[Paper Review] On the quantum origin of a small positive cosmological constant

Saurya Das, Rajat K. Bhaduri|arXiv (Cornell University)|Dec 18, 2018

Dark Matter and Cosmic Phenomena4 citations

TL;DR

The paper proposes that a Bose-Einstein condensate (BEC) of ultralight bosonic dark matter generates a quantum potential that naturally induces a small, positive cosmological constant Λ matching the observed value. This quantum origin explains why Λ is tiny, positive, and why dark matter and dark energy densities are approximately equal today.

ABSTRACT

We show that Dark Matter consisting of ultralight bosons in a Bose-Einstein condensate induces, via its quantum potential, a small positive cosmological constant which matches the observed value. This explains its origin and why the densities of Dark Matter and Dark Energy are approximately equal.

Motivation & Objective

• To resolve the cosmological constant problem by deriving a small positive Λ from quantum mechanics rather than postulating it.
• To explain why the densities of dark matter and dark energy are approximately equal today (the coincidence problem).
• To show that a BEC of ultralight bosons (mass ~10−22 eV/c²) can generate a quantum potential that induces Λ via the de Broglie-Bohm interpretation of quantum mechanics.
• To demonstrate that the observed value of Λ ≈ 10−123 ℓPl⁻² arises naturally from the quantum potential of the coherent BEC wave function.
• To provide a mechanism where Λ is not a fundamental constant but emerges dynamically from the quantum state of dark matter.

Proposed method

• Model dark matter as ultralight bosons (m ≈ 10−22 eV/c²) that form a Bose-Einstein condensate (BEC) in the early universe due to low critical temperature.
• Use the one-particle Schrödinger equation with a harmonic oscillator potential to describe the macroscopic BEC wave function Ψ = R(a) e^{−r²/σ²}, where σ² = 2ℏ/mω.
• Apply the de Broglie-Bohm formulation to derive the quantum potential VQ = −(ℏ²/2m)(∇²R)/R, which modifies Newtonian gravity and introduces an effective cosmological constant.
• Treat the scale factor a(t) as slowly varying, with DM density ρDM ∝ 1/a³ and BEC wave function normalized via R(a) ∝ a⁻³/².
• Calculate the quantum potential explicitly as VQ = (3/2)ℏω − (1/2)mω²r², leading to an effective repulsive force that mimics a positive Λ.
• Derive the quantum-corrected Friedmann equation: ä/a = −(4πGρcrit)/3 + Λc²/3, showing that the quantum potential stabilizes the universe at a = 1 with Λ = 3H₀²/(2c²).

Experimental results

Research questions

• RQ1 Can the small positive value of the cosmological constant Λ be derived from quantum mechanics rather than being postulated?
• RQ2 Why is the observed Λ so small (~10−123 ℓPl⁻²), and why is it positive?
• RQ3 Why are the current densities of dark matter and dark energy approximately equal (the coincidence problem)?
• RQ4 Can the quantum potential of a coherent dark matter BEC wave function generate an effective cosmological constant?
• RQ5 What is the role of the de Broglie-Bohm quantum potential in modifying classical cosmology to yield a stable, accelerating universe?

Key findings

• The quantum potential of the BEC wave function induces a cosmological constant Λ = 3H₀²/(2c²), which matches the observed value of Λ ≈ 10−123 ℓPl⁻².
• The induced Λ has energy density ρΛ = ρcrit/2, equal to the current dark matter density ρDM ≈ ρcrit, solving the coincidence problem.
• The quantum potential is intrinsically positive due to the uncertainty principle, explaining why Λ is positive without fine-tuning.
• The model predicts that ρDM decreases as 1/a³ while ρΛ remains constant, leading to ρΛ > ρDM for a > 1, consistent with cosmic acceleration.
• The required boson mass is m ≈ 10−22 eV/c², which is consistent with known constraints on structure formation and BEC formation.
• The model naturally explains the smallness of Λ: it arises from the small critical density ρcrit ≈ 10−26 kg/m³ and quantum effects in a macroscopic BEC wave function.