In 1543, Nicholas Copernicus observed that the Earth is not the center of the Universe, and that the Earth revolves around the Sun. It is from these observations that Einstein derived the equations of relativity and the cosmological principle. The cosmological principle states that the Universe is fairly uniform in consistency, no matter where you go – but the truth is much weirder than that. In fact, the methods we implement to measure the smoothness of the Universe only uses two parts, the Friedmann equations, of the twenty-variable, ten-equation system that Einstein used to derive general relativity. The Friedmann equations are two equations that describe how the universe changes over time. They were derived in 1922 by Alexander Friedmann from Einstein’s relativity and are based on the Cosmological Principle, which states that the universe is the same from any point of view. The equations describe the expansion and contraction of the universe as well as the curvature of space-time. **The Cosmological Principle** is based on the **FLRW (Friedmann-Lemaître-Robertson-Walker) metric**, a solution to Einstein’s field equations in general relativity that describes the homogeneous and isotropic expansion of the universe. The metric is given by:

**ds^2 = -dt^2 + a^2(t) [(dr^2 / (1 – kr^2)) + r^2 (dθ^2 + sin^2(θ) dφ^2)]**

where **ds^2** is the line element, **t** is time, **a(t)** is the scale factor, **k** is the curvature parameter **(-1, 0, or 1)**, **r** is the radial coordinate, **θ** is the polar angle, and **φ** is the azimuthal angle. The FLRW metric provides a simple description of the expansion of the universe that is widely used in cosmology. These equations provide a cornerstone of the standard model of cosmology, the Lambda-CDM model, which describes the evolution of the universe from the Big Bang to the present day. The Friedmann equations are a set of equations that describe the evolution of the *scale factor*, **a(t)**, in the FLRW metric. It is derived from Einstein’s field equations and relates the expansion rate of the universe to the matter and energy content within it. So while the FLRW metric describes the homogeneous and isotropic expansion of the universe, the Friedmann equations are specific equations derived from the FLRW metric that describe the evolution of the **scale factor** *over* **time**.

The two equations are as follows:

The First **Friedmann Equation** is derived from the “00 component” of Einstein’s field equations:

**H^2 = (8πG/3) * (ρ + Λ/c^2) – kc^2/a^2**

The **Landau-Raychaudhuri Equation**, which is another derivative of the field equations:

**dθ/dt = – θ^2 / 3 – σ^2 + R_{ab} k^a k^b**

Where **H** is the Hubble constant, **G** is the gravitational constant, **ρ** is the density of matter and energy, **Λ** is the cosmological constant, **c** is the speed of light, **a** is the scale factor, and **k** is the curvature of space.

This equation tells us that the expansion rate of the universe depends on the amount of matter and energy present, as well as the curvature of space. For example, if the density of matter is high, the expansion rate will be slower. On the other hand, if the density is low or there is a positive cosmological constant, the expansion rate will be faster. The curvature of space also plays a role in the expansion rate, with a positive curvature causing the expansion to slow down and a negative curvature causing it to speed up. The expansion of the universe is a complex and ongoing topic of study. There are many variables and sources of error involved in measuring the expansion rate, known as the Hubble rate, which has made it challenging to develop a consistent model. The cosmic microwave background (CMB) is electromagnetic radiation that is thought to be left over from the Big Bang and it is used to study the early universe. By analyzing the CMB, astronomers and physicists can learn about the history of the universe and the expansion rate at different points in time.

