Let V be the relative speed along the line of sight between the source and the observer, ${\nu }_e,\nu$ are emitted and observed frequencies correspondingly. With these terms the relativistic redshift z is described by the well-known formula:

$z=\sqrt{\frac{1+\frac{V}{c}}{1-\frac{V}{c}}}-1$, ${\nu }_e=\nu \sqrt{\frac{c+V}{c-V}}$, $\frac{{\nu }_e}{\nu }=1+z$ (9)

where V is assumed to be a time-dependent variable. The time derivative of the observed frequency is defined by

$\frac{\text{d}\nu }{\text{d}t}=-\frac{{\nu }_e^{2}c}{\nu {\left(c+V\right)}^2}\frac{\text{d}V}{\text{d}t}=-{\nu }_e\left(1+z\right)\frac{c}{{\left(c+V\right)}^2}\frac{\text{d}V}{\text{d}t}$ (10)

For a non-relativistic case ( $V\ll c$ ( $z\ll 1$)) we have

$\frac{\text{d}\nu }{\text{d}t}\approx -\frac{{\nu }_e}{1+z}\frac{\text{d}z}{\text{d}t}\approx -{\nu }_e\left(1-z\right)\frac{\text{d}z}{\text{d}t}$, $z=\frac{V}{c}$ (11)

It is evidently that $\frac{\text{d}\nu }{\text{d}t}<0$ because z > 0, and we have a decrease in frequency.

In this case the speed V can be reversed to -V in (9). We have

$z=\sqrt{\frac{1-\frac{V}{c}}{1+\frac{V}{c}}}-1$, ${\nu }_e=\nu \sqrt{\frac{c-V}{c+V}}$ (12)

In the non-relativistic case, we have the same expression for the time derivative of the observed frequency:

$\frac{\text{d}\nu }{\text{d}t}\approx -\frac{{\nu }_e^{2}c}{\nu {\left(c-V\right)}^2}\frac{\text{d}V}{\text{d}t}\approx -{\nu }_e\left(1-z\right)\frac{\text{d}z}{\text{d}t}$, $z=-\frac{V}{c}$ (13)

The derivative $\frac{\text{d}\nu }{\text{d}t}>0$, because $z<0$. Thus, we have an increase in frequency.

We consider the photon, acted upon by a gravitational force, generated by mass M with a spherical distribution of matter density in the weak-field approximation. It can be described by the Hamilton-Jacobi equation in general relativity in the form of the relativistic dispersion equation:

$g^{\alpha \beta }{P}_{\alpha }{P}_{\beta }=m_{\gamma }^2{c}^2$ (14)

Here $g^{\alpha \beta }$-contravariant components of the metric tensor of the gravitational field, $P_{\alpha },P_{\beta }$ are components of 4-vector of the momentum. In the meantime, we have the Schwarzschild metric in spherical coordinates ( $ct,r,\theta ,\phi$) with $r$ as the radial distance from the center [19] [29] [30] :

$\text{d}{s}^2=\left(1-\frac{r_g}{r}\right)c^2\text{d}{t}^2-{\left(1-\frac{r_g}{r}\right)}^{-1}\text{d}{r}^2-r^2\left(\text{d}{\theta }^2+{\sin }^2\theta \text{d}{\phi }^2\right)$ (15)

The contravariant components in Equation (14) in the weak-field approximation ( $r_g\ll r$ with $r_g=2GM/c^2$ being a gravitational radius) are:

$g^{00}={\left(1-\frac{r_g}{r}\right)}^{-1}\approx 1+\frac{r_g}{r},g^{11}=-\left(1-\frac{r_g}{r}\right),g^{22}=-\frac{1}{r^2},g^{33}=\frac{1}{r^2{\sin }^2\theta }$ (16)

