EFFECTIVE COSMOLOGICAL CONSTANT AND THE DARK SECTOR

Transcription

1 EFFECTIVE COSMOLOGICAL CONSTANT AND THE DARK SECTOR Jack Ng University of North Carolina at Chapel Hill Bahamas Advanced Study Institute & Conferences (March 2017) References: Ng, IJMP D1, 145 (1992); Ng & van Dam, IJMP D10, 49 (2001). Ng, PL B657, 10 (2007); Ho, Minic and Ng, PR D85, (2012). Edmonds, Farrah, Minic, Ng, and Takeuchi, Review article on MDM (2017). & references contained therein

2 CONTENTS Spacetime foam & holography From holography to effective cosmological constant Λ Λ via unimodular gravity and causal set theory From Λ to modified dark matter (MDM) [brief; see D Edmonds talk] Infinite statistics and quanta of the dark sector Take-home message: Possibly the dark sector has its origin in quantum gravity; effective Λ 3H 2. May explain why dark energy and dark matter are so different from ordinary matter (their quanta are extended) and may explain why it is so difficult to detect dark matter. Units (most of the time): c = 1, = 1,k B = 1. Notation: l P G/c 3 is the Planck length.

3 Quantum foam (due to fluctuations of spacetime) Measuring quantum fluctuations of spacetime with a gedanken experiment δl l 1/3 l 2/3 P scaling Consistent with Holographic principle (heuristic argument)

4 John Wheeler: Quantum foam or spacetime foam a foamy structure of spacetime arising from quantum fluctuations. Quantum foam [Ng & van Dam; Karolyhazy] To examine how large the fluctuations are, consider a gedankan experiment (to measure δl, the accuracy with which distance l can be measured) in which 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:

5 From the jiggling of the clock s position alone, the uncertainty principle yields: δl ( ) 2l c = δl+ 2l 1 c m 2δl δl2 l mc For the clock not to collapse into a black hole, general relativity requires: d Gm c δl Gm 2 c (since d δl) 2 QM + GR: δl l 1/3 l 2/3 P Shortly we will rederive this scaling of δl by another method which can be generalized to the case of an expanding universe. Can heuristically show (see next slide) that this scaling of δl is exactly what the holographic principle demands, according to which the maximum amount of information stored in a region of space (of size l 3 ) scales as the area ( l 2 ) of its two-dimensional surface, like a hologram.

6 Spacetime foam holographic principle Consider partitioning l 3 into cubes as small as physical laws allow, so intuitively one degree of freedom is associated with each small cube. l δl δl δl l l δl No. of degrees of freedom inside l 3 = # small cubes = ( ) l 3 δl l 2 by holographic principle = δl l 1/3 l 2/3 P l 2 P Reversing argument, δl l 1/3 l 2/3 P h.p. [Ng & van Dam]

7 From holography to cosmological constant Λ Space-time foam / holography via the mapping the geometry of spacetime Effective cosmological constant Λ 3H 2

8 Space-time Foam via Mapping the geometry of spacetime [Lloyd & Ng] Use a global positioning system: Fill space with a swarm of clocks, exchanging signals with the other clocks and measuring the signals time of arrival How accurately can these (many) clocks (of total mass M) map out a volume of space-time with radius l over time l/c it takes light to cross the volume Ticks & clicks of clocks in spacetime volume l 4 /c

9 Process of mapping the geometry of spacetime is a computational operation. (1). Margolus-Levitin theorem (rate of operations E/ ) # operations < (E/ ) time = Mc2 (2). To prevent black-hole formation M < lc2 G l c (1) + (2) # ops or events (i.e., # spacetime cells ) < l 2 c3 G = l2 l 2 P For max. spatial resolution, each clock ticks only ONCE Each cell occupies l 3 l 2 /l 2 P = ll 2 P Separation of cells l1/3 l 2/3 P Interpretation: δl l 1/3 l 2/3 P ; consistent with holography Maximum spatial resolution (requires max.) energy density ρ 3 8π (ll P) 2 ; yields # of bits l 2 /l 2 P.

10 Holography dark energy / effective cosmological constant Let us generalize the above discussion to the case of an expanding universe by substituting l by 1/H (with H being Hubble parameter). 2 main features: (H,R H = Hubble parameter, radius) ( ) 2 critical cosmic energy ρ = 3 H 8π l P (RH l P ) 2 Universe contains I (R H /l P ) 2 bits of info ( for current epoch) Average energy carried by each bit/ particle is ρrh 3 /I R 1 H. We interpret such long-wavelength particles as constituents of dark energy, contributing a more or less uniformly distributed cosmic energy density and acting as a dynamical effective cosmological constant Λ 3H 2. On the average, each bit flips once over the course of the cosmic history (corresponding to each clock ticking only once). These bits are passive and inert. (That s why they are dark?) They supply the energy to accelerate the cosmic expansion (which is a relatively simple task, computationally speaking).

