In a recent paper On the time dependency of inlineformulatexmath idTM0003 notationLaTeXa0texmathinlineformula the authors claim that they have tested one of the predictions of the Scale Invariant Vacuum SIV theory on MOND by studying the dependence of the Modified Newtonian Dynamics MOND acceleration at two data sets lowz inlineformulatexmath idTM0005 notationLaTeX3.2times 104le zle 3.2times 102texmathinlineformula and highz inlineformulatexmath idTM0007 notationLaTeX0.5le zle 2.5texmathinlineformula. They claim both samples show a dependency of inlineformulatexmath idTM0008 notationLaTeXa0texmathinlineformula from z. Here the work mentioned above is revisited. The explicit analytic expression for the zdependence of the inlineformulatexmath idTM0011 notationLaTeXa0texmathinlineformula within the SIV theory is given. Furthermore the first estimates of the inlineformulatexmath idTM0012 notationLaTeXOmega mtexmathinlineformula within SIV theory give inlineformulatexmath idTM0013 notationLaTeXOmega m0.28pm 0.04texmathinlineformula using the lowz data only while a value of inlineformulatexmath idTM0014 notationLaTeXOmega m0.055texmathinlineformula is obtained using both data sets. This much lower inlineformulatexmath idTM0015 notationLaTeXOmega mtexmathinlineformula leaves no room for nonbaryonic matter Unlike in the mentioned paper above the slope in the zdependence of inlineformulatexmath idTM0017 notationLaTeXA0log 10a0texmathinlineformula is estimated to be consistent with zero Zslope for the two data sets. Finally the statistics of the data are consistent with the SIV predictions in particular the possibility of change in the sign of the slopes for the two data sets is explainable within the SIV paradigm however the uncertainty in the data is too big for the clear demonstration of a zdependence yet.
['General Relativity and Quantum Cosmology', 'Astrophysics - Cosmology and Nongalactic Astrophysics']
Elucidating the zdependence of the MOND acceleration aSUB0SUB within the scale invariant vacuum SIV paradigm
{'Elucidating the z -dependence of the MOND acceleration ( a 0 ) within the Scale Invariant Vacuum (SIV) paradigm': 'Vesselin G. Gueorguiev 1 , 2 ⋆  \n- 1 Institute for Advanced Physical Studies, Sofia, Bulgaria\n- 2 Ronin Institute for Independent Scholarship, Montclair, NJ, USA  \nAccepted 2024 September 6. Received 2024 July 16; in original form 2024 April 17', 'ABSTRACT': "In a recent paper: 'On the time dependency of a 0 ' the authors claim that they have tested 'one of the predictions of the Scale Invariant Vacuum (SIV) theory on MOND' by studying the dependence of the Modified Newtonian Dynamics (MOND) acceleration at two data sets, lowz (3 . 2 × 10 -4 ≤ z ≤ 3 . 2 × 10 -2 ) and highz (0 . 5 ≤ z ≤ 2 . 5). They claim 'both samples show a dependency of a 0 from z '. Here, the work mentioned above is revisited. The explicit analytic expression for the z -dependence of the a 0 within the SIV theory is given. Furthermore, the first estimates of the Ω m within SIV theory give Ω m = 0 . 28 ± 0 . 04 using the low-z data only, while a value of Ω m = 0 . 055 is obtained using both data sets. This much lower Ω m leaves no room for non-baryonic matter! Unlike in the mentioned paper above, the slope in the z -dependence of A 0 = log 10 ( a 0 ) is estimated to be consistent with zero Z-slope for the two data sets. Finally, the statistics of the data are consistent with the SIV predictions; in particular, the possibility of change in the sign of the slopes for the two data sets is explainable within the SIV paradigm; however, the uncertainty in the data is too big for the clear demonstration of a z -dependence yet.  \nKey words: cosmology - dark matter - cosmological parameters - cosmology: theory", '1 INTRODUCTION': "Modern physics is well understood based on two main contemporary pillars: Einstein's General Relativity (EGR) and the Relativistic Quantum Field Theory. However, there are some perplexing observations about the motion of stars within galaxies and clusters. Within the popular current model of Cosmology and Astrophysics, the resolution of these perplexing phenomena is often associated with concepts such as Dark Matter and Dark Energy Bertone & Hooper (2018). However, for over 30 years, there has not been a definite detection of any new particles or fields. An alternative to the Dark Matter approach to resolving the observational discrepancies in galaxies and clusters of galaxies is the idea of the Modified Newtonian Dynamics (MOND, Milgrom (1983)) that has steadily gained support in the Astrophysics communities. While the concept of Dark Matter is a natural continuation of the matter paradigm into a non-luminous matter to explain the observational fact of the flat rotational curves, the MOND idea does not need extra matter 1 ; instead, it mod-  \n- ⋆ E-mail: Vesselin at MailAPS.org; Orcid ID: 0000-0002-20226432;\n- 1 Some fashionable models of gravity are trying to explain MOND and its fundamental acceleration a 0 (Pazy 2013; Verlinde 2017), but these models have failed the observational tests by Lelli et al. (2017), while the relativistic implementation of MOND suggests imperceptible variation of a 0 'to redshift unity or even beyond it' (Bekenstein & Sagi 2008).  \n©  \nifies the dynamics once the observed acceleration g = v 2 /r falls below the certain cut-off value a 0 ≫ g . In this deep MOND regime, one expects scale invariance to be present in the system under study (Milgrom 2009).  \nScale invariance is an old idea introduced by Weyl as early as 1918 Weyl (1922, 1970) as a gauge invariant gravity, where along with the metric tensor g µν there is a connexion vector κ µ controlling the length change dl = lκ µ dx µ , and a scalar field λ that describes the gauge freedom g µν → λ 2 g µν . The shortcomings of the original Weyl geometry pointed out by Einstein (1918) were addressed by the introduction of the Weyl Integrable Geometry (WIG) (Eddington 1923), where the connexion vector satisfies κ µ = -∂ µ ln λ . Consequently, Dirac (1973) and Canuto et al. (1977) have applied the idea to formulate scale invariant cosmology and tried to fix λ based on Dirac's Large Numbers Hypothesis (Dirac 1974). The recent reincarnation of the notion of scale invariance was introduced by Maeder (2017a,b), where the scalar field λ was fixed to be only time-dependent by the requirement of homogeneity and isotropy of space. In doing so, the specific functional form of λ ( t ) is determined by the requirement that the macroscopic vacuum must be scale invariant and thus introducing the Scale Invariant Vacuum (SIV) paradigm (Maeder & Gueorguiev 2023). This new approach has been explored only in the past few years by Gueorguiev & Maeder (2024) as a potential alternative to the standard cosmological model of dark energy plus cold dark matter paradigm (ΛCDM) Planck Collaboration: Ade et al. (2016). This new alterna-  \nTable 1. Results from simple statistical analysis for the two sets of redshifts; the first set of lowz data with 0 . 00032 ≤ z ≤ 0 . 032 (Marra et al. 2020) and the second highz data with 0 . 5 ≤ z ≤ 2 . 5 (Del Popolo & Chan 2024c).  \npossible connection to dark matter and dark energy (Maeder & Gueorguiev 2020a). It has been shown recently by Maeder (2023) that the MOND fundamental acceleration a 0 could be derived within the SIV-paradigm, and the result depends on the cosmological parameters such as Hubble constant H 0 and the total current mass fraction Ω m = ρ m /ρ c , where ρ c = 3 H 2 0 / (8 π G ) is the critical density.  \nBy taking this result at face value along with the epochdependent scale factor λ ( t ), it is natural to expect that the SIV-derived MOND acceleration a 0 may have an epochdependent value, just as it is the case for the mass content of the Universe ρ m , and the Hubble parameter as well ( H = ˙ a/a , where a ( t ) is the usual FLRW expansion factor).  \nIn this respect, the recent papers by Del Popolo & Chan (2024a,c) have initiated interesting research about testing the connection between MOND by Milgrom (1983) and its possible justification within the SIV paradigm by Maeder (2023). In doing so, they studied the z -dependence of a 0 using observational data but didn't derive the explicit z -dependence, nor did they discuss the relevant SIV model parameters for Ω m . As a new model different from ΛCDM, one should expect that some of the standard cosmological parameters may have different values within the SIV model. In this case, Ω m is a model parameter to be determined, while the Hubble constant H 0 is a model-constraining observational parameter.  \nIn what follows, I will present my analyses of the z -dependency of the MOND acceleration a 0 along with the specific z -dependent expression of a 0 within SIV. Furthermore, the results of the statistical analyses will be utilized to perform one of the first determinations of the SIV parameter Ω m representing the fraction of the total matter-energy content of the Universe. The results will illustrate a puzzling situation that needs a better understanding of the data or the model utilized.", '2 RESULTS FROM STATISTICS': 'Before going into more detail about the SIV theory, it is important to note that a simple statistical analysis of the two main variables A 0 = log 10 ( a 0 ) with a 0 in km/s 2 and Z = log 10 ( z ) based on data reported by Del Popolo & Chan (2024c) gives averaged values ¯ A 0 = -13 . 07 ± 0 . 06 and ¯ Z = -2 . 49 ± 0 . 04 (see Table 1) for the lowz data set from Marra et al. (2020).  \nFor notational and pragmatic reasons, I maintain the choice of the main variables to be the dimensionless A 0 and Z . Using, the log 10 on a 0 in km/s 2 keeps the corresponding range of A 0 between -15 and -10, while the log 10 on z is in the range -4 to 1. Since a 0 should be in km/s 2 for evaluating log 10 ( a 0 ) to compute A 0 , that is, A 0 = log 10 ( a 0 / (km/s 2 )) for  \nTable 2. Values of the Z-slopes and intercepts for the current work and the values deduced in ref. (Del Popolo & Chan 2024c).  \na 0 in arbitrary units, then the units of A 0 are dimensionless, and so are the corresponding average values ¯ A 0 .  \nThe highz data shown in Del Popolo & Chan (2024c) is based on the work done by Nestor Shachar et al. (2023). It contains only 17 highz Galaxies for which the inferred a 0 is less than 1 . 2 × 10 -13 km/s 2 ; that is, A 0 < -12 . 92. Such selection criteria cut a large data segment from the highz data, while it has not been applied to the lowz data, thus introducing a bias. In my opinion, a fair, unbiased, and appropriate data selection procedure, if any, should be applied to both sets. Here, I disagree with such a selection criteria applied to highz data and not to the lowz data, so I use all 100 data points in Nestor Shachar et al. (2023) as seen in Fig. 1, both data sets have a compatible spread of A 0 -values. The relevant values follow the calculation of a 0 using Eq. (5) in Del Popolo & Chan (2024c) based on the data from Nestor Shachar et al. (2023). In doing the calculations, an error was noticed in the initial evaluations by Del Popolo & Chan (2024a) due to units conversion; upon communicating with the authors, this was later recognized by Del Popolo & Chan (2024b) and corrected in the subsequent version of their paper Del Popolo & Chan (2024c); the proper conversion is necessary to obtain a 0 in km/s 2 for the correct evaluation of the corresponding A 0 . It is worth noticing that the corresponding simple statistical analysis gives ¯ A 0 = -12 . 60 ± 0 . 03 and ¯ Z = 0 . 17 ± 0 . 02 for all of the 100 highz data points (see Table 1). Thus, the Z -slope based on these two aggregated data sets is ∆ ¯ A 0 / ∆ ¯ Z = ( -12 . 6+13 . 07) / (0 . 17+2 . 49) ≈ 0 . 18 ± 0 . 