Energy conditions in $$f(Q, L_m)$$ gravity

Abstract We are experiencing a golden age of experimental cosmology, with exact and accurate observations being used to constrain various gravitational theories like never before. Alongside these advancements, energy conditions play a crucial theoretical role in evaluating and refining new proposals in gravitational physics. We investigate the energy conditions (WEC, NEC, DEC, and SEC) for two $$f(Q, L_m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</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> gravity models using the FLRW metric in a flat geometry. Model 1, $$f(Q, L_m) = -\alpha Q + 2L_m + \beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Q</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:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:math> , features linear parameter dependence, satisfying most energy conditions while selectively violating the SEC to explain cosmic acceleration. The EoS parameter transitions between quintessence, a cosmological constant, and phantom energy, depending on $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> . Model 2, $$f(Q, L_m) = -\alpha Q + \lambda (2L_m)^2 + \beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>Q</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:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mi>α</mml:mi> <mml:mi>Q</mml:mi> <mml:mo>+</mml:mo> <mml:mi>λ</mml:mi> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:msub> <mml:mi>L</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:mi>β</mml:mi> </mml:mrow> </mml:math> , introduces nonlinearities, ensuring stronger SEC violations and capturing complex dynamics like dark energy transitions. While Model 1 excels in simplicity, Model 2’s robustness makes it ideal for accelerated expansion scenarios, highlighting the potential of $$f(Q, L_m)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>f</mml:mi> <mml:mo>(</mml:mo> <mml:mi>Q</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> gravity in explaining cosmic phenomena.