The starting point of this work is the original Einstein action, sometimes called the Gamma squared action. Continuing from our previous results, we study various modified theories of gravity following the Palatini approach. The metric and the connection will be treated as independent variables leading to generalized theories, which may contain torsion or non-metricity or both. Due to our particular approach involving the Einstein action, our setup allows us to formulate a substantial number of new theories not previously studied. Our results can be linked back to well-known models, such as Einstein–Cartan theory and metric-affine theories, and also links to many recently studied modified gravity models. In particular, we propose an Einstein–Cartan type modified theory of gravity, which contains propagating torsion, provided our function depends non-linearly on a boundary term. We also can state precise conditions for the existence of propagating torsion. Our work concludes with a brief discussion of cosmology and the role of cosmological torsion in our model. We find solutions with early-time inflation and late-time matter dominated behavior. No matter sources are required to drive inflation, and it becomes a purely geometrical effect.

## I. INTRODUCTION

The complete formulation of the Einstein field equations was followed, almost immediately, by a variational formulation due to Hilbert. Perhaps surprisingly, the simplest curvature scalar that can be defined on a Lorentzian manifold, namely, the Ricci scalar, acts as the Lagrangian when deriving the Einstein field equations using the calculus of variations. The gravitational action based on this Lagrangian is the Einstein–Hilbert action. In many ways, the Einstein–Hilbert action is somewhat unusual when compared to other field theories. The action contains second derivatives of the dynamical variables, which are the components of the metric tensor. The resulting field equations are nonetheless of second order because it is possible to write all these second derivative terms in the form of a total derivative, which will not contribute to the equations of motion.

When this total derivative or boundary term is subtracted from the Ricci scalar, one finds an action quadratic in the Christoffel symbol components, which is known as the Einstein action.^{1–3} While more natural than the Einstein–Hilbert action, when compared to other field theories, this action is no longer a coordinate scalar. It differs from a coordinate scalar by the boundary term, which was subtracted. In our previous work,^{4} we presented a comprehensive study of modified theories of gravity that are based on this decomposition of the Ricci scalar into two distinct terms. We often refer to those as the bulk and the boundary terms.

It is natural to extend this work in the context of metric-affine theories of gravity (see Refs. 5 and 6). By this, we mean to treat the metric and the connection as independent variables; this is also often called the Palatini approach. When considering the Einstein–Hilbert action and assuming a manifold with curvature and torsion, one will naturally arrive at Einstein–Cartan theory^{7} (also see Ref. 8 for a recent review). In this model, matter is the source of curvature, while spin acts as the source of torsion. A well-known feature of Einstein–Cartan theory is that torsion does not propagate, with its field equation being algebraic. This means regions of spacetime without torsion sources cannot contain torsion.

The approach put forward in this work will allow us to formulate a new Einstein–Cartan type modified theory of gravity, which contains propagating torsion. In fact, we show that Einstein–Cartan theory is the unique, minimally coupled, diffeomorphism invariant theory in which torsion does not propagate (for non-minimal coupling considerations, see Refs. 9–11). In passing, we state the precise conditions for the existence of propagating torsion within our models, shown to be consistent with other theories with propagating torsion.^{12–18}

Our approach leads to a large variety of different modified theories of gravity, many of which have not been studied before. This is due to the decomposition of the Ricci scalar, which is the starting point of our considerations. This is different to many other modified theories of gravity (see, for example, Refs. 6 and 19–36 for various publications covering a plethora of theories and approaches). Probably, the most obvious difference between many previously studied models and ours is the decomposition of the Ricci scalar into diffeomorphism breaking terms. These terms appearing in this decomposition are not true coordinate scalars; they are not invariant under coordinate transformations, in general, and we often refer to such terms as pseudo-scalars. We can therefore construct models that are distinctly different and cannot be directly linked back to previous studies due to the breaking of diffeomorphism invariance. Using, for example, a Born-Infeld type approach,^{37–40} one could construct models that break diffeomorphism invariance on very small scales, where classical physics breaks down.

