# Massless Einstein-Aether Gravity with Dark Matter, Dark Energy and Stiff matter

Emmanuel Kanambaye

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Emmanuel Kanambaye. Massless Einstein-Aether Gravity with Dark Matter, Dark Energy and Stiff matter. 2025. hal-04820635v5

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# Massless Einstein-Aether Gravity with Dark Matter, Dark Energy and Stiff matter

Emmanuel Kanambaye, Independent Researcher

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

Postal Code: BP 104 Bamako

E-mail: wadouba@gmail.com

**Abstract.** Einstein-aether gravity is an interesting framework permitting to achieve amongst other things dark matter from gravitation as prove "mimetic tensor-vector gravity" or "mimetic tensor-vector-scalar gravity" etc...

However usually, the aether field in Einstein-aether gravity is a massive vector field, while the gotten dark matter is a perfetc pressulress fluid.

In this short paper I show how to get a Einstein-aether gravity in which the aether field is massless and for which the dark matter has pressure capable of preventing from caustic formation while yielding dark energy and stiff matter.

Said otherwise we propose among the simplest Einstein-aether gravity with dark energy, stiff matter and dark matter free from caustics.

## 1. Introduction

Einstein-aether gravity [1] and specially its "mimetic tensor-vector" model [2] and/or "mimetic tensor-vector-scalar" models [3, 4] are interesting framework capable of yielding dark matter from gravitation (see the papers [2, 3, 4] for further details).

However, the gotten dark matter is usually a perfect pressurless fluid; likewise the aether field is generally a massive field with roughly complicated source.

In this short paper; we propose among the simplest Einstein-aether gravity yielding caustics free scalar dark matter [5] as well as dark energy and stiff matter [6].

Indeed as we will see, our Einstein-aether gravity in addition of having massless aether field and yielding scalar field dark matter [5], yield also a cosmological constant playing the role of dark energy beside a stiff density [6].

The signature convention (––+) will be assumed for the metric; likewise Einstein’s summation convention as well as natural unit for the speed of light in vacuum i.e. c = 1 will be assumed.
## 2. The Model

To get a ghost free and caustic free scalar field dark matter version/extension of the model proposed in [1, 3], let's consider the below action:

$$S = \frac{1}{2\kappa} \int \left( R + 2\kappa L_{\Psi} - \frac{1}{4} A_{\mu\nu} A^{\mu\nu} - \lambda \left[ A_{\mu} A^{\mu} + 1 - \frac{\alpha}{f} \right] - \varepsilon[\lambda - f] + L_{\chi,\psi} \right) \sqrt{-g} d^4 x \quad (1)$$

where $\kappa$ is the Einstein gravitational constant; $R$ the Ricci scalar; $L_{\Psi}$ the lagrangian of standard matter; $A_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$ the Maxwell bivector of the $A_{\mu}$ vector field; $\lambda$ and $\varepsilon$ Lagrange multipliers, while $f = \alpha A^{\nu}\partial_{\nu}\psi$; $\psi$ a scalar field, whereas $L_{\chi,\psi}$ denotes a lagrangian reading:

$$L_{\chi,\psi} = \alpha \partial_{\mu} \psi \partial^{\mu} \psi - \beta A_{\mu} \partial^{\mu} \psi - \omega [\partial_{\mu} \chi \partial^{\mu} \chi - 1] - \omega_k [\omega - \alpha] + \alpha \partial_{\mu} \chi \partial^{\mu} \chi - \alpha^{\frac{3}{2}} \chi [A_{\mu} A^{\mu} + 1]^2 \quad (2)$$

where $\chi$ is another scalar field, with $\omega$ and $\omega_k$ Lagrange multipliers, while $\alpha$ and $\beta$ are free constants.

Now as all the elements of our action (1) are specified; then we can apply variational principle.

- Indeed varying (1) with respect to $\delta g^{\mu\nu}$ would after little calculus yield:

$$G_{\mu\nu} = \kappa T_{\mu\nu} + \lambda A_{\mu} A_{\nu} + T_{\mu\nu}^{A} + \psi_{\mu\nu}^{A} + T_{\mu\nu}^{\psi} - \lambda_{\mu\nu} + [\omega - \alpha]\chi_{\mu\nu} + \frac{\omega}{2}g_{\mu\nu} + \alpha_{\mu\nu} \quad (3)$$

where $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_{\Psi})}{\delta g^{\mu\nu}}$ denotes the energy-momentum tensor of standard matter fields; while:

