Astrophysical Tests of Kinematical Conformal Cosmology in Fourth-Order Conformal Weyl Gravity

In this work we analyze kinematical conformal cosmology (KCC), an alternative cosmological model based on conformal Weyl gravity (CG), and test it against current type Ia supernova (SNIa) luminosity data and other astrophysical observations. Expanding upon previous work on the subject, we revise the analysis of SNIa data, confirming that KCC can explain the evidence for an accelerating expansion of the Universe without using dark energy or other exotic components. We obtain an independent evaluation of the Hubble constant, H_{0} = 67.53 km/s Mpc, very close to the current best estimates. The main KCC and CG parameters are re-evaluated and their revised values are found to be close to previous estimates. We also show that available data for the Hubble parameter as a function of redshift can be fitted using KCC and that this model does not suffer from any apparent age problem. Overall, KCC remains a viable alternative cosmological model, worthy of further investigation.

able to explain the origin of some gravitational anomalies, such as the Pioneer Anomaly [16] and the Flyby Anomaly [17].
Both models, the 'standard' CG cosmology by Mannheim and KCC, were critically analyzed by Diaferio et al. [18] and compared to standard ΛCDM cosmology by applying a Bayesian approach to available astrophysical data from type Ia supernovae (SNIa) and gamma-ray bursts. Contrary to the authors' expectations [18], the results of this analysis showed that ΛCDM, Mannheim's CG, and KCC can all describe the current astrophysical data equally well. Therefore, models based on conformal gravity can be considered viable alternatives to ΛCDM and are worthy of further investigation.
In addition, a recent study by Yang et al. [19] has tested Mannheim's CG against recent astrophysical data from SNIa, determinations of the Hubble parameter at different redshift, and in relation to the 'age problem' of the old quasar APM 08279+5255 at z = 3.91. The outcome of this analysis is that CG can describe all these astrophysical data in a satisfactory manner and does not suffer from an age problem, as opposed to the case of ΛCDM.
Following this recent work, the goal of this paper is to test our KCC against the same astrophysical data used in Ref. [19] in order to ascertain whether KCC is still a viable cosmological model. In Sect. II, we begin by reviewing the main results of conformal gravity and KCC. In Sect. III, the main part of our paper, we will constrain the KCC parameters, by using the latest Union 2.1 SNIa data, and show that KCC can produce Hubble plots of the same quality as those obtained with standard ΛCDM. In Sect. IV, we will compare the experimental data for the Hubble parameter, as a function of redshift z, with KCC predictions and also briefly analyze the age problem in the context of KCC.
The fourth-order CG field equations, 4α g W µν = T µν , where W µν is the Bach tensor (see [1], or [14] for full details) were first solved, for the practical case of a static, spherically symmetric source in CG, i.e., the fourth-order analogue of the Schwarzschild exterior solution in General Relativity (GR), by Mannheim and Kazanas in 1989 ([5], [6]). This solution, in the case T µν = 0 (exterior solution), is described by the metric with The parameters in the last equation are as follows: β = GM c 2 (cm) is the geometrized mass, where M is the mass of the (spherically symmetric) source and G is the universal gravitational constant; two additional parameters, γ (cm −1 ) and κ (cm −2 ), are required by CG, while the standard Schwarzschild solution is recovered for γ, κ → 0 in the equations above. The quadratic term −κr 2 indicates a background De Sitter spacetime, which is important only over cosmological distances, since κ has a very small value. Similarly, γ measures the departure from the Schwarzschild metric at smaller distances, since the γr term becomes significant over galactic distance scales.
We will revise and update the values of these parameters in Sect. III by constraining them with recent astrophysical data.
(2)- (3) to perform extensive data fitting of galactic rotation curves without any DM contribution, with the values of γ and κ as in Eq. (4). Although the values of these CG parameters are very small, the linear and quadratic terms in Eq. (3) become significant over galactic and/or cosmological distances.
