A cyclic cosmological model based on the f({\rho}) modified theory of gravity

We consider FLRW cosmological models for perfect fluid (with rho as the energy density) in the frame work of the f(rho) modified theory of gravity [V. N. Tunyak, Russ. Phys. J. 21, 1221 (1978); J. R. Ray, L. L. Smalley, Phys. Rev. D. 26, 2615 (1982) ]. This theory, with total Lagrangian R-f(rho), can be considered as a cousin of the F(R) theory of gravity with total Lagrangian F(R)-rho. We can pick proper function forms f(rho) to achieve, as the F(R) theory does, the following 4 specific goals, (1) producing a non-singular cosmological model (Ricci scalar and Ricci tensor curvature are bounded); (2) explaining the cosmic early inflation and late acceleration in a unified fashion; (3) passing the solar system tests; (4) unifying the dark matter with dark energy. In addition we also achieve goal number (5): unify the regular matter/energy with dark matter/energy in a seamless fashion. The mathematics is simplified because in the f(rho) theory the leading terms in Einstein's equations are linear in second order derivative of metric wrt coordinates but in the F(R) theory the leading terms are linear in fourth order derivative of metric wrt coordinates.

Various theories are developed in an effort to explain the cosmic early inflation and late accelerated expansion. The   FR modified theories of gravity (see [35,36,37] for reviews) have recently [38] become one of the leading popular candidates in (1)  In this paper we consider cosmological models based on the (less well known)   f  modified theories of gravity for perfect fluid [39,40] . We show that, like the   FR theories, the   f  theory can also accomplish the same 4 goals.
In addition we show that with   f  theory we can accomplish goal number (5): unifying the regular matter/energy with dark matter/energy in a seamless fashion.
One added benefit is that the mathematics is simplified in   Where the effective energy density and the effective pressure are given by: In (2.3b) 0   , 0 p  are the energy density and the pressure of the perfect fluid. Throughout this paper, they are always nonnegative. The effective equation of state parameter,   eff w  , is then given by: We remark that the conservation of the enegy-momentum becomes: The conventional formula for the conservation of the enegy-momentum becomes the low-energy approximation of (2.5): One consequence of using this effective perfect fluid concept is that the following energy conditions may or may not be satisfied in general.  .
We are amazed, as we show later, that Planck's magic black-body radiation formula would show its charm again, after over 100 years, in shedding some light on solving the UV divergence problem (intrinsic singularity inside of black hole or at naked singularity, etc) in Einstein's general relativity theory of gravity as well.
Later on we find out that the exponential in   So for given   p  as long as we pick function   f  such that: We can deduce from (2.5) that The black-body radiation inspired formulas  

III. The Friedmann equations
In Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmology, the comoving (infalling) spherical symmetric line element is given by: [42]         (the last entry is from [44] ). In Figure 3 below, we will show that the effective equation of state     eff wt  for a new cyclic universe model does vary from less than -1 to larger than +1 in a continuous fashion.
where 0 a is a constant of integration. Substituting these results into (3.1a), we can rewrite the Friedmann equation (3.1a) as: This is probably as far as we can go without specifying   f  and   p  . We would prefer to work with Friedman equations in the format of (3.4) and (3.5) instead of the Friedman equations in the format of (3.2a) and (3,3b). This is because, as we show in the next section, that we do not need to specify the Equation of State    0 p   to obtain major features of cosmology models when   f  is given.
We could bundle (3.2a) and (3.3b) together and write the Friedmann equations in the following different formats as: We would consider the Friedmann equations in terms of This set of Friedmann equations are clearly different from but also closely related to various (modified/generalized) Friedmann equations in the literature [45,46,47,48,49,50,51] . f(ρ) theory of gravity Shi 10 For example [45] considered a dark fluid with Equation of State like, (3.7) In notation of the current work, the Einstein's equations are given by The Friedmann equations in this case become, We can clearly see that (3.9) is similar to but different from (3.6).
(i) If we identify Other examples are considered in [46] where the Friedman equations and Equation of State go like, We can also show that (3.10) is similar to but different from (3.6).
. It then starts a new big bang -big crunch process. So this cosmological model is a cyclic universe model. This model also covers the eras (like bouncing at both ends which are) beyond the early cosmic inflation and late cosmic accelerated expansion. for the cyclic universe model does vary from less than -1 to larger than +1 in a continuous fashion. Our cyclic universe model with W-shaped potential well do incorporate the cosmic early inflation era as well as the cosmic late accelerated expansion, thanks to two downhill slopes (point A to point B and point C to point D in Figure 1). This cyclic universe model is thus different from those in [56,57,58] where there is only one downhill slope because their potential   U  is a U-shaped.
We now define     The time, T , it takes to go from big-band at point A to big-stop at point E is given by: We now assume be determined by the fitting of this model with observational data. This is beyond our current capability.
We now go through our check list to see if we can achieve 5 goals mentioned in the abstract with the cyclic universe model with 7-parameter M-shaped   f  of (3.5).
Goal number (1)  Goal number (2) explaining the cosmic early inflation and late acceleration in a unified fashion. See Figure 1 and the description right after.
Goal number (3) passing the solar system tests. As long as , large enough and  is tiny enough, we have   f   and we can pass the solar system tests.
Goal number (4) unifying the dark matter with dark energy. See Figures 1 and 2 and the description in between.
Goal number (5) unifying the regular matter/energy with dark matter/energy in a seamless fashion. Unlike other dark energy (+ regular matter) theory, there is only one material (one energy density  and one pressure p ) in our   f  theory based cyclic universe model. This single material plays both the role of regular matter/energy when needed in the FLRW era and the role of dark matter/energy when needed in other eras in the cyclic universe model. See Figures 2 and 3 and the description in between for details. It is in this sense that we meant we achieve goal number (5). Maybe regular matter/energy (like perfect fluid) and dark matter/energy are just two aspects of the same material. In other words, we have shown the bright side of the dark matter/energy or the dark side of the regular matter/energy (perfect fluid).
The concept of cyclic universe model itself is not new. For example, any cosmology in general and cyclic cosmology in particular, could be reconstructed in   FR theories of gravity. The corresponding technique is described in [59,60] . Realistic   FR f(ρ) theory of gravity Shi 16 gravity cosmology model has recently been proposed in [61] . This model can also achieve goals (1) through (4) mentioned above.
One unique thing about the   f  theory based cyclic model of (3.6) is that, the mathematics is very much simplified. Without specifying   It remains to be seen if anything significant can be deduced from (5.3).
We would look at the relation between   f  theory and    Table 3 below.
We start with (5.4b), the trace of the original Einstein's equation for dust, We observe that as long as R and  go to infinity at the same speed, the trace equation R   can still be satisfied. We think that this is the cause of the intrinsic singularity.
(i) One way to break up this running away (to infinity) situation is to replace , it is straight forward to show from (5.5b) that R is bounded from above and below and so is  .
(ii) Another way to break up this running away (to infinity) situation is to replace Lagrangian of (5.4a) in Einstein's theory, show that  and R are bounded from above and below as well.
Because of the higher order derivative term in (5.6b), a similar analysis is doable but less straight forward.

