We continue the presentation of an alternative cosmology based on conformal gravity, following our kinematical approach to the subject introduced in a recent paper. In line with the assumptions of our model, which proposes a closed-form expression for the cosmic scale factor 𝑅(𝑑), we revise the Hubble and deceleration parameters and introduce modified cosmological distances, analyzing in particular the case of the luminosity distance. Our kinematical conformal cosmology is then able to explain the anomalous acceleration of the Pioneer spacecraft, as due to a local region of gravitational blueshift. From the reported data of the Pioneer anomaly, we also compute the current value of our first fundamental parameter, 𝛾0=1.94Γ—10βˆ’28cmβˆ’1, in line with the original estimate by P. Mannheim of this quantity. Our second fundamental parameter, 𝛿0=3.83Γ—10βˆ’5, interpreted as the current value of a cosmological time variable, is derived from a detailed fitting of type Ia supernovae β€œgold-silver” data, producing Hubble plots of the same quality of those obtained by standard cosmology, but without requiring any dark matter or dark energy contribution. If further experiments will confirm the presence of an anomalous frequency blueshift in the outer region of the solar system, as described by our model, kinematical conformal cosmology might become a viable alternative to standard cosmological theories.

1. Introduction

This paper is the second part of a project aimed at introducing an alternative cosmology based on conformal gravity (CG), as originally proposed by Weyl [1–3] and recently revisited by Mannheim and Kazanas [4–6]. In the first paper on the subject [7] (paper I, in the following), we presented the mathematical foundations of our new kinematical approach to conformal cosmology. This was based on a critical reanalysis of fundamental astrophysical observations, starting with the cosmological redshift, and on the fact that modern metrology defines our common units of length and time using nongravitational physics, that is, through emission, propagation, and absorption of electromagnetic waves or similar phenomena.

Since the laws of electromagnetism are notoriously invariant under a conformal transformation, we argued that on a cosmological scale a conformal β€œstretching” of the space-time might be present and might yield to an effective change in wavelength or frequency of electromagnetic radiation, equivalent to the observed cosmological redshift (or blueshift, if any). As the origin of this presumed conformal stretching of the metric in the Universe can only be gravitational, we searched its connection with existing theories of gravity that allow this possible conformal symmetry. Our attention was focused on Weyl’s conformal gravity, since it is the simplest known conformal generalization of Einstein’s general relativity (GR). Weyl’s theory is also based on the same principles and assumptions of GR, such as the equivalence principle and other foundational concepts.

The complexity of CG, in particular its fourth-order field equations, as opposed to Einstein’s second-order equations, has rendered this theory quite intractable until Mannheim-Kazanas (MK) found the first complete solutions, such as the exterior solution for a static, spherically symmetric source [4, 5], and they also showed that it reduces to the classic Schwartzschild solution in the limit of no conformal stretching. In addition, the MK solution is able to interpolate smoothly between the classic Schwartzschild solution and the Robertson-Walker (RW) metric, through a series of coordinate transformations based on the conformal structure of the theory.

It was precisely this ability to transform from the conformal MK solution to the standard RW metric, used to describe the cosmological expansion, which convinced us that a universal conformal stretching might be able to mimic the expansion of the Universe. The gravitational origin of this conformal stretching should also lead to the change of observed wavelength frequency of cosmic radiation. The principles of general relativity, which still apply to its conformal extension, naturally propose such a mechanism: the gravitational redshift.

We have shown in our first paper how the original MK potential can support this explanation and how the chain of transformations, from static standard coordinates used in the MK solution to the Robertson Walker coordinates, can lead to a unique expression of the cosmic scale factor 𝑅. In this way, the conformal symmetry of the universe is β€œkinematically” broken, and the precise amount of stretching at each space-time point can be determined, once certain parameters of the original MK potential are measured.

In this second paper, we will show how we can determine these fundamental parameters using astrophysical data, such as the luminosities of type Ia supernovae (SNe Ia) and others. In this way, our kinematical conformal cosmology might become a viable alternative model for the description of the Universe, with the advantage of avoiding most of the controversial features of the standard model, such as dark energy, dark matter, and inflationary phases and so forth.

In the next section, we will review our cosmological solutions from paper I, then we will obtain expressions for the Hubble constant and deceleration parameter and also revise the definitions of the standard cosmological distances. In Section 3, we will fit current astrophysical data in order to compute our cosmological parameters and check the consistency of our model. Finally, in Section 4, we will explore the immediate consequences of our model, in terms of the behavior of fundamental constants and other physical quantities.

2. Kinematical Conformal Cosmology

2.1. Summary of Results from Paper I

In our first paper [7], we essentially worked with two sets of space-time coordinates. We started with static standard coordinates (SSC) (π‘Ÿ,𝑑,πœƒ,πœ™) which are used to express the MK solution for a static, spherically symmetric source, and then we have shown how, far away from massive sources, the MK metric can be transformed into the RW one, by employing a new set of space-time coordinates, denoted in bold type (𝐫,𝐭,πœƒ,πœ™), where the angular coordinates are not affected by the transformations. The cosmic scale factor can be introduced as a function of both time coordinates as βˆšπ‘(𝐭)=𝑅(𝑑)/|π‘˜|, where π‘˜ is a cosmological parameter, with dimensions of an inverse square length, originally introduced in the MK solution and whose value we will also determine in this work.

All the space-time coordinates can be turned into dimensionless quantities (𝐫 is already dimensionless) if we use the following definitions: 𝛼=2||π‘˜||ξ”π‘Ÿ,πœ’=||π‘˜||𝑐𝑑0ξ€Έ,π‘ξ€·π­βˆ’π‘‘πœ=0ξ€Έβˆ’π­π‘ξ€·π­0ξ€Έ,(1) where we use a look-back time (𝑑0βˆ’π‘‘) or (𝐭0βˆ’π­), since we usually observe radiation emitted in the past at coordinates (π‘Ÿ,𝑑) or (𝐫,𝐭), reaching us at the spatial origin and at our current time (π‘Ÿ=0,𝑑0) or (𝐫=𝟎,𝐭0).

In our first paper, we used the MK metric as a source of a cosmological gravitational redshift, associated with a redshift parameter 𝑧 and a cosmic scale factor 𝑅, using the SSC π‘Ÿ coordinate. By considering null geodesics and the other coordinate transformations detailed in [7], we were able to write the cosmic scale factor in any of the variables described above, obtaining the following expressions: 1+𝑧=𝑅(0)=1𝑅(π‘Ÿ)1+π›Ώπ›Όβˆ’4ξ€·1βˆ’π›Ώ2𝛼2ξ‚„βˆ’1/2,𝑅𝑑1+𝑧=0𝑅(𝑑)=coshπœ’βˆ’π›Ώsinhπœ’,1+𝑧=𝐑(𝟎)=βˆšπ‘(𝐫)1+𝐫2π‘ξ€·π­βˆ’π›Ώπ«,1+𝑧=0𝐑=ξƒ¬ξ‚€βˆš(𝐭)cos1βˆ’π›Ώ2πœξ‚+π›Ώβˆš1βˆ’π›Ώ2ξ‚€βˆšsin1βˆ’π›Ώ2πœξ‚ξƒ­βˆ’1.(2)

To avoid possible misunderstandings, the cosmic scale factor 𝑅 is considered a function of the time coordinate (𝑑 or the associated 𝐭) as in standard cosmology, but it is expressed also as a function of the radial coordinate π‘Ÿ (or 𝐫) simply because information from past times is brought to us by light emitted at those radial positions. We used in paper I a gravitational redshift mechanism, based on the static MK potential described in terms of π‘Ÿ, to explain the cosmological redshift, and therefore we have β€œimproperly” defined the scale factor as a function of radial coordinates, as shown in the previous equation.

The solutions in (2) were obtained for the particular case √𝐀=π‘˜/|π‘˜|=βˆ’1, which was found to be the only one associated with a possible redshift of gravitational origin (the other two cases 𝐀=0,+1 did not allow for the observed redshift). The detailed analysis of these solutions can be found again in our paper I. Here, we recall that the solutions in (2) are expressed in terms of another cosmological parameter 𝛿 defined as𝛾𝛿≑2||π‘˜||.(3) The additional quantity 𝛾 was also introduced in the original MK solution, and the main objective of this paper is to determine the values of these three cosmological parameters (𝛿,𝛾, and π‘˜) linked together by the previous equation.

Mannheim was able to fit galactic rotation curves without the need of dark matter and to estimate the current value of 𝛾 as 𝛾Mannheim=3.06Γ—10βˆ’30cmβˆ’1 [8]. In our first paper we argued that the current value of 𝛾 is probably close to Mannheim’s estimate but needs to be computed from more β€œlocal” observations. In addition, π‘˜ should have a negative value (since √𝐀=π‘˜/|π‘˜|=βˆ’1) while 𝛿 is necessarily limited by βˆ’1<𝛿<+1, so that its current value is probably small and positive (see again paper I for details).

Another hypothesis, introduced in our first paper, is to assume that 𝛿 (as well as 𝛾 and π‘˜) are probably time-varying quantities, over cosmological ages. In fact, we have proposed that the dimensionless parameter 𝛿 might constitute an effective cosmological time, varying from βˆ’1 to +1, so that (2) represents the evolution of the Universe as seen at our β€œcurrent time” 𝛿=𝛿(𝑑0). The most general description is obtained by letting 𝛿 vary in the allowed interval, in all the preceding formulas. If this interpretation is correct, it is possible to write the scale factor directly as a function of the variable 𝛿 and of its current value 𝛿(𝑑0) as follows: 𝑅𝛿𝑑1+𝑧=0𝑅=ξƒŽ(𝛿)1βˆ’π›Ώ2𝑑0ξ€Έ1βˆ’π›Ώ2,(4) a simple β€œsemicircular” evolution illustrated in Figure  2 of paper I. The complete connections between all these variables are also discussed in details and summarized in Table  1 of our first paper.

However, at this point, the possible time variation of our cosmological parameters is only a heuristic consideration which needs further investigation. The main parts of this paper (Sections 2 and 3) are devoted to the determination of these parameters regardless of their possible variation over cosmological times. Only in Section 4, we will briefly analyze the consequences of a possible nonconstancy of the parameters.

Therefore, the next step is to check our model against current astrophysical data, in order to establish it as a viable alternative to current cosmology. The cosmological parameters introduced above also need to be evaluated and connected to standard cosmological quantities such as the Hubble constant and the deceleration parameter.

2.2. The Hubble Constant and the Deceleration Parameter

One of the goals of standard cosmology is to determine, both theoretically and experimentally, the Hubble constant and the deceleration parameter which are essential to describe the evolution of the Universe. Since in our model the cosmic scale factor 𝑅 is determined explicitly by (2), it is not difficult to obtain these important parameters.

We recall that in general the Hubble parameter is defined as ̇𝐻(𝑑)=𝑅(𝑑)/𝑅(𝑑) and the deceleration parameter as Μˆπ‘ž(𝑑)=βˆ’π‘…(𝑑)/𝑅(𝑑)𝐻2ΜˆΜ‡π‘…(𝑑)=βˆ’π‘…(𝑑)𝑅(𝑑)/2(𝑑), with their current-time values denoted by 𝐻0 and π‘ž0. Standard cosmology measurements of the Hubble constant are usually reported as [9] 𝐻0=100β„Žkmsβˆ’1Mpcβˆ’1=3.24Γ—10βˆ’18β„Žsβˆ’1,(5) where β„Ž is a number between 0.5 and 1.

Following the model discussed in paper I and briefly reviewed above, we can write 𝐻(𝑑) and π‘ž(𝑑) by using our fundamental solutions, and we can also express these quantities in terms of either one of the two time coordinates 𝑑 or 𝐭, introduced previously. As already explained in Section  4.2 of [7], our preference goes to the simpler 𝑑 coordinate, which makes direct contact with our units of time, but we will also consider the other coordinate 𝐭 in the following.

We start by using the SSC time coordinate 𝑑, which is connected to the dimensionless look-back time βˆšπœ’=|π‘˜|𝑐(𝑑0βˆ’π‘‘) as in (1). In paper I, we have seen in (80)–(82) how to express the first, and second-order time derivatives of 𝑅 in terms of πœ’, or directly in terms of the redshift parameter 𝑧. As a consequence, we can easily write 𝐻(𝑑) and π‘ž(𝑑) also as a function of πœ’ or 𝑧, 𝐻(𝑑)=||π‘˜||𝑐sinhπœ’βˆ’π›Ώcoshπœ’ξ‚Άξ”coshπœ’βˆ’π›Ώsinhπœ’=Β±||π‘˜||𝑐(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ2ξ€Έ,ξ‚΅(1+𝑧)π‘ž(𝑑)=coshπœ’βˆ’π›Ώsinhπœ’ξ‚Άsinhπœ’βˆ’π›Ώcoshπœ’2=βˆ’2(1+𝑧)2(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ2ξ€Έβˆ’2,(6) where, as in the preceding equations, we use the β€œcurrent” value 𝛿=𝛿(𝑑0) (or the value at the time the observations were made). The current-time values of the Hubble and deceleration parameters are obtained in the limit for πœ’β†’0 or 𝑧→0, 𝐻𝑑0𝛾=βˆ’2𝛾𝑐,𝐻(𝑧=0)=Β±2π‘žξ€·π‘‘π‘,0ξ€Έ1=π‘ž(𝑧=0)=𝛿2βˆ’2.(7)

The signs of the quantities in (6) and (7) can be explained with the help of the red-solid curve in Figure  5 of paper I, which represents the ratio 𝑅(πœ’)/𝑅(πœ’0), or equivalently 𝑅(𝑑)/𝑅(𝑑0), over different cosmological epochs. This bell-shaped curve was plotted for a positive value 𝛿=𝛿(𝑑0) and shows a local blueshift area in the β€œpast” evolution of the Universe, extending back to a time 𝑑rs, followed by a redshift region which extends indefinitely to past times and which should represent the observed cosmological redshift from past cosmological epochs.

