Anthropic predictions for vacuum energy and neutrino masses

It is argued that the observed vacuum energy density and the small values of the neutrino masses could be due to anthropic selection effects. Until now, these two quantities have been treated separately from each other and, in particular, anthropic predictions for the vacuum energy were made under the assumption of zero neutrino masses. Here we consider two cases. In the first, we calculate predictions for the vacuum energy for a fixed (generally non-zero) value of the neutrino mass. In the second we allow both quantities to vary from one part of the universe to another. We find that the anthropic predictions for the vacuum energy density are in a better agreement with observations when one allows for non-zero neutrino masses. We also find that the individual distributions for the vacuum energy and the neutrino masses are reasonably robust and do not change drastically when one adds the other variable.Comment: 9 pages, 4 figure

Multiple universes, cosmic coincidences, and other dark matters

Even when completely and consistently formulated, a fundamental theory of physics and cosmological boundary conditions may not give unambiguous and unique predictions for the universe we observe; indeed inflation, string/M theory, and quantum cosmology all arguably suggest that we can observe only one member of an ensemble with diverse properties. How, then, can such theories be tested? It has been variously asserted that in a future measurement we should observe the a priori most probable set of predicted properties (the ``bottom-up'' approach), or the most probable set compatible with all current observations (the ``top-down'' approach), or the most probable set consistent with the existence of observers (the ``anthropic'' approach). These inhabit a spectrum of levels of conditionalization and can lead to qualitatively different predictions. For example, in a context in which the densities of various species of dark matter vary among members of an ensemble of otherwise similar regions, from the top-down or anthropic viewpoints -- but not the bottom-up -- it would be natural for us to observe multiple types of dark matter with similar contributions to the observed dark matter density. In the anthropic approach it is also possible in principle to strengthen this argument and the limit the number of likely dark matter sub-components. In both cases the argument may be extendible to dark energy or primordial density perturbations. This implies that the anthropic approach to cosmology, introduced in part to explain "coincidences" between unrelated constituents of our universe, predicts that more, as-yet-unobserved coincidences should come to light.Comment: 18 JCAP-style pages, accepted by JCAP. Revised version adds references and some clarification

Anthropic prediction in a large toy landscape

The successful anthropic prediction of the cosmological constant depends crucially on the assumption of a flat prior distribution. However, previous calculations in simplified landscape models showed that the prior distribution is staggered, suggesting a conflict with anthropic predictions. Here we analytically calculate the full distribution, including the prior and anthropic selection effects, in a toy landscape model with a realistic number of vacua, $N \sim 10^{500}$. We show that it is possible for the fractal prior distribution we find to behave as an effectively flat distribution in a wide class of landscapes, depending on the regime of parameter space. Whether or not this possibility is realized depends on presently unknown details of the landscape.Comment: 13 page

Is Our Universe Natural?

It goes without saying that we are stuck with the universe we have. Nevertheless, we would like to go beyond simply describing our observed universe, and try to understand why it is that way rather than some other way. Physicists and cosmologists have been exploring increasingly ambitious ideas that attempt to explain why certain features of our universe aren't as surprising as they might first appear.Comment: Invited review for Nature, 11 page

Anthropic predictions for vacuum energy and neutrino masses in the light of WMAP-3

Anthropic probability distributions for the cosmological constant and for the sum of neutrino masses are updated using the WMAP-3 data release. The new distribution for Lambda is in a better agreement with observation than the earlier one. The typicality of the observed value, defined as the combined probability of all values less likely than the observed, is no less than 22%. We discuss the dependence of our results on the simplifying assumptions used in deriving the distribution for Lambda and show that the agreement of the anthropic prediction with the data is rather robust. The distribution for the sum of the neutrino masses is peaked at 1 eV, suggesting degenerate masses, but a hierarchical mass pattern is still marginally allowed at a 2 sigma level.Comment: References added; 11 pages, 6 figure

Anthropic prediction for a large multi-jump landscape

The assumption of a flat prior distribution plays a critical role in the anthropic prediction of the cosmological constant. In a previous paper we analytically calculated the distribution for the cosmological constant, including the prior and anthropic selection effects, in a large toy ``single-jump'' landscape model. We showed that it is possible for the fractal prior distribution we found to behave as an effectively flat distribution in a wide class of landscapes, but only if the single jump size is large enough. We extend this work here by investigating a large ($N \sim 10^{500}$) toy ``multi-jump'' landscape model. The jump sizes range over three orders of magnitude and an overall free parameter $c$ determines the absolute size of the jumps. We will show that for ``large'' $c$ the distribution of probabilities of vacua in the anthropic range is effectively flat, and thus the successful anthropic prediction is validated. However, we argue that for small $c$, the distribution may not be smooth.Comment: 33 pages, 7 figures Minor revisions made and references added. arXiv admin note: substantial text overlap with arXiv:0705.256

What does inflation really predict?

