[Paper Review] Dark Energy as Evidence for Extra Dimensions

Kimball A. Milton|arXiv (Cornell University)|Oct 10, 2002

Cosmology and Gravitation Theories3 citations

TL;DR

The paper proposes that dark energy arises from quantum vacuum fluctuations in extra compactified spatial dimensions, with the cosmological constant scaling as $1/a^4$ where $a$ is the size of the extra dimensions. It finds that $a \approx 100\ \mu\text{m}$ balances observed dark energy with experimental constraints from Newton's law, suggesting extra dimensions may be discovered or ruled out soon.

ABSTRACT

It is argued that fluctuations of quantum fields in four-dimensional space do not give rise to dark energy, but are rather a negligible contribution to dark matter. By (relativistic) dark matter we mean that the relation between pressure and energy density is $p=\frac13 u$, while dark energy is characterized by $p=-u$. A possible source of dark energy are the fluctuations in quantum fields, including quantum gravity, inhabiting extra compactified dimensions. These fluctuations have been computed for some simple geometries, such as $S^2$, $S^4$, and $S^6$. If the extra dimensions are too small, they would give rise to a dark energy larger than that observed, whereas if they are too large they would be in conflict with experimental tests of Newton's law. This notion suggests that the size of the extra dimensions is of order 100 $μ$m. If the limit on the size of extra dimensions becomes lower than this bound, extra dimensions probably do not exist, and another source for cosmological dark energy will have to be found.

Motivation & Objective

• To resolve the cosmological constant problem by proposing extra dimensions as a source of dark energy.
• To explain why the observed dark energy density is so small compared to the naive quantum field theory prediction.
• To reconcile the observed value of the cosmological constant with constraints from laboratory tests of gravity.
• To determine the viable size range for extra dimensions that would produce the observed dark energy without violating experimental bounds.

Proposed method

• Modeling quantum vacuum fluctuations in compactified extra dimensions using Casimir energy calculations on spheres $S^N$.
• Computing the divergent and finite parts of the effective action for gravity and matter fields in $S^{2}, S^{4}, S^{6}$, and other geometries.
• Applying the relation $u_{\text{Casimir}} \propto 1/a^4$ to estimate the vacuum energy density from extra dimensions.
• Using the critical density $\rho_c \sim 10^{-5}\ \text{GeV/cm}^3$ as an upper bound on the Casimir energy to constrain $a$.
• Comparing theoretical predictions with experimental limits on deviations from Newton’s law, particularly at $\sim 200\ \mu\text{m}$.
• Evaluating the role of field content (gravity, scalars, fermions, vectors) in determining the total Casimir energy and its sign.

Experimental results

Research questions

• RQ1 Can quantum vacuum fluctuations in extra dimensions naturally produce the observed dark energy density?
• RQ2 What size must extra compactified dimensions be to yield a cosmological constant consistent with observations?
• RQ3 How do experimental constraints on deviations from Newton’s law at submillimeter scales constrain the size of extra dimensions?
• RQ4 Why do standard 4D quantum field theory vacuum fluctuations fail to produce dark energy, while extra-dimensional fluctuations can?
• RQ5 Can fine-tuning or cancellation mechanisms (e.g., supersymmetry) in extra dimensions explain the smallness of the observed cosmological constant?

Key findings

• Quantum fluctuations in 4D space-time do not produce dark energy but contribute negligibly to dark matter, with $p = \frac{1}{3}u$.
• Fluctuations in extra compactified dimensions, particularly gravity, can generate a Casimir energy density scaling as $u \propto 1/a^4$, consistent with dark energy.
• For $S^2$, $S^4$, and $S^6$, the Casimir energy is finite and positive for gravity, yielding a lower bound of $a > 84\ \mu\text{m}$ to avoid exceeding the critical density.
• Using divergent terms in even dimensions, the constraint tightens to $a > [\alpha \ln(a/L_{\text{Pl}})]^{1/4} \times 80\ \mu\text{m}$, with $a < 200\ \mu\text{m}$ required by Newton’s law experiments.
• The size of extra dimensions is constrained to $a \approx 100\ \mu\text{m}$, where the predicted dark energy matches observations and experimental limits are satisfied.
• If future experiments lower the size limit below $\sim 100\ \mu\text{m}$, extra dimensions are ruled out, and another origin for dark energy must be found.