  Theoretical and observational challenges to standard cosmology such as the
cosmological constant problem and tensions between cosmological model
parameters inferred from different observations motivate the development and
search of new physics. A less radical approach to venturing beyond the standard
model is the simple mathematical reformulation of our theoretical frameworks
underlying it. While leaving physical measurements unaffected, this can offer a
reinterpretation and even solutions of these problems. In this spirit, metric
transformations are performed here that cast our Universe into different
geometries. Of particular interest thereby is the formulation of cosmology in
Minkowski space. Rather than an expansion of space, spatial curvature, and
small-scale inhomogeneities and anisotropies, this frame exhibits a variation
of mass, length and time scales across spacetime. Alternatively, this may be
interpreted as an evolution of fundamental constants. As applications of this
reframed cosmological picture, the naturalness of the cosmological constant is
reinspected and promising candidates of geometric origin are explored for dark
matter, dark energy, inflation and baryogenesis. An immediate observation
thereby is the apparent absence of the cosmological constant problem in the
Minkowski frame. The formalism is also applied to identify new observable
signatures of conformal inhomogeneities, which have been proposed as
simultaneous solution of the observational tensions in the Hubble constant, the
amplitude of matter fluctuations, and the gravitational lensing amplitude of
cosmic microwave background anisotropies. These are found to enhance redshifts
to distant galaxy clusters and introduce a mass bias with cluster masses
inferred from gravitational lensing exceeding those inferred kinematically or
dynamically.