03 indicating a change in the MOND acceleration as seen from Table 1. This is an overly simplified estimate of the Z -slop based on both sets, marked in Table 2 with an asterisk.  \nThe statistical analyses can be taken further, as done in the paper by Del Popolo & Chan (2024c), where they do a linear fit to the two sets and derive a Z -slope of 0 . 6 ± 0 . 2 with an intercept of A 0 = -11 . 6 ± 0 . 5 for the lowz data. Simple linear regression on the same data gives agreement with the zero slope since the result is a Z -slope of 0 . 12 ± 0 . 13 with a different intercept of A 0 = -12 . 8 ± 0 . 3. Such intercept should be regarded as related to Z = 0; thus, to the value of a 0 at z =1 and not at z = 0. Therefore, this corresponds to about 11% or only 2% change of A 0 from ¯ A 0 = -13 . 07 near z ≈ 0 to A 0 = -11 . 6 or A 0 = -12 . 8 near z ≈ 1.  \nRegarding the highz data, the Z -slope of -0 . 2 ± 0 . 4 is consistent with zero while the intercept of A 0 = -9 . 61 ± 0 . 08 for the highz data is questionable due to the applied data selection criteria and the possible error of their analyses mentioned earlier. The simple linear regression on the full data set derived from Nestor Shachar et al. (2023) gives again agreement with the zero Z -slope since the result is Z -slope is 0 . 01 ± 0 . 2 with an intercept of A 0 = -12 . 603 ± 0 . 05. Notice', 'MOND acceleration at varius redshift values': 'Figure 1. Lowz and highz data sets are discussed in the text. For this figure only, I have dropped the outlier related to the NGC2976 Galaxy data in the lowz data set since its value A 0 = -18 . 32 is too low compared to the other values shown; however, the data point has been used in the various data analyses presented. Unlike in Del Popolo & Chan (2024c), where only 17 highz data points have been used, here I have shown all 100 data points related to the data in Nestor Shachar et al. (2023). For the relevant error bars, the reader is referred to the paper by Del Popolo & Chan (2024c).  \n<!-- image -->  \nthat for the current analysis, the lowz and highz data sets have intercepts that agree with each other unlike those in Del Popolo & Chan (2024c). There must be an agreement between these two intercepts since they should reflect the value of the MOND acceleration at z = 1 .', '3 SIV FRAMEWORK': "Within SIV the fundamental MOND acceleration a 0 can be related to the Hubble constant H 0 and the current matter content of the Universe Ω m Maeder (2023); Maeder & Gueorguiev (2020b). For the derivations and formulas to be used in this section, I will denote the MOND fundamental acceleration a 0 by a M whenever appropriate to avoid confusion with the expansion scale factor a but will use a 0 in the absence of such a problem. This is done to avoid confusion with the FLRW expansion scale factor a , which by convention should be denoted by a 0 at the current epoch; and to also avoid awkward notation a 00 for the current value of the MOND acceleration.  \nWithin SIV, there is an extra velocity-dependent term, denoted as dynamical acceleration Maeder & Gueorguiev (2020a):  \nd 2 -→ r dt 2 = -G t M ( t ) r 2 -→ r r + κ ( t ) d -→ r dt , (1)  \nwhere κ = -˙ λ/λ is the time component of the SIV connexion vector, with a simple functional form κ = 1 /t within the SIV gauge, where the SIV cosmic time t is a dimensionless parameter such that t ∈ [ t in , t 0 = 1]. Here, t in is the moment of the Bing Bang when the FLRW scale factor satisfies  \na ( t in ) = 0, happening near t in = Ω 1 / 3 , while at the current epoch a ( t 0 ) = 1 and time is set so that t 0 = 1. Within SIV the conformal-scale factor is λ = 1 /t and is used to perform Weyl transformation g ' µν = λ 2 g µν that relates EGR metric g ' µν to the metric g µν within the WIG framework (Maeder & Gueorguiev 2020a).  \nTo arrive at an expression for the MOND acceleration a 0 , one considers the ratio of the magnitudes of the Newtonian acceleration g N = GM/r 2 to the additional acceleration κ ( t ) v in (1), where v denotes the magnitude of the velocity -→ v = d -→ r /dt :  \nx = κvr 2 GM . (2)  \nNow, one can use the relation given by the instantaneous radial acceleration v 2 /r = GM/r 2 to eliminate the speed v from the expression of x given by (2); then using g N = GM/r 2 to remove GM , one arrives at:  \nx = κvr 2 GM = κ √ r 3 GM = κ √ r g N . (3)  \nWhen the dynamic acceleration dominates over the Newtonian acceleration ( x ≫ 1), one has:  \ng = g N + xg N ≈ xg N = κ √ rg N . (4)  \nTherefore, one arrives at the MOND type relation g ∼ √ a 0 g N from which one can deduce an expression for a 0 :  \na 0 ≈ κ 2 r. (5)  \nThe above expression indicates a possible r dependence of a 0 , thus demoting a 0 of its fundamental parameter status within MOND, which may be testable with future high-precision  \ndata. Such dependence may explain the variance of a 0 . To restore the fundamental character of the above expression (5) as in MOND, one could consider the limit r → r H , where the Hubble radius reflects the influence of the Universe causally connected to the object studied. Thus, the time-dependent MOND acceleration within SIV is the upper bound of (5) given by the expression:  \nκ 2 r → κ 2 r H = κ 2 c/H = a M ( t ) , (6)  \nDuring the matter-dominated epoch, SIV has an analytic form for expansion scale-factor a ( t ) (Jesus 2017; Maeder & Gueorguiev 2020a):  \na ( t ) = ( t 3 -Ω m 1 -Ω m ) 2 / 3 ⇒ H = 2 t 2 t 3 -Ω m . (7)  \nThus, from (6) with SIV time in units t ∈ [ t in , t 0 = 1], one obtains for the MOND fundamental acceleration:  \na M ( t ) = c ( t 3 -Ω m ) 2 t 4 . (8)  \nTo express a M in the usual time units, where τ in = 0 at the Big Bang when the scale factor a = 0 while the age of the Universe now is τ 0 = 13 . 8 billion years, one has to use the chain rule for differentiation, that is:  \na M ( τ ) = c ( ˙ λ λ ) 2 a ˙ a = cκ ( τ ) κ ( t ) H ( t ) = ( dt dτ ) a M ( t ) . (9)  \nThe value of dt/dτ is assessed based on the assumption that the following relation provides a connection between the two time-scales ( t -t in ) / ( t 0 -t in ) = ( τ -τ in ) / ( τ 0 -τ in ) where t in = Ω 1 / 3 m , t 0 = 1 , τ in = 0 , and τ 0 = 13 . 8 billion years (Maeder & Gueorguiev 2020a). Thus, one has:  \ndt/dτ = (1 -t in ) /τ 0 , (10)  \nand therefore:  \na M ( τ ) = c ( 1 -Ω 1 / 3 m τ 0 ) ( t 3 -Ω m 2 t 4 ) =  \n= c 2 b ( 1 -b τ 0 )( x 3 -1 x 4 ) ,  \n(11) (12)  \nwhere x = t/b , Ω m = t 3 in = b 3 . Set ˜ a = a ( 1 -b 3 b 3 ) 2 / 3 and revisit (7) to get:  \na = ( b 3 1 -b 3 ) 2 / 3 ( x 3 -1 ) 2 / 3 ⇒ ˜ a = ( x 3 -1 ) 2 / 3 . (13)  \nThus, upon utilization of (12) and the above substitution, the MOND acceleration (9), as a function of the scale factor, becomes:  \na M ( τ ) = c 2 b ( 1 -b τ 0 ) ˜ a 3 / 2 (˜ a 3 / 2 +1) 4 / 3 . (14)  \nNext, look at the z -dependence, and use a = 1 / ( z +1); thus, when z is 0, or 1, and even 2, then a is 1, or 1 / 2, and correspondingly 1 / 3. Therefore, one has:  \na M ( z ) = a M ( z = 0) × ( 1 z +1 ) 3 / 2 ( ˜ a 3 / 2 0 +1 ) 4 / 3 ( ( 1 z +1 ) 3 / 2 ˜ a 3 / 2 0 +1 ) 4 / 3 (15)  \n= a M 0 × ( z +1) -3 / 2 ( (1 -Ω m ) ( z +1) 3 / 2 +Ω m ) -4 / 3 (16)  \nwhere ˜ a 0 = ( 1 -Ω m Ω m ) 2 / 3 was utilized. The above expression (16) provides the formula for the explicit z -dependence of the MOND fundamental acceleration within the SIV framework. It can be used to test this SIV prediction against the observational data. When evaluated at z = 1 and 2, one finds that a M ( z ) is about 79% and 58% of the current value a M 0 = a M ( z = 0) for Ω m = 0 . 3. Thus, the z -dependence of the MOND acceleration is weak, and it is likely buried within the scatter of the current observational data and its uncertainty (see Del Popolo & Chan (2024c)); as one can see later, the value of a M 0 , at best, is accurate within 13% while the 1 σ error of the data on A 0 reported by Del Popolo & Chan (2024c) usually translates in more than 29% uncertainty for observationally deduced MOND acceleration data points.  \nThe value of a M 0 could be used to assess the parameter Ω m within the SIV theory. That is, by using (12) one has:  \na M 0 = c (1 -b )(1 -b 3 ) 2 τ 0 = c (1 -Ω 1 / 3 m )(1 -Ω m ) 2 τ 0 . (17)  \nOne can solve for Ω m by taking log 10 of (17):  \nA 0 -log 10 ( c 2 τ 0 ) = log 10 [( 1 -Ω 1 / 3 m ) (1 -Ω m ) ] , (18)  \nfor lowz data with ¯ A 0 = -13 . 07 this gives Ω m = 0 . 28. Such a value of Ω m aligns with the comparative study of the SIV and ΛCDM cosmologies that demonstrated only light adjustments in Ω m if the expansion factors a ( t ) of these models are to be very similar Maeder (2017a). However, it contrasts with the MOND idea of dark matter redundancy. Notably, the range of values of A 0 based on (18) for Ω m given by 0 . 01 , 0 . 1 , 0 . 3 , 0 . 6 , 0 . 8 are -12 . 57 , -12 . 78 , -13 . 098 , -13 . 67 , -14 . 31.  \nBased on (17), the corresponding fractional uncertainties are related in the following way:  \n∆ a M 0 a M 0 = ∆Ω m Ω m   Ω m (1 -Ω m ) + 1 3 Ω 1 / 3 m ( 1 -Ω 1 / 3 m )   . (19)  \nThe slope of the z -dependence, m = d log 10 a M ( z ) dz when taking log 10 of (16), is then:  \nm = 1 -(1 + 3(1 + z ) 3 / 2 )Ω m (1 + z )(1 + ((1 + z ) 3 / 2 -1)Ω m ) ln(100) . (20)  \ngiving m 0 = (1 -4Ω m ) / ln(100) at z = 0, which is positive only for Ω m < 0 . 25. While for the Z -slope one has:  \ndA 0 dZ = ( dA 0 dz )( dz dZ ) = m ln(10) z = m ln(10)10 Z , (21)  \nwhere dz/dZ = (exp( Z ln(10))) ' is utilized. Based on the data provided: A 0 = log 10 a M 0 = -13 . 067 ± 0 . 056 for a M 0 in km/s 2 . The fractional uncertainty is ∆ A 0 / | A 0 | = 0 . 0043 = ∆( a M 0 ) / ( | A 0 | a M 0 ln(10)). That is, ∆( a M 0 ) /a M 0 = 0 . 13; therefore, using (19) one has Ω m = 0 . 28 ± 0 . 04. The z -slope is then m 0 = -0 . 01 ± 0 . 3, while the Z -slope is -0 . 0001 using Z = -2 . 49. This results practically in a horizontal line that changes very little from being -13 . 0667 at Z = -4 to -13 . 0671 at Z = 2. This change is well within the current error ( ± 0 . 06) for the lowz . For the highz data, one can now evaluate m 1 to be -0 . 11; therefore, the Z -slope is -0 . 25 in agreement with the Z -slope of -0 . 2 ± 0 . 4 reported by Del Popolo & Chan (2024c).", '4 DISCUSSION AND CONCLUSION': "It is still inconclusive about the z -dependence of the MOND fundamental acceleration, but such dependence is present within the SIV theory (16); furthermore, the SIV expression (20) does suggest that there is a change in the sign of the slope when going through z = (1 / 3 × (1 -Ω m ) / Ω m ) 2 / 3 -1 with m 0 positive for Ω m < 0 . 25 and negative otherwise; this could explain the change in the sign of the slope of the two data sets as noticed by Del Popolo & Chan (2024c). However, the corresponding value of A 0 at z = 1, which is the intercept, is about the same as ¯ A 0 based on low-z with ¯ A 0 = -13 . 07, but Del Popolo & Chan's highz intercept differs significantly from the corresponding ¯ A 0 values around A 0 = -12 . 6. However, if one is to embrace the matching values of A 0 at z = 1, that is, to use A 0 = -12 . 6 ± 0 . 05 along with (17) in (16), Then, the value of Ω m is significantly lower; that is, one gets Ω m = 0 . 055. In this case, the sign change of the slopes will be happening around z = 2 . 7. For Ω m = 0 . 055, equation (18) gives A 0 = -12 . 69, which is lower than A 0 = -12 . 6 but is bigger than A 0 = -13 . 07. Note that such a low value for Ω m does not leave much room for any dark matter. Such a result is better aligned with the MOND view about dark matter but seems to be a drastic departure from the need for dark matter and dark energy as required by ΛCDM. This approach may be favorable as a method of determining Ω m since it relies on the two data sets and their consistent Z -intercepts, which is in contrast to the first method presented that utilized only the lowz data and assumed that ¯ A 0 = -13 . 07 at ¯ Z = -2 . 