# A New Type of Gravity Theory with Auxiliary Field

Emmanuel Kanambaye

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Emmanuel Kanambaye. A New Type of Gravity Theory with Auxiliary Field. 2025. hal-05202180

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# A New Type of Gravity Theory with Auxiliary Field

Emmanuel Kanambaye, Independent Researcher

Adress: Librairie BAH SARL, Hall du Grand Hôte de Bamako-Mali;

Postal Code: BP 104 Bamako

E-mail: wadouba@gmail.com

**Abstract.** Gravity theories with auxiliary fields are excellent way to circumvent Lovelock theorem insofar as they permit modification of Einstein Field Equations without introduction of additional degrees of freedom; what offers the possibility to go beyond Einstein's general relativity without invoking new dynamical physical fields or additional spatial dimension.

Unfortunately as shown by Pani, Sotiriou, Ventagli and Vernieri, the most famous kind of gravity theories with auxiliary fields, like Palatini $f(R)$ gravities and Eddington-inspired Born-Infeld (EiBI) theory have shortcomings insofar as their field equations contain higher-order derivatives of the matter fields making these theories singular on the surface of stars and/or incompatible with standard model of particle physics unless the parameters of the theory are severely constrained; a severe constraint what would make the theory phenomenologically uninteresting compared to general relativity.

In the present paper, we propose a new type of gravity theory with auxiliary field which not introduces higher-order derivatives of the matter fields in its field equations and which can be very interesting at large scale and/or short scale compared to general relativity.

## 1. Introduction

As well known, the advantage of gravity theories with auxiliary fields [1, 2] is their capacity to circumvent Lovelock's theorem [3] without introduction of new physical degrees of freedom.

In other words, gravity theories with auxiliary fields could give rise to deviation from general relativity [4] capable of handling phenomena like dark matter and dark energy and this without invoking new/additional propagating degrees of freedom.

Unfortunately, usual gravity theories with auxiliary fields [1] like Palatini $f(R)$ gravities [5, 6] and Eddington-inspired Born-Infeld theory (EiBI) [7] etc... are argued to have weakness among which formation of spacetime singularities on the surface of stars and incompatibility with standard model of particle physics when the parameters of the theory are not severely constrained [1, 2, 8, 9].

These weakness come from the fact that usual gravity theories with auxiliary fields (like Palatini $f(R)$ gravities and EiBI) involve in their field equations, derivatives of
the matter stress-energy tensor or equivalently higher-order derivatives of the matter fields; higher-order derivatives which if the parameters of these theories are not severely constrained, lead to the formation of singularities on the surface of stars and/or to incompatible with standard model of particle physics [1, 2].

The problem is that when the parameters of a usual gravity theory with auxiliary field (like Palatini $f(R)$ gravities and EiBI) are severely constrained; the theory loses its interest/advantage compared to general relativity.

This is why the question arises as to whether a gravity theory with auxiliary field, which is compatible with standard model of particle physics and interesting compared to general relativity, can be achieved.

Fortunately, as we are going to show in the present paper, the response to this question is positive; more precisely we are going to show that a new kind of gravity theory with auxiliary field which satisfies the weak equivalence principle and which not involves higher-order derivatives of the matter fields in its field equations is not only possible but also that the gotten theory can be very interesting at large scale and/or short scale. For example, we will see that the proposed new paradigm easily gives amongst other things non-singular bouncing FLRW cosmology.

The paper is organised as follow, in the section 2, I introduce/derive the new type of gravity theory with auxiliary field from variational principle; In the section 3, I show that the field equations of the model can be free from higher-order derivatives of the matter fields and compatible with standard model, after that I show that the model yields amongst the simplest non-singular bouncing cosmology, while I devote Section 4 for conclusion.

The signature convention (–+++) will be assumed for the metric; likewise Einstein’s summation convention will be assumed.

## 2. The New Model: Derivation From Variational Principle

In the paper [1], it has been shown that for usual gravity theories with auxiliary fields; the equations of motion once auxiliary fields have been eliminated, can write as [1, 2]:

$$G_{\mu\nu} = \kappa T_{\mu\nu} + S_{\mu\nu} \quad (1)$$

where $T_{\mu\nu}$ denotes the energy-momentum tensor of standard matter, and $S_{\mu\nu}$ a divergence free tensor that depends on the matter fields, the metric, and their derivatives.

