.495 22.83 0.669 23.50 23.16 22.83 22.50 22.16 21.83 21.49 21.16 20.82 20.49 20.15 1195as 0.498 23.71 0.668 24.38 24.04 23.71 23.38 23.04 22.71 22.37 22.04 21.71 21.37 21.04 1997cj 0.5 23.29 0.667 23.96 23.62 23.29 22.96 22.62 22.29 21.96 21.62 21.29 20.96 20.62 1997H 0.526 23.15 0.655 23.81 23.48 23.15 22.82 22.49 22.17 21.84 21.51 21.18 20.86 20.53 1997L 0.55 23.51 0.645 24.16 23.83 23.51 23.19 22.86 22.54 22.22 21.90 21.57 21.25 20.93 1996cf 0.57 23.27 0.637 23.91 23.59 23.27 22.95 22.63 22.31 22.00 21.68 21.36 21.04 20.72 1996I 0.57 23.42 0.637 24.06 23.74 23.42 23.10 22.78 22.46 22.15 21.83 21.51 21.19 20.87 1997af 0.579 23.48 0.633 24.11 23.80 23.48 23.16 22.85 22.53 22.21 21.90 21.58 21.26 20.95 1997F 0.58 23.46 0.633 24.09 23.78 23.46 23.14 22.83 22.51 22.19 21.88 21.56 21.24 20.93 1997aj 0.581 23.09 0.633 23.72 23.41 23.09 22.77 22.46 22.14 21.82 21.51 21.19 20.88 20.56 1997K 0.592 24.42 0.628 25.05 24.73 24.42 24.11 23.79 23.48 23.16 22.85 22.54 22.22 21.91 1997S 0.612 23.69 0.620 24.31 24.00 23.69 23.38 23.07 22.76 22.45 22.14 21.83 21.52 21.21 17 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Riess et al (1998) et al (1998) p. 1020 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Perlmutter et al (1999) et al (1999) p. 570 Riess et al (1998) et al (1998) p. 1020 1995ax 0.615 23.19 0.619 23.81 23.50 23.19 22.88 22.57 22.26 21.95 21.64 21.33 21.02 20.71 1997J 0.619 23.8 0.618 24.42 24.11 23.80 23.49 23.18 22.87 22.56 22.26 21.95 21.64 21.33 1996H 0.62 23.31 0.617 23.93 23.62 23.31 23.00 22.69 22.38 22.08 21.77 21.46 21.15 20.84 1995at 0.655 23.27 0.604 23.87 23.57 23.27 22.97 22.67 22.36 22.06 21.76 21.46 21.16 20.85 1996ck 0.656 23.57 0.604 24.17 23.87 23.57 23.27 22.97 22.66 22.36 22.06 21.76 21.46 21.15 1997R 0.657 23.83 0.604 24.43 24.13 23.83 23.53 23.23 22.92 22.62 22.32 22.02 21.72 21.42 1997G 0.763 24.47 0.567 25.04 24.75 24.47 24.19 23.90 23.62 23.34 23.05 22.77 22.48 22.20 1996cl 0.828 24.65 0.547 25.20 24.92 24.65 24.38 24.10 23.83 23.56 23.28 23.01 22.74 22.46 1997ap 0.83 24.32 0.546 24.87 24.59 24.32 24.05 23.77 23.50 23.23 22.95 22.68 22.41 22.13 1997ck 0.97 24.78 0.508 25.29 25.03 24.78 24.53 24.27 24.02 23.76 23.51 23.26 23.00 22.75 Table 7: Data for FIG1 through FIG.3. Riess et al. (1998) and Perlmutter et al. (1999) APPENDIX A 1. More complete examination of possible mechanism of subtraction of 1/ (1+z) Perlmutter et al. (1999) used the same template (SCP1997) as described by Goldhaber et al. (2001), but used the SNe Ia light-curve width-luminosity relation in Equation 1. (8) mBeff  mB   corr  K BR  AR 18 The correction function corr , as seen in Equation 8 normalizes supernovae peak magnitudes. Thus, each SNeIa is corrected as if they each individually had the light-curve width of the template, where the stretch factor s=1. Note: Value of the scale factor  =1. (http://www.astro.ucla.edu/~wright/cosmo_02.htm) corr   ( s  1) (9) In order that the cancellation process of the cosmic time dilation factor can be followed, the subtraction of the 1 /(1+z) correction factor, which is actually done with a timedilated template, will instead be mathematically depicted in the following equations: Goldhaber et al. (2001) defined the time-axis width factor w . w  s(1  z ) (10) Therefore, the stretch factor s can be defined as the width factor w times 1/ (1+z). Note that Blondin et al. (2008) p.730 states that calculations are preferentially done in 1/(1+z) space rather than (1+z) space, because it avoids asymmetric errors. sw 1 (1  z ) (11) As seen previously, the stretch factor s is part of the correction function corr   ( s  1) (12) The new definition of s can be substituted into the correction function corr   ( w 1  1) (1  z ) (13) With cosmic time dilation factor present When the cosmic time dilation correction function, 1/ (1+z) is subtracted by the timelengthened template to form a rest-frame, in Perlmutter et al. (1999), it becomes part of the correction function. corr   ( w 1 1   1) (1  z ) (1  z ) (14) The two cosmic time dilation factors cancel themselves out, leaving only the expected equation defining the effective peak magnitude, seen with  =1 and w “corrected” as it would appear if the width of the template s=1 (Perlmutter et al. (1999). 