A theoretical physicist models dark energy density as a constant force field with strength $ \rho = k/ r^2 $, where $ r $ is distance from a point source. If $ \rho = 64 $ units at $ r = 2 $, what is $ \rho $ at $ r = 4 $?

["Title: Unraveling Dark Energy Modeling: How a Constant Force Field Influences Space", "Introduction\nIn theoretical physics, understanding the nature of dark energy remains one of the most profound challenges. Recent models propose that dark energy density behaves not like a uniform cosmological constant, but rather as a spatially varying constant force field—specifically, one where the energy density $ \rho $ depends on distance $ r $ from a point source according to $ \rho = \frac{k}{r^2} $. This approach offers a compelling new perspective on cosmic acceleration and gravitational dynamics. Here, we explore a key implication of this model: how the measured density of dark energy changes with distance.", "What is Dark Energy Density in This Model?\nAccording to the model, the dark energy density is governed by:\n[\n\rho(r) = \frac{k}{r^2}\n]\nwhere $ r $ is the distance from the assumed point source emitting this energy field, and $ k $ is a constant representing the total energy flux. This form describes a decaying density proportional to the inverse square of distance, characteristic of influence spreading over expanding space.", "Given experimental or observational data that $ \rho = 64 $ units when $ r = 2 $, we can solve for $ k $:", "[\n64 = \frac{k}{2^2} = \frac{k}{4}\n\Rightarrow k = 64 \ imes 4 = 256\n]", "Calculating $ \rho $ at $ r = 4 $\nUsing $ k = 256 $, the density at $ r = 4 $ is:", "[\n\rho(4) = \frac{256}{4^2} = \frac{256}{16} = 16\n]", "Conclusion\nThis model demonstrates that the strength of a point-like dark energy field weakens with distance, but not linearly—rather, as the inverse square of $ r $. At $ r = 2 $, $ \rho = 64 $; by $ r = 4 $, the value drops to 16 units—a natural outcome of the $ 1/r^2 $ dependence. This insight deepens our conceptual framework for dark energy, suggesting a dynamic, localized origin rather than a fixed cosmological constant. Ongoing research continues to refine such models to reconcile theory with observational data from distant supernovae and cosmic microwave background measurements.", "Keywords: dark energy, constant force field, $ \rho = k/r^2 $, theoretical physics, cosmological models, point source field."]