Towards a holographic theory of cosmology -- threads in a tapestry
aa r X i v : . [ g r- q c ] M a y Towards a holographic theory of cosmology – threads in a tapestry
Y. Jack Ng ∗ Institute of Field Physics, Department of Physics and Astronomy,University of North Carolina, Chapel Hill, NC 27599, U.S.A.
This Essay received Honorable Mention in the 2013 Essay Competition of the GravityResearch FoundationDedicated to the memory of Hendrik van Dam (1934-2013)
Abstract
In this Essay we address several fundamental issues in cosmology: What is thenature of dark energy and dark matter? Why is the dark sector so different fromordinary matter? Why is the effective cosmological constant non-zero but so incrediblysmall? What is the reason behind the emergence of a critical acceleration parameterof magnitude 10 − cm/sec in galactic dynamics? We suggest that the holographicprinciple is the linchpin in a unified scheme to understand these various issues. ∗ [email protected] Introduction: A holographic theory of cosmology
Some alert readers may have already noticed a resemblance between the title of this Es-say and that of S. Glashow’s Nobel Lecture [1] “Towards a unified theory – threads in atapestry.” This resemblance is not a coincidence, for like elmentary particle physics, thestudy of cosmology is like a patchwork quilt. But whereas the patchwork quilt has becomea tapestry for the former, the various threads have yet to be coherently woven for the latter.However now there is reason for optimism: we may have found a powerful guiding principlebehind nature’s intricate design, yielding (eventually) a beautiful tapestry of gravity andmatter. We are referring to the holographic principle [2, 3], an important by-product of thesynthesis of quantum mechanics and general relativity, according to which, the maximumamount of information stored in a region of space scales as the area of its two-dimensionalsurface, like a hologram. The holographic principle is arguably the most important conceptin quantum gravity, playing a role similar to the gauge principle in particle physics.In this Essay we will apply the holographic principle to address a few fundamental issuesin gravity and cosmology. One of the key issues in cosmology is to understand the nature ofdark energy and dark matter and why the dark sector is so different from ordinary matter.Another issue is to explain the twin puzzles of why our universe is at or very close to itscritical density and why the (effective) cosmological constant is nonzero and so small. Atthe (smaller) galactic scale, there are the issues of the observed flat rotation curves andthe emergence of a critical acceleration parameter separating the regime where Newtoniandynamics works well from that where it appears to fail. We liken the resolution of all theseissues to finding the right threads in a tapestry — interwoven coherently, with one threadlogically leading to another. Λ As will be shown shortly, all the aforementioned issues are linked to the quantum natureof spacetime. Thus it behooves us to start by examining how foamy spacetime is, or, inother words, how large the quantum fluctuations of spacetime are.[4, 5] Let us considermapping out the geometry of spacetime for a spherical volume of radius l over the amountof time 2 l/c it takes light to cross the volume.[6] One way to do this is to fill the space withclocks, exchanging signals with the other clocks and measuring the signals’ times of arrival.The total number of operations, including the ticks of the clocks and the measurementsof signals, is bounded by the Margolus-Levitin theorem [7] which stipulates that the rateof operations cannot exceed the amount of energy E that is available for the operationdivided by π ~ /
2. This theorem, combined with the bound on the total mass of the clocksto prevent black hole formation, implies that the total number of operations that can occurin this spacetime volume is no bigger than 2( l/l P ) /π , where l P = p ~ G/c is the Plancklength. To maximize spatial resolution, each clock must tick only once during the entire time2eriod. If we regard the operations as partitioning the spacetime volume into “cells”, thenon the average each cell occupies a spatial volume no less than ∼ l / ( l /l P ) = ll P , yieldingan average separation between neighboring cells no less than ∼ l / l / P . [8] This spatialseparation can be interpreted as the average minimum uncertainty in the