49 is sufficiently close to z = 0 even though z → 0 would imply Z → -∞ . Therefore, further studies are needed to determine the correct Ω m values within the SIV framework. Thus, more precise data analyses are needed along with improved uncertainties of the observational data points (for the currently used data see Fig. 1) to confirm the z -dependence of the MOND fundamental acceleration and to potentially test the SIV theory via its model prediction for a 0 ( z ) as well as to deduce the relevant SIV model parameters.  \nIn conclusion, the long-standing mystery of galactic rotation curves has fueled the development of Modified Newtonian Dynamics (MOND). This work presents a significant contribution by providing the first explicit analytic expressions for the z-dependence of the fundamental MOND acceleration ( a 0 ) within the framework of the Scale Invariant Vacuum (SIV) theory (16). This novel approach goes beyond previous studies Del Popolo & Chan (2024a,c). Furthermore, we leverage existing observational data to perform the firstever estimation of the cosmological matter density parameter (Ω m ) within the SIV framework. The current analysis yields a value of Ω m = 0 . 28 ± 0 . 04 based on low-z data and Ω m = 0 . 055 based on the consistency of both data sets at z = 1, potentially removing the need for dark matter entirely. The above is a puzzling result as to why the two methods presented to determine the value of Ω m within SIV result in relevant values within the ΛCDM model.  \nOn the one hand, the SIV value 28% for Ω m deduced by using only the dataset with z ≈ 0 is close to the ΛCDM model of about 30% (Planck Collaboration: Ade et al. 2016), while on the other hand, the value 5 . 5% deduced by using both datasets (via the A 0 intercept at z = 1) is close to the baryon matter value within the ΛCDM model of about 5%. This could be just a numerical coincidence, or there may  \nbe some deeper reason for why the values are like that. For example, it may be related to the transition from a matterdominated epoch to a cosmological constant (dark-energy) dominated epoch within the ΛCDM model. However, within the SIV paradigm, one does not expect dark-matter and darkenergy components. For example, the energy density Ω Λ E due to the Einstein Cosmological Constant Λ E , which within the ΛCDM model is estimated to be 70%, does not exist within SIV but is replaced by Ω λ = -2 H ˙ λ λ that also compliments Ω m to 1 (assuming flat Universe Ω k = 0) within the SIV paradigm (Maeder & Gueorguiev 2020a).  \nInterestingly, the data suggests an almost flat z -dependence of A 0 = log 10 ( a 0 ), contrasting with previous claims by Del Popolo & Chan (2024a,c). While the current data limitations prevent the definitive confirmation of the z -dependence (16), the observed trends are consistent with SIV predictions. SIV offers a unique explanation for the potential sign change in the slopes previously indicated across different redshift ranges. Future higher precision data will be crucial for definitively resolving the presence or absence of z-dependence in a 0 .", 'ACKNOWLEDGMENTS': 'The author is grateful for the moral and financial support by particularly close private parties during the various stages of the research presented and to Prof. A. Maeder for the long and fruitful scientific collaboration over the years. This research does not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. However, the publishing charges were waived graciously by the Author Support, Oxford Journals, at Oxford University Press - for which the author is very grateful!', 'DATA AVAILABILITY': 'No new data was generated or analyzed in support of this research.', 'REFERENCES': "Bekenstein J. D., Sagi E., 2008, Phys. Rev. D, 77, 103512  \nBertone G., Hooper D., 2018, Reviews of Modern Physics, 90,  \n- 045002\n- Canuto V., Adams P. J., Hsieh S. H., Tsiang E., 1977, Phys. Rev. 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