While we will explain the local blueshift region later in this paper (in Section 3.1), we simply remark here that this red-solid curve in Figure  5 of paper I is symmetric around the point of maximum. Therefore, for each value of 𝑧, that is, for each value of 𝑅(πœ’)/𝑅(πœ’0), we have two corresponding values of the Hubble parameter (except at the maximum, for 𝑧min=√1βˆ’π›Ώ2βˆ’1, where 𝐻=0), and the two related points on the curve, at the same redshift level, will have equal and opposite expansion rates. This yields the double sign in the previous expressions for 𝐻, when given as a function of 𝑧.

This argument applies also to the 𝑧=0 case, corresponding to the current time 𝑑0 at which 𝐻(𝑑0)=βˆ’(𝛾/2)𝑐 is negative, but also referring to a time in the past (𝑑rs) at which we start observing the cosmological redshift, with 𝐻(𝑑rs)=+(𝛾/2)𝑐, a positive quantity. This does not contradict the current estimates of 𝐻0 as a positive quantity. They are based on redshift observations of light coming from galaxies at times in the past 𝑑≲𝑑rs; therefore, what we call 𝐻0 in standard cosmology should be actually indicated as 𝐻(𝑑rs)=+(𝛾/2)𝑐, again a positive quantity.

In Section 3.1, we will evaluate the current value of the gamma parameter, by analyzing the local blueshift in the region of our solar system, corresponding to a negative 𝐻(𝑑0)=βˆ’(𝛾/2)𝑐. In this way, we will be able to estimate the gamma parameter as 𝛾(𝑑0)β‰…1.94Γ—10βˆ’28cmβˆ’1, and this will allow us to evaluate also the Hubble parameter at the beginning of the redshift region (time 𝑑rs), by using the same value of 𝛾 and the positive sign in (7), or equivalently assuming by symmetry 𝛾(𝑑rs)β‰…βˆ’1.94Γ—10βˆ’28cmβˆ’1 and using 𝐻(𝑑rs)=βˆ’(𝛾(𝑑rs)/2)𝑐. Numerically, we estimate 𝐻𝑑0𝛾𝑑=βˆ’0ξ€Έ2π‘β‰…βˆ’2.91Γ—10βˆ’18sβˆ’1,𝐻𝑑rs𝛾𝑑=+0ξ€Έ2𝛾𝑑𝑐=βˆ’rsξ€Έ2𝑐≅+2.91Γ—10βˆ’18sβˆ’1,(8) where again the details of this analysis will be presented later in this paper.

The connection with the standard cosmology value of 𝐻0 in (5) is immediate; all the standard astrophysical observations which led to the existing estimates of 𝐻0 were done by observing celestial objects in the redshift region, therefore for 𝑑≲𝑑rs or π‘Ÿβ‰³π‘Ÿrs, where π‘Ÿrs is the distance at which we start observing the cosmological redshift. Therefore, 𝐻0𝑑=𝐻rs𝛾𝑑=+0ξ€Έ2𝑐≅+2.91Γ—10βˆ’18sβˆ’1=100β„Žrskmsβˆ’1Mpcβˆ’1,β„Žrsβ‰…0.897,(9) in line with the current estimates of 𝐻0 [9]. Our value, β„Žrsβ‰…0.897, is the direct estimate of the β„Ž parameter based on the Pioneer anomaly data (see Section 3.1) and is close to recent determinations by the Wilkinson Microwave Anisotropy Probe (WMAP) [10] β„ŽWMAPβ‰…0.73Β±0.03, by the Hubble Space Telescope Key Project (HST Key Project) [11] β„ŽHSTβ‰…0.72Β±0.08, and others [12].

The deceleration parameter at current time 𝑑0 or at time 𝑑rs (in both cases 𝑧=0) is given by (7) as a function of the dimensionless 𝛿, which cannot be estimated from the Pioneer data. We cannot use results for π‘ž or similar acceleration parameters coming from standard cosmology, such as those obtained from type Ia supernovae analysis, as they are based on totally different assumptions. We will have to analyze and reinterpret the concepts of standard candle, luminosity distance, and so forth, before we can estimate 𝛿(𝑑0) and therefore π‘ž(𝑑0). This will be done in the following sections.

Before we proceed in this direction, we also study the expressions for the Hubble and deceleration parameters which can be obtained by using other variables. For example, we can recalculate our expressions using the RW time variable 𝐭, that is, define the Hubble parameter as ̇𝐇(𝐭)=𝐑(𝐭)/𝐑(𝐭) and the deceleration parameter, ̈πͺ(𝐭)=βˆ’π‘(𝐭)/𝐑(𝐭)𝐇2ΜˆΜ‡π‘(𝐭)=βˆ’π‘(𝐭)𝐑(𝐭)/2(𝐭), where the boldface symbols denote quantities related to the RW metric, as discussed in the previous section and in paper I. It is straightforward to obtain expressions similar to those in (6), π‘βˆšπ‡(𝐭)=βˆ’1βˆ’π›Ώ2𝐑𝐭0ξ€ΈΓ—βŽ‘βŽ’βŽ’βŽ£ξ‚€βˆšπ›Ώcos1βˆ’π›Ώ2πœξ‚βˆ’βˆš1βˆ’π›Ώ2ξ‚€βˆšsin1βˆ’π›Ώ2πœξ‚βˆš1βˆ’π›Ώ2ξ‚€βˆšcos1βˆ’π›Ώ2πœξ‚ξ‚€βˆš+𝛿sin1βˆ’π›Ώ2πœξ‚βŽ€βŽ₯βŽ₯βŽ¦π‘=±𝐑𝐭0(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ2ξ€Έ,πͺ⎑⎒⎒⎣√(𝐭)=1βˆ’π›Ώ2ξ‚€βˆšcos1βˆ’π›Ώ2πœξ‚ξ‚€βˆš+𝛿sin1βˆ’π›Ώ2πœξ‚ξ‚€βˆšπ›Ώcos1βˆ’π›Ώ2πœξ‚βˆ’βˆš1βˆ’π›Ώ2ξ‚€βˆšsin1βˆ’π›Ώ2πœξ‚βŽ€βŽ₯βŽ₯⎦2=ξ€·1βˆ’π›Ώ2ξ€Έ(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ2ξ€Έ,(10) where again 𝛿=𝛿(𝑑0). We use here the dimensionless variable 𝜁=𝑐(𝐭0βˆ’π­)/𝐑(𝐭0), or we express the Hubble and deceleration parameters directly in terms of 𝑧.

The current-time values of these two parameters are obtained in the limit for πœβ†’0 or 𝑧→0, respectively,𝐇𝐭0𝑐=βˆ’π‘ξ€·π­0𝑐𝛿,𝐇(𝑧=0)=±𝐑𝐭0ξ€Έπͺ𝐭𝛿,0ξ€Έ1=πͺ(𝑧=0)=𝛿2βˆ’1,(11) with the same interpretation which was given for (7). These expressions for the Hubble and deceleration parameters in the two temporal variables are obviously connected. It follows from the definitions that 𝐇(𝐭)=𝐻(𝑑)(𝑑𝑑/𝑑𝐭) and πͺ(𝐭)=π‘ž(𝑑)βˆ’(𝑑2𝑑/𝑑𝐭2)/𝐻(𝑑)(𝑑𝑑/𝑑𝐭)2, with the derivatives between time variables given by𝑑𝑑=βˆšπ‘‘π­1βˆ’π›Ώ2||π‘˜||𝐑𝐭0ξ€ΈΓ—1ξ‚ƒβˆš1βˆ’π›Ώ2ξ‚€βˆšcos1βˆ’π›Ώ2πœξ‚ξ‚€βˆš+𝛿sin1βˆ’π›Ώ2𝜁=(1+𝑧)||π‘˜||𝐑𝐭0ξ€Έ,𝑑2𝑑𝑑𝐭2=𝑐1βˆ’π›Ώ2||π‘˜||𝐑2𝐭0ξ€ΈΓ—ξ‚ƒξ‚€βˆšπ›Ώcos1βˆ’π›Ώ2πœξ‚βˆ’βˆš1βˆ’π›Ώ2ξ‚€βˆšsin1βˆ’π›Ώ2πœξ‚ξ‚„ξ‚ƒβˆš1βˆ’π›Ώ2ξ‚€βˆšcos1βˆ’π›Ώ2πœξ‚ξ‚€βˆš+𝛿sin1βˆ’π›Ώ2πœξ‚ξ‚„2=βˆ“π‘(1+𝑧)||π‘˜||𝐑2𝐭0(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ2ξ€Έ,(12) so that the connecting formulas can be easily derived.

Similarly, we could also write Hubble and deceleration parameters in terms of space variables π‘Ÿ, 𝛼, or 𝐫, since we have in (2) the cosmic scale factor 𝑅 expressed in all these variables, but these expressions would not be very useful because the experimental values of 𝐻0 and π‘ž0 are usually referred only to the temporal variables. We will return in Section 3.1 on the connection with experimental observations.

Finally, from (4), it is possible to introduce 𝐻 and π‘ž directly as a function of our cosmological time 𝛿,1𝐻(𝛿)=𝑅(𝛿)𝑑𝑅𝛿𝑑𝛿=βˆ’1βˆ’π›Ώ2,ξ€·π‘‘π‘ž(𝛿)=βˆ’2𝑅/𝑑𝛿2𝑅(𝛿)(𝑑𝑅/𝑑𝛿)2=1𝛿2,(13) and the expressions in (6) and (10) will reduce to those in (13), using the transformations outlined in Table  1 of paper I.

2.3. Luminosity Distance and Other Cosmological Distances

Before we make contact with experimental data, especially with standard candle measurements, we need to review the definitions of the cosmological distances, following the new principles of our kinematical conformal cosmology as outlined above and in paper I.

Several distances are usually introduced in standard cosmology and used in conjunction with astronomical observations in order to establish the β€œcosmological distance ladder,” that is, the different steps employed to measure astronomical distances and the size of the Universe (for an extensive introduction to the subject, see [13, 14]). This process started historically at the time of Greek astronomy with the determination of the size and radius of our planet and with the first approximate measurements of the astronomical unit and other distances within the solar system.

These estimates, based mainly on parallax methods, were later refined by modern astronomers and extended to parallax measurements of nearby stars. These geometrical methods led to the introduction of distances such as the parallax distance 𝑑𝑃, whose definition is not affected by our alternative approach to cosmology. As defined in Weinberg’s books [15, 16],π‘‘π‘ƒβ‰‘π‘πœƒξ€·π­=𝐑0ξ€Έπ«βˆš1βˆ’π€π«2𝐭=𝐑0ξ€Έπ«βˆš1+𝐫2,(14) where 𝑏 is the impact parameter of light reaching the observer from a (parallax) distance 𝑑𝑃 with parallax angle πœƒ, and we have used our preferred value 𝐀=βˆ’1 in the right-hand side of the last equation (see [9, 14, 15] or [16], for full details on all these distances).

Two more fundamental distances are usually introduced in cosmology, the comoving distance 𝑑𝐢 (sometimes also called coordinate or effective distance) and the proper distance 𝑑prop: 𝑑𝐢𝐭≑𝐑0𝑑𝐫,propξ€œβ‰‘π‘(𝐭)π«πŸŽπ‘‘π«ξ…žβˆš1βˆ’π€π«ξ…ž2⎧βŽͺ⎨βŽͺ⎩⎫βŽͺ⎬βŽͺ⎭,=𝐑(𝐭)πœ’=𝐑(𝐭)arcsin𝐫;𝐀=+1𝐫;𝐀=0arcsinh𝐫;𝐀=βˆ’1,(15) where 𝑑𝐢 usually refers to the current-time expansion factor 𝐑(𝐭0), while 𝑑prop can refer to any cosmological time 𝐭 being considered. These definitions are also unchanged in our cosmology and can be rewritten in terms of other variables using the transformations in paper I.

