Towards a possible solution for the coincidence problem: f(G) gravity as background
aa r X i v : . [ g r- q c ] N ov Towards a possible solution for the coincidence problem: f(G) gravity as background
Prabir Rudra Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India.Department of Mathematics, Pailan College of Management and Technology, Bengal Pailan Park, Kolkata-700104, India.
Abstract
In this article we address the well-known cosmic coincidence problem in the framework of the f(G) gravity. In order to achieve this, an interaction between dark energy and dark matter isconsidered. A set-up is designed and a constraint equation is obtained which generates the f(G) models that do not suffer from the coincidence problem. Due to the absence of a universallyaccepted interaction term introduced by a fundamental theory, the study is conducted over threedifferent forms of logically chosen interaction terms. To illustrate the set-up three widely knownmodels of f(G) gravity are taken into consideration and the problem is studied under the designedset-up. The study reveals that the popular f ( G ) gravity models does not approve of a satisfactorysolution of the long standing coincidence problem, thus proving to be a major setback for themas successful models of universe. Finally, two non-conventional models of f(G) gravity have beenproposed and studied in the framework of the designed set-up. It is seen that a complete solution ofthe coincidence problem is achieved for these models. The study also reveals that the b-interactionterm is much more preferable compared to the other interactions, due to its greater compliancewith the recent observational data.Keywords: Dark energy, Dark matter, Modified gravity, coincidence, interaction. Pacs. No.: 95.36.+x, 95.35.+d
Recent observational evidences from Ia supernovae, CMBR via WMAP, galaxy redshift surveys via SDSSindicated that the universe is going through an accelerated expansion of late [1, 2, 3, 4, 5]. With this discoverythe incompatibility of general relativity (GR) as a self sufficient theory of gravity came into light. Since nopossible explanation of this phenomenon could be attributed in the framework of Einstein’s GR, a propermodification of the theory was required that will successfully incorporate the late cosmic acceleration. As thequest began, two different approaches regarding this modification came into existence.According to the first approach, cosmic acceleration can be phenomenally attributed to the presence of amysterious negative energy component popularly known as dark energy (DE) [6]. Here we modify the right handside of the Einstein’s equation, i.e. in the matter sector of the universe. Latest observational data shows thatthe contribution of DE to the energy sector of the universe is Ω d = 0 .
7. With the passage of time, extensivesearch saw various candidates for DE appear in the scene. Some of the popular ones worth mentioning areChaplygin gas models [7, 8], Quintessence Scalar field [9], Phantom energy field [10], etc. A basic feature ofthese models is that, they violate the strong energy condition i.e., ρ + 3 p <
0, thus producing the observedcosmic acceleration. Recent reviews on DE can be found in [11, 12]A different section of cosmologists resorted to an alternative approach for explaining the expansion. Thisconcept is based on the modification of the gravity sector of GR, thus giving birth to modified gravity theories. [email protected] universe associated with a tiny cosmological constant, i.e. the ΛCDM model served as a prototype for thisapproach. It was seen that the model could satisfactorily explain the recent cosmic acceleration and passed afew solar system tests as well. But with detailed diagnosis it was revealed that the model was paralyzed witha few cosmological problems. Out of these, two major problems that crippled the model till date are the Finetuning problem (FTP) and the Cosmic Coincidence problem (CCP). The FTP refers to the large discrepancybetween the observed values and the theoretically predicted values of cosmological parameters. Numerousattempts to solve this problem can be found in the literature. Among them, the most impressive attemptwas undertaken by Weinberg in [13]. Although the approaches for the solutions are different, yet, almost allof them are basically based on the fact that the cosmological constant may not assume an extremely