**4. Torsion–Neutron Low-Energy Interactions**

Our analysis carried out above may give an impression that after the absorption of torsion by the cosmological constant, the Einstein–Cartan gravitational theory reduces to Einstein's gravitational theory [69]. Such an impression can be real only in case of the absence of fermions. As has been shown in [40–42], there is a huge variety of minimal and nonminimal low-energy torsion–neutron interactions. The torsion tensor field T*σµν*, being a tensor of the third rank and antisymmetric with respect to indices *µ* and *ν*, i.e., T*σµν* = −T*σνµ*, is defined by 24 independent components: (i) four vectors E *<sup>µ</sup>* = (<sup>E</sup> 0 , <sup>E</sup><sup>~</sup> ), (ii) four axial vectors <sup>B</sup> *<sup>µ</sup>* = (<sup>B</sup> 0 , <sup>B</sup><sup>~</sup> ) and (iii) 16 tensors <sup>M</sup>*σµν* [35,36,44] (see also [40–42]). The effective low-energy torsion–neutron potentials are presented in the form of expansion in powers of 1/*m*, where *m* is the neutron mass, and restricted by the terms of order *O*(1/*m*), by using the Foldy–Wouthuysen (FW) canonical transformations [89]. The most interesting effective low-energy torsion–neutron interactions are induced in the rotating coordinate systems, which can be used for

experimental probes of torsion in terrestrial laboratories [48,60]. It is important to emphasize that a part of these effective low-energy torsion–neutron interactions provide a violation of time-reversal invariance [42], which can be probed in the terrestrial laboratories.

According to [42], in the coordinate system rotating with an angular velocity Ω~ the time component E<sup>0</sup> of the 4-vector E *<sup>µ</sup>* of the torsion field induces the T–odd, i.e., violating time reversal symmetry, optical potential

$$\boldsymbol{\Phi}\_{\rm eff}^{(\rm T-odd)} = -i \frac{4}{3} \frac{\mathcal{E}\_0}{m} \vec{\mathcal{S}} \cdot \vec{\Omega} \, \tag{62}$$

where ~*S* = <sup>1</sup> 2 ~*σ* is the operator of the neutron spin and~*σ* are 2 × 2 Pauli matrices [90]. As has been shown in [47], because of the T–odd interaction (Equation (62)), the cross section for low-energy neutron–nucleus scattering, caused by the beam of polarized neutrons passing through a spinning cylinder, should acquire the correction [39]

$$
\Delta\sigma\_{\rm TV}(\Omega, p) = \frac{8\pi}{3\sqrt{2}} \mathcal{E}^0 \mathcal{R}^2 L \frac{\Omega}{p} \,'\,\tag{63}
$$

where *R* and *L* are the radius and length of the spinning cylinder, and *p* is a neutron momentum (for a detailed discussion of the *p*-dependence of ∆*σ*TV(Ω, *p*) we refer to [47] below Equation (5)). The aim of the proposed experiment is a search for the Ω-dependent part of the helicity-dependent part of the difference of the cross sections for neutron–nucleus scattering, caused by neutrons polarized parallel and antiparallel to the neutron beam axis coinciding with the axis of a spinning cylinder. For contemporary experimental abilities, such a T–odd correction allows one to probe the time component of the 4-vector part of the torsion field at the level of sensitivity of about |E0| ∼ <sup>10</sup>−<sup>32</sup> GeV. This is a few orders of magnitude better in comparison to the estimate obtained in [44].

Another part of the effective low-energy torsion–neutron potentials, which is not proportional to 1/*m*, can be used for probes of the components of the torsion field in the qBounce experiments dealing with ultracold neutrons (UCNs) bouncing in the gravitational field of the Earth [49–55] (see also [48]). As an example, we consider the effective low-energy potential of the time-component B<sup>0</sup> (pseudoscalar) of the 4-axial vector B *<sup>µ</sup>* and the time-time-space-components (M<sup>~</sup> )*<sup>k</sup>* <sup>=</sup> <sup>M</sup>00*<sup>k</sup>* of the tensor <sup>M</sup>*σµν* [42]

