Comoving Distance Calculator Z 0.975 Lambda-Cdm

This calculator computes the comoving distance at redshift z = 0.975 in a Lambda-CDM cosmological model. Comoving distance measures the physical separation between two points in the universe that remain at a constant comoving distance as the universe expands.

What is Comoving Distance?

Comoving distance is a measure of proper distance between two points in the universe that remains constant over time as the universe expands. Unlike physical distance, which decreases as the universe expands, comoving distance accounts for the expansion of space.

In cosmology, comoving distance is particularly useful for measuring large-scale structures in the universe. It provides a way to compare distances at different times in the universe's history.

Comoving distance is different from proper distance, which is the actual physical distance between two points at a specific time. Proper distance decreases as the universe expands, while comoving distance remains constant.

Lambda-CDM Cosmological Model

The Lambda-CDM model is the current standard model of cosmology, which includes:

Cold Dark Matter (CDM)
A cosmological constant (Lambda) representing dark energy
Ordinary matter and radiation

The model assumes a flat universe with a critical density of matter and energy. The cosmological constant (Lambda) represents the energy density of the vacuum of space, which is causing the accelerated expansion of the universe.

Calculation Method

The comoving distance at redshift z is calculated using the following formula:

d_c(z) = c/H₀ ∫₀ᶻ (1 + z') / √(Ω_m(1 + z')³ + Ω_k(1 + z')² + Ω_Λ) dz'

Where:

d_c(z) is the comoving distance at redshift z
c is the speed of light
H₀ is the Hubble constant
Ω_m is the matter density parameter
Ω_k is the curvature density parameter
Ω_Λ is the dark energy density parameter

For a flat universe (Ω_k = 0), the formula simplifies to:

d_c(z) = c/H₀ ∫₀ᶻ (1 + z') / √(Ω_m(1 + z')³ + Ω_Λ) dz'

This calculator uses the following default values for a flat Lambda-CDM model:

H₀ = 67.4 km/s/Mpc (Planck 2018)
Ω_m = 0.315
Ω_Λ = 0.685

Example Calculation

Let's calculate the comoving distance at redshift z = 0.975 using the default parameters.

d_c(0.975) = (299792.458 km/s) / (67.4 km/s/Mpc) ∫₀⁰·⁹⁷⁵ (1 + z') / √(0.315(1 + z')³ + 0.685) dz'

The integral is evaluated numerically, resulting in approximately 3,500 Mpc.

This means that at redshift z = 0.975, the comoving distance is approximately 3,500 megaparsecs. This corresponds to a physical distance of about 3.5 billion light-years.

FAQ

What is the difference between comoving distance and proper distance?

Comoving distance remains constant over time as the universe expands, while proper distance decreases. Comoving distance accounts for the expansion of space, while proper distance is the actual physical distance at a specific time.

What is the Lambda-CDM model?

The Lambda-CDM model is the current standard model of cosmology, which includes cold dark matter, a cosmological constant (Lambda) representing dark energy, and ordinary matter and radiation. It assumes a flat universe with a critical density of matter and energy.

What is the Hubble constant?

The Hubble constant (H₀) is a measure of the rate of expansion of the universe. It is typically expressed in units of km/s/Mpc and is currently estimated to be approximately 67.4 km/s/Mpc.