This also means that CG solutions (including those for other types of sources, see discussion in [17]) are not asymptotically flat, thus raising the question of possible 'gravitational redshift' effects at large distances. In fact, this was the main motivation for our 'kinematical approach' to conformal cosmology: in regions far away from massive sources (for r ≫ β(2 − 3βγ)) and also ignoring the term βγ, as suggested by the analysis of galactic rotation velocities, B(r) simplifies to This implies a possible gravitational redshift at large distances, analogous to the one experimentally observed in standard GR near massive sources such as the Earth, the Sun, or white dwarfs. This effect is related to the square-root of the ratio of the time-time components g 00 of the metric at two different locations. In Ref. [14] we considered our current spacetime location (r = 0; t 0 ) in relation to the spacetime location (r > 0; t < t 0 ) of a distant galaxy which emits light at a time t in the past that reaches us at present time t 0 and appears to be redshifted in relation to the standard redshift parameter z.
We then argued that this observed redshift could be due (in part, or totally) to the gravitational redshift effect mentioned above. If this effect were indeed the only source of the observed redshift, with the metric in Eq. (6), we would have: In other words, if the CG metric in Eqs. (2)-(3) has a true physical meaning, as it seems to be the case from the detailed fitting of galactic rotational curves, it should also determine strong gravitational redshift at very large cosmological distances. As far as we are aware, this issue has never been raised in all current CG literature (except, of course, in our previous papers).
The CG metric in Eqs. (2) and (6) is actually conformal to the standard FRW metric (see details in [14]): where a(t) is the standard Robertson-Walker scale factor, k = k/ |k| = 0, ±1 and k = −γ 2 /4 − κ. As in our previous papers, we distinguish here between two sets of coordinates: the Static Standard Coordinates -SSC (r, t) used in Eqs.
This local conformal invariance induces a dependence of the length and time units on the local metric, so that the observed redshift can be interpreted as the ratio between the wavelength λ(r, t) of the radiation emitted by atomic transitions, at the time and location of the source, and the wavelength λ(0, t 0 ) of the same atomic transition measured here on Earth at current time. Since modern metrology defines our common units of length δl and time δt as being proportional respectively to the wavelength and to the period (inverse of the frequency ν) of radiation emitted during certain atomic transitions, we can write the following 'redshift equation' connecting wavelengths λ to unit-lengths δl and frequencies ν to unit-time intervals δt (we also use λν = c, with a constant speed of light c).
Therefore, in KCC the observed redshift is due to the change of length and time units over cosmological spacetime, as opposed to the standard explanation of a pure expansion of the scale factor a. In view of this interpretation, and connecting together Eqs. (7) and (9), KCC is able to derive directly the scale factor as a function of space or time coordinates, without solving the dynamical field equations. In terms of SSC, we have: or, using appropriate coordinate transformations, in terms of FRW coordinates: 1 Similarly, bold type characters will be used for quantities referring to the FRW geometry, while normal type characters will be used with reference to the SSC coordinates. For example, the RW scale factor will be denoted here as a(t) or a(t), respectively, in the two cases. In our previous papers we used R(t) and R(t) for the scale factor, but we now prefer to adopt the more common notation, a(t) or a(t), in this work.
All these scale-factor equations can also be written explicitly in terms of the time coordinates t and t, as is usually done in standard cosmology, by computing the time it takes for a light signal, emitted at radial distance r or r, to reach the observer at the origin.
The detailed expressions for a(t) and a(t), as well as all the connecting formulas between the different variables and conformal parameters, can be found in Ref. [14] (see Table I).
Furthermore, from the plots of the KCC scale factors, such as a(r) from Eq. (11), it can be seen that the observed redshift z > 0 is only possible for k = −1, so that the other two cases, k = 0, +1, are actually ruled out.