11 The Universe cannot contain ordinary matter only Consider Universe as a computer: Given l # operations (l/l P ) 2 (l R H =Hubble radius; T=temp.) If all the info in this huge computer is stored in ordinary matter... Stat. mech. for ordinary matter: Energy density T 4, entropy density T 3 (And energy l requirement) Entropy # bits I (on conventional matter) (#op) 3/4 (l/l P ) 3/2 Each cell occupies spatial vol. l 3 (l/l P ) 3/2 = (l 1/2 l 1/2 P )3 implying δl l 1/2 l 1/2 P l 1/3 l 2/3 P (as found in the gedankan experiment!) Ordinary matter contains only an amount of info dense enough to map out spacetime at a level with much coarser spatial resolution There must be other kinds of matter/energy with which the Universe can map out its spacetime geometry to a finer spatial accuracy than is possible with the use of conventional ordinary matter Dark sector exists! (Side remark: The accuracy to which ordinary matter maps out the geometry of spacetime corresponds to the case of events ( cells ) spread out uniformly in space and time.)

12 Table 1. Random-walk model versus holographic model. The corresponding quantities for the random-walk model (second row) and the holographic model (third row) of spacetime foam (STF) appear in the same columns in the following Table. (Entropy is measured in Planck units.) STF distance entropy energy model fluctuations bound density random- δl l 1/2 l 1/2 P (Area) 3/4 (ll P ) 2 ρ walk l 5/2 l 3/2 P holo- δl l 1/3 l 2/3 P Area ρ (ll P ) 2 graphic

13 Rederiving Λ H 2 by using (A) Unimodular gravity (& Baum-Hawking-Adler argument), and (B) Causal set theory (A) Λ 0 + ; δλδv/g 1 (V being spacetime volume) (B) Λ H 2

14 Unimodular gravity [Anderson & Finkelstein; van der Bij, van Dam & Ng] Metric tensor g µν that has unit determinant: detg µν g = 1. Consider gravity without matter: S unimod = 1 16πG (dx) g[r+l( g 1)], yielding R µν 1 2 gµν R = 1 2 Lgµν with trace R = 2L R µν 1 4 gµν R = 0 (only the traceless combination appears). With matter: R µν 1 4 gµν R = 8πG(T µν 1 4 gµν T λ λ ), with T µν being the conserved matter stress tensor. In conjunction with the Bianchi identity D µ (R µν 1 2 gµν R) = 0, the field equation yields D µ (R 8πGT λ λ ) = 0 (R 8πGTλ λ ) is a constant. Denoting that constant of integration by 4Λ, we find R µν 1 2 gµν R = Λg µν 8πGT µν, the familiar Einstein s equation.

15 Note: Λ is an (arbitrary) integration constant, unrelated to any parameter in the original action. Further note: Since Λ arises as an arbitrary constant of integration, it has no preferred value classically. In the corresponding quantum theory, we expect: the state vector of the universe to be given by a superposition of states with different values of Λ and the quantum vacuum functional to receive contributions from all different values of Λ.

16 Generalized unimodular gravity: [Henneaux and Teitelboim] S unimod = 1 16πG [ g(r+2λ) 2Λ µ T µ ](d 3 x)dt. One of its equations of motion is g = µ T µ, the generalized unimodular condition, with g given in terms of the auxiliary field T µ (with T 0 having the meaning of time). In this theory, Λ/G plays the role of momentum conjugate to the coordinate d 3 xt 0, the spacetime volume V. Hence Λ/G and V are conjugate to each other δvδλ/g 1. Following Baum, Hawking, and Adler, consider the vacuum functional for unimodular gravity given by path- over T µ, g µν, the matter fields φ, and Λ: Z = dµ(λ) d[φ]d[g µν ] d[t µ ]exp{ i[s unimod +S M (φ,g µν )]}, where S M stands for the contribution from matter (including radiation) fields (and dµ(λ) denotes the measure of the Λ integration). The integration over T µ yields δ( µ Λ), which implies that Λ is spacetime-independent (befitting its role as the cosmological constant).

17 Do a Wick rotation to study the Euclidean vacuum functional Z. The integrations over g µν and φ give exp[ S Λ (g µν,φ)] where g µν and φ are the background fields which minimize the effective action S Λ. A curvature expansion for S Λ yields a Lagrangian whose first two terms are the Einstein-Hilbert terms g(r+2λ). (Note that Λ now denotes the fully renormalized cosmological constant after integrations over all other fields have been carried out.) Next make a change of variable from the original (bare) Λ to the renormalized Λ for the integration: Z = dµ(λ)exp ( 1 ) 16πG (dx)[ g(r+2λ)+...] For present and recent cosmic eras, φ is essentially in the ground state, then can follow Baum and Hawking to evaluate S Λ (g µν,0), using R µν = Λg µν.

