Unifying dark matter and dark energy as a Fermi gas in extra dimensions

Abstract We propose a unified framework for dark matter and dark energy based on the thermodynamic properties of a degenerate, relativistic Fermi gas in a higher-dimensional space-time. Extending the standard equation of state to D -dimensional space-time, we derive analytical expressions for the energy density, pressure, number density, sound velocity, and polytropic indices of a zero-temperature Fermi gas. Remarkably, we show that in the limit of vanishing Fermi momentum, the system exhibits a negative vacuum pressure and positive energy density when $$D=(5+4n)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>D</mml:mi> <mml:mo>=</mml:mo> <mml:mo>(</mml:mo> <mml:mn>5</mml:mn> <mml:mo>+</mml:mo> <mml:mn>4</mml:mn> <mml:mi>n</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , with $$n\in \mathbb {N}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>N</mml:mi> </mml:mrow> </mml:math> . This feature provides a natural mechanism for the emergence of a cosmological constant-like component from fermionic matter in extra dimensions. In this context, dark matter arises from the bulk fermions’ rest mass energy, while dark energy emerges from their vacuum contributions. The observed cosmological vacuum energy density can be matched by fixing the compactification length scale, leading to constraints that depend on the fermion mass and the number of extra dimensions. Our results suggest that the dark sector may originate from a common fermionic source embedded in a higher-dimensional space-time, offering a novel pathway toward the unification of dark matter and dark energy.