THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY P. Fern´ andez and J.M. Isidro October 16, 2017 P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Summary 1 Introduction 2 Methods 3 Results and discussion 4 Conclusions 5 References P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Introduction The continuum description of spacetime breaks down at short length scales and/or high curvatures. A continuum description emerges after coarse graining some unknown, underlying degrees of freedom. Thermodynamical approach: ignore large amounts of detailed knowledge, concentrate on a few coarse–grained averages. Emergent approach to spacetime: gravity is an entropic force. P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
We do not know the fundamental degrees of freedom of gravity, but their coarse–grained effect is to drive the system in the direction of increasing entropy. Gravitational equipotential surfaces can be identified with isoentropic surfaces. The (baryonic and dark) matter content of a hypothetical Newtonian Universe is regarded as a density of particles | ψ | 2 , where ψ satisfies the Schroedinger equation H ψ = E ψ Given the gravitational potential U , the expectation value � ψ | U | ψ � measures the gravitational entropy of the Universe when the matter is in the state ψ . P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Methods Newtonian cosmology: Gravity described by the Poisson Eq., ∇ 2 U = 4 π G ρ, matter described by continuity and Euler Eqs. (ideal fluid): ∂ρ ∂ v ∂ t + ( v · ∇ ) v + 1 ∂ t + ∇ · ( ρ v ) = 0 , ρ ∇ p − F = 0 . Hubble’s law: H 0 = Hubble ′ s constant v = H 0 r , This implies a repulsive harmonic potential U Hubble ( r ) = − H 2 2 r 2 0 P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Madelung: Factorising ψ into amplitude and phase, � S + i I � ψ = exp , 2 k B � Schroedinger quantum mechanics becomes a fluid mechanics: ∂ v ∂ t + ( v · ∇ ) v + 1 m ∇Q + 1 m ∇ V = 0 , Q := − � 2 S v = 1 ( ∇ S ) 2 + ∇ 2 S � � , S := , m ∇I 2 m 2 k B Q is the quantum potential, V the external potential in H ψ = E ψ . P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Both Newtonian cosmology and Schroedinger quantum mechanics are fluid mechanics: Euler Madelung volume density ρ exp(2 S ) ∇I / m velocity v ∇ p /ρ ∇Q / m pressure term −∇ V / m external forces F Thus Newtonian cosmology can be regarded as a nonrelativistic quantum mechanics. Mass m V contained within a volume V : � d 3 x | ψ | 2 m V = m V The observable Universe has a (baryonic and dark) mass m within a sphere of radius R 0 . P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
What is the Hamiltonian of the (matter content of a Newtonian) Universe? First approximation: the free Hamiltonian H free = − � 2 2 m ∇ 2 Eigenfunctions: free spherical waves with l = 0 , m l = 0 1 1 √ 4 π R 0 ψ κ 00 ( r , θ, ϕ ) = r exp ( i κ r ) , κ ∈ R , Second approximation: the Hubble Hamiltonian H Hubble = − � 2 2 m ∇ 2 − k eff 2 r 2 , k eff = mH 2 0 governs the Hubble expansion of the Universe. P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Hubble eigenfunctions: H Hubble ψ = E ψ with l = 0 , m l = 0: α ( r , θ, ϕ ) = N (1) � i β 2 r 2 � � 3 � 4 − i α 4 , 3 ψ (1) 2; − i β 2 r 2 √ α exp 1 F 1 2 4 π and α ( r , θ, ϕ ) = N (2) � i β 2 r 2 � � 1 � 1 4 − i α 4 , 1 ψ (2) 2; − i β 2 r 2 √ α r exp 1 F 1 . 2 4 π β 4 := m 2 H 2 α := 2 E 0 , , � 2 � H 0 N (1) N (2) radial normalisations, 1 F 1 confluent hypergeometric α α function. P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Results and discussion The gravitational entropy operator S g := N k B mH 0 R 2 � is suggested by Verlinde’s entropic gravity and by Hubble’s law. N : undetermined dimensionless factor. For the free eigenfunctions: � ψ κ 00 |S g | ψ κ 00 � = 10 123 k B , N = 3 / 2 . 6 This saturates the holographic bound. For the Hubble eigenfunctions: � ψ (1) α |S g | ψ (1) α � = 10 120 k B = � ψ (2) α |S g | ψ (2) α � , N = 1 / 6 Three orders of magnitude below the holographic upper bound. P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
Conclusions The holographic principle: S max ≃ 10 123 k B for the whole Universe. Phenomenological estimates: S measured ≃ 10 104 k B . Gravitational entropy (black holes) are the largest single contributors to the entropy budget. Even without black holes, our toy model captures some key elements: the holographic principle is respected by free waves, Hubble waves do not even saturate it. A fully relativistic description will improve these theoretical estimates. P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
References [1] D. Cabrera, P. Fern´ andez de C´ ordoba and J.M. Isidro, Boltzmann Entropy of a Newtonian Universe , Entropy 19 (2017) 212, arXiv:1703.08082 [quant-ph] . [2] C. Egan and C. Lineweaver, A Larger Estimate of the Entropy of the Universe , Astroph. J. 710 (2010) 1825, arXiv:0909.3983 [astro-ph.CO] . [3] P. Fern´ andez de C´ ordoba and J.M. Isidro, On the Holographic Bound in Newtonian Cosmology , arXiv:1710.00507 [gr-qc] . [4] E. Hubble, A Relation between Distance and Radial Velocity among Extra–Galactic Nebulae , Proc. Nat. Acad. Sci. 15 (1929) 168. P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
[5] S. Perlmutter et al. , Cosmology from Type Ia Supernovae , Bull. Am. Astron. Soc. 29 (1997) 1351, arXiv:astro-ph/9812473 . [6] Planck Collaboration, Planck 2015 Results. XIII. Cosmological Parameters , A&A 594 A13 (2016), arXiv:1502.01589 [astro-ph] . [7] A. Riess et al. , Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant , Astron. J. 116 (1998). [8] E. Verlinde, On the Origin of Gravity and the Laws of Newton , JHEP 1104 (2011) 029, arXiv:1001.0785 [hep-th] . [9] S. Weinberg, Cosmology , Oxford University Press, Oxford (2008). P. Fern´ andez and J.M. Isidro THE HOLOGRAPHIC BOUND IN NEWTONIAN COSMOLOGY
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