Effects of gravity rainbow on scalar bosonic and oscillator fields in Bonnor–Melvin space-time with a cosmological constant

Abstract In this paper, we explore the relativistic quantum motion of spin-zero scalar charge-free particles influenced by rainbow gravity’s (RG’s) in Bonnor–Melvin magnetic space-time, a four-dimensional solution featuring a positive cosmological constant. We solve the Klein–Gordon equation in this scenario by using two sets of rainbow function: (i) $$f(\chi )=\frac{1}{(1-\beta \,\chi )}$$ <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>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>β</mml:mi> <mml:mspace/> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> </mml:mrow> </mml:math> , $$h(\chi )=1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>h</mml:mi> <mml:mo>(</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and (ii) $$f(\chi )=\frac{1}{(1-\beta \,\chi )}=h(\chi )$$ <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>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>β</mml:mi> <mml:mspace/> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mfrac> <mml:mo>=</mml:mo> <mml:mi>h</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>χ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $$0 &lt; \chi \left( =\frac{|E|}{E_p}\right) \le 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>χ</mml:mi> <mml:mfenced> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mo>|</mml:mo> <mml:mi>E</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:mfrac> </mml:mfenced> <mml:mo>≤</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> with $$E_p$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>p</mml:mi> </mml:msub> </mml:math> being the Planck’s energy, E is the particle’s energy, and $$\beta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>β</mml:mi> </mml:math> is the rainbow parameter. Furthermore, we study the relativistic quantum oscillator through the Klein–Gordon oscillator equation in the same Bonnor–Melvin magnetic space-time under RG effects. Employing the first pair of rainbow function, we obtain an approximate eigenvalue solution of the quantum oscillator fields. Notably, we demonstrate that the relativistic energy profile of scalar and oscillator particles are influenced by the topology of the geometry and the cosmological constant which is related with the magnetic field strength lies along the symmetry axis. Additionally, we see the impact of rainbow parameter on these approximate relativistic energy profiles in both quantum systems.