Continuing our brief summary of the cosmological distance ladder, modern parallax techniques can determine star distances up to about 100parsec (1pc=3.086Γ—1018cm), and these distance estimates are usually combined with the measured apparent star luminosities, to obtain their corresponding absolute luminosities. For this purpose, the inverse square law is typically assumed, introducing the luminosity distance 𝑑𝐿 and connecting it to the apparent (𝑙) and absolute bolometric luminosity (𝐿) of a light source as follows: 𝐿𝑙=4πœ‹π‘‘2𝐿,𝑑𝐿≑𝐿=𝐑4πœ‹π‘™2𝐭0𝐭𝐑(𝐭)𝐫=(1+𝑧)𝐑0𝐫.(16)

This definition follows from the consideration that β€œIn a Euclidean space the apparent luminosity of a source at rest at distance 𝑑 would be 𝐿/4πœ‹π‘‘2,…” (see again Weinberg [15, Section  14.4]). In the second line of the previous equation, the factor 𝐑(𝐭0)/𝐑(𝐭)=(1+𝑧) appears due to the standard redshift interpretation, namely, that photons of energy β„Žπœˆ are redshifted to energy β„Žπœˆπ‘(𝐭)/𝐑(𝐭0) and that the time interval of photon emission 𝛿𝑑 is also changed to 𝛿𝑑𝐑(𝐭0)/𝐑(𝐭). The total power emitted (energy per unit time) is therefore redshifted by a combined factor 𝐑2(𝐭)/𝐑2(𝐭0)=(1+𝑧)βˆ’2 which will enter the denominator of the square root of (16), thus resulting in the final (1+𝑧) factor in the equation (see [15] for full details). In other words, the standard redshift effect is assumed to alter the apparent luminosity of a standard candle placed far away, so that the (1+𝑧) factor corrects for this effect, while the absolute luminosity 𝐿 of our candle is considered fixed and constant. We will need to change this view in our alternative interpretation.

Before we consider our revised luminosity distance, we recall that astronomical luminosity measurements are usually expressed in terms of magnitudes (π‘š) using Pogson’s law, π‘š1βˆ’π‘š2=βˆ’2.5log10(𝑙1/𝑙2), for any two apparent luminosities. Traditionally, the absolute bolometric luminosity 𝐿 of a standard candle is defined as the apparent bolometric luminosity of the same object placed at a reference distance of 10pc, so that the distance modulus πœ‡ (difference between the apparent and absolute bolometric magnitudes) will result as follows: πœ‡β‰‘π‘šβˆ’π‘€=5log10𝑑𝐿pcβˆ’5=5log10𝑑𝐿Mpc+25.(17)

In standard cosmology, the luminosity distance on the right-hand side of the last equation is then expressed as a function of 𝑧 and other cosmological parameters, such as the density parameters Ω𝑀 and ΩΛ, that is, 𝑑𝐿=𝑑𝐿(𝑧;Ω𝑀,ΩΛ). The comparison with experimental observations is usually carried out by fitting the luminosity distance expression to the measured distance moduli (πœ‡) for several astrophysical light sources, which can be considered standard candles, that is, assumed having constant intrinsic luminosity. This method has proven to be particularly reliable when applied to type Ia supernovae and will be analyzed in detail in the following sections.

However, our view of the luminosity distance is different from the one outlined above. First of all, the cosmological redshift is related to the intrinsic stretching of the space-time fabric at cosmological distances and over cosmological times. This is realized through the gravitational redshift mechanism described in (2) and not anymore through a Doppler-like shift in photons energy or change in their emission frequency. In this view, the correcting factor (1+𝑧) inserted in (16) and described above is no longer necessary; the photons are emitted at the source with the precise frequency, energy, and rate of emission measured by the Earth’s observer (although these differ from the same quantities measured by an observer near the source).

Therefore, we correct the standard definition of the luminosity distance in (16) by removing the (1+𝑧) factor𝑑𝐿≑𝐿𝑧𝐭4πœ‹π‘™=𝐑0𝐫𝐭=𝐑0𝛿(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ2ξ€Έξ€·1βˆ’π›Ώ2ξ€Έ,(18) where we have inserted, on the right-hand side of the equation, our expression for the coordinate 𝐫, introduced in (76) of our paper I, as a function of our current value 𝛿=𝛿(𝑑0). We also choose the positive sign in front of the square root to select the solution corresponding to past redshift, 𝑧>0 for 𝐫>𝐫rs=2𝛿/(1βˆ’π›Ώ2), which is the correct choice for the subsequent analysis (see [7] for details).

In the previous equation, we have also indicated a dependence on 𝑧 of the absolute luminosity 𝐿𝑧 of the standard candle being considered (indicated by the subscript 𝑧). This is the second fundamental difference in our analysis of the luminosity distance. We have based our discussion in paper I on the hypothesis that the fundamental units of measure, such as the meter, the second, or others, depend on the space-time position in the Universe and differ from the same units of measure at another space-time location. Consequently, we have to assume that the same hypothesis applies to the luminosity of a β€œstandard candle,” that is, we cannot assume anymore that 𝐿 is an invariant quantity, when observing these candles placed at cosmological distances.

In our first paper, we also postulated that space or time intervals are β€œdilated” by the (1+𝑧) factor, when referring to a cosmological location characterized by redshift 𝑧, but we did not introduce any similar dependence for the third fundamental mechanical quantity, that is, mass. In fact, we have no a priori indication of how masses should scale due to our space-time stretching; therefore, we summarize the scaling properties of the three fundamental quantities as follows: 𝛿𝑙𝑧=(1+𝑧)𝛿𝑙0,𝛿𝑑𝑧=(1+𝑧)𝛿𝑑0,π›Ώπ‘šπ‘§=𝑓(1+𝑧)π›Ώπ‘š0.(19)

The first two lines in this equation simply rewrite (37) of our first paper in a simplified notation. Quantities with the zero subscript represent units or intervals (of space, time, or mass) as measured by an observer at his/her location and time (typically at the origin of space and at time 𝑑0 as usual). Quantities with the 𝑧 subscript represent the same units or intervals as β€œseen” by the observer located at the origin, but when these β€œobjects” are placed at a cosmological location characterized by redshift 𝑧. (We have avoided so far this subscript notation (also carefully avoided in paper I) because it is easily confused with the standard cosmology notation, where the zero subscript normally indicates the observed (redshifted) quantity, as opposed to the nonredshifted quantity (usually with no subscript). On the contrary, in our new notation, the β€œredshifted” quantity π‘ž acquires a 𝑧 subscript (π‘žπ‘§β‰‘π‘ž(𝑑), observed at the origin, but with information coming from past time 𝑑), while the β€œnonredshifted” quantity acquires the zero subscript (π‘ž0β‰‘π‘ž(𝑑0), observed at the origin, at current time 𝑑0). Since this subscript notation is much more compact than our previous one, we will adopt it for the rest of this paper. For instance, we will write the current values of our cosmological parameters as 𝛿0≑𝛿(𝑑0), 𝛾0≑𝛾(𝑑0), etc.) Space-time intervals are dilated by the (1+𝑧) factor, simply due to our new interpretation of the cosmological redshift.

In the equation for mass (third line of (19)), we have left the dependence on 𝑧 totally undetermined, assuming that some function 𝑓(1+𝑧) will connect units of mass at different locations in the Universe. We only suppose that 𝑓 will be a function of the redshift factor (1+𝑧), such that for 𝑧=0, it reduces to unity, that is, 𝑓(1)=1, leaving masses unchanged at locations where 𝑧=0. We will determine this function 𝑓 in the following.

The scaling of the fundamental mechanical quantities in (19) will be reflected in similar properties of any other mechanical quantity or unit. We consider, for example, the case of energy, as the luminosity discussed above is just energy emitted per unit time. Since 1erg=1gcm2sβˆ’2, an amount of energy can be written as π›ΏπΈβˆΌπ›Ώπ‘šπ›Ώπ‘™2π›Ώπ‘‘βˆ’2, and it is immediate to check that energies will scale like masses, following the last equation. We can therefore write 𝛿𝐸𝑧=𝑓(1+𝑧)𝛿𝐸0,(20) which implies that the total energy emitted by a standard candle placed at redshift 𝑧 would be measured by the observer at the origin to be different from the total energy emitted by the same candle when placed at the origin. Since these energies are emitted during some (finite) intervals of time, we can write 𝛿𝐸𝑧=βˆ«πΏπ‘§(𝑑𝑧)𝑑𝑑𝑧=𝑓(1+𝑧)𝛿𝐸0∫𝐿=𝑓(1+𝑧)0(𝑑0)𝑑𝑑0, where the luminosities are also labeled like all the other quantities and refer to times connected by the same dilation factor 𝑑𝑧=𝑑0(1+𝑧). (In the standard interpretation, 𝑑0 would be considered as the time 𝑑rest in the candle's restframe of reference and 𝑑𝑧 as the time 𝑑obs measured by the observer who sees the candle β€œmoving” due to the expansion of the Universe. Standard relativistic time dilation would yield 𝑑obs=𝑑rest(1+𝑧). This time dilation effect, which has been observed in the evolution of type Ia SNe [17–20], is also present in our theory although its interpretation is the one given by our fundamental (19).) Since the infinitesimal time intervals are similarly related, 𝑑𝑑𝑧=𝑑𝑑0(1+𝑧), the two luminosities are connected in the following way: 𝐿𝑧𝑑𝑧=𝑓(1+𝑧)𝐿(1+𝑧)0𝑑0ξ€Έ.(21)

Our new definition of luminosity distance will therefore generalize the original definition in (16), by using the actual luminosity 𝐿𝑧, instead of 𝐿0, to correct for the intrinsic changes in the candle’s energy output. To obtain the full expression of this distance, we use (18) and (21) together 𝑑𝐿=𝐿𝑧=ξƒŽ4πœ‹π‘™π‘“(1+𝑧)(𝐿1+𝑧)04πœ‹π‘™=𝐑0𝐫=𝐑0𝛿0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€Έξ€·1βˆ’π›Ώ20ξ€Έ,(22) where we also used our simplified notation for the parameters 𝛿0≑𝛿(𝑑0), 𝐑0≑𝐑(𝐭0), as previously discussed.

Comparing our new expression with the original one in (16), we note that basically we replaced the (1+𝑧) factor with a more complex factor √(1+𝑧)/𝑓(1+𝑧), where again the function 𝑓(1+𝑧) is still undetermined at this point. This is because the standard definition would use the β€œconstant” luminosity 𝐿0, as 𝑑𝐿=√𝐿0√/4πœ‹π‘™=(1+𝑧)/𝑓(1+𝑧)𝐑0𝐫, instead of the β€œvariable” 𝐿𝑧. One could argue that if the 𝑓 function was to be equal to (1+𝑧)βˆ’1, that is, if masses were to scale like (1+𝑧)βˆ’1, we would recover the original definition, but this is not exactly the case. Again, the original definition (16) assumes an invariant value 𝐿=𝐿0 of the standard candle’s luminosity at all cosmological locations, while our new definition (22) is based on the choice of 𝐿𝑧 as the standard candle’s luminosity, and in general this quantity is not invariant anymore but changes according to (21).

In other words, we could have defined instead𝑑𝐿=𝐿0=ξƒŽ4πœ‹π‘™(1+𝑧)𝐑𝑓(1+𝑧)0𝐫=ξƒŽ(1+z)𝐑𝑓(1+𝑧)0𝛿0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€Έξ€·1βˆ’π›Ώ20ξ€Έ,(23) using 𝐿0 instead of 𝐿𝑧, and then supplementing this definition with the information of (21). We will see that this will not affect the subsequent discussion on type Ia supernovae. In any case, we prefer our definition in (22) as it corrects the distance estimates, due to the variability of the candle’s luminosity. For example, if a candle is intrinsically dimmed when placed far away, its distance should be smaller than the one estimated with the original definition, given the same value 𝑙 for the observed apparent luminosity. Our new definition (22) precisely incorporates such corrections and will imply a revision of the current distance estimates based on luminosity measurements.

It will be useful in the following to consider also the spectral energy distribution, 𝐹(πœ†)≑𝑑𝐿/π‘‘πœ†. Following (21), we can write βˆ«πΉπ‘§(πœ†π‘§,𝑑𝑧)π‘‘πœ†π‘§βˆ«πΉ=(𝑓(1+𝑧)/(1+𝑧))0(πœ†0,𝑑0)π‘‘πœ†0, where the meaning of the subscripts is the same as in the preceding equations. The wavelength variables and related differentials are connected as usual, πœ†π‘§=πœ†0(1+𝑧) and π‘‘πœ†π‘§=π‘‘πœ†0(1+𝑧), so that the direct relation between 𝐹𝑧 and 𝐹0 is πΉπ‘§ξ€·πœ†π‘§,𝑑𝑧=𝑓(1+𝑧)(1+𝑧)2𝐹0ξ€·πœ†0,𝑑0ξ€Έ.(24)

After discussing at length the modifications to the luminosity distance, we return for completeness to the other definitions of cosmological distances. Two other distances are usually introduced, the angular diameter distance 𝑑𝐴, when an extended light source of true proper diameter 𝐷 is placed at a (angular) distance 𝑑𝐴 and observed having an angular diameter πœ—, and the proper-motion distance 𝑑𝑀, when proper motions in the direction transverse to the line of sight are considered. Their standard definitions are [15] π‘‘π΄β‰‘π·πœ—π‘=𝐑(𝐭)𝐫=0𝐫,𝑑(1+𝑧)π‘€β‰‘π‘‰βŸ‚Ξ”πœ—/Δ𝑑obs=𝐑0𝐫,(25) where, in the second definition, π‘‰βŸ‚ is the true velocity of the source in the direction perpendicular to the line of sight, and Ξ”πœ— is the change in the (angular) position of the object during the time interval of observation Δ𝑑obs.

