Документ взят из кэша поисковой машины. Адрес оригинального документа : http://www.mso.anu.edu.au/~charley/papers/LineweaverEganParisv2.pdf Дата изменения: Mon Nov 23 04:35:01 2009 Дата индексирования: Tue Oct 2 06:03:17 2012 Кодировка: Поисковые слова: релятивистское движение

Dark Energy and the Entropy of the Observable Universe
Charles H. Lineweavera and Chas A. Eganb
a

Planetary S cinece Institu te, Research School o f Astronomy and Astrophysics, and Research School of Earth Scien ces Austra lian National Un iversity b Research School of Astronomy and Astrophysics, Austra lian Na tional University

Abstract. The initial low entropy of the universe has allowed irrev ersible processes, such as th e read er read ing this abstract, to happen in the universe. This initial low entropy is due to a low value fo r the initial gravitational entropy of the universe. Th e standard CD M co smology has a cosmic event horizon and an associated G ibbons-Hawking entropy . We compute th e en tropy of the universe in cluding the entropy of the current even t horizon and the en tropy of the m atte r and photons within th e co smic ev ent horizon. We estim ate th e en tropy of the current cosmic ev ent horizon to be 2.6 ± 0.3 x 10122 k and find it to be ~ 1019 times larg er th an the next mo st dominant contribution, which is from super m assiv e black holes. We plot an entropy budget as a function of tim e and find that the co smic even t horizon entropy has dominated other sources of entropy sin ce 10-20 seconds after the big bang. See Egan & Lineweaver (2009) for details and discu ssion. Keywords: Entropy, Gravity, D ark En ergy PACS: 96.10, 96.12, 96.55, 98.80

THE INITIAL ENTROPY PROBLEM OF THE UNIVERSE Irreversible things keep happening because the universe is not in equilibrium. Coffee cups keep cooling down, the Sun keeps shining and we all keep going about our business oxidizing carbohydrates, then dying. The second law of thermdynamics, dS 0, quantifies these irreversible changes and provides a direction to time. If the universe had started out in equilibrium, the entropy of the universe, Suni ,would be a maximum; Suni = Smax, and irreversible processes, like cooling coffee cups and life forms, would not happen. Without irreversible processes we would have dS = 0, and this fact would be unobservable since there would be no irreversible processes to observe it. The early universe appears to have been in thermal and chemical equilibrium. Figure 1 is an image of the cosmic microwave background (CMB) and shows the universe as it was about 380,000 years after the big bang. In any one direction, the temperature of the photosphere of the universe, (i.e. the surface of last scattering) is almost identical to the temperature in the opposite direction. Both are ~ 2.725 K (Mather et al 1999). Variations in the temperature of the CMB are a few parts in 105 (Smoot et al 1992) and are too small to be seen on this map. If the early universe were in complete thermal, chemical (and all other types of) equilibrium, it would have been at maximum entropy. Therefore, some low entropy component must be missing, otherwise the entropy of the universe could not have increased as it has over the past 13.7 billion years. Thus, although the CMB appears to suggest that Sinitial ~ Smax, we must have Sinitial << Snow << Smax. Penrose (1979, 1987, 1989 & 2004) has pointed out the significance of this paradox and has suggested that low gravitational entropy (Fig. 2) is the missing component.


FIGURE 1. Full-sky map of the temperature of the cosmic microwave background. Variations in the temperature are a few parts in 105 and are too small to be seen on this map.

FIGUR E 2 . In order to satisfy Sinitial << Snow << Smax , and allow for th e n ear ther mal and chemical equilibriu m of the CMB, Sinitial must be due to grav ity and hav e a very low v alue. Ther e is little consensus about how to d efin e Smax or Sinitial or about whether S (= Smax - Suni ) is increasing or decreasing (Egan & Lin eweaver 2010). Figure from Linew eav er & Eg an (2008).


FIGUR E 3 . Top p anel: when grav itational in ter actions are very w eak , diffusion incr eases the volume an d entropy of the objects (e.g. molecules of perfume in a room). Bo tto m panel: when th e objects ar e large an d gravitational interactions are strong, angular momen tum transfer to some objects and their ejection allows th e gravitational co llapse of oth er objects in to black holes. If th e Hawk ing temper ature of the b lack hole is h igher th an the environmen t, the black hole w ill evaporate as photons and other par ticles. In all of th ese irrev ersib le processes, entropy must incr ease. Dark energy does not clump, so it cannot contribute to the incr easing gravitational entropy of the un iverse. Figure from Lin eweaver & Egan (2008).