smallstatic value at all times during the evolution of the universe (as predicted by GR), but its nature should berather dynamical [14]. These drawbacks reduced the effectiveness of the model, as well as its acceptability, andhence alternative modifications of gravity was sought for. Some of the popular models of modified gravity thatcame into existence in recent times are loop quantum gravity [15, 16], Brane gravity [17, 18, 19], f(R) gravity[20, 21, 22], f(T) gravity [23, 24, 25, 26], etc. Reviews on extended gravity theories can be found in [27, 28].In this work we will consider f(G) model as the theory of gravity [34, 35]. Over the years, several modifica-tions to GR have been achieved, by generalizing the Einstein-Hilbert Lagrangian used in GR. F ( R ) and F ( T )gravities are common examples of such modifications. Gauss-Bonnet (GB) modification to GR is another wayof modifying the Einstein gravity that has gained popularity over the past few years, because it is considered asthe low energy limit of string theory. In this modification, one generally adds quadratic terms, specifically theGB terms, which involve second order curvature invariants. But as it turns out to be, the GB term becomestrivial in 4-dimension, and hence, it is used from another form of modified gravity, namely, the modified GBgravity [34]. Here an arbitrary function of the GB term, f ( G ) is added to the Einstein-Hilbert Lagrangian, tobring about the modification.In [36] Bisabr studied cosmological coincidence problem in the background of f(R) gravity. Motivated byBisabr’s work, we dedicate the present assignment to the study of the coincidence problem in f(G) gravity. Thepaper is organized as follows: Basic equations of fG gravity are furnished in section 2. In section 3, we discussthe coincidence problem. The set-up for the present study is discussed in section 4. We illustrate the designedset-up by a few examples in section 5, and finally the paper ends with a short conclusion in section 6. f(G) gravity The 4-dimensional action in f(G) gravity is given by, S = 1 κ Z √− g (cid:20) R f ( G ) (cid:21) d x + S m (1)where R is the Ricci scalar curvature, f ( G ) is a generic function of the Gauss-Bonnet topological invariant G , κ = 8 πG and S m is the matter action.Varying the above action with respect to the metric one can obtain the field equation as, G µν +8 (cid:20) R µρνσ + R ρν g σµ + 12 ( g µν g σρ − g µσ g νρ ) R − R ρσ g νµ − R µν g σρ + R µσ g νρ (cid:21) ∇ ρ ∇ σ F ′ ( G )+( GF ′ ( G ) − F ) g µν = T mµν (2)where G µν is the Einstein tensor, T mµν is the energy momentum tensor of matter. Here we consider κ = 8 πG = 1and prime denotes ordinary derivative with respect to G . For spatially flat Robertson-Walker metric ds = − dt + a ( t ) dx (3)we have R = 6 (cid:16) ˙ H + 2 H (cid:17) , G = 24 H (cid:16) ˙ H + H (cid:17) (4)where H is the Hubble parameter and dot denotes the time derivative. Considering the universe to be filledwith pressureless dark matter under the assumption of a flat universe the modified Friedmann equations for f(G) gravity are, 3 H = GF ′ ( G ) − F − H ˙ F ′ ( G ) + ρ m (5) − H = − H ˙ F ′ ( G ) + 16 H ˙ H ˙ F ′ ( G ) + 8 H ¨ F ′ ( G ) + ρ m (6) he energy conservation equations are given by,˙ ρ m + 3 Hρ m = Q (7)˙ ρ G + 3 H (1 + ω G ) ρ G = − Q (8)Here ω G = p G ρ G is the EoS parameter of the energy sector and Q is the interaction between the matter andthe energy sector of the universe. The EoS parameter is given by, ω G = − (cid:16) H ¨ G + 16 H ˙ H ˙ G − H ˙ G (cid:17) F ′′ + 8 H ˙ G F ′′′ GF ′ − F − H ˙ GF ′′ (9) The cosmic coincidence problem has been a serious issue in recent times regarding various dark energy models.Recent cosmological observations have indicated that the densities of the matter sector and the DE sector of theuniverse are almost identical in late times. It is known that the matter and the energy component of the universehave evolved independently from different mass scales in the early universe, then how come they reconcile toidentical mass scales in the late universe! This is major problem having its roots in the very formation of thetheory. Almost all the DE models known till date more or less suffer from this phenomenon.Various attempts to solve the coincidence problem can be found in literature. Among them the mostimpressive are the ones which use the concept of a suitable interaction between