Dynamical heredity from f(R)-bulk to braneworld: curvature dynamical constraint and f(R)-unimodular gravity

TL;DR

This paper explores the relationship between f(R) gravity theories in bulk and brane worlds, introducing a curvature dynamical constraint to connect them, and applies this to cosmological models with dark radiation.

Contribution

It develops the concept of curvature dynamical constraint to relate f(R)-brane and bulk theories, and derives a specific case of f(R)-unimodular gravity with cosmological applications.

Findings

f(R)-brane/bulk theories can be related via curvature dynamical constraint

Derived a dynamical version of the Gauss equation trace for f(R) theories

Applied to cosmology, showing dark radiation effects in f(R)-unimodular gravity

Abstract

Recently, Borzou et al. (BSSY) generalized the Shiromizu-Maeda-Sasaki (SMS) formulation to f(R)-bulks. BSSY brane projected equation carries an additional stress tensor, besides SMS correction for Einstein's theory on the brane. If we change this perspective, by requiring BSSY tensor in the geometrical side acting as f(R)-brane generator, it is possible to relate f(R)-brane/bulk theories, by using curvature dynamical constraint (CDC), a concept that we developed. Since that brane and bulk are f(R), 5/4D scalar curvatures also play a dynamical role and, thus, a dynamical version to Gauss equation trace, or CDC, must be offered. We will work yet in a specific case to obtain f(R)-unimodular gravity, formally identical with obtained by Nojiri et al. (NOO). Therefore, two applications which consider cosmological scenarios in f(R)-unimodular gravity with dark radiation correction will be offered.

Equations

$$\text{“if }{\mathcal{Q}}_{\mu \nu }\text{ is in the left side, it generates an effective }f(R)\text{-theory on the brane”.}$$

$$g_{ab}d{x}^{a}d{x}^b=q_{\mu \nu }(x^{\alpha },y)d{x}^{\mu }d{x}^{\nu }+d{y}^2.$$

$$S=\frac{1}{{\kappa }_4^2}\int d^{4}x\left[\sqrt{-g}\left(f\left(R\right)-\pounds \right)+\pounds \right]+S_{fields},{\kappa }_4^2\equiv 8\pi G,$$

$$R_{\mu \nu }{d}_{R}f(R)-\frac{1}{2}\left[f(R)-\pounds \right]g_{\mu \nu }+\left(g_{\mu \nu }\square -{\nabla }_{\mu }{\nabla }_{\nu }\right)d_{R}f(R)={\kappa }_4^2{T}_{\mu \nu },$$

$$\sqrt{-g}=1,$$

$$R_{\mu \nu }{d}_{R}f(R)-\frac{1}{2}f(R)g_{\mu \nu }+\left(g_{\mu \nu }\square -{\nabla }_{\mu }{\nabla }_{\nu }\right)d_{R}f(R)={\kappa }_4^2{T}_{\mu \nu },$$

$$\left[\left(R+3\square \right)d_R-2\right]f(R)={\kappa }_4^{2}T.$$

$$\left(\square -m^2\right)R=m^2{\kappa }_4^{2}T,\text{ with }m\equiv \pm 1/6a^2.$$

$$ϝ\equiv 6H{\partial }_t{\partial }_{t}H+18H^2{\partial }_{t}H-3{\left({\partial }_{t}H\right)}^2=-3m^2{H}^2,$$

$$R_{\mu \nu }-\frac{1}{4}R{g}_{\mu \nu }={\kappa }_4^2\left(T_{\mu \nu }-\frac{1}{4}T{g}_{\mu \nu }\right).$$

$${\nabla }^{\mu }{G}_{\mu \nu }=0={\nabla }^{\mu }{T}_{\mu \nu }:{\partial }_{\mu }\left(R+{\kappa }_4^{2}T\right)=0,$$

$$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }=-{\Lambda }^{(U)}{g}_{\mu \nu }+{\kappa }_4^2{T}_{\mu \nu }.$$

$$R_{abc}{}^f=q^f{}_{e}q{}_a{}^{d}q{}_b{}^{g}q{}_c{}^h{\mathcal{R}}_d{}_{gh}{}^e+2K_{c[a}{K}_{b]}{}^f,$$

$$K_{\mu \nu }=-\frac{1}{2}{k}_5^2\left[{\tau }_{\mu \nu }+\frac{1}{3}\left(\lambda -\tau \right)q_{\mu \nu }\right].$$

$${\mathcal{R}}_{ab}{d}_{\mathcal{R}}\mathcal{F}(\mathcal{R})-\frac{1}{2}\mathcal{F}(\mathcal{R})g_{ab}+\left(g_{ab}\boxminus -D_a{D}_b\right)d_{\mathcal{R}}\mathcal{F}(\mathcal{R})={\kappa }_5^2{T}_{ab},$$

$$T_{ab}=-{\Lambda }_5{g}_{ab}+\left(-\lambda g_{ab}+{\tau }_{ab}\right)\delta (y).$$

$$R_{\mu \nu }-\frac{1}{2}R{q}_{\mu \nu }={\mathcal{J}}_{\mu \nu }+{\mathcal{Q}}_{\mu \nu },$$

