The Descartes Code (Spin Orbital Rotation of Photons)–III. The Cosmological Redshift and the Hubble Constant

A new alternative model to the Friedmann-Lemaître-Robertson-Walker metric is presented based on Descartes’ code—the old color theory published in 1637. In this model, the photon spin-orbital speed is slowly decreasing in the free space of the Universe. The formulae for the wavelength, frequency, local time, momentum, energy, temperature, and photon energy dilation are presented for cosmological photons outside of a gravitational field. The Hubble constant H_⊕ = 2.3195 ∗ 10⁻¹⁸ s⁻¹ (= 71.572 kms⁻¹Mpc⁻¹) is calculated as (G M_⊕)/(R_⊕ c³) ∗ dc_rotational/dt where G is the Newtonian gravitation constant, M_⊕ is the Earth’s mass, R_⊕ is the Earth’s radius, and crotational is the rotational speed of Descartes’ light “globules” without any fitting. The cosmological photons expand their wavelength in the Earth’s gravitational field based on the age of those photons determined via their redshift given as (1 + z). This newly defined Hubble constant can be falsified in the gravitational field of the Moon, and Mars. The simple formulae for the angular diameter distance d_A, the luminosity distance d_L, the distance modulus m—M, the concentration of galaxies at the redshift z can be tested on the recent data. E.g., the distance modulus was several times tested on the cosmological objects till their redshift z = 5 (references to those studies are given). The formula for the angular diameter distance d_A was recently tested for the data taken using the James Webb Space Telescope (JWST) by Lovyagin et al. and the value of d_A followed the data trend better than the standard model. This Descartes’ hypothesis can start a discussion of Descartes’ cosmological photons can newly explain experimental data without fitting in Euclidean space.

Introduction

Einstein’s theory of gravity is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics [1]. Einstein used approximation methods in working out initial predictions of his theory. The exact solution of the Einstein field equations of general relativity was developed independently by Friedmann-Lemaître-Robertson-Walker (FLRW metric) in the 1920s and 1930s. This model is sometimes called the Standard model of modern cosmology, e.g. [2].

This FLRW metric leads to quantitative predictions that were very intensively experimentally tested with impressive accuracy [3], however, fitting of data is necessary based on an unknown dark energy and on an unknown dark matter. Since that time there were published many attempts as alternative scenarios to the original Einstein’s theory of gravity [4] in order to avoid the fitting data based on that unknown dark energy and dark matter. None of these proposals can compete with predictions of the original Einstein’s theory. There was one alternative proposal by Zwicky in 1929 known as the “tired light” scenario [5]. Since that time there were proposed about 60 ad hoc modifications of this model [6] but none was accepted because this model could not interpret all experimental data found for the cosmological photons.

Therefore, it seems almost impossible to appear with an alternative model describing the influence of the gravitational field on cosmological photon properties in the Euclidean space. This contribution brings to the focus an old, forgotten and overlooked theory of colors that might newly quantitatively describe the photon colors—the gravitational redshift of cosmological photons in the gravitational field of an observer.

The Expansion of the Cosmological Photons in the Gravitational Fields of Earth, Moon, and Mars

In this model, the cosmological photons interact with the gravitational field of the Earth [7] and their wavelength expands in the dependence of their age expressed via their redshift. This model can be falsified by the performing of this experiment in the gravitational fields of the Moon, and Mars. This experiment could be very valuable for the estimation of the reality of Descartes’ code. The project ICURE (India, China, United States of America, Russia, and European Union) could bring new data for the properties of cosmological photons in the gravitational field of the Earth, Moon, and Mars. The characteristic expansion parameters for those gravitational fields termed as the Hubble constant are given in Table I. These gravitational fields modify the spin-orbital rotation speed of cosmological photons c_ROT and the characteristic redshift z is measured by the Hubble telescopes located close to the surface of those objects.

Table I. The Hubble Constant for the Gravitational Fields of the Earth, Moon, and Mars

	Earth	Moon	Mars
Mass [kg]	5.9722*1024	7.342*1022	6.4171*1023
Radius [km]	6378.137	1737.4	3389.5
Gravitational effect on the rotational speed of old photons	$H_{\oplus }={\displaystyle \frac{G{M}_{\oplus }}{R_{\oplus }{c}^3}}{\displaystyle \frac{d{c}_{ROT}}{dt}}$	$H_{MOON}={\displaystyle \frac{G{M}_{MOON}}{R_{MOON}{c}^3}}{\displaystyle \frac{d{c}_{ROT}}{dt}}$	$H_{MARS}={\displaystyle \frac{G{M}_{MARS}}{R_{MARS}{c}^3}}{\displaystyle \frac{d{c}_{ROT}}{dt}}$
Expansion parameter of the gravitational field [s−1]	2.3195*10−18	1.0468*10−19	4.6897*10−19
Expansion parameter of the gravitational field[kms−1Mpc−1]	71.572	3.230	14.471
Expansion parameter for cosmological photons	$H\left(z\right)=H_{\oplus }(1+z)$	$H\left(z\right)=H_{MOON}(1+z)$	$H\left(z\right)=H_{MARS}(1+z)$