the matter and the dark energycomponents of the universe, as given in the conservation equations (7) and (8). In this approach, it is consideredthat the two sectors of the universe have not evolved independently from different mass scales. Instead theyevolve together, interacting with each other, allowing a mutual flow of matter and energy between the twocomponents. Due to this exchange, the densities of the two components coincide in the present universe.Although the concept seems to be a really promising one, yet a problem persists. There is no universallyaccepted interaction term, introduced by a fundamental theory known till date.It is known that both dark energy and dark matter are not universally accepted facts, but concepts whichare still at the speculation level. Due to this unknown nature of both dark energy and dark matter, it is notpossible to derive an expression for the interaction term ( Q ) from the first principles. Such a situation, demandsus to use our logical reasoning and propose various expressions for Q that will be reasonably acceptable. Thelate time dominating nature of dark energy indicates that Q must be considered a small and positive value.On the other hand a large negative value of interaction will make the universe dark energy dominated from theearly times, thus leaving no scope for the condensation of galaxies. So the most logical choice for interactionshould contain a product of energy density and the hubble parameter, because it is not only physically but alsodimensionally justified. So Q = Q ( Hρ m , Hρ de ), where ρ de is the dark energy density. Since here we are notplanning to add any dark energy by hand, so the effective density resulting from the extra terms of the modifiedGB gravity, ρ G will replace ρ de . This leads us to three basic forms of interactions as given below [42]: b − model : Q = 3 bHρ m η − model : Q = 3 ηHρ G Γ − model : Q = 3Γ H ( ρ m + ρ G ) , (10)where b , η and Γ are the coupling parameters of the respective interaction models.It is worth mentioning that due to its simplicity the most widely used interaction model is the b -model andis available widely in literature [41, 42, 43, 44]. In this note we address the coincidence problem in f(G) gravity. f(G) gravity has evolved over the past decade asa candidate for modified gravity theory. From the literature it is known that f(G) gravity is itself self competentin producing the late cosmic acceleration without resorting to any forms of dark energy. Therefore in order tokeep it simple and reasonable, we do not consider any separate dark energy components in the present study.The extra terms of the modified GB gravity provides the exotic nature and is considered as the dark energy.We consider the ratio of the densities of matter and dark energy as, r ≡ ρ m /ρ G . Our aim is to devise a set-upthat will aim towards a possible solution to the coincidence problem. We also want to set up a filtering process hat will separate the favorable f ( G ) models, that produce a stationary value of the ratio of the componentdensities, r from the unfavorable ones that do not. The time evolution of r is as follows,˙ r = ˙ ρ m ρ G − r ˙ ρ G ρ G (11)Using eqns. (7), (8) and (11), we obtain ˙ r = 3 Hrω G + Qρ G (1 + r ) (12)Using the b-interaction given in eqn.(10), we get the expression for ˙ r as,˙ r = 3 Hr ( b + br + ω G ) (13)where ω G is given by eqn.(9). Now in order to comply with observations, it is required that universe shouldapproach a stationary stage, where either r becomes a constant or evolves slower than the scale factor. In orderto satisfy this ˙ r = 0 in the present epoch, it leads to the following equation, g ( f, H, r s , q ) = 0 (14)where g ( f, H, r s , q ) = 3 Hr s b + ˙ HH + q + br s + 1 (cid:16) − f [ G ] + Gf ′ [ G ] − H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) f ′′ [ G ] (cid:17) × (cid:16)(cid:16)
384 ˙ HH (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) − H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) + 192 H (cid:16) ... H H + 2 ˙ H (cid:16) ˙ H +6 H (cid:1) + 2 ¨ H (cid:16) HH + 2 H (cid:17)(cid:17)(cid:17) f ′′ [ G ] + 4608 H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) f ′′′ [ G ] (cid:19)(cid:21) (15)and r s is the value of r when it takes a stationary value.Using the η -interaction given in eqn.(10), we get the expression for ˙ r as,˙ r = 3 H [ η + r ( η + ω G )] (16)where ω G is given by eqn.(9). In order to satisfy this ˙ r = 0 in the present epoch, it leads to the followingequation, g ( f, H, r s , q ) = 0 (17)where g ( f, H, r s , q ) = 3 H η + r s ˙ HH + q + η + 1 (cid:16) − f [ G ] + Gf ′ [ G ] − H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) f ′′ [ G ] (cid:17) × (cid:16)(cid:16)
