In a previous paper we postulated that the repulsive force responsible for the universal expansion is associated with the excitation of the empty space (quantum vacuum) and the excitation energy is represented by the energy of the cosmic microwave background (CMB). In this paper, we show that the concept of the repulsive space expanding photon field (i) can successfully be applied to explain the local velocity anomaly of the Milky Way Galaxy as shown by Faber and Burstein (1998) and Tully (1998), (ii) offers a convincing explanation of the still disputed question of the cosmological expansion on local and intergalactic scales discussed by Cooperstock et al. (1998), and (iii) explains the redshift (RS) of the CMB in accordance with the law of energy conservation without the need for dark matter (DM) and dark energy (DE). Probably the most remarkable result of this model (abbreviated as photon/baryon: PB model in the following discussion) is that the individual voids building up the soup-bubble- (SB-) like galaxy distribution are the governing dynamical components of the universal expansion. Further consequence implies that the universe is considerably older than the interpretation of the Hubble constant as expansion velocity suggests.

from gravitational potentials, and cosmological redshifts caused by the universe expanding. The value of a redshift is often denoted by the letter z, corresponding In physics, a redshift is an increase in the wavelength, or equivalently, a decrease in the frequency, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and energy, is known as a blueshift. Three forms of redshift occur in astronomy and cosmology: Doppler redshifts due to the relative motions of radiation sources, gravitational re In physics, a redshift is an increase in the wavelength, or equivalently, a decrease in the frequency, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and energy, is known as a blueshift. Three forms of redshift occur in astronomy and cosmology: Doppler redshifts due to the relative motions of radiation sources, gravitational redshift as radiation escapes from gravitational potentials, and cosmological redshifts caused by the universe expanding. The value of a redshift is often denoted by the letter z, corresponding to the fractional change in wavelength (positive for redshifts, negative for blueshifts), and by the wavelength ratio 1 + z (which is greater than 1 for redshifts and less than 1 for blueshifts). Automated astronomical redshift surveys are an important tool for learning about the large-scale structure of the universe. Redshift and blueshift can also be… In physics, a redshift is an increase in the wavelength, or equivalently, a decrease in the frequency, of electromagnetic radiation (such as light). The opposite change, a decrease in wavelength and increase in frequency and energy, is known as a blueshift. Three forms of redshift occur in astronomy and cosmology: Doppler redshifts due to the relative motions of radiation sources, gravitational redshift as radiation escapes from gravitational potentials, and cosmological redshifts caused by the universe expanding. The value of a redshift is often denoted by the letter z, corresponding to the fractional change in wavelength (positive for redshifts, negative for blueshifts), and by the wavelength ratio 1 + z (which is greater than 1 for redshifts and less than 1 for blueshifts). Automated astronomical redshift surveys are an important tool for learning about the large-scale structure of the universe. Redshift and blueshift can also be related to photon energy and, via Planck's law, to a corresponding blackbody temperature. Examples of strong redshifting are a gamma ray perceived as an X-ray, or initially visible light perceived as radio waves. For these large redshifts, the age of the universe, t(z), is small, meaning that the light was emitted when the universe was young. The cosmological redshift is commonly attributed to stretching of the wavelengths of photons due to the stretching of space. This interpretation can be misleading. As required by general relativity, the cosmological expansion of space has no effect on local physics. There is no term related to expansion in Maxwell's equations that govern light propagation. The cosmological redshift can be interpreted as an accumulation of infinitesimal Doppler shifts along the trajectory of the light. The redshift due to expansion of the universe depends upon the recessional velocity in a fashion determined by the cosmological model chosen to describe the expansion of the universe, which is very different from how Doppler redshift depends upon local