(B) The relation between   f  theory and Nonlinear ElectroDynamics (NED)
We are intrigued by the nonsingular exact black hole solutions with Nonlinear ElectroDynamics (NED) [65,66,67] . As a matter of factor, the inspiration to both the original authors of [39, 65] can be traced back to the famous Born-Infeld theory [68] .
. The square root function is used to break up the running away to infinity situation (at the center of a charged particle). Thus we would guess that        

(C) The relation between   f  theory and Chaplygin Gas
The Chaplygin gas is an exotic perfect fluid (a kind of dark energy) with equation of state [69,70,71] : It is used to explain the cosmic late accelerated expansion.
Notice that the equation of state behaves like: The problem with Chaplygin gas of (5.9) is that it diverges at 0   in the range The   f  based cyclic universe model of (3.5) does not have this kind of divergence problem.

VI. Conclusion
We considered FLRW cosmological model for perfect fluid (with In addition, we also achieve goal number (5) We would like to emphasize that in our   f  theory based cyclic universe model, the single material (energy density  ) plays both the role of regular matter/energy when needed in the FLRW era and the role of dark matter/energy when needed in other eras in the cyclic universe. Thus we guess that the regular matter/energy (like perfect fluid) and dark matter/energy might just be two aspects of the same material. In other words, we have probably shown the bright side of the dark matter/energy or the dark side of the regular matter/energy (perfect fluid).
The apparent unification of the regular matter/energy with dark matter/energy in a seamless fashion and simplification of Einstein's equations (Friedman's equations) in   f  theory are probably the two interesting benefits of using   f  theory. We now follow the standard text book procedure (cf. Hawking and Ellis [72] ) to derive the energy-momentum density tensor using the material Lagrangian   f  for the ideal fluid.
We denote  the mass density,   p  the pressure,    the internal energy density, and    the (total) energy density of the ideal fluid. We remark that in [72] symbol  represents mass density (so it is equivalent to our  here) and symbol  represents (total) energy density (so it is equivalent to our  here).
The energy-momentum tensor is defined as: In [72] , the material Lagrangian is chosen as: Here we choose: Substitution of (A.3) into (A.1), we have: The derivation may be simplified by noting that the conservation of the current ju    may be expressed as: Eq.(A.12b) is equivalent to (2.2)-(2.3) of the main text. So we accomplish the first task in this appendix.
The second task we want to accomplish is to convert the energy-momentum tensor in [40] into the same form as in (A.12). We remark that in [40] symbol  represents mass density (so it is equivalent to our  here).
In our notation the energy-momentum tensor (in the absence of torsion) is given by (cf. (3.4h) of [40]): Notice that (A.13) looks different from (A.12b) in the following ways: (i) the differentiation is wrt to mass density  [in (A.12b), the differentiation is wrt to (total) energy density  ]; (ii) the pressure p does not appear explicitly [in (A.12b), the pressure p does appear explicitly]. We remark that (A.13) is not suitable for practical calculation because pressure p does not appear explicitly.
In [40] ,   F  is eventually chosen as     Substitution of (A.14) into (A.13), we find out that the result is identical to (A.12a).