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 $\theta$ parameter, ${\cal O}(10^{-10})<\theta<{\cal O}(10^{-9})$ has been obtained, using {\tt Planck+WMAP9} best fit cosmological parameters.

In this work we propose an Effective Field Theory of inflation taking into account the spin density of matter which contributes to torsion. We first explicitly show that torsion mimics the role of a scalar field which controls the dynamics of inflation. We have obtained a strict bound on the CP violating θ parameter, O(10 −10 ) < θ < O(10 −9 ), using Planck+WMAP9 best fit cosmological parameters.
It is well known that torsion and curvature of any manifold are related to translation and rotation respectively. When both the symmetries are operative, the gravity sector should contain contributions from the both. More specifically, when the energy momentum tensor is represented by the spin density of matter sources, the torsion contributes to gravity whereas mass density gives rise to curvature. In the context of cosmology, in Ref. [1] it was pointed out that the general features of inflationary scenario can be explained through torsion. Further, in [2] it was shown that torsion can be treated as an alternative to cosmic inflation which is generally believed to be caused by scalar field called 'inflaton'. In the realm of quantum gravity when Asthekar's formalism of quantization of gravity is transcribed to Lagrangian formulation as proposed by Capovilla, Jacobson and Drell (CJD) [3] we can include torsion through the introduction of CP violating θ term in the Lagrangian. In this work our prime motivation is to explicitly show that there is a hidden scalar field in torsion caused by the spin density of matter. From the modified CJD Lagrangian including torsion we develop an effective potential in terms of this hidden scalar field which is instrumental in causing inflation. Using this methodology we give a strict bound on CP violation using Planck+WMAP9 best fit cosmological parameters [4].
At the classical level a generalized theory of gravity incorporating torsion is given by the celebrated Einstien-Cartan-Kibble-Sciama (ECKS) formalism [5] 1 . In this, the affine connection has non-vanishing antisymmetric contribution leading to torsion which can be represented by spin-spin interaction. A dual current-current interaction picture can be developed by translating the SU(2) spin basis into the topological current. Within this prescription the spin-current duality can be explained in terms of a four-vector n µ ∈ SU(2) in the Casimir op-erator basis as: where ψ 1 = (cos θ/2)e iφ/2 , ψ 2 = (sin θ/2)e −iφ/2 . Using this one can construct an SU(2) group element g = n 0 I+ i n. σ, in terms of which the topological current can be expressed as [6]: 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 Eqn.(2) the Kac-Moody like current J µ can be recast in terms of the Chern-Simons secondary characteristic class as [7]: which allows us to define a topological invariant as: which is commonly known as the Pontryagin index 2 . Consideration of the Lagrangian L = − 1 4 Tr µνλσ F µν F λσ which effectively corresponds to the term ∂ µ J µ where J µ is given by eqn. (3) leads to the construction of the current [8]. It can be shown that the axial vector current J 5 µ =ψγ µ γ 5 ψ is related 2 In fact the Pontryagin index is related to the Chern-Simons invariant through the relation: where F is the two-form related to the field strength. to the second component of the current J µ through the relation ∂ µ J The consistency of the current conservation equations implies that: J (2) µ [9]. Consequently, the current-current interaction can be expressed in terms of J (2) µ only. Here we define a duality condition for the component J (2) µ as: where φ(x) is the hidden scalar field and we will later see that it plays the role of inflaton, and νλσ is the rank-3 Levi-Civita tensor. Therefore, the action turns out to be: which actually represents the CP conserving contribution from torsion. In Eqn.(6) Σ represents the current-current interaction strength. Within EFT the effective mass of the scalar degrees of freedom is defined as 3 : where a(t) is the scale factor in FRW space-time. Eqn. (6) suggests that the potential associated with torsion can be written as: The negative sign of the time dependent coupling constant m 2 actually corresponds to the self interaction, when orientation of all the spin degrees of freedom are along the same direction.