velocity. Describing the cosmological expansion origin of redshift, cosmologist Edward Robert Harrison said, "Light leaves a galaxy, which is stationary in its local region of space, and is eventually received by observers who are stationary in their own local region of space. Between the galaxy and the observer, light travels through vast regions of expanding space. Both the photon count rate and the photon energy are redshifted. (See K correction for more details on the photometric consequences of redshift.) Determining the redshift of an object with spectroscopy requires the wavelength of the emitted light in the rest These "redshift-space distortions" can be used as a cosmological probe in their own right, providing information on how structure formed in the universe, and how gravity behaves on large scales. The Hubble law's linear relationship between distance and redshift assumes that the rate of expansion of the universe is constant. However, when the universe was much younger, the expansion rate, and thus the Hubble "constant", was larger than it is today. The DEEP2 Redshift Survey used the Keck telescopes with the "DEIMOS" spectrograph; a follow-up to the pilot program DEEP1, DEEP2 was designed to measure faint galaxies with redshifts 0.7 and above, and it recorded redshifts of over 38,000 objects by its conclusion in 2013. == Effects from physical optics or radiative transfer == The interactions and phenomena summarised in the subjects of radiative transfer and physical optics can result in shifts in the wavelength and frequency of electromagnetic radiation. In such cases, the shifts correspond to a physical energy transfer to matter or other photons rather than being by a transformation between reference frames. Note that the magnitude of the redshifting (blueshifting) effect is not a function of the emitted angle or the received angle of the photon—it depends only on how far radially the photon had to climb out of (fall into) the potential well. It is a natural consequence of conservation of energy and mass–energy equivalence, and was confirmed experimentally in 1959 with the Pound–Rebka experiment. Gravitational blueshift contributes to cosmic microwave background (CMB) anisotropy via the Sachs–Wolfe effect: when a gravitational well evolves while a photon is passing, the amount of blueshift on approach will differ from the amount of gravitational redshift as it leaves the region.

By applying Maxwell’s equations to curved spacetimes, the Planck–Einstein energy–frequency relation for photons, originally formulated in Minkowski space, is generalized for application in Riemann space. According to this relation, photon energy depends not only on the photon frequency but also on the physical speed of photons, which may vary when locally measured in non-inertial static frames. In Minkowski space, the energy of free photons is conserved as neither frequency shifts nor changes in photon speed are observed. In Riemann space, energy of free photons also remains conserved as gravitational redshift is compensated by a corresponding variation in photon speed. The generalized Planck–Einstein relation may have significant astrophysical implications, particularly for gravitational lensing, observations of neutron star mergers, supernovae and quasars, the propagation of light near black holes, and expanding cosmologies.

The quantum theory of the Maxwell free field in Coulomb gauge on the de Sitter expanding universe is completed with the technical elements needed for building a coherent quantum theory of redshift. Paying a special attention to the conserved observables and defining the projection operator selecting the detected momenta it is shown that the expectation values of the energies of the emitted and detected photons comply with the Lemaître rule of Hubble's law. Moreover, the quantum corrections to the dispersions of the principal observables and new uncertainty relations are derived. Recently we proposed an improvement of this approach replacing the special relativity with our de Sitter relativity [ 6 , 7 ]. We obtained thus a redshift formula having a new term combining the cosmological and kinetic contributions in a non-trivial manner [ 8 ]. Moreover, we related the black hole shadow and redshift for the Schwarzschild [ 9 ] and Reissner–Nordstrom [ 10 ] black holes moving freely in the de Sitter expanding universe. The next step might be the quantum theory of redshift but this was never considered because of the real or presumed difficulties in constructing the quantum theory of light in curved backgrounds. Of special interest is the energy operator, which does not commute with the components