Holographic dark energy with varying gravitational constant in Horava-Lifshitz cosmology
aa r X i v : . [ h e p - t h ] J a n Holographic dark energy with varying gravitational constant inHoˇrava-Lifshitz cosmology ∗ M. R. Setare † and Mubasher Jamil ‡ Department of Physics, University of Kurdistan, Pasdaran Ave., Sanandaj, Iran Center for Advanced Mathematics and Physics,National University of Sciences and Technology, Rawalpindi, 46000, Pakistan
Abstract:
We investigate the holographic dark energy scenario with a varyinggravitational constant in a flat background in the context of Hoˇrava-Lifshitz grav-ity. We extract the exact differential equation determining the evolution of the darkenergy density parameter, which includes G variation term. Also we discuss a cos-mological implication of our work by evaluating the dark energy equation of statefor low redshifts containing varying G corrections. I. INTRODUCTION
Observational data indicates that our universe is currently under accelerating expansion[1–3]. This acceleration implies that if Einstein’s theory of gravity is reliable on cosmologicalscales, then our universe is dominated by a mysterious form of energy. This unknown energycomponent possesses some strange features, for example it is not clustered on large lengthscales and its pressure must be negative so that can drive the current acceleration of theuniverse. Since the fundamental theory of nature that could explain the microscopic physicsof DE is unknown at present, phenomenologists take delight in constructing various modelsbased on its macroscopic behavior.Despite the lack of a quantum theory of gravity, we can still make some attempts to probethe nature of dark energy according to some principles of quantum gravity. An interestingattempt in this direction is the so-called “holographic dark energy” (HDE) proposal [4–7]. ∗ M. R. Setare dedicated this paper to the 70 year Jubilee of Professor Farhad Ardalan † Electronic address: [email protected] ‡ Electronic address: [email protected]
The HDE is defined by ρ Λ = 3 c πGL . (1)Here c is a holographic parameter of order unity and G is a gravitational ‘constant’. In thischapter, we shall treat G as a variable. The HDE paradigm has been constructed in the lightof holographic principle of quantum gravity [8], and thus it presents some interesting featuresof an underlying theory of dark energy. Furthermore, it may simultaneously provide asolution to the coincidence problem, i.e why matter and dark energy densities are comparabletoday although they obey completely different equations of motion [6]. The holographic darkenergy model has been extended to include the spatial curvature contribution [9]. Lastly, ithas been tested and constrained by various astronomical observations [10].Recently, a power-counting renormalizable, ultra-violet (UV) complete theory of gravity wasproposed by Hoˇrava in [11–14]. Although presenting an infrared (IR) fixed point, namelyGeneral Relativity, in the UV the theory possesses a fixed point with an anisotropic, Lifshitzscaling between time and space of the form x → ℓ x , t → ℓ z t , where ℓ , z , x and t are thescaling factor, dynamical critical exponent, spatial coordinates and temporal coordinate,respectively.Due to these novel features, there has been a large amount of effort in examining andextending the properties of the theory itself [15–27]. Additionally, application of Hoˇrava-Lifshitz gravity as a cosmological framework gives rise to Hoˇrava-Lifshitz cosmology, whichproves to lead to interesting