18 Based on dimensional considerations alone, Λ = fv 1/2 from which follows S Λ (g µν,φ = 0)] = f2 8πGΛ. For negative Λ, S Λ is positive; the probability, being proportional to exp( S), is exponentially small. For positive Λ, the solution of the Einstein equations is a four-sphere given by R µνρσ = 1 r 2 (g µσ g νρ g µρ g νσ ) with radius r = 3/Λ; yielding S Λ (g µν,0) = 3π/GΛ, so that Z dµ(λ)exp(3π/gλ). This implies that the observed cosmological constant in the present and recent cosmic epochs is essentially zero. But if Λ fluctuates, it fluctuates about Λ = 0 and is positive.

19 Causal set theory [Sorkin] Causal-set theory stipulates that continous geometries in classical gravity should be replaced by causal-sets, the discrete substratum of spacetime; the fluctuation in the number of elements N making up the set is of the Poisson type, i.e., δn N. For a causal set, the spacetime volume V becomes l 4 P N δv l 4 P δn l4 P N l 2 P V = G V. Now recall: (1) δvδλ/g 1; (2) Λ fluctuates about 0 and is positive. Hence, Λ δλ V 1/2 R 2 H H2, as found in the argument using spacetime fluctuations and holography.

20 From Λ to modified dark matter (MDM) See collaborator Doug Edmonds talk. Summary: [DE, Farrah, Ho, Minic, YJN, and Takeuchi] By extending the works of Jacobson and Verlinde on entropic gravity, we show that Λ gives rise to a critical acceleration parameter (a 0 Λ) in galactic dynamics, and this naturally leads to the construction of a (modified) dark matter model in which the dark matter density profile depends on both Λ and ordinary matter. Newton s laws are modified: F entropic = m[ a 2 +a 2 0 a 0] = ma N [1+ 1 π ( a0 a ) 2 ]. (Here a N GM r 2.) Modified dark matter (MDM) behaves like modified Newtonian dynamics (MOND: F = maµ(a/a c ) with µ(x) = 1 for x 1 and µ(x) = x for x 1, and a c = a 0 /(2π)) at galactic scales but like CDM at cluster and cosmic scales. MDM passes observational tests at both the galactic and cluster scales (and plausibly also at cosmological scales).

21 Quanta of the dark sector obey infinite statistics? Quanta of dark energy (dynamical Λ) Infinite statistics Quanta of MDM

22 SPECULATION ON QUANTA OF DARK ENERGY [Ng] Consider dark energy (given by the dynamical effective Λ) composed of long-wavelength particles (according to spacetime foam /holography argument). How different are these particles? Consider N (R H /l P ) 2 such particles obeying Boltzmann statistics in volume V R 3 H at T R 1 H The partition function Z N = (N!) 1 (V/λ 3 ) N Entropy of the system is S = N[ln(V/Nλ 3 )+5/2] with λ T 1 But V λ 3, so S becomes negative unless N 1 which is equally nonsensical Solution: The N inside the log in S, i.e, the Gibbs factor (N!) 1 in Z N, must be absent the N particles are distinguishable! Then S = N[ln(V/λ 3 )+3/2], + ve S N

23 The only known consistent statistics in greater than 2 space dimensions without the Gibbs factor is the quantum Boltzmann statistics, aka infinite statistics A logical speculation: The particles constituting dark energy obey infinite statistics, rather than the familiar Fermi or Bose statistics. This is the overriding difference between DE and conventional matter. In the framework of M-theory, V. Jejjala, M. Kavic and D. Minic [hep-th: ] have made a similar suggestion Infinite statistics is described by the Cuntz algebra (a curious average of the bosonic and fermionic algebras) a i a j = δ ij. Note: Theories of particles obeying statistics are non-local [Fredenhagen; Greenberg] (The fields associated with infinite statistics are not local, neither in the sense that their observables commute at spacelike separation nor in the sense that their observables are pointlike functionals of the fields.) Quanta of particles obeying infinite statistics are extended. Conventional local QFT cannot be used to describe their interactions.

24 INFINITE STATISTICS [Doplicher, Haag, & Roberts; Govorkov; Greenberg;...] q-deformation of the Heisenberg algebra ( 1 q 1) a k a l qa l a k = δ k,l (q = ±1 corresponds to bosons/fermions) Take q = 0 a k a l = δ k,l (for infinite statistics, aka quantum Boltzmann statistics) Any 2 states obtained by acting on 0 > with creation operators in different order are orthogonal to each other < 0 a i1...a in a jn...a j1 0 >= δ i1,j1...δ in,jn implying that particles obeying inf. stat. are virtually distinguishable The partition function is Z = Σe βh, NO Gibbs factor In infinite statistics, all representations of the particle permutation group can occur.

