Updated December 2, 2023

# Redshift (z)

Image: the spectrum of visible light, which goes from infrared to ultraviolet, corresponds to wavelengths from 400 nanometers in the violet to 780 nanometers in the red.
This image shows the shift in wavelengths of a star moving at about 1% of the speed of light.
- Top: the spectrum of the star, if it were immobile.
- In the middle: the spectrum of the star, shifted by around 5 nm towards the red, if it moved away from us.
- Bottom: the spectrum of the star, shifted by around 5 nm towards the blue, it was getting closer to us.
The black lines represent the absorption of wavelengths specific to elements present in the stellar atmosphere.

## Redshift (z)

In astronomy, redshift is an increase in the wavelength of light emitted by a distant celestial object. This results in a red shift of the spectral lines of light, which is visible on a spectrogram.
The redshift is denoted by the letter "z". It is calculated by comparing the observed wavelengths of spectral lines emitted by astronomical objects with the wavelengths expected in a laboratory on Earth.
If the object moves away from us, the spectral lines are shifted towards red, hence the term "redshift".
If the object comes closer, the shift is towards blue, and we then speak of "blueshift".
In other words, the farther away the object is, the higher its redshift (z).

This phenomenon is explained by the Doppler effect.
The Doppler effect is a modification of the frequency of a wave and therefore of its wavelength, when there is a relative movement between the source of the wave and the observer.

Redshift is an essential tool for studying the distant universe. It makes it possible to determine the distance of cosmic objects, to study their evolution over time, and to understand the structure of the universe.

## How to determine the red shift?

Image: Redshift curve.
Example: if z = 6, the age of the observed object corresponds to the time when the light was emitted, i.e. approximately 13 billion years.

The cosmological red shift is a phenomenon observed in light spectra coming from distant objects in the universe.

The red shift is primarily caused by the expansion of the universe, which stretches the wavelengths of photons as they travel through space. This results in a shift of the spectral lines towards longer wavelengths, therefore towards the red in the electromagnetic spectrum.

When an object emits light, its emission spectrum shows characteristic spectral lines. The peaks and valleys of the spectrum are due to the absorption and emission lines of the elements found in the observed object. These lines are associated with specific energy transitions in the atoms or molecules of the object.
For example, when measuring the Lyman Alpha line of hydrogen in the laboratory we see a transition at approximately 121.6 nanometers in the ultraviolet region of the electromagnetic spectrum. The Lyman Alpha line is so named because it represents the transition to the lowest Lyman level (n = 1), passing from level 2 to level 1 energy. The Lyman level refers to a specific set of electronic energy levels in a hydrogen atom. These energy levels are associated with the different orbits allowed for electrons in a hydrogen atom, and they are defined by the principal quantum n.

If we observe an object whose transition is at 480 nm, this means that the observed object is approximately 12 billion years away from us.
The red shift acts as a multiplicative factor. The factor in this example is 4. We will find this factor on all the other characteristics of the spectrum (carbon, silicon, etc.). So all the wavelengths are multiplied by this factor which we call redshit and which we call z.
The redshit is equal to the multiplicative factor -1: z - 1 = 3
If there is no offset z = 0.
The larger z is, the farther away and therefore older the object we observe. We can thus relate the z to the age of the object at the time when the light was emitted.
- z = 0 corresponds to an age of 13.8 billion years.
- z = 1 corresponds to an age of 5.8 billion years.
- z = 3 corresponds to an age of 2 billion years.
- z = 6 corresponds to an age of 800 million years.
- z = 11 corresponds to an age of 400 million years.
- z = 20 corresponds to an age of 200 million years.
- z = 30 corresponds to an age of 100 million years.

By comparing the observed spectrum with the expected spectrum of a nearby object with the same emission spectrum, one can determine the red shift. Red-shifted spectral lines indicate the degree of shift.

## Calculation of redshift for distant objects

Calculating the redshift (z) for a distant object traveling at half the speed of light:
z + 1 = c + v / c - v
z + 1 = 300,000,000 + 150,000,000 / 300,000,000 - 150,000,000
z = 3

Calculation of the speed for a distant object whose redshift is 4:
v = c * ((1 + z)2 -1) / ((1 + z)2 + 1)
v = 276,923,077 m/s or 92% of the speed of light.

## Why is redshift a crucial concept?

Redshift is essential for estimating cosmological distances, for understanding the expansion of the universe, and for exploring the nature of dark energy, a mysterious form of energy that appears to accelerate this expansion.
It allows us to reconstruct the history of the universe. Indeed, measuring the red shift of distant galaxies makes it possible to determine their age and their relative distance. This makes it possible to reconstruct the history of the expansion of the universe and to understand how galaxies were formed and how they evolved.

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