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−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 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 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 . The ap- tion, Λ is related to the number of degrees of freedom by pearance of large numbers in our relations may then be understood, following Dirac [4], as a purely cosmological the holographic principle. As a consequence, one could −1 expect that holography would play a fundamental role in issue. Since H0 is essentially the age of the universe, explaining the coincidence of the large numbers arising the fact that Ω ≫ 1 is just a consequence of the uni- in cosmology and particle physics. A first indication that verse being so old. In addition, it is easy to check that, this intuition may work is provided by Zizzi’s work [21], given formula (5), the scaling transformation for mN is who recovered Eddington number starting with a discrete equivalent to Eq. (1). Therefore, the only coincidence of quantum model for the early universe that saturates the large numbers that needs explanation is the Eddington- holographic bound. The main aim of the present pa- Weinberg relation. per is to prove that the large number hypothesis and the Suppose now that nucleons (or hadronic particles in holographic conjecture are in fact not fully independent. general) can be described as elementary excitations of To be more precise, we will show that, in a homoge- typical size lN in an effective quantum theory. The num- neous, isotropic, and (quasi)flat universe like ours, the ber of physical degrees of freedom in a spatial region of 3 relations between large numbers can be explained by the volume V will be of the order of 3V/(4πlN ). In a cosmo- holographic principle assuming that the present energy logical setting, it seems natural to consider the Hubble density is nearly dominated by Λ. radius as the largest size of the region in which such an The scaling relations that lie behind the large number effective quantum description of particles may exist, be- hypothesis can be expressed in the form cause it provides the scale of causal connection where the microphysical interactions take place. For a homo- lN ≈ ΩlP , (3) geneous and isotropic universe with negligible curvature, 3 like the one we inhabit, the FRW equations imply that to understand the origin of the Eddington-Weinberg re- 2 2 8πGρ0 + c Λ ≈ 3H0 [2]. Given the positivity of ρ0, guar- lation. According to the explanation that we have put anteed by the dominant energy condition, the maximum forward, such a relation does not hold at all times, but Hubble radius is thus close to p3/Λ. For an almost only when the cosmological constant dominates the en- flat FRW universe, the volume of the corresponding spa- ergy density. Although we expect this condition to be tial region is nearly 4πp3/Λ3. As a consequence, the satisfied at present and in the future, it excludes the early maximum number of observable degrees of freedom N stages of the evolution of the universe. In our theoretical in this kind of cosmological scenarios should roughly be framework, the constants of nature G, ¯h, and c do not 3 6 p27/(Λ lN ). Taking into account the holographic N vary with time, and so we do not recover Dirac’s cosmol- bound (2), we then conclude ogy [4].

