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Cosmology Calculator

Calculate cosmological parameters and distances using the expanding universe model

Parameters

ⓘ
km/s/Mpcⓘ
Gyrⓘ
Ω_mⓘ
Show Trail

Controls

xⓘ

Calculated Values

Scale Factor:
0.91;0.91;
Lookback Time:
1.25;Gyr1.25;Gyr
Comoving Distance:
428.27;Mpc428.27;Mpc
Luminosity Distance:
471.10;Mpc471.10;Mpc
Angular Diameter Distance:
389.34;Mpc389.34;Mpc
Critical Density:
0.00;kg/m30.00;kg/m³
Expansion Rate:
77.00;km/s/Mpc77.00;km/s/Mpc
CMB Temperature:
3.00;K3.00;K

Examples

Example 1: Nearby Galaxy

A galaxy at redshift z = 0.1 (about 1.4 billion light years).

  • Scale Factor: 0.910.91
  • Lookback Time: 1.301.30
  • Comoving Distance: 428.60428.60
  • Luminosity Distance: 471.40471.40
  • CMB Temperature: 3.003.00

Example 2: Distant Quasar

A quasar at redshift z = 3.0 (about 11 billion light years).

  • Scale Factor: 0.250.25
  • Lookback Time: 11.5011.50
  • Comoving Distance: 12857.1012857.10
  • Luminosity Distance: 51428.6051428.60
  • CMB Temperature: 10.9010.90

Example 3: CMB Surface

The cosmic microwave background at redshift z = 1100.

  • Scale Factor: 0.000.00
  • Lookback Time: 13.7013.70
  • Comoving Distance: 4714285.704714285.70
  • Luminosity Distance: 5200000000.005200000000.00
  • CMB Temperature: 3000.003000.00

Visualization

Cosmology

Cosmology is the study of the universe as a whole, including its origin, evolution, and ultimate fate. The modern understanding is based on the Big Bang theory, which describes the universe as expanding from an extremely hot and dense initial state about 13.8 billion years ago.

The expansion of the universe is described by Hubble's Law: the recessional velocity of distant galaxies is proportional to their distance from us. The constant of proportionality is called the Hubble constant (H₀), currently measured to be about 70 km/s/Mpc. This expansion causes the redshift of light from distant objects.

Redshift (z) is a measure of how much the wavelength of light has been stretched by the expansion of the universe. It's defined as z = (λ_observed - λ_emitted) / λ_emitted. The larger the redshift, the farther away and older the light source. Redshift is directly related to the scale factor of the universe.

The geometry and fate of the universe depend on the total density of matter and energy. The critical density is the density required for a flat universe. If the actual density is greater than critical, the universe is closed and will eventually collapse. If less, it's open and will expand forever.

Dark energy, discovered in the late 1990s, is causing the expansion of the universe to accelerate. This mysterious form of energy makes up about 70% of the universe's energy density and has negative pressure, acting like a repulsive gravitational force.

Key Concepts

  • Big Bang Theory: Universe began from hot, dense state
  • Hubble's Law: v = H₀d - expansion rate proportional to distance
  • Redshift: z = (λ_obs - λ_emit) / λ_emit - measure of cosmic expansion
  • Scale Factor: a(t) - describes how distances change with time
  • Critical Density: ρ_c = 3H₀²/8πG - density for flat universe
  • Dark Energy: Mysterious energy causing accelerated expansion

Real-World Applications

  • Distance Measurements: Using redshift to determine cosmic distances
  • Age Determination: Calculating the age of the universe
  • Galaxy Evolution: Understanding how galaxies change over cosmic time
  • Dark Energy Studies: Investigating the nature of cosmic acceleration
  • CMB Analysis: Studying the cosmic microwave background radiation

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Physics Equations

Hubble's Law:
v=H0dv = H_0 d
Redshift Definition:
z=λobs−λemitλemitz = \frac{\lambda_{obs} - \lambda_{emit}}{\lambda_{emit}}
Scale Factor:
a(t)=11+za(t) = \frac{1}{1 + z}
Critical Density:
ρc=3H028πG\rho_c = \frac{3H_0^2}{8\pi G}
Luminosity Distance:
dL=d(1+z)d_L = d(1 + z)

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Scale Factor

Find the scale factor from redshift:

Equation:

a=11+za = \frac{1}{1 + z}

Calculation:

a=11+0.1=0.909a = \frac{1}{1 + 0.1} = 0.909

Explanation:

The scale factor describes how much the universe has expanded since the light was emitted.

