Dynamical suppression of spacetime torsion
Abstract
A surprising feature of our present four dimensional universe is that its evolution appears to be governed solely by spacetime curvature without any noticeable effect of spacetime torsion. In the present paper, we give a possible explanation of this enigma through “cosmological evolution” of spacetime torsion in the backdrop of a higher dimensional braneworld scenario. Our results reveal that the torsion field may had a significant value at early phase of our universe, but gradually decreased with the expansion of the universe. This leads to a negligible footprint of torsion in our present visible universe. We also show that at an early epoch, when the amplitude of the torsion field was not suppressed, our universe underwent through an inflationary stage having a graceful exit within a finite time. To link the model with observational constraints, we also determine the spectral index for curvature perturbation (\(n_s\)) and tensor to scalar ratio (r) in the present context, which match with the results of Planck 2018 (combining with BICEP2 KeckArray) data (Akrami et al. in arXiv:1807.06211 [astroph.CO], 2019; Ade et al. in Phys Rev Lett 116:031302 https://doi.org/10.1103/PhysRevLett.116.031302, arXiv:1510.09217 [astroph.CO], 2016).
1 Introduction
A surprising feature of the present universe is that its large scale behaviour appears to be controlled by one type of geometrical deformation only, namely curvature; while we notice practically no effect of another type of deformation, namely torsion. The most straightforward way of including torsion is to add an antisymmetric component to the connection \(\Gamma ^{\alpha }_{\mu \nu }\), which is the essence of the socalled Einstein–Cartan theory [3, 4, 5]. Once torsion enters into the theory in this manner, it can in principle couple with all matter fields having non zero spin. From dimensional argument, it can be easily shown that such interaction terms in general are of dimension 5, and are suppressed by the Planck mass (\(M_p\)), just as in the case of graviton couplings. But there has been no experimental evidence of the footprint of spacetime torsion on the present universe. An example is the Gravity Probe B experiment which was designed to estimate the precession of a gyroscope to observe any signature of spacetime torsion [6]. However, all such probes, within the limit of their experimental precision, have consistently produced negative results and thereby disfavored the presence of the torsion in the spacetime geometry of our (\(3 + 1\)) dimensional visible universe [7, 8, 9]. Therefore the apparent torsion free universe indicates that the torsion field, if exists, must be severely suppressed at the present scale of the universe. Thus the question that naturally arises is : why are the effects of spacetime torsion are less perceptible than the spacetime curvature ? There is no satisfactory answer to this in the domain of four dimensional classical gravity models.
The proposals to remove torsion by quantum effects in four dimensional spacetime have appeared much before in [10], where the authors showed the invisibility of spacetime torsion (on the present energy scale of our universe) through the consideration of quantum corrected (caused by vacuum effects) gravitational action with torsion. There was also attempts to seek an answer to this in the context of higher dimensional braneworld models [11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. In particular, in Randall–Sundrum (RS) scenario [14] which involves one extra compact spacelike dimension with \(S^1/Z_2\) orbifolding along the extra dimension proposed a possible explanation for this suppression of spacetime torsion in four dimension. This kind of scenario postulates gravity in the fivedimensional ‘bulk’, whereas our fourdimensional universe is confined to one of the two 3branes located at the two orbifold fixed points along the compact dimension. However it has been already shown that a rank2 antisymmetric tensor field, generally known as Kalb–Ramond (KR) field (\(B_{MN}\)), can act as a source of spacetime torsion where the torsion is identified with rank3 antisymmetric field strength tensor \(H_{MNL}\) having a relation with \(B_{MN}\) as \(H_{MNL}=\partial _{[M}B_{NL]}\) [22]. In the RS like scenario where both the gravity and the KR field propagate in the bulk, the exponential warping nature of spacetime geometry causes the KR field (or equivalently the torsion) to be diluted on the visible 3brane [11, 23, 24, 25]. Also there is a recent work on spacetime torsion with antisymmetric tensor fields in higher curvature gravity model in the context of both four dimensional and five dimensional spacetime [26], where the authors showed that due to the effect of higher curvature term(s), the amplitude of torsion field gets suppressed in the course of the universe evolution.

How does the Kalb–Ramond field evolve from early era of our universe? Does this evolution lead to an explanation of why the effect of torsion is so much weaker than that of curvature on the present visible brane?