25 Theories of particles obeying statistics are non-local [Fredenhagen; Greenberg] Number operator n i (such that n i a j a j n i = δ i,j a j ) n i = a i a i + k a k a i a ia k + l a l a k a i a ia k a l +..., k and Hamiltonian, etc., are both nonlocal and nonpolynomial in the field operators TCP theorem and cluster decomposition still hold; QFTs with statistics remain unitary Nonlocality in statistics can be a virtue may be related to nonlocality in holographic principle

26 Quanta of MDM obeying infinite statistics Modified Dark Matter via Gravitational Born-Infeld Theory A particularly useful reformulation of MDM is via an effective gravitational dielectric medium, motivated by the analogy between Coulomb s law in a dielectric medium and Milgrom s law for MoND. [E.g., write Milgrom s µ as 1+χ with χ being interpretted as gravitational susceptibility.] MoNDian force law is recovered (as can be shown) if the quanta of MDM obey infinite statistics. So we can expect unusual dynamics and interactions with ordinary matter. Perhaps this explains the difficulty in detecting dark matter. Extended nature of quanta of MDM connects them to such global aspects of spacetime as Λ and H. (Recall MDM mass profile depends on Λ [and ordinary matter].)

27 Modified Dark Matter via Gravitational Born-Infeld Theory It has been known to (Blanchet, Milgrom and) others that the MoNDian force law can be formulated as being governed by a nonlinear generalization of Poisson s equation which describes the nonlinear electrostatics embodied in the Born-Infeld theory. Consider the Born-Infeld (BI) theory defined with the following Lagrangian density (b being a dimensionful parameter; the second form is for the case of B = 0): L BI = b E2 B 2 b 2 ( E B) 2 b 4 b 2 (1 1 E 2 /b 2 ), (a nonlinear electrodynamics motivated by the analogy with relativistic mechanics given by L particle = mc 2 (1 1 v 2 /c 2 ) with c b). Let us set B = 0, concentrate on the Hamiltonian density H, supply an extra overall factor of 1 4π and use the notation D = ǫ E.

28 Then the corresponding gravitational Hamiltonian density reads: H g = b2 4π ( 1+ D2 g b 2 1 Let A 0 b 2 and A b D g, then the Hamiltonian density becomes H g = 1 4π ) ( ) A 2 +A 2 0 A 0 Assume there exists an energy equipartition (which is invoked in the derivation of Newton s law of gravity via the approach of entropic gravity) H g = 1 2 k BT eff, where T eff is an effective temperature associated with the energy. (Note that this energy density is energy per unit volume. But we can regard it as energy per degree of freedom by recalling that volume, which usually scales as entropy S, scales as the number of degrees of freedom N in a holographic setting. Parenthetically S N is one of the features of infinite statistics.)..

29 The Unruh temperature formula T eff = 2πk B a eff implies that the effective acceleration is given by a eff = A 2 +A 2 0 A 0. The equivalence principle suggests that we should identify (at least locally) the local accelerations a and a 0 with the local gravitational fields A and A 0 respectively, viz., a A, a 0 A 0. Then a eff should be identified as: a eff a 2 +a 2 0 a 0, which, in turn, implies that the Born-Infeld inspired force law takes the form (for a given test mass m) ( ) F BI = m a 2 +a 2 0 a 0, which is precisely the MoNDian force law!

30 (Note: For a typical acceleration of order 10 ms 2, the corresponding effective temperature is of order k B T eff ev, much smaller than the mass of any viable cold dark matter candidate.) To be a viable cold dark matter candidate, the quanta of our MoNDian dark matter are expected to be much heavier than k B T eff. Recall that the equipartition theorem in general states that the average of the Hamiltonian is given by H = logz(β), β where β 1 = k B T and Z denotes the partition function. To obtain H = 1 2 k BT per degree of freedom, we require Z to be of the Boltzmann form Z = exp( βh). (The conventional quantum-mechanical Bose-Einstein or Fermi-Dirac statistics would not lead to H = 1 2 k BT per degree of freedom at low temperataure.)

31 Thus the validity of H g = 1 2 k BT eff for very low temperature T eff somehow requires a unique quantum statistics with a Boltzmann partition function. This is precisely what is called the infinite statistics as described by the Cuntz algebra (a curious average of the bosonic and fermionic algebras) a i a j = δ ij. Thus, by invoking infinite statistics, the assumption of energy equipartition H g = 1 2 k BT eff, even for very low temperature T eff, is justified. Recap: (i) the relation between our force law that leads to MoNDian phenomenology and an effective gravitational Born-Infeld theory; (ii) the need for infinite statistics of some microscopic quanta which underly the thermodynamic description of gravity implying such a MoNDian force law. (A side remark: From the Matrix theory point of view, we expect infinite statistics and an effective theory of the gravitational Born-Infeld type to be closely related.)