$$\boldsymbol{\Phi}\_{\rm eff} = \frac{1}{3} \mathcal{B}\_0 \vec{\mathcal{S}} \cdot \left( \vec{\Omega}\_{\oplus} \times \left( \vec{\mathcal{R}}\_{\oplus} + \vec{r} \right) \right) - \frac{1}{2} \vec{\mathcal{S}} \cdot \left( \vec{\mathcal{M}} \times \left( \vec{\Omega}\_{\oplus} \times \left( \vec{\mathcal{R}}\_{\oplus} + \vec{r} \right) \right) \right), \tag{64}$$

where <sup>Ω</sup><sup>~</sup> <sup>⊕</sup> and <sup>~</sup>*R*<sup>⊕</sup> are the angular velocity and the radius vector of the Earth as they are shown in Figure 1. Then,~*r* is the radius–vector of the UCN in the laboratory.

The experiments with UCNs, bouncing in the gravitational field of the Earth, are being performed in the laboratory at Institut Laue Langevin (ILL) in Grenoble. The ILL laboratory is fixed to the surface of the Earth in the northern hemisphere. Following [91–95] we choose the ILL laboratory or the standard laboratory frame with coordinates (*t*, *x*, *y*, *z*), where the *x*, *y* and *z* axes point south, east and vertically upwards, respectively, with northern and southern poles on the axis of the Earth's rotation with the Earth's sidereal frequency <sup>Ω</sup><sup>⊕</sup> <sup>=</sup> <sup>2</sup>*π*/(<sup>23</sup> hr <sup>56</sup> min 4.09 s <sup>=</sup> 7.2921159 <sup>×</sup> <sup>10</sup>−<sup>5</sup> rad/s. The position of the ILL laboratory on the surface of the Earth is determined by the angles *χ* and *<sup>φ</sup>*, where *<sup>χ</sup>* <sup>=</sup> <sup>90</sup><sup>0</sup> <sup>−</sup> *<sup>θ</sup>* is the colatitude of the laboratory, defined in terms of the latitude *<sup>θ</sup>*, and *<sup>φ</sup>* is the longitude of the laboratory measured east of south with the values *θ* = 45.16667◦ N and *φ* = 5.71667◦ E [96], respectively. The beam of UCNs moves from south to north antiparallel to the *x*–direction and with energies of UCNs quantized in the *z*–direction.

**Figure 1.** The position of the ILL laboratory doing the qBounce experiments on the surface of the Earth.

In the qBounce experiments the contributions of interactions beyond the gravitational interaction of the Earth are measured in terms of the transition frequencies *ωpq* = *E<sup>p</sup>* − *E<sup>q</sup>* of the transitions |*q*i → |*p*i between two gravitational states of UCNs |*q*i and |*p*i [49–55] (see also [59,60]). As of the small values of the components of the torsion field the contribution of the~*r*-dependent part of the effective torsion–neutron potential Equation (64), where the vector ~*r* defines a location of the UCN in the coordinate system (*t*, *x*, *y*, *z*), to the transition frequencies between quantum gravitational states of UCNs can be neglected in comparison to the contributions of the terms independent of~*r*. Relative to the axes (*x*, *<sup>y</sup>*, *<sup>z</sup>*) the vectors <sup>Ω</sup><sup>~</sup> <sup>⊕</sup> and <sup>~</sup>*R*<sup>⊕</sup> are equal to <sup>Ω</sup><sup>~</sup> <sup>⊕</sup> = (−Ω<sup>⊕</sup> sin *<sup>χ</sup>*, 0, <sup>Ω</sup><sup>⊕</sup> cos *<sup>χ</sup>*) and <sup>~</sup>*R*<sup>⊕</sup> = (0, 0, *<sup>R</sup>*⊕), respectively. This allows one to transcribe the effective low-energy torsion–neutron potential Equation (64) into the form