2

Step 2: Calculate Lookback Time

Find how long ago the light was emitted:

Equation:

tlookback=tuniverse(1−a)t_{lookback} = t_{universe}(1 - a)

Calculation:

tlookback=13.8×(1−0.909)=1.3 Gyrt_{lookback} = 13.8 \times (1 - 0.909) = 1.3 \text{ Gyr}

Explanation:

This gives the time that has passed since the light was emitted from the distant object.

3

Step 3: Calculate Comoving Distance

Find the distance accounting for expansion:

Equation:

dcomoving=czH0d_{comoving} = \frac{cz}{H_0}

Calculation:

dcomoving=(3×108)(0.1)70=428.3 Mpcd_{comoving} = \frac{(3\times10^8)(0.1)}{70} = 428.3 \text{ Mpc}

Explanation:

This is the distance that would be measured today between us and the object.

4

Step 4: Calculate Luminosity Distance

Find the distance from apparent brightness:

Equation:

dL=dcomoving(1+z)d_L = d_{comoving}(1 + z)

Calculation:

dL=428.3×(1+0.1)=471.1 Mpcd_L = 428.3 \times (1 + 0.1) = 471.1 \text{ Mpc}

Explanation:

This distance accounts for the dimming of light due to redshift.

5

Step 5: Calculate Critical Density

Find the density required for a flat universe:

Equation:

ρc=3H028πG\rho_c = \frac{3H_0^2}{8\pi G}

Calculation:

ρc=3(70×1000/3.1×1022)28π(6.67×10−11)=9.21e−27 kg/m3\rho_c = \frac{3(70 \times 1000 / 3.1\times10^{22})^2}{8\pi(6.67\times10^{-11})} = 9.21e-27 \text{ kg/m}^3

Explanation:

This is the density that would make the universe geometrically flat.

6

Step 6: Calculate Expansion Rate

Find the expansion rate at the time of emission:

Equation:

H(z)=H0(1+z)H(z) = H_0(1 + z)

Calculation:

H(z)=70(1+0.1)=77.0 km/s/MpcH(z) = 70(1 + 0.1) = 77.0 \text{ km/s/Mpc}

Explanation:

The universe was expanding faster in the past when the light was emitted.

7

Step 7: Calculate CMB Temperature

Find the temperature of the cosmic microwave background:

Equation:

T(z)=T0(1+z)T(z) = T_0(1 + z)

Calculation:

T(z)=2.725(1+0.1)=3.0 KT(z) = 2.725(1 + 0.1) = 3.0 \text{ K}

Explanation:

The universe was hotter in the past, so the CMB temperature was higher when the light was emitted.

Frequently Asked Questions (FAQ)

What is redshift and how is it measured?

Redshift is the stretching of light waves due to the expansion of the universe. It's measured by comparing the observed wavelength of light from distant objects to the known wavelength of the same light emitted in laboratories. The larger the redshift, the farther away and older the object.

What is the Hubble constant and why is it important?

The Hubble constant (H₀) measures the current rate of expansion of the universe. It's the slope of the relationship between galaxy distances and their recessional velocities. Its value determines the age and size of the universe, and is crucial for understanding cosmic evolution.

What is the difference between comoving and luminosity distance?

Comoving distance is the distance that would be measured today between two points, accounting for the expansion of the universe. Luminosity distance is the distance inferred from the apparent brightness of an object, which is affected by both the expansion and the redshift of light.

What is dark energy?

Dark energy is a mysterious form of energy that makes up about 70% of the universe's energy density. It has negative pressure and causes the expansion of the universe to accelerate. Its nature is one of the biggest mysteries in modern physics.

How do we know the age of the universe?

The age of the universe is determined by measuring the Hubble constant and the density of matter and energy. The relationship between these parameters in the Big Bang model gives us the time since the initial expansion began, currently estimated at 13.8 billion years.

Practice MCQs

  1. What does Hubble's Law state?
  2. What is the relationship between redshift and scale factor?
  3. What causes the redshift of light from distant galaxies?
  4. What is the critical density of the universe?
  5. What is the cosmic microwave background?

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