Our next step is to find the contribution in the action from curvature, for which we utilize the CJD model, where the action is given by [3,10]: where Ω ij = αβγδ F αβi F γδj with α, β, γ, δ as space time indices, i, j the SU(2) group indices and η is a scalar density. The canonical transformation of SU(2) gauge potential (A ai ) and the corresponding non-abelian fields gives rise to a CP-violating θ term in the CJD Lagrangian so that for a = −1/2 4 the action now reads [3,10]: 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. The association of the second term in Eqn. (9) reveals that this term in 3+1 description corresponds to the contribution of the metric. Consequently Eqn. (9) can be recast as: , and the symbol [· · · ] signifies the boundary value of the coordinates in the affine parameter space. Now from Eqn.(10) we get 5 : Noting that the asymptotic constancy of torsion compensates the bare cosmological constant [11] we can relate the contribution to torsion from the CP violating θ parameter in Eqn. (11) with the effective cosmological constant given by, Here Λ U V signifies the UV cut-off scale of the proposed EFT theory 7 . Thus the expression for the 5 Here we use the following spin-particle duality relations: Here the effective cosmological constant Λ ef f has mass dimension 2. 7 Above the scale Λ U V 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 [12]. However such quantum corrections are extremely hard to compute as it completely belongs to the hidden sector of the theory dominated by heavy fields [13]. String theory and Loop Quantum gravity are the two parallel approaches through which one can probe the various technical issues of such hidden sector. In the trans-Planckian regime the classical gravity sector is corrected by incorporating the effect of higher derivative interactions appearing through the local modifications to GR which plays significant role in this context [14].
In this case the appropriate choice of the co-efficients of the corrections would modify the UV behaviour of gravity. Such local higher derivatives can be renormalizable and only help to explain the UV features of gravity in 4D but they typically contain debris like massive ghosts. This problem can be addressed by incorporating the infinite higher derivatives appearing through non-local corrections to GR. In such a situation the non-local contributions, yielding a ghost-free condition for certain analytic potential from CJD Lagrangian incorporating the CP violating θ term yields: V C (φ) = V 0 + λ 4 φ 4 . Now let us consider a situation where the superspace has Riemann structure. In such a case the contribution to the conserved current can be expressed as: J µ g = 1 2 µνλσ R νλσδ v δ , 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 Eqns.(6,11,13) the total action for the present field theoretic setup can finally be written as: (14) such that the total effective potential is given by: 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 be constrained as: choices of the non-local entire functions [15]. 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 [16]. However below Λ U V the effect of all such quantum corrections are highly suppressed and the heavy fields getting 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]. But such possibility is only significant when the contribution of the primordial gravity waves become negligibly small (see Eqn. (16)). In order to compare the theoretical predictions with the latest observational data we use a numerical code CLASS [18]. 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 Planck + WMAP9 best fit values: h = 0.670, Ω b = 0.049, Ω c = 0.268, Ω Λ = 0.682 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 [19][20][21] 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 efoldings 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: |∆φ| ∼ O(4.1 − 5.9) × Λ U V , 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: Λ U V ∼ O(0.16 − 0.24) M p < M p , which is just below the scale of reduced Planck mass. Finally using Eqn. (17) we get the following bound on the CP violating parameter 8 : Thus once we fix Q P , this will further give an estimate of θ according to the Eqn. (21). In Fig. (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 [23]. 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 [23]. 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 antifermion 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 Eqn. (21) 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 9 giving rise to torsion can be witnessed 8 From experimental measurements of the neutron electric dipole moment, the experimental limit on the CP violating θ parameter is θ ≤ 10 −9 [22], which is consistent with our derived stringent bound on θ. 9 In the context of canonical quantization of gravity it is observed through the small value of θ which is operative at the boundary.
To summarize, we have proposed a methodology for generating hidden scalar field within EFT framework from the spin spin interaction picture. We have explicitly computed the vacuum energy corrected effective potential in sub-Planckian scale through which 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 VEV of the heavy hidden sector fields appearing as vacuum energy correction within EFT.