Symmetric teleparallel cosmology with boundary corrections

General Relativity is a well-tested theory gw1 ; however, it faces challenges on cosmological scales ten1 ; ten1a ; ten2 . Recent cosmological observations rr1 ; Teg ; Kowal ; Komatsu ; suzuki11 suggest that the universe is currently experiencing an acceleration phase driven by an exotic matter source known as dark energy jo1 ; jo . On the other hand, to address the homogeneity and flatness problems, it has been proposed that the universe underwent an inflationary epoch in its early stages driven by the inflaton field inf1 .

The nature and physical properties of the inflaton and dark energy remain subjects of debate among cosmologists. Cosmologists have proposed different models to explain cosmic acceleration, which can be broadly categorized into two groups. In the first category, a matter source with negative pressure is introduced into the field equations of General Relativity ratra ; de1 ; de2 ; de3 ; de4 ; de5 . In the second category of models, cosmologists focus on determining a new gravitational theory by modifying the gravitational Action Integral de01 ; de02 ; de03 .

The consideration of quantum-gravitational effects in the one-loop approximation in gravity df1 ; df2 leads to the introduction of the $R^{2}$ term in the gravitational Lagrangian. This quadratic theory of gravity qua1 ; qua2 ; qua3 has been used as a mechanism to explain inflation star ; barc ; stt1 . Indeed, when the $R^{2}$ term dominates, it leads to a de Sitter expansion bar1 . The success of the $R^{2}$ term in describing acceleration has given rise to a family of theories known as $f(R)$-gravity Buda . For a comprehensive review, we recommend referring to Sotiriou .

The symmetric teleparallel $f(Q)$-theory of gravity has garnered the attention of cosmologists lav2 ; lav3 . $f(Q)$-theory is defined within the framework of symmetric teleparallel theory nester ; nester1 , where the physical space is described by the metric tensor. However, the fundamental connection in this theory is not the Levi-Civita connection but a symmetric and flat connection, leading to the definition of a nonzero non-metricity scalar $Q$. When $f(Q)$ is linear, the gravitational theory becomes equivalent to General Relativity, known as Symmetric Teleparallel General Relativity (STGR) nester . In STGR, the Ricci scalar $R$ and the non-metricity scalar $Q$ differ only by a boundary term $B$, which means that the variation of the boundary term is neglected, resulting in equivalent field equations. There is a plethora of studies in the literature on $f(Q)$-theory; for example, see fq1 ; fq2 ; fq3 ; fq4 ; fq5 ; fq6 ; fq7 ; fq8 ; fq9 ; fq10 ; fq11 , and references therein.

In the process of constructing a gravitational theory, the boundary term $B$ has been utilized to modify the gravitational Action Integral. Inspired by teleparallelism bah , the symmetric teleparallel theory with a boundary term, expressed as $f(Q,B)$-gravity, was introduced in ftc1 ; ftc2 . It was discovered that the introduction of the boundary term further modifies the field equations of the gravitational model. In this study, we aim to investigate the effects of the boundary term on cosmological dynamics.

Because the definition of the symmetric and flat connection is not unique, there can be multiple different connections that describe the same gravitational model. The corresponding non-metricity constants differ by a boundary term, which means that they lead to the same equations in STGR. However, this is not true in the case of a nonlinear Lagrangian. As a result, the definition of the connection affects the boundary term, implying that $f(Q,B)$-gravity depends on the definition of the connection.

Recently, in min , it was proven that $f(Q)$-theory of gravity admits a minisuperspace description, and the theory is of fourth- or sixth-order, depending on the connection used. The higher-order derivatives can be attributed to a scalar field, and the gravitational Lagrangian can be written as that of a higher-dimensional second-order dynamical system. This approach is applied in this study in the context of $f(Q,B)$-gravity for a homogeneous and isotropic geometry. The structure of the paper is as follows.

