Cosmology in the mimetic higher-curvature \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(R,R_{\mu \nu }R^{\mu \nu })$$\end{document}f(R,RμνRμν) gravity

In the framework of the mimetic approach, we study the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(R,R_{\mu \nu }R^{\mu \nu })$$\end{document}f(R,RμνRμν) gravity with the Lagrange multiplier constraint and the scalar potential. We introduce field equations for the discussed theory and overview their properties. By using the general reconstruction scheme we obtain the power law cosmology model for the \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(R,R_{\mu \nu }R^{\mu \nu })=R+d(R_{\mu \nu }R^{\mu \nu })^p$$\end{document}f(R,RμνRμν)=R+d(RμνRμν)p case as well as the model that describes symmetric bounce. Moreover, we reconstruct model, unifying both matter dominated and accelerated phases, where ordinary matter is neglected. Using inverted reconstruction scheme we recover specific \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(R,R_{\mu \nu }R^{\mu \nu })$$\end{document}f(R,RμνRμν) function which give rise to the de-Sitter evolution. Finally, by employing the perfect fluid approach, we demonstrate that this model can realize inflation consistent with the bounds coming from the BICEP2/Keck array and the Planck data. We also discuss the holographic dark energy density in terms of the presented \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$f(R,R_{\mu \nu }R^{\mu \nu })$$\end{document}f(R,RμνRμν) theory. Thus, it is suggested that the introduced extension of the mimetic regime may describe any given cosmological model.


Theoretical model
The main idea behind the mimetic gravity is that the g µν metric (the fundamental variable in gravity) may be expressed by the new degrees of freedom 19 . By doing so, such degrees of freedom can admit wider class of solutions than the standard GR. In particular, the conformal degree of freedom can be isolated by expressing the physical metric in terms of the scalar field φ and the auxiliary metric ĝ µν as follows: Note that the above parametrization posses Weyl symmetry under the conformal transformation, i.e. ĝ αβ = e ω(x) g αβ . It is also worth to remark, that instead of using the physical metric g µν in variation of the gravitational action, one can use variation with respect to the metric ĝ µν and the scalar field φ . As a result, from Eq. (1) the following constraint can be obtained: In order to impose given constraint at the action level we use the Lagrange multiplier formalism 73,74 . In the mimetic regime we need to introduce the Lagrange multiplier ( ) that corresponds to the mimetic constraint 19 . Hence, for the mimetic f (R, R µν R µν ) gravity we have: where f (R, R µν R µν ) is the analytical function of the Ricci scalar R and contraction of the Ricci tensors (the Ricci squared term) R µν R µν . Moreover, V (φ) denotes scalar (mimetic) potential. We note that the equations obtained from the variation with respect to the physical metric g µν with imposed mimetic constraint are fully equivalent to the equations that one can derive by using action written in terms of the auxiliary metric ĝ µν . We emphasize (1) g µν = −ĝ αβ ∂ α φ∂ β φĝ µν .
(3) S = d 4 x −g f (R, R µν R µν ) − V (φ) + (g µν ∂ µ φ∂ ν φ + 1) , www.nature.com/scientificreports/ that standard matter fields are not part of our considerations since one of the core characteristics of the mimetic theory is to mimic matter content 19 . In what follows, the variation of the action (3) with respect to the components of the metric tensor g µν , gives the following field equation: ∂Y , whereas the indices in brackets denote symmetrization ( A (µν) = 1 2 (A µν + A νµ ) ). Next, by varying the action with respect to the scalar field φ the following scalar equation can be obtained: where prime stands for the differentiation with respect to the scalar field ( V ′ (φ) = dV (φ) dφ ). It is important to remark, that the variation of the action with respect to returns mimetic constraint.
Further in this work we assume that the background geometry is described by the FLRW metric, with the following line element: where i = 1, 2, 3 and a(t) is the scale factor. For simplicity, we also assume that the scalar field depends only on time, i.e. φ = φ(t) . By using the FLRW metric (6), the corresponding (t, t) component of the mimetic f (R, R µν R µν ) gravity equations (4) is: while the (i, j) component reads: From Eq. (5) the corresponding scalar equation is given as: Note, that dot denotes differentiation with respect to the cosmic time t and H =ȧ/a is the Hubble rate. For brevity, we note that the Ricci scalar and the Ricci tensor squared in the FLRW spacetime are: Finally, the mimetic constraint for the metric given by Eq. (6) is: and yields: with C being integration constant. Keeping in mind that the scalar field appears only with derivatives, shift symmetry allows for setting constant C = 0 without losing generality. By using φ = t the field and scalar equations (7-9) reduce to: and Hence, Eq. (14) can be used to determine potential V (φ) that generates evolution of the Hubble parameter for the specific f(R, Y) gravity:  (13) and (14), the analytic form of the Lagrange multiplier (t) is provided: This equation allows to obtain the desired evolution in various ways. Given the Hubble parameter H(t) for the specified f(R, Y) function, one can find corresponding multiplier and mimetic potential V (φ) . Also, given the form of the mimetic potential and the Lagrange multiplier , one can solve above equations to find proper f (R, R µν R µν ) gravity model that realizes cosmological scenario of interest.

