Quantum gravity effect in torsion driven inflation and CP violation

We have derived an effective potential for inflationary scenario from torsion and quantum gravity correction in terms of the scalar field hidden in torsion. A strict bound on the CP violating θ parameter, O10−10<θ<O10−9\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{O}\left(1{0}^{-10}\right)<\theta <\mathcal{O}\left(1{0}^{-9}\right) $$\end{document} has been obtained, using Planck+WMAP9 best fit cosmological parameters.


Introduction
The paradigm of cosmic inflation complements the big-bang theory and when combined together it is the best theory compatible with the latest observations. Inflation is generally believed to be driven by a scalar field known as inflaton. Gasperini [1] has pointed out that the inflationary scenario can be well explained through torsion. 1 Later, Poplawski [3] have argued that torsion can be treated as an alternative source of inflation. In this context, it is also to be noted that when torsion is considered to be generated from spin-spin interaction a hidden scalar field can be associated with torsion [4]. It is interesting to know whether the associated scalar field in torsion plays the role of inflaton in the inflationary regime such that "the scalar field driven inflation" as well as the "torsion driven inflation" appear to be equivalent statements. The motivation of the present paper is to show that if in the Einstein-Cartan-Kibble-Sciama (ECKS) theory of gravity, the quantum gravity effect in the early universe is taken into account, we can formulate an effective potential for inflation in terms of the scalar field hidden in the torsion. Besides, the formulation gives rise to a CP violating term. The estimate of the bound on CP violation in the early Universe using Planck+WMAP9 best fit cosmological parameters [5,6] has also been obtained.

Torsion induced potential
To study torsion in terms of the spin-spin interaction we take resort to a spin-current duality relation so that the action for torsion can be developed through a dual currentcurrent interaction. We consider a four vector n µ in terms of the spinorial variables as 1 Also it is important to note that, in ref. [2], the authors have explicitly studied the late-time cosmic acceleration from torision and the emergent scalar degree of freedom arose from the BCS condensation of the fermions.

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where ψ 1 = (cos θ/2)e iφ/2 , ψ 2 = (sin θ/2)e −iφ/2 , (2.2) with σ 0 = I, where I is the identity matrix and σ is the vector of Pauli matrices. Using this one can construct an SU(2) group element in terms of which we can construct the topological current as [7]: where µνλσ is the rank-4 Levi-Civita tensor. Now by demanding that in 4-dimensional Euclidean space the field strength F µν of a gauge potential vanishes on the boundary S 3 of a certain volume Vol 4 inside of which F µν = 0, we can write the gauge potential as A µ = g −1 ∂ µ g ∈ SU(2). Then from eq. (2.4) the Kac-Moody like current J µ can be recast in terms of the Chern-Simons secondary characteristic class as [8]: This gives rise to a topological invariant: which is known as the Pontryagin index. We can construct the Lagrangian from the divergence of the current J µ and write which leads to the construction of the current [9] j µ = µνλσ a ν ⊗ f λσ = µνλσ ∂ ν f λσ (2.8) It can be shown that the axial vector current is related to the second component of the current j µ through the relation The consistency of the current conservation equations implies that [10]:

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Consequently, the current-current interaction can be expressed in terms of j (2) µ only which effectively displays the spin-spin interaction. Now we can write the action for torsion as [11] where M p being the reduced Planck mass, given by M p ≈ 2.43 × 10 18 GeV. It is now observed that there is a hidden scalar field φ in torsion which follows from the relation where νλσ is the rank-3 Levi-Civita tensor. The action now turns out to be: eq. (2.15) suggests that the potential associated with torsion can be written as: The negative sign of the coupling constant m 2 actually corresponds to the self interaction, when orientation of all the spin degrees of freedom are along the same direction.