As an application of our construction, we consider a modified Einstein–Cartan type theory in the context of flat Friedmann–Lemaitre–Robertson–Walker (FLRW) cosmology. Our model contains a simple quadratic non-linear term that is sufficient to give rise to propagating torsion. More importantly, torsion can exist without the need to introduce sources, similar to general relativity where vacuum solution contains curvature. Torsion decays as the universe expands as one would expect. Most interestingly, we find early-time solutions where the Universe undergoes a period of accelerated expansion, i.e., inflation. Our solution does not require the introduction of sources to drive inflation, and it becomes a purely geometrical effect. We carefully study the early-time and late-time asymptotic behavior of our solution and compare our results with those of standard cosmology.

### A. Notation and definitions

Following our previous work,^{4} Greek indices refer to spacetime objects, and we use the signature (−, +, +, +). An overline/overbar generally denotes the “full” object in the context of metric-affine spaces. This means that $\Gamma \u0304$ stands for the complete connection, which contains the usual Levi-Cività part plus contributions from torsion and non-metricity. Where possible and practical, we will follow the notation of Schouten.^{41} The covariant derivative ∇ will always stand for the derivative with respect to the metric-affine connection. In the few instances where a different derivative is used, this is made explicit.

*A*

_{[μν]}= (

*A*

_{μν}−

*A*

_{νμ})/2 and

*A*

_{(μν)}= (

*A*

_{μν}+

*A*

_{νμ})/2. The affine connection can be decomposed into its Levi-Cività part Γ and torsion and non-metricity parts,

## II. MODIFIED METRIC-AFFINE GRAVITY

This entire section contains the basic setup, which underlies our work. At its heart is the decomposition of the metric-affine Ricci scalar and the subsequent split of its Levi-Cività part into a bulk term and a boundary term. As will be emphasized throughout this work, boundary terms play a crucial role in our approach, and it is the particular way in which they are treated, which gives rise to new modified theories of gravity.

### A. Affine curvature decomposition

### B. Alternative forms of the bulk term

The above decomposition of the Ricci scalar is unique in the sense that there is only one canonical boundary term. However, one can write $G\u0304$ in a variety of different equivalent ways, as was already done in (2.6) and (2.8).

Formulation (2.8) turns out to be useful as the field equations will contain the object $E\u0304\mu \nu \lambda $, as will be seen below. Second, in the limit of vanishing non-metricity and vanishing torsion, one immediately sees that $\u2212E\mu \nu \lambda \u2202\lambda g\mu \nu /4=G$. The first term of (2.8) in this limit gives −**G**. This neatly matches Ref. 4 where we wrote $G=12M\mu \nu \lambda \Gamma \mu \nu \lambda $, with *E*^{μνλ} = *M*^{{λμν}}.

### C. Note on boundary terms in teleparallel theories

*T*and

*Q*are the torsion and non-metricity scalars,

*B*

_{T}and

*B*

_{Q}are their respective boundary terms, and

**C**are the torsion-non-metricity cross terms. The quantities

**G**and

**B**are the bulk and boundary terms of the Levi-Cività Ricci scalar

*R*. One can then show that the affine quantities $G\u0304$ and $B\u0304$ are related to the geometric scalars by

*Q*

_{λμν}= 0. These assumptions lead to the following relations between the geometric scalars given in Eq. (2.13):

*b*

_{T}is another boundary term identified in Ref. 4. Likewise, for symmetric teleparallel geometries ($R\u0304\mu \nu \rho \lambda =0$ and

*T*

^{λμν}= 0), one finds

*b*

_{Q}being a boundary term.

^{4}

An interesting observation is that the requirement for these boundary terms *b*_{T} or *b*_{Q} to vanish (i.e., for **G** = −*T* or **G** = −*Q*) is equivalent to requiring $G\u0304=0$, along with flatness $R\u0304=0$. This former requirement is a coordinate dependent condition. The *Teleparallel Equivalents of General Relativity*^{33,47–50} utilize choosing an affine connection that is flat such that the Einstein–Hilbert Lagrangian *R* can be related to the torsion or non-metricity scalars, up to boundary terms. However, from (2.14), it is clear that setting just the bulk part to zero $G\u0304=0$ yields another equivalence between general relativity and theories with torsion and non-metricity. Clearly, this is very much at the heart of the equivalence of the different formulations of general relativity and emphasizes issues such as the choice of tetrads in *f*(*T*) theories or the choice of coordinates in *f*(*Q*) theories. In particular, the condition *b*_{T} = 0 depends on both frames and coordinates.