We recall that the exact form of $S_{\mu\nu}$ will depend on the specific form of the auxiliary field and how it enters the Lagrangian of the theory.

Be that as it may, $S_{\mu\nu}$ generically involves derivatives of the $T_{\mu\nu}$ or equivalently higher-order derivatives of the matter fields [1]:

$$S_{\mu\nu} = \alpha_1 g_{\mu\nu} T + \alpha_2 g_{\mu\nu} T^2 + \alpha_3 T T_{\mu\nu} + \dots + \beta_1 \nabla_{\mu} \nabla_{\nu} T + \beta_2 g_{\mu\nu} \Box T + \beta_3 \Box T_{\mu\nu} + \dots \quad (2)$$
To succeed in circumventing the problem; let's consider the below action (3):

$$S = \frac{M_P^2}{2} \int \left[ R + 2\kappa L_\psi - L_{\chi_0} - L_{\chi_1} - L_{\chi_2} - \mathcal{L}(S) \right] \sqrt{-g} d^4 x \quad (3)$$

where R is the Ricci scalar; $L_\psi$ the lagrangian of normal matter, while $L_{\chi_0}$, $L_{\chi_1}$, $L_{\chi_2}$ and $\mathcal{L}(S)$ are additional lagrangians respectively reading:

$$L_{\chi_0} = \omega_0 \left[ \partial_{\mu} \chi_0 \partial^{\mu} \chi_0 + 1 \right] \quad (4)$$

$$L_{\chi_1} = \omega_1 \left[ \partial_{\mu} \chi_1 \partial^{\mu} \chi_1 + 1 \right] + \varepsilon_1 \left[ \omega_1 - m^2 \right] - m^2 \partial_{\mu} \chi_1 \partial^{\mu} \chi_1 \quad (5)$$

$$L_{\chi_2} = \omega_2 \left[ \partial_{\mu} \chi_2 \partial^{\mu} \chi_2 + 1 \right] + \varepsilon_2 \left[ \omega_2 + m^2 \right] + m^2 \partial_{\mu} \chi_2 \partial^{\mu} \chi_2 \quad (6)$$

$$\mathcal{L}(S) = \frac{\chi_2}{m} \left[ \omega_0 \partial_\mu \chi_0 \partial_\nu \chi_0 - f(\psi) K_\mu K_\nu + \alpha g_{\mu\nu} f(\psi) K_\sigma K^\sigma - S_{\mu\nu}(g, \psi) \right]^2 R^{\mu\nu} R^{\nu\mu} \quad (7)$$

where $\omega_0$, $\chi_0$, $\omega_1$, $\chi_1$, $\varepsilon_1$, $\omega_2$, $\chi_2$ are scalars auxiliary fields; $K_\mu$ a vector auxiliary field; $f(\psi)$ a function of standard model fields; $m$ and $\alpha$ constants; $R_{\mu\nu}$ the Ricci tensor and $S_{\mu\nu}(g, \psi)$ a symmetric tensor that can depend on the matter fields, the metric and/or their derivatives, i.e. a symmetric tensor of which the exact expression are left unspecified.

Our action (3) being given; let's now apply variational principle; indeed:

a) varying (3) with respect to $\delta g^{\mu\nu}$ would yield:

$$G_{\mu\nu} = \kappa T_{\mu\nu} + \frac{1}{\sqrt{-g}} \left[ \frac{\delta(\sqrt{-g}L_{\chi_0})}{\delta g^{\mu\nu}} + \frac{\delta(\sqrt{-g}L_{\chi_1})}{\delta g^{\mu\nu}} + \frac{\delta(\sqrt{-g}L_{\chi_2})}{\delta g^{\mu\nu}} + \frac{\delta(\sqrt{-g}\mathcal{L}(S))}{\delta g^{\mu\nu}} \right] \quad (8)$$

where of course $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_M)}{\delta g^{\mu\nu}}$ denotes the energy-momentum tensor of standard matter, while.