384 ˙ HH (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) − H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) + 192 H (cid:16) ... H H + 2 ˙ H (cid:16) ˙ H + 6 H (cid:17) +2 ¨ H (cid:16) HH + 2 H (cid:17)(cid:17)(cid:17) f ′′ [ G ] + 4608 H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) f ′′′ [ G ] (cid:19)(cid:19)(cid:21) (18)Using the Γ-interaction given in eqn.(10), we get the expression for ˙ r as,˙ r = 3 H (cid:2) Γ r + r (2Γ + ω G ) + Γ (cid:3) (19)where ω G is given by eqn.(9). In this case, in order to satisfy ˙ r = 0 in the present epoch, it leads to the followingequation, g ( f, H, r s , q ) = 0 (20) here g ( f, H, r s , q ) = 3 H Γ + r s Γ + r s ˙ HH + q + 2Γ + 1 (cid:16) − f [ G ] + Gf ′ [ G ] − H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) f ′′ [ G ] (cid:17) × (cid:16)(cid:16)
384 ˙ HH (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) − H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) + 192 H (cid:16) ... H H + 2 ˙ H (cid:16) ˙ H + 6 H (cid:17) +2 ¨ H (cid:16) HH + 2 H (cid:17)(cid:17)(cid:17) f ′′ [ G ] + 4608 H (cid:16) ¨ HH + 2 ˙ H (cid:16) ˙ HH + 2 H (cid:17)(cid:17) f ′′′ [ G ] (cid:19)(cid:19)(cid:21) (21)In our analysis we will consider H , r and q as the present day values of H , r and q respectively. As far as q is concerned, we start from the best fit parametrization obtained directly from observational data. Here we usea two parameter reconstruction function for q ( z ) [45, 46] q ( z ) = 12 + q z + q (1 + z ) (22)On fitting this model to Gold data set, we get q = 1 . +1 . − . and q = − . ± .
43 [46]. We consider z = 0 . q ≈ − .
2. From recent observations, we obtain r ≡ ρ m ( z ) ρ T ( z ) ≈ [47, 48, 49]. The present value of Hubble parameter, H is taken as 72, in accordance with the latest observationaldata. We consider the scale-factor, a as a power-law form of time, t as given below, a = a t n (23)In order to illustrate the above set-up we consider three different f ( G ) gravity models found widely inliterature and test them for the coincidence phenomenon. The three models used are [50, 51, 52]: Model1: F ( G ) = αG m + βG ln G (24)where α , β and m are constants, whose values depend on the cosmographic parameters [52]. Model2: F ( G ) = α G m + b α G m + b (25)where α , α , m , b and b are constants. Model3: F ( G ) = a G m (1 + b G m ) (26)where a , m , m and b are constants.Using the model 1, i.e., (24) and eqn. (23) in eqn. (15), we get the following expression for the dynamicalquantity g , g model = 1 t nr b − n + q + br + − m (cid:18) − n t + n t (cid:19) m α − t n n t − n (cid:16) − n t + n t (cid:17) t - interaction0 2. ´ ´ ´ ´ ´ ´ ´ ´ - ´ - ´ - ´ - t g Variation of g against t Fig.1
Fig 1 : The plot of g ( f , H , r s , q ) against time t for model1 (red), model2 (blue) and model3 (green)using b interaction. The other parameters are considered as q = − . , r = 3 / , α = 1 , β = 4 , n =10 , m = 0 . , b = 1 . , a = − , b = − , a = 2 , b = 0 . , m = 1 . , a = − , b = 0 , m = 1 . , m = 0 . − m ( − m ) m (cid:18) − n t + n t (cid:19) − m α + β (cid:0) − n t + n t (cid:1) ! − (cid:18) − n t + n t (cid:19) βLog (cid:20) (cid:18) − n t + n t (cid:19)(cid:21) +24 (cid:18) − n t + n t (cid:19) − m m (cid:18) − n t + n t (cid:19) − m α + β + βLog (cid:20) (cid:18) − n t + n t (cid:19)(cid:21)!! − × t n n t − n (cid:16) − n t + n t (cid:17) t − m ( − m )( − m ) m (cid:18) − n t + n t (cid:19) − m α − β (cid:0) − n t + n t (cid:1) ! + − n (cid:18) n t − n (cid:0) − n t + n t (cid:1) t (cid:19) t − n (cid:18) n t − n (cid:0) − n t + n t (cid:1) t (cid:19) t + 192 n (cid:18) − n t + n (cid:0) − nt + n t (cid:1) t + n (cid:0) − n t + n t (cid:1) t (cid:19) t − m ( − m ) m (cid:18) − n t + n t (cid:19) − m α + β (cid:0) − n t + n t (cid:1) !! (27)Similarly expressions for g is obtained for the other two f ( G ) models. Expressions for g and g are also foundfor all the three gravity models. As it can be seen from above that the expressions are really lengthy, so we donot include all of them in the manuscript.We have generated plots for g , g and g against cosmic time, t for each of the three models in the figures 1, 2and 3, for all the three forms of interactions, b , η and Γ. Particular numerical values for the involved parametershave been considered which are in accordance with the recent observational data [50, 51, 52]. Moreover in figs4 and 5, we have illustrated two non-conventional models of f ( G ) gravity i.e., the logarithmic model and theexponential model, which gives very interesting results when used in the designed set-up. - interaction0 2. ´ ´ ´ ´ ´ ´ ´ ´ - ´ - ´ - ´ - ´ - ´ - ´ - ´ - t g Variation of g against t Fig.2
