Irregular universe in the Nieh-Yan modified teleparallel gravity

Stimulated by the experimental detections of gravitational waves (GWs) ligo1 ; ligo2 and the developments in the cosmic microwave background radiation (CMB) experiments CMB1 ; CMB2 , parity violating gravities attracted lots of interests in recent years. A famous and frequently studied parity violating gravity model is the so-called Chern-Simons modified gravity CSgravity1 ; CSgravity2 which within the framework of Riemannian geometry modifies general relativity (GR) by a gravitational Chern-Simons term. The Chern-Simons modified gravity predicts the difference between the amplitudes of the left- and right-handed polarized components of gravitational waves, i.e., the so-called amplitude birefringence phenomenon. However, this model was found to suffer from the problem of vacuum instability because one of the circularly polarized components of GWs becomes a ghost at high frequencies CSgravity3 . Further extensions Crisostomi:2017ugk ; Gao:2019liu ; Zhao:2019xmm to this model did not circumvent this difficulty because in these extended models the pathological behavior still appear at high energy scales, as shown in Ref. Bartolo:2020gsh . It is very difficult to have a ghost-free parity violating gravity model within the framework of Riemannian geometry.

Successful parity violating gravity models are available if we go beyond the Riemannian geometry. For example, the Nieh-Yan modified teleparallel gravity (NYTG) PVtele1 ; PVtele2 is constructed within the framework of the teleparallel gravity (TG) Tele ; tele2021 , where the gravity is identified with the spacetime torsion in stead of the curvature. One may have a GR equivalent TG model Maluf:2013gaa (we may call it TGR). The NYTG model PVtele1 ; PVtele2 modifies TGR slightly by the anomalous coupling $\theta\mathcal{T}\widetilde{\mathcal{T}}$ between an axion-like field $\theta(x)$ and the Nieh-Yan density Nieh:1981ww : $\mathcal{T}\widetilde{\mathcal{T}}\equiv(1/2)\varepsilon^{\mu\nu\rho\sigma}\mathcal{T}^{\lambda}_{~{}\mu\nu}\mathcal{T}_{\lambda\rho\sigma}$, where $\mathcal{T}^{\lambda}_{~{}\mu\nu}$ is the torsion tensor, $\varepsilon^{\mu\nu\rho\sigma}$ is Levi-Civita tensor which relates the totally antisymmetric symbol $\epsilon^{\mu\nu\rho\sigma}$ and the determinant of the metric $g$ through the equation $\varepsilon^{\mu\nu\rho\sigma}=\epsilon^{\mu\nu\rho\sigma}/\sqrt{-g}$. The Nieh-Yan density is parity-odd, so at the background with $\partial_{\mu}\theta\neq 0$, the Nieh-Yan coupling term $\theta\mathcal{T}\widetilde{\mathcal{T}}$ violates the parity symmetry spontaneously. The NYTG model has been applied to cosmology in Refs. PVtele1 ; PVtele2 , where it was found that this model predicts a difference between the propagating velocities of the left- and right-handed polarized components of GWs, i.e., the so-called velocity birefringence phenomenon. More importantly, through detailed investigations on the cosmological perturbations, it was shown in Refs. PVtele1 ; PVtele2 that the NYTG model is ghost-free. Recently, this model was found to be compatible with the results of most local tests in the Solar System at the post-Newtonian order Rao:2021azn ; Qiao:2021fwi , the upper limit on its model parameters by the GWs data of LIGO/Virgo Collaboration was obtained in Ref. Wu:2021ndf , and the enhancement of primordial GWs during inflation due to the velocity birefringence of NYTG model and its implications in the air-based GWs experiments were studied in Ref. Cai:2021uup . Other recent studies on parity violating gravities can be found in Refs. Li:2021mdp ; Gong:2021jgg ; Hohmann:2022wrk ; Tong:2022cdz ; Li:2022vtn ; Zhang:2022xmm ; Zhu:2022dfq ; Zhu:2022uoq ; Filho:2022yrk ; Qiao:2022mln ; Cai:2022lec ; Chen:2022wtz .

