Holography and the Large Number Hypothesis Guillermo A. Mena Marug´an1 and Saulo Carneiro1,2 1 I.M.A.F.F., C.S.I.C., Serrano 121, 28006 Madrid, Spain 2 Instituto de F´ısica, Universidade Federal da Bahia, 40210-340, Salvador, Bahia, Brazil Dirac’s large number hypothesis is motivated by certain scaling transformations that relate the parameters of macro and microphysics. We show that these relations can actually be explained in terms of the holographic N bound conjectured by Bousso and a series of purely cosmological observations, namely, that our universe is spatially homogeneous, isotropic, and flat to a high degree of approximation and that the cosmological constant dominates the energy density at present. PACS numbers: 98.80.Hw, 04.20.Cv, 11.10.-z, 04.70.Dy Explaining the value of the constants of nature is one of mological constant [6]. In addition, accurate measure- the most exciting challenges of theoretical physics. Some ments of the angular power spectrum of anisotropies in of these constants play a fundamental role in the foun- the cosmic microwave background (CMB) have shown dations of the scientific paradigms. This is the case of that the curvature of the universe is close to flat [7]. Planck constant ¯h in quantum mechanics, and of New- These CMB and SNe Ia data, together with other cosmo- ton constant G and the speed of light c in general rela- logical information, have been combined in a consistent tivity. These three constants provide a natural system of (nearly) flat FRW model whose values of the cosmologi- units for all physical quantities. For instance, the length cal constant Λ and matter density ρ0 are, approximately, 3 −35 2 2 and mass units are lP = p¯hG/c = 1.6 × 10 m and c Λ=16πGρ0 =2H0 [8]. −8 mP = p¯hc/G =2.2 × 10 kg. In terms of these Planck This value of the cosmological constant poses two puz- units, the other constants of nature become dimension- zles. On the one hand, one would expect that Λ emerged less numbers. from vacuum fluctuations. In a theory of quantum grav- Already in the 20’s, Eddington tried unsuccessfully to ity, these fluctuations would have Planck energy density. deduce the value of all constants of physics from theo- The discrepancy from this theoretical expectations is of retical considerations [1]. Most importantly, he pointed nearly 120 orders of magnitude, since, in Planck units, −60 out the existence of relations between the parameters of H0 ≈ 10 . This is the so-called cosmological constant fields that at first sight seem unconnected, like nuclear problem [9]. The value of Λ, on the other hand, is con- physics and cosmology. Among these, perhaps the most stant, whereas the density of matter decreases with ex- 2 intriguing relation is the apparent coincidence between pansion. As a consequence, the relation 16πGρ0 ≈ c Λ the present number of baryons in the universe, known as is not valid in most of the history of the universe. Why Eddington number, and the squared ratio of the electric is it precisely now that the matter content and Λ pro- to the gravitational force between the proton and the vide similar contributions to the energy? This additional electron. This coincidence between large numbers can puzzle is known as the cosmic coincidence problem [10]. also be expressed in the alternative form [2] A new perspective of the cosmological constant prob- 2 3 lem, which puts the emphasis on fundamental aspects ¯h H0 ≈ GcmN . (1) arXiv:gr-qc/0111034v2 25 Feb 2002 of gravity rather than in purely quantum field theory This approximate identity is sometimes called the (QFT) considerations, has recently emerged with the ad- vent of holography [11]. In an over-simplified version, Eddington-Weinberg relation. Here, mN is the proton mass and H0 ≈ 70 km/(sMpc) is the present value of the the holographic principle states that the entropy S [12] Hubble constant [3]. of a physical system subject to gravity is bounded from Actually, the Hubble parameter is not a true constant, above by a quarter of its boundary area in Planck units, ≤ 2 but varies as the inverse of the cosmological time in stan- S A/(4lp). From this point of view, the physical de- dard Friedman-Robertson-Walker (FRW) cosmology [2]. grees of freedom are not proportional to the volume in This fact led Dirac [4] to put forward the hypothesis that the presence of the gravitational field, but reside in the −1 Newton constant must depend on time as H0, G ∝ t , so bounding surface. that relation (1) is always valid. In spite of its attractive A more rigorous, covariant formulation of the holo- features, Dirac’s large number hypothesis turns out to be graphic