If we choose $\theta =\pi /2$ as the plane of motion of the photon, then we can set $P_2=P_{\theta }=0$. Magnitudes of the next components of 4-vector momentum are $P_0=E/c$, $P_1=P_r$, $P_3=P_{\phi }$. The last one is the generalized angular momentum of the photon which in the weak-field approximation takes the form [14] $P_{\phi }=\gamma m_{\gamma }vr$, where $\gamma$ is the Lorentz -factor: $\gamma =1/\sqrt{1-v^2/c^2}$. Substituting (16) and the components of the 4-vector momentum into (14) we obtain:

$\left(1+\frac{r_g}{r}\right)\frac{E^2}{c^2}-\left(1-\frac{r_g}{r}\right)P_r^2-\frac{P_{\phi }^2}{r^2}=m_{\gamma }^2{c}^2$ (17)

This equation can be rewritten in the following form if we take $E=h\nu$:

$\left(1+\frac{r_g}{r}\right)\frac{{\left(h\nu \right)}^2}{c^2}-m_{\gamma }^2{c}^2=\left(1-\frac{r_g}{r}\right)P_r^2+\frac{m_{\gamma }^2{v}^2}{1-\frac{v^2}{c^2}}$ (18)

Now we can apply equation (2) and definition ${\nu }_0=\frac{m_{\gamma }{c}^2}{h}$ to (18) and bring it down to the form:

${\left(\frac{h\nu }{c}\right)}^2\frac{r_g}{r}{\left(1-\frac{r_g}{r}\right)}^{-1}\approx {\left(\frac{h\nu }{c}\right)}^2\frac{r_g}{r}=P_r^2$ or $P_r=\sqrt{\frac{r_g}{r}}\frac{h\nu }{c}$ (19)

The force acting on the photon can be found by differentiating (19) with respect to the coordinate time:

$F\left(r\right)=\frac{\text{d}{P}_r}{\text{d}t}=\sqrt{\frac{r_g}{r}}\frac{h}{c}\left(\frac{\text{d}\nu }{\text{d}t}-\frac{\nu }{2r}\frac{\text{d}r}{\text{d}t}\right)$ (20)

The first term in parentheses shows the rate of gravitational shift in frequency and the second one consists of the radial derivative—the term describing the rate of change in the radial displacement of the photon while bending its trajectory. Numerical values of the derivatives in (20) depend on specific astrophysical scenarios that we are going to consider below in the text.

If we set $P_{\phi }=0$ in Equation (17) the radial component of the momentum $P_r$, describing a “free fall motion” of the photon, can be found from the equation:

$P_r=\sqrt{\left(1+\frac{2r_g}{r}\right){\left(\frac{h\nu }{c}\right)}^2-\left(1-\frac{r_g}{r}\right)m_{\gamma }^2{c}^2}=\frac{h\nu }{c}\sqrt{1+\frac{2r_g}{r}-\left(1-\frac{r_g}{r}\right){\left(\frac{{\nu }_0}{\nu }\right)}^2}$ (21)

In the case of massless photons ( ${\nu }_0=0$), it boils down to the much simpler approximate equation

$P_r=\frac{h\nu }{c}\left(1+\frac{r_g}{r}\right),$ (22)

For the gravitational field near the Earth surface ( $r=6.374\times {10}^6\text{m}$) with $r_g=9.067\times {10}^{-3}\text{m}$ the ratio $r_g/r=1.422\times {10}^{-9}$ can be ignored compared with 1 and we simply have $P_r=h\nu /c$—the result that could have been used straightforwardly. Taking time derivative of the momentum, we can find the force of Earth gravity (weight) acting of the photon:

$F=2.209\times {10}^{-42}\frac{\text{d}\nu }{\text{d}t}$ (23)

This formula is the same as (5) and (8).

Historically that was the second confirmed prediction of general relativity. The testing was performed by two English groups of astronomers during the solar eclipse on May 29 of 1919. They have observed stars from Hyades cluster in Taurus constellation. The results obtained by both groups are summarized in the article [17] . We’ll try to estimate the magnitude of the force acting on the photon emitted by the star ${\kappa }^1$ Tauri which is classified as A7 IV spectral type in The SkyLive [31] with the effective temperature T_eff = 8748 ⁰K. According to Rhee [32] , spectral type of the star is A7 IV-V and T_eff = 9000 ⁰K, but Kaler [33] points to T_eff = 8290 ⁰K. For the effective temperature in 9000 ⁰K, the peak frequency of radiation (if the photon is massless) for the star, according to Wien’s displacement law, is

${\nu }_{\max }=5.879\times {10}^{10}{T}_{eff}\left(\text{Hz}/\text{K}\right)=5.291\times {10}^{14}\text{Hz}$ (24)

This frequency is used below in the text as the emitted frequency ${\nu }_e$.