Inflationary modeling with the CP violating term
Now we analyse the contribution from quantum gravity. To this end we utilize the model of Capovilla, Jacobson and Drell (CJD) [12,13], where the action is given by [12,13]: where with α, β, γ, δ as space time indices, i, j the SU(2) group indices and η is a scalar density. In ref. [12,13] it has been shown that in 3+1 decomposition this action yields Ashteker action directly provided we have a = − 1 2 and the determinant of the magnetic field B i a is non zero and as such the equivalence to the Einstein's theory is established. The equivalence to the Einstein's theory can also be shown when the space time metric is found to be given by The constraint that is obtained when the CJD action is varied with respect to the Lagrangian multiplier η is actually the Hamiltonian constraint

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This implies that Ψ ≈ 0 and H ≈ 0 are equivalent statements provided detB = 0. The canonical transformation of SU(2) gauge potential (A ai ) and the corresponding non-abelian fields (E a i , B a i ): gives rise to a CP-violating θ term in the CJD Lagrangian so that for a = −1/2 the action now reads [4,[12][13][14][15]: In the first term the parameter θ essentially corresponds to the measure of CP violation which contributes to torsion and the rest is curvature contribution. Consequently eq. (3.7) can be recast as: , and the symbol [· · · ] signifies the boundary value of the coordinates in the affine parameter space. Now from eq. (3.8) we get: 2 It may be mentioned here that the first term on the right hand side incorporates the Pontryagin index given by eq.(2.6) which is a topological term arising from a total divergence. This does not contribute classically but has the effect in the quantum mechanical formulation. From eq. (2.15) and eq. (3.9), we note that the action for torsion (curvature) when expressed in terms of the φ field involves the term φ 2 (φ 4 ). This indicates that the anisotropies associated with the torsion are much suppressed in comparison to the contribution from curvature for large values of φ. It is noted that the the expression of curvature in terms of the scalar field arises when we use CJD Lagrangian. In this sense the scalar field does not arise from gravitation as such, but it originates from the torsional degrees of freedom associated with the spin density.
Noting that the asymptotic constancy of torsion compensates the bare cosmological constant [16] we can define a small but non-vanishing cosmological constant in terms of the Pontryagin index as Here we use the following spin-particle duality relations:

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where M p coresponds to the Planck mass. We can define the vacuum energy V 0 through the relation Here Λ U V signifies the UV cut-off scale of the proposed EFT theory. 3 Below Λ U V the effect of all quantum corrections are highly suppressed and the heavy fields from the hidden sector gets their VEV. Such VEV is one of the possible sources of vacuum energy correction in the spin-current dominated EFT picture which uplifts the scale of inflationary potential and the contributions of the VEV become significant upto a scale Λ C ≤ Λ U V . But at very low scale, Λ low Λ C , one can tune the vacuum energy correction, V 0 ≈ 0 for which the contributions of the VEV can be neglected [17,18]. Such possibility is only significant when the contribution of the primordial gravity waves become negligibly small (see eq. (5.1)). Thus the expression for the potential from CJD Lagrangian incorporating the CP violating θ term yields: (3.12)

The effective potential
Now in the background of a space-time manifold having Riemannian structure the contribution to the conserved current can be expressed as: where v δ is an arbitrary vector and Riemann curvature tensor can be expressed as: As a result the gravitational part of the action can be written in terms of gravitational current-current interaction in the Riemann space as: Now clubbing the contributions from eqs. (2.15), (3.9), (4.3) the total action for the present field theoretic setup, taking into account quantum gravity correction, can finally be written as: 3 Above the scale ΛUV it is necessarily required to introduce the higher order quantum corrections to the usual classical theory of gravity represented via Einstein-Hilbert term, as the role of these corrections are significant in trans-Planckian scale to make the theory UV complete [19,20]. However such quantum corrections are extremely hard to compute as it completely belongs to the hidden sector of the theory dominated by heavy fields [21,22]. In the trans-Planckian regime the classical gravity sector is corrected by incorporating the effect of higher derivative interactions appearing through the modifications to GR which plays significant role in this context [23][24][25][26][27][28]. On the other hand in trans-Planckian regime quantum corrections of matter fields and their interaction between various constituents modify the picture which are appearing through perturbative loop corrections [29][30][31][32].