32 SUMMARY Magnitude of spacetime fluctuations is consistent with holographic principle. Applied to cosmology, the cosmic energy density has the critical value ρ (H/l P ) 2 ; the universe of Hubble size R H contains (R H /l P ) 2 bits/particles. Long-wave constituents of dark energy act as a dynamical cosmological constant with the observed small but non-zero value Λ 3H 2. Unimodular gravity and causal set arguments also yield Λ 3H 2. The effective Λ gives rise to a critical acceleration parameter in galactic dynamics, and this naturally leads to the construction of a dark matter model in which the dark matter density profile depends on both Λ and ordinary matter. Speculation: particles constituting DE & DM obey statistics (hence, extended quanta); novel particle phenomenology; difficult to detect dark matter An intriguing question: Can quantum gravity be the origin of particle statistics with the underlying statistics being infinite statistics (and ordinary particles being collective degrees of freedom?)

33 The Margolus-Levitin Theorem Consider the maximum speed of dynamical evolution for a given physical system. Assume that the system has a discrete energy spectrum E n,n = 0,1,2... with the lowest energy chosen to be E 0 = 0. Write an arbitrary state ψ as a superposition of energy eigenstates, with coefficients c n at time t = 0. Let ψ 0 evolve for a time t to become ψ t. Consider the transition amplitude S(t) = ψ 0 ψ t. We want to find the smallest value of t such that S(t) = 0. Note that Re(S) = n c n 2 cos(e n t/ ). Using the inequality cosx 1 (2/π)(x+sinx), valid for x 0, we get Re(S) 1 2Et/(π )+2Im(S)/π, where E denotes the average energy of the system. But S(t) = 0 implies both Re(S) = 0 and Im(S) = 0. So this inequality becomes 0 1 4Et/h, where h = 2π. Thus the earliest that S(t) can possibly equal zero is when t = h/4e. Applied to a computer, this result implies that the maximum speed of computation is given by 4/h times the energy available for computation.

34 Unimodular gravity is actually very well motivated on physical grounds. Following Wigner for a proper quantum description of the massless spin-two graviton, the mediator in gravitational interactions, we naturally arrive at the concept of gauge transformations. Without loss of generality, we can choose the graviton s two polarization tensors to be traceless (and symmetric). But since the trace of the polarization states is preserved by all the transformations, it is natural to demand that the graviton states be described by traceless symmetric tensor fields. The strong field generalization of the traceless tensor field is a metric tensor g µν that has unit determinant: detg µν g = 1, hence the name unimodular gravity. It is worth mentioning: (1) Unlike the ordinary theory, the Lagrangian for unimodular gravity can be expressed as a polynomial of the metric field. (2) Conformal transformations g µν = C 2 g µν in the unimodular theory of gravity are very simple, the unimodular constraint fixes the conformal factor C to be 1.

35 Since Λ arises as an arbitrary constant of integration, it has no preferred value classically. In the corresponding quantum theory, we expect the state vector of the universe to be given by a superposition of states with different values of Λ and the quantum vacuum functional to receive contributions from all different values of Λ. Possible shortcomings: The Euclidean formulation of quantum gravity is plagued by the conformal factor problem, due to divergent path-integrals. But, in our defense, we have used the effective action in the Euclidean formulation at its stationary point only. We should also recall that the conformal factor problem is arguably rather benign in the original version of unimodular gravity (as pointed out above), so perhaps it is not that serious even in the generalized version that we have just employed. Since part of the above argument bears some resemblance to Coleman s wormhole approach, one may worry that some of the objections to Coleman s argument (on top of the conformal factor problem) may also apply here. But it appears that they do not (e.g., we have a simple instead of Coleman s double exponential in the integrand for the vacuum functional). To the extent that our argument is valid, we have understood why the observed cosmological constant is so small and positive.

36 Back-up slides: MDM: construction, tests (λ term in M ) MDM: strong lensing, cosmology; future work holographic principle statement derivation of holographic principle a disclaimer Unimodular gravity and causal set arguments (brief)

37 From Λ to modified dark matter (MDM) Review of Verlinde s entropic gravity (for Λ = 0) Constructing MDM via entropic gravity (for Λ 0)

38 Constructing MDM via (Verlinde s) entropic gravity Verlinde s recipe ( recipe ): Verlinde derives (I) Newton s 2nd law F = m a, by using (1) First law of thermodynamics entropic force F entropic = T S x, and invoking Bekenstein s original arguments concerning the entropy S of black mc holes: S = 2πk B x. (2) The formula for the Unruh temperature, k B T = a 2πc, associated with a uniformly accelerating (Rindler) observer. We will generalize the T a relation for Λ 0 case. (II) Newton s law of gravity a = GM/r 2 by considering an imaginary quasi-local (spherical) holographic screen of area A = 4πr 2 with temperature T, and using (1) Equipartition of energy E = 1 2 Nk BT with N = Ac 3 /(G ) being the total number of degrees of freedom (bits) on the screen; (2) The Unruh temperature formula and the fact that E = Mc 2. We will generalize the T M relation for Λ 0 case.