$${\mathcal{J}}_{\mu \nu }=-{\Lambda }_4{q}_{\mu \nu }+{\kappa }_4^2{\tau }_{\mu \nu }+\frac{6{\kappa }_4^2}{\lambda }{\pi }_{\mu \nu }-{\mathcal{E}}_{\mu \nu }.$$

$${\pi }_{\mu \nu }=\frac{1}{12}\tau {\tau }_{\mu \nu }-\frac{1}{4}{\tau }_{\mu \sigma }{\tau }_{\nu }^{\sigma }+\frac{1}{24}(3{\tau }_{\sigma \delta }{\tau }^{\sigma \delta }-{\tau }^2)q_{\mu \nu },\tau ={\tau }_{\mu }{}^{\mu }.$$

$${\Lambda }_4=\frac{1}{2}{k}_5^2\left[{\Lambda }_5+\frac{1}{6}{k}_5^2{\lambda }^2\right],\text{ with }{\kappa }_4^2=\frac{\lambda }{6}{\kappa }_5^4,$$

$${\mathcal{Q}}_{\mu \nu }={\left[F(\mathcal{R})q_{\mu \nu }+\frac{2}{3}\frac{D_a{D}_b\left(d_{\mathcal{R}}\mathcal{F}\right)}{d_{\mathcal{R}}\mathcal{F}}\left({\delta }_{\mu }^a{\delta }_{\nu }^b+n^a{n}^b{q}_{\mu \nu }\right)\right]}_{y=0},$$

$$F(\mathcal{R})=-\frac{4}{15}\frac{\boxminus \left(d_{\mathcal{R}}\mathcal{F}\right)}{d_{\mathcal{R}}\mathcal{F}}-\frac{\mathcal{R}}{10}\left(\frac{3}{2}+d_{\mathcal{R}}\mathcal{F}\right)+\frac{1}{4}\mathcal{F}-\frac{2}{5}\boxminus \left(d_{\mathcal{R}}\mathcal{F}\right).$$

$$R=-\mathcal{Q}-\mathcal{J},$$

$$\mathcal{J}=-4{\Lambda }_4+{\kappa }_4^2\tau +\frac{6{\kappa }_4^2}{\lambda }{\pi }_{\mu }{}^{\mu }\text{ and }\mathcal{Q}={\left[4F(\mathcal{R})+\frac{2}{3}\frac{D_a{D}_b\left(d_{\mathcal{R}}\mathcal{F}\right)}{d_{\mathcal{R}}\mathcal{F}}\left({\delta }^{ab}+4n^a{n}^b\right)\right]}_{y=0}.$$

$${\nabla }^{\mu }{\mathcal{Q}}_{\mu \nu }={\nabla }^{\mu }\left({\mathcal{E}}_{\mu \nu }-\frac{6{\kappa }_4^2}{\lambda }{\pi }_{\mu \nu }\right),$$

$$G_{\mu \nu }^{[f(R)]}\equiv G_{\mu \nu }^{[R]}-{\mathcal{Q}}_{\mu \nu }={\mathcal{J}}_{\mu \nu }.$$

$$R_{\mu \nu }-\frac{1}{2}R{q}_{\mu \nu }-{\mathcal{Q}}_{\mu \nu }=R_{\mu \nu }{d}_{R}f-\frac{1}{2}f{q}_{\mu \nu }+\left(q_{\mu \nu }\square -{\nabla }_{\mu }{\nabla }_{\nu }\right)d_{R}f,$$

$$R_{\mu \nu }{d}_{R}f-\frac{1}{2}f{q}_{\mu \nu }+\left(q_{\mu \nu }\square -{\nabla }_{\mu }{\nabla }_{\nu }\right)d_{R}f={\mathcal{J}}_{\mu \nu },$$

$$\left[\left(R+3\square \right)d_R-2\right]f(R)+R=-\mathcal{Q},$$

$$\left[\left(R+3\square \right)d_R-2\right]f(R)=\mathcal{J}.$$

Full text

I Introduction

By following pioneer models rubakov ; RS99a ; RS99b ; gog1 ; gog2 , SMS provided projected Einstein equations onto $3$-thin braneworld sms ; sms2 from $5D$ bulk. If we choose anti-de Sitter bulk, Randall-Sundrum model (RS) with infinite extra dimension RS99b  can be obtained by using the structure of SMS. A lot of these ideas were inspired on string theory advances since RS scenario can be gotten from Horava-Witten’s theory witten . SMS formulation gives two corrections for usual gravity theory: projected Weyl tensor $\mathcal{E}_{\mu\nu}$ and high-energy term $\pi_{\mu\nu}$. Object $\pi_{\mu\nu}$ must be considered in the early universe when the quadratic matter-energy density would can overcome the brane tension sms . Despite $\pi_{\mu\nu}$ changes $4D$ Einstein’s theory in the matter-energy presence, $\mathcal{E}_{\mu\nu}$ modifies vacuum theory, for instance black holes theories dadhich ; aliev ; CFM ; neves . If we consider cosmological scenarios, both $\pi_{\mu\nu}$ and $\mathcal{E}_{\mu\nu}$ lead to corrections in the Friedmann equation bruni ; govender , as also in the black hole metric with electromagnetic radiation chamblin . Actually, $\mathcal{E}_{\mu\nu}$ generates an effective radiation on the brane (dark radiation), which its source can be, for example, Schwarzschild-anti-de Sitter bulk mukohyama . This radiation can generate LTB branes, as shown in neves . Possible connection of $\mathcal{E}_{\mu\nu}$ and $\pi_{\mu\nu}$ with AdS/CFT duality can be seen in shiromizu ; papantonopoulos .