$$\frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_0})}{\delta g^{\mu\nu}} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 - \frac{\omega_0}{2} g_{\mu\nu} \left[ \partial_{\sigma} \chi_0 \partial^{\sigma} \chi_0 + 1 \right] \quad (9)$$

$$\frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_1})}{\delta g^{\mu\nu}} = [\omega_1 - m^2] \left[ \partial_{\mu} \chi_1 \partial_{\nu} \chi_1 - \frac{1}{2} g_{\mu\nu} \partial_{\sigma} \chi_1 \partial^{\sigma} \chi_1 \right] - \frac{\omega_1}{2} g_{\mu\nu} \quad (10)$$

$$\frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_2})}{\delta g^{\mu\nu}} = [\omega_2 + m^2] \left[ \partial_{\mu} \chi_2 \partial_{\nu} \chi_2 - \frac{1}{2} g_{\mu\nu} \partial_{\sigma} \chi_2 \partial^{\sigma} \chi_2 \right] - \frac{\omega_2}{2} g_{\mu\nu} \quad (11)$$
$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g}\mathcal{L}(S))}{\delta g^{\mu\nu}} = \frac{\chi_2}{m} \left[ 2 \left[ R^{\mu\nu} \frac{\delta \mathcal{Y}_{\mu\nu}}{\delta g^{\mu\nu}} + \mathcal{Y}_{\mu\nu} \frac{\delta R^{\mu\nu}}{\delta g^{\mu\nu}} \right] - \frac{1}{2} g_{\mu\nu} \mathcal{Y}_{\rho\sigma} R^{\rho\sigma} \right] \mathcal{Y}_{\rho\sigma} R^{\rho\sigma} \quad (12) $$

where for notation simplicity we introduced the tensor $\mathcal{Y}_{\mu\nu}$ reading:

$$ \mathcal{Y}_{\mu\nu} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 - f(\psi) K_{\mu} K_{\nu} + \alpha g_{\mu\nu} f(\psi) K_{\sigma} K^{\sigma} - S_{\mu\nu}(g, \psi) \quad (13) $$

b) varying (3) with respect to $\delta\omega_0$ and $\delta\chi_0$ would respectively yield:

$$ \partial_{\mu} \chi_{0} \partial^{\mu} \chi_{0} + 1 + 2 \frac{\chi_{1}}{m} \mathcal{Y}_{\mu\nu} R^{\mu\nu} R^{\nu\mu} \frac{\partial \mathcal{Y}_{\mu\nu}}{\partial \omega_{0}} = 0 \quad (14) $$

$$ \nabla^{\mu} \left[ 2\omega_0 \partial_{\mu} \chi_0 + 2 \frac{\chi_1}{m} \mathcal{Y}_{\mu\nu} R^{\mu\nu} R^{\nu\mu} \frac{\partial \mathcal{Y}_{\mu\nu}}{\partial(\partial^{\mu} \chi_0)} \right] = 0 \quad (15) $$

c) varying (3) with respect to $\delta\omega_1$, $\delta\varepsilon_1$ and $\delta\chi_1$ would respectively yield:

$$ \partial_{\mu} \chi_{1} \partial^{\mu} \chi_{1} + 1 + \varepsilon_{1} = 0 \quad (16) $$

$$ \omega_1 - m^2 = 0 \quad (17) $$

$$ \nabla^{\mu} [2(\omega_1 - m^2) \partial_{\mu} \chi_1] = 0 \quad (18) $$

d) varying (3) with respect to $\delta\omega_2$, $\delta\varepsilon_2$, $\delta\chi_2$ and $\delta K_{\mu}$ would respectively yield:

$$ \partial_{\mu} \chi_{2} \partial^{\mu} \chi_{2} - 1 + \varepsilon_{2} = 0 \quad (19) $$

$$ \omega_2 + m^2 = 0 \quad (20) $$

$$ \nabla^{\mu} [2(\omega_2 + m^2) \partial_{\mu} \chi_2] - \frac{1}{m} [\mathcal{Y}_{\mu\nu} R^{\mu\nu}]^2 = 0 \quad (21) $$

$$ 2 \frac{\chi_2}{m} \mathcal{Y}_{\mu\nu} R^{\mu\nu} R^{\nu\mu} \frac{\partial \mathcal{Y}_{\mu\nu}}{\partial K_{\mu}} = 0 \quad (22) $$

From there, it is clear that because of (20); the above equation (21) would enforce:

$$ \frac{1}{m} [\mathcal{Y}_{\mu\nu} R^{\mu\nu}]^2 = 0 \quad (23) $$

Now since $m$ is a constant and that $R^{\mu\nu}$ is not always zero; then one infers that the
above (23) enforces in its turn:

$$ \mathcal{Y}_{\mu\nu} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 - f(\psi) K_{\mu} K_{\nu} + \alpha g_{\mu\nu} f(\psi) K_{\sigma} K^{\sigma} - \mathcal{S}_{\mu\nu}(g, \psi) = 0 \quad (24) $$