In Section II, we present the definition of symmetric teleparallel gravity and its generalization, the $f(Q)$-theory, where $Q$ represents the non-metricity scalar. The extension of symmetric teleparallel theory with a boundary term $B$ is discussed in Section III. For the $f(Q,B)$-theory, we discuss when it is equivalent to teleparallel $f(T,B_{T})$ theory and when it approaches the limit of $f(R)$-gravity. In Section IV, we present the gravitational field equations for a homogeneous and isotropic universe. Specifically, we employ Lagrange multipliers to demonstrate that the field equations in $f(Q,B)$-theory can be described within a minisuperspace framework. It’s important to note that the definition of the symmetric and flat connection in a FLRW universe is not unique. Therefore, we provide the field equations for all four different families of connections. One novel aspect of the Lagrange multiplier approach is the introduction of scalar fields to account for the higher-order derivatives in the gravitational model. Consequently, $f(Q,B)$-cosmology is a fourth-order gravitational theory, equivalent to teleparallel $f(T,B_{T})$-cosmology, when the symmetric and flat connection is defined in the coincidence gauge. However, when the connection is defined in the non-coincidence gauge, the field equations become eighth-order and are described by three scalar fields.

Furthermore, in Section V, we focus on the particular case of $f(Q,B)=Q+F(B)$ cosmology, where the gravitational Action Integral is modified by nonlinear terms of the boundary. For this specific gravitational model, the order of the gravitational field equations is reduced by two in the non-coincidence gauge. Consequently, the geometric fluid is described by two scalar fields. In Section VI, we present a new analytic solution for symmetric teleparallel cosmology with a boundary term in the non-coincidence gauge. This new solution is capable of describing the $\Lambda$CDM universe at the present time and possesses a de Sitter attractor. Finally, in Section VII, we summarize our findings and draw our conclusions.

We examine a gravitational model described by the four-dimensional metric tensor $g_{\mu\nu}$ and the covariant derivative $\nabla_{\lambda}~{}$ which is defined using the generic connection $\Gamma_{\mu\nu}^{\kappa}\,$, such that the autoparallels are defined as auto

Connection  $\Gamma_{\mu\nu}^{\kappa}$ determines the nature of the geometry.

For the general connection we can define the the Riemann tensor

the torsion tensor

and the non-metricity tensor Eisenhart

In General Relativity, $\Gamma_{\mu\nu}^{\kappa}$ is recognized as the Levi-Civita connection, denoted as $\tilde{\Gamma}{\mu\nu}^{\kappa}$. Consequently, in this framework, $\mathrm{T}{\mu\nu}^{\lambda}=0$ and $Q_{\lambda\mu\nu}=0$. Therefore, the primary scalar in General Relativity is the Ricci scalar $R$.

On the other hand, in the Teleparallel Equivalent of General Relativity (TEGR) de03 , the connection $\Gamma_{\mu\nu}^{\kappa}$ is replaced by the antisymmetric Weitzenb”ock connection, resulting in $R_{;\lambda\mu\nu}^{\kappa}=0$ and $Q_{\lambda\mu\nu}=0$. In this context, the torsion scalar $T$ takes on the role of the fundamental geometric object in teleparallel gravity.

In the theory under consideration, which is STGR, $\Gamma_{\mu\nu}^{\kappa}$ possesses the property of being both flat and torsionless. This implies that $R_{;\lambda\mu\nu}^{\kappa}=0$ and $\mathrm{T}{\mu\nu}^{\lambda}=0$. Additionally, it inherits the symmetries of the metric tensor $g{\mu\nu}$. Thus, the nonmetricity scalar $Q$ defined as nester

is the fundamental geometric quantity of gravity.

Tensor $P^{\lambda\mu\nu}$ is defined as nester

which is written with the help of the traces¹¹1Parenthesis in the indices denote symmetrization, that is, $A_{(\mu\nu)}=\frac{1}{2}\left(A_{\mu\nu}+A_{\nu\mu}\right)$; and $\delta_{\;\nu}^{\mu}$ is the Kroncker delta. $Q_{\mu}=Q_{\mu\nu}^{\phantom{\mu\nu}\nu}$ and $\bar{Q}_{\mu}=Q_{\phantom{\nu}\mu\nu}^{\nu\phantom{\mu}\phantom{\mu}}$.

The Ricci scalar $R$ correspond to the Levi-Civita connection $\tilde{\Gamma}_{\mu\nu}^{\kappa}$ of the metric tensor $g_{\mu\nu}$, and the nonmetricity scalar $Q$ for a symmetric and flat connection $\Gamma_{\mu\nu}^{\kappa}$ differ by a boundary term $B,$defined as $B=R-Q~{}$.

The gravitational Action Integral of STGR reads nester

from where it follows that STGR is dynamically equivalent to GR.

