dS4

dS4 commonly denotes four-dimensional de Sitter space, a maximally symmetric Lorentzian manifold of constant positive curvature. It occurs as a solution to Einstein's field equations with a positive cosmological constant and serves as a model of a universe dominated by vacuum energy. In four dimensions, dS4 is the spacetime geometry underlying rapid exponential expansion and is used in cosmology to approximate late-time acceleration or to model the inflationary epoch in the early universe.

Mathematically, dS4 can be described as the hyperboloid -X0^2 + X1^2 + X2^2 + X3^2 + X4^2 = α^2 embedded in five-dimensional Minkowski space with metric diag(-,+,+,+,+). The constant α is related to the cosmological constant Λ by Λ = 3/α^2, and the scalar curvature is R = 12/α^2. The isometry group of dS4 is SO(1,4) (or its covering group), reflecting its high degree of symmetry.

Different coordinate representations highlight its geometric properties. Global coordinates yield a metric of the form ds^2 = -dt^2 + α^2 cosh^2(t/α) dΩ3^2, while flat slicing gives ds^2 = -dt^2 + e^{2t/α} d𝑥^2. Static coordinates cover a causally restricted region with a cosmological horizon at r = α; observers in this region experience a Gibbons-Hawking temperature T = 1/(2π α).

In physics, dS4 is central to discussions of cosmic inflation, dark energy–driven acceleration, and holographic ideas like dS/CFT, which conjecture a dual conformal field theory description at future infinity. It contrasts with AdS4 due to its positive curvature and causal structure.