Clearly, because of this above constraint (24); the equations (14), (15) and (22) would respectively simplify as:

$$ \partial_{\mu} \chi_{0} \partial^{\mu} \chi_{0} + 1 = 0 \quad (25) $$

$$ \nabla^{\mu} \left[ 2\omega_{0} \partial_{\mu} \chi_{0} \right] = 0 \quad (26) $$

$$ 0 = 0 \quad (27) $$

In the same way, because of (17), (20), (24) and (25); the equations (9), (10), (11) and (12) would simplify as:

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_0})}{\delta g^{\mu\nu}} = \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 \quad (28) $$

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_1})}{\delta g^{\mu\nu}} = -\frac{1}{2} m^2 g_{\mu\nu} \quad (29) $$

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} L_{\chi_2})}{\delta g^{\mu\nu}} = +\frac{1}{2} m^2 g_{\mu\nu} \quad (30) $$

$$ \frac{1}{\sqrt{-g}} \frac{\delta(\sqrt{-g} \mathcal{L}(S))}{\delta g^{\mu\nu}} = 0 \quad (31) $$

whence the gravitational equation (8) would simplify as:

$$ G_{\mu\nu} = \kappa T_{\mu\nu} + \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 \quad (32) $$

From there, since the constraint (24) implies also:

$$ \omega_0 \partial_{\mu} \chi_0 \partial_{\nu} \chi_0 = f(\psi) K_{\mu} K_{\nu} - \alpha g_{\mu\nu} f(\psi) K_{\sigma} K^{\sigma} + \mathcal{S}_{\mu\nu}(g, \psi) \quad (33) $$

then we will finally get the gravitational equation reading:

$$ G_{\mu\nu} = \kappa T_{\mu\nu} + f(\psi) K_{\mu} K_{\nu} + \mathcal{S}_{\mu\nu}(g, \psi) \quad (34) $$

Also, it is worth mentioning that because of the enforced constraint (24); we can verify that the Euler-Lagrange equation of motion of any given field $\psi$ of the standard model of particle physics would remain the usual one:
$$ \nabla^{\mu} \left[ 2\kappa \frac{\partial L_{\psi}}{\partial(\partial^{\mu}\psi)} - \frac{\partial L(S)}{\partial(\partial^{\mu}\psi)} \right] - 2\kappa \frac{\partial L_{\psi}}{\partial\psi} + \frac{\partial L(S)}{\partial\psi} = \nabla^{\mu} \left[ 2\kappa \frac{\partial L_{\psi}}{\partial(\partial^{\mu}\psi)} \right] - 2\kappa \frac{\partial L_{\psi}}{\partial\psi} = 0 \quad (35) $$

i.e. standard model of particle physics remains unchanged in our paradigm (3), we will see this more deeply later.

Now, since as we clearly see, none of the auxiliary variable $\omega_0$, $\chi_0$, $\omega_1$, $\chi_1$, $\varepsilon_1$, $\omega_2$, $\chi_2$ and $\varepsilon_2$, contribute in the equation of motion (35) of standard matter, while only $K_\mu$ contribute explicitly in the gravitation equation (34); then excepted $K_\mu$, all the other auxiliary variables $\omega_0$, $\chi_0$, $\omega_1$, $\chi_1$, $\varepsilon_1$, $\omega_2$, $\chi_2$ and $\varepsilon_2$ could be seen as just pure mathematical fields (i.e. non-physical fields) helping to achieve the Euler-Lagrange equations of motion (34) and (35).