The new CG dimensionless parameter δ = already mentioned above. 2 It can be shown that |δ| < 1 and that, for k = −1, Eq. (11) yields the following direct relation between r and z: The plus-minus sign in the last equation indicates that there are two locations where z = 0: at the origin r = 0, and at a particular radial location r rs = 2δ 1−δ 2 which becomes of physical significance for δ > 0. In fact, in this particular case, there is a region of negative redshift (i.e., a blueshift) for 0 < r < r rs , followed by a standard redshift region at larger 2 In our previous papers, we considered the possibility that all these CG parameters might also be changing with spacetime coordinates. In particular, we supposed that the δ parameter might play the role of a universal time and we used the zero subscript to denote the current values of all these parameters (i.e., δ 0 , γ 0 , etc.). In this paper, we are just considering the current values of these parameters, so we simply write δ, γ, κ, etc. radial distances, for r > r rs = 2δ 1−δ 2 . This suggests that the (current) value of δ should be small and positive, so that the supposed blueshift region would be a small (practically undetectable) region around the observer: for example, a small region of the size of the Solar System, or similar.
In two of our previous papers ( [15], [16]) we actually suggested that this local blueshift region could have been the origin of the Pioneer Anomaly (PA -for a review, see [29]) since 'blueshifted' signals coming from the Pioneer spacecraft would appear to be equivalent to the observed anomalous acceleration. In view of this possible connection, the value of the γ parameter in Eq. (5) was directly inferred from the Pioneer anomalous acceleration ( [15], [16]); the value of the δ parameter was then computed [15] from the fitting of the SNIa data available at the time, and the values of the parameters k and κ were obtained through Eqs. (12) and (13). In summary, the values of the CG parameters were determined as follows (see also Table 1 in Ref. [15]): Although it is still possible that the PA might have a gravitational origin, i.e., due to modifications of GR, it is now widely accepted that the cause of this anomaly is probably more mundane [30]: thermal recoil forces originating from the spacecraft radioactive thermoelectric generators. Therefore, in the following sections we will perform a new computation of the CG parameters in Eq. (15), without using any more data related to the PA. We will begin, in the following section, by constraining our parameters using updated SNIa data.

III. KCC AND TYPE IA SUPERNOVAE
In order to constrain the CG parameters with recent SNIa data we need to redefine the luminosity distance in KCC, since this is the main cosmological distance used in this context. In this section we will expand upon concepts already introduced in Ref. [15] (more details about the definitions of distances in KCC can be found in this reference). We start by noticing that the new interpretation of the redshift discussed in the previous section (in particular, in Eq. (9)) implies that lengths and time intervals scale with redshift z as: where the subscript 0 indicates intervals of the given quantity associated with objects which share the same spacetime location of the observer at the origin (namely, here on Earth at r = 0 and at our current time t 0 ), while the subscript z indicates intervals of the same quantity associated with objects at redshift z = 0, as seen or measured by the same observer at the origin.
It should be emphasized that this change in lengths, or time intervals (as well as wavelengths, frequencies, and all other kinematical quantities derived from lengths and times), is due to the spacetime location of the object being studied (as measured by the redshift parameter z) and not to the 'cosmic expansion' as in the standard cosmological model.
It is natural to assume that masses, energies, luminosities, and other dynamical quantities will follow similar scaling laws, but not necessarily the same as the one in Eq. (16). In Ref.
As a consequence of these scaling laws, the 'absolute luminosity' L, or energy emitted per unit time, will scale as where the meaning of the subscripts is the same as described above for the other quantities.
Thus, KCC postulates a change in the absolute luminosity of a 'standard candle,' which is intrinsically due to its spacetime location, while standard cosmology assumes an invariable absolute luminosity L of the standard candle being considered.
Standard cosmology defines the luminosity distance as d L = L 4πl = a 0 r(1 + z), with L and l being the absolute and apparent luminosities of the standard candle being used as a distance indicator; a 0 denotes the current value of the scale factor and the (1 + z) factor on the right-hand side of the equation originates from a (1 + z) 2 dimming factor under the square root. This factor is due to the standard redshift of the photon frequency and also to a time dilation effect of the emission interval of photons.
KCC considers instead this (1 + z) 2 dimming factor as unphysical, so the (1 + z) factor on the right-hand side of the standard luminosity distance equation is completely eliminated.
In view also of our scaling law for luminosities in Eq. (18), and of Eq. (14), we then define the luminosity distance in KCC as: 4 Since this definition assumes an intrinsic dimming of the luminosity L z with redshift z, it leads to distance estimates which are dramatically different from those of standard cosmology for different values of z (see the first three columns in Table 2 of Ref [15]).