### D. Variations with respect to metric and connection

*Q*

_{μ}or

*T*

_{μ}completely unconstrained, again due to the projective invariance. One retrieves GR by assuming either of these vectors to vanish.

^{4}except with the affine connection being used instead of the Levi-Cività connection.

The connection equation of motion is proportional to Palatini tensor; recall that $G\u0304$ is also algebraic in torsion and non-metricity. Therefore, the connection equation of motion is not dynamical in either torsion or non-metricity, in agreement with the analogous results in the standard Einstein–Palatini variations. For example, it is well-known that torsion does not propagate in Einstein–Cartan theory, which, in particular, implies the absence of torsional gravitational waves in vacuum. This is in stark contrast to metric perturbations, which can propagate through vacuum regions of spacetime.

In order to obtain dynamical behavior in either of these geometric quantities, we must consider a modified action that includes the boundary term $B\u0304$. It is precisely those boundary terms that contain the derivatives of torsion and non-metricity. Using the calculus of variations, the required integration by parts introduces additional derivative terms that ultimately give rise to differential as opposed to algebraic equations.

It is interesting to note that making the Palatini variations of $f(G\u0304)$ and then choosing the Levi-Cività connection (consistent with solving the connection field equation with vanishing hypermomentum) leads back to our metric *f*(**G**) gravity theory.^{4} However, performing the same procedure with $f(G\u0304,B\u0304)$ does not yield the metric *f*(**G**, **B**) model. This is perhaps unsurprising as the same situation occurs in Palatini *f*(*R*) gravity.^{21,22,52,53} The reason is clear from the form of (2.24): if one assumes the Levi-Cività connection, the Palatini tensor $P\mu \nu \lambda $ vanishes identically. Therefore, none of the content of the connection equation of motion of the *f*(**G**) model is lost. In contrast, the fourth-order terms coming from the *f*(…, **B**) field equation are “lost” in the connection equation of motion. Specifically, the Levi-Cività connection includes derivatives of the metric, which means an integration by parts would be performed on the *∂*_{λ}*f*_{,B} term, leading to the fourth-order terms in the Levi-Cività variation of the *f*(**G**, **B**) action.

### E. Diffeomorphisms and conservation equations

^{4}Note that this object is not a tensor. The transformation above can be seen more easily when the bulk term is written in the form (2.11); however, the calculation is still quite tedious.

*δ*

_{ξ}

*S*= 0, up to boundary terms. Hence, we obtain the identity

^{4}where performing the same calculation leads to the contracted Bianchi identity, albeit written in a way that is not manifestly covariant. The above identity (2.31) is just the metric-affine version of the contracted Bianchi identity. To see this written in a more conventional form, take the metric and connection variations (2.19), and let this be generated by the infinitesimal transformation

*ξ*, which leads to

*f*(

*R*) gravity and Palatini $f(R\u0304)$ gravity.

^{54,55}Instead, we will take the former approach, as in Eq. (2.30), where we obtain a conservation law from the infinitesimal transformations of $G\u0304$ and $B\u0304$ that takes a more compact form. Under an infinitesimal diffeomorphism, the modified action transforms as

*ξ*leaves the expression

*g*

_{μν}and the choice of coordinates

*x*

^{μ}. If we also include minimally coupled matter in our total action, we have the following on-shell conservation law:

*S*

_{tot}=

*S*

_{mod}+

*S*

_{matter}, we require Eq. (2.38) to be satisfied. This is the most general case as we do not consider non-minimal couplings here. If the matter action

*S*

_{matter}is a coordinate scalar, then we must impose that the first of these equations (2.37) is satisfied. In these cases, one obtains the usual metric-affine matter-hypermomentum conservation laws,

More generally, one could conceive scenarios where the matter action is also non-covariant, and it is only the total action that remains invariant under diffeomorphisms. Some of these possibilities are discussed in more detail for the Levi-Cività case in Ref. 4. This could be implemented by allowing for non-minimal couplings between the pseudo-scalars $G\u0304$ and $B\u0304$ and matter, in which case (2.38) would need to be modified.