Reconstruction of the FLRW cosmologies
In the following section, we use reconstruction methods in order to obtain various cosmological scenarios in the mimetic f (R, R µν R µν ) gravity. The procedure of cosmological reconstruction is widely used in the context of modified theories of gravity and helps with the extraction of physical implications in the theory of interest 35,71,[75][76][77][78][79][80] . In general, obtaining solutions for the GR's extensions may be troublesome, since even small modifications can drastically increase difficulty of equations. The reconstruction scheme used in the extensions of the GR relies on the fact that arbitrary function (or potential) is used in the definition of the modified gravity. Once the Hubble factor H is specified, the field equations are solved to obtain the model of interest. This technique is a good way to survey modified theories of gravity, since satisfying well established cosmological models in gravity's extensions is desired. Moreover, this technique yields difference between the GR and its modifications, namely specific class (or classes) of the extended gravity may describe any given spacetime 10 . In this section, once the function f(R, Y) is given, we obtain the Lagrange multiplier and potential V satisfying specified evolution. In general, by using this technique one can achieve model describing any given Hubble rate such as inflation or transitions between phases for the mimetic f (R, R µν R µν ) approach.
In the present study, we assume specific form of the f (R, R µν R µν ) function, where Ricci squared term enters with arbitrary power p, i.e.: Thus, the rearranged equations for this choice of the f(R, Y) gravity are: and As an example, let us consider the power-law scale factor and the corresponding Hubble parameter: The solutions of this type are very useful in cosmology, especially in describing various phases in the history of the Universe 62 . We note that for 0 < n < 1 , the decelerated universe occurs where for dust dominated Universe n = 2 3 and for the radiation era n = 1 2 . The corresponding Ricci scalar and Ricci squared are: By using this expressions in the field equations, one can reconstruct the Lagrange multiplier and the mimetic potential for the f (R, R µν R µν gravity. The general expressions obtained for the Lagrange multiplier and potential are given in the Appendix. For the dust n = 2/3 and the inverse p = −1 power of Y, one gets:  www.nature.com/scientificreports/ It is worth to mention, that the Ricci squared contribution leads to the higher-order terms in the expressions for V and . In the absence of this term ( d = 0 ), the potential takes form 8 3 t −2 that is consistent with the result obtained in the standard mimetic gravity, i.e. V (t) = C 0 t −2 for constant C 0 = 8/3 19 .
As an another example we consider the exponential symmetric bounce cosmology, associated with the scale factor and the Hubble rate: One of the main characteristics of the bounce cosmologies is the absence of the initial singularity. These theories constitute also interesting alternatives to the well-established inflationary paradigm 65,81 . For these scenarios, the Universe contraction decreases the effective radius of the Universe to the minimal size. Then, accelerated expansion occurs. We also want to note that, the symmetric bounce of Eq. (25) should be combined with other cosmological model to complete this scenario 82 . Authors of 83 have pointed out that the Hubble horizon at t → −∞ never decreases and is increasing. This means that primordial modes cannot originate from a times before the bounce happened. Even if those modes correspond to the time near the point of bounce, horizon after t > 0 is decreasing, thus modes cannot reenter the horizon. However, our focus in this work is purely on the realization of various cosmological models, with the exponential bounce as an example of the bouncing Universe. In the exponential bounce case, the Ricci scalar and Ricci tensor squared are: The obtained Lagrange multiplier and mimetic potential take form: when again we use p = −1 and the general form of the potential and the Lagrange multiplier is listed in the Appendix.
In the following part, we provide reconstruction of the model that describes transition between matterdominated and accelerated phases of the Universe's history. In this scenario, the Hubble rate is given by 32 : It is important to remark, that for small t, H ≈ g 1 t and the Universe is filled with the perfect fluid with the EoS (equation of state) parameter w = −1 + 2/3g 1 . We note that recent results of the Planck observations constrained the equation of state of the Dark Energy models. In our case, the relationship between g 1 and ω DE takes the following form 32 : . The corresponding constraints on the ω DE from the recent Planck observations are ( 95% of confidence): −1 < ω DE < −0.95 84 . Thus, for the large g 1 , the equation of state consistent with the recent data can be obtained 84 . On the other hand, for the large t, the Hubble rate approaches H → g 0 and the Universe looks like the de-Sitter one 10,32 . We note that in this scenario, there is no real matter and the contribution of the mimetic f(R, Y) gravity plays the role of the standard matter content.