$$\boldsymbol{\Phi}\_{\text{eff}} = \boldsymbol{\Omega}\_{\text{\oplus}} \boldsymbol{R}\_{\text{\oplus}} \sin \chi \left( \frac{1}{3} \boldsymbol{\mathcal{B}}\_{0} \boldsymbol{\mathcal{S}}\_{\text{\oplus}} + \frac{1}{2} \boldsymbol{\mathcal{M}}\_{2} \boldsymbol{\mathcal{S}}\_{\text{\mathbb{Z}}} - \frac{1}{2} \boldsymbol{\mathcal{M}}\_{1} \boldsymbol{\mathcal{S}}\_{\text{\mathbb{Z}}} \right) = 1.1 \times 10^{-6} \left( \frac{1}{3} \boldsymbol{\mathcal{B}}^{0} \boldsymbol{\mathcal{S}}\_{\text{\mathbb{Z}}} + \frac{1}{2} \boldsymbol{\mathcal{M}}\_{2} \boldsymbol{\mathcal{S}}\_{\text{\mathbb{Z}}} - \frac{1}{2} \boldsymbol{\mathcal{M}}\_{1} \boldsymbol{\mathcal{S}}\_{\text{\mathbb{Z}}} \right), \tag{65}$$

where (*Sx*, *Sy*, *Sz*) are operators of the neutron spin ~*S*–operator components. Thus, measuring the transition frequencies of spin-flip transitions between gravitational states |*q* ↓i → |*p* ↑i one may measure the contributions of the pseudoscalar B<sup>0</sup> and tensor M*<sup>x</sup>* = −M00*<sup>x</sup>* and M*<sup>z</sup>* = −M00*<sup>z</sup>* components of the torsion field. A predictable power of the qBounce experiments we may demonstrate by example of the estimate of the contribution of the pseudoscalar component B<sup>0</sup> of the torsion field coupled to UCNs. Indeed, according to Lämmerzahl [43], the value of the pseudoscalar component of the torsion field is constrained by |B0<sup>|</sup> <sup>&</sup>lt; <sup>2</sup> <sup>×</sup> <sup>10</sup>−<sup>18</sup> GeV. Its contribution to the transition frequencies between quantum gravitational states of UCNs <sup>|</sup>*<sup>q</sup>* ↓i → |*<sup>p</sup>* ↑i is of about <sup>|</sup>∆*ωp*↑*q*<sup>↓</sup> <sup>|</sup> <sup>&</sup>lt; <sup>7</sup> <sup>×</sup> <sup>10</sup>−<sup>16</sup> eV. This value is at the level of current experimental sensitivity ∆*E* < 10−<sup>15</sup> eV [55] and the sensitivity of a nearest future, which is of about ∆*E* < 10−<sup>17</sup> eV and even ∆*E* < 10−<sup>21</sup> eV [49,97]. The experiments

discussed in this sections and and many others, which could be carried out by using effective low-energy potentials of torsion–neutron interactions derived in [40–42], might make reliable the geometrical origin of the cosmological constant or the relic dark energy, induced by torsion [39].

### **5. Discussion**

We would like to emphasize that our analysis of the chameleon field as a candidate for quintessence is carried out within the classical Einstein–Cartan gravitational theory with the Einstein-Hilbert action linear in the Ricci scalar curvature. By definition [4], quintessence is a hypothetical form of dark energy described by a canonical scalar field for an explanation of the observable acceleration of the Universe's expansion. The most important that quintessence should be a hypothetical form of dark energy. In this connection in the Einstein–Cartan gravitational theory, when the cosmological constant or the relic dark energy density has the geometrical origin, caused by torsion, the chameleon field possesses no chance to be a hypothetical form of dark energy. In other words having provided a geometrical origin for the cosmological constant or the relic dark energy torsion deprives the chameleon field to have a chance to be quintessence. As a result, the chameleon field is able only to evolve above the relic background of the dark energy, caused by torsion, but not to originate it. Then, as a consequence of conservation of the total energy–momentum of the system, the chameleon field can affect the dark energy dynamics and as well as the Universe's expansion even also the late-time acceleration. We have shown that such an influence of the chameleon field on the acceleration of the Universe's expansion retains also even if the conformal factor, relating Einstein's and Jordan's frames and defining the interaction of the chameleon field with its ambient matter, is equal to unity (see Equations (60) and (61). This result is closely related to our proof that for the system, including the chameleon field, radiation and matter (dark and baryon matter), the Friedmann–Einstein equation for *a*˙ <sup>2</sup>/*a* 2 is the first integral for the Friedmann–Einstein equation for *a*¨/*a*.