Finally, relating to the non-covariance of these theories, we note the interesting possibility of restoring the full diffeomorphism invariance of the theory by using the Stückelberg trick.^{56} In this sense, the theory can be thought of as a gauge-fixed version of a fully covariant theory, where extra degrees of freedom should become apparent. This is exactly the case for the Levi-Cività version of *f*(**G**) gravity, which turns out to give rise to the modified symmetric teleparallel theories *f*(*Q*) gravity when covariance is restored.^{57} There, the extra degrees of freedom can be attributed to the independent components of the symmetric teleparallel connection, the coordinate functions often denoted by *ξ*(*x*). These are precisely the Stückelberg fields, which trivialize in the coincident gauge *ξ*^{μ}(*x*) = *x*^{μ}, where *f*(*Q*) reduces to *f*(**G**) (see Refs. 4, 57, and 58). This is also reminiscent of the diffeomorphism-invariant formulation of classical unimodular gravity, which introduces Stückelberg fields to restore the full symmetry of the theory.^{59,60}

This would be an interesting avenue to explore in the future as it is not obvious what type of theory this procedure would lead to in this metric-affine case, nor how these extra degrees of freedom would manifest. For the remainder of this work, we will assume that (2.37) is satisfied by using an appropriate choice of coordinates.

## III. NEW MODELS AND RELATIONS TO OTHER THEORIES

### A. General field equations with source terms

### B. General relativity and its metric-affine generalizations

^{7}follows from the above when setting non-metricity to zero. However, as explained, this procedure cannot be directly put into these equations as the metric and connection variations are no longer fully independent, and one would arrive at incorrect field equations. Let us supplement our action by the following Lagrange multiplier:

*λ*

^{μνλ}=

*λ*

^{μλν}to match the symmetry properties of non-metricity. We now choose as independent variables ${g,\Gamma \u0304,\lambda}$ and any matter fields present. Writing out this covariant derivative explicitly and using integration by parts, we can write the constraint as

*Q*= 0 in (2.12), one finds

*P*

_{μ(νρ)}= 0 as this object is skew-symmetric over the last two indices. Hence, the symmetric part of (3.14) yields the Lagrange multiplier to be

*λ*

_{μνρ}= −Δ

_{μ(νρ)}, while the skew-symmetric part of this equation gives

*P*

_{μνρ}= 2

*κ*Δ

_{μ[νρ]}as expected from Einstein–Cartan theory. Finally, substitution of these results back into the first field equation (3.13) gives the desired field equations of Einstein–Cartan theory,

Let us make a brief comment regarding the need to use Lagrange multipliers in the metric-affine setting. We found explicitly that *λ*_{μνρ} = −Δ_{μ(νρ)}. For arbitrary matter sources, Δ_{μ(νρ)} ≠ 0, in general, so that *λ*_{μνρ} ≠ 0, in general. Consequently, one cannot simply set non-metricity to zero after performing the variations since one would lose the second term on the right-hand side of (3.16).

*τ*

_{μνλ}≔ Δ

_{μ[νλ]}.

### C. Palatini *f*(*R*) gravity

Palatini $f(R\u0304)$ gravity is also included in our approach and achieved by simply setting $f(G\u0304,B\u0304)=f(G\u0304+B\u0304)$. This follows from the basic decomposition of the Ricci scalar that underlies this work. Here, we work in the general affine setting with torsion and non-metricity present.

Taking the trace of the metric equation (3.20) leads to the well known algebraic relation $\u22122f(R\u0304)+f\u2032(R\u0304)R\u0304=\kappa T$. This implies an algebraic relation between the Ricci scalar $R\u0304$ and the trace of the energy–momentum tensor *T*. Contrast this with the $f(G\u0304,B\u0304)$ equations (3.1), where the trace leads to a dynamical equation as opposed to an algebraic one. This leads to a number of interesting possibilities beyond what can be found in the metric-affine $f(R\u0304)$ theories.

### D. A modified Einstein–Cartan theory

*λ*

_{μνρ}= Δ

_{μ(νρ)}and $P\mu \nu \rho f,G\u0304=2\kappa \Delta \mu [\nu \rho ]$. Hence, we can write the first and second field equations in the form

### E. A modified Einstein–Cartan theory with boundary terms

*λ*

_{μνρ}= −Δ

_{μ(νρ)}is the same as before. The skew-symmetric part now includes an additional dynamical term

The presence of the boundary terms in the form $f,B\u0304$ in the second field equation make this model distinctly different to the previous modified Einstein–Cartan type theories. In Eqs. (3.17) and (3.27), the vanishing of the source term, that is, Δ_{μ[νλ]} = 0, implied the vanishing of the Palatini tensor, which, in turn, implies the vanishing of the torsion tensor. One often says that torsion is non-dynamical in Einstein–Cartan theory.

