The density parameter and the Anthropic Principle

TL;DR

The paper develops a framework for predicting the distribution of the density parameter $Ω$ in quasi-open inflation models with two slow-roll fields. By combining a tunneling/volume factor with an anthropic weight for galaxy formation, the authors derive a tractable probability measure $P(Ω)$ that depends on a single dimensionless parameter $μ = 1/(24 π G f^2)$. They show that, for viable ranges of $μ$, the most probable $Ω$ can be nonzero, but compatibility with CMB observations requires either a small slow-roll parameter $ε$ or an alternative source of CMB fluctuations. This approach provides a principled way to constrain particle physics models of inflation using cosmological data and offers a potential explanation for the observed proximity of the curvature-dominated epoch to the present era, while highlighting tensions in simple two-field realizations that may be alleviated by generalizations or additional fields.

Abstract

In the context of open inflation, we calculate the probability distribution for the density parameter $Ω$. A large class of two field models of open inflation do not lead to infinite open universes, but to an ensemble of inflating islands of finite size, or ``quasi-open'' universes, where the density parameter takes a range of values. Assuming we are typical observers, the models make definite predictions for the value $Ω$ we are most likely to observe. When compared with observations, these predictions can be used to constrain the parameters of the models. We also argue that obsevers should not be surprised to find themselves living at the time when curvature is about to dominate.

Figures (5)

• Figure 1: The coefficient $\kappa$ which relates the variable $y$ to the density parameter $y= \kappa x = \kappa (1-\Omega)/\Omega$, depends on $\sigma_{rec}$, the value of the density contrast at the time of recombination. Our ability to infer $\sigma_{rec}$ from CMB observations is limited by the fact that $\Omega_0$ in our observable "subuniverse" is not known very precisely. In the figure we plot the inferred value of $\kappa$ for various assumed values of $\Omega_0$. The value of $\sigma_{rec}$ depends moreover on the scale $R_{gal}$ corresponding to objects of galactic mass. The curve is plotted for two different values of this scale (see Appendix B). The parameter $\kappa$ depends on the temperature at which we observe $\Omega$. Here we have taken $T_{CMB} = 2.7 K$ .
• Figure 2: The probability distribution (\ref{['distributiony']}) as a function of $y$, for various values of $\mu$. Also represented is the fraction of clustered matter $\nu(y)$ as a function of $y$.
• Figure 3: Peak of the probability distribution (\ref{['distributiony']}) (curve $a$). The approximate value of $y_{peak}$ given by (\ref{['ypeak']}) is represented by the curve $b$. Curve $c$ represents the possible effect of helium line cooling failure, as discussed in Appendix C.
• Figure 4: The coefficient $K_l$ for various values of the density parameter
• Figure 5: The probability distribution for $\Omega$ is sensitive to the fact that objects which collapse at very late times have very low density, and therefore may be unsuitable for life. Neglecting these "selection" effects, frame (a) shows the probability distribution for $\Omega$ for various values of $\mu$ (The value of $\Omega$ is the one measured at the temperature $T_{CMB}=2.7 K$). In this case, the anthropic factor $\nu(\Omega)$ (also shown in the plot) is just proportional to the total fraction of matter that clusters on the galactic mass scale in the entire history of a particular region. In frame (b) we disregard matter which clusters after the time when helium line cooling becomes inefficient, so that the collapsed galactic mass objects cannot fragment into stars. Finally, as a more extreme case, in frame (c) we disregard matter that clumps after the time $t_*\approx 3 \cdot 10^{9} {\rm Yr}$, since we do not see many giant galaxies forming at redshifts lower than $z=2$.