In the small t limit, the Lagrange multiplier and the mimetic potential are equal to: while in the large t limit one gets: where again we have chosen p = −1 . Note also that the full form of the obtained potential and the Lagrange multiplier can be found in the Appendix. Thus, in principle, the mimetic f (R, R µν R µν ) gravity is able to unify phases dominated by matter with the transition to the late time accelerated evolution even in the absence of real matter for the appropriate mimetic potential and Lagrange multiplier.
To this end, we reconstruct the particular class of f (R, R µν R µν ) gravity that satisfy mimetic field equations, once particular mimetic potential and the Lagrange multiplier are assumed. This is inversion of the previously used reconstruction. In this approach, as an example, we consider the de-Sitter space as specified by the constant Hubble parameter: www.nature.com/scientificreports/ We remark that this spacetime usually serves as a description of the accelerated expansion phase and the early Universe inflation 62 . By assuming the constant potential V and the multiplier as: we obtain the following partial differential equation (PDE): This PDE has the following solution: where F is an arbitrary function.
In conclusion, the reconstruction technique can work in both ways: one can assume specific functional form of the mimetic f (R, R µν R µν ) and cosmological evolution to obtain the mimetic potential and the Lagrange multiplier and vice versa. In general, by using this procedure one can obtain any cosmological scenario for the mimetic f (R, R µν R µν ) gravity.

Inflation
Inflation is believed to be one of the fundamental building blocks of the modern cosmology that may solve some of its problems 61,69 . This is to say, the main goal of the inflationary theory is to explain primordial fluctuations i.e. the density variations occurring in very early stages of the cosmic evolution. Since inflation is the most plausible scenario for the early Universe, the modified theories of gravity should be able to properly describe this phase of the cosmic history. Moreover, the deviations from GR can be interpreted as a quantum-induced corrections or motivated by the ultra-violet (UV) behavior of the quantum gravity, playing an important role in the inflationary phase of the early Universe 85 . Thus, viable description of this stage in any modified gravity is desired. Therefore, in the following section we analyse inflation in the context of the mimetic f (R, R µν R µν ) gravity to explore ability to obtain model comparable with the recent BICEP2/Keck observations 63,85 . Note, that the cosmological reconstruction can also be successfully applied to the discussed scenario. In this section we obtain the mimetic f (R, R µν R µν ) gravity that describes inflationary model which is consistent with the recent observational data. Again, the functional form of Eq. (18) is assumed.
While discussing the inflationary cosmology it is worth to use the number of e-foldings N (intervals for which the scale factor grows by the N factors of e) instead of the cosmic time t. The relation between the scale factor and the e-folding number N is following, e N = a a 0 , where a 0 is the initial value of the scale factor in the initial time instance. For brevity, we list transformation rules for the time derivatives with respect to the e-foldings number: We note that prime symbols ( ′ ) correspond to the derivatives dH(N) dN . In order to study the slow-roll indices we use perfect fluid approach developed in 86 . In this approach extra terms in the gravitational action (1) can be regarded as the perfect fluid. In order to achieve this, an extra contributions in the action may be associated with the modeling of the perfect fluid, while mimetic field will enter the equations through the pressure-less fluid. This formalism has been successfully applied to the other extensions of the mimetic theory, such as the unimodular f(R) or the f (G ) gravity 50,56 . This formalism allows to obtain the spectral indices independently from the model. According to this approach, the slow-roll parameters are given as a functions of the Hubble rate H: We note that the above procedure is valid for η and ε ≪ 1 . By using the above, the spectral index of the curvature perturbations n s and the scalar-to-tensor ratio r in terms of the slow-roll parameters can be provided: The parameters introduced in Eq. (39) are constrained by the recent Planck collaboration results 63 : .
(39) n s = −6ε + 2η + 1, r = 16ε. www.nature.com/scientificreports/ We remark that the spectral index describes variation of the density fluctuations with respect to the scale. On the other hand, the scalar-tensor ratio relates spectras of the scalar and tensor perturbations. By using Eqs. (16), (17) and (37), the field equations in terms of the e-foldings number N are: where we have used relations from Eq. (37). Thus, one can obtain the Lagrange multiplier and potential corresponding to the scale factor H(N), effectively reconstructing any given inflationary model. We note, that inverse reconstruction method is also possible once (N) and V(N) are specified.
