Kinematical Conformal Cosmology: Fundamental Parameters from Astrophysical Observations

Abstract

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 , 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, , in line with the original estimate by P. Mannheim of this quantity. Our second fundamental parameter, , interpreted as the current value of a cosmological time variable, is derived from a detailed fitting of type Ia supernovae “gold-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 “stretching” 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 . In this way, the conformal symmetry of the universe is “kinematically” 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) () 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 (), 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 , where 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 ( is already dimensionless) if we use the following definitions: where we use a look-back time or , since we usually observe radiation emitted in the past at coordinates or , reaching us at the spatial origin and at our current time or .

In our first paper, we used the MK metric as a source of a cosmological gravitational redshift, associated with a redshift parameter and a cosmic scale factor , using the SSC 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:

To avoid possible misunderstandings, the cosmic scale factor is considered a function of the time coordinate ( or the associated ) as in standard cosmology, but it is expressed also as a function of the radial coordinate (or ) 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 , to explain the cosmological redshift, and therefore we have “improperly” 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 , which was found to be the only one associated with a possible redshift of gravitational origin (the other two cases 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 defined as The additional quantity 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 (, and ) 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 as [8]. In our first paper we argued that the current value of is probably close to Mannheim’s estimate but needs to be computed from more “local” observations. In addition, should have a negative value (since ) while is necessarily limited by , 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 (as well as and ) are probably time-varying quantities, over cosmological ages. In fact, we have proposed that the dimensionless parameter might constitute an effective cosmological time, varying from −1 to , so that (2) represents the evolution of the Universe as seen at our “current time” . The most general description is obtained by letting 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 and of its current value as follows: a simple “semicircular” 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 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 and the deceleration parameter as , with their current-time values denoted by and . Standard cosmology measurements of the Hubble constant are usually reported as [9] where is a number between 0.5 and 1.

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

We start by using the SSC time coordinate , which is connected to the dimensionless look-back time as in (1). In paper I, we have seen in (80)–(82) how to express the first, and second-order time derivatives of in terms of , or directly in terms of the redshift parameter . As a consequence, we can easily write and also as a function of or , where, as in the preceding equations, we use the “current” value (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 or ,

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 , or equivalently , over different cosmological epochs. This bell-shaped curve was plotted for a positive value and shows a local blueshift area in the “past” evolution of the Universe, extending back to a time , 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 , that is, for each value of , we have two corresponding values of the Hubble parameter (except at the maximum, for , where ), 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 , when given as a function of .

This argument applies also to the case, corresponding to the current time at which is negative, but also referring to a time in the past () at which we start observing the cosmological redshift, with , a positive quantity. This does not contradict the current estimates of as a positive quantity. They are based on redshift observations of light coming from galaxies at times in the past ; therefore, what we call in standard cosmology should be actually indicated as , 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 . In this way, we will be able to estimate the gamma parameter as , and this will allow us to evaluate also the Hubble parameter at the beginning of the redshift region (time ), by using the same value of and the positive sign in (7), or equivalently assuming by symmetry and using . Numerically, we estimate where again the details of this analysis will be presented later in this paper.

The connection with the standard cosmology value of in (5) is immediate; all the standard astrophysical observations which led to the existing estimates of were done by observing celestial objects in the redshift region, therefore for or , where is the distance at which we start observing the cosmological redshift. Therefore, in line with the current estimates of [9]. Our value, , is the direct estimate of the 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] , by the Hubble Space Telescope Key Project (HST Key Project) [11] , and others [12].

The deceleration parameter at current time or at time (in both cases ) is given by (7) as a function of the dimensionless , which cannot be estimated from the Pioneer data. We cannot use results for 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 and therefore . 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 , that is, define the Hubble parameter as and the deceleration parameter, , 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), where again . We use here the dimensionless variable , or we express the Hubble and deceleration parameters directly in terms of .

The current-time values of these two parameters are obtained in the limit for or , respectively, 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 and , with the derivatives between time variables given by so that the connecting formulas can be easily derived.

