# A note on a mimetic scalar–tensor cosmological model

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## Abstract

A specific Hordenski scalar-gravity mimetic model is investigated within a FLWR space-time. The mimetic scalar field is implemented via a Lagrangian multiplier, and it is shown that the model has equations of motion formally similar to the original simpler mimetic matter model of Chamseddine–Mukhanov–Vikman. Several exact solutions describing inflation, bounces, and future-time singularities are presented and discussed.

### Keywords

Dark Matter Dark Energy Hubble Parameter Standard Cosmological Model Diffeomorphism Invariance## 1 Introduction

It is well known that General Relativity (GR) with additional suitable cosmological positive constant and Dark Matter, describes quite well a large part of the history of the Universe, including Dark Energy era. With an additional scalar degree of freedom, this model may also describe the primordial inflationary period, which is necessary for solving the horizon and flatness problem. This is essentially the so-called \(\Lambda \)-CDM model, or standard cosmological model, and it has been recently tested with high accuracy [1, 2].

*H*(

*t*) is the usual Hubble parameter. When the solution of Eq. (2), namely \(\rho (t)=\rho _0 a(t)^{-3(1+\omega )}\), is taken into account, one has a non-linear second order differential equation for

*a*(

*t*). Of course, in GR, one may make use of the Friedmann equation to arrive directly at the explicit solution for

*a*(

*t*), a textbook result. In the presence of a scalar field, things are not so simple, and, in general, it is not easy to find exact solutions (see, for example [3, 4]).

Furthermore, in the standard cosmological model, Dark Matter and Dark energy issues are still under investigation, since one has no clue what they are. In fact, it is well known that the dark energy effect is well parametrized by the inclusion of a tiny positive cosmological constant, but the coincidence and cosmological constant problems arise. In the physics of elementary particles there exist a few candidates for dark matter, but experimental verification is still lacking, and other alternatives are possible.

For these reasons, in this paper, we would like to consider a generalization of the so called mimetic dark matter-gravity models [5, 6, 7]. This proposal may be considered a minimal modification of GR, in which cosmological dark matter may be described. Soon it had also been realized [8, 9] that this class of models are related to GR by singular disformal transformations. In particular the models [5, 6, 7] have been introduced by making use of a singular conformal transformation.

In this paper, in order to deal with the mimetic field \(\phi \), we shall follow the Lagrange multiplier approach.

The outline of the paper is the following. In Sect. 2, the mimetic model is introduced. In Sect. 3 addresses a solution in the absence of matter, while in Sect. 4 matter or radiation is included. The paper ends with the conclusions and an Appendix.

We use units in which the reduced Planck mass is \(M_P^2=1\).

## 2 Mimetic scalar–tensor gravity model

When the constants \(\beta \), \(\alpha \), and \(\gamma \) are vanishing, and in the absence of matter, the above model reduces to the original mimetic gravity proposed by Chamseddine and Mukhanov [5]. When \(\alpha \) and \(\beta \) are vanishing, the model reduces to the one studied in [24, 25]; see also [20] for other aspects.

*b*(

*t*) is an arbitrary dynamical variable which takes the value \(b=0\) after variations. The action (5) may be written as a functional of

*a*(

*t*),

*b*(

*t*), and \(\lambda \). Variation with respect to \(\lambda \) and assuming \(\phi \) to depend only on

*t*give

*b*gives the generalized Friedmann equation and reads

*a*leads to

A remark is in order. The equation of motion (10) in this Hordenski mimetic model does not contain the Lagrange multipliers, and it is similar to the one valid in GR plus ordinary matter. However, here the mimetic potential *V* appears in a very peculiar way, and this helps a lot in the search for exact solutions. In fact, one is dealing with a non-linear first order Riccati differential equation.

Furthermore, as in GR, one may have the de Sitter solution \(H=H_0\) if and only if the potential is a constant, and \(\omega =-1\) or \(\rho =0\). Furthermore, one may have the de Sitter solution with vanishing potential and in the absence of matter, but with the constant \(\beta \ne 0\). The effective cosmological constant depends on the ratio \(\frac{\beta }{c_1}\). Thus, in the presence of a non-trivial potential, one may have only a quasi-de Sitter solution, and inflation and the current acceleration may be described. With regard to other solutions, there exist several possibilities.

## 3 Absence of matter

*b*a real parameter. In this case, the bounce solution is

Other exact solutions have been presented in [6].

## 4 Presence of matter

*H*becomes

*C*is an integration constant. Furthermore, if \(\omega +1 \) is non-vanishing, one has

*C*is the contribution associated with the mimetic dark matter. The term depending on \(\frac{\beta }{c_1}\) acts again as an effective cosmological constant. Another constant contribution may be obtained by the simplest choice for the potential \(V=V_0\), namely a constant potential. In this case, we have

*N*to the de Sitter space-time.

*g*(

*N*) a known function. In this case, the solution is

*b*. To simplify the discussion, assume \(C=\beta =\rho _0=0\), and \(\frac{ V_0}{ c_1}>0\) Thus

## 5 Conclusions

In this paper, a specific cosmological Hordenski scalar-gravity mimetic model has been investigated within a FLWR space-time. The mimetic scalar field has been implemented making use of a Lagrange multiplier, and it has been shown that the model leads to equations of motion formally similar to the original simpler mimetic matter model of the so-called mimetic matter model [5]. Several exact solutions describing inflation, bounces, and future-time singularities have been presented and discussed.

It should be interesting to investigate a spherically symmetric static solution of this generalized mimetic model along the lines of Ref. [32].

