PRAMANA c Indian Academy of Sciences Vol. 84, No. 4 — journal of April 2015 physics pp. 503–516 Ghost quintessence in fractal gravity HABIB ABEDI1 and MUSTAFA SALTI2,∗ 1Department of Physics, University of Tehran, North Kargar Ave, Tehran, Iran 2Physics Department of Science Faculty, Dicle University, 21280, Diyarbakir Turkey ∗Corresponding author. E-mail: [email protected] MS received 24 October 2013; revised 24 February 2014; accepted 13 May 2014 DOI: 10.1007/s12043-014-0850-2; ePublication: 15 November 2014 Abstract. In this study, using the time-like fractal theory of gravity, we mainly focus on the ghost dark energy model which was recently suggested to explain the present acceleration of the cosmic expansion. Next, we establish a connection between the quintessence scalar field and fractal ghost dark energy density. This correspondence allows us to reconstruct the potential and the dynamics of a fractal canonical scalar field (the fractal quintessence) according to the evolution of ghost dark energy density. Keywords. Ghost dark energy; quintessence; fractal gravity. PACS Nos 04.50.+h; 95.35.+d 1. Introduction Recent cosmological observations give very important evidences in favour of the present acceleration of cosmic expansion. It is commonly believed that our Universe has a phase transition [1] from decelerating phase to accelerating phase and expands with accelerat- ing velocity. To explain this acceleration, in the context of modern cosmology, we need an antigravity fluid with negative pressure [2]. This interesting feature of the Universe is caused by the mysterious dark components: dark energy, dark matter, and dark radiation. The cosmological constant, as vacuum energy density, is the best instrument to identify this nature of the Universe. This, with the equation-of-state parameter ω =−1, repre- sents the earliest and the simplest theoretical candidate for dark energy, but causes some constraints like fine-tuning and cosmic-coincidence puzzle [3]. The former cosmologists question why the vacuum energy density is so small, [4] and the latter ones explain why the vacuum energy and dark matter are nearly equal [5]. Furthermore, according to Type Ia supernovae observations, it is now known that the time-varying dark energy models give better fits compared with a cosmological constant [2] and in particular, the value of the equation-of-state parameter of dark energy (ωD) gives three different phases: vac- uum (ωD =−1), phantom (ωD < −1) and quintessence (ωD > −1). Also, many other Pramana – J. Phys., Vol. 84, No. 4, April 2015 503 Habib Abedi and Mustafa Salti candidates (tachyon, K-essence, quintessence, dilaton, Chaplygin gas, modified gravity) have been proposed to explain the nature of dark energy [6,7], but yet the nature of dark Universe is completely unknown [8]. A good review about the dark energy problem, including a survey of some theoretical models, is given by Li et al in 2011 [9]. In previous works, a very interesting interpretation on the origin of dark energy is sug- gested, without new degrees of freedom, with the dark energy of the right magnitude to obtain the observed expansion [10–14]. Among various models, the new model of dark energy called Veneziano ghost dark energy [15] is supposed to solve the U(1)A problem in low-energy effective theory of QCD [16–18], but it is completely decoupled from the physical sector [19–21]. The Veneziano ghost field is unphysical in the quantum field the- ory in Minkowski space-time, but exhibits an important non-trivial physical influence in the expanding Universe and this remarkable effect gives rise to the vacuum energy density ∼ 3 ∼ −3 4 ∼ −33 ∼ ρD HQCD (10 eV) (with H 10 eV and QCD 100 eV we have the right magnitude for the force accelerating the Universe) [22]. This numerical coincidence is noteworthy and also means that the ghost energy model overcomes the problem of fine tuning. On the other hand, scalar fields can be regarded as an effective way of describing the dark Universe which naturally arise in particle physics, including the string/M theory and supersymmetric field theories. Hence scalar fields are expected to reveal the dynam- ical mechanism and the nature of the dark Universe [2]. Fundamental theories such as string/M theory provide many possible scalar field candidates, but they do not predict their potential V(φ)uniquely. In this work, we are interested to consider the ghost dark energy model as the under- lying theory of dark energy in time-like fractal gravity and how the low-energy effective scalar field model can be used to describe it. For this purpose, we reconstructed the poten- tial and the dynamics of the fractal quintessence according to the results obtained for the ghost dark energy. We can establish a correspondence between the ghost dark energy and quintessence scalar field in time-like fractal gravity, and describe ghost dark energy effectively by making use of quintessence. 