In such circumstance, does the four dimensional universe undergo an accelerating expansion at early epoch? If such an inflationary scenario is allowed, then what is the dependence of the duration of inflation on the KR field energy density? Moreover what are the values of the spectral index (\(n_s\)) and tensor to scalar ratio (r) in the present context?
Our paper is organized as follows: the model is described in Sect. 2, while Sect. 3 is reserved for presenting the cosmological field equations and their possible solutions from the perspective of four dimensional effective theory. Their implications and possible consequences are discussed in the remaining part of the paper.
2 The model
2.1 Effective action for 5D Einstein–Hilbert term
2.2 Effective action for bulk scalar field (\(\Psi \)): radion potential
2.3 Effective action for KR field action
At this stage it deserves mentioning that the energy scale, or the compactification scale, of the five dimensional bulk is \(\sim \) Planck scale. However as mentioned earlier that here we are interested on inflation on our 4D visible universe, where the energy scale (or the inverse of the duration of inflation) comes with \(\sim 10^{10}\) GeV which is consistent with the Planck observations as has been described later. Thus the 4D inflationary energy scale is lesser compared to the 5D bulk scale and we can consider the 4D effective action where the extra dimensional component of 5D metric i.e the modulus appears as radion field. Thus the approach here is motivated by the calculation of the effective action proposed by Goldberger and Wise in [35, 36].
3 Effective cosmological equations and their possible solutions
4 Beginning of inflation
At this stage, it deserves mentioning that the parameters \(v_v\) and \(h_0\) controls the strength of the radion field and the KR field energy density respectively. Therefore the interplay between the radion field and the KR field fixes whether the early universe evolves through an accelerating stage or not. However in order to solve the flatness and horizon problems (for a review, we refer to [28, 29]), the universe must passes through an accelerating stage at early epoch and from this requirement, here we stick to the condition shown in Eq. (47).
5 End of inflation and reheating
In the previous section, we show that the very early universe expands with an acceleration and this accelerating stage is termed as the inflationary epoch. In this section, we check whether such acceleration of the scale factor has an end in a finite time or not.
Therefore it is clear that the inflation comes to an end in a finite time. In order to estimate the duration of inflation explicitly, one needs the value of the parameters \(h_0\), \(\xi _0\) and \(v_v\), which can be determined from the expressions of spectral index and tensor to scalar ratio as discussed in the next section.
6 Spectral index, tensor to scalar ratio and number of efoldings
Figures 2 and 3 clearly demonstrate that for \(34<\xi _0<38\) (in Planckian unit), both the observable quantities \(n_s\) and r remain within the constraints provided by Planck 2018 [1, 2].
Further with the estimated values of \(v_v\), \(h_0\) and \(\xi _0\), the duration of inflation (\(t_ft_0\), see Eq. (51)) comes as \(10^{10}\)(Gev)\(^{1}\) if the ratio m / k (bulk scalar field mass to bulk curvature ratio) is taken as 0.2 [35]. We also determine the number of efoldings, defined by \(N = \int _{0}^{\vartriangle t}H dt\) (\(\vartriangle t = t_ft_0\), duration of inflation), numerically and lands with \(N \simeq 58\) (with \(\xi _0 = 36\), in Planckian unit).
Estimated values of various quantities for \(\kappa v_v = \frac{\sqrt{h_0}}{M^2} \simeq 10^{7}\) and \(\xi _0 = 36\)
Parameters  Estimated values 

\(n_s\)  0.969 
r  0.100 
\(t_ft_0\)  \(10^{10}\,(\hbox {GeV})^{1}\) 
N  58 
Table 1 clearly indicates that the present model may well explain the inflationary scenario of the universe in terms of the observable quantities \(n_s\) and r as per the results of Planck 2018.
where we use the relation \(\xi (t)=\sqrt{\frac{24M^3}{k}}e^{k\pi T(t)}\). Figure 4 clearly reveals that the interbrane separation increases with time and saturates at \(\frac{4k^2}{m^2}\ln {[v_h/v_v]}\) (\(= k\pi \langle T\rangle \), see Eq. (44)) asymptotically. It may be mentioned that for \(v_h/v_v = 1.5\) and \(m/k = 0.2\), \(k\pi \langle T\rangle \) acquires the value \(\simeq 36\) – required for solving the gauge hierarchy problem. Further Fig. 5 demonstrates that the early universe starts from an accelerating stage with a graceful exit at a finite time.
7 Solution for Kalb–Ramond extra dimensional wave function
Figure 6 clearly demonstrates that in the regime \(\phi \simeq \pi \), the KR wave function monotonically decreases with time and the decaying time scale (\(\tilde{t}\simeq 25\)) is less than the exit time of the inflation (\(\tilde{t} = 55\)). This may explain why the present universe carries practically no observable signatures of the rank two antisymmetric Kalb–Ramond field (or equivalently the torsion field).
Figure 7 reveals that the zeroth mode of KR wave function \(\chi ^{(0)}(t,\phi )\) decreases with time in the whole five dimensional bulk i.e for \(0 \le \phi \le \pi \). However for a fixed t, \(\chi ^{(0)}(t,\phi )\) has different values (in Planckian unit) on hidden and visible brane and such hierarchial nature of \(\chi ^{(0)}(t,\phi )\) (between the two branes) is controlled by the constant b.
8 Conclusion