conjecture has been elaborated by Bousso, pro- incompatible with the experimental bounds that exist on viding in principle an entropy bound on null hypersur- the time variation of G [5]. Therefore, the explanation of faces [13, 14]. Other less general holographic proposals the Eddington-Weinberg relation still remains a mystery. that find straightforward application to spatial volumes Recently, the determination of cosmological parame- in cosmology have also been suggested [15, 16, 17]. In ters has experienced a considerable revolution. The ob- this respect, an issue of debate has been the largest re- servation of type Ia supernovae (SNe Ia) at high red- gion of the universe in which an entropy bound may be shift has provided evidence in favor of a positive cos- feasible. Fischler and Susskind [15] originally proposed 2 −1 to consider the particle horizon, at least for adiabatic mN ≈ Ω mP , (4) evolution, but other possibilities that appear more nat- −1 3 lU ≡ cH0 ≈ Ω lP , (5) ural were soon suggested. One such possibility is the 3 m ≈ Ω m . (6) use of the cosmological apparent horizon, which bounds U P 19 20 an anti-trapped region and has an associated notion of The scale Ω has the value 10 –10 . Here, mN and lN gravitational entropy [13, 16]. Another proposal that has are the mass and radius of a nucleon, e.g. the proton. found considerable support is the restriction to the Hub- −1 The symbol lU denotes the observable radius of the uni- ble radius cH0 [17], since this supplies the scale of causal verse, that we define as the distance that light can travel −1 connection beyond which gravitational perturbations on in a Hubble time H0 . This time is roughly the age of a flat background cannot grow with time. It is worth our universe. Finally, the mass of the universe mU is the noting, anyway, that for a flat FRW model like the one energy contained in a spatial region of radius lU . that possibly describes our universe, the apparent and In fact, relations (3) and (4) are not independent. For Hubble horizons do in fact coincide [16]. an elementary particle governed by quantum mechanics, For any spacetime with a positive cosmological con- the typical effective size should be of the order of its stant, Bousso [18] has argued that the holographic prin- Compton wavelength, lN ≈ ¯h/(cmN ). It therefore suf- ciple leads to the prediction that the number of degrees −1 fices to explain, for instance, why mP mN is of order Ω. of freedom N available in the universe is related to Λ by Something similar happens with the scaling laws (5) and (6). Assuming homogeneity and isotropy, mU is de- 3π 3 T T 2 N = . (2) fined as 4πl ρ0 /3. Here, ρ0 ≡ ρ0 + c Λ/(8πG) is the to- Λl2 ln 2 U P tal energy density. Hence, given the relation between lU and lP , formula (6) amounts to the approximate equality The observable entropy S is then bounded by N ln 2. 2 ρT ≈ ρC , where ρC ≡ 3H /(8πG) is the critical density This conjecture is called the N bound. Under quantiza- 0 0 0 0 of a FRW model at present. In a universe like ours, the tion, the system would be describable by a Hilbert space scaling equation for m is thus a consequence of Eq. (5) of finite dimension (equal to 2N ). Bousso’s conjecture U and spatial flatness. is largely influenced by Banks’ ideas about the cosmo- Examining relations (3)–(6), a length scale lS of order logical constant [19]. According to Banks, Λ should not 2 Ω in Planck units appears to be missing. Roughly, this be considered a parameter of the theory; rather, it is scale corresponds to the size of stellar gravitational col- determined by the inverse of the number of degrees of lapse determined by Chandrasekhar limit (or any other freedom. From this viewpoint, the cosmological constant similar mass limit) [22]. Actually, for such stellar-mass problem disappears, because N can be regarded as part black holes, the formulas of the Schwarzschild radius and of the data that describe the system at a fundamental the Chandrasekhar mass [2] lead to level. Based also on holography, other possible expla- 2 2 nations have been proposed for the value of Λ that are lS ≈ Ω lP , mS ≈ Ω mP . (7) closer in spirit to the standard methods of QFT [20]. Since the cosmological constant affects the large scale At this stage of our discussion, the only scaling laws structure of the universe but should originate from effec- that remain unexplained are relations (4) and (5). In tive local vacuum fluctuations, it may provide a natural fact, one of these approximate identities can be viewed connection between macro and microphysics. In addi- as the definition of Ω, e.g. the equation for lU .