On another hand, dynamical extensions for General Relativity are technicaly viables. Several possible modifications are given by $f(R)$ theories bergmann ; ruzmaikina ; breizman ; buchdahl ; felice ; nojiri . Such theories substitute Einstein-Hilbert curvature term by a generalized form $f(R)$, with least action principle yielding alternative field equations. Dynamical scalar equation controlling an additional freedom degree is derived by taking in account the trace of these field equations. Sometimes, this freedom degree is called of scalaron field. We can recognize scalaron with inflaton field, since our gravity theory is like-Starobinsky: $f(R)=R+aR^{2}$ Starobinsky . By considering $\mathcal{F(R)}$-bulk and using SMS procedure, in borzou , authors obtained new tensor components on the brane: $\mathcal{Q}_{\mu\nu}$ (BSSY tensor). This tensor carries $\mathcal{F(R)}$ functions and together with $\mathcal{E}_{\mu\nu}$ and $\pi_{\mu\nu}$ establish the full correction in the Einstein equations. Cosmological application is given in borzou . In Ref. larranaga , it has been analized topological braneworld black hole with constant scalar curvature. Since trace of $\mathcal{Q}_{\mu\nu}$ appears in the metric of topologically charged black holes as effective cosmological constant on the brane, solutions have been studied by analising classical tests of General Relativity in roldao .

By wanting to developed some mechanism that makes the projection $\mathcal{F(R)}\Rightarrow f(R)$, we will consider BSSY equations, as well as to announce additionally:

[TABLE]

If we consider this assumption, brane-bulk scalaron equation is obtained naturally, in which $\mathcal{Q=Q}_{\mu}{}^{\mu}$ plays fundamental role. Since $\mathcal{Q}$ inherits the information of the $\mathcal{F(R)}$ functions, $f(R)$ solution of our mixed scalaron equation is directly influenced by the $\mathcal{F(R)}$ dynamics. There are two stages to compute:

First: $f(R)$-brane can be obtained by expressing some particular equation to $\mathcal{Q}$, where we do not need the previous knowledge of $\mathcal{F(R)}$.

Second: if we choose some specific bulk, it implies an extra equation which reports, dynamically, extrinsic with intrinsic curvature.

Second stage is necessary to find explicitly $\mathcal{F(R)}\Rightarrow f(R)$. If taken in account the decomposition $\mathcal{Q}_{\mu\nu}=q_{\mu\nu}Q$, we will observe yet that BSSY theory implies $f(R)$-unimodular gravity, which is formally identical with obtained in noo ; noo2 . Nowadays, unimodular gravity anderson ; unruh gives hope to solve the cosmological constant problem weinberg ; carroll . This problem emerges due to cosmological constant value obtained from General Relativity or Quantum field theory, since each result does not agree with each another. Key point is that unimodular gravity generates a cosmological constant unrelated directly with the usual constant. Strictly speaking, we will show that our unimodular gravity is obtained by applying traceless differential operator in $f(R)\neq R$. Taking in account Friedmann-Robertson-Walker (FRW) ansatz, finally, we will obtain cosmological expressions.

In the section 2 we will review quickly the $f(R)$-unimodular gravity and BSSY thin brane formulation. Already in the section 3 we will promote our main ideas about effective $f(R)$-branes. A general approach will be elaborate in the section 4, where we will obtain the $f(R)$-unimodular gravity. In the section 5 we will conclude our study. In order to fix the notation, hereupon, $\{\theta_{\mu}\}$, with $\mu=0,1,2,3$ [and $\{\theta_{a}\}$, with $a=0,1,2,3,5$] denotes a basis for the cotangent bundle on a braneworld, embedded in the $5D$ bulk. Furthermore, $\{e_{a}\}$ is its dual basis and $\theta^{a}=dx^{a}$, when a coordinate chart is chosen. Let $n=n_{a}\,\theta^{a}$ be a timelike covector field normal to the brane and $y$ the associated Gaussian coordinate. In particular, $n_{a}\,dx^{a}=dy$ on the hypersurface defined by $y=0$. The brane metric $q_{\mu\nu}$ and the corresponding components of the bulk metric $g_{ab}$ are in general related by $g_{ab}=q_{ab}+n_{a}\,n_{b}$. With these choices it follows that $g_{55}=1$ and $g_{\mu 5}=0$, the $5D$ bulk metric

[TABLE]

II Short Reviews: $f(R)$-Unimodular Gravity and BSSY Brane Formulation

II.1 $f(R)$-Unimodular Gravity

The combination of $f(R)$ theories with unimodular gravity has been done in noo ; noo2 . If we take $4D$ action given by

[TABLE]

and so, by varying it with respect to the metric, we obtain

[TABLE]

where $S_{fields}$ and $\pounds$ are, respectively, physical fields action and Lagrange multipler function (for details noo ). Still, $g$ is the metric determinant and $G$ is the usual Newton gravitation constant. We will use the notation $d_{R}\equiv d/dR$.

