A new Rényi holographic dark energy model and its cosmological implications

Abstract We develop a generalized holographic dark energy model based on the Rényi entropy, which introduces a logarithmic deformation of the Bekenstein–Hawking entropy and is characterized by a non-extensivity parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> . By adopting the future event horizon as the infrared cutoff, we formulate the New Rényi Holographic Dark Energy (NRHDE) scenario and derive a modified holographic energy density that reduces smoothly to the standard HDE limit for $$\alpha \rightarrow 0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>→</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> . Starting from the Rényi entropy formalism, we obtain a closed and self-consistent set of evolution equations for the dark energy density parameter $$\Omega _d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> , the equation-of-state parameter $$w_d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> </mml:msub> </mml:math> , and the deceleration parameter q . We perform a detailed numerical investigation of the background dynamics over a physically reasonable range of the holographic parameter c and the Rényi deformation parameter $$\alpha $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> , and show that the NRHDE model predicts a late-time phantom regime over an extended region of the $$(c,\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> parameter space, with a smooth approach toward the cosmological-constant boundary $$w_d=-1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>w</mml:mi> <mml:mi>d</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> as either parameter increases. We further provide a global characterization of the parameter space by means of two-dimensional maps of the present-day equation-of-state parameter and the transition redshift, which clarify the joint impact of $$(c,\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> on the late-time cosmological evolution. Finally, a qualitative comparison between the NRHDE background predictions and observational Hubble data from cosmic chronometers is presented as a consistency check of the model at the background level. The NRHDE framework therefore constitutes a minimal and thermodynamically motivated extension of holographic dark energy, offering a flexible platform for future quantitative tests with late-time expansion data.