However, when nonlinear components of the non-metricity scalar $Q$ are introduced in the gravitational Action, as in $f(Q)$-gravity, this equivalence is lost. Nevertheless, the corresponding gravitational theory does not possess any dynamical equivalence with General Relativity or its generalization, $f(R)$-gravity.

The Action Integral in symmetric teleparallel $f(Q)$-gravity is defined lav2 ; lav3

The resulting field equations are

Equations (7) represent the modified Einstein field equations in $f(Q)$-gravity, while equation (8) corresponds to the equation of motion for the connection. . When equation (8) holds true for a specific connection at all times, that connection is referred to as the “coincidence gauge”. However, if the equation (8) is not always satisfied for a particular connection, then that connection is defined within the so-called “non-coincidence gauge”as discussed in lav3 .

Recently, a generalization of the $f(Q)$ theory was introduced ftc1 ; ftc2 by incorporating a boundary term into the gravitational Action Integral. More precisely, this extended framework includes the gravitational Action Integral as follows

where $B=R-Q$.

In the notation of ftc2 , the boundary term is noted as $C$. This gravitational theory is equivalent to General Relativity, when $f\left(Q,B\right)$ is a linear function, that is, $f\left(Q,B\right)=f_{1}Q+f_{2}B-2\Lambda$; the limit of $f\left(Q\right)$ is recovered when $f\left(Q,B\right)=f\left(Q\right)+f_{2}B$; while the fourth-order $f\left(R\right)$ theory of gravity is recovered when $f\left(Q,B\right)=f\left(Q+B\right)$.

The gravitational field equations which correspond to the Action Integral (9) are

$f\left(Q,B\right)$ has been inspired by the teleparallel boundary gravity $f\left(T,B_{T}\right)~{}$bah , where now $B_{T}$ is the boundary term relates the Ricciscalar $R$ and the torsion scalar $T$ for the Weitzenböck connection Weitzenb23 . In a similar way, $f\left(T,B_{T}\right)$ recovers $f\left(R\right)$-gravity when $f\left(T,B_{T}\right)=f\left(T+B_{T}\right)~{}$bah , while the limit of GR is recovered when $f\left(T,B_{T}\right)=f_{1}T+f_{2}B_{T}-2\Lambda;$ and $f\left(T\right)$-theory follows when $f\left(T,B_{T}\right)=f\left(T\right)+f_{2}B_{T}$. There are many astrophysical and cosmological applications of $f\left(T,B_{T}\right)$-theory in the literature ftb1 ; ftb2 ; ftb3 ; ftb4 ; ftb5 ; ftb6 . From these results it is clear that the boundary $B_{T}$ plays an important role in geometric description of dark energy. Thus, the generalization if symmetric teleparallel theory seems natural.

$f\left(T,B_{T}\right)$ theory of gravity is a fourth-order theory, similar to $f\left(R\right)$-theory, while when $f\left(T,B_{T}\right)=f\left(T\right)+f_{2}B_{T}$ the resulting field equations are of second-order ftb7 . The order of symmetric teleparallel $f\left(Q\right)$-theory depends on the connection which is used for the definition of the nonmetricity scalar $Q$. For the connection defined in the coincidence gauge $f\left(Q\right)$ is a second-order theory, while for a connection in the non-coincidence gauge $f\left(Q\right)$-theory is a sixth-order theory of gravity.

To comprehend the degrees of freedom within $f(Q,B)$-theory, we turn our attention to a cosmological model representing an isotropic and homogeneous universe. Employing the Lagrange multipliers method, we introduce scalar fields that account for the dynamical degrees of freedom within the $f(Q,B)$-theory. Consequently, our analysis reveals that the theory introduces either one or three scalar fields.

The isotropic and homogeneous universe is described by the FLRW line element

in which $N\left(t\right)$ is the lapse function and $a\left(t\right)$ is the scale factor denotes the radius of the universe. Hence, $H=\frac{1}{N}\frac{\dot{a}}{a}$, where $\dot{a}=\frac{da}{dt}$ is the Hubble function. Parameter $k$ is the spatial curvature, for $k=0$, the universe is spatially flat, $k=+1$ corresponds to a closed FLRW geometry and $k=-1$ describes an open universe.

FLRW spacetime admits a sixth-dimensional Killing algebra consisted by the vector fields