When $S$ is varied by $\pounds$, the called unimodular constraint

[TABLE]

is derived. In the Refs. noo ; noo2 , have been shown several physical applications. For instance, in noo has been studied inflationary scenarios while Newton law behavior in noo ; noo2 .

If we want to derive $f(R)$ theories bergmann ; ruzmaikina ; breizman ; buchdahl ; felice ; nojiri from (3), we must choice $\pounds=0$, i. e.,

[TABLE]

$f(R)$ theories are scalar-tensor theories, in the sense that trace of (5) provides a scalar dynamical equation:

[TABLE]

Expression (6) has as source the stress tensor trace: $T$. For example, by putting $f(R)=R+aR^{2}$ in (6), it becomes

[TABLE]

Therefore, curvature scalar $R$ satisfies Klein-Gordon equation with associated mass $m\equiv 1/6a^{2}$ and source $\kappa_{4}^{2}T$. Therewith, several authors sometimes have called $R$ (or $d_{R}f(R)$) of scalaron field.

If taken into account Friedmann-Robertson-Walker (FRW) metric and $T=0$, expression (5) yields (for details felice )

[TABLE]

with $H\equiv a^{-1}\partial_{t}a$ being the Hubble function. Result (8) composes Starobinsky theory proposed in 1980 Starobinsky and it is first Friedmann equation in this case. In the inflation epoch, $\digamma\mathfrak{\simeq}18H^{2}\partial_{t}H\simeq-3m^{2}H^{2}$, so that $H\simeq H_{0}-(m^{2}/6)(t-t_{0})$ leads to an inflationary scale factor $a\simeq a_{0}\exp[H_{0}(t-t_{0})-(m^{2}/12)(t-t_{0})^{2}]$ where $H_{0}$ and $a_{0}$ are defined in the start of the inflation $t_{0}$.

Unimodular gravity anderson ; unruh is obtained by choosing $f(R)=R$ in (3) and so getting its trace: $2\pounds=\kappa_{4}^{2}T-R$, such that we can rewrite (3) as follows

[TABLE]

Taking the divergence of (9), we have

[TABLE]

which implies $\Lambda^{(U)}\equiv R+\kappa_{4}^{2}T$ constant. If now we substitute $T$ in (9), we obtain then the usual Einstein theory given by

[TABLE]

Since $\Lambda^{(U)}$ does not related with cosmological constant found in Einstein theory, it is unrelated to the vacuum energy regarded in Quantum field theory. Thus, unimodular gravity “solves” the cosmological constant problem weinberg ; carroll .

II.2 BSSY Brane Formulation

By generalizing RS model RS99a ; RS99b , SMS obtained effective Einstein equations on thin braneworlds sms ; sms2 (for a review maartens ). In a similar way, BSSY equations borzou are obtained by taking Gauss equations

[TABLE]

and Israel junction

[TABLE]

$\mathcal{R}_{d}{}_{gh}{}^{e}$ is the five dimensional Riemann tensor and $R_{abc}{}^{f}$ its four dimensional version. $K_{\mu\nu}$ is the brane extrinsic curvature in $y=0$ where $K_{\mu\nu}\sim\partial_{y}q_{\mu\nu}$ at Gaussian coordinates. Equation (13) is obtained if $\mathbb{Z}_{2}$-symmetry is assumed. $5D$ gravitation field equations taken by BSSY are

[TABLE]

with $5D$ stress tensor given by

[TABLE]

We have used the notation: $d_{\mathcal{R}}\equiv d/d\mathcal{R}$. $D_{a}$ is covariant derivative with respect to bulk metric $g_{ab}$ and $\boxminus\equiv D^{a}D_{a}$. $\lambda$ and $\tau_{\mu\nu}$ are, respectively, brane tension and stress tensor of fields on the brane. The function $\delta(y)$ provides thin brane localization: $y=0$. Original SMS formulation is derived by taking $\mathcal{F}(\mathcal{R})=\mathcal{R}$ in (14).

If we combine (12), (13) and (14), these equations yield the projected equations on the brane borzou

[TABLE]

where

[TABLE]

$\mathcal{J}_{\mu\nu}$ is the SMS stress tensor with $\pi_{\mu\nu}$ given by

[TABLE]

Object $\pi_{\mu\nu}$ represents a high energy correction while $\mathcal{E}_{\mu\nu}=$ $q_{\mu}{}^{a}q_{\nu}{}^{b}\mathcal{C}^{d}{}_{acb}n_{d}n^{c}$ originates from $5D$ Weyl tensor: $\mathcal{C}^{d}{}_{acb}$.

Since $\mathcal{E}_{\mu\nu}$ is traceless, it behaves as an effective radiation on the brane, which carries informations about the bulk geometry sms ; maartens ; maartens2 . Several physical implications of $\mathcal{E}_{\mu\nu}$ are given in the Refs. dadhich ; aliev ; CFM ; neves ; bruni ; govender ; chamblin ; mukohyama ; shiromizu ; papantonopoulos . $\Lambda_{5}$ is the bulk cosmological constant while the effective cosmological constant $\Lambda_{4}$ is given by

[TABLE]

where $\kappa_{4}$ ($\kappa_{5}$) is the four (five) dimensional gravitation constant. Randall-Sundrum model RS99b  is obtained by taking in account $\Lambda_{4}=0$, since that $\Lambda_{5}<0$ from (19).