### References

- 1.R. Adam et al. (Planck Collaboration), arXiv:1502.01582 [astro-ph.CO]
- 2.P. Ade et al. (Planck Collaboration), arXiv:1502.01589 [astro-ph.CO]
- 3.A.Y. Kamenshchik, E.O. Pozdeeva, A. Tronconi, G. Venturi, S.Y. Vernov, Class. Quantum Gravity
**31**, 105003 (2014). arXiv:1312.3540 [hep-th]ADSCrossRefGoogle Scholar - 4.A.Y. Kamenshchik, E.O. Pozdeeva, A. Tronconi, G. Venturi, S.Y. Vernov, Class. Q. Grav.
**33**, 015004 (2016). arXiv:1509.00590 [gr-qc] - 5.A.H. Chamseddine, V. Mukhanov, JHEP
**1311**, 135 (2013). arXiv:1308.5410 [astro-ph.CO]ADSCrossRefGoogle Scholar - 6.A.H. Chamseddine, V. Mukhanov, A. Vikman, JCAP
**1406**, 017 (2014). arXiv:1403.3961 [astro-ph.CO]ADSCrossRefGoogle Scholar - 7.L. Mirzagholi, A. Vikman, JCAP
**1506**(06), 028 (2015). arXiv:1412.7136 [gr-qc] - 8.
- 9.A.O. Barvinsky, JCAP
**1401**(01), 014 (2014). arXiv:1311.3111 [hep-th] - 10.J.D. Bekenstein, Phys. Rev. D
**48**, 3641 (1993). arXiv:gr-qc/9211017 - 11.F. Arroja, N. Bartolo, P. Karmakar, S. Matarrese, JCAP
**1509**, 051 (2015). arXiv:1506.08575 [gr-qc]ADSCrossRefGoogle Scholar - 12.
- 13.E.A. Lim, I. Sawicki, A. Vikman, JCAP
**1005**, 012 (2010). arXiv:1003.5751 [astro-ph.CO] - 14.S. Capozziello, J. Matsumoto, S. Nojiri, S.D. Odintsov, Phys. Lett. B
**693**, 198 (2010). arXiv:1004.3691 [hep-th]ADSCrossRefGoogle Scholar - 15.S. Nojiri, S.D. Odintsov, Mod. Phys. Lett. A
**29**(40), 1450211 (2014). arXiv:1408.3561 [hep-th] - 16.J. Matsumoto, S.D. Odintsov, S.V. Sushkov, Phys. Rev. D
**91**(6), 064062 (2015). arXiv:1501.02149 [gr-qc] - 17.S.D. Odintsov, V.K. Oikonomou, arXiv:1508.07488 [gr-qc]
- 18.R. Myrzakulov, L. Sebastiani, S. Vagnozzi, Eur. Phys. J. C
**75**, 444 (2015). arXiv:1504.07984 [gr-qc] - 19.M. Raza, K. Myrzakulov, D. Momeni, R. Myrzakulov, arXiv:1508.00971 [gr-qc]
- 20.R. Myrzakulov, L. Sebastiani, S. Vagnozzi, S. Zerbini, Fund. J. Mod. Phys.
**8**, 119 (2015). arXiv:1505.03115 [gr-qc] - 21.G.W. Horndeski, Int. J. Theor. Phys.
**10**, 363 (1974)CrossRefGoogle Scholar - 22.C. Deffayet, X. Gao, D.A. Steer, G. Zahariade, Phys. Rev. D
**84**, 064039 (2011). arXiv:1103.3260 [hep-th]ADSCrossRefGoogle Scholar - 23.A. De Felice, T. Kobayashi, S. Tsujikawa, Phys. Lett. B
**706**, 123 (2011). arXiv:1108.4242 [gr-qc]ADSCrossRefGoogle Scholar - 24.S. Nojiri, S.D. Odintsov, Phys. Lett. B
**691**, 60 (2010). arXiv:1004.3613 [hep-th]ADSMathSciNetCrossRefGoogle Scholar - 25.G. Cognola, E. Elizalde, L. Sebastiani, S. Zerbini, Phys. Rev. D
**83**, 063003 (2011). arXiv:1007.4676 [hep-th]ADSCrossRefGoogle Scholar - 26.R. Myrzakulov, L. Sebastiani, S. Zerbini, Eur. Phys. J. C
**75**(5), 215 (2015). arXiv:1502.04432 [gr-qc] - 27.R. Myrzakulov, L. Sebastiani, Astrophys. Space Sci.
**352**, 281 (2014). arXiv:1403.0681 [gr-qc]ADSCrossRefGoogle Scholar - 28.T. Biswas, T. Koivisto, A. Mazumdar, JCAP
**1011**, 008 (2010). arXiv:1005.0590 [hep-th]ADSCrossRefGoogle Scholar - 29.T. Biswas, A.S. Koshelev, A. Mazumdar, S.Y. Vernov, JCAP
**1208**, 024 (2012). arXiv:1206.6374 [astro-ph.CO]ADSCrossRefGoogle Scholar - 30.S. Nojiri, S.D. Odintsov, S. Tsujikawa, Phys. Rev. D
**71**, 063004 (2005). arXiv:hep-th/0501025 - 31.G. Acquaviva, L. Bonetti, G. Cognola, S. Zerbini, Phys. Rev. D
**88**(12), 124024 (2013). arXiv:1309.6950 [gr-qc] - 32.R. Myrzakulov, L. Sebastiani, Gen. Relativ. Gravit.
**47**(8), 89 (2015). arXiv:1503.04293 [gr-qc] - 33.H.R. Lewis, J. Math. Phys.
**9**, 1976 (1968)ADSCrossRefGoogle Scholar - 34.H.R. Lewis, W.B. Riesenfeld, J. Math. Phys.
**10**, 1458 (1969)ADSCrossRefGoogle Scholar - 35.C.J. Eliezer, A. Gray, SIAM
**30**, 463 (1976)Google Scholar

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