Is Inflation Compatible with Thermodynamics?

Philosophy, Physics  · 

Our best current understanding of the origin of the universe is described by the Big Bang theory: the universe arose from a singularity. The inflation theory accounts for several problems with the standard Big Bang model, including the horizon problem, the flatness problem and the magnetic monopole problem; and it explains the origin of large-scale structures of the universe. While the theory is successful in making sense of the evolution of the early universe, the initial conditions of the inflation remains a controversial topic among cosmologists. In this essay, we will examine the inflation theory from a thermodynamical perspective.

In statistical mechanics, the thermodynamic asymmetry is indubitable – heat flows from what is hot to what is cold. For instance, sugar cube dissolves in tea, but never the reverse. The second law of thermodynamics thus defines an arrow of time – the direction in which entropy increases1. When seeking an answer in the philosophy of the asymmetry, physicists segregate into different views. A distinguished view on this asymmetry consists the following:

A. If we impose that the early universe had very low entropy, then the second law of thermodynamics is readily imbedded in the time-symmetric microphysics.

B. The second law of thermodynamics is a consequence of phase space geometry – the patch corresponding to equilibrium is much larger than other regions in phase space. Thus, unless the initial conditions of a system are extremely special, it will evolve towards a state of equilibrium (Goldstein, 2001).

Putting above postulates into the context of our discussion, (A) largely prevail on the low-entropy early universe while (B) demands that the proto-inflationary universe was in a rather “special” state2. In the following sections, I will discuss separately the above requirements.

The evolution of the early universe from inflation theorists’ point of view follows the timeline (unit: seconds): Planck Epoch (${0} < t < {10}^{-43}$), in which the all fundamental forces were unified; Grand Unification Epoch (${10}^{-43}<t<{10}^{-36}$), during which gravitation were separated from the unified forces; Inflationary Epoch (${10}^{-36}<t<{10}^{-32}$), in which the strong force was separated from the unified forces producing baryon asymmetry as observed today and the universe expended exponentially by the order of ${10}^{26}$. During the pre-inflationary period, the universe was dominated by the inflation potential $V\left(\varphi\right)$, this allows the space to expand exponentially. At this point, all of the energy density was stored in one degree of freedom – the ‘slow row’ inflaton field $\varphi$3. The spacetime was then able to equilibrate into de Sitter space, providing the essential properties for the early universe. The ‘slowly rolling’ inflation potential ends with the $\varphi$ speeding up and driving $V\left(\varphi\right)$ to decay rapidly, allowing the universe to ‘reheat’ into a thermal state of ordinary matters.

One way to examine the second law with regards to the inflation is to compare the entropy of the universe at different stages with respect to the inflationary epoch. If the universe we observe today is in a low entropy state, then the post-inflationary universe must have had even lower entropy. However, observations of the cosmic microwave background (CMB) suggest the universe was left in a state near thermal equilibrium after inflation.

To reconcile this phenomenon and the second law, we need to examine the gravitational entropy in the early universe. For a kinetically dominated system, particles would diffuse until the system reaches thermal equilibrium, leaving a smooth distribution of matter. However, for a gravitational dominated system, a homogeneous distribution of matter corresponds to a minimal gravitational entropy state: at the initial singularity $t\rightarrow0$, the Weyl conformal tensor is constrained to near diagonal (Penrose, 1980), this implies low initial gravitational entropy (Penrose, 2004). During the reheating stage of inflation, false vacuum energy was disposed as a smooth distribution of relativistic particles. The expansion and cooling of the universe then produced a smooth distribution of clumpable matter, leaving the universe in a low gravitational entropy state (Patel & Lineweaver, 2017).

Following the CMB, the emergence of self-gravitating protostars means an apparent decrease in homogeneity. But such systems would have negative microcanonical specific heat. The formation of stars implies that energy was dissipated during the process, hence increasing entropy. Blackholes come into existence from further gravitational collapses, increasing entropy drastically (Bekenstein, 1973). The highest contributors to the overall entropy of the universe are supermassive blackholes at the centre of galaxies. And such blackholes will eventually evaporate through Hawking radiations and further increase entropy (Page, 2005).

The post-inflationary entropy issue is resolved by the realisation that the post-inflation homogeneous CMB has low gravitational entropy. The overall entropy will continue to increase while the overall homogeneity of the universe reduces.

The second law would also require the proto-inflationary universe to be in a lower entropy state relative to later stages. It can be shown that this is the case for inflation by considering the entropy of the universe at different stages. Inflation theory itself is consistent with the second law, so we need to examine only the proto-inflation and post-inflation entropies.