Both definitions need to be reconsidered in our new interpretation. In the first definition, the β€œtrue” diameter of the light source will depend on 𝑧 as for all the other lengths, 𝐷𝑧=(1+𝑧)𝐷0, but using the RW metric, the proper distance across the source is 𝐷𝑧=𝐑(𝐭)π«πœ— for small angular diameters πœ—β‰ͺ1, which leads essentially to the same expression as in the standard treatment, just with 𝐷 replaced by 𝐷𝑧. Therefore, our definition of 𝑑𝐴 is similar to the standard one, but needs to be supplemented by the scaling equation of the source diameter π‘‘π΄β‰‘π·π‘§πœ—π‘=𝐑(𝐭)𝐫=0𝐫,𝐷(1+𝑧)𝑧=(1+𝑧)𝐷0.(26) In this way, the right-hand side of the equation connecting 𝑑𝐴 to the cosmological quantities is unaffected, but when we make contact with observations, that is, we use the left-hand side of the equation, the change of the true diameter with 𝑧 must be included. Since 𝐷𝑧 increases with (positive) 𝑧, using the old definition, which assumes a fixed 𝐷=𝐷0, would result in underestimating the diameter distances by a factor (1+𝑧).

Similar care has to be taken in revising the proper motion distance 𝑑𝑀. A moving source will have a transverse velocity π‘‰βŸ‚ which is unaffected by our (19), since it is a ratio between quantities which scale in the same way. In the standard theory, the time interval of observation Δ𝑑obs is thought to be different from the time interval of motion Δ𝑑mot due to the usual redshift factor, Δ𝑑mot=Δ𝑑obs/(1+𝑧). However, in our view, the observed time Δ𝑑obs is precisely the true time interval during which the object moved at a redshift 𝑧: Δ𝑑obs=Δ𝑑𝑧. The proper distance travelled is Δ𝐷𝑧=π‘‰βŸ‚Ξ”π‘‘π‘§=π‘‰βŸ‚Ξ”π‘‘obs=𝐑(𝐭)π«Ξ”πœ—, using the RW metric as in the treatment of the angular distance above, and our revised expression of the proper motion distance isπ‘‘π‘€β‰‘π‘‰βŸ‚Ξ”πœ—/Δ𝑑obs𝐑=𝐑(𝐭)𝐫=0𝐫(1+𝑧).(27)

In this case, we had to modify the right-hand side of the equation (compared to the standard definition), but the left-hand side is unaffected (in particular, Δ𝑑obs=Δ𝑑𝑧 is still simply the observation time interval). Finally, we note that our corrected expressions for 𝑑𝐴 and 𝑑𝑀 are essentially the same (both lead to the expression 𝐑(𝐭)𝐫) since the geometry is completely equivalent (𝐷𝑧 is equivalent to π‘‰βŸ‚Ξ”π‘‘obs, πœ— to Ξ”πœ—).

Equations (14), (15), (22), (26), and (27) are our revised expressions for the classic distances used in cosmology. As in the case of their standard counterparts, they are all approximately equal to each other for 𝑧β‰ͺ1 and 𝐫β‰ͺ1, that is, 𝑑𝐢≃𝑑prop(𝐭0)≃𝑑𝑃≃𝑑𝐿≃𝑑𝐴≃𝑑𝑀≃𝐑0𝐫. We also note that three of the six definitions of distance were modified by our new kinematical conformal cosmology, thus potentially affecting current astronomical distance estimates.

One final consideration is needed regarding the SSC space coordinate π‘Ÿ. This dimensionful coordinate can be used to measure distances in the static standard coordinates, which are in general different from the cosmological distances described above. However, for small distances, or for 𝑧β‰ͺ1, this coordinate π‘Ÿ should also reduce to 𝐑0𝐫 as for all the other distances. We recall from paper I that the coordinate transformation between π‘Ÿ and 𝐫 is 1π‘Ÿ=||π‘˜||𝐫√1+𝐫2βˆ’π›Ώ0π«βŸΆπ«ξ”||π‘˜||𝐭≃𝐑0𝐫,(28) where the limit in the last expression is for 𝐫β‰ͺ𝟎 and 𝛿0≃0. The previous equation implies that the current value of the scale factor is simply related to the parameter π‘˜ (if the current value of 𝛿0 is close to zero),𝐑0𝐭=𝐑0≃1||π‘˜0||,(29) where, from now on, we will also denote with a zero subscript (π‘˜0) the current value of the π‘˜ parameter. Equation (27) is particularly important to simplify the connection between the Hubble constants 𝐇 and 𝐻, in view of (6)–(12). It is easy to check that, using the previous equation, the general connection simplifies to 𝐇(𝐭)≃𝐻(𝑑)(1+𝑧),(30) so that for 𝑧=0 the two Hubble constants basically coincide, that is, 𝐇0≃𝐻0. Again, these results are valid only for 𝛿0≃0, but we will show in the next section that this is actually the case of interest.

It is beyond the scope of this paper to attempt a full revision of the β€œcosmological distance ladder,” in view of our changes in the distance definitions. While we leave this analysis to future work on the subject, we will concentrate our efforts in the next section on applying our revised definition of the luminosity distance to type Ia supernovae.

3. Cosmological Parameters

The central part of our analysis will deal with the evaluation of the cosmological parameters in our model: the dimensionless 𝛿 parameter, the 𝛾 and π‘˜ parameters (or the original πœ… quantity, see paper I [7]) all measured with reference to the current time 𝑑0 if these quantities are considered time-varying parameters. (We have already remarked in Section 2.1 that the determination of these parameters is independent of their possible time variation, that is, the results in the following subsections (Sections 3.1, 3.2, and 3.3) are valid also in the case of no time variation of the parameters. In this case, 𝛿, 𝛾, and π‘˜ should be considered simply as universal constants.)

We have already introduced 𝛾0=𝛾(𝑑0)β‰…1.94Γ—10βˆ’28cmβˆ’1 in Section 2.2, but we still have to show how this value was obtained and compare it to the original evaluation of 𝛾 done by Mannheim. We will proceed first to estimate 𝛾0 in the next subsection, later we will obtain 𝛿0=𝛿(𝑑0) from supernovae data, and finally, all the other parameters will be derived from these two quantities.

3.1. Cosmological Blueshift and the Pioneer Anomaly

In Section 2, we have summarized all the fundamental expressions of our cosmology and outlined the reasons why we consider the 𝐀=βˆ’1 solution as a possible description of the evolution of the Universe. Although this solution can explain the observed cosmological redshift, it has an additional new feature. It requires the existence of a blueshift region in the immediate vicinity of our current space-time position in the Universe.

This could be a serious setback for our model, since we normally do not observe blueshift of nearby astrophysical objects except for the one caused by the peculiar velocities of nearby galaxies, which is presumably due to standard Doppler shift. However, experimental evidence has recently accumulated regarding a possible local region of blueshift, related to the so-called Pioneer anomaly [21–31].

This is a small anomalous frequency drift, actually a blueshift, which was observed analyzing the navigational data from the Pioneer 10/11 spacecraft, received from distances between 20and70 astronomical units from the sun, while exploring the outer solar system. This β€œblueshifted” frequency is uniformly changing at a rate of β‹…πœˆπ‘ƒβ‰…6Γ—10βˆ’9Hz/s and can be interpreted as a constant sunward acceleration, reported as π‘Žπ‘ƒ=βˆ’(8.74Β±1.33)Γ—10βˆ’8cm/s2 (radial inward acceleration) or as a β€œclock acceleration” π‘Žπ‘‘β‰‘π‘Žπ‘ƒ/𝑐=βˆ’(2.92Β±0.44)Γ—10βˆ’18s/s2, resulting in a frequency drift of about 1.5 Hz every 8 years [22].

Preliminary findings indicate the possibility of detecting such anomaly also in the radiometric data from other spacecraft traveling at the outskirts of the solar system, such as the Galileo and Ulysses missions [22]. These discoveries have prompted an extended reanalysis of all the historical navigational data from these space missions, which is currently underway [25–28, 32], in order to determine additional characteristics of the anomaly, such as its precise direction, the possible temporal evolution, and its dependence on heliocentric distance. Future-dedicated missions are also being planned [33–35] to test directly this puzzling phenomenon.

Meanwhile, the origin and nature of this anomaly remains unexplained; all possible sources of systematic errors have been considered [22–24, 27, 28, 36], but they cannot account for the observed effect. Possible physical origins of the signal were studied, including dark matter, modified gravity, or other nonconventional theories, but no satisfactory explanation has been found so far (see details in [22, 29, 30, 32]). On the contrary, we can analyze the Pioneer anomaly with the model outlined in the previous sections and use the data reported above to estimate the local values of our cosmological parameters.

The phenomenology of the Pioneer anomaly is related to the exchange of radiometric data between the tracking station on Earth (Deep Space Networkβ€”DSN) and the spacecraft, using S-band Doppler frequencies (in the range 1.55–5.20 GHz). More precisely, an uplink signal is sent from the DSN to the spacecraft at 2.11 GHz, based on a very stable hydrogen maser system, the Pioneer then returns a downlink signal at a slightly different frequency of about 2.29 GHz, to avoid interference with the uplink signal. This is accomplished by an S-band transponder which applies an exact and fixed turn-around ratio of 240/221 to the uplink signal.

This procedure, known as a two-way Doppler coherent signal, allows for very precise tracking of the spacecraft since the returning signal is compared to the original one, as opposed to a one-way Doppler signal (fixed signal source on spacecraft, whose frequency cannot be directly monitored for accuracy). This type of tracking system and the navigational capabilities of the Pioneer spaceship (spin-stabilized spacecraft and power source of special design) allowed for a great acceleration sensitivity of about 10βˆ’8cm/s2, once the influence of solar-radiation-pressure acceleration decreased below comparable levels (for distances beyond about 20AU from the Sun).

After a time delay of a few minutes or hours, depending on the distance involved, the DSN station acquires the downlink signal, and any difference from its expected frequency is interpreted as a Doppler shift due to the actual motion of the spacecraft. Modern-day deep space navigational software can also predict with exceptional precision the expected frequency of the signal returned from the Pioneer, which should coincide with the one observed on Earth. On the contrary, a discrepancy was found, corresponding to the values indicated above, whose origin cannot be traced to any systematic effect due to either the performance of the spacecraft or the theoretical modeling of its navigation.

Moreover, the signal analysis performed so far [23, 26] indicates an almost constant value of the anomalous acceleration or frequency shift reported above (temporal and space variation of π‘Žπ‘ƒ within 10%), over a range of heliocentric distances ~20–70 AU, and possibly at even closer distances ≲10 AU.

The Pioneer phenomenology corresponds exactly to the simplest experiment we might conceive to check the validity of our model. In principle, it would be sufficient to set up a lab experiment in which we emit some radiation of known wavelength πœ†(𝑑0) at time 𝑑0, keep this radiation from being absorbed for a long enough time, and then compare its wavelength πœ†(𝑑0) with the radiation emitted by the same source at a later time πœ†(𝑑1), with 𝑑1>𝑑0. In our model, we would expect the two wavelengths to be different and, if we are already in a phase of universal contraction as illustrated by the red-solid curve of Figure  5 in our paper I [7], we would have πœ†(𝑑0)<πœ†(𝑑1). In terms of frequencies, 𝜈(𝑑0)>𝜈(𝑑1), that is, the radiation from time 𝑑0 would appear to be blueshifted, compared to the same radiation emitted by the same source at a later time 𝑑1. We will now proceed to interpret the Pioneer anomaly in a similar way, but we will return in Section 4 on the feasibility of detecting wavelength variation in lab experiments.

It is useful to introduce a new time coordinate 𝑑: let 𝑑=0 be the time at which the radiometric signal is sent from Earth to the spacecraft, which is then received at time 𝑑 and immediately retransmitted down to the DSN, arriving back on Earth at time 2𝑑. In the standard analysis, the model for the signal coming back to Earth is based on the relativistic Doppler effect. We will denote as 𝜈mod the frequency of the expected signal following this model. Its ratio with the reference frequency 𝜈ref of the signal (the uplink frequency of about 2.11 GHz, since we consider the two-way system) is therefore given by the standard relativistic Doppler formula (see (2.2.2) in [15]) 𝜈mod𝜈ref=√1βˆ’π―2/𝑐21+π‘£π‘Ÿπ‘£/𝑐≃1βˆ’π‘Ÿπ‘,(31) where π‘£π‘Ÿ is the source radial velocity, and the approximation on the right-hand side holds to first order in π‘£π‘Ÿ/𝑐.