If the inflaton potential has multiple minima, as may be expected in, e.g., the string theory "landscape", inflation predicts a probability distribution for the cosmological parameters describing spatial curvature (Omega_tot), dark energy (rho_Lambda, w, etc.), the primordial density fluctuations (Omega_tot, dark energy (rho_Lambda, w, etc.). We compute this multivariate probability distribution for various classes of single-field slow-roll models, exploring its dependence on the characteristic inflationary energy scales, the shape of the potential V and and the choice of measure underlying the calculation. We find that unless the characteristic scale Delta-phi on which V varies happens to be near the Planck scale, the only aspect of V that matters observationally is the statistical distribution of its peaks and troughs. For all energy scales and plausible measures considered, we obtain the predictions Omega_tot ~ 1+-0.00001, w=-1 and rho_Lambda in the observed ballpark but uncomfortably high. The high energy limit predicts n_s ~ 0.96, dn_s/dlnk ~ -0.0006, r ~ 0.15 and n_t ~ -0.02, consistent with observational data and indistinguishable from eternal phi^2-inflation. The low-energy limit predicts 5 parameters but prefers larger Q and redder n_s than observed. We discuss the coolness problem, the smoothness problem and the pothole paradox, which severely limit the viable class of models and measures. Our findings bode well for detecting an inflationary gravitational wave signature with future CMB polarization experiments, with the arguably best-motivated single-field models favoring the detectable level r ~ 0.03. (Abridged)Comment: Replaced to match accepted JCAP version. Improved discussion, references. 42 pages, 17 fig

The Mathematical Universe

I explore physics implications of the External Reality Hypothesis (ERH) that there exists an external physical reality completely independent of us humans. I argue that with a sufficiently broad definition of mathematics, it implies the Mathematical Universe Hypothesis (MUH) that our physical world is an abstract mathematical structure. I discuss various implications of the ERH and MUH, ranging from standard physics topics like symmetries, irreducible representations, units, free parameters, randomness and initial conditions to broader issues like consciousness, parallel universes and Godel incompleteness. I hypothesize that only computable and decidable (in Godel's sense) structures exist, which alleviates the cosmological measure problem and help explain why our physical laws appear so simple. I also comment on the intimate relation between mathematical structures, computations, simulations and physical systems.Comment: Replaced to match accepted Found. Phys. version, 31 pages, 5 figs; more details at http://space.mit.edu/home/tegmark/toe.htm

Testing the Cosmological Constant as a Candidate for Dark Energy

It may be difficult to single out the best model of dark energy on the basis of the existing and planned cosmological observations, because many different models can lead to similar observational consequences. However, each particular model can be studied and either found consistent with observations or ruled out. In this paper, we concentrate on the possibility to test and rule out the simplest and by far the most popular of the models of dark energy, the theory described by general relativity with positive vacuum energy (the cosmological constant). We evaluate the conditions under which this model could be ruled out by the future observations made by the Supernova/Acceleration Probe SNAP (both for supernovae and weak lensing) and by the Planck Surveyor cosmic microwave background satellite.Comment: 6 pages, 2 figures, revtex

Sinks in the Landscape, Boltzmann Brains, and the Cosmological Constant Problem

This paper extends the recent investigation of the string theory landscape in hep-th/0605266, where it was found that the decay rate of dS vacua to a collapsing space with a negative vacuum energy can be quite large. The parts of space that experience a decay to a collapsing space, or to a Minkowski vacuum, never return back to dS space. The channels of irreversible vacuum decay serve as sinks for the probability flow. The existence of such sinks is a distinguishing feature of the string theory landscape. We describe relations between several different probability measures for eternal inflation taking into account the existence of the sinks. The local (comoving) description of the inflationary multiverse suffers from the so-called Boltzmann brain (BB) problem unless the probability of the decay to the sinks is sufficiently large. We show that some versions of the global (volume-weighted) description do not have this problem even if one ignores the existence of the sinks. We argue that if the number of different vacua in the landscape is large enough, the anthropic solution of the cosmological constant problem in the string landscape scenario should be valid for a broad class of the probability measures which solve the BB problem. If this is correct, the solution of the cosmological constant problem may be essentially measure-independent. Finally, we describe a simplified approach to the calculations of anthropic probabilities in the landscape, which is less ambitious but also less ambiguous than other methods.Comment: 42 pages, 5 figures, the paper is substantially extended, a section on the cosmological constant is addeed; the version published in JCA