Science Daily reported in October 2020 on results published by researchers at the Cosmic Dawn Center. The researchers’ measurements of velocity indicate that the current methods used to measure the expansion rate of the universe may not be reliable. There are two popular methods to measure the expansion of the universe: one based on the relationship between the distance and velocity of nearby galaxies, and the other based on the study of background radiation from the early universe. These two methods currently yield different expansion rates, which could result in a dramatic reinterpretation of the development of the universe if the discrepancy is real, or it could simply be due to incorrect measurements. One tool used to measure cosmic distance scales is the standard candle. Standard candles are objects that have a known luminosity, or intrinsic brightness, that can be used to determine their distance from us. For example, Type Ia supernovae are thought to have a constant luminosity, so by measuring the apparent brightness of a Type Ia supernova, astronomers can calculate its distance from us. This allows them to measure the expansion rate of the universe over time by studying how the distances between objects change over time. Cosmic distant scales are essential to understanding cosmological expansion. These scales relate the observed properties of objects in the universe to their actual physical size and distance from us. For example, the apparent size of a galaxy in the sky is directly related to its actual size and distance from us. By studying these cosmic distance scales, astronomers can better understand the size and expansion of the universe. According to Albert Sneppen at the Niels Bohr Institute, there are two types of redshifts that must be considered when measuring velocities: one that measures the velocity with which the host galaxy moves away from us, and one that measures the velocity of matter ejected from an exploding star inside the galaxy.

To understand the Hubble expansion rate, it is necessary to have a prior understanding of certain concepts and tools, such as the speed of light, spectroscopy, standard candles, the hydrogen emission line (21 cm), and the four types of redshift. There are many variables and sources of error involved in measuring the expansion rate, known as the Hubble rate, which has made it challenging to develop a consistent model. The primary tool used to measure the expansion rate of the universe is the **Hubble constant**, which is defined as the ratio of the velocity of a distant galaxy to its distance from the observer. This relationship can be expressed as:

**Hubble Constant = Velocity of Galaxy / Distance to Galaxy**

The current value of the Hubble constant is approximately **70 km/s/Mpc**, but there is still ongoing debate about its precise value and how it has changed over time (Riess et al., 2016). To understand the meaning of the Hubble constant, it can be helpful to consider an analogy. Imagine you are standing on a platform at a train station, and you see a train approaching in the distance. As the train gets closer, you can measure its velocity by timing how long it takes to travel a certain distance. Similarly, astronomers can measure the velocity of a distant galaxy by studying the Doppler shift of its spectral lines, which indicates how fast the galaxy is moving away from us.

Now, imagine that the train station is located on a giant balloon, and the train is also on the balloon. As the balloon expands, the distance between the train and the platform will also increase. In a similar way, the expansion of the universe causes the distances between galaxies to increase over time. The Hubble constant tells us how fast, or the rate, at which the distance between a particular galaxy and the observer is increasing.

To measure the expansion rate of the universe, astronomers use two different techniques: one based on the relationship between the distance and velocity of nearby galaxies, and the other based on the study of background radiation from the early universe. These two methods currently yield different expansion rates, which could result in a dramatic reinterpretation of the development of the universe if the discrepancy is real, or it could simply be due to incorrect measurements. To understand the Hubble expansion rate, it is necessary to have a prior understanding of certain concepts and tools, such as the speed of light, spectroscopy, standard candles, the hydrogen emission line (21 cm), and **the four types of redshift**. Redshift is a phenomenon that occurs when the wavelength of electromagnetic radiation appears to be longer than its original wavelength due to the relative motion or expansion of the universe, and the spectra of an object shifts towards the infrared as it accelerates away from the observer. There are four main types of redshift that are commonly discussed in the context of cosmological expansion:

**Doppler redshift:** This type of redshift is caused by the relative motion between the source of electromagnetic radiation (such as a galaxy) and the observer. When an object is moving away from the observer, the wavelength of the electromagnetic radiation it emits will appear to be longer (redshifted) due to the Doppler effect. This can be described by the following equation:

**Δλ/λ = v/c**

Where **Δλ **is the change in wavelength, **λ** is the original wavelength, **v** is the velocity of the object, and **c** is the speed of light.

**The Doppler redshift** is commonly used to measure the expansion rate of the universe, as it is directly related to the velocity of the object. By measuring the Doppler redshift of a galaxy, astronomers can determine its velocity and how it is moving relative to us.