measurement of adistance l , that is, δl & l / l / P , in agreement with the result found in the Wigner-Saleckergedanken experiment to measure the fluctuation of a distance l .We make two observations: [11, 12] First, maximal spatial resolution (corresponding to δl ∼ l / l / P ) is possible only if the maximum energy density ρ ∼ ( ll P ) − is available to mapthe geometry of the spacetime region, without causing a gravitational collapse. Secondly,since, on the average, each cell occupies a spatial volume of ll P , a spatial region of size l cancontain no more than ∼ l / ( ll P ) = ( l/l P ) cells. Hence, this result for spacetime fluctuationscorresponds to the case of maximum number of bits of information l /l P in a spatial regionof size l , that is allowed by the holographic principle[2, 3].It is straightforward to generalize [11] the above discussion for a static spacetime regionwith low spatial curvature to the case of an expanding universe by the substitution of l by H − in the expressions for energy and entropy densities, where H is the Hubble parameter.(Henceforth we adopt c = 1 = ~ for convenience unless stated otherwise for clarity.) Thus,applied to cosmology, the above argument leads to the prediction that (1) the cosmic energydensity has the critical value ρ ∼ ( H/l P ) , and (2) the universe of Hubble size R H contains ∼ HR H /l P ∼ ( R H /l p ) bits of information. It follows that the average energy carried byeach particle/bit is ρR H /I ∼ R − H . Such long-wavelength constituents of dark energy giverise to a more or less uniformly distributed cosmic energy density and act as a dynamicalcosmological constant with the observed small but nonzero value Λ ∼ H . [8] Later wewill show that these “particles”/bits have exotic statistical properties. Λ to MoNDian dark matter The dynamical cosmological constant (originated from quantum fluctuations of spacetime)can now be shown to give rise to a critical acceleration parameter in galactic dynamics.The argument [14] is based on a simple generalization of E. Verlinde’s recent proposal ofentropic gravity [15, 16]. Consider a particle with mass m approaching a holographic screen One way to detect this minute fluctuation is to look for blurry images of distant quasars in powerfultelescope interferometers. [9] In the Wigner-Salecker experiment [10, 4], a light signal is sent from a clock to a mirror (at a distance l away) and back to the clock in a timing experiment to measure l . From the jiggling of the clock’s positionalone, the uncertainty principle yields ( δl ) & ~ l/mc , where m is the mass of the clock. On the other hand,the clock must be large enough not to collapse into a black hole; this requires δl & Gm/c . We concludethat the fluctuation of a distance l scales as δl & l / l / P . [4, 5] Here we will not address the old cosmological constant problem of why it is not of the Planck scale. SeeRef. [13] for possible solutions.
3t temperature T . Using the first law of thermodynamics to introduce the concept of entropicforce F = T ∆ S ∆ x , and invoking Bekenstein’s original arguments [17] concerning the entropy S of black holes, ∆ S = 2 πk B mc ~ ∆ x , we get F = 2 πk B mc ~ T . In a deSitter space with cosmologicalconstant Λ, the net Unruh-Hawking temperature, [18, 19, 20] as measured by a non-inertialobserver with acceleration a relative to an inertial observer, is T = ~ πk B c [ p a + a − a ],[21] where a ≡ p Λ /
3. Hence the entropic force (in deSitter space) is given by F = m [ p a + a − a ]. For a ≫ a , we have F/m ≈ a which gives a = a N ≡ GM/r , the familiarNewtonian value for the acceleration due to the source M . But for a ≪ a , F ≈ m a a , so the terminal velocity v of the test mass m in a circular motion with radius r should bedetermined from ma / (2 a ) = mv /r . In this small acceleration regime, the observed flatgalactic rotation curves ( v being independent of r ) now require a ≈ ( a N a ) . But thatmeans F ≈ m √ a N a . This is the celebrated modified Newtonian dynamics (MoND) scaling[22, 23, 24], discovered by Milgrom who introduced the critical acceleration parameter a byhand to phenomenologically explain the flat galactic rotation curves. Lo and behold, we haverecovered MoND with the correct magnitude for the critical galactic acceleration parameter a ∼ − cm/s . From our perspective, MoND is a classical phenomenological consequenceof quantum gravity (with the ~ dependence in T ∝ ~ and S ∝ / ~ cancelled