Understanding distances in Cosmology

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In summary, There are several definitions for distance in cosmology, including proper distance, comoving distance, conformal time, and lookback time/distance. These can be calculated using integrals involving the Hubble parameter. A helpful resource for understanding these distance measures is the paper linked above.
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Arman777
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I am trying to understand this graph but I am confused about the distance definitions. So there's an object located at a comoving distance ##r##.

The proper distance of the object at ##z'## can be written as
$$d(z{'}) = \frac{1}{1+z{'}}\int_0^{z{'}} \frac{dz{'}}{H(z{'})}$$

In this case,

$$d_1(z) = \frac{1}{1+z}\int_0^z \frac{dz}{H(z)}$$

and

$$d_0(z) = \int_0^z \frac{dz}{H(z)}$$

And we can also define the conformal time where it represents the distance taken by photon from the distant galaxy to us and can be written as

$$\eta(z) = \int_0^z \frac{dz}{H(z)}$$

There is also lookback time/distance but I am not sure it's same as the conformal time or not...

Are these definition that I have made are correct ?
 
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Related to Understanding distances in Cosmology

1. What is the Hubble constant and why is it important in understanding distances in cosmology?

The Hubble constant is a measure of the rate at which the universe is expanding. It is important in understanding distances in cosmology because it allows us to calculate the distance to faraway galaxies based on their observed redshift. This helps us to understand the size and age of the universe.

2. How do we measure distances in cosmology?

Distances in cosmology are measured using a variety of methods, including parallax, standard candles, and redshift. Parallax involves measuring the apparent shift of an object against a background of distant stars as the Earth orbits the sun. Standard candles, such as Type Ia supernovae, have a known luminosity and can be used to calculate their distance. Redshift, the stretching of light from distant galaxies, can also be used to measure distances.

3. What is the difference between comoving distance and proper distance?

Comoving distance is the distance between two objects in the expanding universe, taking into account the expansion of space. Proper distance, on the other hand, is the distance between two objects at a specific moment in time, without accounting for the expansion of space. Proper distance is often used to measure the distance between objects in the local universe, while comoving distance is used for objects at larger distances.

4. How does the expansion of the universe affect distance measurements?

The expansion of the universe causes objects to appear farther away than they actually are. This is due to the stretching of space between objects as the universe expands. As a result, the further away an object is, the faster it appears to be moving away from us. This is known as the Hubble Law and is an important factor in calculating distances in cosmology.

5. Can we accurately measure distances in the early universe?

Due to the expansion of the universe, distances in the early universe are more difficult to measure accurately. However, scientists use a variety of methods, such as the cosmic microwave background radiation and the abundance of light elements, to estimate the size and age of the universe in its early stages. These measurements are continually refined as our understanding of the universe improves.

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