19 (15) mBeff  mB  K BR  AR Without cosmic time dilation factor present: However, if cosmic time dilation isn’t present in the observer-frame data, then when the correction factor 1/ (1+z) is subtracted by the template, the cosmic time dilation factor (1+z) is not there to cancel it. Therefore, the uncancelled cosmic time dilation correction factor remains in the correction function as seen in Equation 9. corr   ( w  1  1) (1  z ) (16) Substituting the correction function into the effective peak magnitude equation, it now includes the uncancelled cosmic time dilation correction factor, seen when  and w =1. mBeff  mB  1  K BR  AR (1  z ) (17) APPENDIX B Because the 1/ (1+z) factor is non-linear, its subtraction is complex in that it causes both absolute and relative effects. The relative effects have been well defined. Fig.3 in the main paper demonstrated how subtracting the non-linear 1/ (1+z) correction factor to form a rest-frame, makes it appear that it increases the high redshift SNeIa peak magnitude, even though it actually lowers it. This is because the non-linear subtraction reduces the high z SNeIa peak magnitude less than it reduces the low SNeIa peak magnitude. Note that subtraction of the factor lowers the peak magnitude of both the high and low z SNe1a. It just lowers the low z SNe1a more. Fig.1 in this appendix has two curves: one with no subtracted 1/ (1+z) factors and the other with a total of five subtracted 1/ (1+z) factors. Together they show how the subtraction is misleading. As seen in the figure, the subtraction of 1/ (1+z) factors from Riess et al. (1998) and Perlmutter et al. (1999 data appears to increase the relative slope of the peak magnitude of the high redshift SNeIa but actually each subtraction lowers the total peak magnitude of both ranges, as would be expected by a subtraction. Only the high z SNE1a magnitudes appear to increase because the 1/ (1+z) factor is nonlinear and therefore subtracts more from the peak magnitude of the low z SNeIa than it does from the peak magnitude of the high z SNeIa. Note: Data for Fig.5 is available in Table 3 on page 9. FIG. 5 illustrates both relative and absolute effects of the subtraction of the non-linear factor using data seen in FIG 1 in the main paper. Note that the data in the lower curve has had four additional 1/ (1+z) factors (total of five) subtracted as compared to the 20 upper curve which has had one factor added to cancel the factor subtracted by Riess et al (1998) and Perlmutter et al (1999), which gave it zero factors. The comparison of these two curves in FIG.5 demonstrates both the absolute drop in value of MB and the relative increase in slope of the MB in the high z SNe1a. 30.00 25.00 MB 20.00 Zero Five 15.00 10.00 5.00 0.00 0.01 . 0.1 1 10 Redshift Fig.5: Comparison of the absolute decrease in peak magnitude plus increase in slope of the high z SNeIa in the Riess et al. (1998) and Perlmutter et al. (1999) data when zero and five 1/ (1+z) factors are subtracted APPENDIX C Table 8 gives the procedure of how we determined the graphs and slopes of the high and low ranges of SNeIa seen in the previous Table 6 in order to assist and facilitate rechecking our work. Step # 1 2 3 4 5 6 7 8 Step Copy contents of Table 9 to a Microsoft™ Excel ™ spreadsheet Referring to Table 5, remove all of the outliers listed. Open the chart function and then select the (xy) scatter option Select the “x” values as “z”, from column 3 in Table 9 starting on page 18. Select the “y” values from column 6 “zero” to column 15 “Five” depending on the number of 1/(1+z) correction factors you want to be in the data. Select “Format axis”, then select “Scale”. Check the box for “logarithmic scale” and set log range to 10. Once you have a finished graph, use the “Source data” option to reopen the data option. Referring to the range data in Table 6, restrict the range in the “y” values to fit the range you want, either high redshift SNeIa range or low redshift SNeIa range. 21 9 10 11 12 Add a trend line and select the option to display equations The numbers before the x in the equation are the slope. Repeat steps 7-10 for each range. Repeat steps 5-11 for each column. Table 8: Instructions to determine the slopes of the high and low ranges of SNeIa REFERENCES Blondin, S, et al, 2008,ApJ, 736, 682 Chang, H.-Y. 2001, ApJ, 557, L85 Deng, M., & Schaefer, B.E. 1998, ApJ, 502, L109 Foley, et al, 2005, ApJ, 626, L11 Goldhaber, G, et al, 2001,558, 359 Hawkins, M.R. S., 2010, Mon. Not. R. Astron. Soc. 000, Howell, D. A., et al, 2007, ApJ, 667, L37 Perlmutter, S, et al, 1999,ApJ, 517, 565 Riess, et al, 1997, arXiv:astro-ph/9707260v1 Riess, A, et al, 1998,AJ, 166, 1009 22