A density-responsive scalar-field framework for singularity regularization and dynamical dark energy

Abstract We present a covariant scalar-field framework that unifies the space-time singularity regularization with dynamical dark energy. The theory extends general relativity by introducing a scalar field Φ whose potential couples to the Lorentz-invariant quantity <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>≡</mml:mo> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>α</mml:mi> </mml:mrow> </mml:msub> <mml:msub> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>β</mml:mi> </mml:mrow> </mml:msub> <mml:msubsup> <mml:mi>T</mml:mi> <mml:mrow> <mml:mrow> <mml:mi>matter</mml:mi> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mi>β</mml:mi> </mml:mrow> </mml:msubsup> </mml:mrow> </mml:math> , ensuring manifest covariance. The resulting density-responsive scalar energy <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:msub> </mml:mrow> </mml:math> exhibits dual behavior: (i) in high-density regimes, it saturates at <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:msub> <mml:mtext>⩽</mml:mtext> <mml:mi>A</mml:mi> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>P</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , providing a Planck-scale upper bound on the total energy density that regularizes classical singularities; (ii) in low-density regimes, it approaches a constant <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>ρ</mml:mi> <mml:mi mathvariant="normal">Φ</mml:mi> </mml:msub> <mml:mo accent="false" stretchy="false">→</mml:mo> <mml:mi>A</mml:mi> <mml:msubsup> <mml:mi>M</mml:mi> <mml:mi>U</mml:mi> <mml:mn>4</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> , driving cosmic acceleration as dynamical dark energy. A natural renormalization group evolution with an anomalous dimension γ ≈ 0.501 connects the Planck scale to the meV dark energy scale without fine-tuning. The model makes distinctive, testable predictions: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>w</mml:mi> <mml:mn>0</mml:mn> </mml:msub> <mml:mo>≈</mml:mo> <mml:mo>−</mml:mo> <mml:mn>0.99</mml:mn> </mml:mrow> </mml:math> and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>a</mml:mi> </mml:msub> <mml:mo>≈</mml:mo> <mml:mo>+</mml:mo> <mml:mn>0.03</mml:mn> </mml:mrow> </mml:math> , where the positive w a distinguishes it from ΛCDM and standard quintessence models. Despite the novel interaction terms, the fifth forces are suppressed by <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>β</mml:mi> <mml:mtext>eff</mml:mtext> </mml:msub> <mml:mo>∝</mml:mo> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:msubsup> <mml:mi>ρ</mml:mi> <mml:mi>m</mml:mi> <mml:mn>2</mml:mn> </mml:msubsup> </mml:mrow> </mml:math> , yielding factors below 10 −58 in laboratory environments, and ensuring compatibility with all precision gravity tests. This framework demonstrates how a single quantum field theory mechanism can simultaneously address UV singularities and IR dark energy, providing concrete predictions for future Stage-IV cosmological surveys.