For our purpose we can consider the photons traveling from infinity to the Sun and passing it by at a distance r (impact parameter). This situation corresponds to the blueshift of the emitted radiation while it is entering the solar gravitational field despite the radiation itself could have been redshifted if an emitting source is moving away from the Sun (we consider the event in the reference frame centered in the Sun). The deflection angle for massive photons was calculated by Lowenthal [14] and can be obtained straightforwardly from the deflection formula for ultra-relativistic particles [34] as it is mentioned in our work [15] .

Here we consider only weak-field approximation of general relativity near the gravitational center that leads to the Newtonian regime characterized by the condition $r_g\ll r$ ( $r_g=2GM/c^2$ is the Schwarzschild (or gravitational) radius, M is the mass of a gravitating center), and the slight blueshift in the frequency of an approaching massless photon, which can be described by the formula, well-known from any course of general relativity (see for example [19] ):

$\nu ={\nu }_e\left(1-\frac{\Delta \phi }{c^2}\right)$, (25)

where $\nu ,{\nu }_e$ are observed and emitted frequencies, $\Delta \phi =\phi =-GM/r$ is the difference between gravitational potentials at a distance $r$ from the center and at infinity where the potential is zero.

The light deflection (and gravitational lensing) can be considered as a scattering of the photon on a gravitational potential, approaching the center of gravity from infinity and going to infinity (or to the observer) restoring its initial frequency. When the photon approaches mass M, the change in frequency takes place on the short arclength $\Delta l$ near the turning point where deflection of the photon’s straight pathway occurs (detailed description of the light ray trajectories in Schwarzchild metric is given by [29] (fig. 6.1), [35] ), and the particle displaces slightly toward the gravitating mass in a radial direction by $\Delta r$. If ${\theta }_E=4GM/r{c}^2=2r_g/r$ is the Einstein’s deflection angle, then $\Delta l\approx r\theta /2\approx r_g$, and the characteristic time of change in frequency ( $\Delta \nu =\nu -{\nu }_e=-{\nu }_e\phi /c^2={\nu }_e{r}_g/2r$) is $\Delta t~\Delta l/c=r_g/c$. In the weak-field approximation $r_g\ll r$ and $\Delta \nu \ll {\nu }_e$. For the estimate of the first derivative in (20), we have:

$\frac{\text{d}\nu }{\text{d}t}~\frac{\Delta \nu }{\Delta t}=\frac{{\nu }_{e}c}{2r}$ (26)

For the second derivative $\text{d}r/\text{d}t\approx \Delta r/\Delta t$ we have, considering that the deflection angle is extremely small, $\Delta r\approx r{\theta }_E^2/2$ and $\Delta r/\Delta t\approx 2r_{g}c/r$. Thus, $\left(\nu /2r\right)\left(\text{d}r/\text{d}t\right)=\nu r_{g}c/r^2$, where the observed frequency can be written as $\nu ={\nu }_e\left(1-r_g/2r\right)\approx {\nu }_e$. We can see that the second term in (20) is significantly small compared with $\text{d}\nu /\text{d}t$ and we can ignore it. It enables us to rewrite (20) in the form

$F=\sqrt{\frac{r_g}{r}}\frac{h}{c}\frac{{\nu }_{e}c}{2r}\text{N}\cdot {\text{s}}^2=2.209\times {10}^{-42}\sqrt{\frac{r_g}{r}}\frac{{\nu }_{e}c}{2r}$ (27)

We use (x = 0.334, y = 0.472) coordinates of ${\kappa }^1$ Tauri star on the photographic plate (as it is measured by the Eddington’s group in conventional units, unit = 50') to estimate its angular distance from the Sun. It turned out to be equal to 28’.911, or $r=1.841R_0$, where $R_0=6.95\times {10}^8\text{m}$ is the radius of the Sun. Substituting all these numbers with the speed of light $c$ and (24) for ${\nu }_e$ into (27), we obtain for the force acting on the massless photon in the visible part of the spectrum: $F=1.444\times {10}^{-31}\text{N}$.