Similarly, we could also write Hubble and deceleration parameters in terms of space variables , , or , since we have in (2) the cosmic scale factor expressed in all these variables, but these expressions would not be very useful because the experimental values of and 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 and directly as a function of our cosmological time , 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 “cosmological 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 , whose definition is not affected by our alternative approach to cosmology. As defined in Weinberg’s books [15, 16], where is the impact parameter of light reaching the observer from a (parallax) distance with parallax angle , and we have used our preferred value 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 (sometimes also called coordinate or effective distance) and the proper distance : where usually refers to the current-time expansion factor , while can refer to any cosmological time 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 parsec (), 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 and connecting it to the apparent () and absolute bolometric luminosity () of a light source as follows:

This definition follows from the consideration that “In a Euclidean space the apparent luminosity of a source at rest at distance would be ” (see again Weinberg [15, Section  14.4]). In the second line of the previous equation, the factor appears due to the standard redshift interpretation, namely, that photons of energy are redshifted to energy and that the time interval of photon emission is also changed to . The total power emitted (energy per unit time) is therefore redshifted by a combined factor which will enter the denominator of the square root of (16), thus resulting in the final 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 factor corrects for this effect, while the absolute luminosity 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 () using Pogson’s law, , for any two apparent luminosities. Traditionally, the absolute bolometric luminosity of a standard candle is defined as the apparent bolometric luminosity of the same object placed at a reference distance of , so that the distance modulus (difference between the apparent and absolute bolometric magnitudes) will result as follows:

In standard cosmology, the luminosity distance on the right-hand side of the last equation is then expressed as a function of and other cosmological parameters, such as the density parameters and , that is, . The comparison with experimental observations is usually carried out by fitting the luminosity distance expression to the measured distance moduli () 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 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 factor where we have inserted, on the right-hand side of the equation, our expression for the coordinate , introduced in (76) of our paper I, as a function of our current value . We also choose the positive sign in front of the square root to select the solution corresponding to past redshift, for , which is the correct choice for the subsequent analysis (see [7] for details).

In the previous equation, we have also indicated a dependence on of the absolute luminosity of the standard candle being considered (indicated by the subscript ). 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 “standard candle,” that is, we cannot assume anymore that 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 “dilated” by the factor, when referring to a cosmological location characterized by redshift , 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:

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 as usual). Quantities with the subscript represent the same units or intervals as “seen” by the observer located at the origin, but when these “objects” are placed at a cosmological location characterized by redshift . (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 “redshifted” quantity acquires a subscript (, observed at the origin, but with information coming from past time ), while the “nonredshifted” quantity acquires the zero subscript (, observed at the origin, at current time ). 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 , , etc.) Space-time intervals are dilated by the 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 totally undetermined, assuming that some function will connect units of mass at different locations in the Universe. We only suppose that will be a function of the redshift factor , such that for , it reduces to unity, that is, , leaving masses unchanged at locations where . We will determine this function 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 , an amount of energy can be written as , and it is immediate to check that energies will scale like masses, following the last equation. We can therefore write which implies that the total energy emitted by a standard candle placed at redshift 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 , where the luminosities are also labeled like all the other quantities and refer to times connected by the same dilation factor . (In the standard interpretation, would be considered as the time in the candle's restframe of reference and as the time measured by the observer who sees the candle “moving” due to the expansion of the Universe. Standard relativistic time dilation would yield . 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, , the two luminosities are connected in the following way:

Our new definition of luminosity distance will therefore generalize the original definition in (16), by using the actual luminosity , instead of , 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 where we also used our simplified notation for the parameters , , as previously discussed.

Comparing our new expression with the original one in (16), we note that basically we replaced the factor with a more complex factor , where again the function is still undetermined at this point. This is because the standard definition would use the “constant” luminosity , as , instead of the “variable” . One could argue that if the function was to be equal to , that is, if masses were to scale like , we would recover the original definition, but this is not exactly the case. Again, the original definition (16) assumes an invariant value of the standard candle’s luminosity at all cosmological locations, while our new definition (22) is based on the choice of 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 instead using instead of , 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 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, . Following (21), we can write , where the meaning of the subscripts is the same as in the preceding equations. The wavelength variables and related differentials are connected as usual, and , so that the direct relation between and is

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 , when an extended light source of true proper diameter is placed at a (angular) distance and observed having an angular diameter , and the proper-motion distance , when proper motions in the direction transverse to the line of sight are considered. Their standard definitions are [15] where, in the second definition, is the true velocity of the source in the direction perpendicular to the line of sight, and is the change in the (angular) position of the object during the time interval of observation .

Both definitions need to be reconsidered in our new interpretation. In the first definition, the “true” diameter of the light source will depend on as for all the other lengths, , but using the RW metric, the proper distance across the source is for small angular diameters , which leads essentially to the same expression as in the standard treatment, just with replaced by . Therefore, our definition of is similar to the standard one, but needs to be supplemented by the scaling equation of the source diameter In this way, the right-hand side of the equation connecting 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 must be included. Since increases with (positive) , using the old definition, which assumes a fixed , would result in underestimating the diameter distances by a factor .