For the case of the Pioneer spacecraft, we use π‘£π‘Ÿ=2𝑣mod(𝑑), where 𝑣mod(𝑑) is the expected velocity of the spacecraft, according to the theoretical navigation model, at time 𝑑, when the spaceship receives and immediately retransmits the signal. The factor of two arises from the double Doppler shift involved (two-way system). With this radial velocity, (31) to first order in π‘£π‘Ÿ/𝑐 becomes𝜈modξ€·π‘‘ξ€Έβ‰ƒπœˆref1βˆ’2𝑣mod𝑑𝑐,(32) and this frequency should be observed with high precision, due to the reported excellent navigational control of the spacecraft.

The anomaly comes from observing a different frequency 𝜈obs(𝑑)>𝜈mod(𝑑), involving an additional unexplained blueshift. Over the range of the observed anomaly, the frequency difference is reported asΞ”πœˆ=𝜈obsξ€·π‘‘ξ€Έβˆ’πœˆmodξ€·π‘‘ξ€Έβ‰ƒΜ‡πœˆπ‘ƒ2𝑑,Μ‡πœˆπ‘ƒβ‰…6Γ—10βˆ’9Hz/s.(33) We point out here that the cited references adopt a rather confusing β€œDSN sign convention” for the frequency difference in (33) (see [23, 26] and reference [36] of [21]), resulting in a sign change in most of their equations. We prefer to use here our definition of Ξ”πœˆ as given in the previous equation.

An alternative way of reporting the anomaly is the following. As in (32), we can also write the observed frequency to first order in π‘£π‘Ÿ/𝑐 as𝜈obsξ€·π‘‘ξ€Έβ‰ƒπœˆref1βˆ’2𝑣obs𝑑𝑐,(34) where the β€œobserved” velocity of the spacecraft always refers to the time of interest 𝑑. Combining together these last three equations, we write the frequency difference asπœˆΞ”πœˆ=βˆ’2ref𝑐𝑣obsξ€·π‘‘ξ€Έβˆ’π‘£mod𝑑.(35) We then multiply and divide the last equation by 𝑑, so that we can introduce π‘Žπ‘ƒ=[𝑣obs(𝑑)βˆ’π‘£mod(𝑑)]/𝑑, the residual Pioneer acceleration of unknown origin. This is the change of velocity over time 𝑑 and not over the double time 2𝑑 as in (33). With this residual acceleration, the Pioneer anomaly is usually reported asπœˆΞ”πœˆ=βˆ’2refπ‘π‘Žπ‘ƒπ‘Žπ‘‘,𝑃=βˆ’(8.74Β±1.33)Γ—10βˆ’8cms2,(36) where the negative sign of the residual acceleration indicates its sunward direction; therefore, we have a positive frequency shift corresponding to a local unexplained blueshift of radiation emitted by the spacecraft.

Given our discussion in the preceding sections, we can now explain (36) or (33) with our new interpretation. The Pioneer data represent the equivalent of a local measurement of the current blueshift predicted in Section 2.1; therefore, they can be used to find the value of the parameter 𝛾 at the time when the Pioneer radiation was emitted (a few years ago), but this can be considered to be the current value, due to the slow variation of the cosmological parameters.

Let us consider the current time 𝑑0 as the time of arrival of the Pioneer radiation on Earth. The uplink signal was therefore sent at 𝑑=𝑑0βˆ’2𝑑 and retransmitted by the spacecraft, as a downlink signal, at 𝑑=𝑑0βˆ’π‘‘. We need to re-interpret (32) and (34) as, in our view, the anomaly is not due to a change in velocity of the spacecraft (we assume 𝑣mod(𝑑)=𝑣obs(𝑑)=𝑣(𝑑)), but just related to a shift of the reference frequency. The reference frequency in (34) is the one emitted in the past 𝜈ref=𝜈(𝑑=𝑑0βˆ’2𝑑), while the reference frequency in (32) is the one at current time 𝜈ref=𝜈(𝑑=𝑑0), so these two equations are modified, respectively, as follows𝜈obsξ€·π‘‘ξ€Έξ€·β‰ƒπœˆπ‘‘=𝑑0βˆ’2𝑑1βˆ’2𝑣𝑑𝑐,𝜈modξ€·π‘‘ξ€Έξ€·π‘‘β‰ƒπœˆ01βˆ’2𝑣𝑑𝑐.(37)

Following (33), the frequency difference, to first order in 𝑣/𝑐, is nowξ€Ίξ€·π‘‘Ξ”πœˆ=𝜈(𝑑)βˆ’πœˆ01βˆ’2𝑣𝑑𝑐𝑅𝑑=𝜈(𝑑)1βˆ’0𝑅(𝑑)1βˆ’2π‘£π‘‘ξ€Έπ‘ξƒ­β‰ƒΜ‡πœˆπ‘ƒξ€·π‘‘0ξ€Έ,βˆ’π‘‘(38) having used our fundamental (37) from paper I [7] and also (𝑑0βˆ’π‘‘)=2𝑑. We can simplify the above equation by introducing additional approximations. From the second line of (2), we have 1βˆ’π‘…(𝑑0√)/𝑅(𝑑)=1βˆ’{cosh[|π‘˜|𝑐(𝑑0βˆšβˆ’π‘‘)]βˆ’(𝛾/2√|π‘˜|)sinh[|π‘˜|𝑐(𝑑0βˆ’π‘‘)]}β‰…(𝛾/2)𝑐(𝑑0βˆ’π‘‘), since we can assume √|π‘˜|𝑐(𝑑0βˆ’t)β‰ͺ1 (The time delay for a two-way Pioneer signal at the maximum distance of 70AU is about 20 hours; we can assume √|π‘˜|βˆΌπ›ΎβˆΌ10βˆ’28βˆ’10βˆ’30cmβˆ’1, therefore √|π‘˜|𝑐(𝑑0βˆ’π‘‘)∼10βˆ’13βˆ’10βˆ’15β‰ͺ1.). The reference frequency is 𝜈ref=𝜈(𝑑)β‰…πœˆ(𝑑0)=2.11GHz, corresponding to the original uplink frequency (In some of the references cited, the reference frequency 𝜈ref is taken as the downlink frequency of 2.29GHz, since the downlink signal is compared directly with this value. We prefer to use the uplink value, since this is the frequency used at the original time 𝑑=𝑑0βˆ’2𝑑.), and we can also neglect the ratio 2𝑣(𝑑)/𝑐β‰ͺ1, since the typical Pioneer speed in the outer solar system was π‘£β‰ˆ12km/s [23]. With these approximations, the last equation becomesΞ”πœˆβ‰…πœˆref𝛾2𝑐𝑑0ξ€Έξ‚„βˆ’π‘‘β‰…Μ‡πœˆπ‘ƒξ€·π‘‘0ξ€Έβˆ’π‘‘.(39)

The clock acceleration π‘Žπ‘‘, the Pioneer acceleration π‘Žπ‘ƒ, the frequency shift Μ‡πœˆπ‘ƒ, and the reference frequency 𝜈ref are all related together byπ‘Žπ‘‘=π‘Žπ‘ƒπ‘=βˆ’Μ‡πœˆπ‘ƒπœˆref,(40) which follows from (33) and (36). Combining these last two equations and using the reported values of the Pioneer anomaly, we can finally obtain our estimate of the current local value of the cosmological parameter 𝛾,𝛾0𝑑=𝛾0ξ€Έβ‰…2π‘Μ‡πœˆπ‘ƒπœˆref2=βˆ’π‘2π‘Žπ‘ƒ2=βˆ’π‘π‘Žπ‘‘=1.94Γ—10βˆ’28cmβˆ’1,(41) which represents the best estimate of 𝛾 at our current time (although the Pioneer data are a few years old). (A more complete discussion of the derivation of our cosmological parameters, based only on the Pioneer anomaly, can be found in our latest work [37].) This is the value which was quoted in Section 2.2 and led to our evaluation of the Hubble constant. (Since the discovery of the Pioneer anomaly, many researchers have noticed the numerical β€œcoincidence” between the Hubble constant and the value of the Pioneer acceleration π‘Žπ‘ƒ divided by 𝑐 and proposed many different explanations for this. This coincidence is even more striking if one uses the value cited in [23] as the experimental value for Pioneer 10 data before systematics, π‘Žπ‘ƒ=βˆ’7.84Γ—10βˆ’8cmsβˆ’2, thus obtaining 𝐻0=80.7kmsβˆ’1Mpcβˆ’1 and 𝛾0=1.74Γ—10βˆ’28cmβˆ’1. Although this choice would result in a value of the Hubble constant closer to standard cosmology evaluations, we prefer to base all our analysis on the usually quoted value of π‘Žπ‘ƒ as in (36).)

In the previous equation, we connected 𝛾 to the measured Pioneer acceleration π‘Žπ‘ƒ (or the clock acceleration π‘Žπ‘‘) simply because such was the way these data were reported in the literature cited. However, it should be clear from the discussion above that we explain the Pioneer anomaly in terms of our cosmological-gravitational blueshift (equivalent in a way to the clock acceleration mentioned above). In this view, there is no real dynamic acceleration of the Pioneer spacecraft (or of any other object in the solar system) oriented toward the sun, due to some new gravitational force or modification of existing gravity. In fact, π‘Žπ‘ƒ=[𝑣obs(𝑑)βˆ’π‘£mod(𝑑)]/𝑑=0 in our analysis, since we assume there is no difference between the two velocities 𝑣obs(𝑑) and 𝑣mod(𝑑).

In this way, we overcome the objection, reported, for example, in [23], that β€œthe anomalous acceleration is too large to have gone undetected in planetary orbits, particularly for Earth and Mars,” since β€œNASA’s Viking mission provided radio-ranging measurements [38] to an accuracy of about 12m,” which should have shown the effect of the anomalous acceleration on the orbits of these two planets.

In fact, there cannot be any anomalous acceleration for the Earth or Mars if we measure distances with round-trip time of flight of radio signals transmitted from Earth to the Viking spacecraft on the Mars surface [39]. On the contrary, we would observe a similar effect for Earth, Mars, or any other object in the solar system, if we were to study its motion through Doppler frequency ranging, because of the intrinsic differences in frequency or wavelength for light emitted at different space-time positions, due to our new cosmological effects.

We will return on this difference between experimental techniques for estimating distances in a later section. We conclude this section by noting that if the Pioneer anomaly is indeed caused by our new cosmological effects and not by modification of the gravity from the sun, the β€œdirection” of the anomalous β€œacceleration” should be pointing towards the terrestrial observer and not towards the sun. (Since our cosmology is based on (2), which was derived in paper I for regions in space far away from mass sources, i.e., independent of the location and size of massive objects in the Universe, the anomalous acceleration does not need to point in the direction of a massive source, such as the Sun, the center of our galaxy, or others. The effect should be seen in the same way from every point of observation in the Universe, and the observed anomalous acceleration would seem to be pointing simply towards the point of observation (Earth, in this case).) This is currently being studied (see [26] for details) by using data from a period of time when the spacecraft was much closer to the Earth and the sun, so that a clear direction of the anomaly can be determined, but no results from this new analysis are available yet.

3.2. Kinematical Conformal Cosmology and Type Ia Supernovae

The determination of 𝛿0 can be done with the powerful astrophysical tool represented by type Ia supernovae (SNe Ia) used as standard candles (see β€œThe High-Z SN Search” website [40], or the β€œHigh Redshift Supernova Search Supernova Cosmology Project” web page [41] for an introduction to the topic, see also [42–44] for reviews on the subject). Since the original discoveries made by these two leading groups [45, 46], the observational data were recently expanded to the so-called β€œgold” and β€œsilver” sets ([47, 48] and references therein) including the highest redshift supernovae known, at 𝑧≳1.25. From these data, we will determine the value of 𝛿0 and then obtain all the other parameters.

The use of type Ia SNe as standard candles involves the revision of the concept of luminosity distance and related quantities, in view of our alternative cosmology. The observational techniques must also be carefully considered, since they are based on the current standard cosmology. We have seen in the preceding section how to generalize the luminosity distance in line with our new interpretation; we will now apply those concepts to the case of type Ia SNe.

Equations (19)–(24) have shown that quantities such as the absolute luminosity of standard candles and others depend on their cosmological location, simply described by the redshift parameter 𝑧. We want to stress once more that in our new interpretation all the characteristic quantities of emission and absorption of radiation, such as frequencies, wavelengths, time intervals between radiative events, and energies, are intrinsically dependent on the space-time location of the events being observed. No receding motion is needed, and no cosmological Doppler shift is inducing redshifts in the observed spectra, but care is to be used to evaluate the different quantities at the space-time location of interest, and the proper corrections will enter the formulas only when we compare the same quantity at different cosmological locations.