The first time the effect of gravity on radiation frequency was experimentally confirmed by Pound, Rebka [18] . In their experiment the distance between the source and the receiver was taken to be $l=22.5\text{m}$ in vertical direction. The radiation of 14.4 keV energy was emitted by the iron isotope ⁵⁷Fe. This is equivalent to the frequency ${\nu }_e=3.482\times {10}^{18}\text{Hz}$. The difference in gravitational potential between the two locations is $gl$, where $g=9.81\text{m}/{\text{s}}^{\text{2}}$ is the free fall acceleration. It produces the expected relative gravitational red/blueshift equal to $\Delta \nu /{\nu }_e\approx 2.45\times {10}^{-15}$ which has been successfully measured with 10% of accuracy. Then the accuracy was improved up to 1% [36] and 0.007% [37] .

Despite the effect is conventionally interpreted in terms of gravitational time dilation there is no restriction to interpret it in terms of force acting on the photon in its free fall motion near the Earth (see also [16] ). The time it takes for the light to travel the distance of 22.5 m is $\Delta t=l/c=7.5\times {10}^{-8}\text{s}$. It gives us the estimate for the time derivative of the frequency:

$\frac{\text{d}\nu }{\text{d}t}\approx \frac{\Delta \nu }{\Delta t}=1.137\times {10}^{11}{\text{s}}^{-2}$ (28)

Because the Earth’s gravitational field is weak and homogeneous we can estimate the force acting on a massless gamma-ray photon by substituting the number in (28) into (8). We obtain $F=2.512\times {10}^{-31}\text{N}$. This force is practically immeasurable, making 10⁻⁸ of the experimental threshold 4.2 × 10⁻²³ N [38] . For the frequency of the photon in the visible part of the spectrum used above, we find $F=3.534\times {10}^{-35}\text{N}$.

The same “weight” of the photon can be found straightforwardly, if we formally multiply its mass equivalent $m=h\nu /c^2$ by the acceleration of gravity

$F=\frac{h\nu }{c^2}g=7.222\times {10}^{-50}\nu$ (29)

For the same frequency of gamma radiation we obtain $F=2.515\times {10}^{-31}\text{N}$ and for the visible radiation with ${\nu }_e=5.0\times {10}^{14}\text{Hz}$ we find $F=3.611\times {10}^{-35}\text{N}$ as above in the text with slight difference. Comparing (5) and (26), we can write for a photon near the Earth gravity:

$\frac{\text{d}\nu }{\text{d}t}=3.27\times {10}^{-8}\nu$ (30)

Formula (29) can be derived from the expression in 3-form of the force acting on a particle in the homogeneous gravitational field [19] (see also discussion in [16] ):

$F=\frac{m_{\gamma }{c}^2}{\sqrt{1-\frac{v^2}{c^2}}}\left\{-\nabla \ln \left(\sqrt{1+\frac{2\phi }{c^2}}\right)+\sqrt{1+\frac{2\phi }{c^2}}\frac{v}{c}\times \left(\nabla \times g\right)\right\}$ (31)

The last term in (31) is equivalent to the Coriolis force and can be dropped in the first approximation. In the weak field approximation $2\phi /c^2\ll 1$ and $\text{ln}\sqrt{1+2\phi /c^2}\approx \phi /c^2=-GM/r{c}^2$. From the optical dispersion in vacuum, it follows that the Lorentz factor in (31) can be written simply as $\nu /{\nu }_0$, and because $m_{\gamma }/{\nu }_0=h/c^2$, (31) takes the form:

$F=\frac{h\nu }{c^2}\frac{GM}{r^2}$. (32)

which is the law of universal gravitation applied to the interaction between a photon of mass (mass equivalent) $h\nu /c^2$ and the Earth. For the Earth surface, if $r$ is taken equal to the radius of the planet, (32) turns into (29).