behavior [28, 29]. In particular, one can examine specific solutionsubclasses [30–32], the perturbation spectrum [33–37], the gravitational wave production[38, 39], the matter bounce [40–42], the black hole properties [43–48], the dark energyphenomenology [49–52], the astrophysical phenomenology [53, 54] etc. However, despitethis extended research, there are still many ambiguities if Hoˇrava-Lifshitz gravity is reliableand capable of a successful description of the gravitational background of our world, as wellas of the cosmological behavior of the universe [19–21].In the present work we are interested to study the Holographic dark energy in framework ofHoˇrava-Lifshitz gravity. We extend our analysis with considering the time variable Newton’sconstant G . Until now, in most the investigated dark energy models a constant Newton’s“constant” G has been considered. However, there are significant indications that G canby varying, being a function of time or equivalently of the scale factor [55]. In particular,observations of Hulse-Taylor binary pulsar [56, 57], helio-seismological data [58] and astro-seismological data from the pulsating white dwarf star G117-B15A [59] lead to (cid:12)(cid:12)(cid:12) ˙ G/G (cid:12)(cid:12)(cid:12) / . × − yr − , for z . . G has some theoretical advantagestoo, alleviating the dark matter problem [61], the cosmic coincidence problem [62] and thediscrepancies in Hubble parameter value [63].The plan of the paper is as follows: In the second section we shall present a brief overviewof Hoˇrava-Lifshitz cosmology. In third section we construct the holographic dark energywith varying gravitational constant and extract the differential equation that determinesthe evolution of dark energy parameter. In section four, we use these expressions in orderto calculate the corrections to the dark energy equation of state for low redshift. Finally insection five, we briefly discuss our results. II. HO ˇRAVA-LIFSHITZ COSMOLOGYA. Dark-matter field formulation
We begin with a brief review of Hoˇrava-Lifshitz gravity. The dynamical variables are thelapse and shift functions, N and N i respectively, and the spatial metric g ij (roman lettersindicate spatial indices). In terms of these fields the full metric is ds = − N dt + g ij ( dx i + N i dt )( dx j + N j dt ) , (2)where indices are raised and lowered using g ij . The scaling transformation of the coordinatesreads ( z = 3): t → l t and x i → lx i . (3)
1. Detailed Balance
Decomposing the gravitational action into a kinetic and a potential part as S g = R dtd x √ gN ( L K + L V ), and under the assumption of detailed balance [13] (the extensionbeyond detail balance will be performed later on), which apart form reducing the possibleterms in the Lagrangian it allows for a quantum inheritance principle [11] (the D + 1 di-mensional theory acquires the renormalization properties of the D-dimensional one), the fullaction of Hoˇrava-Lifshitz gravity is given by S g = Z dtd x √ gN (cid:26) κ ( K ij K ij − λK ) −− κ w C ij C ij + κ µ w ǫ ijk √ g R il ∇ j R lk − κ µ R ij R ij ++ κ µ − λ ) (cid:20) − λ R + Λ R − (cid:21)(cid:27) , (4)where K ij = 12 N ( ˙ g ij − ∇ i N j − ∇ j N i ) , (5)is the extrinsic curvature and C ij = ǫ ijk √ g ∇ k (cid:0) R ji − Rδ ji (cid:1) , (6)is the Cotton tensor, and the covariant derivatives are defined with respect to the spatialmetric g ij . ǫ ijk is the totally antisymmetric unit tensor, λ is a dimensionless constant and Λis a negative constant which is related to the cosmological constant in the IR limit. Finally,the variables κ , w and µ are constants with mass dimensions −