_{μ[νλ]}= 0 in Eq. (3.33). Using the expression of the Palatini tensor in terms of torsion, one can solve for the torsion tensor and finds the neat expression

As was already remarked at various occasions, these boundary terms play a crucial role in these modified theories of gravity. Models depending on non-linear functions of these boundary terms show distinctly different properties than models with a linear term, which is perhaps expected as an arbitrary function of a boundary term is no longer a boundary term.

## IV. COSMOLOGY

### A. Brief introduction

*a*(

*t*). One often works with an arbitrary lapse function

*N*(

*t*), which we generally set to one,

*N*(

*t*) = 1. One can verify that this choice is consistent by noting that the conservation equation (2.37) is satisfied for this model for a general lapse function

*N*(

*t*) in these coordinates. The spatially flat FLRW line element is given by

^{66}The only allowed components of a cosmological torsion tensor are given by

*a*

^{−3}was included. Let us now consider our Eq. (3.40), assuming a cosmological setting where all objects are functions of time only. Then, the right-hand side of (3.40) identically vanishes if all indices take spatial values {

*x*,

*y*,

*z*}. Therefore, this source free equation immediately yields $T123=T231=T312=0$ so that we can set

*k*(

*t*) = 0, and one is left with only a vector torsion contribution

*h*/

*a*

^{3}.

**B**and a torsional contribution

*B*

_{T}[see Eq. (2.15)]. These are given by

*a*

^{−3}in (4.3). The torsion field equation now becomes

### B. Quadratic boundary term model

*p*=

*wρ*. The connection equation is

*Riccati*[a Riccati equation is a non-linear ordinary differential equation (ODE) of the form

*y*′(

*x*) =

*q*

_{0}(

*x*) +

*q*

_{1}(

*x*)

*y*+

*q*

_{2}(

*x*)

*y*

^{2}, where

*q*

_{1}= 0] differential equation relating torsion, the scale factor, and the matter content

*β*> 0. The chosen sign will become clear in what follows. The task at hand is to solve the system of equations (4.11) and (4.12), together with the Riccati equation (4.14) and continuity equation for matter. Since the boundary term depends on torsion and the scale factor, one naturally has a system of two equations in two unknowns. Note that matter satisfies the standard conservation equation in our model so that

*ρ*=

*ρ*

_{0}/

*a*

^{3(1+w)}. It turns out that one can eliminate torsion from the resulting equations and arrive at a single first order equation in the scale factor. For simplicity, we will only state this equation for

*w*= 0, but the general equation is of similar form

*Y*simplifies the presentation of this key equation substantially. The main observation is that the right-hand side is a function of the scale factor only. Therefore, this equation is, in principle, separable. Perhaps surprisingly, the resulting integral can be expressed in terms of elementary functions. This equation cannot be solved for

*a*(

*t*) explicitly but gives a closed form implicit solution. Some additional detail is given in Appendix C.

*a*(

*t*) ≫ 1, which corresponds to the late-time universe, and arrive at

*β*, which strongly suggests that the boundary term affects the early-time dynamics of the universe only.

*a*(

*t*) ≪ 1, which gives

*a*=

*a*

_{0}exp(

*λt*) for the early universe, one finds the very neat result

*β*. Let us also note that the previous equation can easily be found for general

*w*, which gives the expression

*λ*is affected by the matter equation of state

*w*, we can safely state that reasonable matter choices have no qualitative impact on this inflationary epoch.

This early-time inflation will nonetheless yield a late-time matter dominated universe whose expansion is independent of *β*. It is reasonable to expect that the introduction of the standard cosmological constant into this model will yield late-time accelerated expansion. Self-accelerating solutions with propagating torsion have been found in other theories with higher powers of curvature scalars even without spin/hypermomentum sources, e.g., in the context of massive gravity.^{67–69} However, given the simplicity of the model studied here, this is a most surprising result.