In this section we are going to consider a gravitational blueshift of the photon’s frequency in the reference frame connected with the center of inertia of a galaxy cluster. In this case, while estimating the magnitude of the force, acting on the photon approaching the cluster, we can ignore the effect of cosmological expansion in further considerations. The frequency of the photon, approaching (or entering) the galaxy cluster, is determined by the relative radial velocity between an emitting source and the cluster and can easily be calculated for specified objects. In this paper it is assumed to be equal (for demonstration purposes only) to the standard frequency in the visible part of the spectrum we used above in the text.

The gravitational lensing, produced by galaxy clusters, is accompanied by a blueshift in the frequency of the radiation emitted by the background source subject to focusing. Regarding this problem, we must recall, that the dynamics of galaxy clusters is assumed to be determined by dark matter (DM) that constitutes a very significant fraction of the mass of clusters (80% - 90%) [39] , whose density profile is traditionally described by NFW model (Navarro, Frenk, White [40] [41] ; see also [42] for observations and models). DM is the main cause of gravitational lensing in these systems, and, as such, is the leading contributor in the gravitational potential of the cluster and the light frequency shift compared with the baryonic matter contribution. It is important to mention that NFW model was confirmed by other studies based on gravitational lensing methodology (see, for example, [43] - [45] ).

As the standard emitted frequency, we choose ${\nu }_e=5.0\times {10}^{14}\text{Hz}$ in the visible part of the spectrum. As an example of a typical cluster of galaxies we choose Abell 370 with z = 0.375 that shows we are dealing with the relativistic object (its recession velocity is V = 92432 km/s) [46] . For our estimates instead of the radius of the cluster (which is not clearly defined parameter yet) we are going to take the virial radius $R_{vir}=r_{200}~2.55\text{Mpc}$ [47] [48] as the characteristic size of the system. The subscript “200” points to the ratio of the average density of the cluster to the critical density of the Universe ${\rho }_{crit}$ at the shown distance and the cluster redshift [40] . Although the deflection of light by a galaxy cluster is usually explained in terms of spacetime curvature we can formally interpret this phenomenon in terms of gravitational attraction of the photon by the cluster in flat spacetime in compliance with the equivalence principle.

Gravitational potential of DM distribution with NFW density profile [41]

$\rho \left(r\right)=\frac{{\rho }_{crit}{\delta }_c}{\frac{r}{R_s}{\left(1+\frac{r}{R_s}\right)}^2}$ (33)

at the distance $r$ from its center can be found by solving Poisson’s equation $\Delta \Phi =4\pi \varrho$. Here ${\delta }_c$ is a characteristic (dimensionless) density of the cluster, $R_s$ is a scale radius. For an infinitely extended halo in the case of spherical symmetry, the potential is defined by the integral relation

$\Phi \left(r\right)=-4\pi G\left[\frac{1}{r}{\displaystyle {\int }_0^r{r^{\prime }}^2\rho \left(r^{\prime }\right)\text{d}{r}^{\prime }}+{\displaystyle {\int }_r^{\infty }{r}^{\prime }\rho \left(r^{\prime }\right)\text{d}{r}^{\prime }}\right]$ (34)

Substituting (33) in (34) and performing simple integration one can obtain:

$\Phi \left(r\right)=-\frac{4\pi G{\rho }_0{R}_s^3}{r}\ln \left(1+\frac{r}{R_s}\right)$, (35)

where ${\rho }_0={\rho }_{crit}{\delta }_c$ is called a characteristic density and $R_s$ is the scale radius. From (35) it follows that

$\underset{r\to \infty }{\lim }\Phi \left(r\right)=0$, $\underset{r\to 0}{\lim }\Phi \left(r\right)=-4\pi G{\rho }_0{R}_s^2$ (36)