39 A particle with mass approaches a part of the holographic screen

40 A particle with mass m near a spherical holographic screen

41 Constructing MDM Need a generalization of Verlinde s proposal to de Sitter (ds) space with +ve cosmological constant Λ in an accelerating universe. Note: Unruh-Hawking temperature, as measured by an inertial observer, is T ds = 1 Λ 2πk B a 0 where a 0 = 3 H (numerically). Net temperature as measured by the non-inertial observer (due to some matter sources that cause the acceleration a) is* T T ds+a T ds = 1 2πk B [ a 2 +a 2 0 a 0]. *Expression for T ds+a as given by Deser and Levine, and by Jacobson.

42 (I) Verlinde s approach the entropic force in de Sitter space is F entropic = T x S = m[ a 2 +a 2 0 a 0]. For a a 0, we have F entropic ma. For a a 0 : F entropic m a2 2a 0, so the terminal velocity v of the test mass m should be determined from ma 2 /(2a 0 ) = mv 2 /r. For the small acceleration a a 0 regime: The observed flat galactic rotation curves (v is independent of r) and the observed Tully-Fisher relation (v 4 M) now require (recall a N = GM/r 2 ) that a ( 2a N a 3 0/π ) 1 4. But that means F entropic m a2 2a 0 = F Milgrom m a N a c. We have recovered MoND provided we identify a 0 2πa c, with the (observed) critical galactic acceleration a c Λ/3 H 10 8 cm/s 2. Thus from our perspective, MoND is a phenomenological consequence of quantum gravity.

43 (II) For an imaginary holographic screen of radius r, Verlinde s argument ( ) ( ) 2Ẽ M 2πk B T = 2πk B = 4π = G M Nk B A/G r, 2 where M represents the total mass enclosed within the volume V = 4πr 3 /3. M = M +M where M is some unknown mass, i.e., dark matter; consistency M = 1 π ( a0 a ) 2 M. [ F entropic = m[ a 2 +a 20 a 0] = ma N 1+ 1 π ( a0 For a a 0, F entropic ma ma N, and hence a = a N. (M 0) For a a 0, F entropic m a2 2a 0 ma N (1/π)(a 0 /a) 2, yielding a = ( 2a N a 3 0/π ) 1 4, as required. (M ( Λ/G) 1/2 M 1/2 r, relating dark matter (M ), dark energy (Λ) and ordinary matter (M).) a ) 2 ]

44 Observational tests of MDM In galaxies In clusters

45 Fitting rotation curves with MDM mass profiles Modified Dark Matter: [ F entropic = m[ a 2 +a 20 a 0] = ma N 1+ 1 π ( a0 a ) 2 ] To determine rotation curves: F entropic = mv 2 /r We fit rotation curves for 30 local spiral (HSB as well as LSB) galaxies. Next 4 slides: Samples of rotation curves and dark matter density profiles. Data - black squares; Stars - blue line; Gas - green line. MDM - red line; CDM - black dashed line (using NFW profile). [Sanders & Verheijen] Fitting parameters: MDM (1): mass-to-light ratio M/L; CDM (3): c,v 200, M/L. MDM uses the minimum number of parameters: hence the name Minimal DM.

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48 We fit rotation curves for 30 local spiral galaxies, providing the first astrophysical test of MDM IT PASSED!

49 Dark matter density profiles for 30 local spiral galaxies (HSB/LSB)

50 MDM IN CLUSTERS (dynamical mass versus observed mass) The total gravitating mass in Newtonian, MOND, and MDM dynamics: Newtonian dynamical mass: In Newtonian dynamics, ( ) the ( mass ) enclosed within radius r out may be estimated: M N = r out kt dlnρ G µm p dlnr, where [ ] 1.5β, ρ = ρ 0 1+ r2 r with observed values β c ;rc 0.25 Mpc; µ the mean molecular weight (0.62); m p the mass of the proton. M N 4.4M obs the old missing mass problem. MOND dynamical mass: M MOND = M N 1+(ac /a) 2 (Here we use µ(x) = x ( 1+x 2) 1/2 for the interpolating formula.) MDM dynamical mass: M MDM = M N 1+λ(a 0 /a)+(a 0 /a) 2 /π (The observed (effective) acceleration is given by a obs = a 2 +a 2 0 a 0. With the aid of the more general expression for the MDM profile, we have a obs = GM MDM r {1+λ ( a 0 ) 2 a + 1 π We also recall that a obs = GM N /r 2 for Newtonian dyanmics. ) ( a0 a ) 2}.

