entropy Article Solutions to the Cosmic Initial Entropy Problem without Equilibrium Initial Conditions Vihan M. Patel 1 and Charles H. Lineweaver 1,2,3,* 1 Research School of Astronomy and Astrophysics, Australian National University, Canberra 2600, Australia; [email protected] 2 Planetary Science Institute, Australian National University, Canberra 2600, Australia 3 Research School of Earth Sciences, Australian National University, Canberra 2600, Australia * Correspondence: [email protected] Received: 30 June 2017; Accepted: 7 August 2017; Published: 10 August 2017 Abstract: The entropy of the observable universe is increasing. Thus, at earlier times the entropy was lower. However, the cosmic microwave background radiation reveals an apparently high entropy universe close to thermal and chemical equilibrium. A two-part solution to this cosmic initial entropy problem is proposed. Following Penrose, we argue that the evenly distributed matter of the early universe is equivalent to low gravitational entropy. There are two competing explanations for how this initial low gravitational entropy comes about. (1) Inflation and baryogenesis produce a virtually homogeneous distribution of matter with a low gravitational entropy. (2) Dissatisfied with explaining a low gravitational entropy as the product of a ‘special’ scalar field, some theorists argue (following Boltzmann) for a “more natural” initial condition in which the entire universe is in an initial equilibrium state of maximum entropy. In this equilibrium model, our observable universe is an unusual low entropy fluctuation embedded in a high entropy universe. The anthropic principle and the fluctuation theorem suggest that this low entropy region should be as small as possible and have as large an entropy as possible, consistent with our existence. However, our low entropy universe is much larger than needed to produce observers, and we see no evidence for an embedding in a higher entropy background. The initial conditions of inflationary models are as natural as the equilibrium background favored by many theorists. Keywords: entropy; gravity; inflation; initial conditions 1. The Entropy of the Universe, the Second Law, the Past Hypothesis, and the Cosmic Initial Entropy Problem As stars shine and black holes accrete, the entropy of the universe goes up [1]. Galaxies with shining stars and accreting black holes are distributed relatively homogeneously across the universe. This observed homogeneity of the universe on large scales implies an approximately zero net flow of entropy between volumes larger than a scale of a few hundred million light years. These large representative volumes of the universe are effectively closed. Thus, the second law of thermodynamics applies to the entire universe, and its entropy does not decrease: dS ≥ 0 [2–4]. In particular, the entropy of the observable universe Suni (defined as the entropy of the comoving volume of our current particle horizon) does not decrease [5–7]. Thus, the entropy of the universe was smaller in the past [8,9] and will be larger in the future. This requirement of low entropy conditions in the early universe is often called the ‘past hypothesis’ [10–13]. The increasing entropy of the universe will eventually approach a maximum entropy state: Smax ([1,14,15], but see [16] for an opposing view). The second law and the past hypothesis predict that the early universe was at low entropy. However, the photons in the cosmic microwave background (CMB) have temperature deviations of DT −5 only T ∼ 10 [17] around their average blackbody temperature of 2.7 K [18]. The entropy of a Entropy 2017, 19, 411; doi:10.3390/e19080411 www.mdpi.com/journal/entropy Entropy 2017, 19, 411 2 of 9 Entropy 2017, 19, 411 2 of 9 only ~ 10 [16] around their average blackbody temperature of 2.7 K [17]. The entropy of a given givencomoving comoving volume volume of blackbody of blackbody photons photons in an in anexpanding expanding universe universe (including (including deSitter deSitter universes) universes) remainsremains constantconstant [[1,18]1,19] andand correspondscorresponds toto maximalmaximal entropyentropy forfor thethe photons.photons. Thus,Thus, thethe cosmiccosmic microwave background radiation is remarkablyremarkably closeclose toto anan equilibriumequilibrium blackbodyblackbody spectrumspectrum andand revealsreveals anan apparentlyapparently highhigh entropyentropy universeuniverse closeclose toto thermalthermal andand chemicalchemical equilibrium.equilibrium. TheThe cosmiccosmic initialinitial entropyentropy problem problem is is illustrated illustrated in in Figure Figure1a. Solutions1a. Solutions to the to cosmicthe cosmic initial initial entropy entropy problem problem (IEP) need(IEP) toneed explain to explain both