The mass within the virial radius $R_{vir}$ can easily be obtained by direct integration:

$M\left(R_{vir}\right)=4\pi {\rho }_0{R}_s^3\left[\ln \left(1+c\right)-\frac{c}{1+c}\right]$, (37)

where $c$ is the concentration parameter such that $c=R_{vir}/R_s$. From (35) and (37) we can deduce

$\Phi \left(r\right)=-\frac{GM\left(R_{vir}\right)}{r}\frac{\ln \left(1+\frac{r}{R_s}\right)}{\ln \left(1+c\right)-\frac{c}{1+c}}$ (38)

For many values of the concentration parameter the logarithmic term in (37) is of the order of unity, varying between 0.5 and 5.5 [49] . As we stated in section 4.1, the change in frequency caused by a gravitating mass takes place along a short arclength where the light deflection takes place. Thus, the time derivative of the frequency of the upcoming radiation is described by the formula like (26).

We use the same description of the light deflection influenced by gravity near a spherically symmetric gravitating mass as we did in section 4.2 (ignoring microlensing effects) and assume that the largest $r\approx r_E~R_{vir}$, where $r_E$—the radius of the Einstein’s ring [50] . Outside a sphere of this radius light travels in a straight line in compliance with the law of inertia. In this approximation we can apply formula (27) to estimate the force acting on the photon entering the galaxy cluster from a background source. The change in frequency due to the change in the gravitational potential from zero to $\Phi \left(R_{vir}\right)$ is

$\Delta \nu =2{\nu }_e\frac{\Phi \left(R_{Vir}\right)}{c^2}=\frac{2{\nu }_{e}GM\left(R_{vir}\right)}{R_{vir}{c}^2}=\frac{{\nu }_e}{R_{vir}}{r}_g$ (39)

Thus, we have for the estimate of the derivative $\text{d}\nu /\text{d}t$ in the weak-field approximation:

$\frac{\text{d}\nu }{\text{d}t}\approx \frac{\text{Δ}\nu }{\text{Δ}t}=\frac{{\nu }_{e}c}{2R_{vir}}$, $\Delta t=2r_g/c$ (40)

As we have stated above, the positive sign of the derivative shows an increase in frequency of radiation (gravitational blueshift) as the background photon approaches and enters the galaxy cluster. If we take for the mass of the galaxy

cluster typical number $M\left(R_{vir}\right)~{10}^{15}{M}_0~{10}^{45}\text{kg}$ ( $M_0$—mass of the Sun) [40] [43] [44] [51] , we obtain $\frac{\text{d}\nu }{\text{d}t}\approx 0.954{\text{s}}^{-2}$. By substituting all above mentioned numbers into (27) we find for the magnitude of the force $F~{10}^{-44}\text{N}$. Of course, a real situation with gravitational lensing is more complicated and uncertainties in the parameters we just used, can change the estimate of the magnitude of the force presented above.

The redshift of a galaxy is a time dependent variable for any model of the Universe, uniformly expanding or accelerating. The accelerated expansion may be caused by some type of a repulsive force generated, for example, by dark energy in the late-time accelerating models of the Universe [52] or another nature [53] . We can state that time dependence of the redshift is caused by time dependence of the observed frequency because an emitted frequency from recessing source can be considered constant. This situation can be interpreted as conditioned by a force acting on the photon even if its rest mass is zero. The force manifests itself not through the change in speed (which is impossible), but through the time-dependent change in frequency. Our goal is to estimate the magnitude of the force.