In all the previous studies of the cosmological applications of the NYTG model, both the metric and the affine connection of the background universe are required to be homogeneous and isotropic at the beginning. The spacetime under this strong symmetry constraint is called the regular universe in this paper. The background solutions of the regular universe have been well studied within the TG framework Hohmann:2019nat ; Hohmann:2020zre ; Coley:2022qug , and are universally applicable to almost all TG models. In fact these solutions have been frequently adopted by different authors, e.g., Myrzakulov:2010vz ; Cai:2011tc ; Cai:2015emx ; Hohmann:2022wrk ¹¹1 Actually, the cosmological background solution whose tetrad is $e^{A}_{~{}\mu}=\mathrm{diag}(1,a,a,a)$ or $e^{A}_{~{}\mu}=\mathrm{diag}(a,a,a,a)$ under the Weitzenböck gauge is the regular flat universe. However, most of the earlier literature did not clearly point out that the selection of such a tetrad under the Weitzenböck gauge actually requires the connection to satisfy the same symmetry of the metric..

However, the cosmological principle only needs the metric of the background universe to meet the high symmetry requirement. In the Riemannian geometry, once we impose this symmetry requirement on the metric, the connection (i.e., the Christoffel symbol) satisfies the same symmetry requirement automatically. In TG models, the symmetry constraint on the affine connection is independent of the one on the metric. If one drops this extra constraint on the connection and leaves it arbitrary at the beginning, there will be final solutions for which the connection is neither homogeneous nor isotropic. We call the universe which has a homogeneous and isotropic metric and a non-homogeneous and non-isotropic affine connection the irregular universe. So far the irregular universe has rarely aroused research interest, only a few literatures have initially studied the flat irregular universe (or irregular Minkowski spacetime) solutions in $f(\mathbb{T})$ gravity models Bejarano:2017akj ; Bejarano:2019fii ; BeltranJimenez:2020fvy ; Golovnev:2020nln . The irregular universe does not violate the cosmological principle, but questions are in coming: What features and new physical phenomena could exist in the irregular universe? Or might the irregular universe have properties that are clearly contradictory to experiments so that only the regular universe is physically feasible? These questions deserve detailed studies for any TG models.

In this paper, we will study the irregular universe in the NYTG model. Firstly, we will obtain a more general flat universe solution than those in Refs. PVtele1 ; PVtele2 by solving the equations of motion of the NYTG model directly under the condition that only the metric is required to be homogeneous and isotropic. By analyzing the symmetry of the connection, we will show that the flat universe we obtain is generally an irregular flat universe, and in special cases it reduces back to a regular universe. We will also show that even in the irregular flat universe, the background equations in the NYTG model are exactly the same as those in GR. Secondly, we will study the linear cosmological perturbations around the irregular flat universe. We will find that tensor perturbations and scalar perturbations are coupled at the linear perturbation level. This is a peculiar feature that distinguishes the irregular universe from the regular universe in the NYTG model. We speculate that this peculiar feature is caused by the fact that the interior space does not satisfy the homogeneity and isotropy in the irregular universe. Finally, we will study the primordial fluctuations generated by slow-roll inflation in the regular and irregular flat universes. We will show that the primordial fluctuations of left- and right-handed GWs are different whether in the regular universe or in the irregular universe. We will also show that there is a strong statistical correlation between primordial scalar fluctuations and primordial tensor fluctuations generated by slow-roll inflation in the irregular universe.

This paper is organized as follows. In Sec. II, we briefly introduce the TG theory and the NYTG model. In Sec. III, we study spatially flat cosmological background solutions that only requires the metric to be homogeneous and isotropic in the NYTG model. In Sec. IV, through the quadratic actions for scalar, vector, and tensor perturbations, we investigate linear perturbations around the regular and irregular flat universes. In Sec. V, we apply our result to the early universe and discuss briefly the primordial perturbations generated by slow-roll inflation.

In this paper, we adopt the unit $8\pi G=1$, and use the signature $(+,-,-,-)$ for the metric. The tensor indices of the interior space are denoted by $A,B,C,...=0,1,2,3$ and by $a,b,c,...=1,2,3$ when limiting to spatial components. They are lowered and raised by the Minkowski metric $\eta_{AB}$ and its inverse $\eta^{AB}$. The spacetime tensor indices are denoted by Greek $\mu,\nu,\rho,...=0,1,2,3$ and by Latin $i,j,k,...=1,2,3$ when limiting to spatial components. They are lowered and raised by the spacetime metric $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. The antisymmetric symbol $\epsilon^{\mu\nu\rho\sigma}$ has the properties: $\epsilon^{0ijk}=\epsilon^{ijk}\equiv\epsilon_{ijk}$, and $\epsilon^{123}=1$. In addition, we distinguish the spacetime affine connection $\hat{\Gamma}^{\rho}_{~{}\mu\nu}$ and its associated covariant derivative $\hat{\nabla}$ from the Levi-Civita connection ${\Gamma}^{\rho}_{~{}\mu\nu}$ and its associated covariant derivative ${\nabla}$ respectively.