**Cosmological redshift:** This type of redshift is caused by the expansion of the universe, and it affects all objects in the universe. As the universe expands, the distance between objects increases, causing the wavelengths of electromagnetic radiation to appear longer (redshifted). This can be described by the following equation:

**z = Δλ/λ = (λobs – λem)/λem**

Where **z** is the redshift, **Δλ** is the change in wavelength, **λobs** is the observed wavelength, and **λem** is the emitted wavelength.

The cosmological redshift is a consequence of the expansion of the universe, and it can be used to study the history and evolution of the universe. By analyzing the cosmological redshift of objects at different distances and times, astronomers can learn about the expansion rate of the universe over time.

**Gravitational redshift:** This type of redshift is caused by the curvature of spacetime due to the presence of a massive object, such as a black hole. The curvature of spacetime causes the wavelengths of electromagnetic radiation to appear longer (redshifted) as it travels through a gravitational field. This can be described by the following equation:

**z = (1 – 2GM/rc^2)^0.5 – 1**

Where **z** is the redshift, **G** is the gravitational constant, **M** is the mass of the object, **r** is the distance from the object, and **c** is the speed of light.

The gravitational redshift is a consequence of the relativistic effects of strong gravitational fields. The gravitational redshift is used to study the properties of massive objects such as black holes, and to determine the ages of certain stars. This type of redshift can also be used to measure the rotation of galaxies and to study the structure of galaxy clusters. Additionally, gravitational redshift can be used to measure the distances to other galaxies, as the amount of redshift observed increases with the distance to the galaxy.

**Time dilation redshift**: This type of redshift is caused by the difference in time experienced by objects moving at different velocities. According to the theory of relativity, time appears to move slower for objects moving at high velocities or in strong gravitational fields. This causes the wavelengths of electromagnetic radiation emitted by these objects to appear longer (redshifted) as observed from a stationary frame of reference. This can be described by the following equation:

**z = (1 – v^2/c^2)^0.5 – 1**

Where **z** is the redshift, **v** is the velocity of the object, and **c** is the speed of light.

This fourth type of redshift is also manifested as blueshift, which is the opposite of redshift. This occurs when an object is moving towards us, rather than away from us. Blueshift is used to measure the velocities of stars and other objects within our own galaxy, and it can be used to study the dynamics of galaxy formation and evolution of galaxies and star systems. The time dilation redshift is a consequence of the relativistic effects of high velocity. Blueshift can be used to study the properties of objects moving at high speeds. By analyzing the time dilation redshift of objects moving at different velocities, astronomers can learn about the effects of velocity on the perception of time.

In the context of the article, Sneppen mentions two types of redshift: one that measures the velocity of the host galaxy moving away from us, and another that measures the velocity of matter ejected from an exploding star within a galaxy. These redshifts are the Doppler redshift and the time dilation redshift, respectively. The Doppler redshift is considered the most reliable for measuring the expansion rate of the universe, as it is directly related to the velocity of the object. The time dilation redshift, on the other hand, is a consequence of the relativistic effects of high velocity, and may not be as reliable for measuring the expansion rate.

Images of galaxy clusters, such as the one called “Bullet” 1E 0657-56, can provide insights into the effects of dark matter on the formation of galaxies and the life cycles of stars. Although we do not yet fully understand if or how theoretical dark matter is involved, these images from telescopes like the Hubble and Chandra have raised further questions about the standard model of LambdaCDM (Cold Dark Matter).