Figure 1 shows the evolution of the scale factor and the Hubble parameter for a matter dominated universe, in agreement with the above asymptotic discussions. We also show the relevant results of standard cosmology, as dashed lines. Contrary to standard cosmology where the Hubble function diverges as *t* → 0, we find that *H* → *λ* as *t* → −∞. This discussion can also be repeated for a radiation equation of state where the epoch of early-time inflation would then be followed by a radiation dominated epoch.

Next, let us turn to the behavior of torsion. Since the late-time behavior of the scale factor and matter agree with the GR results, we can assume *ρ* ∝ *a*^{−3} ∝ *t*^{−2} and *a* ∝ *t*^{2/3} for late times, again assuming *w* = 0. Using this input in (4.14), we can solve the Ricatti equation, explicitly or numerically. For the matter dominated epoch, we find −*h*/*a*^{3} ∝ 1/*t*^{3} ∝ *ρ*^{3/2}, which, using the form of *a*(*t*), also yields $\u2212h\u221d\rho $. This is confirmed in Fig. 2 where we display a log–log plot to emphasize the scaling behavior of the late-time solution.

Likewise, we can also study the Ricatti equation assuming *a* ∝ exp(*λt*). As before, one has *ρ* ∝ *a*^{−3} because the equation of state is unchanged. This gives −*h*(*t*) ∝ exp(3*λt*/2), which we can again verify using the numerical solutions, shown in Fig. 3. Rather interestingly, torsion again scales with the scale factor, and perhaps even more noteworthy, it thus scales with the matter. In particular, one finds $\u2212h(t)\u221d1/\rho $, which is the inverse relation when compared to the late-time solution.

## V. CONCLUSIONS AND DISCUSSIONS

The primary objective of this work was to generalize our previous results in the metric-affine or Palatini way. By this, we mean gravity models where the metric and the connection are treated as *a priori* independent variables. Within this framework, one can recover theories such as Einstein–Cartan theory or general relativity via the introduction of suitable Lagrange multipliers. From our point of view, this is the most elegant approach to consider particular models without torsion or without non-metricity. Teleparallel models can also be studied by setting curvature to zero, again via a Lagrange multiplier. While other approaches can work, directly setting some quantities to zero in the action can prove problematic. This can be seen succinctly in Sec. III B. One might be tempted to set non-metricity and the Lagrange multiplier to zero; however, this is inconsistent because the Lagrange multiplier is, in fact, non-vanishing, as already commented on in Sec. III B.

The theory proposed here, based on the non-covariant decomposition of the affine Ricci scalar, breaks diffeomorphism invariance and, therefore, allows for a number of interesting new gravitational models to be studied. To maintain the consistency of the variational methods procedure, we enforced that the total action be invariant under arbitrary coordinate transformations. This introduces a set of “constraint equations” or conservation equations that must be satisfied for all solutions, which can alternatively be viewed as picking out appropriate coordinates. Here, as well as the examples studied in our previous work,^{4} the gravitational sector satisfied these equations, implying the usual conservation laws for the matter energy–momentum tensor.

After setting up the entire framework so that metric-affine theories can be studied, we were particularly interested in models in an Einstein–Cartan geometry with propagating torsion. We demonstrated that torsion can exists in certain settings, even in the absence of sources. This was achieved by considering actions containing non-linear boundary terms, at which point they can no longer be seen as boundary terms. The explicit conditions needed to produce propagating torsion were derived, and the presence of these modified boundary terms were found to be essential.

Using a simple quadratic term, we considered cosmological solutions assuming the standard FLRW framework. In this setting, the metric only contains a single unknown function, while the torsion tensor is reduced to two independent components, a vector torsion piece and an axial torsion piece. In our specific model, the axial torsion component had to vanish identically to satisfy the cosmological field equations, leaving us with only one additional function when compared to general relativity. The resulting field equations are relatively easy to deal with, and it is straightforward to study their asymptotic behavior for early and late times. We found late-time solutions in agreement with standard cosmology, while the early-time solutions were affected by our modified non-linear terms. In particular, we found an early-time epoch of inflation, driven solely by our geometric torsion component. Our solutions offers a smooth transition from an epoch of accelerated expansion to an epoch of matter (or radiation) domination. This is achieved using a relatively simple modified theory of gravity based primarily on the Einstein action, without the need of additional fields.

