On an extension of Entangled Relativity with a positive cosmological constant

Abstract We present a non-linear $$f(R,\mathcal {L}_m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>R</mml:mi> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> extension of General Relativity that possesses the full solution space of General Relativity with a positive cosmological constant $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> for matter fields whose on shell Lagrangian satisfies $$\mathcal {L}_m= T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> , where T is the trace of the stress-energy tensor. The theory belongs to the generalized Entangled Relativity family recently identified in the literature, and retains its Machian property that forbids the existence of vacuum solutions. Remarkably, despite being attracted toward General Relativity with a positive cosmological constant during the cosmological expansion of the Universe, the theory still involves one fewer fundamental parameter than General Relativity with $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> . Moreover, the dimensionful constant $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> is replaced by a dimensionless parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , from which an effective cosmological constant emerges in the field equations. Since the Universe is not entirely composed of matter fields satisfying $$\mathcal {L}_m=T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> on shell—notably due to non-relativistic neutrinos that are such that $$\mathcal {L}_m=0\ne T$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> <mml:mo>≠</mml:mo> <mml:mi>T</mml:mi> </mml:mrow> </mml:math> on shell—we argue that this framework naturally predicts small deviations from $$\Lambda $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Λ</mml:mi> </mml:math> CDM, potentially relevant for the interpretation of current cosmological tensions. For the same reason, the theory also lifts the degeneracy between different dark matter candidates and therefore can be used to make strong predictions on the dark matter sector.