If we put $\mathcal{F}(\mathcal{R})=\mathcal{R}$ in (14), we have $\mathcal{Q}_{\mu\nu}=0$ in (16), such that the term $\mathcal{Q}_{\mu\nu}$ acts as a stress tensor on the brane providing new corrections besides SMS. Consequently, we have three corrections in the Einstein equations for $\mathcal{F}(\mathcal{R})\neq\mathcal{R}$ case. Explicitly, $\mathcal{Q}_{\mu\nu}$ is given by

[TABLE]

with $F(\mathcal{R})$ being

[TABLE]

We should note that $\mathcal{Q}_{\mu\nu}$ encapsulates the $\mathcal{F(R)}$ effects on the brane.

By taking the trace of (16), we have

[TABLE]

with

[TABLE]

Null divergence requeriment becomes

[TABLE]

since that $\nabla^{\mu}\left(\mathcal{J}_{\mu\nu}+\mathcal{Q}_{\mu\nu}\right)=0$. Onto a vacuum conformally flat bulk, we have $\nabla^{\mu}\mathcal{Q}_{\mu\nu}=0$, so that we can identify $\mathcal{Q}_{\mu\nu}$ as a kind of matter. Several approaches projecting $f(R)$-bulk in thin braneworlds can be seen in the Refs. carames ; haghani ; Chakraborty .

III $f(R)$-branes and Curvature Dynamical Constraint

III.1 $f(R)$-branes

We have seen that $\mathcal{F(R)}$-bulk yields an additional extra stress tensor in Einstein brane. Our proposal is to create a mechanism in which $\mathcal{F(R)}$-bulk generates an effective $f(R)$-brane.

Instead to consider the original idea (16), we will require that $\mathcal{Q}_{\mu\nu}$ operates on the left side of (16) in the following way

[TABLE]

Notation $G_{\mu\nu}^{[f(R)]}$ denotes Einstein tensor component for some four dimensional $f(R)$ theory. Therefore, we can rewrite (16) and (25) as

[TABLE]

[TABLE]

with each respective trace, respectively, yielding

[TABLE]

[TABLE]

Expression (28) can be obtained directly of (29), by using (22). Equation (29) establishes a SMS version to scalaron dynamical equation (6). Explicitly in the vacuum, we have not any difference with scalaron theory without extra dimension. This result gives the information which scalaron field does not explicitly affected by $\mathcal{E}_{\mu\nu}$. If $\tau\neq 0$, we must consider $\pi_{\alpha}{}^{\alpha}$.

If we want to obtain an effective $f(R)$-brane, let us consider (28). Since that

[TABLE]

with $\mathcal{R}(R)$ dictated by the Gauss equations trace (12):

[TABLE]

equation (28) can be rewritten as

[TABLE]

where we have used (23). $\mathcal{O}$, $\Pi^{\left[f(R)\right]}$ and $\Pi^{\left[\mathcal{F}\left(\mathcal{R}\right)\right]}$ are respectively

[TABLE]

[TABLE]

If we take the trace of (14), we obtain the $5D$ scalaron theory

[TABLE]

so that $\Pi^{\left[f(R)\right]}$ (by (29)) and $\Pi^{\left[\mathcal{F}\left(\mathcal{R}\right)\right]}$ are, respectively, brane (bulk) scalaron operators. Thus, expression (32) provides a relation between $5D$ scalaron theory with its $4D$ version. It is easy to see that if we consider $\mathcal{F}(\mathcal{R})=\mathcal{R}$ ($\mathcal{Q}=0$), then $f(R)=R$. This result gives the original SMS theory.

Now, let us work non-trivial examples. If the equation $3\square d_{R}f(R)=-\mathcal{Q}$ is satisfied in (28), we must solve $\left(-Rd_{R}+2\right)f=1$, which implies a “Starobinsky-Shiromizu-Maeda-Sazaki” brane (SSMS brane). In fact

[TABLE]

with $\mathfrak{a}$ an arbitrary constant. Therefore, SSMS brane can be represented by the equations

[TABLE]

[TABLE]

Thus, hybrid theory which combines inflationary Starobinsky model with SMS theory is obtained from (37) and (38). If $\mathfrak{a}=0$, we recover SMS formulation. By using (37) and (38), we obtain alternative forms to SSMS scalaron theory, i.e.,

[TABLE]

with $m^{2}\equiv 1/6\mathfrak{a}$. $\Delta_{KG}^{\left(\pm\right)}\equiv\square\pm m^{2}$ is the positive (negative) Klein Gordon operator.

Others two examples are given by

[TABLE]

[TABLE]

where $\mathfrak{f}=f_{0}/R_{0}^{2}$ and $\mathfrak{c}$ are constants. In this stage, we do not work bulk theory. In the follows, we will developed a mecanism which relates brane with bulk theories.