1, 0 and 1, respectively.In order to add the dark-matter content in a universe governed by Hoˇrava gravity, a scalarfield is introduced [28, 29], with action: S m ≡ S φ = Z dtd x √ gN " λ −
14 ˙ φ N + m m φ ∇ φ −− m φ ∇ φ + 12 m φ ∇ φ − V ( φ ) (cid:21) , (7)where V ( φ ) acts as a potential term and m i are constants. Although one could just followa hydrodynamical approximation and introduce straightaway the density and pressure of amatter fluid [20], the field approach is more robust, especially if one desires to perform aphase-space analysis.Now, in order to focus on cosmological frameworks, we impose the so called projectabilitycondition [19] and use an FRW metric, N = 1 , g ij = a ( t ) γ ij , N i = 0 , (8)with γ ij dx i dx j = dr − kr + r d Ω , (9)where k = − , , φ ≡ φ ( t ). By varying N and g ij , we obtainthe equations of motion: H = κ − (cid:20) −
14 ˙ φ + V ( φ ) (cid:21) ++ κ − (cid:20) − κ µ k λ − a − κ µ Λ λ − (cid:21) ++ κ µ Λ k λ − a , (10)˙ H + 32 H = − κ − (cid:20) −
14 ˙ φ − V ( φ ) (cid:21) −− κ − (cid:20) − κ µ k λ − a + 3 κ µ Λ λ − (cid:21) ++ κ µ Λ k λ − a , (11)where we have defined the Hubble parameter as H ≡ ˙ aa . Finally, the equation of motion forthe scalar field reads: ¨ φ + 3 H ˙ φ + 23 λ − dV ( φ ) dφ = 0 . (12)At this stage we can define the energy density and pressure for the scalar field responsiblefor the matter content of the Hoˇrava-Lifshitz universe: ρ m ≡ ρ φ = 3Λ −
14 ˙ φ + V ( φ ) (13) p m ≡ p φ = 3Λ −
14 ˙ φ − V ( φ ) . (14)Concerning the dark-energy sector we can define ρ Λ ≡ − κ µ k λ − a − κ µ Λ λ −
1) (15) p Λ ≡ − κ µ k λ − a + 3 κ µ Λ λ − . (16)The term proportional to a − is the usual “dark radiation term”, present in Hoˇrava-Lifshitzcosmology [28, 29]. Finally, the constant term is just the explicit (negative) cosmologicalconstant. Therefore, in expressions (15),(16) we have defined the energy density and pressurefor the effective dark energy, which incorporates the aforementioned contributions.Using the above definitions, we can re-write the Friedmann equations (10),(11) in thestandard form: H = κ λ − h ρ m + ρ Λ i + βka , (17)˙ H + 32 H = − κ λ − h p m + p Λ i + βk a . (18)In these relations we have defined κ = 8 πG and β ≡ κ µ Λ8(3 λ − , which is the coefficient of thecurvature term. Additionally, we could also define an effective Newton’s constant and aneffective light speed [28, 29], but we prefer to keep κ − in the expressions, just to makeclear the origin of these terms in Hoˇrava-Lifshitz cosmology. Finally, note that using (12)it is straightforward to see that the aforementioned dark matter and dark energy quantitiesverify the standard evolution equations:˙ ρ m + 3 H ( ρ m + p m ) = 0 , (19)˙ ρ Λ + 3 H ( ρ Λ + p Λ ) = 0 . (20) III. HOLOGRAPHIC DARK ENERGY WITH VARYING GRAVITATIONALCONSTANT IN A FLAT BACKGROUND
Let us construct holographic dark energy scenario allowing for a varying Newton’s con-stant G . The space-time geometry will be a flat Robertson-Walker: ds = − dt + a ( t ) ( dr + r d Ω ) , (21)with a ( t ) the scale factor and t the comoving time. As usual, the first Friedmann equationreads: H = 8 πG λ − ρ (22)with H the Hubble parameter, ρ = ρ m + ρ Λ , ρ m = ρ m a − , where ρ m and ρ Λ stand respectivelyfor matter and dark energy densities and the index 0 marks the present value of a quantity.Furthermore, we will use the density parameter Ω Λ ≡ πG H ρ Λ , which, imposing explicitly theholographic nature of dark energy according to relation , becomesΩ Λ = c H L . (23)Finally, in the case of a flat universe, the best choice for the definition of L is to identify itwith the future event horizon [6], that is L ≡ R h ( a ) with R h ( a ) = a Z ∞ t dt ′ a ( t ′ ) = a Z ∞ a da ′ Ha ′ . (24)In the following we will use ln a as an independent variable. Thus, denoting by dot thetime-derivative and by prime the derivative with respect to ln a , for every quantity F weacquire ˙ F = F ′ H . Differentiating (23) using (24), and observing that ˙ R h = HR h −