To avoid this issue, an alternative definition could be employed, which would retain the concept of an invariable luminosity L 0 of a standard candle, while including the other aspects of KCC. We can obtain this alternative luminosity distance d L by modifying the previous equation as follows: so that the right-hand side of the equation now depends explicitly on the still unknown function f (1 + z). In Table 2 of Ref. [15], it was shown that distances estimated using d L are very close to those of standard cosmology (compare the values in the fourth column of this table with those in the third or fifth columns), so the KCC definition in Eq. (20) more closely agrees with the luminosity distance of standard cosmology.
We will see in the following that both definitions, in Eqs. (19) and (20), lead to the same results when applied to SNIa data, but they differ conceptually: the former assumes a variable absolute luminosity L z of a standard candle, while the latter assumes an invariable absolute luminosity L 0 , which is more in line with the standard interpretation.
Before we can apply these definitions to the analysis of SNIa data, we need to obtain an explicit form for the f (1 + z) function, which enters most of the KCC equations above.
Expanding upon the arguments discussed in our previous work [15], we can assume the following properties for this function: 1. f is some arbitrary function of (1 + z), with a 'fixed point' at 1, that is, f (1) = 1, or 2. f is a dimensionless quantity, so that Eqs. (17)-(20) are dimensionally correct.
3. f is a function possibly built out of other expressions of KCC, which also depend on the factor (1 + z).
Although the last property in the list above is just an educated guess, it suggests that the function f might depend on the following KCC factor: constructed as the (dimensionless) ratio between the luminosity distance in Eq. (19) and the reference distance which corresponds to the value r rs of the radial coordinate (other than the origin) where we have z = 0 (see discussion after Eq. (14)). Therefore, as it was argued also in Ref. [15], Following the discussion above, the most general form of the function f (1 + z) that we will consider is: where α and β are coefficients to be determined from SNIa data fitting.
A. SNIa data fitting In our previous work, we determined the CG parameters by using the SNIa data available at the time (292 SNIa data of the 'gold-silver' set, see [15] for details) and by considering the value of the Pioneer anomalous acceleration. As already mentioned, we will not use the PA data in this study, but we will use the latest compilation of SNIa data: the 580 supernovae from the Union 2.1 data set ( [31], [32], [33]).
The distance modulus µ (difference between the apparent magnitude m and the absolute magnitude M) is usually computed, using Pogson's law, in terms of the logarithm of the ratio between the apparent luminosity l z (at redshift z) and the reference apparent luminosity . We have: where the subscript z refers to quantities evaluated at redshift z = 0, while the subscript REF indicates the 'reference' value of the quantity, i.e., when the standard candle is placed at the reference distance.
As explained before, we have two possible choices for this reference distance: the traditional distance of 10 pc (since usually the absolute luminosity L of a 'standard candle' is defined as the apparent luminosity of the same object placed at 10 parsec) and the KCC reference distance d REF in Eq. (22) above, since this is the only location, other than the origin, where z = 0.
Using this latter choice for the reference distance and combining Eq. (24) with Eqs. (18), (19), (21), and (23), we obtain explicitly: an expression which can be used directly to fit SNIa data and determine the value of the three free parameters α, β, and δ.
Using this last equation as a fitting formula for the Union 2.1 SNIa data, we obtained the following 'best-fit' values for the free parameters: Assuming that α and β are likely to be integer numbers, due to their role in the definition of the function f (1 + z) in Eq. (23), and close to the values reported in the previous equation, we repeated the fitting procedure, first by setting β = 1: then by fixing both α and β as follows: All these fits have good statistical quality (R 2 = 0.996) and clearly confirm the results of our past SNIa data fitting [15], where it was a priori postulated that α = 2, β = 1, and δ was found to be as in Eq. (15). It can also be shown that our fitting formula in the second line of SNIa data fitting procedure is valid even if we use d L instead of d L , which is equivalent to using a luminosity distance whose estimates are very close to those of standard cosmology.