Let $a\left(t\right)$ be the cosmological scale factor presented in the Freedman-Robertson-Walker metric the numerical value of which is linked to the model of inflationary Universe [54] - [56] . As we know, $\stackrel{\dot{}}{a}\left(t\right)/a\left(t\right)=H\left(t\right)$, where $H\left(t\right)$ is the Hubble parameter, the current value of which will be taken equal to $H_0=70$ km/s Mpc as we stated above in the text, and $a(t_0)=1$, $t_0$—present time. If ${\nu }_e$ and ν are emitted and observed frequencies with the redshift $z$, then we can write the following well known relations

$a\left(t\right)=\frac{1}{1+z}$, $z=\frac{{\nu }_e}{\nu }-1$, (41)

from which we have

$\stackrel{\dot{}}{a}=-\frac{\text{d}z/\text{d}t}{{\left(1+z\right)}^2}=-a{\left(t\right)}^2\frac{\text{d}z}{\text{d}t}=a{\left(t\right)}^2\frac{{\nu }_e}{{\nu }^2}\frac{\text{d}\nu }{\text{d}t}=\frac{1}{{\nu }_e}\frac{\text{d}\nu }{\text{d}t}$ (42)

Hence it follows

$\frac{\text{d}\nu }{\text{d}t}=\frac{{\nu }^2}{{\nu }_e}\frac{H\left(t\right)}{a\left(t\right)}=\nu H\left(t\right)$ (43)

For an arbitrary cosmological epoch with $z=z\left(t\right)$ we have the well-known expression for the Hubble parameter, derived from Friedmann equations within standard ΛCDM—model of the Universe which is more preferable model [57] , [58] [59] , where the contribution of radiation is ignored:

$H\left(z\right)=H_0\sqrt{{\Omega }_m{\left(1+z\right)}^3+{\Omega }_k{\left(1+z\right)}^2+{\Omega }_{\Lambda }}$ (44)

Here $H_0$ is the current Hubble constant, ${\Omega }_m,{\Omega }_k,{\Omega }_{\Lambda }$ are correspondingly matter, curvature and dark energy density parameters, Λ is the cosmological constant. For the flat space-time ${\Omega }_k=0$ (according to [24] ${\Omega }_k\approx -0.002$) and (44) can be reduced to

$H\left(z\right)=H_0\sqrt{{\Omega }_m{\left(1+z\right)}^3+{\Omega }_{\Lambda }}$ (45)

The density parameters satisfy the condition ${\Omega }_m+{\Omega }_{\Lambda }=1$. As their estimates we will use the average of measurements conducted within [24] [26] [60] ): ${\Omega }_m=0.3$ and ${\Omega }_{\Lambda }=0.7$. For $z\ll 1$ expression (45) can be simplified:

$H\left(z\right)\approx H_0\left(1+\frac{3}{2}{\Omega }_{m}z\right)$ (46)

Then we have from (43) and (8)

$\text{d}\nu /\text{d}t=\nu H_0\left(1+\frac{3}{2}{\Omega }_{m}z\right)$, $F=2.209\times {10}^{-42}\nu H_0\left(1+\frac{3}{2}{\Omega }_{m}z\right)\left(\text{N}\cdot {\text{s}}^2\right)$ (47)

This formula shows that the magnitude of the force acting on the photon, as was previously stated, is proportional to its frequency and increases with an increase in redshift. For Markarian galaxy Mrk 421 $z=0.031$ [61] . The galaxy is the BL Lac type object being a very strong source of γ-rays with the frequency $\nu ~{10}^{26}\text{Hz}$ [21] [62] [63] . By substituting these numbers into (47) we obtain $F~{10}^{-34}\text{N}$. For visible radiation with $\nu =5\times {10}^{14}\text{Hz}$ $F~{10}^{-45}\text{N}$, which is of the same order of magnitude as we found in section 4.

It follows from the above that a fictious/(or real force) acting upon the photon (massive or massless) and causing the change in its frequency always exists in the expanding Universe, no matter if the Universe is accelerating or not. The radial speed of a recessing galaxy increases with time as the galaxy moves farther away from us according to Hubble’s law.

In this section we are going to consider the effect of massiveness of a traveling photon on the change of its frequency. We are going to find the difference in change of frequency shift between the massless and massive photon.