Fig 2 : The plot of g ( f , H , r s , q ) against time t for model1 (red), model2 (blue) and model3 (green)using η interaction. The other parameters are considered as q = − . , r = 3 / , α = 1 , β = 4 , n =10 , m = 0 . , η = 1 . , a = − , b = − , a = 2 , b = 0 . , m = 1 . , a = − , b = 0 , m = 1 . , m = 0 . G- interaction0 2. ´ ´ ´ ´ ´ ´ ´ g Variation of g against t Fig.3
Fig 3 : The plot of g ( f , H , r s , q ) against time t for model1 (red), model2 (blue) and model3(green) using Γ interaction. The other parameters are considered as q = − . , r = 3 / , α = 1 , β =4 , n = 10 , m = 0 . , Γ = 5 × , a = − , b = − , a = 2 , b = 0 . , m = 1 . , a = − , b = 0 , m =1 . , m = 0 . - interaction Η- interaction G- interactioncoincidence5 10 15 20 - g Variation of g against t for the model f H G L = Σ Log @ G D +Τ Fig.4
Fig 4 : The plot of g against time t for different forms of interaction for the Logarithmic model, f ( G ) = σLog ( G ) + τ . The other parameters are considered as q = − . , r = 3 / , σ = 5 , τ = 0 . , n =3 , b = 1 . , η = 1 . , Γ = 1 . . b - interaction Η- interaction G- interactioncoincidence5 10 15 20 - g Variation of g against t for the model f H G L = Λ e ∆ G Fig.5
Fig 5 : The plot of g against time t for different forms of interaction for the Exponential model, f ( G ) = λ exp( δG ) . The other parameters are considered as q = − . , r = 3 / , λ = 2 , δ = 0 . , n =3 , b = 1 . , η = 1 . , Γ = 1 . . Discussion and conclusion
From the figures it is evident that the g vs t curves become asymptotic near the time axis, when the cosmictime corresponds to the age of the universe, i.e. 14 × years. As a result of this, g never reaches the zerolevel. Hence ˙ r = 0, makes the realization of a stationary phase extremely difficult. The asymptotic nature ofthe curves are indicative of the fact that as time evolves the trajectories move closer and closer to the time axis.Therefore for the given models, the coincidence problem is substantially alleviated with the evolution of time,but never ever gets solved. This is truly a set back for the models which are known to satisfy most of the solarsystem tests.But from the set-up that we have designed in this assignment, we can generate as well as filter various modelsof f ( G ) gravity which are completely free from the coincidence problem. Two such models have been illustratedin the figs.4 and 5. In fig.4, we have generated the plot of g vs t , for the logarithmic model ( f ( G ) = σLog ( G )+ τ )for all the three interaction terms. It can be seen from the plot that g reaches the zero level for all the interactionterms at around t = 8. Particularly for the b-interaction, the stationary scenario is realized for t >
11. In fig.5,a similar plot has been generated for the exponential model ( f ( G ) = λ exp( δG )). From the plot, it is evidentthat a stationary scenario is achieved at around t = 6 for all the interactions. Particularly for the b-interactiona continuous stationary scenario is realized for t ≥
7. So from the above discussion it is quite clear that forthe logarithmic and the exponential models a complete solution for the coincidence problem can be achievedfollowing the set-up that we have designed in the present assignment. Looking at the plots 4 and 5, it must alsobe mentioned that the b-interaction term is much more preferable compared to the other interactions, since ithelps us to realize a continuous stationary phase between dark energy and dark matter after a certain point oftime in the cosmological time-line, thus complying with the recent observational data.
Acknowledgement:
The author sincerely acknowledges the anonymous referee for his or her constructive comments which helpedthe author to improve the quality of the manuscript.
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