It is also worth to mention that because of (35); applying the Bianchi identity $\nabla^{\mu}G_{\mu\nu} = 0$ to (34) would impose:

$$ \nabla^{\mu} \left[ f(\psi) K_{\mu} K_{\nu} - \alpha g_{\mu\nu} f(\psi) K_{\sigma} K^{\sigma} + S_{\mu\nu}(g, \psi) \right] = 0 \quad (36) $$

which is the equation that enforces the auxiliary variable $A_\mu$; an equation which of course could also be obtained from (26) by considering (25) and (33); whence the gravitational theory yielded by our action (3), would in summary be:

$$ G_{\mu\nu} = \kappa T_{\mu\nu} + f(\psi) K_{\mu} K_{\nu} - \alpha g_{\mu\nu} f(\psi) K_{\sigma} K^{\sigma} + S_{\mu\nu}(g, \psi) \quad (37) $$

$$ \nabla^{\mu} \left[ f(\psi) K_{\mu} K_{\nu} - \alpha g_{\mu\nu} f(\psi) K_{\sigma} K^{\sigma} + S_{\mu\nu}(g, \psi) \right] = 0 \quad (38) $$

where of course $G_{\mu\nu}$ denotes the Einstein tensor; $\kappa$ the Einstein gravitational constant; $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_{\psi})}{\delta g^{\mu\nu}}$ the energy-momentum tensor of standard matter; $f(\psi)$ a function of standard matter; $K_\mu$ a vector auxiliary field enforced by (38); $\alpha$ a free constant, and $S_{\mu\nu}(g, \psi)$ a to be specified symmetric tensor that can depend on the matter fields, the metric and/or their derivatives.

Now, what is interesting with this gravitational theory (37-38), is that, it is not only a "gravitational theory" derivable from variational principle but also, it is (as we are going to see in the next section), a gravity theory with auxiliary field which is capable to not involve higher-order derivatives of the matter fields in its field equations, while being competitive.
### 3. Absence of higher-order derivatives of matter fields and interest compared to general relativity

If in the case of usual gravity theories with auxiliary fields [1] (like Palatini $f(R)$ gravities and Eddington-inspired Born-Infeld theory), the auxiliary field which is a scalar field could after some algebraic manipulation be eliminated in a way giving the field equation (1) where $S_{\mu\nu}$ is a divergence-free tensor that only depends from the matter fields, the metric and/or their derivatives; in the case of our above (37-38) gravity theory with vector auxiliary field, things are more complicated.

Indeed, there is neither trivial nor general algebraic manipulation capable in the case of (37-38) to eliminate the vector field $K_\mu$ in a way permitting to get the field equation (1) where $S_{\mu\nu}$ is a divergence-free tensor that only depends from the matter fields, the metric and/or their derivatives.

Said otherwise there is no general algebraic manipulation permitting to put (37-38) in the form (1) where $S_{\mu\nu}$ is a divergence-free tensor that only depends from the matter fields, the metric and/or their derivatives.

This is due to the fact that in the case of (37-38); the auxiliary field is a vector field and as such cannot in a general manner be eliminated in a way permitting to put (37-38) in the form (1).

This limitation makes (37-38) a gravity theory with auxiliary field fundamentally different from the usual gravity theories with auxiliary fields like Palatini $f(R)$ gravities and Eddington-inspired Born-Infeld theory which can be put in the form (1).

In other words, (37-38) is well and truly a new kind of gravity theory with auxiliary field; a gravity theory of which one of the principal interest/advantage is to be able to not involve in its field equations (38) higher-order derivatives of matter fields.

Indeed the equations (37-38) not requires that $S_{\mu\nu}(g, \psi)$ be a divergence-free tensor or a tensor involving higher-order derivatives of the matter fields; instead what it is required is:

$$ \nabla^{\mu} S_{\mu\nu}(g, \psi) = [1 - \alpha] \left[ 2f(\psi)K_{\nu}\nabla^{\mu}K_{\mu} + K_{\nu}K_{\mu}\nabla^{\mu}f(\psi) \right] \quad (39) $$

a requirement which leaves the freedom to choose for $S_{\mu\nu}(g, \psi)$ expression which not involves higher-order derivatives of the matter fields.

Because of this freedom (37-38) could for appropriate $S_{\mu\nu}(g, \psi)$ be not only compatible with standard model but also it could yield interesting cosmology among which non-singular bouncing cosmology, dark matter, dark energy etc...

To be convinced let's consider a simplified version of (37-38) by assuming for example $f(\psi) = -1$, $\alpha = -\frac{2}{3}$ and $S_{\mu\nu}(g, \psi) = 0$.