In the last decades many scholars experimentally determined the value of the Hubble constant, some reviews analyzed the so-called Hubble tension, e.g. [8]–[12]. Wang et al. [13] statistically analyzed 216 measurements of the Hubble constant during the period 2012–2022. We have made a simple arithmetic average of those 216 measurements organized in the histogram in Fig. 1.

Fig. 1. Histogram of 216 measurements of the Hubble constant in the period 2012–2022 from reference [13].

The actual state of the art offers many procedures how to evaluate the present value of the Hubble constant H₀. The analysis is very complicated and it is difficult to select the technique leading to the true value of H₀. The simple arithmetic average of those 216 measurements gives the value H₀ = (70.99 ± 3.17) kms⁻¹Mpc⁻¹. The effect of the Earth’s gravitational field on the expansion of the cosmological photons gives the value H_⊕ = 2.3195*10⁻¹⁸ s⁻¹ (= 71.572 kms⁻¹Mpc⁻¹). In this scenario, there is no fitting parameter to get the value of the Hubble constant. The value H_⊕ is determined from the Earth’s mass and radius, the Newtonian gravitational constant G, and the light speed c.

The Descartes Code: Expansion Data from Cosmic Chronometers

The very old (1637) Descartes’ color theory [14] based on the rotation of “light globules” can be the “lost key” on how to crack the mysterious behavior of cosmological photons. This Descartes’ color theory was rejected by Newton in 1672 and was forgotten and overlooked in the coming centuries [15]–[25]. During the last year there were several important contributions for the description of color in physics that were newly re-interpreted based on the Descartes color theory, [26]–[28]. The recent strong growing interest in the Spin Angular Momentum (SAM) and the Orbital Angular Momentum (OAM) of photons brings many new experimental data, e.g. [29]–[36]. It could be very interesting to study the rotation of cosmological photons in the gravitational field as well.

Descartes’ model is based on the idea that cosmological photons expand their wavelength in the gravitational field of an observer. This wavelength expansion of cosmological photons increases with their age and with the gravitational field intensity at the observer site. This model can be tested on experimental data from cosmic chronometers published by authors given in Table II. The data in Table II represent the standard set for the analysis of behavior of cosmological photons based on the cosmic chronometers.

Table II. The Expansion Data of Cosmological Photons in the Earth’s Gravitational Field from Cosmic Chronometers

Redshift z	H(z) km s−1 Mpc−1	Reference
0.07	69.0 ± 19.6	Zhang et al. [37]
0.09	69.0 ± 12.0	Jimenez et al. [38]
0.12	68.6 ± 26.2	Zhang et al. [37]
0.17	83.0 ± 8.0	Simon et al. [39]
0.179	75.0 ± 4.0	Moresco et al. [40]
0.199	75.0 ± 5.0	Moresco et al. [40]
0.2	72.9 ± 29.6	Zhang et al. [37]
0.27	77.0 ± 14.0	Simon et al. [39]
0.28	88.8 ± 36.6	Zhang et al. [37]
0.352	83.0 ± 14.0	Moresco et al. [40]
0.38	83.0 ± 13.5	Moresco et al. [41]
0.4	95.0 ± 17.0	Simon et al. [39]
0.4	77.0 ± 10.2	Moresco et al. [41]
0.425	87.1 ± 11.2	Moresco et al. [41]
0.45	92.8 ± 12.9	Moresco et al. [41]
0.47	89.0 ± 34.0	Ratsimbazafy et al. [42]
0.478	80.9 ± 9.0	Moresco et al. [41]
0.48	97.0 ± 62.0	Stern et al. [43]
0.593	104.0 ± 13.0	Moresco et al. [40]
0.68	92.0 ± 8.0	Moresco et al. [40]
0.75	98.8 ± 33.6	Borghi et al. [44]
0.781	105.0 ± 12.0	Moresco et al. [40]
0.8	113.1 ± 28.5	Jiao et al. [45]
0.875	125.0 ± 17.0	Moresco et al. [40]
0.88	90.0 ± 40.0	Stern et al. [43]
0.9	117.0 ± 23.0	Simon et al. [39]
1.037	154.0 ± 20.0	Moresco et al. [40]
1.3	168.0 ± 17.0	Simon et al. [39]
1.363	160.0 ± 33.6	Moresco [46]
1.43	177.0 ± 18.0	Simon et al. [39]
1.53	140.0 ± 14.0	Simon et al. [39]
1.75	202.0 ± 40.0	Simon et al. [39]
1.965	186.5 ± 50.4	Moresco [46]

