Cent. Eur. J. Phys. • 12(4) • 2014 • 245-255 DOI: 10.2478/s11534-014-0441-3 Central European Journal of Physics Planck-scale corrections to Friedmann equation Research Article Adel Awad∗1,2,3, Ahmed Farag Ali† 1,4 1 Centre for Fundamental Physics, Zewail City of Science and Technology Sheikh Zayed, 12588, Giza, Egypt 2 Department of Physics, Faculty of Science, Ain Shams University, Cairo, 11566, Egypt 3 Center for Theoretical Physics, British University of Egypt, Sherouk City 11837, P.O. Box 43, Egypt 4 Department of Physics, Faculty of Science, Benha University, Benha 13518, Egypt Received 16 November 2013; accepted 18 January 2014 Abstract: Recently, Verlinde proposed that gravity is an emergent phenomenon which originates from an entropic force. In this work, we extend Verlinde’s proposal to accommodate generalized uncertainty principles (GUP), which are suggested by some approaches to quantum gravity such as string theory, black hole physics and doubly special relativity (DSR). Using Verlinde’s proposal and two known models of GUPs, we obtain modifications to Newton’s law of gravitation as well as the Friedmann equation. Our modification to the Friedmann equation includes higher powers of the Hubble parameter which is used to obtain a cor- responding Raychaudhuri equation. Solving this equation, we obtain a leading Planck-scale correction to Friedmann-Robertson-Walker (FRW) solutions for the p = ωρ equation of state. PACS (2008): 04.60.Bc;04.60.Cf;04.70.Dy Keywords: quantum gravity • entropic force • thermodynamics • black holes © Versita sp. z o.o. 1. Introduction mutation relations between position coordinates and mo- menta [1–8]. This can be understood in the context of string theory since strings can not interact at distances smaller than their size. The GUP is represented by the following One of the intriguing predictions of various frame works form [9–11]: of quantum gravity, such as string theory and black ~ hole physics, is the existence of a minimum measurable ∆xi∆pi ≥ [1 + β (∆p)2+ < p >2 2 length. This has given rise to the so-called generalized un- 2 2 + 2β ∆pi + < pi > ] ; (1) certainty principle (GUP) or equivalently, modified com- `2 p2 P p p β β / M c 2 β p M where = j j , = 0 ( p ) = 0 2 , p = Planck ∗ j ~ E-mail: [email protected] † E-mail: [email protected] (Corresponding author) mass, and Mpc2 = Planck energy. The inequality (1) is 245 Brought to you by | CERN library Authenticated Download Date | 10/4/17 1:58 PM Planck-scale corrections to Friedmann equation equivalent to the following modified Heisenberg algebra scales are only a few orders of magnitude less than Planck [9]: scale. [xi; pj ] = i~(δij + βδij p2 + 2βpipj ) : (2) Let us review Verlinde’s proposal [38] on the origin of grav- ity, where he suggested that a gravitational force might This form ensures, via the Jacobi identity, that be of an entropic nature. This assumption is based on the x ; x p ; p [ i j ] = 0 = [ i j ] [10]. relation between gravity and thermodynamics [40–42]. Ac- cording to thermodynamics and the holographic principle, Recently, a new form of GUP was proposed in [12, Verlinde’s approach results in Newton’s law of gravitation, 13], which predicts maximum observable momenta as well the Einstein Equations [38] and the Friedmann Equations as the existence of minimal measurable length, and is con- [43]. The theory can be summarized as sistent with doubly special relativity theories (DSR) [14– at the temperature T , the entropic force F of a gravita- 19], string theory and black holes physics [1–7]. It ensures tional system is given as: [xi; xj ] = 0 = [pi; pj ], via the Jacobi identity. F∆x = T ∆S; (4) h p p x ; p i δ − α pδ i j [ i j ]= ~ ij ij + p where ∆S is the change in the entropy such that, at a i x + α2 p2δij + 3pipj ; (3) displacement ∆ , each particle carries its own portion of entropy change. From the correspondence between the entropy change ∆S and the information about the bound- α α /Mpc α `p/~;Mp `p where = 0 = 0 = Planck mass, = ary of the system and using Bekenstein’s argument [40– Mpc2 Planck length, and = Planck energy. In a series 42], it is assumed that ∆S = 2πkB, where ∆x = ~/m and of papers, various applications of the new model of GUP kB is the Boltzmann constant. have been investigated [20–33]. mc The upper bounds on the GUP parameter α has been ∆S = 2πkB ∆x; (5) derived in [22]. It was suggested that these bounds can ~ be measured using quantum optics and gravitational wave where m is the mass of the elementary component, c is techniques in [34, 35]. Recently, Bekenstein [36, 37] pro- speed of light and ~ is the Planck constant. posed that quantum gravitational effects could be tested The