In this case, (37-38) will simplify as:

$$ G_{\mu\nu} = \kappa T_{\mu\nu} - K_{\mu}K_{\nu} - \frac{2}{3}g_{\mu\nu}K_{\sigma}K^{\sigma} \quad (40) $$
$$ \nabla^{\mu} K_{\mu} = 0 \qquad (41) $$

This model (40-41) not only excludes modification of standard model of particle physics since $K_{\mu}$ has no coupling with standard model fields, but also yields for example non-singular bouncing FLRW cosmology.

Indeed in the case of spatially homogeneous and isotropic Friedmann metric of line-element [10]:

$$ ds^2 = dt^2 - \gamma_{ij}dx^i dx^j \qquad (42) $$

where $\gamma_{ij}$ denotes the three dimensional spatial metric; only the temporal component $K_0$ of $K_{\mu}$ matters.

As consequence by assuming that our standard matter energy-momentum tensor $T_{\mu\nu}$ is the one of a perfect fluid, reading:

$$ T_{\mu\nu} = [\rho_{\Psi} + P_{\Psi}]u_{\mu}u_{\nu} - P_{\Psi}g_{\mu\nu} \qquad (43) $$

where $\rho_{\Psi}$, $P_{\Psi}$ and $u_{\mu}$ are respectively the energy-density, the pressure and four-velocity of standard matter; then we will for (40-41) get the following friedmann equations:

$$ 3H^2 + 3\frac{K}{a^2} = \kappa\rho_{\Psi} - \frac{1}{3}K_0K_0 \qquad (44) $$

$$ 2\dot{H} + 3H^2 + 3\frac{K}{a^2} = -\kappa P_{\Psi} + \frac{2}{3}K_0K_0 \qquad (45) $$

where $H = \frac{\dot{a}}{a}$ denotes the Hubble parameter; a the scale factor; $\frac{K}{a^2}$ the spatial curvature; $\rho_{\Psi}$ and $P_{\Psi}$ are respectively the energy-density and pressure of standard matter; $K_0$ the temporal component of the auxiliary vector field $K_{\mu}$.

Likewise to get $K_0$ it suffices to consider the equation (41), which would simplify as:

$$ \frac{1}{a^3} \partial^0 [a^3 K_0] = 0 \qquad (46) $$

what yields $K_0 = \frac{C}{a^3}$ with $C$ a real constant of integration.

Therefore, by remembering that $\dot{H} = \frac{\ddot{a}}{a} - H^2$; we could rewrite (44-45) as:

$$ H^2 = \frac{\kappa}{3}\rho_{\Psi} - \frac{K}{a^2} - \frac{1}{9}\frac{C^2}{a^6} \qquad (47) $$

$$ \frac{\ddot{a}}{a} = -\frac{\kappa}{6}[\rho_{\Psi} + 3P_{\Psi}] - \frac{K}{a^2} + \frac{7}{18}\frac{C^2}{a^6} \qquad (48) $$
From there, it is clear that for standard matter or any matter content scaling as $a^{-3}$ or $a^{-4}$ or $a^{-6}$; we will have $\frac{\ddot{a}}{a} > 0$ at $H = 0$ specially in the case of flat universe $K = 0$; whence (40-41) yields non-singular bouncing FLRW cosmology.

Better, (40-41) is among the simplest modified gravity theory capable of yielding non-singular bouncing FLRW cosmology; what makes (37-38) a quite interesting type of modified gravity, specially that depending from the exact expression of $S_{\mu\nu}(g, \psi)$ or by replacing $S_{\mu\nu}(g, \psi)$ by $S_{\mu\nu}(g, \psi, K) = g_{\mu\nu} \sum_{p=1}^{n} [\alpha_p f(\psi) K_\sigma K^\sigma]^{T_p} + S_{\mu\nu}(g, \psi)$ where $r_p$ is a real number, we could handle others problems like dark matter or dark energy etc...; we will prove this in next papers.

## 4. Conclusion

In this paper, we succeeded in showing that, it is clearly possible to achieve from variational principle a gravity theory with auxiliary field, more precisely a gravity theory with vector auxiliary field which not only is capable to not involve higher-order derivatives of matter fields in its field equations but also which can be very interesting at large scale and/or short scale compared to general relativity.

As example we have shown that the paradigm can yield among the simplest non-singular bouncing cosmology.

We will see in next papers that the paradigm could also yield dark matter and dark energy for appropriate expression for $S_{\mu\nu}(g, \psi)$; what makes our gravity theory with vector auxiliary field a paradigm quite deserving consideration.

## 5. References

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