Fig. 2 shows the dependence of the expansion of cosmological photons in the gravitational field of the Earth. The value H_⊕ is determined from the Earth’s mass and radius, the Newtonian gravitational constant G, and the light speed c. The value of the redshift can be extracted from the measured spectra with a very good accuracy. The red line in Fig. 2 allows no fitting of data.

Fig. 2. Expansion of cosmological photons in the gravitational field of the Earth based on the cosmic chronometers. Red line is not fitted to data.

The Descartes Code and the Properties of the Cosmological Photons in the Free Space

The Descartes’ code describes newly the properties of cosmological photons far from the gravitational fields of planets, stars and galaxies on their long journey through the free space to an observer—Table III.

Table III. The Descartes Code—Cosmological Photon Properties in the Free Space

Descartes’code—cosmological photon properties in the free space
$c_{rotational}={\displaystyle \frac{c_{longitudinal}}{(1+z)}}=c/(1+z)c=299792458m{s}^{-1}$
$c_{longitudinal}=c=\lambda \nu ={\lambda }_0{\nu }_0=299792458m{s}^{-1}$
$\nu ={\displaystyle \frac{{\nu }_0}{1+z}}frequencyofphoton$
$\lambda ={\lambda }_0(1+z)wavelengthofphoton$
$t=t_0(1+z)localtimeofphoton$
$p_z={\displaystyle \frac{h}{{\lambda }_0}}1/(1+z)momentumofphoton$
$E_z=h{\nu }_z=h{\nu }_{0}1/(1+z)energyofphoton$
${\displaystyle \frac{E_z}{t_z}}={\displaystyle \frac{h{\nu }_z}{t_z}}={\displaystyle \frac{h{\nu }_0}{t_0}}{\displaystyle \frac{1}{{(1+z)}^2}}photonenergydilation$
$T=T_{0}1/(1+z)temperatureofphoton$
$L_{\mathrm{\Omega }}^{\circ }={\displaystyle \frac{M^{\circ }}{\pi }}=\sigma /\pi T_z^4=\sigma /\pi T_0^4{\displaystyle \frac{1}{{(1+z)}^4}}Stefan-Boltzmannlaw$

It was discovered that the light curves of distant supernovae of type Ia were stretched in proportion to the redshift factor (1 + z), e.g. [47]–[53]. This effect is interpreted here as the local time of cosmological photons that stretched the width of observed light curves. The Tolman test and the prediction (1 + z)⁻⁴ of the surface brightness decrease with the fourth power of (1 + z) is explained as the result of the Stefan-Boltzmann law, e.g. [54],[55].

The Descartes Code and the Cosmological Photons in the Earth’s Gravitational Field

Descartes’ code describes newly the properties of cosmological photons in the gravitational field of the Earth. From Zwicky’s formula, we can determine the angular diameter distance d_A. From the photon energy dilation and the angular diameter distance we can determine the luminosity distance d_L. Table IV summarizes formulae important for the description of cosmological events.

Table IV. The Cosmological Photons Observed in the Earth’s Gravitational Field

Angular diameter distance dA
$1+z=\exp \left({\displaystyle \frac{H_{\oplus }}{c}}{d}_A\right)\to d_A={\displaystyle \frac{c}{H_{\oplus }}}\ln (1+z)$
Luminosity distance dL
$I\left(z\right)={\displaystyle \frac{L}{4\pi }}{\left[{\displaystyle \frac{H_{\oplus }}{c(1+z)\ln (1+z)}}\right]}^2\to d_L={\displaystyle \frac{c}{H_{\oplus }}}(1+z)\ln (1+z)$
Distance modulus m–M
$m-M=5\log [(1+z)\ln (1+z)]+25+5\log \left({\displaystyle \frac{{\displaystyle \frac{c}{H_{\oplus }}}}{Mpc}}\right)$
Number of galaxies in the volume with radius dA
$N_z=N_0\ast 4\pi /3d_A^3=N_0\ast 4\pi /3{\left({\displaystyle \frac{c}{H_{\oplus }}}\right)}^3{[\ln (1+z)]}^3$
Number of galaxies at z
${\displaystyle \frac{d{N}_z}{dz}}=4\pi /3{\left({\displaystyle \frac{c}{H_{\oplus }}}\right)}^3{\displaystyle \frac{3{[\ln (1+z)]}^2}{1+z}}max\left\{{\displaystyle \frac{3{[\ln (1+z)]}^2}{1+z}}\right\}={\displaystyle \frac{12}{e^2}}atx=e^2-1$