We find that the Kalb–Ramond energy density (\(\rho _{KR}\)) on our visible universe depends on the onbrane scale factor a(t) as \(\rho _{KR} \propto 1/a^6\) (see Eq. (33)). As we may observe that \(\rho _{KR}\) monotonically decreases as the universe expands with time, which leads to a negligible footprint of the KR field on the present universe. However Eq. (33) also entails that the energy density of the KR field may be significant in early universe. This points us to explore the dynamical evolution of the KR field from very early phase of the universe. For this purpose, we solve the coupled Freidmann equations for the radion field (\(\xi (t)\)) and the scale factor (a(t)) during initial era and the solutions are given in Eqs. (39) and (40)) respectively. It is demonstrated in Fig. 4 that the interbrane separation increases with time and saturates at a constant value (\(\langle T\rangle \)) asymptotically. It is also found that without any fine tuning of the parameters, the asymptotic value of the modulus can address the gauge hierarchy problem. On the other hand, the solution of the scale factor corresponds to an accelerating expansion of the early universe and the rate of expansion depends on the parameters \(v_v\) and \(h_0\) (with \(v_v\) and \(h_0\) controls the energy density of the bulk scalar field and the KR field respectively). At this stage, it deserves mentioning that in absence of the bulk scalar field (\(\Psi \)), the radion field becomes constant while the Hubble parameter varies as \(H \propto 1/a^3\). This is expected because for \(\Psi \rightarrow 0\) (or \(v_v = 0\)), the potential \(V(\xi )\) goes to zero and thus the radion field has no dynamics which in turn makes the variation of the Hubble parameter as \(H \propto 1/a^3\) (solely due to the KR field having equation of state parameter \(=1\)). The duration of inflation (\(t_ft_0\)) is obtained in Eq. (51) which reveals that the accelerating phase of the universe terminates within a finite time. Further we also discuss the possible effects of the KR field on the reheating in the present context. We explained that the presence of Kalb–Ramond field makes the reheating time (the time interval after which the production of new particles becomes effective) lesser in comparison to the case when the KR field is absent. However, this is expected because the KR field corresponds to a deceleration of the universe i.e due to the appearance of KR field the Hubble parameter (H(t)) decreases with a faster rate by which H(t) reaches to \(\Gamma \) (the decay amplitude) more quickly relative to the situation where the KR field is absent.

In order to test the model with the observations of Planck 2018 (combining with BICEP2 KeckArray data), it is crucial to calculate the spectral index of curvature perturbation (\(n_s\)) and tensor to scalar ratio (r), which are defined in terms of the slowroll parameter (\(\epsilon \)). Using these definitions, the expressions of \(n_s\) and r are explicitly determined in the present context and as a result, we find that for suitable values of the parameters (\(v_v\), \(h_0\), \(\xi _0\)), \(n_s\) and r remain within the constraints provided by Planck 2018 [1, 2] (see Table 1). Moreover the duration of inflation comes as \(10^{10}\) (GeV)\(^{1}\) if the ratio m / k (bulk scalar field mass to bulk curvature ratio) is taken as 0.2 [35].

However the overlap of the zeroth mode KR wave function (\(\chi ^{(0)}(t,\phi )\)) with the visible brane actually fixes the coupling strengths of KR field with various Standard Model fields on the brane. Keeping this in mind, we solve \(\chi ^{(0)}(t,\phi )\) on the visible brane, numerically, as plotted in Fig. 6. It is clearly demonstrated that at \(\phi =\pi \) the KR wave function monotonically decreases with time and the decaying time scale is less than the exit time of the inflation. Further we also determine the numerical solution for the KR wave function in the whole bulk (see Fig. 7), which reveals that the effect of \(\chi ^{(0)}(t,\phi )\) decreases with time in the full five dimensional bulk i.e. for \(0 \le \phi \le \pi \). However it may be mentioned that the dynamics of \(\chi ^{(0)}(t,\phi )\) is controlled by the evolution of the radion field and it turns out that for \(T(t)=\langle T\rangle \), \(\chi ^{(0)}(t,\phi )\) acquires a constant value throughout the bulk as obtained in Eq. (69). Consequently we determine the coupling strengths of KR field with various matter fields on our present visible universe. As a result, such interaction strengths come with a heavily suppressed factor over the usual gravitymatter coupling \(1/M_p\). This may provide a natural explanation why the large scale behaviour of our present universe is solely governed by gravity and carries practically no observable footprints of spacetime torsion.

The second rank antisymmetric Kalb–Ramond field is related to a pseudoscalar field, known as axion field (Z(x)) given by \(H^{\mu \nu \alpha } = \epsilon ^{\mu \nu \alpha \beta }\partial _{\beta }Z\). It may be mentioned that there exist some dark matter models where the axion field was considered as a possible candidate to solve the mystery of dark matter [44, 45, 46]. However an experimental program named ABRACADABRA is designed to search for axion dark matter and the first results of ABRACADABRA is recently published in [47] where the authors, through estimating the axionphoton coupling, have found no evidence for axionlike cosmic dark matter with 95 percentage C.L. This is consistent with the results of our present paper i.e the present universe carries no evidence of axionlike dark matter coming from Kalb–Ramond field.
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