In KCC, the values of the CG parameters γ and δ are also connected to the current value of the Hubble parameter: H 0 = c a 0 δ in SSC or FRW coordinates, respectively, but with H 0 ≃ H 0 for |δ| ≪ 1 [15]. Since in this work we are not relying any longer on the PA data, we can now derive the value of γ directly from the Hubble constant, using the previous equation.
The Union 2.1 SNIa data are consistent with the Hubble constant estimate by Riess et al. [34], H 0 = (73.8 ± 2.4) km s −1 Mpc −1 , from which we obtain γ = 2 c H 0 = (1.596 ± 0.052) × 10 −28 cm −1 . However, the most commonly used estimate of the Hubble constant is from the Planck collaboration 2013 results [35]: therefore, in the rest of this paper we will consider the value of γ above as the current KCC estimate.
It could be argued that, since the Union 2.1 SNIa data are based on the standard definitions for the luminosity distance, standard candles, etc., it might be more appropriate to use d REF = 10 pc as a reference distance. This leads to a slightly different fitting formula, in view also of Eqs. (19) and (29): which also includes the 'normalized Hubble constant' h as a fitting parameter. This dimensionless quantity is related to H 0 as follows: As in our previous fitting formula (25), we now have the option of leaving all four parameters (α, β, δ, and h) completely free, or to fix some of them, for example, by choosing integer values for α and β. If we leave all four parameters free, our best fit to Union 2.1 SNIa data yields: Comparing our results for δ, in Eqs. (28) and (34), we see that our two possible fitting formulas (25) and (31) produce consistent results for δ ≃ 3.36 − 3.37 × 10 −5 , in line also with our previous determinations from Ref. [15], or in Eq. (15). In addition, our analysis confirms that the f (1 + z) function in Eq. (23) should be considered with α = 2 and β = 1, i.e., and the function f (1 + z) is given in Eq. (35). In the next section we will plot our results and compare them with those of standard cosmology.

B. Union 2.1 data and KCC plots
As already mentioned at the beginning of Sect. III A, our new KCC fits were performed with the latest Union 2.1 SNIa data 5 ( [31], [32], [33]). The Supernova Cosmology Project "Union2.1" SNIa compilation is an update of the previous "Union2" compilation, bringing together data for 833 supernovae, drawn from 19 datasets. Of these, 580 SNe pass usability cuts and are included in the data set. In Fig. 1  In Fig. 2 we reproduce the same data and the same fitting curves as in Fig. 1, but in the form of a standard Hubble plot, with logarithmic axis for redshift z. In this way, all the fitting curves become almost straight lines and the differences between them can be better appreciated. Again, the two main KCC fits (red-solid and black-long dashed) are almost indistinguishable and only slightly different from the equivalent standard cosmology prediction (blue, short-dashed).
Similarly, Fig. 3 presents the same information in the form of residual values ∆µ, with the baseline represented by our main KCC fit (red-solid, with parameters as in Eq. (28)).
In this figure it is easier to notice the small differences between our two KCC fits and the standard cosmology prediction. It is also evident that most of the SNIa data points fall within the α = 1.9 − 2.1 band.
The last study we performed, in connection with the Union 2.1 data, was related to the low-z behavior of our fitting formulas. As already discussed at length in [15], we cannot Also shown (dotted-green curves) is the range of our KCC fitting curves for a variable α = 1.9−2.1.
effectively expand in powers of z our luminosity distance d L in Eq. (19), due to the very small value of the δ parameter. Therefore, we just discard terms containing δ in the same expression for d L and retain only the leading term depending on z:  µ(z) = 2.5(2 + α) which becomes our "low-z" fitting formula.
To check this expression we selected 179 SNIa data from the Union 2.1 set with z 0.1 and applied our fitting formula (38) to this data subset. Fig. 4 shows the results of this low-z fitting: our main KCC fit (red-solid curve), for a fixed α = 2, yields δ = In our previous work (see Sect. 3.2 in [15]) we also remarked that our low-z distance modulus expression in Eq. (38), for α = 2, can be rewritten as µ(z) ≃ 10 log 10 √ 2z 2δ = 5 log 10 z 2δ 2 , so that it corresponds perfectly to the first terms of the standard cosmology expansion µ(z) ≃ 25 + 5 log 10 cz H 0 = 5 log 10 10 5 cz H 0 , neglecting higher-order terms in z. Comparing the right-hand sides of these two 'low-z' expressions, we find a direct connection between the Hubble constant and the KCC δ parameter: having used our best estimate for δ in Eq. (28) and with the speed of light given as c = 299792.458 km s −1 .