For example, in our view, the β€œabsolute” luminosity of a standard candle at position 𝐫 emitting light at a past time 𝐭 would be measured differently by two observers, one placed at the origin 𝐫=0 observing the radiation at current time 𝐭0 and the other at the source 𝐫 observing the radiation at time 𝐭. (The absolute luminosity is the total power emitted by the source. Obviously only apparent luminosities, i.e., power per unit area, can be measured by observers far away from the source. The absolute luminosity 𝐿𝑧 therefore refers to an ideal measurement, as if we could measure all the energy flowing through a spherical surface of radius 𝐫, centered around the source.) Following our notation mentioned above, the former observer would measure the luminosity 𝐿𝑧, which was used in the previously mentioned equations to derive the luminosity distance. The latter, observing the radiation at the source, would measure what we denote by 𝐿0, that is, the equivalent of placing the standard candle near the origin and observing it at our current time. The two luminosities 𝐿𝑧 and 𝐿0 are then related by (21).

To further clarify this issue with an example, a standard candle of β€œintrinsic” total luminosity 𝐿0=1erg/s and β€œintrinsic” spectral color, for example, a β€œblue” candle, will always have the same luminosity and always show its blue color when seen by an observer placed near the candle, employing the appropriate local units of space, time, energy, and so forth. On the contrary, when placed at some cosmological distance corresponding to some definite 𝑧 value (say 𝑧=1, e.g.), it would be observed as a β€œred” candle and having a different luminosity 𝐿𝑧=𝑓(1+𝑧)𝐿0/(1+𝑧) following (21), using units proper to the observer at position 𝐫=0 and time 𝐭0. All these effects are just due to the different intrinsic units used by the two observers, as related to (19), and are not in any way connected to relativistic Doppler shifts, which might be only additional corrections due to the peculiar motion of the source relative to the observer.

The only missing piece in our luminosity definition is the explicit expression of the function 𝑓(1+𝑧). We might expect this function to be related to the usual (1+𝑧) factor as discussed before, but we have no reason to assume a simple dependence such as the one followed by space-time intervals in (19). In choosing the form for the function 𝑓(1+𝑧), we are guided by the following considerations. The current theory of standard candles, as already remarked before, considers a reference distance 𝑑ref=10pc, at which the apparent luminosity of the candle is taken as the absolute luminosity. This leads to the standard expression of the luminosity distance as in (17). Since our luminosity 𝐿𝑧 depends on (1+𝑧), it seems more correct to place the candle at a position where 𝑧=0, in order to estimate its β€œabsolute” luminosity. Instead of having the candle at the origin of the space coordinates (a rather impracticable choice if the candle is a supernova), we can place it at a distance 𝑑rs=𝐑0𝐫rs=𝐑0(2𝛿0/(1βˆ’π›Ώ20)), at the beginning of the β€œredshift region,” where it is also 𝑧=0, as discussed at length in our first paper. In other words, there are only two positions where a standard candle’s luminosity is unaffected by the universal conformal stretching: one is at the origin and the other, more conveniently used as a reference distance, is this special distance 𝑑rs, which also depends on the current value 𝛿0 of our fundamental parameter.

The second hypothesis that will be made is on the form of the function 𝑓(1+𝑧). In this, we are guided again by the classic definition of the luminosity distance, based on the inverse square law. In fact, we propose a generalization of the original inverse square law 𝑙=𝐿/4πœ‹π‘‘2 which assumed 𝐿 to be invariant, to a generalized form 𝑙=𝐿0/4πœ‹π‘‘π‘Ž where now 𝐿0 is measured near the source, therefore constant by definition, but we generalize the power dependence on the distance to account for the luminosity variations with 𝑧. Since this form of the revised inverse-square law would be dimensionally incorrect for π‘Žβ‰ 2, we further refine it by including the reference distance 𝑑ref=𝑑rs as follows: 𝐿𝑙=𝑧𝑑𝑧4πœ‹π‘‘2𝐿=𝐿0𝑑0ξ€Έ4πœ‹π‘‘2𝐿𝑑𝐿/𝑑refξ€Έπ‘Ž,𝐿𝑧𝑑𝑧=𝐿0𝑑0𝑑𝐿/𝑑refξ€Έπ‘Ž,𝐿𝑓(1+𝑧)=(1+𝑧)𝑧𝑑𝑧𝐿0𝑑0ξ€Έ=(1+𝑧)𝑑𝐿/𝑑refξ€Έπ‘ŽβŽ‘βŽ’βŽ’βŽ’βŽ£=(1+𝑧)2𝛿0𝛿0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€ΈβŽ€βŽ₯βŽ₯βŽ₯βŽ¦π‘Ž.(42)

In the equation above, we show in the first line our β€œinverse-power law” generalization of the apparent luminosity-distance relationship, with an exponent π‘Ž to be determined by fitting the type Ia SNe data. This generalization implies the dependence on 𝑧 of the absolute luminosity 𝐿, as given in the second line, and by comparison with the (21), it determines the form of the unknown function 𝑓(1+𝑧), as given in the last line of (42).

We want to emphasize here that our choice of the function 𝑓(1+𝑧) or of the generalized β€œinverse-power law” is just an educated guess, which will lead to a good fit of the SNe data in the following. At this stage, we cannot justify it on the basis of our kinematical conformal cosmology. Therefore, going back to our (19) which details the scaling properties of units of length, time, and mass, while the first two lines are fundamentally based on our new interpretation, the last expression with the function 𝑓(1+𝑧) of (42) is currently our best hypothesis but might need to be revised in the future.

In any case, assuming our current hypothesis to be a reasonable one, we can finally obtain the equivalent of (17), which will express the distance modulus πœ‡(𝑧) directly as a function of the redshift parameter. Combining together (18), (22), and (42), we obtainπœ‡bol𝑧;π‘Ž,𝛿0ξ€Έ=π‘šbol𝑧,π‘‘π‘§ξ€Έβˆ’π‘€bol𝑧=0,𝑑0ξ€Έ=βˆ’2.5log10𝑙bol𝑧,𝑑𝑧/𝑙bol𝑧=0,𝑑0ξ€Έξ€»=2.5(2+π‘Ž)log10𝑑𝐿/𝑑refξ€Έ=2.5(2+π‘Ž)log10Γ—βŽ‘βŽ’βŽ’βŽ’βŽ£π›Ώ0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€Έ2𝛿0⎀βŽ₯βŽ₯βŽ₯⎦.(43)

In this equation, we have indicated that all magnitudes, luminosities, and so forth, refer to bolometric quantities, that is, are measured over all wavelengths of emitted radiation. We will consider later the effect of observing this radiation with particular filters. We have also indicated the time dependence of the observed quantities, where as mentioned before 𝑑𝑧=𝑑0(1+𝑧). This is due to the fact that type Ia SNe has a temporal evolution [17–20] and the luminosities are usually referred to the peak values. The time at which a Supernova reaches its peak luminosity differs if observed near the source (𝑑peak0) or if it is seen from a cosmological distance, in which case 𝑑peak𝑧=𝑑peak0(1+𝑧).

In (43), the (bolometric) distance modulus πœ‡bol(𝑧;π‘Ž,𝛿0) is an explicit function of the redshift variable 𝑧 and of the two parameters π‘Ž and 𝛿0, whose values will be determined by fitting this formula to the experimental data of type Ia SNe. Our treatment therefore parallels the standard cosmology model where usually πœ‡bol(𝑧;Ω𝑀,ΩΛ) is a function of the density parameters Ω𝑀, ΩΛ. The comparison with experimental observations is usually carried out by fitting these expressions to the measured distance moduli πœ‡bol, for several types of Ia SNe as observed by the most recent surveys [47, 48].

The determination of the bolometric magnitudes from the astrophysical measurements is quite complex, involving conversions from observations performed in precise wavelength bands (π‘ˆ, 𝐡, 𝑉, 𝑅, 𝐼 filters, for ultraviolet, blue, visible, red, and infrared bands resp.) to the total (bolometric) luminosities and corresponding magnitudes. This involves precise β€œK-corrections” to transform the observations in the different bands to the bolometric quantities, plus other corrections involving the extinction of the SNa light in the host galaxy as well as in our own galaxy, resulting in a complex procedure which is also able to discriminate the overall β€œquality” of the candidate Supernova. A more detailed analysis of some of these procedures will be given in the appendix, in order to ascertain that they are consistent with our new cosmological interpretation and with (43).

We have considered the best current compilation of existing data, given by the 292 SNe of the β€œgold-silver set” taken from [48, Table  4] and also available in a machine-readable form in [49]. In this table, as well as in similar data compilations, the normalization of the distance moduli is usually arbitrary, since only the relative distances are needed to obtain the dynamical cosmological parameters, such as Ω𝑀, ΩΛ. The overall normalization of the data is linked to the value of the Hubble constant, but this is usually treated as a nuisance parameter in the fitting procedure, therefore not explicitly determined.

This approach follows from the standard cosmology definition of the luminosity distance [50] as a function of 𝑧, of the density parameters Ω𝑀, ΩΛ, and of the Hubble constant 𝐻0,𝑑𝐿𝑧;Ω𝑀,ΩΛ,𝐻0ξ€Έ=𝑐(1+𝑧)𝐻0βˆšπ‘†Γ—ξ‚»βˆš|πœ…|ξ€œ|πœ…|𝑧01+π‘§ξ…žξ€Έ2ξ€·1+Ξ©π‘€π‘§ξ…žξ€Έβˆ’π‘§ξ…žξ€·2+π‘§ξ…žξ€ΈΞ©Ξ›ξ‚„βˆ’1/2π‘‘π‘§ξ…žξ‚Ό,(44) where for Ω𝑀+ΩΛ>1, 𝑆(π‘₯)≑sin(π‘₯) and πœ…=1βˆ’Ξ©π‘€βˆ’Ξ©Ξ›; for Ω𝑀+ΩΛ<1, 𝑆(π‘₯)≑sinh(π‘₯) and πœ… as above; for Ω𝑀+ΩΛ=1, 𝑆(π‘₯)≑π‘₯ and πœ…=1.

Since 𝑑𝐿 is inversely proportional to 𝐻0, following (17), the standard expression of the distance modulus becomes: πœ‡(𝑧)=π‘š(𝑧)βˆ’π‘€=5log10𝑑𝐿𝑧;Ω𝑀,ΩΛ,𝐻0ξ€Έ+25=5log10π’ŸπΏξ€·π‘§;Ω𝑀,Ξ©Ξ›ξ€Έβˆ’5log10𝐻0ξ€Έ+25,(45) introducing the so-called β€œHubble constant-free” luminosity distance π’ŸπΏ(𝑧;Ω𝑀,ΩΛ)≑𝐻0𝑑𝐿. The term containing the Hubble constant is then usually summed together with 𝑀 and the other constants in the previous equation and then integrated upon in the standard fitting procedure.

However, as noted in [49], the data of the latest β€œgold-silver” sets can be reconciled with the latest Cepheid-SNe-based Hubble constant value [12] (𝐻0=(73Β±6.4)kmsβˆ’1Mpcβˆ’1) by subtracting 0.32mag from all distance moduli in [49, Table  4]. By performing this adjustment, the distance moduli πœ‡ become consistent with 𝐻0β‰…73kmsβˆ’1Mpcβˆ’1, which is however different from our value in (9), 𝐻0β‰…89.7kmsβˆ’1Mpcβˆ’1.

Since standard cosmology assumes that the luminosity distance 𝑑𝐿 is inversely proportional to 𝐻0 as noted above, it is easy to check that subtracting from all data another common value equal to 5log10(89.7/73)β‰…0.45, related to the ratio of the Hubble constants reported above, will bring all the distance moduli in line with our preferred value of the Hubble constant.

Therefore, we have performed this β€œdouble correction,” subtracting the factor 0.32+0.45=0.77 from all the distance moduli of the β€œgold-silver” sets, to bring them in line with our assumptions. This is the best option we have to fix the normalization of the existing data in the literature, since we do not have access to the normalization algorithm used by the SNe groups (see also the discussion in the appendix).