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such that the total effective potential is given by: (4.5)

Estimate on the CP violation term
The effective potential is dominated by the vacuum energy correction term which determines the scale of inflation. To obtain the scale of inflation at k * ≈ k cmb , we express V 0 in terms of inflationary observables as: where r is the tensor-to-scalar ratio defined as: r = A T /A S with (A T , A S ) being the amplitudes of the power spectra for scalar (S) and tensor (T ) modes at k = aH ≈ k * . The effective cosmological constant or equivalently the CP violating parameter θ can then be constrained as: In order to compare the theoretical predictions with the latest observations we use a numerical code CLASS [33]. In this code we can directly input the shape of the potential along with the model parameters. Then for a given cosmological background the code provides the estimates for different CMB observables. In the code we set the momentum pivot at k * = 0.05 Mpc −1 and used the Planck + WMAP9 best fit values: for background cosmological parameters. In this work we scan the parameter space within the following window: As a result, the CMB observables are constrained within the following range: Within the present context the field excursion [34][35][36][37][38][39][40][41][42][43] is defined as: where |∆φ| = |φ * − φ f |, in which φ * and φ f represent the field value corresponding to CMB scale and end of inflation respectively. Also N cmb is the number of e-foldings at CMB scale which is fixed at N cmb ≈ 50 − 70 to solve the horizon problem associated with inflation. Subsequently we get the following constraint on the field excursion: which implies to make the EFT of inflation validate within the prescribed setup for which we need to constrain the UV cut-off of the EFT within the following window: Thus once we fix Q P , this will further provide an estimate of θ according to the eq. (5.9). In figure (1) we have explicitly shown the constraint on θ from the proposed EFT picture which is obtained by using Planck + WMAP9 best fit cosmological parameters. To exemplify we have prescribed the bound on θ for different integer values of Q P lying within 1 ≤ Q P ≤ 10. From the plot it is easy to see that as the value of Q P increases the bound on the parameter θ converges to a very small value. This suggests that θ will converge to a constant value beyond a certain value of Q P . It may be mentioned that the Pontryagin index can be taken to correspond to the fermion number [45,46]. Indeed a fermion can be realized as a scalar particle encircling a vortex line which is topologically equivalent to a magnetic flux line and thus represents a skyrmion [45,46]. The monopole charge µ = 1/2 corresponding to a magnetic flux line is related to the Pontryagin index through the relation Q P = 2µ. In view of this, one may note that Q P represents the fermion number which is the topological index carried by a fermion. For an anti-fermion Q P takes the negative value.
In any system the effective fermion number is given by the difference between the number of fermions and anti-fermions. Thus we can quantify the fermionic matter and hence the spin density through the total accumulated value of Q P . As Q P increases we have the increase of fermions implying the increase in spin density. So from eq. (5.9) we note that for a fixed volume when Q P increases indicating the increase in spin density, the bound on the parameter θ converges to a small value representing the residual effect of torsion residing at the boundary. Thus the remnant of CP violation 5 giving rise to torsion can be witnessed through the small value of θ which is operative at the boundary.
To summarize, we have derived an effective potential for inflationary scenario, taking into account the quantum gravity effect, in terms of the hidden scalar field associated with torsion along with a CP violating term.Using this we give an estimate of inflationary CMB observables by constraining the model parameters-vacuum energy, mass and selfcoupling from Planck + WMAP9 best fit values of the cosmological parameters. Finally, for the first time we constrain the CP violating topological θ parameter from the vacuum energy correction within EFT.

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regular participants of weekly student discussion meet "COSMOMEET" from Department of Theoretical Physics and Department of Astronomy and Astrophysics, Tata Institute of Fundamental Research for their strong support. Last but not the least, we would all like to acknowledge our debt to the people of India for their generous and steady support for research in natural sciences, especially for theoretical high energy physics, string theory and cosmology.
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