Predicting the cosmological constant from the causal entropic principle

We compute the expected value of the cosmological constant in our universe from the causal entropic principle. Since observers must obey the laws of thermodynamics and causality, the principle asserts that physical parameters are most likely to be found in the range of values for which the total entropy production within a causally connected region is maximized. Despite the absence of more explicit anthropic criteria, the resulting probability distribution turns out to be in excellent agreement with observation. In particular, we find that dust heated by stars dominates the entropy production, demonstrating the remarkable power of this thermodynamic selection criterion. The alternative approach—weighting by the number of “observers per baryon”—is less well-defined, requires problematic assumptions about the nature of observers, and yet prefers values larger than present experimental bounds.

Figure 1

The probability distribution of the vacuum energy measured by typical observers, computed from the causal entropic principle, is shown as a solid curve. The values consistent with present cosmological data, in the shaded vertical bar, are well inside the $1\sigma$ region (shown in white), and hence, not atypical. For comparison, the dashed line shows the distribution derived by estimating the number of observers per baryon. Unlike our curve, it assumes that galaxies are necessary for observers; yet, the observed value is very unlikely under this distribution. For more details about both curves, see Figs. 2, 8.

Figure 2

Weighted by “observers per baryon,” the probability distribution for ${\rho }_{\Lambda }$ depends strongly on specific assumptions about conditions necessary for life. Three curves are shown, corresponding to different choices for the minimum required mass of a galaxy: $M_{*}=({10}^7,{10}^9,{10}^{12})M_{\odot }$. In neither case is the observed value (vertical bar) in the preferred range. The choice $M_{*}={10}^7{M}_{\odot }$ (also shown in Fig. 1) corresponds to the smallest observed galaxies. The choice $M_{*}={10}^{12}{M}_{\odot }$ minimizes the discrepancy with observation but amounts to assuming that only the largest galaxies can host observers. By contrast, the causal entropic principle does not assume that observers require structure formation, let alone galaxies of a certain mass; yet its prediction is in excellent agreement with the observed value (see Fig. 8).

Figure 3

A causal diamond is the largest spacetime region over which matter can interact. It is delimited by a future light cone from a point on the reheating surface (orange/light), and by a past light cone from a late-time event (blue/dark); in the case of de Sitter vacua this is the cosmological horizon. A vacuum should be weighted by the number of observations made in this spacetime region. Since observation requires free energy, we expect that on average, this number will be proportional to the amount of entropy, $\Delta S$, produced in the causal diamond. Entropy entering through the bottom cone (bottom arrow), such as the CMB, does not contribute to this entropy difference.

Figure 4

The lower plot shows comoving volume in the causal diamond, $V_{\mathrm{c}}$, as a function of time for ${\rho }_{\Lambda }=(0.1,1,10)$ times the observed value given in Eq. (1.1). The kink in the comoving volume corresponds to the “edge” of the causal diamond, where the top and bottom cone meet (see Fig. 3). The upper plot shows the rate of entropy production computed in Sec. 3c, which peaks around 2 to 3.5 Gyr. As explained in Sec. 4b, the causal entropic principle prefers values of ${\rho }_{\Lambda }$ such that the (${\rho }_{\Lambda }$-dependent) peak of the comoving volume coincides with the (nearly ${\rho }_{\Lambda }$-independent) peak of the entropy production rate (see Fig. 9).

Figure 5

The star formation rate as a function of time from Nagamine et al. [54], labeled N, and from Hopkins and Beacom [55], labeled H. They are peaked at $t_{\star }\sim 1.7$ and 2.8 Gyr, respectively. We have normalized both SFRs such that the stellar luminosity density calculated below agrees with the bolometric luminosity observed today. It should be noted, however, that our result is not sensitive to this renormalization.

Figure 6

The entropy production rate in our universe as a function of time, from dust heated by starlight. The two curves shown correspond to different models of the star formation history of the universe [54, 55]; see Fig. 5.

Figure 7

The entropy production rate of Fig. 6 depends only mildly on the vacuum energy. Hence, dependence on the vacuum energy enters our calculation mainly through the size of the causal diamond (see Figs. 4, 9). The rate of Ref. 54 is shown here for ${\rho }_{\Lambda }=(0.1,1,10)$ times the observed value (from top to bottom) using the approximation of Eq. (3.23).

Figure 8

The probability distribution over $\log ({\rho }_{\Lambda })$ computed from the causal entropic principle. The two curves shown arise from two different models of the star formation rate (see Figs. 5, 6). Their differences hint at the systematic uncertainties in our calculation that arise since the history of entropy production is not known to arbitrary precision. These uncertainties are apparently irrelevant to our main conclusion: either way, the observed value of ${\rho }_{\Lambda }$ (vertical bar) is not unlikely.

Figure 9

This cartoon demonstrates how the causal entropic principle leads to a preferred value of the cosmological constant. The horizontal band represents the rate of entropy production; darker areas correspond to a higher rate. Vacua are weighted by $\Delta S$, the total amount of “darkness” inside a causal diamond. Vacua with large cosmological constant are plentiful in the landscape, but they lead to small causal diamonds, which capture virtually no entropy production (right). For some smaller value, the diamond will be just large enough to capture the bulk of the entropy production (center). This is the preferred cosmological constant. Larger diamonds may capture slightly more $\Delta S$ (left), but not in proportion to their size. They correspond to vacua with very small cosmological constant which are much rarer in the landscape. Therefore they will be suppressed.