Modified Jeans instability and Friedmann equation from generalized Maxwellian distribution

We study Jeans instability with generalized Maxwellian distribution. The results reveal two significant features of the modified Jeans instability. First, the Jeans wavelength of the system covers the original <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>λ</m:mi> </m:mrow> <m:mrow> <m:mi>J</m:mi> </m:mrow> </m:msub> </m:math> {λ }J when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> k=1 . Second, as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k approaches 0, the modified Jeans wavelength approaches infinity. This means that the system is always gravitationally stable. Furthermore, we examine the implications of the modified Maxwellian distribution on the Friedmann equation. Our analysis suggests that the effective gravitational constant should incorporate the contribution of temperature <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> </m:math> T in order to describe the system dynamics.