The main goal of this part of our study was to reconstruct proper inflationary model in the mimetic f (R, R µν R µν ) theory. For more detailed discussion of other inflationary scenarios in the mimetic gravity, we refer reader to 50 . We conclude that extension of the GR presented here is comparable with other mimetic gravities and may constitute an interesting alternative 50,56 . Mimetic f (R, R µν R µν ) gravity and the holographic dark energy density Our discussion ends with some remarks on the holographic dark energy model 87 , which has been recently studied in the various modified gravity scenarios 88 . The holographic dark energy arises from applying the holographic principle to the problem of the dark energy 89 . In this regard, we will briefly discuss possibility of expressing the mimetic f(R, Y) gravity in terms of the holographic dark energy density, in accordance with the recent work of Nojiri and collaborators 88 . The main idea behind that is to express cut-offs in terms of the particle horizons and derivatives and relate them to the corresponding higher order modified gravity. For the holographic model, one gets from the Friedmann equation 88 : where L IR denotes the infrared cutoff, which usually is associated with the particle Horizon L p or the future event horizon L f . The infrared cutoff then is given by 90 : In terms of the particle horizon (51), the Hubble parameter will take form: The Ricci scalar and the Ricci tensor squared for the Hubble factor (52) are given by: (48) n s = 1 2(γ + αe βn )(3γ + α(β + 3)e βn ) 2 − 3(β 2 + 6β − 6)γ 3 + 2α 3 (β + 3) 2 (2β + 1)e 3βn + 2α 2 (2β 3 + 16β 2 + 39β + 27)γ e 2βn + α(2β 3 + 3β 2 + 12β + 54)γ 2 e βn ,   where Thus, we have associated the mimetic f(R, Y) theory with the holographic energy density 88 . This shows that the mimetic f (R, R µν R µν ) gravity may be successfully used for modelling the holographic dark energy models and constitute an interesting choice for the future works regarding the following subject.

Summary
In this work, we presented mimetic extension of the f (R, R µν R µν ) gravity using the Lagrange multiplier formalism with the mimetic scalar potential added to the Lagrangian, where mimetic field isolates conformal degree of freedom. Introduced extension was implemented by using the well-known reconstruction technique in order to obtain models of interest. Assuming functional form as f (R, Y ) = R + dY p , we obtained the Lagrange multiplier and the mimetic potential U(φ) that satisfy the power law evolution of the Universe. Another reconstructed model described symmetric bounce cosmology. Moreover, we have reconstructed model that satisfy transition between matter dominated and accelerated phase in the history of the Universe, showing that this extension of the mimetic gravity is capable to unify various stages of cosmic history. Additionally, using the inverse reconstruction method and choosing appropriate forms of and V we have obtained f(R, Y) model describing the de-Sitter model with constant Hubble factor H 0 . Our work is closed with discussion of the inflationary cosmology in a given regime. By using the Hubble parameter H(N) = αe βN + γ we get Lagrange multiplier and the mimetic potential which successfully describes inflationary model. The reconstructed inflationary model is phenomenologically viable for the wide range of the parameters when confronted with BICEP2/Keck data 50,63 . Moreover, we have shown that the theory presented here can be expressed in terms of the holographic dark energy density. This fact opens new possibilities and applications regarding the proposed theory. More sophisticated analysis of the holographic dark energy for the mimetic f (R, R µν R µν ) gravity will be conducted in the future works.
In conclusion, the mimetic extension of the f (R, T, R µν T µν ) can describe any given early or late-time cosmological model in particularly clear way. In general, the procedure presented here can be extended to other higher-curvature modifications of the general relativity or theories that include coupling with the trace of the energy-momentum tensor such as the f (R, T, R µν T µν ) or f (G , T) approaches 18,76 . We also remark that the matter fields were not used in our considerations, since main feature of the mimetic gravity is to mimic cosmological behavior driven by the matter fields 19,57 . In the approach presented here, the mimetic condition is supported by the higher curvature terms of the f (R, R µν R µν ) theory, such that the geometry of the considered theory incarnates matter. Moreover, the mimetic condition can play an important role in further considerations of the various higher order extensions of the general relativity and can be extended by inclusion of the unimodular concept in the considerations 57 . We also wish to point out that by choosing an appropriate functions and parameters one can obtain analytical results in the mimetic f (R, R µν R µν ) approach, which generally is hard task in the context of the modified theories of gravity 5 . Since there are many other proposals of the modified gravities that satisfy realistic models with sufficient degree of accuracy, the local constraints should be employed, such as the Post-Newtonian or Solar-System tests 91 . Moreover, recently the higher curvature theories supported by Lagrange multipliers have been studied in the context of eliminating the ghosts 92 . It was shown, that constraints provided by the multiplier can lead to the absence of the ghosts. The mimetic constraint introduced to the action by the Lagrange multiplier may lead to the proper behaviour of the theory. In case of the mimetic f (R, R µν R µν ) gravity, the absence of the ghosts is expected since twin f(R) and f (R, G ) gravities supported by the Lagrange multiplier posses this desired property 92 . Future studies should be devoted to these topics, by addressing not only mimetic extension presented here, but also other proposals such as mimetic versions of the f(R, T), f(R) or f (G ) gravity.

Appendix: Full form of the Lagrange multipliers and mimetic potentials
The power law evolution. (54)