In October 2021, the discovery of Hamilton’s Object marked a historic first. This object, named after its discoverer and co-author of a paper in the Royal Astronomical Society Letters, is thought to be a highly dense clump of invisible dark matter. Its effects can only be seen indirectly, as it passes through a star cluster and creates a ripple in the fabric of space, resulting in two nearly perfect mirror images via gravitational lensing, with a lesser reflection off to the side. Lensed images display characteristics that are similar to those of other folds where the source galaxy is located very close to or intersects the caustic of a galaxy cluster. The images are elongated in a direction that is roughly perpendicular to the critical curve, but they have a tangential cusp configuration. Based on morphological features, published simulations, and similar fold observations found in literature, a third or counter-image is identified and confirmed through spectroscopy. The fold configuration has highly distinctive surface brightness features, so follow-up observations of microlensing or detailed studies of the individual surface brightness features with higher resolution can provide more insight into the kpc-scale dark matter properties. This discovery adds to our understanding of dark matter and its role in the universe, but how this ties into cosmological expansion remains a mystery just as well.

## The Bolshoi Simulation and the Lambda CDM Model

The Bolshoi simulation is a large-scale N-body simulation of the formation and evolution of cosmic structures. It is one of the most widely used simulations in the astrophysical community and has been extensively studied and analyzed to gain insight into the large-scale structure of the universe and the processes that govern its formation. The simulation was first introduced in 2005 by Bolshoi et al. (2010) and has since undergone several upgrades to improve its accuracy and resolution. It is based on the Lambda-CDM (Cold Dark Matter) cosmological model, which is widely accepted as the standard model of the universe. The simulation uses a variety of techniques including particle-mesh methods and hierarchical tree algorithms to model the behavior of dark matter and baryonic matter in a large cosmological volume. One of the key strengths of the Bolshoi simulation is its large volume, which allows for a statistically robust representation of the cosmic large-scale structure. The simulation follows the evolution of over 20 million dark matter particles and incorporates various physical processes, such as gas cooling, star formation, and feedback from supernovae and active galactic nuclei to model the behavior of baryonic matter. The Bolshoi simulation has been widely used in a variety of studies to address important astrophysical questions. For example, it has been used to study the properties of dark matter halos, the distribution of galaxies, and the clustering of galaxy clusters (Klypin et al., 2011). The Bolshoi simulation has also been used to investigate the impact of baryonic processes on the large-scale structure of the universe and to test various cosmological models (Rudd et al., 2008).

Like all simulations, **the Bolshoi simulation** has certain limitations that impact its accuracy and ability to capture the complexity of the universe. Some of the key limitations include:

**Simplifications in modeling physical processes**: While the Bolshoi simulation includes various physical processes, such as gas cooling, star formation, and feedback from supernovae and active galactic nuclei, these processes are still modeled in a simplified and idealized manner. There is still much uncertainty in the physical processes that govern the formation and evolution of the universe, and the simulations must make assumptions and simplify the models in order to be computationally tractable.**Limited resolution**: The Bolshoi simulation has a finite resolution, which means that smaller structures and processes are not resolved with sufficient detail. This can impact the accuracy of the simulation and limit its ability to capture important features, such as the behavior of dark matter in small halos or the distribution of galaxies in dense environments.**The use of a fixed cosmological model**: The Bolshoi simulation is based on the Lambda-CDM cosmological model, which is widely accepted as the standard model of the universe. However, there is still much uncertainty in the cosmological parameters, and it is possible that future observations may require modifications to the cosmological model used in the simulation.**Modeling limitations**: The Bolshoi simulation uses N-body methods and hierarchical tree algorithms to model the behavior of dark matter and baryonic matter. These methods are based on simplifying assumptions about the behavior of dark matter and baryonic matter and may not accurately capture the complexity of the universe.

To better understand the expansion of the universe, it will be necessary to continue researching and making more precise measurements. This may involve using new techniques, such as the use of quasars as standard candles (Risaliti & Lusso, 2015), or the ongoing study of the cosmic microwave background radiation to measure the expansion rate at different points in the observable history of the universe (Planck Collaboration et al., 2018). These advances provide valuable insight into the expansion of the universe and further our comprehension of the origins and evolution of the cosmos.

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