51 Galactic Clusters: the sample White, Jones & Forman (1997, MNRAS 292) tabulated observed temperatures and mass estimates of the hot gas for 207 clusters from X-ray data collected by the Einstein satellite. Mass of stars is estimated using the rough correlation found by David et al. (1990, ApJ, 356). M gas /M stars 0.5T kev h David s correlation and beta-models are imprecise for clusters with small outer radius. We therefore consider only clusters with outer radius 0.75 Mpc. We are left with 93 clusters. We have adapted Sanders approach (for MOND) to the case of MDM (to compare MOND with MDM, formerly known as MONDian dark matter). (Note: Preliminary results. Work in progress.)

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53 A comparison of mass profiles Proposed form of MDM mass: [ M = λ ( a0 a ) + 1 π ( a0 a ) 2 ] M, In 2014 ApJ paper, we chose λ = 0, which gave us good fits to galactic rotation curves. Fits to the cluster data suggests λ 1/2 So we re-fit the galactic rotation curves with λ = 1/2. The fits are nearly identical. The fitting parameter M/L is reduced by about 35%, and the values are still physically reasonable.

54 ( While the Modified Dark Matter profile given by M = 1 a0 ) 2 π a M reproduces the correct force laws (to the leading order) in both regimes of a a 0 and a a 0, we expect a more generic profile of the form [ ( M a0 ) = λ + 1 ( a0 ) ] 2 M, a π a with λ > 0 (and of order 1) which ensures that M > 0 when a a 0. (One can easily check that this more generic expression for M will also lead to the correct predictions for force laws in both regimes of a a 0 and a a 0 ; for the former regime, a a N +(1+λ)a 0.) As a function of r, the dark matter profile now reads (for a a 0 ): [ ( ) ( ) ( ) ] 2 M a0 1 λ + a N π λ(1+λ) a0 M. a N

55 A comment on strong lensing Strong lensing: the formation of multiple images of background sources by the central regions of some clusters. Critical surface density required for strong lensing is Σ c = 1 4π F 10, typical observations. ch 0 G F(z l,z s ), with Deep MOND limit: Σ MOND a c /G Numerical values: a c ch 0 /6 So, as noted by Sanders, MOND cannot produce strong lensing on its own: Σ c 5Σ MOND But MDM mass distribution appears to be sufficient for strong lensing: a 0 = ch 0 = 2πa c 6a c

56 Cosmology: Friedmann s Equations (short version) In a fully relativistic situation, we should use the active gravitational (Tolman-Komar) mass, i.e., replace a non-relativistic source of gravity by a fully relativistic source Friedmann s Equations: and R R = 4πG 3 (ρ+3p)+ Λ 3, H 2 = 8πG 3 ρ+ Λ 3. I.e., One can in principle have Einstein s gravity together with a(n additional) Modified Dark Matter source. If we naively use MoND at the cluster scale, we would be missing the pressure and cosmological constant terms which could be significant. This may explain why MoND doesn t work well at the cluster scale, despite the CDM-MoND duality realized at the galactic scale.

57 Cosmology: Friedmann s Equations. (long version) The FRW metric: ds 2 = dt 2 +R(t)(dr 2 +r 2 dω 2 ), where R(t) is the scale factor. Assume that the matter sources in the universe form a perfect fluid, with four velocity u µ (= (1, 0) in rest frame of fluid). Consider Verlinde s imaginary holographic screen of comoving radius r (with physical radius r = rr(t)). In a fully relativistic situation, we replace M by the active gravitational (Tolman-Komar) mass M = 1 4πG dv Rµν u µ u ν, and by Einstein s field equation, ] M = 2 dv ( T µν 1 2 Tg µν + Λ 8πG g µν Friedmann s Equations: Friedmann s Equations: and ) u µ u ν = ( 4 3 πr3 R 3)[ (ρ+3p) Λ 4πG R R = 4πG 3 (ρ+3p)+ Λ 3, H 2 = 8πG 3 ρ+ Λ 3. Thus one can in principle have Einstein s gravity together with a(n additional) MoNDian dark matter source.

58 The departure from MoND happens when (we replace M with M, i.e. when) a non-relativistic source is replaced by a fully relativistic source. In that case a2 +a 2 0 a 0 = GM r, where r = rr(t) is the physical radius, i.e., 2 a 2 +a 2 0 a 0 = G(M(t)+M (t)) r 2 +4πGp r Λ 3 r. If we naively use MoND at the cluster scale, we would be missing 4πGp r Λ 3 r which could be significant. This may explain why MoND doesn t work well at the cluster scale, despite the CDM-MoND duality realized at the galactic scale.