$$T_{\mu\nu}^{A} = [A_{\mu\sigma}A_{\nu}^{\sigma} - \frac{1}{4}g_{\mu\nu}A_{\rho\sigma}A^{\rho\sigma}];$$

$$\psi_{\mu\nu}^{A} = \beta [A_{\mu} \partial_{\nu} \psi - \frac{1}{2} g_{\mu\nu} A_{\sigma} \partial^{\sigma} \psi];$$

$$T_{\mu\nu}^{\psi} = -\alpha [\partial_{\mu} \psi \partial_{\nu} \psi - \frac{1}{2} g_{\mu\nu} \partial_{\sigma} \psi \partial^{\sigma} \psi];$$

$$\chi_{\mu\nu} = \partial_{\mu} \chi \partial_{\nu} \chi - \frac{1}{2} g_{\mu\nu} \partial_{\sigma} \chi \partial^{\sigma} \chi$$

$$\lambda_{\mu\nu} = \lambda \frac{\delta}{\delta g^{\mu\nu}} \left[ \frac{\alpha}{f} \right] + \varepsilon \frac{\delta f}{\delta g^{\mu\nu}} + \frac{1}{2} g_{\mu\nu} \left[ \lambda A_b A^b + \lambda - \lambda \frac{\alpha}{f} + \varepsilon[\lambda - f] \right]$$

$$\alpha_{\mu\nu} = \alpha^{\frac{3}{2}} \chi [A_b A^b + 1] A_{\mu} A_{\nu} - \frac{1}{2} \alpha^{\frac{3}{2}} \chi g_{\mu\nu} [A_b A^b + 1]^2 - \frac{1}{2} g_{\mu\nu} \omega_k [\omega - \alpha].$$

- varying (1) with respect to $\delta\lambda$, $\delta\varepsilon$ and $\delta\psi$ would respectively yield:

$$A_{\mu} A^{\mu} + 1 - \frac{\alpha}{f} + \varepsilon = 0 \quad (4)$$

$$\lambda = f = \alpha A^{\nu} \partial_{\nu} \psi \quad (5)$$
$$ \nabla^{\mu} \left[ -2\alpha \partial_{\mu} \psi + \beta A_{\mu} + \lambda \frac{\alpha}{f^2} \frac{\partial f}{\partial(\partial^{\mu} \psi)} - \varepsilon \frac{\partial f}{\partial(\partial^{\mu} \psi)} \right] = 0 \quad (6) $$

- Likewise varying (1) with respect to $\delta A^{\mu}$ would yield:

$$ \nabla^{\mu} A_{\mu\nu} - 2\lambda A_{\nu} - \beta \partial_{\nu} \psi - \lambda \frac{\alpha}{f^2} \frac{\partial f}{\partial A^{\mu}} + \varepsilon \frac{\partial f}{\partial A^{\mu}} - 4\alpha^{\frac{3}{2}} \chi A_{\nu} [A_{\mu} A^{\mu} + 1] = 0 \quad (7) $$

- while varying (1) with respect to $\delta\omega$, $\delta\omega_k$ and $\delta\chi$ would respectively yield:

$$ \partial_{\mu} \chi \partial^{\mu} \chi - 1 + \omega_k = 0 \quad (8) $$

$$ \omega - \alpha = 0 \quad (9) $$

$$ \nabla^{\mu} [2\omega \partial_{\mu} \chi - 2\alpha \partial_{\mu} \chi] - \alpha^{\frac{3}{2}} [A_{\mu} A^{\mu} + 1]^{2} = 0 \quad (10) $$

From there, it is clear that because of (9); the equation (10) would simplify into:

$$ \alpha^{\frac{3}{2}} [A_{\mu} A^{\mu} + 1]^{2} = 0 \quad (11) $$

what enforces:

$$ A_{\mu} A^{\mu} + 1 = 0 \quad (12) $$

This latter (12) would in its turn because of (4), yield for the lagrange multiplier $\varepsilon$, the solution:

$$ \varepsilon = \frac{\alpha}{f} \quad (13) $$

which (13) would in its turn because of (5), imply:

$$ \lambda \frac{\alpha}{f^2} \frac{\partial f}{\partial(\partial^{\mu} \psi)} - \varepsilon \frac{\partial f}{\partial(\partial^{\mu} \psi)} = 0 \quad (14) $$

$$ \lambda \frac{\alpha}{f^2} \frac{\partial f}{\partial(\partial^{\mu} \varphi)} - \varepsilon \frac{\partial f}{\partial(\partial^{\mu} \varphi)} = 0 \quad (15) $$

