At the time of the last scattering epoch (at a redshift...

At the time of the last scattering epoch (at a redshift $z=1000$ ), a. what was the average density of normal matter, if today it is about $3 \times 10^{-31} \mathrm{kg} / \mathrm{L} ?$ b. how large a sphere would have contained $10^{11} M_{\odot}$, about as much normal matter as in the Milky Way Galaxy? Express your answer in parsecs.

At the time of the last scattering epoch (at a redshift $z=1000$ ),
a. what was the average density of normal matter, if today it is about $3 \times 10^{-31} \mathrm{kg} / \mathrm{L} ?$
b. how large a sphere would have contained $10^{11} M_{\odot}$, about as much normal matter as in the Milky Way Galaxy? Express your answer in parsecs.

Key Concepts

Parsec

A parsec is an astronomical unit of distance used to measure large distances to objects outside the solar system. One parsec is defined as the distance at which one astronomical unit subtends an angle of one arcsecond. It is a key unit in astrophysics that facilitates the conversion of distances calculated from mass-density relationships into a more intuitive and standardized measure for comparing cosmic scales.

Density Evolution in an Expanding Universe

The density of matter in the universe evolves as the universe expands. For non-relativistic matter, the density scales inversely with the cube of the scale factor (or equivalently, with (1+z)^3). This means that if we know the density today and the redshift in the past, we can calculate the density during that epoch by multiplying the current density by (1+z)^3. This relation is crucial for understanding how matter was distributed in earlier epochs compared to the present day.

Mass-Density-Volume Relationship

The relationship between mass, density, and volume is fundamental in astrophysics. Given a known density and total mass, one can estimate the volume that would have contained that mass using the formula mass = density × volume. For spherical volumes, this often involves rearranging the volume formula of a sphere (volume = (4/3)?r^3) to solve for the radius. This concept is used to estimate the spatial size of regions like galaxies in different epochs of the universe.

Cosmological Redshift

Cosmological redshift is a measure of the stretching of light as the universe expands. It quantifies how much the wavelength of light from a distant source is increased due to the expansion, and it is directly related to the scale factor of the universe. A redshift value (like z=1000) indicates that the universe was much smaller in the past, thereby affecting both the density and temperature of matter and radiation at that epoch.

Scale Factor

The scale factor is a parameter in cosmology that describes how the size of the universe changes over time. It is inversely related to the redshift by the relation a = 1/(1+z). Through this relation, we understand that a redshift of 1000 implies that the universe was 1/1001 its current size, which in turn means that lengths were shorter and volumes were much smaller, affecting density measurements across cosmic time.

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