III.2 Projecting $\mathcal{F}(\mathcal{R})\Rightarrow f(R)$: Curvature

Dynamical Constraint

By introducing the concept of curvature dynamical constraint (CDC), we will develop a mechanism which provides an effective $f(R)$-brane from $\mathcal{F(R)}$-bulk. Equation (31) will be called of curvature geometrical constraint (CGC). CDC is necessary if we want to know explicitly the $\mathcal{F(R)}$-bulk. The philosophical idea is that if we require any gravitation theory with specific dynamics on the brane, we must offer also a CDC besides CGC. For example, if we have a bulk with $\mathcal{F(R)}$ dynamics which projects a $f(R)$ dynamics on the brane, it is necessary to provide an extra relation between the curvature objects.

It is convenient to define the objects $\Pi_{\ast}^{\left[f\right]}$, $\Pi_{\ast}^{\left[\mathcal{F}\right]}$ and $\Theta$ as follow

[TABLE]

With (42), the triad of equations (28), (29) and (35) assume the compact forms

[TABLE]

while (32) is rewritten as

[TABLE]

We are now ready to introduce the CDC concept.

Formally, CDC is obtained if the bulk scalaron term $\Pi_{\ast}^{\left[\mathcal{F}\right]}\mathcal{F}$ taken in $y=0$, follows the prescription

[TABLE]

If we put (45) in (44), we obtain

[TABLE]

Object $D\left(\mathcal{K},R\right)$ is some dynamical term relating the curvatute scalars. Therefore, (46) is our generic CDC. Equation (46) relates only two dynamical variables, since CGC eliminates one of them.

Let us elaborate an example which considers a $\left(\mathcal{R}+\mathfrak{b}\mathcal{R}^{2}\right)$-bulk projecting a $\left(R+\mathfrak{a}R^{2}\right)$-brane. By acting $\Pi_{\ast}$-operators on their respective theories, we have

[TABLE]

If we take into account (47), (31) and so put them in (44), we obtain the following equation

[TABLE]

We have used $\boxminus=\square+D^{y}D_{y}$ and

[TABLE]

By comparing left/right sides of (48), we note the relation $\mathfrak{a}=8\mathfrak{b}/15$. Since that $[\ldots]_{y=0}$ term is equal to zero, we find explicitly our first CDC:

[TABLE]

Our result can be formulate as follows: the projection

[TABLE]

which has the bulk/brane scalaron theories

[TABLE]

contains the additional relation (50). It is viable to note that $5D/4D$ masses, associated with $5D/4D$ scalaron fields, are related by $m_{5}^{2}=(3/5)m_{4}^{2}=3/16\mathfrak{b}$.

Let us consider other possible projections which involve $\mathfrak{f}R^{2}$/$\mathfrak{h}\mathcal{R}^{2}$-theories. Their $\Pi_{\ast}$-operators acting on $\mathfrak{f}R^{2}$/$\mathfrak{h}\mathcal{R}^{2}$ are

[TABLE]

If we take (52) and (47), and to consider their full combinations, posteriorly putting them in (44), we obtain separately each projection with its respective CDC. In resume, we have

[TABLE]

In what follows, we will obtain a generalized unimodular if taken into account a specific case.

IV Emergent $f(R)$-Unimodular Gravity: $\mathcal{Q}_{\mu\nu}=Q(\mathcal{R})q_{\mu\nu}$ case

In the present section, we will explore the case: $\mathcal{Q}_{\mu\nu}=Q(\mathcal{R})q_{\mu\nu}$. This decomposition is possible if

[TABLE]

In this specific case, we can rewrite (16) and (43), respectively, as

[TABLE]

with $Q$ given by

[TABLE]

Before we obtain $f(R)$-unimodular gravity, let us understand the traceless Einstein tensor basics for the case (54).

We will denote by the symbol ∘, the composition:

[TABLE]

for any generic tensor $A_{\mu\nu}$. We should note that $A_{\mu\nu}^{\circ}$ is traceless. By (54), we have the traceless Einstein tensor:

[TABLE]

Expression (57) is obtained directly from tensor equation given in (54). It is easy verify that (57) does not contain $\Lambda_{4}$ predicted in SMS formulation, since $q_{\mu\nu}^{\circ}=0$. Therefore, we have two corrections in the traceless stress-tensor: $\pi_{\mu\nu}^{\circ}$ and $\mathcal{E}_{\mu\nu}^{\circ}$, which are

[TABLE]

In the vacuum, equation (57) is driven by $\mathcal{E}_{\mu\nu}$ 111In the usual case: $R_{\mu\nu}^{\circ}=0$., i.e.,

[TABLE]

If we want to obtain $f(R)$-unimodular gravity on the brane, we must combine the equations (54) and, posteriorly, to put $\mathcal{J}_{\mu\nu}$ in (27).

By performing the mentioned procedures, we derive generalized unimodular gravity:

[TABLE]

with $\Delta_{\mu\nu}\equiv\nabla_{\mu}\nabla_{\nu}$, so that $\Delta_{\mu\nu}^{\circ}$ is our $4D$ traceless operator. Theory (60) is supplemented by the equations (57). Mathematically, $f(R)$-unimodular gravity is stated by the pair:

[TABLE]

If $f(R)\rightarrow R$, left expression assumes right equation form.

