Curvature constraints from the causal entropic principle

Current cosmological observations indicate a preference for a cosmological constant that is drastically smaller than what can be explained by conventional particle physics. The causal entropic principle (Bousso et al.) provides an alternative approach to anthropic attempts to predict our observed value of the cosmological constant by calculating the entropy created within a causal diamond. We have extended this work to use the causal entropic principle to predict the preferred curvature within the “multiverse.” We have found that values larger than ${\rho }_k=40{\rho }_m$ are disfavored by more than 99.99% peak value at ${\rho }_{\Lambda }=7.9\times {10}^{-123}$ and ${\rho }_k=4.3{\rho }_m$ for open universes. For universes that allow only positive curvature or both positive and negative curvature, we find a correlation between curvature and dark energy that leads to an extended region of preferred values. Our universe is found to be disfavored to an extent depending on the priors on curvature. We also provide a comparison to previous anthropic constraints on open universes and discuss future directions for this work.

Figure 1

Top left panel: Probability density for ${\rho }_{\Lambda }$ with fixed curvature of ${\rho }_k=10{\rho }_m$ (dashed) and ${\rho }_k=0$ (solid). Top center panel: The comoving volume (in units of ${10}^{12}{\mathrm{Mpc}}^3$) for ${\rho }_{\Lambda }={10}^{-123}$ (larger) and ${\rho }_{\Lambda }={10}^{-122}$ (smaller) are shown for ${\rho }_k=10{\rho }_m$ (dashed) and ${\rho }_k=0$ (solid). Top right panel: The star formation rate in units of $\frac{M_{\odot }}{{\mathrm{Mpc}}^3\mathrm{yr}}$. The upper curve for each value of curvature represented in solid and dashed, respectively, is ${\rho }_{\Lambda }={10}^{-122}$ and the lower is ${\rho }_{\Lambda }={10}^{-123}$. The bottom row is the probability density, comoving volume, and star formation rate for ${\rho }_k=-50{\rho }_m$. The blue/dashed curve is ${\rho }_{\Lambda }={10}^{-119}$ and the black/solid curve is ${\rho }_{\Lambda }={10}^{-120}$.

Figure 2

The 68.27% (dark grey), 95.44% (light grey), 99.73% (inner white), and 99.99% (outer white) contours for a multiverse that only admits negative curvature universes. The solid blue line that cuts through the other four contours is the anthropic bound from 17 for smaller sized galaxies.

Figure 3

The 68.27% (dark grey), 95.44% (light grey), 99.73% (inner white), and 99.99% (outer white) contours for a multiverse that only admits positive curvature universes.

Figure 4

The 68.27% (dark grey), 95.44% (light grey), 99.73% (inner white), and 99.99% (outer white) contours for a multiverse that admits both positive and negative curvature universes.

Figure 5

This is a zoomed-in plot of Fig. 4 providing a closer view of the full span range of curvature values consistent with $\mathrm{WMAP}+\mathrm{HST}$ ($-0.06{\rho }_m\le {\rho }_k\le 0.016{\rho }_m$) and ${\rho }_{\Lambda }=1.25\times {10}^{-123}$. These values are all within the 95.44% C.L. (light grey).