The TG theory can be considered as a constrained metric-affine theory. It is formulated in a spacetime endowed with a metric $g_{\mu\nu}$ and an affine connection $\hat{\Gamma}^{\rho}_{~{}\mu\nu}$, which is curvature free and metric compatible,

Without curvature and nonmetricity, in the TG theory the gravity is identified with spacetime torsion $\mathcal{T}^{\rho}_{~{}\mu\nu}=2\hat{\Gamma}_{\,[\mu\nu]}^{\rho}$. One can also describe the TG theory using the language of the tetrad $e^{A}_{~{}\mu}$ and the spin connection $\omega^{A}_{~{}B\mu}$. They relates the metric $g_{\mu\nu}$ and the affine connection $\hat{\Gamma}^{\rho}_{~{}\mu\nu}$ through the following relations

The torsion tensor is written as

The teleparallel constraints (1) dictate that the spin connection can be in general expressed as

where $\Lambda^{A}_{~{}B}$ is arbitrary element of Lorentz transformation matrix which is position dependent and satisfies the relation $\eta_{AB}\Lambda^{A}_{~{}C}\Lambda^{B}_{~{}D}=\eta_{CD}$ at any spacetime point. Therefore, the tetrad $e^{A}_{~{}\mu}$ and the Lorentz matrix $\Lambda^{A}_{~{}B}$ can be regarded as the basic variables of the TG theory. In this way, the teleparallel constraints (1) are automatically satisfied.

The TGR model, as the GR equivalent TG model, has the following action,

where ${|e|}=\sqrt{-g}$ is the determinant of the tetrad, $\mathbb{T}$ is the torsion scalar, and $\mathcal{T}_{\mu}=\mathcal{T}^{\alpha}_{~{}~{}\mu\alpha}$ is the torsion vector. Since we have the identity $-{R}(e)=\mathbb{T}+2{\nabla}_{\mu}\mathcal{T}^{\mu}$, the action (5) is identical to the Einstein-Hilbert action up to a surface term, where the curvature scalar $R(e)$ is defined by the Levi-Civita connection and considered as being fully constructed from the metric, and in turn from the tetrad. Since the surface term in the action does not affect the equations of motion, we say that the TGR is equivalent to GR at the level of the equations of motion.

The NYTG model PVtele1 ; PVtele2 modifies the TGR model by introducing the coupling

between an axion-like field $\theta$ and the Nieh-Yan density $\mathcal{T}\widetilde{\mathcal{T}}$. The coupling constant $c$ is dimensionless. Generally we should also consider its own dynamics of the axion-like field and take other matter into account, so the full action of the NYTG model is

Other matter with the action $S_{m}$ is assumed to be coupled to spacetime minimally through the tetrad. At the background in which the axion-like field has non-zero spacetime derivatives, the Nieh-Yan coupling term breaks parity spontaneously. Because only the first-order derivatives of the basic variables appears in the action, the NYTG model can avoid the Ostrogradski ghost mode, which is expected to be originated from higher-order derivatives in the action Woodard:2006nt .

As with most modified TG theories, the NYTG model apparently has two kinds of gauge symmetries: diffeomorphism invariance and local Lorentz invariance. The latter transformation makes the following change:

where $L^{A}_{~{}B}(x)$ are the element of Lorentz matrix. We would like to use different notations to distinguish two kinds of Lorentz matrices: $\Lambda^{A}_{~{}B}(x)$ is used to express the spin connection as in Eq. (4), but $L^{A}_{~{}B}(x)$ represents the local transformation that makes a shift from one local frame to another. Transformation (8) can be expressed in terms of tetrad and spin connections as

It is easy to prove that the metric $g_{\mu\nu}$ and torsion tensor $\mathcal{T}_{~{}\mu\nu}^{\rho}$ are invariant under the local Lorentz transformation (8), as is the action (7). Due to the local Lorentz invariance, one can choose the gauge $\Lambda^{A}_{~{}B}=\delta^{A}_{~{}B}$, i.e., $\omega^{A}_{~{}B\mu}=0$. This is the Weitzenböck connection, which has been frequently adopted in the literature. In addition, there is another symmetry hidden in the NYTG model. The Nieh-Yan term (6) can be integrated by parts as