Formulae in Table IV were derived without any fitting and postulating some additional phenomena as e.g. the dark energy or the dark matter. These formulae can be tested on valuable experimental data taken recently by the James Webb Space Telescope (JWST) and compare with the standard cosmological model predictions. Lovyagin et al. [56] tested the formula of the angular diameter distance derived from the Zwicky model together with some traditional formulae and found a very interesting tendency of the Zwicky formula to follow the experimental data better than the traditional cosmological models. Gupta [57] used the Lovyagin et al. data [56] and added to this analysis several cosmological models with fitting of data—“hybrid expansion models.” Fig. 3 is reprinted from the Gupta [57] paper with his fitted “hybrid models” together with Lovyagin et al. [56] curves for the tired light prediction and the ΛCDM model.

Fig. 3. Cosmological model tests with JWSTdata reprinted from Gupta’s inspirative paper [57]. From Lovyagin et al. [56] are data for the tired light prediction (cyan curve) and the ΛCDM prediction (violet curve). From Gupta [57] are his proposed “hybrid expansion models” (blue, green and black dotted curves). The cyan curve based on the angular diameter distance derived from the Zwicky model keeps the trend of data better than the traditional cosmological models based on the expansion of the Universe.

If we replace the rejected Zwicky’s interpretation of the “tired light” by the Descartes’ rotating “light globules” we can introduce the “wavelength expansion” of cosmological photons in the Earth’s gravitational field without any fitting of data. The future analysis of the JWST data can clarify the reality of the Descartes’ model.

The Distance Modulus Formula without Fitting of Data

The distance modulus formula was derived without fitting data by several independent scholars. This formula was tested on many sets of redshifted objects till the redshift z = 5. In our model we have calculated the Hubble constant for the Earth’s gravitational field as H_⊕ = 2.3195*10⁻¹⁸ s⁻¹ (= 71.572 kms⁻¹Mpc⁻¹). Therefore, this version of the distance modulus is without any fitting. Table V gives brief information about those activities with references to the original papers with those analyses. It will be interesting to test the recent data from JWST with this formula.

Table V. The Distance Modulus Formula—the Formula without Fitting of Data

$m-M=5\log [(1+z)\ln (1+z)]+25+5\log \left({\displaystyle \frac{{\displaystyle \frac{c}{H_{\oplus }}}}{Mpc}}\right)$
The expanding spacetime theory	Masreliez	[58]
Machian general relativity	Booth	[59]
Static Universe	Ostermann	[60]
Geometrical interpretation of “c”	Vigoureux et al.	[61]
Tired light hypothesis	Sorrell	[62]
Tenable cosmology	Traunműller	[63]
Exponential tired light	Marosi	[64]

Conclusion

This contribution is based on the old, forgotten, and rejected Descartes’ color theory based on the spin orbital rotation of “light globules”. This Descartes’ model of cosmological photon spinning in the gravitational fields leads to simple but powerful formulae. Cosmological photons in the gravitational fields expand their wavelength—the age of photons increases this expansion, the growing intensity of the gravitational field at the observer site increases this expansion, too.

1. The Descartes’ model of rotating light “globules” (based on the spin-orbital rotation of photons) was selected as the potential candidate to interpret newly effects of the gravitational fields on cosmological photons.
2. The basic properties of cosmological photons (wavelength, frequency, local time, momentum, energy, photon energy dilation) were defined.
3. The value of the Hubble constant for the Earth’s gravitational field was calculated without fitting as H_⊕ = 2.3195*10⁻¹⁸ s⁻¹ (= 71.572 kms⁻¹Mpc⁻¹). The value H_⊕ is determined from the Earth’s mass and radius, the Newtonian gravitational constant G, and the light speed c. We can term this constant as the “geocentric” Hubble constant.
4. The standard set of redshift data from the cosmic chronometers was used to document the expansion of cosmological photons in the Earth’s gravitational field.
5. The simple formulae for the angular diameter distance, the luminosity distance, the distance modulus, the number of galaxies at the redshift z can be tested on numerous available data.
6. The formula for the distance modulus without any fitting was derived.
7. The angular diameter distance formula derived from Zwicky’s model follows the trend in the James Webb Space Telescope (JWST) data better than the traditional cosmological models based on the expansion of the Universe (based on Lovyagin et al. [56] and Gupta [57]).