of the conserved momentum generating new uncertainty relations [ 13 ]. On the other hand, the conformal coordinates are different from the physical ones which are of the Painlevé type [ 14 ] being related to the conformal ones through coordinate transformations depending on time. However, in the quantum theory these transformations change the time evolution picture as we have shown in Refs. [ 15 , 16 , 17 ]. Therefore, for avoiding this difficulty, we restrict ourselves to the conformal coordinates setting the initial conditions at the time \(t_0\) when the scale factor \(a(t_0)=1\) and the physical and conformal space coordinates coincide. Under such circumstances the physical effects may be studied by using exclusively the conserved one-particle operators. In addition, we pay attention to a pair of sensitive technical problems which are crucial in our approach. The first one is related to the momentum-dependent phase of the plane wave solutions of the Maxwell equations which determines the form of the energy operator. In the last part of this section we show how the wave packets can be measured by choosing a suitable projection operator for selecting the momenta of the modes which contribute to the expectation values of the principal conserved observables. The next section is devoted to the quantum redshift for which we derive the new quantum corrections and uncertainty relations. Finally we present \frac{\partial x_{\mathrm{c}}^i}{\partial d^j}\right| _{\mathbf{d}=0}=\delta ^i_j=\omega _{\mathrm{H}}\left( k^i_{(0,j)}+k^i_{(i,4)}\right) ,\nonumber \\ \end{aligned}$$ (14) where \(k^i_{(AB)}=g^{ij}(x_{\mathrm{c}})\,k_{(AB)\,j}\) result from Eq. ( 10 ). This gives rise to the conserved momentum of the classical approach and to the momentum operator of the quantum theory. We shall see in the next section that these isometries transform the energy, angular momentum and dual momentum but preserve the conserved momentum. 3 Null geodesics and redshift We consider now the null geodesics of the photons (with \(m=0\) ) denoting the conserved quantities along these geodesics with capital letters. If we know that the photon is emitted in \(\mathbf{x}'(t_0)=0\) with the momentum \(\mathbf{k}=-\mathbf{n}\,k\) and energy \(k^0=k\) we may ask what the energy and momentum are of this photon measured in the origin O at the final time \(t_f\) when the photon reaches this point. For solving this problem we look first for the conserved momentum that is the same in the points \(O'\) and O , $$\begin{aligned} \mathbf{P}'=\mathbf{P}=\mathbf{k}\,e^{\omega _M t_0}~ ~\rightarrow ~~P=k\,e^{\omega _M t_0},~~~\mathbf{n}_P=-\mathbf{n}, \end{aligned}$$ (29) since this is invariant under translations being associated to their generators. Furthermore, we observe that the choice of the initial time $$\begin{aligned} t_{\mathrm{c}0}=-\frac{1}{\omega _{\mathrm{H}}} ~~\rightarrow ~~ t_0=0, \end{aligned}$$ (30) when \(a=1\) and, consequently, the conformal and physical space coordinates coincide . This simplifies the calculations, allowing us to find the quantities measured by the observers O and \(O'\) at this moment derived from Eqs. ( 23 )–( 25 ). The results are presented in the next table where we introduce the intuitive notations for the initial, \(E_i\) , and final, \(E_f\) , photon energies which are the physical quantities involved in the redshift. These results allow us to recover the Lemaître expression of Hubble’s law giving the redshift z as $$\begin{aligned} \frac{1}{1+z}=\frac{E_{\mathrm{f}}}{E_\mathrm{i}}=1-\omega _{\mathrm{H}} d=1-\frac{d}{l_{\mathrm{H}}}, \end{aligned}$$ (33) and the physical observer–source distance at the time \(t_{\mathrm{f}}\) , $$\begin{aligned} d_{\mathrm{f}}=\frac{d}{-\omega _{\mathrm{H}} t_{\mathrm{cf}}}=\frac{d}{1-\omega _{\mathrm{H}} d}, \end{aligned}$$ (34) which was increasing because of the space expansion during the photon propagation. Thus we revisited the redshift in the particular case when \(O'\) does not have a peculiar velocity. 7 Concluding remarks We presented the complete classical and quantum theory of the Maxwell field minimally coupled to the gravity of the de Sitter expanding universe focusing on the principal effect due to the space expansion, namely the redshift for which we derived the quantum corrections and the principal uncertainty relations. In the actual expanding universe the quantum corrections and the limits of the uncertainty relations are extremely small, since the actual value of \(\omega _{\mathrm{H}}\) (or \(\hbar \,\omega _{\mathrm{H}}\) in SI units) is of the order \(10^{-33} eV\) such that it is less probably identified in astronomic observations.