We have plotted these β€œdouble-corrected” distance moduli πœ‡ as a function of the observed redshift 𝑧 in Figure 1 (yellow circular points for β€œgold” data, grey triangular points for β€œsilver” SNe). In the same figure, we fit the expression in (43) to the experimental data, choosing the β€œgold set” only for our fit, due to the greater reliability of these data over the β€œsilver set.” (We have performed similar fits including both β€œgold” and β€œsilver” data for completeness. We obtained basically the same results as in the case of β€œgold” only, with a slightly worse statistical quality of the fit.) Our fit for the β€œgold” SNe has a good statistical quality (πœ’2=0.0534; 𝑅2=0.996), and we obtain the best fit parameters as follows: π›Ώπ‘Ž=2.014Β±0.018,0=(3.951Β±0.167)Γ—10βˆ’5.(46)

The resulting fit is shown as a black-dotted curve in the figure. Given our hypothesis of an β€œinverse power law,” our fit seems to suggest an integer value π‘Ž=2 for the exponent of the factor (𝑑𝐿/𝑑ref) in (42). If we perform a new fit of the same data, keeping π‘Ž=2 fixed, we obtain the continuous red curve in Figure 1 (virtually equivalent to the black-dashed one) and a slightly different value for 𝛿0,π›Ώπ‘Ž=2,0=(3.827Β±0.014)Γ—10βˆ’5.(47)

Since this is also a good-quality fit of the data (πœ’2=0.0533; 𝑅2=0.996), we are inclined to consider the values given in (47) as our best estimates of these parameters. In particular, we will use in the following our inverse power law with an integer exponent π‘Ž=2. In Figure 1, we also plot the curves (green dashed) for π‘Ž=1.9–2.1, keeping 𝛿0 as in (47), to show how our fitting solution depends on the parameter π‘Ž. It can be seen also that the majority of the experimental points lie within this range, confirming our hypothesis of π‘Žβ‰ƒ2.

We also show for comparison the standard cosmology fitting curve (orange, short dashed), following (44)-(45), for a flat Universe with Ω𝑀=0.27, ΩΛ=0.73, and 𝐻0=89.7kmsβˆ’1Mpcβˆ’1 to bring it in line with our value of the Hubble constant. Our preferred fit (in red solid) is practically equivalent to the standard cosmology curve, except for a slight difference at highest redshift values, thus showing that our model can produce Hubble plots of the same quality of those obtained by current cosmology, but without requiring any dark matter or dark energy contribution.

Our analysis of type Ia SNe therefore confirms the applicability of the kinematical conformal cosmology to standard candle luminosity measurements and proposes a small positive value 𝛿0≃3.83Γ—10βˆ’5 for our fundamental cosmological parameter, as noted previously in this work and also in paper I.

We have to mention again that the set of β€œgold-silver” SNe data and particularly the distance moduli πœ‡(𝑧) from [48] that we used in our analysis were originally obtained through a rather complex calibration algorithm (called MLCS2k2, see [47, 48] for complete details) which takes into consideration the wavelength bands being used (π‘ˆ,𝐡,𝑉,𝑅,𝐼,etc.) and related K-corrections, the current value of the Hubble constant, the extinction and reddening effects, the zero point luminosity calibration, and absolute magnitude of type Ia SNe.

This calibration method originated in the early papers of the two leading groups, as the β€œmulticolor light-curve shape” (MLCSβ€”[51, 52]) then was revised into the latest MLCS2k2, and the template fitting method-Ξ”π‘š15 [53]. In these early works, it was still possible to find a step-by-step description of the methods being used and the values of almost all the corrections employed, in order to check our normalization procedure.

On the contrary, in the latest papers based on the MLCS2k2 method [12, 47, 48], which produced the β€œgold-silver set” of distance moduli that we used in Figure 1, the complexity of the parametrization and fitting procedure makes the comparison with our luminosity normalization procedure very difficult. While we will analyze the K-corrections in the appendix, we make no attempt to revise the other corrections and normalization procedures for the distance moduli (except for the β€œdouble correction” we used above to bring the data in line with our preferred value of the Hubble constant).

In Figure 2, we present the same data and similar fitting curves as in Figure 1, but in the form of residual values Ξ”πœ‡, where the baseline is represented by our fit (in solid red) with the values of (47). It can be seen again how most of the SNe data fall within the π‘Ž=1.9–2.1 band.

Before proceeding to study the other cosmological parameters, we want to analyze briefly the low-z case and the related fit to supernova data, to confirm the feasibility of our approach also at low redshift. In standard cosmology, the expression of the luminosity distance as a function of 𝑧, such as the one described in (44), is not easily integrated, so that the low-z behavior is usually studied by expanding it in Taylor series around 𝑧=0. This procedure yields the well-known low-z expansion [15]𝑑𝐿=𝑐𝐇01𝑧+2ξ€·1βˆ’πͺ0𝑧2𝑧+π’ͺ3ξ€Έξ‚„.(48)

The first-order term corresponds to the original Hubble’s law, π‘£π‘Ÿβ‰ƒπ‘π‘§β‰ƒπ‡0𝑑𝐿, where π‘£π‘Ÿ is the recessional velocity of the galaxy, following Hubble’s original explanation based on a pure Doppler effect. The equivalent expansion for the distance modulus isπœ‡(𝑧)=25βˆ’5log10𝐇0+5log10ξ€·(𝑐𝑧)+1.0861βˆ’πͺ0𝑧𝑧+π’ͺ2ξ€Έ,(49) where 𝐇0 is measured as usual in kmsβˆ’1Mpcβˆ’1 while 𝑐=299792.458kmsβˆ’1. These expressions need to be corrected, due to our new luminosity distance 𝑑𝐿=𝐑0𝐫 from (22), as opposed to the standard cosmology expression 𝑑𝐿=(1+𝑧)𝐑0𝐫. We also need to include the reference distance 𝑑rs=𝐑0𝐫rs=𝐑0(2𝛿0/(1βˆ’π›Ώ20)) at which we start observing the redshift.

Since our expression for 𝑑𝐿 in (22) is an explicit function of 𝑧, we can directly expand this function and obtain the following result (We note that if we were to correct the standard expansion formula, as given in [15] or [16], we would obtain 𝑑𝐿=(𝑐/𝐇rs)[π‘§βˆ’(1/2)(1+πͺrs)𝑧2+π’ͺ(𝑧3)] = 𝐑0[(1/|𝛿0|)𝑧 βˆ’ (1/2|𝛿0|3)𝑧2 + π’ͺ(𝑧3)]. This is the same result we would get by expanding our luminosity distance for a negative value 𝛿0<0. This is because standard cosmology does not include the distance 𝑑rs in the derivation, and this assumption is equivalent to considering the current value of 𝛿 as negative.): 𝑑𝐿=𝐑02𝛿01βˆ’π›Ώ20+1+𝛿20𝛿0ξ€·1βˆ’π›Ώ20ξ€Έ1π‘§βˆ’2𝛿30𝑧2𝑧+π’ͺ3ξ€Έξƒ­,(50) from which we could derive an expansion for the distance modulus πœ‡(𝑧). The problem with this type of expression is that it has a very limited range of validity in our revised cosmology, due to the small value of 𝛿0≃3.83Γ—10βˆ’5 (or due to the large value of πͺ0=1/𝛿20βˆ’1≃6.82Γ—108). It is easy to check that, in our new expression for 𝑑𝐿 in the last equation, we could neglect the 𝑧2 and higher-order terms only for 𝑧≲10βˆ’9, which is a range of no practical interest. Therefore, a low-z analysis similar to the one of standard cosmology is actually not feasible in our new approach.

However, due to the low value of 𝛿0≃3.83Γ—10βˆ’5, we can expand (22) in powers of 𝛿0 around 𝛿0=0 (or just consider (22) for 𝛿0β†’0). With a Taylor expansion, we obtain𝑑𝐿=𝐑0ξ‚ƒβˆšπ‘§2+2𝑧+(1+𝑧)𝛿0𝛿+π’ͺ20ξ€Έξ‚„,πœ‡(𝑧)=2.5(2+π‘Ž)log10Γ—ξƒ―1βˆ’π›Ώ202𝛿0ξ‚ƒβˆšπ‘§2+2𝑧+(1+𝑧)𝛿0𝛿+π’ͺ20ξ€Έξ‚„ξƒ°.(51) The leading terms at low redshift of the expressions in the previous equation are the following (neglecting the 1βˆ’π›Ώ20 factor in the second line, due to the small value of 𝛿0): 𝑑𝐿≃𝐑0√2𝑧,πœ‡(𝑧)≃2.5(2+π‘Ž)log10ξƒ©βˆš2𝑧2𝛿0ξƒͺ.(52)

This equation can be considered our β€œlow-z” expression, and we can check its validity by using the second line to fit the β€œgold” SNe data of [48] in the low-z regime. For this purpose, we selected the β€œgold” data with 𝑧≲0.1, with the same β€œdouble-correction procedure” mentioned above, and we used the last equation as a fitting formula. The results obtained are the following: π›Ώπ‘Ž=2.041Β±0.084,0=(4.159Β±0.693)Γ—10βˆ’5,(53) (πœ’2=0.0430; 𝑅2=0.975) leaving π‘Ž variable. If we constrain π‘Ž to our preferred value, we obtain insteadπ›Ώπ‘Ž=2,0=(3.836Β±0.023)Γ—10βˆ’5,(54) which is also a good-quality fit of the data (πœ’2=0.0425; 𝑅2=0.975). These results are shown in Figure 3, where the fit with the parameters of (54) is shown as a blue continuous curve, while the black-dotted line illustrates the fit with variable π‘Ž. As in the previous figures, these two curves are virtually equivalent, confirming our hypothesis of an integer value for π‘Ž.

In this figure, we also show (dashed green curves) our low-z fitting curve of (52) for π‘Ž=2 and for the range 𝛿0=3.5βˆ’4.5Γ—10βˆ’5, just to illustrate how sensitive our fitting formula is to the value of 𝛿0. Considering also the previous figures, we can see that the value of the parameter π‘Ž determines the slope (or the shape of the curves in a logarithmic plot), while the value of 𝛿0 basically determines the normalization of the curves, as it will also be shown in the following paragraphs.

Comparing these expressions with the standard cosmology ones in (48)-(49), it seems at first that our results do not yield the standard Hubble’s law, π‘£π‘Ÿβ‰ƒπ‘π‘§β‰ƒπ‡0𝑑𝐿, but as already remarked before, this law was introduced by Hubble following the original interpretation of the redshift as a pure (relativistic) Doppler shift. This interpretation was later generalized into the cosmological expansion, but this view is not shared by our model, and therefore we do not need to recover the original Hubble’s law in our approach.

On the contrary, we can rewrite our β€œlow-z” expression for the distance moduli in (52), with π‘Ž=2, as πœ‡(𝑧)≃10log10ξƒ©βˆš2𝑧2𝛿0ξƒͺ=5log10𝑧2𝛿20ξƒͺ,(55) so that we have a perfect correspondence between our expression for πœ‡(𝑧) in the last equation and the classical one from (49), which can also be rewritten as πœ‡(𝑧)≃25+5log10𝑐𝑧𝐇0ξ‚Ή=5log10ξ‚Έ105𝑐𝑧𝐇0ξ‚Ή,(56) neglecting higher-order terms in 𝑧. Since both expressions (55) and (56) fit the experimental data and the standard expression can be used to measure the Hubble constant 𝐇0 (with 𝐇0≃𝐻0, see the discussion related to (30)) comparing them together and using the value of 𝛿0 from (54), we obtain𝐇0≃𝐻0≃2Γ—105𝑐𝛿20=88.2kmsβˆ’1Mpcβˆ’1.(57)

This value is very close to the Hubble constant we obtained in Section 2.2 (see (9)) and that we used for our calibration of the β€œgold-silver” data. This result shows that our calibration procedure of the SNe data was essentially correct and links directly the Hubble constant to the fundamental parameter 𝛿0, as shown in the last equation. (We also tried fitting the original low-z β€œgold” data, based on the standard value of 𝐻0=73kmsβˆ’1Mpcβˆ’1, proposed by Riess et al. in [12]. This resulted in a slightly different value of 𝛿0≃3.461Γ—10βˆ’5, which is consistent with the Hubble constant used, in view of (57).)

In addition, (57) can be combined with (9) to give a direct relation between 𝛿0 and 𝛾0,𝛾0≃1.296Γ—10βˆ’19𝛿20cmβˆ’1,(58) where the numerical factor in the previous equation is a consequence of the different units used to measure the Hubble constant. Using the value of 𝛿0 from (54) or (47) into the last equation, we obtain 𝛾0≃1.9Γ—10βˆ’28cmβˆ’1, as introduced in (41) and based on the discussion of the Pioneer anomaly. (We also performed a fit of the gold supernova data using (51), the β€œlow-𝛿” expansion formula. Setting π‘Ž=2, we obtained in this case 𝛿0=3.868Γ—10βˆ’5, which placed in (57) and (58) yields, respectively, 𝐻0=89.7kmsβˆ’1Mpcβˆ’1 and 𝛾0=1.94Γ—10βˆ’28cmβˆ’1, the exact values on which we base our calibration of supernova data. In this sense, our β€œlow-𝛿” expansion formula (51) should probably take the place of the β€œlow-z” expression (52) as being more relevant to our analysis.)

Finally, in Figure 4, we reproduce the same data and fitting curves as in the previous figures, but in the form of a standard linear Hubble plot, to show that our approach can yield also these type of graphs, with the same quality of those of standard cosmology.