holographic principle assumes that for a region en- experimentally, suggesting“a tabletop experiment which, closed by some surface gravity can be represented by the given state of the art ultrahigh vacuum and cryogenic degrees of freedom on the surface itself and independent technology, could already be sensitive enough to detect of the its bulk geometry. This implies that the quantum Planck scale signals” [36]. This would enable several gravity can be described by a topological quantum field quantum gravity predictions to be tested in the Laboratory theory, for which all physical degrees of freedom can be [34, 35]. This is considered as a milestone in the field of projected onto the boundary [44]. The information about quantum gravity phenomenology. the holographic system is given by N bits forming an ideal Motivated by the above arguments, we investigate pos- gas. It is conjectured that N is proportional to the entropy sible effects of GUP on the Friedmann equation. We use of the holographic screen, the entropic force approach suggested by Verlinde [38] to calculate corrections to Newton’s law of gravitation and the Friedmann equations for two models of GUP, which 4S N = ; (6) are mentioned above. We found that Planck-scale cor- kB rections to the Friedmann equation include higher pow- ers of the Hubble parameter which are suppressed by then according to Bekenstein’s entropy-area relation [40– Planck length. Using these corrections we construct the 42] corresponding Raychaudhuri equations, which we then kBc3 S = A: (7) solve to obtain a leading Planck-scale correction to the 4G~ Friedmann-Robertson-Walker (FRW) solutions with equa- Therefore, one gets tion of state p = ωρ. Deriving the effects of GUPs on Newton’s law of gravitation through the entropic force Ac3 4πr2c3 N = = ; (8) approach was initially reported in [39] where a modi- G~ G~ fied Newton’s law of gravitation was calculated. A possi- ble application of these Planck-scale corrections is early where r is the radius of the gravitational system and A = cosmology, in particular inflation models where physical 4πr2 is area of the holographic screen. It is assumed that 246 Brought to you by | CERN library Authenticated Download Date | 10/4/17 1:58 PM Adel Awad, Ahmed Farag Ali each bit emerges out of the holographic screen i.e. in one 2. GUP-quadratic in ∆p and BH dimension. Therefore each bit carries an energy equal thermodynamics to kBT/2. Using the equipartition rule to calculate the energy of the system, one gets In this section, we review the modified thermodynamics of the black hole which yields a modified entropy due to πc3r2 E 1 Nk T 2 k T Mc2: GUP [56–62]. Using the holographic principle, we get a = B = G B = (9) 2 ~ modified number of bits N which yields quantum gravity corrections to Newton’s law of gravitation and the Fried- By substituting Eq. (4) and Eq. (5) into Eq. (9), we get mann equations. Black holes are considered as a good laboratories for the Mm clear connection between thermodynamics and gravity, so F G ; = r2 (10) black hole thermodynamics will be analyzed in this sec- tion. We then make an analysis of BH thermodynamics p making it clear that Newton’s law of gravitation can be if the GUP-quadratic in ∆ that was proposed in [1–11] derived . is taken into consideration. With Hawking radiation, the emitted particles are mostly photons and standard model Recently, a modified Newton’s law of gravity due to (SM) particles. Using the Hawking-Uncertainty Relation, Planck-scale effects through the entropic force approach the characteristic energy of the emitted particle can be was derived by one of the authors [39]. Derivation of identified [56–60]. It has been found [25, 26], assuming Planck scale effects on the Newton’s law of gravity are some symmetry conditions from the propagation of Hawk- based on the following procedure: modified theory of grav- ing radiation, that the inequality that would correspond ity → modified black hole entropy→ modified holographic to Eq. (1) can be written as follows: surface entropy → Newton’s law corrections . This proce- ~ p2 x p ≥ 5 µ β `2 ∆ : dure has been followed with other approaches like non- ∆ ∆ 1 + (1 + ) 0 p (11) 2 3 ~2 commutative geometry in [45–48]. In this paper, we take 2:821 2 into account the quantum gravity corrections due to GUP where µ = π . in the entropic-force approach, following the same proce- By solving the inequality (11) as a quadratic equation in dure as [39], and extend our study to calculate the modified ∆p, we obtain Friedmann and Raychaudhuri equations. To calculate the quantum gravity corrections to the Friedmann equations, s we use the procedure that has been followed in [49, 50].