Let ${\nu }_e$ be the frequency of radiation emitted by a galaxy with recession velocity V and $\nu$ is the frequency recorded by an observer. If the motion is relativistic then the redshift of the galaxy is defined by the standard expression (9) shown above, if the photon is massless. The massless photon always travels with the invariant speed of light, no matter if its source is at rest or is set into motion. It can easily be seen from the relativistic composition law of velocities. We are going to refer to this law for the massive photon. Let $v_{\gamma }$—the velocity of the emitted massive photon in the rest frame of the source, moving with a velocity V with respect to the observer. The velocity $u_{\gamma }$ measured by the observer can be found from the relativistic composition law:

$u_{\gamma }=\frac{V+v_{\gamma }}{1+\frac{V{v}_{\gamma }}{c^2}}$ (48)

We can write two vacuum dispersion relations:

$v_{\gamma }=c\sqrt{1-{\left({\nu }_0/{\nu }_e\right)}^2}$, $u_{\gamma }=c\sqrt{1-{\left({\nu }_0/\nu \right)}^2}$ (49)

If we substitute these expressions into (48) we will arrive at the equation:

$\sqrt{1-{\nu }_0^2/{\nu }^2}=\frac{\sqrt{1-{\nu }_0^2/{\nu }_e^2}-\frac{V}{c}}{1-\sqrt{1-{\nu }_0^2/{\nu }_e^2}\frac{V}{c}}$, (50)

which can be simplified if we consider that in most cases of astrophysical interest the following condition holds: ${\nu }_0\ll {\nu }_e,\nu$. Thus, we can approximate

$\sqrt{1-{\nu }_0^2/{\nu }^2}\approx 1-\frac{{\nu }_0^2}{2{\nu }^2}$ and $\sqrt{1-{\nu }_0^2/{\nu }_e^2}\approx 1-\frac{{\nu }_0^2}{2{\nu }_e^2}$ (51)

and rewrite (50) in the form

$\frac{{\nu }_e^2}{{\nu }^2}\approx \frac{1+\frac{V}{c}}{1-\frac{V}{c}}\left(1-\frac{{\nu }_0^2}{2{\nu }_e^2}\frac{V}{c-V}\right)$ (52)

The first fraction on the right side in (52 is simply ${\left(1+z\right)}^2$, where $z$ is the redshift of a massless photon. If we introduce a variable $z^{\prime }$ for the redshift of a massive photon, then we can rewrite (52) in the following way

${\left(z^{\prime }+1\right)}^2\approx {\left(z+1\right)}^2\left(1-\frac{{\nu }_0^2}{2{\nu }_e^2}\frac{V}{c-V}\right)$, (53)

from which, after taking square root of both sides, we approximately obtain:

$\text{Δ}z\approx z^{\prime }-z\approx -\left(z+1\right)\frac{{\nu }_0^2}{4{\nu }_e^2}\frac{V}{c-V}$ (54)

We can conclude that the massiveness of the photons reduces the redshift (heavy photons are slightly more reluctant in getting reddish than massless photons). This effect is extremely weak and virtually unnoticeable. If we use extreme numbers (lowest and highest frequencies) for ${\nu }_0~{10}^{-3}\text{Hz}$, ${\nu }_e~{10}^{26}\text{Hz}$, presented in publications [21] [64] [65] , and the highest recession velocity ever recorded for galaxies (galaxy JADES-GS-z14.3 [66] [67] ) with the redshift of 14.32 (which makes $V=297454.44\text{km}/\text{s}$) then we will come to the estimate of $\left|\text{Δ}z\right|~{10}^{-58}$. Despite this result points to the additional slight component in the observed Doppler redshifts caused exclusively by “massiveness” of the photons it is far below experimental and observational possibilities to be measured (existing uncertainties and nominal precision in redshifts of galaxies are ~10⁻⁴ and ~10⁻⁶ respectively [68] [69] ). The fact that the difference in redshifts between massive and massless photons depends on the square of the ratio of the rest and emitted frequencies keeps the number unmeasurable even for the ratio of much higher order of magnitudes. The photon acceleration phenomenon works like a process of energy transfer from one part of the spectrum to another [9] much like the refraction angle (coefficient of refraction and the speed) of the radiation, passing through a spectral prism, depends on its own frequency.