out). [14] As abonus, we have also recovered the observed Tully-Fisher relation ( v ∝ M ).Having generalized Newton’s 2nd law, we [14] can now follow the second half of Verlinde’sargument [15] to generalize Newton’s law of gravity a = GM/r by considering an imaginaryquasi-local (spherical) holographic screen of area A = 4 πr with temperature T . Invokingthe equipartition of energy E = N k B T with N = Ac / ( G ~ ) being the total number ofdegrees of freedom (bits) on the screen, as well as the Unruh temperature formula and thefact that E = M total c , we get 2 πk B T = G M total /r , where M total = M + M d representsthe total mass enclosed within the volume V = 4 πr /
3, with M d being some unknownmass, i.e., dark matter. For a ≫ a , consistency with the Newtonian force law a ≈ a N implies M d ≈
0. But for a ≪ a , consistency with the condition a ≈ ( a N a ) requires M d ≈ (cid:0) a a (cid:1) M ∼ ( √ Λ /G ) / M / r . This yields the dark matter mass density ρ d profilegiven by ρ d ( r ) ∼ M / ( r v )( √ Λ /G ) / /r , for an ordinary (visible) matter source of radius r v with total mass M ( r v ). Thus dark matter indeed exists! And the MoND force law derived above, at the galacticscale, is simply a manifestation of dark matter ! [25] Dark matter of this kind can behave as if there is no dark matter but MoND. Therefore, we call it “MoNDian dark matter”.Intriguingly the dark matter profile we have obtained relates, at the galactic scale, darkmatter ( M d ), dark energy (Λ) and ordinary matter ( M ) to one another. Moreover, ourtheory, unlike the MoND scheme, is compatible with cosmology, if one properly uses a fullyrelativistic source (including MoNDian dark matter) at the cluster and cosmic scales.[14] This result can be compared with the distribution associated with an isothermal Newtonian sphere inhydrostatic equilibrium (used by some dark matter proponents): ρ ( r ) = σ ( r + r ) − . Asymptotically thetwo expressions agree with σ identified as ∼ M / ( r v )( √ Λ /G ) / . Infinite statistics for the dark sector
Why is the dark sector so different from ordinary matter? The reason, as we will show inthis section, is that the quanta constituting the dark sector obey, not the familiar Fermi orBose statistics as for ordinary matter, but rather an exotic statistics known as the infinitestatistics . [12]First consider the N ∼ ( R H /l P ) “particles” constituting dark energy at temperature T ∼ R − H (the average particle energy) in a volume V ∼ R H that is the whole Hubble volume.Let us assume that the ”particles” obey the familiar Boltzmann statistics. A standardcalculation (for the relativistic case) yields the partition function Z N = ( N !) − ( V /λ ) N ,where λ = ( π ) / /T , and the entropy S = N [ ln ( V /N λ ) + 5 / V ∼ λ ,the entropy S becomes nonsensically negative unless N ∼ N ∼ ( R H /l P ) ≫
1. However, if the N inside the log term for S somehow isabsent, then we have a manifestly non-negative S ∼ N ∼ ( R H /l P ) . That is the case if the“particles” are distinguishable and nonidentical, for then the Gibbs 1 /N ! factor is absentfrom the partition function Z N . But the only known consistent statistics in greater than twospace dimensions without the Gibbs factor is infinite statistics (sometimes called “quantumBoltzmann statistics”) [26, 27], as described by the Cuntz algebra (a curious average of thebosonic and fermionic algebras) a i a † j = δ ij . Thus the “particles” constituting dark energyobey infinite statistics. [12, 28] Next, to show that the quanta of MoNDian dark matter also obey this exotic statistics,we [30] first reformulate MoND via an effective gravitational dielectric medium, motivated bythe analogy [31] between Coulomb’s law in a dielectric medium and Milgrom’s law for MoND.We start with the nonlinear electrostatics embodied in the Born-Infeld theory [32], and writethe corresponding gravitational Hamiltonian density as H g = b (cid:16) q D g /b − (cid:17) / (4 π ),where D stands for the electric displacement vector and b is the maximum field strength inthe Born-Infeld theory. With A ≡ b and ~A ≡ b ~D g , the Hamiltonian density becomes H g = (cid:16) p A + A − A (cid:17) / (4 π ). If we invoke energy equipartition ( H g = k B T eff ) and the Unruhtemperature formula ( T eff = ~ π k B c a eff ), and apply the equivalence principle (in identifying,at least locally, the local accelerations ~a and ~a with the local gravitational fields ~A and ~A