Equations (61) are formally identical with (3). In fact, by taking the trace of (3) to isolate $\pounds$ and, putting it in (3), we obtain (61) with $\kappa_{4}^{2}T_{\mu\nu}^{\circ}$ instead of $\mathcal{J}_{\mu\nu}^{\circ}$. Consequently, our result provides corrections for NOO theory. For instance, let us consider $\Delta T_{\mu\nu}$ a deviation of (3). Since we have now $T_{\mu\nu}+\Delta T_{\mu\nu}$, we find

[TABLE]

If we compare (62) with (61), we can identify $\Delta T_{\mu\nu}^{\circ}$ with $\pi_{\mu\nu}^{\circ}$ and $\mathcal{E}_{\mu\nu}$ or, in general,

[TABLE]

Therefore, $\Delta T_{\mu\nu}^{\circ}$ provides extra dimension signature if compared with NOO theory. On the vacuum, any deviation can be speculated as dark radiation manifestation, since $\Delta T_{\mu\nu}^{\circ}\sim-\mathcal{E}_{\mu\nu}$.

Similarly as has been done in the subsection II.A, we obtain from (57) the continuity relations

[TABLE]

in accordance with (22) and (24). If $\partial_{\mu}Q=0$, we have $\mathcal{J}+R=-4Q$ constant: $c\equiv 4\Lambda^{\left(Q\right)}$. By isoling $\mathcal{J}$ and, posteriorly, by substituting it in (57), we obtain

[TABLE]

Thus, being $Q$ constant, we can identify it as the cosmological constant obtained from unimodular theory, while the effective cosmological constant is $\Lambda_{4}^{eff}\equiv\Lambda_{4}+\Lambda^{\left(Q\right)}$.

Let us define $\mathfrak{A}(R)$ as follows

[TABLE]

so that $\mathfrak{A}$ excludes $f(R)=R$ model on the brane. Thus, expression (61) can be rewritten as

[TABLE]

It is useful to define also the objects $\Psi_{\mu\nu}$ and $\Phi$:

[TABLE]

By rewriting (28) with (66) and, simultaneously, taking in account $\Psi_{\mu}{}^{\mu}$, we obtain

[TABLE]

such that our brane theory has been codificade in $\Psi_{\mu\nu}$. Therefore, if we know the information contained in $\Psi_{\mu\nu}$, the pair $\left[q^{\mu\nu},\circ\right]$ generates the equations of our $f(R)$-unimodular gravity, i.e.,

[TABLE]

with $\Psi_{\mu\nu}^{\circ}=R_{\mu\nu}^{\circ}$. We stress that in general $\Psi_{\mu\nu}\neq R_{\mu\nu}$. Object $\Psi_{\mu\nu}$ was defined in the following way (68), by wanting that its traceless version coincides with $R_{\mu\nu}^{\circ}$ while its trace gives $\Phi\neq R$. In general $R_{\mu\nu}=\Psi_{\mu\nu}+q_{\mu\nu}\varphi$, with $4\varphi\equiv R-\mathfrak{A}^{-1}\square\mathfrak{A}$. If $\varphi=0$, the inequalities change to equalities.

Let us work in vacuum. If $\tau_{\mu\nu}=0$, the objects to put in (70) are

[TABLE]

First expression is obtained by comparing (59), (67) with definition of $\Psi_{\mu\nu}$, while second is generated by taking (22) in the vacuum. In the follows, we will apply the formulation developed here.

We will consider FRW ansatz, i.e.,

[TABLE]

with $a(t)$ being the scale factor. $\delta_{ij}$ is Kronecker delta component. Traceless gravity can be written as follows

[TABLE]

where we have used (67), (70) and right expression of (71). If we consider (72) and, then, if we take into account $00$ and $ii$ components, we deduce

[TABLE]

which yields

[TABLE]

On another hand, scalaron equation (right expression of (70)) provides

[TABLE]

If we combine (75) with (76), we derive

[TABLE]

Equation (77) is our guide to study universe expansion for generic case. Hubble function which satisfies (77) is

[TABLE]

with $C$ constant.

Of course that $H$, $\mathfrak{A}$ and $\Psi$ of (78) must be compatible with $R=6(\partial_{t}H+2H^{2})$ obtained due to definition of $R$, since $\mathfrak{A}$ is given by (66). If we restrict (77) considering $f(R)=R+\mathfrak{a}R^{n}$ with $n\geq 2$ and, then, by taking vacuum brane and RS fine-tuning, we have

[TABLE]

If we take yet SSMS brane, with $n=2$ and $m_{4}^{2}=1/6\mathfrak{a}$, and if consider too $R=-4Q=6(\partial_{t}H+2H^{2})$, we will obtain from (79) our modified first Starobinsky-Friedmann equation:

[TABLE]

If we choice yet Starobinski bulk/brane, (80) yields

[TABLE]

In general $\mathfrak{\digamma}^{bulk}\mathfrak{\propto}\Theta_{\ast}-(1/4)\Pi_{\ast}^{\left[\mathcal{F}\right]}\mathcal{F}$, such that we can think $\mathfrak{\digamma}$ as remnant from $5D$ scalaron field.