$$ \lambda_{\mu\nu} = -\frac{1}{2}\alpha g_{\mu\nu} \quad (16) $$

$$ \alpha_{\mu\nu} = 0 \quad (17) $$
whence in summary, by considering (9) and (12) which enforces $\nabla^\mu A_\mu = 0$; the equations (3), (4), (6) and (7) would respectively simplify/write as:

$$G_{\mu\nu} = \kappa T_{\mu\nu} - \alpha A_{\mu} \partial_{\nu} \psi + T_{\mu\nu}^{A} + \psi_{\mu\nu}^{A} + T_{\mu\nu}^{\psi} + \alpha g_{\mu\nu} \quad (18)$$

$$A_{\mu} A^{\mu} + 1 = 0 \qquad (19)$$

$$\nabla^{\mu} \nabla_{\mu} \psi = 0 \qquad (20)$$

$$\nabla^{\mu} A_{\mu\nu} + [2\alpha - \beta] \partial_{\nu} \psi = 0 \qquad (21)$$

where of course $T_{\mu\nu} = -\frac{2}{\sqrt{-g}} \frac{\delta(\sqrt{-g}L_{\Psi})}{\delta g^{\mu\nu}}$ denotes the energy-momentum tensor of standard matter; $T_{\mu\nu}^{A} = [A_{\mu\sigma}A_{\nu}^{\sigma} - \frac{1}{4}g_{\mu\nu}A_{\rho\sigma}A^{\rho\sigma}]$; $\psi_{\mu\nu}^{A} = \beta[A_{\mu} \partial_{\nu} \psi - \frac{1}{2}g_{\mu\nu}A_{\sigma} \partial_{\nu} \psi]$; $T_{\mu\nu}^{\psi} = -\alpha[\partial_{\mu} \psi \partial_{\nu} \psi - \frac{1}{2}g_{\mu\nu} \partial_{\sigma} \psi \partial^{\sigma} \psi]$; .

Let's also note that for a given standard matter field $\Psi$; its Euler-Lagrange equation of motion would be the usual:

$$\nabla^{\mu} \left[ 2\kappa \frac{\partial L_{\Psi}}{\partial(\partial^{\mu} \Psi)} \right] - 2\kappa \frac{\partial L_{\Psi}}{\partial \Psi} = 0 \qquad (22)$$

As we see, the field $\chi$ as well as the lagrange multiplier $\omega_k$ contribute neither in the gravitation equation (18), nor in the equations of motion for the fields $A_\mu$, $\psi$ and $\Psi$; whence they can be seen as just auxiliary/mathematical fields (i.e. non-physical fields) helping to achieve the Euler-Lagrange equations of motion (18), (19), (20) and (21) etc...

In the same way, the Lagrange multiplier $\varepsilon$ also is excluded from the equations of motion of the theory thanks to the constraints (12) and (13), while $\lambda$ and $\omega$ participate in some sense through respectively $f$ and $\alpha$ see (5) and (9).

Be that as it may, what is interesting with our gravitational theory (18-21); is that it is not only a "theory" derivable from variational principle, but also, it is among the simplest Einstein-aether gravity [1] capable to yield caustic free scalar field dark matter candidate, plus stiff matter [6] and dark energy through $\alpha g_{\mu\nu}$.

Indeed, as we are going to see in the next section, it could be verified that in cosmology, we would get interesting dark matter candidate; likewise contrary to what happen in usual Einstein-aether gravity [2, 3, 4]; the aether field $A_\mu$ is in our model a massless field, see equation (21).
### 3. Dark Matter, Stiff Matter and Dark Energy

In the case of homogeneous and isotropic Friedman metric of line-element [10]:

$$ds^2 = dt^2 - \gamma_{ij}dx^i dx^j \qquad (23)$$

where $\gamma_{ij}$ denotes the three dimensional spatial metric.

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

$$T_{\mu\nu} = [\rho_{\Psi} + P_{\Psi}]u_{\mu}u_{\nu} - P_{\Psi}g_{\mu\nu} \qquad (24)$$

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

$$3H^2 + 3\frac{K}{a^2} = \kappa\rho_{\Psi} - \alpha + \rho^A + [\beta - \alpha]A_0\dot{\psi} - \alpha\frac{1}{2}\dot{\psi}^2 \qquad (25)$$

$$2\dot{H} + 3H^2 + 3\frac{K}{a^2} = -\kappa P_{\Psi} - \alpha - P^A - \frac{1}{2}\beta A_0 \dot{\psi} + \alpha \frac{1}{2} \dot{\psi}^2 \qquad (26)$$

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 fields; $\rho^A$ and $P^A$ respectively the energy-density and pressure from the tensor $T^A_{\mu\nu}$; $A_0$ the temporal component of $A_{\mu}$ and $\dot{\psi}$ the time derivative of $\psi$.