It can be seen that the Nieh-Yan term (6) is invariant under the following transformation

where $L^{A}_{~{}B}(\theta)$ is Lorentz matrix that depends only on axion-like field $\theta$. Note that $\Lambda^{A}_{~{}C}e^{C}_{~{}\mu}$ is invariant under transformation (8). Due to the Lorentz symmetry (8), the transformation (11) can always be attributed to the fact that the tetrad $e^{A}_{~{}\mu}$ remains unchanged while the Lorentz matrix $\Lambda^{A}_{~{}B}$ undergoes a Lorentz transformation. Obviously the metric and the action of TGR model are invariant under such a transformation. So the total action of the NYTG model is invariant under the transformation (11).

The equations of motion follow from the variation of the action (7) with respect to $e^{A}_{~{}\mu}$ and $\Lambda^{A}_{~{}B}$ separately

where $N^{\mu\nu}=(c/2)\varepsilon^{\mu\lambda\rho\sigma}\partial_{\lambda}\theta\,\mathcal{T}^{\nu}_{~{}\rho\sigma}$, $G^{\mu\nu}$ is the Einstein tensor, $T^{\mu\nu}=-(2/\sqrt{-g})(\delta S_{m}/\delta g_{\mu\nu})$ and $T^{\mu\nu}_{\theta}=[V(\theta)-\nabla_{\alpha}\theta\nabla^{\alpha}\theta/2]g^{\mu\nu}+\nabla^{\mu}\theta\nabla^{\nu}\theta$ are the energy-momentum tensors for the matter and the axion-like field $\theta$ respectively. Similar to most modified TG models, the equation of motion (13) from the variation of $\Lambda^{A}_{~{}B}$ is not independent of Eq. (12), it is just the antisymmetric part of the latter. As explained in Ref. PVtele2 , this is due to the local Lorentz invariance of the action, any change caused by $\delta\Lambda^{A}_{~{}B}$ can always be equivalent to the change caused by $\delta e^{A}_{~{}\mu}$, so requiring the action to take the extremum under $\delta e^{A}_{~{}\mu}$ already includes the case where the action takes the extremum under $\delta\Lambda^{A}_{~{}B}$. There is another equation following from the variation of the action (7) with respect to $\theta$,

where ${\square}=g^{\mu\nu}{\nabla}_{\mu}{\nabla}_{\nu}$ and $V^{(n)}=\mathrm{d}^{n}V(\theta)/\mathrm{d}\theta^{n}$. All of these equations of motion are consistent with the Bianchi identity $\nabla_{\mu}G^{\mu\nu}=0$ and the covariant conservation law $\nabla_{\mu}T^{\mu\nu}=0$.

Also in Refs. PVtele1 ; PVtele2 , the cosmological perturbations of the NYTG model were analyzed in detail. It was found that the NYTG model makes a difference between the propagating velocities of the left- and right-handed polarized components of GWs, but makes no difference between their amplitudes. This phenomenon is called velocity birefringence, which is a clear physical signal of parity violation. More importantly, the NYTG model was confirmed to be ghost free through the quadratic action of cosmological perturbations.

It is worth mentioning that the Nieh-Yan density $\mathcal{T}\widetilde{\mathcal{T}}$ is not the only parity-odd term within the TG framework. A more general model including all the parity-odd terms which are quadratic in the torsion tensor was considered in Ref. PVtele3 . But then it was found in Ref. PVtele4 that this more general model suffers from the problem of ghost instability again, unless it completely reduces to the NYTG model. Therefore, within the TG framework, for all parity-odd terms which are quadratic in the torsion tensor, only the Nieh-Yan density $\mathcal{T}\widetilde{\mathcal{T}}$ can avoid the ghost instability. This means the NYTG model is robust to some extent.

So far all the studies on the cosmological applications of the NYTG model only considered the regular universe as the background, that means both the metric and the affine connection are constrained to be homogeneous and isotropic. This constraint may be too strong, after all the cosmological principle which is supported by current observations only needs the metric of the background spacetime to meet the high symmetry requirement. In this paper, we will drop the symmetry requirement on the connection and leave it arbitrary at the beginning. After this relaxation, it is expected that the NYTG model will have more interesting cosmological background solutions. We are interested in the irregular universe solutions in which the metric homogeneous and isotropic but the connection is neither homogeneous nor isotropic. For simplicity, we will only consider the spatially flat universe.