2. Correspondence between ghost and quintessence We assume the ghost dark energy is accommodated in a flat Friedmann–Robertson– Walker ds2 =−dt2 + a2(t) dr2 + r2(dθ 2 + sin2 θdφ2) . (1) Here a(t) is the cosmic scale factor and it measures the expansion of the Universe. In four-dimensional time-like fractal gravity, we have [23,24] S = SG + Sm, (2) where 1 √ S = dξ −g[R − η∂ υ∂μυ], (3) G 2κ2 μ √ Sm = dξ −g£m, (4) 504 Pramana – J. Phys., Vol. 84, No. 4, April 2015 Ghost quintessence in fractal gravity where g, R and £m are the determinants of metric gμν , Ricci scalar and the matter part of total Lagrangian, respectively. Also, we have κ2 = 8πG, where G denotes the grav- itational constant. υ and η are two quantities known as the fractal function and fractal parameter, respectively. It is important to mention here that dξ(x) is the Lebesgue– Stieltjes measure generalizing the standard four-dimensional measure d4x. The dimension of ξ is [ξ]=−Dα, where α is a positive parameter. The theory of fractal gravity is power- counting, renormalizable, free from ultraviolet divergence and a Lorentz invariant [25]. Recently, Calcagni [23,24] worked on the quantum gravity in a fractal Universe and has discussed cosmology in that framework. Considering a time-like fractal profile [24] υ = t−β (where β = 4(1 − α) is the fractal dimension) in four-dimensional (D = 4) fractal gravity, we recover the following Friedmann equation: ηβ 2 1 H 2 − βHt−1 + = (ρ + ρ ), (5) 2(β+1) 2 G m 6t 3MP where ρG and ρm are the densities of the ghost dark energy and dark matter inside the Universe, respectively. Here, we assume a pressureless dark matter pm = 0, MP is the −2 = =˙ reduced Planck mass (MP 8πG) and H a/a is the Hubble parameter. Accordingly, it is known that β = 0 describes the infrared regime while β = 2 implies the ultraviolet regime. On the other hand, the continuity equation is written as ρ˙ + (3H − βt−1)(ρ + p) = 0, (6) where ρ and p are the total energy and pressure density, respectively. Nonetheless, the gravitational constraint [24] in a flat fractal Friedmann–Robertson– Walker space-time is 3η βH β(β + 1) ηβ(2β + 1) H˙ + 3H 2 + 2 + − − = 0. (7) t2β t t2 t2β+2 Note that in the infrared regime, eq. (5) gives the corresponding relation in Einstein’s theory of general relativity (there is no gravitational constraint). Also, the gravitational constraint in the ultraviolet regime becomes 3η 2H 6 10η H˙ + 3H 2 + 2 + − − = 0. (8) t4 t t2 t6 Solving this equation gives [24] 22η (15/4; 17/4; 3η/2t4) H(t) =−2t−1 − , (9) 13t5 (11/4; 13/4; 3η/2t4) 11 13 3η a3(t) = t−6 ; ; . (10) 4 4 2t4 Here (also denoted as 1F1 or M) is Kummer’s confluent hypergeometric function of the first kind: +∞ (b) (a + n) xn (a; b; x) ≡ . (11) (a) (b + n) n! n=0 Pramana – J. Phys., Vol. 84, No. 4, April 2015 505 Habib Abedi and Mustafa Salti Defining the following dimensionless density parameters, ρ = G , (12) G 2 2 3H MP ρ = m , (13) m 2 2 3HMP β H ηβ = − , (14) f H 2 t 6t2(β+1) we can rewrite the fractal Friedmann equation as 1 = G + m + f. (15) Here, the parameter f represents the fractal contribution to the total density. Moreover, the Friedman equation can also be rewritten in a very elegant form as i ≡ 1, (16) i=G,m,f where i ≡ (G,f,m). (17) 2.1 Non-interacting case For this case, the conservation equations read as β ρ˙ + 3H − ρ = 0, (18) m t m β ρ˙ + 3H − (1 + ω )ρ = 0, (19) G t G G where ωG = pG/ρG. The ghost energy density is proportional to the Hubble parame- ter [26] ρG = λH. (20) 3 ∼ Here λ is a constant of order QCD and QCD 100 MeV is the QCD mass scale. Taking a time derivative in both sides of relation (20) and using Friedmann eq. (5), we obtain ηβ 2(β + 1) βH ρ ρ˙ = λ(2H −βt−1)−1 − − G (3H −βt−1)(1+ω +) , G 2β+3 2 2 G 3t t 3MP (21) where ρ = m . (22) ρG Inserting this relation in continuity eq. (19) and using expressions of dimensionless density parameters, we find ηβ 2(β+1) − β − − β − − 3H 2t2β+3 Ht2 3H t (1 G f) ωG =−1 + . (23) − β − − β 3H t G 2 t 506 Pramana – J. Phys., Vol. 84, No. 4, April 2015 Ghost quintessence in fractal gravity In the fractal infrared regime, considering eq. (23), we get IR = − −1 ωG (G 2) .