59 FUTURE WORK: 1. Gravitational lensing; Can it distinguish MDM from CDM? 2. Study interactions of MDM (quanta obeying infinite statistics) with ordinary (baryonic) matter particle phenomenology. The Bullet Cluster; How strongly coupled is MDM to baryonic matter? How does MDM self-interact? 3. Tests at cosmological scales (acoustic oscillations measured in the CMB...); Simulations of structure formation? 4. Stars made of quanta obeying infinite statistics? 5. Can quantum gravity be actually the origin of particle statistics and the underlying statistics is infinite statistics in that ordinary particles obeying Bose or Fermi statistics are actually some sort of collective degrees of freedom? (What are the effects on the idea of grand unification?) STAY TUNED FOR FUTURE DEVELOPMENTS

60 Holographic principle (h.p.) The Universe which we perceive to have 3 spatial dimensions can be encoded on a 2-dimensional surface, like a hologram. Max. amount of information in region is bounded by area I.e., # of degrees of freedom area in l 2 P [ thooft; Susskind] Origin of h.p.: Black holes are hot: Entropy ( r s l P ) 2 area [Bekenstein; Hawking]

61 A heuristic derivation of Holographic Principle In essence, the holographic principle ( thooft, Susskind...) stipulates that although the world around us appears to have three spatial dimensions, its contents can actually be encoded on a two-dimensional surface, like a hologram. In other words, the maximum entropy of a region of space is given (aside from multiplicative factors of order 1) by its surface area in Planck units. This result can be derived by appealing to black hole physics and the second law of theromodynamics as follows. Consider a system with entropy S 0 inside a spherical region Γ bounded by surface area A. Its mass must be less than that of a black hole with horizon area A (otherwise it would have collapsed into a black hole). Now imagine a spherically symmetric shell of matter collapsing onto the original system with just the right amount of energy so that together with the original mass, it forms a black hole which just fills the region Γ. The black hole so formed has entropy S A/l 2 P. But according to the second law of thermodynamics, S 0 S. It follows immediately that S 0 A/l 2 P, and hence the maximum entropy of a region of space is bounded by its surface area, as asserted by the holographic principle.

62 A Cautionary Disclaimer (speaking on my own behalf only) In a recent paper New Constraints on Quantum Gravity from X-ray and Gamma-Ray Observations by Perlman, Rappaport, Christiansen, Ng, DeVore, and D. Pooley (ApJ. 805, 10 (2015)), it was claimed: We reassess previous proposals to use astronomical observations of distant quasars and AGN to test models of spacetime foam. We show explicitly how wavefront distortions on small scales cause the image intensity to decay to the point where distant objects become undetectable when the path-length fluctuations become comparable to the wavelength of the radiation. (C)onstraints can be set utilizing detections of quasars at GeV energies with Fermi, and at TeV energies with ground-based Cherenkov telescopes: (they seem to rule out the holographic spacetime foam model.)

63 There are, however, a number of caveats to this idea for constraining models of spacetime foam. E.g., In this work the authors have considered the instantaneous fluctuations in the distance between the location of the emission and a given point on the telescope aperture. Perhaps one should average over both the huge number of Planck timescales during the time it takes light to propagate through the telescope system, and over the equally large number of Planck squares across the detector aperture. It is then possible that the fluctuations the authors have been calculating are vanishingly small, but at the moment there is no formalism for carrying out such averages.

64 Quantum (Generalized Unimodular) Gravity and (Dynamical) Λ (short version) Let us rederive the dynamical cosmological constant by using another method based on quantum gravity. The idea makes use of the theory of unimodular gravity (van der Bij, van Dam & Ng, Henneaux & Teitelboim, etc.), more specifically its generalized action given by S unimod = (16πG) 1 [ g(r+2λ) 2Λ µ T µ ](d 3 x)dt. In this theory, Λ/G plays the role of momentum conjugate to the coordinate d 3 xt 0 which can be identified as the spacetime volume V. Hence the fluctuations of Λ/G and V obey a quantum uncertainty principle, δvδλ/g 1. Next we borrow an argument due to Sorkin, drawn from the causal-set theory, which stipulates that continous geometries in classical gravity should be replaced by causal-sets, the discrete substratum of spacetime.

65 In the framework of the causal-set theory, the fluctuation in the number of elements N making up the set is of the Poisson type, i.e., δn N. For a causal set, the spacetime volume V becomes lp 4 N. It follows that δv lp 4 δn P l4 N l 2 P V = G V, and hence δλ V 1/2. By following an argument due to Baum and Hawking, we argued that, in the framework of unimodular gravity, Λ vanishes to the lowest order of approximation and that its first order correction is positive (at least for the the current and recent cosmic epochs). We conclude that Λ is positive with a magnitude of V 1/2 R 2 H, contributing a cosmic energy density ρ given by: ρ 1, which is of the order of lp 2 R2 H the critical density as observed!