In flat universe, the metric can be expressed in rectangular coordinate as

where $a=a(\eta)$ is the scale factor of the universe, $\eta$ is the conformal time. This is the Friedmann-Robertson-Walker (FRW) metric. There are 6 Killing vector fields $\{\xi_{I}^{\mu},I=1,2...6\}$ in flat universe, which can be expressed as

where $\xi^{\mu}_{1},\xi^{\mu}_{2},\xi^{\mu}_{3}$ are Killing vector fields representing the symmetry of spatial translation, and $\xi^{\mu}_{4},\xi^{\mu}_{5},\xi^{\mu}_{6}$ are Killing vector fields representing the symmetry of spatial rotation. One can prove that the FRW metric satisfies the condition: $\mathcal{L}_{\xi_{I}}g_{\mu\nu}=0$, where $\mathcal{L}_{\xi_{I}}$ is the Lie derivative along the Killing vector field $\xi^{\mu}_{I}$. This reflects the result that the metric is homogeneous and isotropic. One can also prove that $\mathcal{L}_{\xi_{I}}\Gamma^{\rho}_{~{}\mu\nu}=0$ for the Levi-Civita connection $\Gamma^{\rho}_{~{}\mu\nu}$, which is automatically homogeneous and isotropic. This is why we do not need to pay extra attention to the symmetry of the connection within the framework of Riemannian geometry.

In the previous section we studied the flat universe solution of the NYTG model that only requires the metric to be homogeneous and isotropic. We found that the Nieh-Yan term has no effect even on the irregular flat universe background. In order to explore the effect of the Nieh-Yan term on the irregular flat universe, we study the linear cosmological perturbations around the irregular flat universe (40) in this section. For simplicity, we only consider the case of $F(\vec{v}\cdot\vec{x})=\vec{v}\cdot\vec{x}$, which is equivalent to requiring that the coefficients of the equations of linear perturbations do not depend on the spatial coordinates $\vec{x}$ (see below for details). And we also ignore other matter so that $S_{m}=0$.

We use the following parametrization for perturbed tetrad Izumi:2012qj :

So the perturbed metric components have the familiar forms:

where $B=\chi-\beta$ and $B^{V}_{i}=\chi^{V}_{i}-\beta^{V}_{i}$. Besides the familiar scalar perturbations ($A,B,\psi,\alpha$), vector perturbations ($B_{i}^{V},\alpha^{V}_{i}$), and tensor perturbations $h^{T}_{ij}$ in the metric, the parametrization of tetrad brings six extra variables, which are scalar perturbation $\lambda,\chi+\beta$ and vector perturbation $\lambda_{i}^{V},\chi_{i}^{V}+\beta_{i}^{V}$. All the vector perturbations are transverse and denoted by the superscript $V$, both the tensor perturbations are transverse and traceless and denoted by the superscript $T$. In addition, the scalar field $\theta$ is decomposed as $\theta(\eta,\vec{x})=\bar{\theta}(\eta)+\delta\theta(\eta,\vec{x})$.

Although we can perform a similar decomposition on the Lorentz matrix $\Lambda^{A}_{~{}B}$ following the parametrization in Ref. PVtele2 , we do not need to do so in this paper. Because we can always transform the perturbed Lorentz matrix into the background Lorentz matrix in Eq. (40) through the infinitesimal Lorentz transformation (8). In other words, we can always absorb the perturbations of the Lorentz matrix $\Lambda^{A}_{~{}B}$ into the perturbations of the tetrad $e^{A}_{~{}\mu}$ through the infinitesimal Lorentz transformation (8), so that we only need to deal with the perturbations of the the tetrad.

Due to the diffeomorphism invariance, it is safe to take the unitary gauge $\delta\theta=0,~{}\alpha=0,~{}\alpha_{i}^{V}=0$. This simplifies the calculations, for example, the gauge invariant scalar perturbation $\zeta=-(\psi+\mathcal{H}\delta\theta/\theta^{\prime})$ representing the curvature perturbation of the hypersurfaces of constant $\theta$ reduces to $-\psi$ under the unitary gauge. Since both $\alpha$ and $\alpha_{i}^{V}$ are perturbations which enter the metric, the perturbations $\alpha$, $\alpha_{i}^{V}$ and $\delta\theta$ are invariant under the infinitesimal Lorentz transformation (8). Therefore, the unitary gauge is compatible with the operation of absorbing the perturbations of the Lorentz matrix into the perturbations of the tetrad.