We have found that local conservation of the total energy–momentum of the system, including the chameleon field, radiation and matter (dark and baryon matter), leads to the equations of the evolution of the radiation and matter densities, corrected by the conformal factor. Of course, since for radiation the evolution equation defines an evolution of the product *ρr*(*a*)*f*(*φ*), where the conformal factor *f*(*φ*) is a function of the expansion parameter *a*, such a product *ρr*(*a*)*f*(*φ*) does not deviate from the standard behavior *ρr*(*a*)*f*(*φ*) ∼ *a* −4 . Since the radiation density *ρr*(*a*) enters to the Friedmann–Einstein equations only in the form of the product *ρr*(*a*)*f*(*φ*) one may not probably separate the contribution of the conformal factor above the standard shape proportional to *a* −4 . In turn, for the matter density *ρm*(*a*) the contribution of the conformal factor leads to a deviation from the standard behavior *ρm*(*a*) ∼ *a* −3 [69]. However, such a deviation might be, in principle, noticeable only during the matter-dominated era. In the dark energy–dominated era that is in our time of the late-time acceleration of the Universe, where the expansion parameter is equal to *a*<sup>0</sup> = *a*(*t*0) for the Hubble time *t*<sup>0</sup> = 1/*H*<sup>0</sup> [69], the contributions of the conformal factor to the radiation and matter densities in comparison to the standard values *ρr*(*a*0) = 3*M*<sup>2</sup> PlH<sup>2</sup> <sup>0</sup>Ω*<sup>r</sup>* and *<sup>ρ</sup>m*(*a*0) = 3*M*<sup>2</sup> PlH<sup>2</sup> <sup>0</sup>Ω*<sup>m</sup>* are practically unobservable. This agrees well with the constraints on the deviations of the radiation and matter densities from their values at our time to a few parts per million [98], which can be obtained from the constraints on the fifth force caused by the chameleon field in the Galaxy and the Solar system.

The cosmological constant Λ*C*, induced by torsion [39], we have included additively to the potential of the self-interaction of the chameleon field as a background of the relic dark energy: *V*(*φ*) = *ρ*<sup>Λ</sup> + Φ(*φ*). In the chameleon field theory [1,2] the relic dark energy density *ρ*<sup>Λ</sup> is defined as follows: *ρ*<sup>Λ</sup> = Λ<sup>4</sup> , where the scale Λ = <sup>4</sup> q 3*M*<sup>2</sup> PlH<sup>2</sup> <sup>0</sup>Ω<sup>Λ</sup> = 2.24(1) meV is calculated for the relative dark energy density Ω<sup>Λ</sup> = 0.685(7) [70]. The *φ*-dependent part of the potential of the self-interaction of the chameleon field Φ(*φ*) is arbitrary to some extent, i.e., model-dependent, and demands a special analysis similar to that carried out in [4,10,16,88]. However, such an analysis goes beyond the scope of our paper. We would like to emphasize that a specific analysis of a dynamics of the chameleon field such as different mechanisms of chameleon screening and a formation of a fifth force, for example, in the

Galaxy and the Solar system is related also to a special choice of the potential of the self-interaction of the chameleon field [16,98]. Such an analysis has been carried out by Brax et al. [16] and Jain et al. [98]. The repetition of such an analysis goes beyond the scope of this paper.

As regards the assertion by Wang et al. [67] and Khoury [68] that since the conformal factor is practically constant during the Hubble time, so the chameleon field is not responsible for the late-time acceleration of the Universe, one may argue that the conformal factor might be, in principle, practically constant (or better to say unity), but such a behavior of the conformal factor does not prohibit the chameleon field, evolving above the relic dark energy background induced by torsion, to take a certain part in dark energy dynamics and, correspondingly, in the acceleration of the Universe's expansion (see Equations (60) and (61)) and even so in the late-time acceleration of the Universe's expansion.