It is very remarkable that our KCC model and the related SNIa data fitting are able to obtain an estimate for the Hubble constant which is very close to the 2013 Planck collaboration value. We want to emphasize that our value for δ in Eq. (28) came from the fitting formula in Eq. (25), which is independent of any assumed value for H 0 .
Therefore, our value of H 0 in Eq. (39) represents KCC's direct evaluation of the Hubble constant, in agreement with current best estimates. We can recompute the value for γ using H 0 = 67.53 km s −1 Mpc −1 as γ = 2 c H 0 = 1.460×10 −28 cm −1 , which is essentially equivalent to our previous estimate in Eq. (30), based on the 2013 Planck collaboration value for H 0 .

IV. KCC AND HUBBLE PARAMETER DATA
Another important test of our KCC model can be performed in relation with observed data for the Hubble parameter H(z), measured as a function of redshift. As it was done by Yang et al. in their recent analysis [19] of Mannheim's CG, we will use here all the available data for H(z), obtained from different sources and with different methods, as reported in Table I. 6 Although different methods were used to obtain the data in this table, the most common argument relies on the fact that the Hubble parameter depends on the differential age of the Universe, as a function of redshift, in the form: Therefore, a determination of dz dt , or more practically of the ratio ∆z ∆t between finite intervals of redshift and time, will lead to a direct measurement of H(z).
In order to measure the time interval ∆t, we need to identify and use so-called 'cosmic chronometers,' i.e., astrophysical objects, such as a galaxies, whose evolution follows a known fiducial model, so that these objects behave as 'standard clocks' in the Universe.
Once this population of standard clocks has been found and dated, the 'differential-age' technique can be used: the age difference ∆t, and the corresponding redshift difference ∆z, between two of these cosmic chronometers can be measured, thus determining H(z) in view of Eq. (40). This differential age (DA) method has the advantage of not using  any integrated cosmological quantity (such as the luminosity distance, which is expressed through an integral in standard cosmology), since these quantities depend on the integral of the expansion history, thus yielding less direct measurements of the expansion history itself.
Since the original proposal of this DA method ( [44], [38]), the best choice of 'cosmic chronometers' was found to be a population of 'red-envelope' galaxies: massive galaxies, harbored in high-density regions of galaxy clusters and containing the oldest stellar populations, which are now evolving only passively (i.e., with very limited new star formation).
The age of these passively evolving galaxies can then be used in connection with the DA technique explained above to measure H(z) ( [38], [39], [40]). A similar approach, also based on passively evolving galaxies, but more centered on a differential spectroscopic evolution of early-type galaxies as a function of redshift, was introduced by Moresco et. al. ([45], [36]), yielding more data points, followed by the more recent work by Zhang et al. [41].
A different approach [37] to the measurement of H(z) considered instead the baryon acoustic oscillations (BAO) peak position as a standard ruler in the radial direction. This BAO method was later connected to the Alcock-Paczynski distortion from galaxy clustering (GC) in the WiggleZ Dark Energy Survey [42], and one additional data point was recently obtained [43] by using galaxy clustering data. All the measured data points for H(z) are reported in Table I; we will now interpret these data in view of our kinematical conformal cosmology.
In KCC, the Hubble parameter is directly related to z as follows (see Eq.(10) in [15]): in view also of Eq. (29) and assuming δ > 0. At first, it seems impossible to fit the observational Hubble data (OHD) in Table I  such as powers of (1 + z) and/or f (1 + z), into our fitting formula (41).