respectively), then the effective acceleration a eff is identified as a eff ≡ p a + a − a . Butthis, in turn, implies that the Born-Infeld inspired force law takes the form (for a given testmass m ) F BI = m (cid:16) p a + a − a (cid:17) , which is precisely the MoNDian force law!To be a viable cold dark matter candidate, the quanta of the MoNDian dark matter mustbe much heavier than k B T eff since T eff , with its quantum origin (being proportional to ~ ),is a very low temperature. Now recall that the equipartition theorem in general states thatthe average of the Hamiltonian is given by h H i = − ∂ log Z ( β ) ∂β , where β − = k B T . To obtain Our result for the N ∼ ( R H /l P ) quanta of dark energy obeying infinite statistics has received supportfrom Ref. [29] which shows that the entropy bound of infinite statistics obeys the area law. H i = k B T per degree of freedom, even for very low temperature, we require the partitionfunction Z to be of the Boltzmann form Z = exp( − β H ) . But this is precisely the case ofinfinite statistics. [30] In summary, by examining the microscopic fluctuations of spacetime we have found that ouruniverse is naturally at or close to its critical density. The application of the holographicprinciple then yields an effective dynamical cosmological constant of the observed value.Next we have provided an entropic/holographic interpretation behind Milgrom’s modifica-tion of Newton’s laws and have uncovered a critical galactic acceleration parameter of thecorrect magnitude whose value is intimately related to the dynamical cosmological constant.We have also explained how Milgrom’s MoND can be viewed as a phenomenological mani-festation of dark matter with a curious mass profile that connects, at the galactic scale, thedark matter content to the ordinary matter content and dark energy. In principle this darkmatter mass profile can be checked by observations. [33] Last but not least, we have shownthat the quanta of the dark sector obey infinite statistics; this may explain why the darksector is so different from ordinary matter.The last result could be profound. But, if true, it also makes an analysis of the darksector considerably more difficult. The reason is that a theory of particles obeying infinitestatistics, unlike ordinary quantum field theories, is not local . [27] On the other hand, sucha theory of MoNDian dark matter would be fundamentally quantum gravitational and thuswould give very unusual and distinct yet-to-be explored particle phenomenology.Now if indeed the quanta of the dark sector obey infinite statistics, then we may won-der whether quantun gravity is actually the origin of particle statistics and whether theunderlying statistics is infinite statistics. Here is an intriguing thought [30]: Is it possiblethat ordinary particles that obey Bose or Fermi statistics are actually some sort of collectivedegrees of freedom? (For a discussion of constructing bosons and fermions out of particlesobeying infinite statistics, see Ref. [34] and [35].)Using the holographic principle as our beacon, we have taken some small yet tightlylogical steps towards a comprehensive understanding of how our universe works – from thefoaminess of spacetime to the critical cosmic energy density, from the dynamical cosmologicalconstant via the holographic principle to the critical acceleration parameter in local galacticdynamics, and from dark matter with MoNDian scaling to the dark sector obeying infinitestatistics. All these various issues of cosmology have been found to be inter-related – likeinter-connecting patches of a quilt, woven together. Yet this is obviously work in progress. The fields are not local, neither in the sense that their observables commute at spacelike separation norin the sense that their observables are pointlike functionals of the fields. The expression for the numberoperator is both nonlocal and non-polynomial in the field operators, and so is the Hamiltonian.
Acknowledgments:
This Essay is partly based on work done in collaboration with C.M.Ho and D. Minic, with S. Lloyd, and with M. Arzano and T. Kephart. I thank them all. Iwould like to dedicate this Essay to the memory of my friend and longtime collaborator H.van Dam who passed away recently, in witness of my appreciation for him. This work wassupported in part by the US Department of Energy and the Bahnson Fund of UNC-CH.
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