[Paper Review] Towards a non-anthropic solution to the cosmological constant problem

Andrei Linde, Vitaly Vanchurin|arXiv (Cornell University)|Oct 31, 2010

Cosmology and Gravitation Theories34 references6 citations

TL;DR

This paper proposes a non-anthropocentric solution to the cosmological constant problem by leveraging an exponentially divergent probability measure in the string landscape. It argues that such a measure, when applied to vacuum states, uniquely selects one vacuum with an extremely small cosmological constant due to runaway pressure, thereby fixing fundamental parameters without relying on observer-based selection.

ABSTRACT

Many probability measures in the multiverse depend exponentially on some observable parameters, giving rise to potential problems such as youngness bias, Q-catastrophe etc. In this paper we explore a possibility that the exponential runaway dependence should be viewed not as a problem, but as a feature that may help us to fix all parameters in the landscape, including the value of the cosmological constant, without using anthropic considerations.

Motivation & Objective

• To resolve the cosmological constant problem without invoking anthropic reasoning.
• To investigate whether an exponentially divergent probability measure in the multiverse can uniquely select a vacuum state.
• To explore whether such a measure can simultaneously fix not only the cosmological constant but also other fundamental parameters.
• To assess the viability of this approach under realistic assumptions about the number of vacua and the distribution of physical parameters.

Proposed method

• Assumes a finite but large number $ N $ of vacua in the landscape with a normalizable prior distribution $ P({\bf x}) $ over observable parameters.
• Introduces a positive definite weighting function $ w({\bf x}) $ with exponential dependence on at least one parameter, modeling strong multiverse pressure.
• Uses the expectation value formula $ \langle{\bf x}\rangle = \frac{\sum_i {\bf x}_i w({\bf x}_i)}{\sum_i w({\bf x}_i)} $ to compute the most probable vacuum state.
• Analyzes the limit of large $ N $, showing that the runaway behavior in $ w({\bf x}) $, particularly in the form $ e^{C_1/x + C_2/y + \cdots} $, forces the expectation value of the most sensitive parameter (e.g., $ \Lambda $) to approach zero.
• Demonstrates that for finite $ N \sim 10^{500} $ or $ 10^{120} $, the expectation value of the runaway parameter is slightly shifted from zero, but remains extremely small.
• Considers the physical plausibility of such measures, especially those dependent on $ \Lambda $, and argues that $ \Lambda $-dependence is robust at cosmological scales.

Experimental results

Research questions

• RQ1 Can a non-anthropic, exponential runaway measure in the multiverse uniquely select a vacuum with a small cosmological constant?
• RQ2 How does the presence of an exponentially divergent weighting function $ w({\bf x}) $ affect the expectation value of physical parameters in the landscape?
• RQ3 To what extent does the finite number of vacua $ N $ affect the predicted value of the cosmological constant in this framework?
• RQ4 Can this mechanism simultaneously fix other fundamental parameters beyond $ \Lambda $, such as inflaton mass or coupling constants?
• RQ5 Is the resulting selection mechanism robust against modifications to the observer-dependent assumptions typically used in anthropic reasoning?

Key findings

• The cosmological constant $ \Lambda $ is predicted to be extremely small—not because of anthropic selection, but due to exponential pressure from the weighting function $ w({\bf x}) $.
• In the limit of infinite $ N $, the expectation value of the runaway parameter $ x $ (e.g., $ \Lambda $) is exactly zero, provided $ w({\bf x}) \propto e^{C_1/x} $ with $ C_1 > 0 $.
• For finite $ N \sim 10^{500} $ or $ 10^{120} $, the expectation value of $ \Lambda $ is shifted from zero but remains suppressed by an exponential factor $ e^{-N} $, making other vacua overwhelmingly unlikely.
• The remaining parameters (e.g., inflaton mass) are determined solely by the prior distribution $ P({\bf x}) $, once the runaway parameter is fixed.
• The method does not require assumptions about the rarity of observers or the uniqueness of carbon-based life, making it independent of anthropic observer selection.
• The analysis suggests that the hierarchy between scales—such as $ H_I $ and $ \Lambda^{1/2} $—is naturally explained by the exponential runaway mechanism, linking it to the total number of vacua $ N $.