IV.1 Cosmological Solutions

We have obtained formal aspects from the theory presented. Now, we will study two possible cosmological solutions associated with it. In the first case, we will work the $f(R)$ Starobinsky theory. In the second application, we will investigate the NOO equation which governs the cosmological evolution to obtain an example with dark radiation correction. We will choice the Starobinsky scale factor for this and, by applying the method of the reconstruction, we will obtain which $f(R)$ generates it.

IV.1.1 $f(R)$ Starobinsky theory

We have seen that the cosmological evolution in Starobinsky model satisfies the expression (80). Let us rewrite (80) as follows

[TABLE]

The object $\iota$ can assume the values $1$ or [math]. By comparing (82) with (8), if $\iota=0$ we recover the Starobinsky theory without extra dimension while $\iota=1$ provides our correction. In inflationary epoch we have $\digamma\simeq 18H^{2}\partial_{t}H$, such that putting it in (82), we obtain the following solution

[TABLE]

We must note that if $\iota=0$ we obtain the usual Starobinsky inflation for $H_{-}^{0}$. The scale factor associated with $H_{\pm}^{\mathcal{E}}$ is given by

[TABLE]

Again, we recover the Starobinsky scale factor if $a_{-}^{0}$. Thus, (84) describes the cosmological evolution in the inflation epoch for $f(R)$ Starobinsky theory.

IV.1.2 Starobinsky scale factor: application of the reconstruction

method with dark radiation influence

We will assume the unimodular constraint for induced metric, i. e., $\sqrt{-q}=1$, by taking the line element as follows

[TABLE]

In noo has been shown that on this metric, the equation which governs the cosmological evolution is given by

[TABLE]

with $h(\tau)=a(\tau)^{-1}\partial_{\tau}a(\tau)$. Since we have a dark radiation correction given by $\Delta T_{\mu\nu}^{\circ}=-\mathcal{E}_{\mu\nu}$, we rewrite $\Delta\rho=-\mathcal{E}_{00}$ and $\Delta P=-\mathcal{E}_{ii}$ in (86), where we will consider now $\rho+\Delta\rho$ and $P+\Delta P$. Usually, in the SMS brane/bulk, the dark radiation can be express as follows mukohyama2

[TABLE]

with $q_{\mathcal{E}}$ the dark radiation tidal charge. Let us consider (87) generalized to

[TABLE]

Therefore, (86) can be rewritten as

[TABLE]

By following the Ref. noo , we will choice

[TABLE]

to obtain the $f(R)$ gravity which generates (90). The quantities $m_{4}$, $t_{0}$ and $H_{0}$ are arbitrary constants. On the conformal time $\tau$, where $d\tau=a(t)^{3}dt$, the scale factor and $h(\tau)$ become

[TABLE]

where we have used $\mathcal{N}=6H_{0}+m^{2}t_{0}$. As have been shown in noo , the values $H_{0}=0,00000016$, $m_{4}=0,053$, $t_{0}=10^{-3}$sec and $a_{0}=1$ lead for the spectral index $n_{s}\simeq 0,966$, which is compatible with the Planck result. By using (91) in (89) and considering $\rho=0=P$, we obtain

[TABLE]

The equation (92) provides the following solution

[TABLE]

We have taken the integration constants equal to one. The relationship between the conformal time and the curvature scalar is given by noo

[TABLE]

By putting (94) in (93), we obtain

[TABLE]

Finally, if we integrate (95) in $R$, we can find the $f\left(R\right)$ theory which generates the scale factor (91). Explicitly, we find

[TABLE]

with $\widetilde{f}\left(R\right)$ given by

[TABLE]

The NOO case is obtained simply by taking $q_{\mathcal{E}}=0$.

Let us analyze specific situations of (97) for $n\geq 0$. For instance, if $n=0$ we have the theories $\sim R^{2}$ and $\sim-R^{-2}$. For $n=1,2,4$, the first term in (97) diverges. Actually, if $n=4$ we obtain

[TABLE]

by taking $n=4$ in (95). We must note that for $n\geq 5$, the first term becomes negative. Thus, if $q_{\mathcal{E}}>0$, our correction provides a negative contribution for NOO theory and, if $q_{\mathcal{E}}<0$, this contribution becomes positive. In the follows, we provide a table detailing some possible scenarios for $0\leq n\leq 9$:

[TABLE]

V Conclusion

We have developed two different aspects associated with effective $f(R)$ branewords. The first has been general rules to create it from $\mathcal{F(R)}$-bulk. By wanting to generate a “copy” of the bulk dynamics, we have developed the CDC concept, in which geometrical reduction does not affect dynamical structure of the left side, such that $\mathcal{F(R)}$ in $5D$ is fully analog to $f(R)$ in $4D$. Similar idea to generate effective $f(R)$-brane was developed in aguilar , with projection stipulated by applying CGC directly in $\mathcal{F}$: $\mathcal{F(R)}=f(R+\mathcal{K})$. Thus, we have obtained a new dynamical projection class: $\mathcal{F(R)}\Rightarrow f(R)$. As future study, must be found a physical principle which provides the correct projection, as done in no to solve brane predictability problem.

In the second step, we have taken a specific case to obtain $f(R)$-unimodular gravity. We showed that the derived equations are formally identical with obtained in noo . Two applications were obtained and a full study about these solutions will be examined in a future paper.