In view also of the explicit form of f (1 + z) in Eq. (35), we generalize our fitting formula for H(z) as: where l and m are free parameters to be determined with our fitting procedure. Of course, these new parameters l and m are not necessarily related to the similar α and β parameters used before in the SNIa data fitting. In the last equation we also used our direct connection in Eq. (39) between H 0 and δ to avoid over-parametrizing this fitting formula.
We then used our revised formula (42) to fit the OHD in Table I, allowing up to three dimensionless parameters: δ, l, and m. However, leaving all three parameters completely free does not lead to a satisfactory fit of the data, so we simply set δ to our preferred value: δ = 3.36 × 10 −5 . Our best fit, considering l and m as free parameters, is: l = 1.288 ± 0.084, m = 1.092 ± 0.006, δ = 3.36 × 10 −5 , and is shown in Fig. 5 (red-solid curve), together with all the OHD from Table I. If we fix l to be an integer value, close to the previous estimate, we obtain instead: which is also shown in Fig. 5 (black, long-dashed curve). In the same figure, the stan-  combining Eqs. (40) and (41), we have: where the (1 + z) factor on the right-hand side is due to the rescaling of the time intervals. In particular, age estimates are typically sensitive to the distance scale (see discussion in Ref. [46], pp. 62-63): a fractional change δd/d in distance estimates will produce a change δL/L = −2δd/d in absolute luminosities and thus a fractional change δt/t ≈ +2δd/d in age estimates, since the absolute luminosity of stars at the turn-off point in the main sequence is roughly inversely proportional to the age of the globular cluster being studied.
In KCC the change in luminosity distance δd L is due to the difference between the revised Finally, we wish to comment on the 'age problem' analyzed in Ref. [19], which was related to Mannheim's CG. The issue being studied was a possible age problem for the old quasar APM 08279+5255 at z = 3.91, as well as the current estimates of the age of the Universe. As already remarked in Sect. I, it was shown that CG does not suffer from an age problem, as it might be the case instead for standard cosmology (see again [19] and references therein). For a cosmological model where H(z) is known explicitly, all age estimates are essentially obtained by integrating Eq. (40). For instance, the current age of the Universe t 0 is: assuming z = ∞ at time zero and z = 0 at current time. More generally, the age of an astrophysical object (such as the old quasar mentioned above) which is observed at redshift z, but whose formation occurred at earlier times, corresponding to a formation redshift z f > z, is computed as: In ΛCDM cosmology, using the standard expression for H(z) with Ω M ∼ = 0.3, Ω Λ ∼ = 0.7, is computed as t 0 = 14.0 Gyr, in line with estimates based on globular clusters, or other astrophysical objects. On the contrary, the quasar APM 08279+5255 is observed at z = 3.91, with an estimated formation redshift z f = 15 [19]. Using Eq. (46), the standard cosmology age for this quasar would be T SC (3.91, 15) = 1.34 Gyr, causing a possible age problem, since the best estimated age for this quasar is 2.1 Gyr, with a 1σ lower limit of 1.8 Gyr and an absolute lowest limit of 1.5 Gyr [19].
As discussed at length in our previous work (see Sect. 4.5 in Ref. [14]), in KCC we have two possible time coordinates: the static standard coordinate t related to our local unit of time, as opposed to the FRW coordinate t, where the former is essentially the conformal time of the latter. When using the former coordinate t, the Universe does not appear to have initial or final singularities (thus, the age of the Universe would be infinite, if measured using this coordinate), while both singularities appear when using the latter coordinate t.
However, if we use FRW coordinates to estimate ages, i.e., if we use H(z) as in Eq. (41) in Eqs. (45) KCC was also tested against OHD for H(z) and in relation with the age of the Universe and of old quasars. As in the case of luminosity distance determinations, it was found that age determinations in KCC need to be corrected by using the same scale factors which are at the basis of our model. With these scale corrections, KCC can effectively accommodate the existing H(z) data, and does not show any apparent age problem, including the case of quasar APM 08279+5255.
Therefore, our final conclusion is that kinematical conformal cosmology is still a viable alternative cosmological model, although surely not as popular as other models based on conformal gravity, or standard ΛCDM cosmology. Further studies will be needed to check this model against other astrophysical data in order to see if it remains a possible alternative cosmology.