4 −1 1 6 In obtaining relation (8), we have actually supposed l ≈ (l Λ ) / . (8) N P that the total number of degrees of freedom N available in the universe is roughly of the same order as the max- Using that lN mN ≈ lP mP , a relation that we have already justified, we immediately obtain imum number of degrees observable in its baryonic con- tent. It should be clear that this assumption does not 3 3 2 1/2 conflict with the fact that the present energy density is mN ≈ mP (lP Λ) . (9) not dominated by baryonic matter. More importantly, This approximate identity reproduces Eq. (1) provided since the number of baryonic degrees of freedom cannot −1 −1/2 4 −1 1/6 that the present Hubble radius cH0 is close to Λ . exceed N, the quantity (lP Λ ) provides, in any case, Therefore, the so-far unexplained Eddington-Weinberg a lower bound to the typical size of nucleons lN . Further relation can be understood from a holographic perspec- discussion of this point will be presented elsewhere. tive, assuming an almost flat FRW cosmology, if and only The length scale (8) has also been deduced by Ng, al- if the cosmological constant has a nearly dominant con- though replacing Λ−1 with the square of the observable tribution to the present energy density. This is ensured, radius of the universe [23]. However, he has proposed e.g., by cosmic coincidence. to interpret l as the minimum resolution length in the 2 ≈ 2 N Note that the result c Λ H0 can be regarded as presence of quantum gravitational fluctuations, instead a partial solution to the cosmological constant problems of as the typical size of particles in the effective QFT (the value of Λ and cosmic coincidence) in our (quasi)flat that describes the baryonic content. From our viewpoint, universe if, adopting a different viewpoint, we take for this scale does not provide a fundamental length limiting granted Bousso’s proposal and Eq. (1). Alternatively, if the resolution of spacetime measurements, but rather re- 2 ≈ 2 we use the Eddington-Weinberg relation and c Λ H0 , stricts the number of degrees of freedom available in the the arguments given above about the relation between effective QFT. Concerning the value of lN , Ng proposes N and lN allow us to reach an approximate version of two ways to deduce it. In one of them, a spatial region the N bound for our spacetime. Thus, we see that in a is considered as a Salecker-Wigner clock able to discern nearly homogeneous, isotropic and flat universe like ours, distances larger than its Schwarzschild radius [23]. The the cosmological constant problems, the N bound, and question arises whether this interpretation is applicable the coincidence of large numbers are interrelated. to the , because its Schwarzschild and In our application of the N bound, we have argued Hubble radii are of the same order of magnitude. The that the Hubble radius is the largest scale in which mi- other line of reasoning employs holographic arguments crophysics can act. Nonetheless, our conclusions would related to those presented here. Nevertheless, since Ng not have changed if, as proposed in Ref. [16] for cosmic uses the present size of the universe instead of Λ−1/2, it holography, we had employed the cosmological apparent is not clear whether the resolution scale that he obtains horizon instead of the Hubble radius, because they are must be viewed as time independent. approximately equal in quasiflat FRW models. We have also made use of the fact that, for this kind of models, Let us return to expression (5) for the present Hub- the maximum Hubble radius is nearly p3/Λ if Λ is pos- ble radius, which we have interpreted as the definition itive. This is also the size of the cosmological horizon of Ω. We have argued that the fact that Ω ≫ 1 can be of the de Sitter space with the same value of Λ. In (al- regarded as a consequence of the old age of the universe, most) flat FRW cosmologies with a dominant Λ-term at which is a cosmological problem and not a numerical co- late times, a situation that apparently applies to our uni- incidence between microscopic and macroscopic param- verse, any observer has a future event horizon that tends eters. Nonetheless, using the N bound and the present asymptotically to such a de Sitter horizon. Hence, our dominance of Λ, it is actually possible to explain the ap- results would neither have been altered had we replaced pearance of the large scale Ω along very similar lines to the maximum Hubble radius with the asymptotic event those proposed by Banks for the resolution of the cos- horizon in all our considerations. mological constant problem [19]. As we have seen, when The fact that the N bound provides an effective length the energy density is nearly dominated by Λ, the Hub- scale for microphysics, given by Eq. (8), has played a ble radius is close to p3/Λ. In addition, the N bound central role in our arguments. This fact has allowed us implies that this latter length is equal to lP pN ln 2/π. 4

Recalling Eq. (5), we then obtain intriguing that SS matches relatively well what seems to be the actual entropy of the universe, S0. The main con- 1 6 Ω ≈ N / . (10) tribution to this entropy comes from super-massive black holes in galactic nuclei. Assuming that a typical 11 12 So, Ω is a large number because our universe contains a contains 10 –10 stellar masses mS and that the mass 6 7 huge amount of degrees of freedom. From this perspec- of its central black hole is 10 –10 mS, it is straightfor- 3 tive, the value of Ω is fixed by N, which can be considered ward to find that S0 ≈ 1–10 SS. an input of the theory that describes our world. Finally, we want to present some brief comments about Summarizing, we have proved that, in the light of the the entropy of the universe. If the only entropic contri- holographic principle, the relations between large num- bution were baryonic, we could estimate it as Sb ≈ nN . bers constructed from microscopic and cosmological pa- Here, we have supposed that each baryon has an associ- rameters are not independent of other fine-tuning and ated entropy of order unity, and nN is Eddington num- coincidence problems that have a purely cosmological ber, that can be calculated as the ratio of the baryonic nature. More explicitly, provided that the universe can mass of the universe to the typical mass of a nucleon. In be approximately described by a spatially homogenous, a rough approximation (valid for our estimation of or- isotropic, and flat cosmological model and that the main ders of magnitude), we can identify the matter and the contribution to the present energy density comes from baryonic energy densities. Taking into account cosmic the cosmological constant, it is possible to explain all the −1 coincidence, we can then approximate nN by mU m . scaling relations that motivated Dirac’s large number hy- 4 N In this way, we get Sb ≈ nN ≈ Ω . This is much less pothesis appealing exclusively to basic principles and to than the maximum allowed entropy, which, from rela- the N bound conjecture. tion (10) and the definition of N, is of the order of Ω6. An intermediate entropic regime would be reached if the G.A.M.M. acknowledges DGESIC for financial support matter of the universe collapsed into stellar-mass black under Research Project No. PB97-1218. S.C. was par- holes. As we have commented, this regime corresponds tially supported by CNPq. The authors want to thank 2 to the length scale lS ≈ Ω lP . One can check that, in also L.J. Garay and P.F. Gonz´alez-D´ıaz for helpful com- 5 this case, the entropy would be SS ≈ Ω . It is rather ments.

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