🗂️ Cosmology & Space · View mindmap

Proper Distance

Proper distance is the distance between two points in space measured at a specific instant of cosmic time. In an expanding universe, this distance changes over time as the scale factor $a(t)$ evolves. It is distinct from comoving distance, which factors out the expansion of the universe, and luminosity distance or angular diameter distance, which are observational measures dependent on redshift.

Key Characteristics

• Instantaneous Measurement: Proper distance requires a hypothetical ruler laid out at a fixed cosmic time $t$. It represents the physical separation between objects at that specific moment.
• Time Dependence: Because the universe expands, the proper distance between two comoving objects increases with time. $D_p(t)=a(t)\cdot \chi$, where $\chi$ is the comoving distance.
• Horizon Limits: The proper distance to the particle horizon defines the radius of the Observable Universe.

Relation to Observable Universe Size

A common point of confusion arises regarding the age of the universe versus the size of the observable universe.

• Age vs. Diameter: The universe is approximately 13.8 billion years old, yet the observable universe has a proper diameter of roughly 92–93 billion light-years.
• Metric Expansion: This discrepancy is resolved by understanding that space itself has expanded during the time light has been traveling to us. Light emitted from distant galaxies 13 billion years ago has traveled for 13 billion years, but the space it traversed has stretched, placing the source objects much farther away in proper distance today than the light-travel distance suggests.
• Source Integration:
  • See Observable Universe Expansion: Age vs. Current Size Explained for a detailed breakdown of this paradox.
  • Dr. Don Lincoln (Fermilab) explains that the “width” of the observable universe refers to the current proper distance to the edge of the observable region, not the distance light has traveled.

Mathematical Context

The proper distance $D_p$ at time $t$ is related to the comoving distance $\chi$ by the scale factor $a(t)$:

$D_p(t)=a(t)\chi$

For the current epoch ($t_0$), $a(t_0)=1$, so the proper distance equals the comoving distance. However, for past epochs, $a(t)<1$, meaning proper distances were smaller.

References

• Observable Universe Expansion: Age vs. Current Size Explained