Similar care has to be taken in revising the proper motion distance . A moving source will have a transverse velocity 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 is thought to be different from the time interval of motion due to the usual redshift factor, . However, in our view, the observed time is precisely the true time interval during which the object moved at a redshift : . The proper distance travelled is , using the RW metric as in the treatment of the angular distance above, and our revised expression of the proper motion distance is

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, is still simply the observation time interval). Finally, we note that our corrected expressions for and are essentially the same (both lead to the expression ) since the geometry is completely equivalent ( is equivalent to , to ).

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 and , that is, . 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 . 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 , this coordinate should also reduce to as for all the other distances. We recall from paper I that the coordinate transformation between and is where the limit in the last expression is for and . The previous equation implies that the current value of the scale factor is simply related to the parameter (if the current value of is close to zero), where, from now on, we will also denote with a zero subscript () the current value of the parameter. Equation (27) is particularly important to simplify the connection between the Hubble constants and , in view of (6)–(12). It is easy to check that, using the previous equation, the general connection simplifies to so that for the two Hubble constants basically coincide, that is, . Again, these results are valid only for , 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 “cosmological 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 parameter, the and parameters (or the original quantity, see paper I [7]) all measured with reference to the current time 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, , , and should be considered simply as universal constants.)

We have already introduced in Section 2.2, but we still have to show how this value was obtained and compare it to the original evaluation of done by Mannheim. We will proceed first to estimate in the next subsection, later we will obtain 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 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 astronomical units from the sun, while exploring the outer solar system. This “blueshifted” frequency is uniformly changing at a rate of and can be interpreted as a constant sunward acceleration, reported as (radial inward acceleration) or as a “clock acceleration” , 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 Network—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 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 , once the influence of solar-radiation-pressure acceleration decreased below comparable levels (for distances beyond about 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 within ), 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 at time , keep this radiation from being absorbed for a long enough time, and then compare its wavelength with the radiation emitted by the same source at a later time , with . 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 . In terms of frequencies, , that is, the radiation from time would appear to be blueshifted, compared to the same radiation emitted by the same source at a later time . 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 : let be the time at which the radiometric signal is sent from Earth to the spacecraft, which is then received at time and immediately retransmitted down to the DSN, arriving back on Earth at time . In the standard analysis, the model for the signal coming back to Earth is based on the relativistic Doppler effect. We will denote as the frequency of the expected signal following this model. Its ratio with the reference frequency 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 () in [15]) where is the source radial velocity, and the approximation on the right-hand side holds to first order in .

For the case of the Pioneer spacecraft, we use , where is the expected velocity of the spacecraft, according to the theoretical navigation model, at time , 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 becomes 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 , involving an additional unexplained blueshift. Over the range of the observed anomaly, the frequency difference is reported as We point out here that the cited references adopt a rather confusing “DSN sign convention” for the frequency difference in (33) (see [23, 26] and reference of [21]), resulting in a sign change in most of their equations. We prefer to use here our definition of 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 as where the “observed” velocity of the spacecraft always refers to the time of interest . Combining together these last three equations, we write the frequency difference as We then multiply and divide the last equation by , so that we can introduce , the residual Pioneer acceleration of unknown origin. This is the change of velocity over time and not over the double time as in (33). With this residual acceleration, the Pioneer anomaly is usually reported as 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 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 as the time of arrival of the Pioneer radiation on Earth. The uplink signal was therefore sent at and retransmitted by the spacecraft, as a downlink signal, at . 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 ), but just related to a shift of the reference frequency. The reference frequency in (34) is the one emitted in the past , while the reference frequency in (32) is the one at current time , so these two equations are modified, respectively, as follows

Following (33), the frequency difference, to first order in , is now having used our fundamental (37) from paper I [7] and also . We can simplify the above equation by introducing additional approximations. From the second line of (2), we have , since we can assume (The time delay for a two-way Pioneer signal at the maximum distance of is about 20 hours; we can assume , therefore .). The reference frequency is , corresponding to the original uplink frequency (In some of the references cited, the reference frequency is taken as the downlink frequency of , 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 .), and we can also neglect the ratio , since the typical Pioneer speed in the outer solar system was [23]. With these approximations, the last equation becomes

The clock acceleration , the Pioneer acceleration , the frequency shift , and the reference frequency are all related together by 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 , which represents the best estimate of 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 “coincidence” between the Hubble constant and the value of the Pioneer acceleration divided by 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, , thus obtaining and . 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 as in (36).)