PARTICLE AND EVENT HORIZONS AS NORMALIZING VOLUMES We would like to quantify the entropy of the universe. This can be done by considering the comoving entropy density normalized to some large representative volume (such as the current paricle horizon) or the time dependent volume inside the current event horizon. These are referred to respectively as " scheme1 " and " scheme 2 " in Egan and Lineweaver (2009). Here we focus on scheme 2. The radius of the particle horizon R
PH

is: (Eq. 1)

Inserting the Friedmann equation for a(t ) including the dark energy term of the standard CDM model, yields RPH ( to) = 47 Glyr. The radius of the cosmic event horizon RCEH is: (Eq. 2) Inserting the Friedmann equation for a(t ) with the values of the standard CDM model yields RCEH ( to) = 15.7 Glyr. If we let t approach infinity we get RCEH ( t ) = 16.4 Glyr, which is only slightly larger than 15.7 Glyr. This can be seen in the top panel of Fig. 4 where the 15.7 Glyr and 16.4 Glyr radii of the event horizon can be read off the proper distance on the x-axis. Notice that in comoving space (middle panel) the event horizon is shrinking and as it does, comoving galaxies move outside the event horizon. The entropy of the cosmic event horizon according to Gibbon & Hawking (1977) is: (Eq. 3)


FIGURE 4. Th e size of the observable univ erse is the particle horizon (Eq. 1 ). The comoving size of the cu rrent p article horizon is colored grey in all th ree pan els. The event horizon (Eq. 2) as a function of tim e is d istinct from the Hubble sph ere (wh ere th e velocity of recession equals th e speed of light). In the top pan el th e dashed comoving lines of the g alaxies show th e expan sion of the universe w ith respect to proper d istan ce on th e x-axis. In the m iddle and bottom panels we u se comoving sp ace as the x axis, so th e expansion is scaled out. In th e bottom pan el temporal "expan sion" is scaled out by the u se of conform al time = tim e/a.

Inserting the value for R

CEH

derived above yields: (Eq.4)

SCEH = 2.6 x 10122 k

Since (RCEH (t ) / RCEH (to))2 ~ 1.1, the ultimate entropy of the cosmic event horizon is only about 10% larger than the entropy associated with it today. THE ENTROPY BUDGET OF THE UNIVERSE We have used recent measurements of the supermassive black hole mass function to find that, after the entropy of the cosmic event horizon, supermassive black holes are the largest contributor to the entropy of the universe -- contributing at least an order of magnitude more entropy than previously estimated (Frampton & Kephart 2008). We have also made the first tentative estimate of the entropy of dark matter within the cosmic event horizon.


TABLE 1. Th e entropy of the universe including the G ibbons-H awking entropy of the co smic even t horizon as well as th e entropy of the dominant components contained w ithin the co smic event horizon. See Egan & Lin eweaver (2009) for details.

THE ENTROPIC MEANING OF BLACK HOLE AND COSMIC EVENT HORIZONS The entropy of the cosmic event horizon dominates the budget of entropy of the universe. It is unclear what significance this has. Is the Gibbons-Hawking entropy of the cosmic event horizon due to the loss of information to the observer as comoving objects go across the horizon? Davis, Davies and Lineweaver (1999) suggest that in order to obey the generalized second law, the entropy in the matter and photons that passes outside the horizon and is lost, is compensated for by the increasing entropy of the cosmic event horizon. Yet the growth of the horizon depends on the energy crossing the horizon, not the entropy. This led Bekenstein (1981) to propose the Bekenstein bound on the entropy of a system. Lloyd (2002) relates the amount of information I, in bits, to the logarithm of the number of distinct quantum states available to the system: I = (Smax / k ln 2). The monotonic growth of the cosmic event horizon (it cannot shrink because its entropy must not decrease) seems to rule-out phantom equations-of-state, where w < -1; these would result in a shrinking cosmic event horizon. To interpret the CEH entropy itself as a maximum entropy is to invoke the holographic bound on the volume of the cosmic event horizon. Fig. 6 shows the holographic bound being violated between 10-40 and 10-20 seconds after the big bang, when the entropy of radiation is larger than the entropy of the cosmic event horizon. During this time interval the radiation within the CEH contained more entropy than the CEH itself. This happens because the holographic bound is not expected to hold on volumes much larger than the Hubble sphere. One can see in Fig. 4 that for very early times, the event horizon is much larger than the Hubble sphere. Any volume larger than the Hubble sphere is more dense than a black hole the same size. Violations of the standard holographic bound, such as the one seen here, motivated Bousso (1999) to propose a modified holographic bound, the Covariant Entropy Conjecture, which is not violated in the same way. ACKNOWLEDGMENTS We acknowledge useful discussions with Tamara Davis and Paul Davies.


FIGUR E 5. The Gibbons-Hawking entropy of the cosmic event horizon dominates th e entropy budget of the conten ts of the even t horizon. We hav e assumed an epoch of inflation and reheating at th e Planck time, 10-43 seconds after the big b ang. Although, the en tropy of the even t horizons of super-massive black holes ("SMBHs") is 1019 times smaller than the en tropy of the current cosmic ev ent horizon, the en tropy from SMBH s dominates all other forms of en tropy, in cluding stellar mass b lack holes and th e entropy of the cosmic microw ave b ackground and th e neutrino b ackground.

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