In particular, we see in this figure that our normal fit (in red, upper-solid curve) following (43) and the β€œlow-z” fit (in blue, lower-solid curve) following (52) for a fixed π‘Ž=2 are almost equivalent at low redshift as expected, but they become consistently different for 𝑧≳0.1. Since our normal fit, based on 𝛿0=3.827Γ—10βˆ’5, can describe in a better way all the supernova data in our gold set, we will prefer this value over the low-z evaluation of (54). We will therefore use 𝛿0≃3.83Γ—10βˆ’5 in the summary of the fundamental parameters presented in the following section.

3.3. The Other Cosmological Parameters

Using the results from the previous sections we are finally able to report our best estimates of the parameters which enter our kinematical conformal cosmology. These parameters are shown in Table 1, where all the quantities either refer to their current value (subscript zero) or to the values at the location where the redshift starts being observed (subscript rs). We used for all these estimates the value of 𝛿0 from (47).

As it was previously mentioned, it is beyond the scope of this paper to perform a full revision of the β€œcosmological distance ladder,” in view of the changes proposed by our new approach. However, we want to show here the difference between our luminosity distance estimates and the standard results, for different redshift values. Table 2 illustrates the results of this computation for different values of the redshift parameter: in the second column, we use our new definition of luminosity distance following (18), while in the third column, we employ the standard cosmology formula in (44) with Ω𝑀=0.27, ΩΛ=0.73, 𝐻0=89.7kmsβˆ’1Mpcβˆ’1, to bring it in line with our preferred value of the Hubble constant.

As we can see from the table, there is a quite large difference between the estimates in these two columns, a difference of about three orders of magnitude just at low redshift, for π‘§βˆΌ0.01–0.1. This is mainly due to our hypothesis of a change in the β€œabsolute” luminosity of standard candles in (21), resulting in dramatically smaller revised distances.

However, if we were to redefine our new luminosity distance as in (23), that is, considering as in standard cosmology an invariable absolute luminosity 𝐿0 of the source and using corrections due to an expansion of the Universe similar to standard cosmology, we would obtain𝑑𝐿=𝐿0=√4πœ‹π‘™(1+𝑧)/𝑓(1+𝑧)𝐑0𝐫=𝐑0𝛿0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€Έξ‚Ή22𝛿0ξ€·1βˆ’π›Ώ20ξ€Έ,(59) where we used our explicit form of the function 𝑓(1+𝑧) in (42) with π‘Ž=2.

In the fourth column of Table 2, we used the previous equation to compute the distances, and we notice that these values are close to those of the standard cosmology for both 𝐻0=89.7kmsβˆ’1Mpcβˆ’1 (third column) or for the more standard value 𝐻0=73kmsβˆ’1Mpcβˆ’1 (values in the last column of the table). This is of course expected, since the distance in (59) would also fit the supernova data, if we were to assume an expansion equivalent to that of standard cosmology. In other words, (59) is our equivalent of the β€œstandard” luminosity distance and yields to virtually the same distance estimates as in standard cosmology. However, in our interpretation, (18) is to be considered the correct distance, since it includes the intrinsic dimming of the source.

We wanted to introduce also this β€œstandard-equivalent” luminosity distance in (59), in order to make a comment on the so-called β€œTolman surface brightness test,” which is usually employed in cosmology to distinguish between standard expansion theories and alternative models of redshift, such as the β€œtired light” explanation (see discussion in Section  1.7 of [16] and recent experimental results in [54]). This test is based on the ratio between the angular diameter distance 𝑑𝐴 and the luminosity distance 𝑑𝐿. Using standard cosmology distances from (16) and (25), this ratio is simply 𝑑𝐴/𝑑𝐿=(1+𝑧)βˆ’2, so that the surface brightness 𝐡 of a luminous object (defined as the apparent luminosity per solid angle—𝑙/Ξ©) will result in 𝐡≑𝑙/Ξ©=(β„’/4πœ‹)(𝑑𝐴/𝑑𝐿)2=(β„’/4πœ‹)(1/(1+𝑧)4), where β„’ is the intrinsic absolute luminosity per unit of proper area of the source (see [16] for details). As shown in the above equation, this quantity should scale like (1+𝑧)βˆ’4, and this prediction is recovered (within certain limits, due to the evolution of the galactic light sources) in experimental studies [54]. Tired light theories would require 𝐡 to scale as (1+𝑧)βˆ’1 and are essentially ruled out by these experimental evidences.

In our model, the ratio 𝑑𝐴/𝑑𝐿, constructed using (18) and (26), does not seem to scale as required by the Tolman brightness test, that is, 𝑑𝐴/𝑑𝐿=(1+𝑧)βˆ’2, but rather as (1+𝑧)βˆ’1, yielding a surface brightness scaling as (1+𝑧)βˆ’2. However, experimental tests of this effect, such as the one reported in [54], are based on a standard approach using invariable luminosities and invariable diameters of the light sources being studied. Following the discussion in the previous paragraphs, this amounts to using our β€œstandard-equivalent” luminosity distance in (59), instead of the one in (18). We can see from the fourth column in our Table 2 that distances computed with this β€œstandard-equivalent” expression are similar to those calculated by standard cosmology. Therefore, we infer that current tests of the Tolman effect would not be in disagreement with our model. We will leave to future work a more detailed analysis of this effect.

In any case, if our approach is correct and we use our new luminosity distance in (18), other distance estimates in the cosmological ladder might also need to be revised. We mentioned in Section 2.3 that several other distances need to be changed and that, for example, our new angular diameter distance would imply larger distances than previously thought; therefore, the overall reduction in distance estimates might be less dramatic than the one illustrated in Table 2, comparing just the second and third columns.

This is related to the final point we want to address in this section, that is, the apparent discrepancy between our estimate of the parameter 𝛾0=1.94Γ—10βˆ’28cmβˆ’1 and the value proposed by Mannheim, 𝛾Mannheim=3.06Γ—10βˆ’30cmβˆ’1 [8]. We recall that this value was obtained by Mannheim using a sample of eleven galaxies, where the rotational motion data were fitted by the conformal gravity theory over a range of radial distances of a few kiloparsec, from the center of each galaxy. The non-Keplerian effects are manifest beyond the peak value at π‘Ÿpeak=2.2π‘Ÿ0, with π‘Ÿ0 ranging from 0.48kpc to 4.48kpc, for the sample of eleven galaxies. The average π‘Ÿpeak is about 4.27kpc; therefore, the measured 𝛾0 was obtained by fitting the original Mannheim-Kazanas potential over radial distances π‘Ÿβ‰³π‘Ÿpeak-ave=4.27kpc from the reference point of observation (the center of each galaxy). The global redshift of each galaxy was already subtracted from the rotational data; therefore, the measured 𝛾0 refers to the intrinsic scale of the galaxies being considered (kiloparsec scale) and should be in line with the value we propose.

However, Mannheim’s analysis was based on standard cosmology estimates of the distances to all these galaxies. We have seen above that our new interpretation of the luminosity distance implies rather smaller distances than those previously estimated. The ratio between the two estimates of the gamma parameter 𝛾0/𝛾Mannheim=63.4(60) could be explained in terms of a similar ratio between distance estimates in the standard theory versus our new approach. This is due to the fact that, as mentioned in our paper I, galactic rotational curve was fitted by Mannheim using a potential proportional to the quantity π›Ύπ‘Ÿ, so that the overestimation of the distances π‘Ÿ would result in an underestimation of the 𝛾 parameter. We will also leave to future work a more detailed analysis of this issue.

4. Consequences of the Model

To conclude the analysis of our kinematical conformal cosmology, we want to summarize in this section the scaling properties of all the dimensionful quantities and link these results to the more general problem of the time variation of physical constants. This is followed by a similar analysis of the dimensionless parameters and constants, which might establish a direct connection between our parameter 𝛿 and the fine-structure constant 𝛼em of the electromagnetic theory.

4.1. Scaling Properties of Dimensionful Quantities

In our approach to cosmology, all physical quantities with dimensions (of length, time, mass, etc.) are affected by the general β€œscaling” properties detailed by (19), with the function 𝑓(1+𝑧) defined as in (42), where we use π‘Ž=2 as our preferred value for the inverse-power law generalization. In particular, using the relations in Table I of our first paper [7], we can express all these scaling factors as a function of 𝑧 and 𝛿0, or as a function of the β€œcosmological time” 𝛿 and its current value 𝛿0, since we haveξƒŽ1+𝑧=1βˆ’π›Ώ201βˆ’π›Ώ2,𝑓(1+𝑧)=4𝛿20(1+𝑧)𝛿0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€Έξ‚Ή2=4𝛿20𝛿0+||𝛿||ξ€»2ξƒŽ1βˆ’π›Ώ21βˆ’π›Ώ20.(61)

These equations imply that when we observe the Universe at a certain fixed value of the cosmological time (e.g., the current value 𝛿0) all dimensionful quantities and constants appear to scale with the redshift parameter 𝑧 (or with 𝛿 varying from βˆ’1 to +1). This will also affect the fundamental constants of physics (with the exception of the speed of light, as already discussed in our paper I).

For example, Planck’s constant, whose current value is β„Ž0=6.626Γ—10βˆ’27ergs, would be perceived as scaling like the product of a mass and a length, resulting in the following: β„Ž=β„Ž0(1+𝑧)𝑓(1+𝑧)=β„Ž04𝛿20(1+𝑧)2𝛿0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€Έξ‚Ή2=β„Ž04𝛿20𝛿0+||𝛿||ξ€»2.(62) This equation implies that if we could observe an experiment designed to measure β„Ž and performed at a location characterized by redshift 𝑧, the outcome of this experiment would be Planck’s constant as given by (62) and not our current standard value β„Ž0.

Another example of these scaling properties is related to the elementary charge 𝑒0=4.803Γ—10βˆ’10stat𝐢 (current value in cgs units). Whenever we observe the Universe at a fixed cosmological time, it is natural to assume that all dimensionless quantities should also be considered fixed at the particular values they have at that cosmological time. In particular, the fine-structure constant of electromagnetism is defined as the (dimensionless) quantity 𝛼em=𝑒2/ℏ𝑐 (with ℏ=β„Ž/2πœ‹) and has a current value of 𝛼em0=7.297Γ—10βˆ’3. This definition of 𝛼em implies that the square of the elementary charge, 𝑒2, should scale as Planck’s constant (since 𝑐 does not scale and 𝛼em is assumed to be a fixed dimensionless quantity). Therefore, the elementary charge 𝑒 should scale as𝑒=𝑒0√(1+𝑧)𝑓(1+𝑧)=𝑒02𝛿0(1+𝑧)𝛿0(1+𝑧)+(1+𝑧)2βˆ’ξ€·1βˆ’π›Ώ20ξ€Έξ‚Ή=𝑒02𝛿0𝛿0+||𝛿||ξ€»,(63) and similar scaling properties can be obtained for other physical constants of interest.

4.2. Dimensionless versus Dimensionful Quantities

Using the equations outlined in the previous sections we can compute the scaling properties of all quantities with physical dimensions, as seen from an observer at the current cosmological time 𝛿0. However, we argue that the origin of the variability of all these quantities should be found in the values of dimensionless parameters and constants. In fact, we have shown above how the scaling properties can be described in terms of a variable dimensionless 𝛿 parameter.

An alternative but equivalent way of describing these effects is to connect them to a variable fine-structure constant 𝛼em, that is, to assume that a dimensionless quantity such as 𝛼em will vary over cosmological times, following changes of the 𝛿 parameter. We recall that the wavelength of emitted radiation, in a first-order approximation, can be related to 𝛼em as follows: β„Žπœ†βˆΌπ‘šπ‘π›Ό2em,(64) since the Rydberg constant for infinite nuclear mass is π‘…βˆž=π‘šπ‘’π‘π›Ό2em/2β„Ž, and the wavelength of emitted radiation is inversely proportional to π‘…βˆž, as stated by the simple Balmer’s formula. In (64), π‘š can be considered to be the electron mass π‘šπ‘’, or the reduced mass of the atomic system emitting the radiation.

If we assume that the cosmological redshift or blueshift is due to the intrinsic changes of a dimensionless quantity such as 𝛼em, we are induced to consider the dimensionful constants such as β„Ž and π‘š as fixed, since their changes are only apparent when we observe them at a fixed value of the dimensionless 𝛿 parameter. In this line of reasoning, we can rewrite the first expression in (61) asξƒŽ1+𝑧=1βˆ’π›Ώ201βˆ’π›Ώ2=πœ†πœ†0=𝛼2em0𝛼2em,(65) having used (64) with the dimensionful quantities considered fixed.

We therefore obtain how the fine-structure constant changes with 𝑧 or 𝛿,𝛼em=𝛼em0√1+𝑧=𝛼em04ξƒŽ1βˆ’π›Ώ21βˆ’π›Ώ20,(66) which is illustrated in Figure 5, where the dependence of 𝛼em on our parameter 𝛿 is shown, given the current values of these two quantities, 𝛼em0=7.297Γ—10βˆ’3 and