The Entropy of the Universe

First, what is entropy? Is it a philosophers dream?

"Acqua Alta - L'amnésie contagieuse de l'amante religieuse." by jef safi

The chaos is not a shapeless state, or a confused and inert mixing, but rather the place of a plastic and dynamic becoming... Philosophy, science and art are "drawing plans" on the chaos, they are the three chords. Philosophy on its plane of immanence of ideas and concepts, science on its plane of consistency of variables and functives, and art on its plane of composition of affects and percepts." -Gilles Deleuze, Qu'est-ce que la philosophie?

That was about as quantitative as dancing about architecture. Let us define entropy as a measurement of disorder in a system. The entropy of a system can be defined as proportional to (the natural log of) the number of microstates corresponding to the observed system macrostate. Entropy is usually defined by big S.

Where Ω is the number of accessible microstates and where k_B = 1.381×10⁻²³ J K⁻¹ (through out the rest of this post k will represent Boltzmann constant and K the unit of Kelvin) is the Boltzmann constant in units of Joules per Kelvin. Entropy has always intrigued me so when I saw this paper on entropy of the entire Universe I was delighted.

A Larger Estimate of the Entropy of the Universe

Authors: Chas Egan, Charles Lineweaver

Abstract: Using recent measurements of the supermassive black hole mass function we find that supermassive black holes are the largest contributor to the entropy of the observable Universe, contributing at least an order of magnitude more entropy than previously estimated. The total entropy of the observable Universe is correspondingly higher, and is S_obs = 3.1×10¹⁰⁴ k. We calculate the entropy of the current cosmic event horizon to be S_CEH = 2.6±0.3×10¹²² k, dwarfing the entropy of its interior, S_{CEH int}= 1.2×10¹⁰³ k. We make the first tentative estimate of the entropy of dark matter within the observable Universe, S_dm = 10^88±1 k. We highlight several caveats pertaining to these estimates and make recommendations for future work.

Entropy is important in the Universe because it is inexorably linked to the arrow of time and understanding why the Universe began in such a low entropy state, namely the big bang, is an open question (considering the anthropic explanation unsatisfactory or at least not fully quantitative). A better quantifiable question that we can answer is the current total value of entropy in our Universe and the constituent contributions from major astrophysical phenomena. The authors explain that the increase of the entropy budget of our Universe is associated with all irreversible process including gravitational clustering, accretion disks, supernovae, stellar fusion, terrestrial weather, chemical, geological and biological processes. The authors have assumed a flat Universe with standard cosmological parameters (Ω_k=0, h=.705, Ω_b = 0.0224, Ω_m= 0.136 and T_cmb=2.725 K) and applied the second law of thermodynamics to the determine the entropy contribution of the most dominate processes in the entropy budget. The exact volume of the Universe is the dominating error in some of the estimates. In fact the entire definition of the volume of the Universe is questionable. In the case of the observable Universe we are considering a volume with a moving boundary in which matter or information may flow in and out, however we may attempt to dispel this concern that our system is not closed because as the authors state, 'the system is effectively isolated because large-scale homogeneity and isotropy imply no net flows of entropy into or out of the comoving volume'. In the appendix of the paper they explain how to calculate the volume of the Universe in a very simple manner if you are familiar with cosmography. The volume of the observable Universe is 3.65×10⁸⁰ m³ or s 43.2×10⁴ glyr³.

Figure 1 from the paper illustrates the particle and cosmic event horizons. At the origin on the x-axis is a vertical dashed line representing our galaxy. In the top panel the x-axis is the comoving distance, x=D/a where a is the cosmic scale factor. In the bottom panel the x-axis is the proper distance D. The region inside the particle horizon is the observable Universe. The comoving volume that corresponds to the observable Universe today is filled grey.

The largest contributor to the entropy budget of our Universe is super massive black holes (BH or SMBH). In the Schwarzchild case:

where , G is the gravitational constant, h (or rather h-bar) is Planck's constant, c is the speed of light, A is area, and M is mass. The largest contributor to the entropy of the Universe besides SMBH is certainly the cosmic microwave background which can be calculated succinctly using the equation of blackbody from Kolb and Turner (1990):

Where Tγ is the photon temperature and g_γ is the number of photon spin states, namely 2. The results for all phenomena in the entropy budget considered are tabulated in the table below:

Table 1 from the paper. The bracketed numbers refer to previous literature references.

It was interesting to hear the discussion on neutrino and graviton entropy. The total entropy of the neutrino background is a little tricky to compute because the neutrino entropy cannot be calculated directly because the temperature of the cosmic neutrino background has not been measured (to young scientists out there, go measure it and report back for a guaranteed Noble Prize). Additionally, infall of neutrinos into nonlinear structure with significant gravitational potentials may alter the neutrino entropy and the authors see this as possible future work. Then there is the relic graviton radiation, and its entropy contribution which is again insignificant compared to SMBH, but it is interesting to note that by reversing the relationship between the current graviton temperature and photon temperature it may enable, 'calculating the number of relativistic degrees of freedom at the Planck time using future measurements of the graviton background temperature'. Dark matter is the final twist. The authors present the first ever tentative estimates by interpreting it as a weakly-interacting superpartner to conclude its contribution is minimal.

Ultimately the entire entropy budget is dominated by black holes and critically the masses of black holes in our Universe. Entropy increases when gravitons are produced. The contribution of super massive black holes to entropy comes from the production of gravitons. Take for example, not a black hole, but another extreme gravitational system of two black holes or neutron stars inspiralling towards each other; gravitational waves are emitted from the system extracting orbital energy and therefore entropy allowing the system to contract. The authors find the BH entropy to be the dominating factor and larger by at least an order of magnitude compared to previous estimates which have different BH initial mass functions (IMF). Indeed the flaw in any work that attempts to make estimates of populations of stars, galaxies, or BHs faces the issue of ambiguous IMFs for the objects. The primary source of uncertainty that I perceive here is the lack of understanding of the IMF. Regardless if anyone was wondering what the entropy of the observable Universe is, and I know I was, it is 3.1×10¹⁰⁴ k .

References:

Chas A. Egan, & Charles H. Lineweaver (2010). A Larger Estimate of the Entropy of the Universe ApJ arXiv: 0909.3983v3