As regards another canonical scalar fields which can be introduced for an explanation of an origin of dark energy and an influence on acceleration of the Universe's expansion such as symmetron and dilaton, we may say the following. Since the dynamics of the symmetron field differs from the dynamics of the chameleon one only by the shape of the potential of self-interaction [99], our conclusion concerning an identification of the chameleon field with quintessence is fully applicable to the symmetron one. In other words the symmetron field cannot be quintessence to full extent. Moreover, we have to mention that an existence of the symmetron field and its importance for an evolution of the Universe might seem to be rather questionable after the qBounce experiments [55] on the transition frequencies between quantum gravitational states of UCNs, which have excluded the existence of the symmetron field.

Then, we have to confess that our analysis of an identification of a canonical scalar field with quintessence, carried out within the classical Einstein–Cartan gravitational theory, can say practically nothing concerning dilaton. Indeed, unlike the gravitational theories with the chameleon and symmetron fields the gravitational theories with dilaton are based on string theory in terms of the non-riemannian structure of space-time [100–107]. Of course, since an inclusion of dilaton as a canonical scalar field is closely related to a requirement of scale invariance of the action of the dynamical system under consideration, the classical Einstein–Cartan gravitational theory can be, in principle, modified by a requirement of scale invariance [108]. However, in such a modified Einstein–Cartan gravitational theory the problem of the geometrical origin of the cosmological constant or the relic dark energy caused by torsion demands a special analysis, which goes beyond the scope of this paper. Nevertheless, if in such a modified Einstein–Cartan gravitational theory torsion would be a geometrical origin of the cosmological constant or the relic dark energy density, our conclusion concerning an impossibility to identify dilation with quintessence might have been valid only within such a modified Einstein–Cartan gravitational theory.

Robust support fpr the geometrical origin of the cosmological constant or the relic dark energy could be experimental observations of torsion in terrestrial laboratories in terms of its contributions to observables of different physical processes. In Section 4 we have discussed two of these experiments, which can be carried out by using beams of polarized UCNs. We mean the contribution of the T–odd torsion–neutron low-energy interaction to the cross section for the scattering of the beam of polarized neutrons by nucleus in the end of spinning cylinder. This allows one to estimate the value of the time component <sup>E</sup>0o f the 4-vector part of the torsion field at the level of about |E0| ∼ <sup>10</sup>−<sup>32</sup> GeV. Another experiment on the probe of torsion can be carried out at ILL by the French–Austrian qBounce Collaboration by using Gravity Resonance Spectroscopy (GRS), a new measuring technique combining quantum measurements and gravity experiments [49–55]. The qBounce experiments, measuring transition frequencies between quantum gravitational states of UCNs, allow one to probe torsion with a sensitivity of about <sup>∆</sup>*<sup>E</sup>* <sup>&</sup>lt; <sup>10</sup>−<sup>17</sup> eV and even <sup>∆</sup>*<sup>E</sup>* <sup>∼</sup> <sup>10</sup>−<sup>21</sup> eV [49,97]. This should improve the existing upper bound on the pseudoscalar B<sup>0</sup> component of the torsion field by a few orders of magnitude and give new constraints on the tensor M00*<sup>k</sup>* components of the torsion field [44].

**Author Contributions:** A.N.I.: Conceptualization, Methodology, Data Curation, Investigation, Supervision, Writing—Original draft preparation. M.W.: Formal analysis, Investigation, Methodology, Software, Validation, Writing—Original draft preparation. All authors have read and agreed to the published version of the manuscript.

**Funding:** The work of A. N. Ivanov was supported by the Austrian "Fonds zur Förderung der Wissenschaftlichen Forschung" (FWF) under the contracts P31702-N27, P26636-N20 and P33279-N, and "Deutsche Förderungsgemeinschaft" (DFG) AB 128/5-2. The work of M. Wellenzohn was supported by the MA 23 (FH-Call 16) under the project "Photonik— Stiftungsprofessur für Lehre".

**Acknowledgments:** We are grateful to Hartmut Abele for stimulating discussions and to Philippe Brax and Alkistis Pourtsidou for fruitful discussions.

**Conflicts of Interest:** The authors declare no conflict of interest.