In the same way, one can verify that the equations (19) and (20) would respectively yield:

$$A_0 = 1 \qquad (27)$$

$$\dot{\psi} = \frac{\psi_0}{a^3} \qquad (28)$$

where $\psi_0$ is a constant of integration; what means we will have:

$$\rho^A = 0 = P^A \qquad (29)$$

whence (26) and (27) would simplify/rewrite as:

$$3H^2 + 3\frac{K}{a^2} = \kappa\rho_{\Psi} + [\beta - \alpha]\frac{\psi_0}{a^3} - \alpha\frac{\dot{\psi}_0^2}{a^6} - \alpha \qquad (30)$$

$$2\dot{H} + 3H^2 + 3\frac{K}{a^2} = -\kappa P_{\Psi} - \beta \frac{\psi_0}{a^3} + \alpha \frac{\dot{\psi}_0^2}{a^6} - \alpha \qquad (31)$$
where $[\beta - \alpha] \frac{\psi_0}{a^3} = \rho_{dm}$ denotes the dark matter energy-density and $\beta \frac{\psi_0}{a^3} = P_{dm}$ the dark matter pressure capable of precluding caustics formation.

It is worth mentioning that we should consider $\alpha < 0$, which would play the role of dark energy.

It is also clear that the pressure $P_{dm}$ of our dark matter depends from the free constant $\beta$; whence for $\beta \ll -\alpha$; we would have $P_{dm} \ll \rho_{dm}$ i.e. our dark matter would mimick cold dark matter, what means that we could reproduce $\Lambda$CDM model [7] since the stiff density $\frac{\psi_0^2}{a^6}$ would become negligible in late universe [6].

Moreover, as on the one hand, our dark matter candidate is a scalar field dark matter [5] since coming from the scalar field $\psi$, and that on the other hand it has non-zero pressure $P_{dm}$; then it becomes quite possible to handle problems like small satellite problem [8] and/or cups-core problem [9] by assuming either sufficiently light dark matter particle or appropriate pressure through $\beta$.

## 4. Conclusion

We proposed in this short paper a massless Einstein-Aether gravity capable of yielding viable dark matter candidate and dark energy.

Indeed as show the equations (30) and (31); the model is quite capable of reproducing $\Lambda$CDM phenomenology for any appropriate values for the constant $\alpha$ and $\beta$.

Better by yielding scalar field dark matter; the model could more easily explain small satellite problem and/or cups-core problem if we assume a sufficiently light scalar dark matter particle.

## 5. References

[1] Jacobson, Ted; (2008). "Einstein-aether gravity: A status report". arXiv:gr-qc/0801.1547.

[2] A. O. Barvinsky (2014). "Dark matter as a ghost free conformal extension of Einstein theory". JCAP01, 014. arXiv:hep-th/1311.3111.

[3] M. Chaichian; J. Kluson; M. Oksanen; A. Tureanu (2014). "Mimetic dark matter, ghost instability and a mimetic tensor-vector-scalar gravity". JHEP12, 102. arXiv:hep-th/1404.4008.

[4] D. Benisty; M. Chaichian; J. Kluson; M. Oksanen (2023). "Mimetic Tensor-Vector-Scalar Cosmology: Incorporating Dark Matter, Dark Energy and Stiff Matter". arXiv:gr-qc/2107.12161.

[5] Jae-Weon Lee (2018). "Brief History of Ultra-light Scalar Dark Matter Models". EPJ Web of Conferences 168, 06005. arXiv:astro-ph.CO/1704.05057.

[6] P. H. Chavanis (2015). "Cosmology with a stiff matter era". Phys. Rev. D92, 103004. arXiv:gr-qc/1412.0743.

[7] B. C. Leslie (2020). "COSMOLOGY NOW: Lambda-cdm model, What we know and How we know it". ISBN 979-8654852007.

[8] J. S. Bullock (2010). "Notes on the Missing Satellites Problem". arXiv:astro-ph.CO/1009.4505.

[9] W. J. G. de Blok (2009). "The core-cusp problem". Advances in Astronomy, 1-14. arXiv:astro-ph.CO/0910.3538.

[10] Nemiroff, Robert J.; Patla, Bijnath (2008). "Adventures in Friedmann cosmology: A detailed expansion of the cosmological Friedmann equations". American Journal of Physics. 76 (3): 265-276. arXiv:astro-ph/0703739.