1, weobtain: Ω ′ Λ Ω = 2Ω Λ h − − ˙ HH + √ Ω Λ c i . (25)Until now, the varying behavior of G has not become manifested. However, the next step isto eliminate ˙ H . This can be obtained by differentiating Friedman equation, leading to˙ H = 2 π λ − H ( ˙ Gρ + G ˙ ρ ) , (26)where G is considered to be a function of t . Using˙ G = G ′ H, (27)and the energy conservation equation˙ ρ = − H (1 + ω ) ρ, (28)where ω is ω = ω Λ ρ Λ ρ = ω Λ Ω Λ λ − . (29)The EoS parameter for HDE is given by ω Λ = − (cid:16)
13 + 2 √ Ω Λ c (cid:17) (30)Eq. (26) becomes 2 ˙ HH = ∆ G − ω ) , (31)where ∆ G ≡ G ′ /G is a dimensionless number.2 ˙ HH = ∆ G − h − Ω Λ λ − (cid:16)
13 + 2 √ Ω Λ c (cid:17)i , (32)Using (32) in (25) we get finallyΩ ′ Λ Ω = 2Ω Λ h − ∆ G √ Ω Λ c + Ω Λ λ −
1) + Ω / c (3 λ − i (33) IV. COSMOLOGICAL IMPLICATIONS
What we are interested in most is the prediction about the equation of state at thepresent time. Since we have extracted the expressions for Ω ′ Λ , we can calculate w ( z ′ ) forsmall redshifts z ′ , performing the standard expansions of the literature. In particular, since ρ Λ ∼ a − w ) we acquire after expanding ρ Λ asln ρ Λ = ln ρ + d ln ρ Λ d ln a ln a + 12 d ln ρ Λ d (ln a ) (ln a ) + . . . , (34)where the derivatives are taken at the present time a = 1 (and thus at Ω Λ = Ω ). Then, w (ln a ) is given as w (ln a ) = − − (cid:20) d ln ρ Λ d ln a + 12 d ln ρ Λ d (ln a ) ln a (cid:21) , (35)up to second order. In addition, we can straightforwardly calculate w ( z ′ ), replacing ln a = − ln(1 + z ′ ) ≃ − z ′ , which is valid for small redshifts, defining w ( z ) = − − (cid:18) d ln ρ Λ d ln a (cid:19) + 16 (cid:20) d ln ρ Λ d (ln a ) (cid:21) z ′ ≡ w + w z ′ . (36)Since ρ Λ = 3 H Ω Λ / (8 πG ) = Ω Λ ρ m / Ω m = ρ m Ω Λ / (1 − Ω Λ ) a − , the derivatives are easilycomputed using the obtained expressions for Ω ′ Λ . Hence we get ω = − − Ω ′ Λ Ω Λ (1 − Ω Λ ) , (37) ω = 16 h Ω ′′ Λ Ω Λ (1 − Ω Λ ) − Ω ′ (1 − Ω Λ ) (1 + 2Ω Λ + 2Ω ) i . (38)Using Eq. (33) in (37) and (38), we get ω = − − − Ω Λ h − ∆ G + √ Ω Λ c + Ω Λ λ −
1) + Ω / c (3 λ − i . (39) ω = 16 h − Ω Λ (cid:16) − ∆ G + √ Ω Λ c + Ω Λ λ −
1) + Ω / c (3 λ − (cid:17) × (cid:16) − ∆ G + 3 √ Ω Λ c + Ω Λ λ − / c (3 λ − (cid:17) − Ω (1 − Ω Λ ) (1 + 2Ω Λ + 2Ω ) × (cid:16) − ∆ G + √ Ω Λ c + Ω Λ λ −
1) + Ω / c (3 λ − (cid:17)i . (40)Eqs. (39) and (40) together determine the linear equation of state of dark energy. In orderto choose their numerical values and fit them to the observational data, we require a suitablevalue of λ . None of the numerical values i.e. λ = 1 , / ∞ gives a good estimate. Hencewe think that the appropriate value of λ should be deduced from the observational data. V. CONCLUSIONS
Astrophysical observations suggest that dark energy state parameter must be variableand changing over cosmic time. In this connection, a proposed dark energy candidate isthe holographic dark energy. The HDE naturally represents a variable form of dark energyby involving a parameter c in it. The choice c > c >
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