Studies of the Inhomogeneous Cosmology in Higher Dimensional Space-Time with a Cosmological Constant

We study an inhomogeneous cosmology in Kaluza–Klein space-time with a positive cosmological constant in a dust dominated era ( $$p=0$$ ). Depending on an integration constant, we have derived two types of solutions. The dimensional reduction of the extra-dimensional scale factor is possible due to an inhomogeneity depending on the curvature of the metric for the positive cosmological constant in all solutions. The high value of entropy in the present observable universe and the possible matter leakage in 4D world due to reduction of the extra dimension are also discussed. Our solutions show an early decelerating and late accelerating nature of the universe. The findings are verified by the well-known Raychaudhuri equation.