Particle Physics and Cosmology: The Expanding Universe

Citation: The content below is based on the most recent edition of University Physics with Modern Physics.

Important Formulas

Hubble’s Law:
$v = H_0 d$
Friedmann Equation:
$\left(\frac{\dot{a}}{a}\right)^2 = \frac{8 \pi G}{3} \rho - \frac{k}{a^2} + \frac{\Lambda}{3}$
Redshift Formula:
$1 + z = \frac{\lambda_{\text{observed}}}{\lambda_{\text{emitted}}}$
Critical Density:
$\rho_c = \frac{3 H_0^2}{8 \pi G}$
Scale Factor Relation:
$d(t) \propto a(t)$
Age of the Universe (for flat universe with $\Lambda = 0$ ):
$t_0 = \frac{1}{H_0}$

Key Terms and Definitions

Hubble’s Constant ( $H_0$ ): The rate of expansion of the universe, expressed in units of velocity per distance (e.g., $\text{km/s/Mpc}$ ).
Friedmann Equation: A fundamental equation in cosmology describing the dynamics of the universe’s expansion based on general relativity.
Redshift ( $z$ ): The observed increase in the wavelength of light from distant objects, indicating their motion away from the observer due to cosmic expansion.
Scale Factor ( $a(t)$ ): A dimensionless quantity that describes how the distances between objects in the universe change over time.
Critical Density ( $\rho_c$ ): The density at which the universe is perfectly flat; a key value in cosmology for understanding the fate of the universe.
Cosmic Horizon: The maximum distance from which light emitted in the past can reach an observer today, defining the observable universe.

Example

If a galaxy has a redshift of $z = 0.5$ and the current Hubble constant is $70 \, \text{km/s/Mpc}$ , calculate the observed wavelength of a spectral line originally emitted at $500 \, \text{nm}$ .

Using the redshift formula:

$1 + z = \frac{\lambda_{\text{observed}}}{\lambda_{\text{emitted}}}$

Substituting values:

$1 + 0.5 = \frac{\lambda_{\text{observed}}}{500 \, \text{nm}}$

$\lambda_{\text{observed}} = 750 \, \text{nm}$

Result:

The observed wavelength is $750 \, \text{nm}$ , corresponding to a shift into the red part of the spectrum.

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