- #1
Ranku
- 423
- 18
For Ω=1, κ=0. Does the value of κ simply follow from the value of Ω, or can its value have an independent existence? So if Ω>1, does κ have to be 1?
Yes, by definition. Since Ω=1 is the critical density - the density at which the universe is flat - any other value of Ω necessitates that the k parameter is not 0 and has the same sign as Ω-1.Ranku said:So if Ω>1, does κ have to be 1?
See the discussion in the General Metric section of https://en.wikipedia.org/wiki/Friedmann–Lemaître–Robertson–Walker_metric, for example.mathman said:Define k for me.
The density parameter, often denoted as Ω (Omega), is a dimensionless quantity that measures the ratio of the actual density of the universe to the critical density needed for the universe to be flat. It helps determine the overall geometry and fate of the universe.
The curvature index, usually denoted as k, indicates the geometry of the universe and is directly related to the density parameter. If Ω = 1, the universe is flat (k = 0). If Ω > 1, the universe is closed and has positive curvature (k > 0). If Ω < 1, the universe is open and has negative curvature (k < 0).
The critical density is the theoretical density of the universe that would create a flat geometry (k = 0). It is dependent on the Hubble constant and is approximately 9.47 x 10^-27 kilograms per cubic meter. The actual density of the universe is compared to this critical density to determine the density parameter.
Measurements of the Cosmic Microwave Background (CMB) provide detailed information about the early universe, including its density fluctuations and temperature variations. By analyzing these data, scientists can estimate the total density of the universe and hence the density parameter, contributing to our understanding of the universe's geometry and evolution.
If Ω > 1, the universe is closed and will eventually stop expanding and collapse in a "Big Crunch." If Ω = 1, the universe is flat and will expand forever at a decreasing rate. If Ω < 1, the universe is open and will expand forever, potentially leading to a "Big Freeze" where galaxies drift apart and stars burn out. These scenarios depend on the balance between the actual density and the critical density.