It should again be emphasized that our model generally breaks diffeomorphism invariance, with a new set of constraint equations resulting from requiring the action to be invariant under infinitesimal coordinate transformations. We interpreted these equations as constraints on the allowed coordinates of our metric and connection equations. In the cosmological model studied above, using Cartesian coordinates and the standard FLRW symmetries, these constraint equations are automatically satisfied. Consequently, the energy–momentum tensor is covariantly conserved, despite the lack of invariance of the action.

The cosmological solutions were studied in the Einstein–Cartan framework with vanishing non-metricity, but the full metric-affine theory could easily be incorporated into the study. Similarly, our general field equations and conservation equations allow for the possibility of connection source terms, the hypermomentum current. This may lead to a more interesting, albeit more complicated, set of solutions. One could also generalize the approach to allow for non-minimal couplings between the geometrical pseudo-scalars and the matter content, at which point the usual energy–momentum conservation laws would be modified. While non-minimal couplings between matter and geometry are straightforward to formulate at the level of the action, it is difficult to make qualitative statements about the effects of such terms without performing a detailed investigation. Moreover, we lack the knowledge of any guiding principles to select certain couplings over others, other than perhaps a preference for simpler models. Having said this, the minimally coupled theory presented here can be seen as the most obvious modification based on the natural decomposition of the Ricci scalar into its bulk and boundary parts within the metric-affine framework.

## ACKNOWLEDGMENTS

We thank Yuri Obukhov for valuable discussions. Erik Jensko was supported by the EPSRC Doctoral Training Program (Grant No. EP/R513143/1).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Christian G. Böhmer**: Supervision (lead); Writing – original draft (equal); Writing – review & editing (equal). **Erik Jensko**: Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

### APPENDIX A: EXPLICIT VARIATIONAL CALCULATIONS

#### 1. Derivation of Einstein field equations

*δg*

_{αβ}

*∂*

_{ν}

*g*

^{αβ}= −

*δg*

^{σγ}

*g*

_{σα}

*g*

_{γβ}

*∂*

_{ν}

*g*

^{αβ}=

*δg*

^{σγ}

*∂*

_{ν}

*g*

_{σγ}. Introducing the shorthand notation $\Gamma \u0303\kappa \lambda \nu \u2254\Gamma \u0304\mu \lambda \mu \delta \kappa \nu \u2212\Gamma \u0304\kappa \lambda \nu $ and expanding the partial derivatives, we have

#### 2. Derivation of modified field equations

*f*is a linear function of $B\u0304$, then this term is a pure boundary term and does not contribute to the equations of motion. Plugging these into the variations (A11) and discarding boundary terms, we obtain

### APPENDIX B: PROJECTIVE TRANSFORMATIONS

*P*

_{μν}≔

*P*

_{μ}

*P*

_{ν}− ∇

_{μ}

*P*

_{ν}and setting

*c*

_{1}=

*c*

_{2}= 1, the above (Levi-Cività) transformation can be expressed as

^{41}

*c*

_{1}= 0, the Ricci scalar is invariant.

*P*

^{λ}

*P*

_{λ}term remains; this, along with the total divergence term ∇

_{κ}

*P*

^{κ}in the

**B**transformation below, immediately gives the Schouten result (B8)]

*c*

_{1}= 0. From this, we can deduce that any action comprised of these objects will share this invariance (see Ref. 52). Importantly, this implies that the connection variations will lead to equations that are trace free over these indices. The variation of the $f(G\u0304,B\u0304)$ action with respect to the connection leads to the equation

*ν κ*,

### APPENDIX C: COSMOLOGICAL SOLUTION

*t*

_{0}is the constant of integration. Using the expression of

*y*in terms of

*a*gives an implicit formula for

*a*(

*t*).

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*Relativitätstheorie*

*Gravitation: Foundations and Frontiers*

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*Modified and Quantum Gravity: From Theory to Experimental Searches on All Scales*

*f*(

*R*) theories of gravity

*Teleparallel Gravity: An Introduction*

*f*(

*T*) teleparallel gravity and cosmology

*Extensions of f(R) Gravity: Curvature-Matter Couplings and Hybrid Metric-Palatini Theory*

*Quantum Riemannian Geometry*

*Ricci-Calculus*

*Teleparallel Gravity: An Introduction*

*f*(

*R*) theories of gravity

*f*(

*R*) theories and beyond

*f*(

*R*) gravity