theboth initial the initial low entropylow entropy of the of universe the universe (required (required by the by secondthe second law), law), and and the apparentthe apparent high high entropy entropy of the of observedthe observed CMB. CMB. Figure 1. TheThe Initial Initial Entropy Entropy Problem. Problem. (a) The (a) Thesecond second law and law the and past the hypothesis past hypothesis make a makelow entropy a low entropyprediction prediction for the early for the universe. early universe. However, However, observations observations of the cosmic of the microwave cosmic microwave background background (CMB) (CMB)show a show universe a universe at thermal at thermal and chemical and chemical equilibrium, equilibrium, i.e., i.e.,maximum maximum entropy. entropy. The The problem problem is isresolved resolved in in (b ()b when) when we we include include the the low low gravitational gravitational en entropytropy of of the the homogeneous distribution ofof matter in the early universe and definedefine a new maximum entropy that includes gravitational entropy: =, +,. Also, we require , ≫, . Thus, the inclusion of SMAX = SMAX, grav + SMAX, CMB. Also, we require SMAX, grav SMAX, CMB. Thus, the inclusion of gravitational entropy resolves the discrepancydiscrepancy betweenbetween our expectationsexpectations of a lowlow entropyentropy beginningbeginning andand thethe observedobserved highhigh entropyentropy ofof thethe CMB.CMB. 2. Gravitational Entropy and Penrose’s WeylWeyl CurvatureCurvature HypothesisHypothesis 2.1. Kinetically vs. Gravitationally-DominatedGravitationally-Dominated Systems InIn kinetically-dominated kinetically-dominated systems, systems, any any concentrated concentrated particles particles (e.g., (e.g., perfume perfume in a bottle) in a diffuse bottle) diffuseuntil the until particles the particles are aredistributed distributed relatively relatively homogeneously. homogeneously. A A homogeneous homogeneous distributiondistribution correspondscorresponds to aa statestate ofof thermalthermal equilibriumequilibrium andand maximummaximum entropy. PenrosePenrose [[6,19]6,19] hashas suggestedsuggested thatthat gravitationally-dominated gravitationally-dominated systems systems behave behave in inthe the opposite opposite way: way: a smoot a smoothh distribution distribution of matter of mattercorresponds corresponds to minimal to minimal gravitational gravitational entropy. entropy.In the panels In the of panelsFigure of2b, Figurestarting2b, from starting a close-to- from ahomogeneous close-to-homogeneous distribution, distribution, the entropy the increases entropy with increases gravitational with gravitational collapse to collapseblack holes to black[14]. holesThe highest [14]. gravitational The highest entropies—as gravitational entropies—aswell as the largest well ascontributions the largest to contributions the entropy toof thethe entropyuniverse—are of the in universe—are the supermassive in the black supermassive holes at blackthe centre holes of at many the centre galaxies of many[4,7]. galaxiesThe eventual [4,7]. TheHawking eventual evaporation Hawking evaporationof the black of holes the black [20] holes increases [20] increases the entropy the entropy even further even further [21–23]. [21 –The23]. Thegravitational gravitational entropy entropy ( (S) gravof this) of progression this progression from froma smooth a smooth distribution distribution of matter, of matter, to black to blackholes holesand then and to then the tophotons the photons from fromblack blackhole evaporation, hole evaporation, cannot cannot yet be yet expressed be expressed and quantified and quantified in an inequation an equation of the of form the form Sgrav=(= (f,(P ()k),, t))where, where (,P( k), tis) isthe the time-dependent time-dependent power power spectrum spectrum ofof large-scalelarge-scale structure. In order for for Sgrav toto solve solve the the initialinitial entropyentropy problem,problem, gravitationalgravitational collapsecollapse andand thethe resulting change in in Sgrav hashasto to bebe thethe sourcesource ofof allall entropyentropy increase since the end of of inflation. inflation. Entropy 2017, 19, 411 3 of 9 Entropy 2017, 19, 411 3 of 9 Figure 2. Comparison of the entropic evolution of (a) a kinetically-dominated system undergoing Figure 2. Comparison of the entropic evolution of (a) a kinetically-dominated system undergoing diffusiondiffusion and and ( b)) a gravitationally-dominated system system un undergoingdergoing collapse collapse and and evaporation via via HawkingHawking radiation. radiation. The The relationship relationship between between gravitat gravitationalional collapse collapse and the and increase the increase in entropy in entropy is not wellis not established, well