**De Sitter space** is a solution to Einstein's field equations of general relativity that describes a universe with a positive cosmological constant, leading to a spacetime with constant positive curvature. It is named after the Dutch astronomer [[Willem de Sitter]], who first studied its properties. De Sitter space is of great interest in cosmology and theoretical physics, particularly in the context of the accelerated expansion of the universe and in string theory. ### Key Features of De Sitter Space 1. **Positive Curvature**: - De Sitter space has constant positive curvature, meaning it is a hyperbolic space that expands exponentially. - It can be thought of as the maximally symmetric solution to Einstein's equations with a positive cosmological constant. 2. **Metric**: - In four dimensions, the metric of de Sitter space in static coordinates can be written as: $ds^2 = -\left(1 - \frac{r^2}{L^2}\right) dt^2 + \left(1 - \frac{r^2}{L^2}\right)^{-1} dr^2 + r^2 d\Omega^2$ - where - $L$ is the de Sitter radius related to the cosmological constant $\Lambda$ by $L^2 = 3/\Lambda$, - $r$ is the radial coordinate, - $t$ is the time coordinate, and - $d\Omega^2$ represents the metric on the 2-sphere. 3. **Cosmological Implications**: - De Sitter space describes a universe dominated by a positive cosmological constant, leading to exponential expansion. This is relevant for both the early universe (inflation) and the current accelerated expansion of the universe. ### Importance in Cosmology 1. **Inflationary Cosmology**: - The concept of inflation involves a phase of accelerated expansion in the early universe, which can be modeled by de Sitter space. During inflation, the universe expands exponentially, smoothing out any initial irregularities and leading to the large-scale structure observed today. 2. **Dark Energy and Accelerated Expansion**: - Observations of distant supernovae and the cosmic microwave background suggest that the universe is currently undergoing accelerated expansion, likely driven by dark energy. De Sitter space serves as a simple model for this accelerated phase, providing insights into the nature of dark energy. 3. **Cosmological Horizons**: - De Sitter space features a cosmological horizon, similar to the event horizon of a black hole. This horizon represents the limit beyond which events cannot affect an observer, due to the exponential expansion of space. ### Mathematical Properties 1. **Symmetry and Isometries**: - De Sitter space is maximally symmetric, with the same number of symmetries as flat space. Its isometry group is $SO(4,1)$ in four dimensions, which includes rotations and boosts in a higher-dimensional embedding space. 2. **Geodesics**: - The geodesics in de Sitter space, which describe the paths of free-falling particles, exhibit unique properties due to the space's positive curvature and expansion. These geodesics can converge or diverge depending on their initial conditions. ### Applications in Theoretical Physics 1. **String Theory**: - De Sitter space is studied in the context of string theory, particularly in the search for realistic cosmological solutions. The challenge is to find stable de Sitter vacua within the landscape of string theory compactifications. 2. **Quantum Field Theory**: - Quantum field theory in de Sitter space provides insights into the behavior of fields in an expanding universe. It helps in understanding particle production during inflation and the generation of cosmological perturbations. 3. **AdS/CFT Correspondence**: - Although the AdS/CFT correspondence specifically relates Anti-de Sitter space to conformal field theories, exploring analogous correspondences for de Sitter space (dS/CFT) is an active area of research. This involves understanding how quantum gravity in de Sitter space might relate to a lower-dimensional theory. ### Conclusion De Sitter space is a fundamental construct in cosmology and theoretical physics, providing a simple yet profound model of an exponentially expanding universe. Its relevance to inflationary cosmology and the current accelerated expansion of the universe makes it a key focus of research. Additionally, its properties and implications extend to quantum field theory and string theory, offering insights into the fundamental nature of spacetime and the dynamics of the universe. # References ```dataview Table title as Title, authors as Authors where contains(subject, "de Sitter space") sort title, authors, modified, desc ```