The post-inflationary entropy can be calculated from existing data of the CMB. The current observed temperature of the CMB is $T_{CMB}=2.725\ K$. The photon number density is given by:

$n=16\pi\zeta\left(3\right)\left(\frac{k_BT_{CMB}}{hc}\right)^3=4.105\times{10}^8\ m^{-3}\tag{1}$

where the Riemann Zeta function has $\zeta\left(3\right)=1.202$. The most recent data from the Planck Observatory estimates the age of the universe to be $t_0=13.799\times{10}^9$ yr (Planck Collaboration, 2016). The comoving distance from the Earth to the edge of the observable universe is $R_{horizon}=46.6\times{10}^9$ light years (Gott, J. Richard III, et al., 2005), thus the volume of the observable universe is:

$V=\frac{4\pi}{3}{R_{horizon}}^3=3.589\times{10}^{80}\ m^3\tag{2}$

Hence the total number of photons from the CMB is:

$N=n\times V=1.473\times{10}^{89}\tag{3}$

The average entropy of each photon is $s_\gamma=3.6k_B$. So, entropy of CMB is:

$S_{CMB}=s_\gamma\times N=5.303\times{10}^{89}\ k_B\tag{4}$

An estimate of proto-inflationary entropy was made by Carroll and Chen4:

$S_{proto-inflation}=7.243^{34} \times k_B\tag{5}$ (Carroll & Chen, 2005)

Comparing above results we find the entropy of the universe at the beginning of inflation is far less than the post-inflationary entropy. The proto-inflationary and post-inflationary entropies are thus in perfect accordance with the second law of thermodynamics. Applying the results from microphysics, we have established that the universe did start off from a low entropy state while abiding to the second law of thermodynamics, hence (A) is satisfied.

The question remains, should we expect appropriate proto-inflationary conditions within the randomly fluctuating early universe? Seeking an answer to this question to its full extent is beyond the scope of this essay, what we should ask instead is, does the second law allow the imposition of such a condition? While inflation requires a small fluctuation5 in the early inflaton field to produce a universe like ours, its exponential extension is generic and abides by the second law. The arrow of time does not rule out the possibility of such a fluctuation, and if we assert that inflation did occur, its initial conditions is automatically satisfied, without conflicting the second law of thermodynamics. The motivation of inflation or other models of this its kind is not to impose completely generic initial conditions on the early universe.

Reference

Albrecht, A., 2003. Cosmic Inflation and the Arrow of Time. In: J. D. Barrow, P. Davies & C. H. eds., eds. Science and Ultimate Reality: From Quantum to Cosmos. Cambridge: Cambridge University Press, pp. 363-401.

Bekenstein, J., 1973. Black Holes and Entropy. Physics Review, D(7), pp. 2333-2346.

Carroll, S. M. & Chen, J., 2005. Does Inflation Provide Natural Initial Conditions for the Universe?. International Journal of Modern Physics, D(15), pp. 2335-2340.

Fixsen, D., 2009. The Temperature of the Cosmic Microwave Background. The Astrophysical Journal, 707(2), pp. 916-920.

Goldstein, S., 2001. Boltzmann’s approach to statistical mechanics. In: J. Bricmont, et al. eds. Chance in Physics: Foundations and Perspectives. Berlin: Springer, pp. 39-54.

Gott, J. Richard III, et al., 2005. A Map of the Universe. The Astrophysical Journal, 624(2).

Page, D., 2005. Hawking Radiation and Black Hole Thermodynamics. New Journal of Physics, 7(203).

Patel, V. & Lineweaver, C., 2017. Solutions to the Cosmic Initial Entropy Problem without Equilibrium Initial Conditions. Entropy, 19(411).

Penrose, R., 1980. Singularities and time asymmetry. In: S. W. Hawking & W. E. Israel, eds. General Relativity: An Einstein Centenary Volume. Cambridge: Cambridge University Press, pp. 581-638.

Penrose, R., 2004. The Big Bang and its Thermodynamic Legacy. In: Road to Reality: A Complete Guide to the Laws of the Universe. London: Johnathan Cape, pp. 686-743.

Planck Collaboration, 2016. Planck 2015 results XIII. Cosmological parameters. Astronomy & Astrophysics, XIII(594).

  1. Hawking presented a short and elegant thought experiment in chapter 9 of his book “A Brief History of Time” to demonstrate this definition of the arrow of time. ↩

  2. For inflation to begin from randomly fluctuating early universe, there would have been a small fluctuation in inflaton field to produce a region like the universe we live in. ↩

  3. In the beginning of inflation, the scalar field \varphi starts out in the rather flat part of the inflation potential, varying slowly with time. ↩

  4. This is a calculation of the entropy of inflaton field \varphi, it relies on the estimate of the inflationary energy scale. ↩

  5. The proto-inflationary initial conditions are only special in a small region of the inflaton field, not the entire state of the universe (Albrecht, 2003). ↩