Astrophysics

The expanding cosmos

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ABSTRACT

 

For centuries, mankind’s understanding of the universe was mainly informed through religious beliefs. It was not until the 20th century, when Einstein published his paper on General relativity, in 1917, that humanity began to understand the nature of the universe in scientific terms. Einstein’s work not only formed the backbone of physical cosmology but also inspired others to begin similar exploration. Subsequently, extraordinary minds such as Alexander Friedmann, George Lemaitre, and Alan Guth added their ideas to the field, some of which were experimentally proven by researchers such as Edwin Hubble. Within a decade, mankind went from understanding little about the universe to not only observing its expansion but also confirming the rate of that expansion.

 

This paper will trace the evolution of the scientific ideas that emerged in the field of cosmology in the 20th century and were tested by experimentation and observation. These ideas include Friedmann-Lemaitre’s expansion and the three models of the universe, Lemaitre’s big bang, Guth’s inflationary cosmology, and Perlmutter’s discovery of dark energy. The paper will conclude with a discussion of the hypothetical futures that await our universe, namely The Big Crunch, The Big Rip, and The Big Freeze.

INTRODUCTION

Cutting across race, religion, and even centuries, humankind has long been united in its desire to understand and explain the universe around us – how it works, how it came into being, and how it will end. For centuries, people believed the notion that ‘God’ created a static universe, one that will always exist, even billions of years after their death.

Albert Einstein, while writing his paper on general relativity, found that according to his model, all the galaxies in the universe must be pulled towards each other, and the universe must contract. He, in effect, introduced a cosmological constant that countered the effects of gravity [1]. However, even he – one of the most brilliant minds in scientific history – failed to see that the universe may be dynamic.

Alexander Friedmann and George Lemaître were among the first theorists to hypothesize the expansion of the universe. They presented their models, which were quite controversial, and many scientists, including Einstein, criticized them until they were proven.

In the 1920s, many astronomers observed the spectra of stars in distant galaxies using the modern telescope. They realized that the light coming from the stars was red-shifted. Astronomer Edwin Hubble concluded this bizarre observation to be the result of galaxies moving away.[2] This discovery meant that the universe was expanding, and this also led Einstein to abandon his cosmological constant and proclaim it as his life’s biggest mistake.

Alexander Friedmann had assumed that the universe was homogeneous and isotropic on larger scales (later proven and given the name ‘Cosmological principle”)[3]. He had created a set of equations on this principle, which are still an integral part of cosmology, and his works can be considered the core principles of cosmology.

George Lemaître later presented the idea that the universe had started from a point of infinite density and infinite curvature[4]. He traced the expansion backwards and thus hypothesized a beginning of the universe. This idea was also subject to ridicule until, in the 1960s, the discovery of Cosmic Background Radiation forced all the physicists to accept it. Afterwards, in 1981, Alan Guth gave the inflation theory, which further deepened our understanding of the cosmos.

In 1998, the world of cosmology took a steep turn by a surprising discovery. Scientists found that the universe was expanding at an accelerating rate[5]. Looking as far as they can, cosmologists have hypothesized three possible outcomes of the expansion and dynamic nature of our universe: the Big Crunch, the Big Freeze, and the Big Rip. Each of these models presents a different state of the universe in the distant future, and currently, we cannot rule out any of them. In this paper, the cosmological understanding of the dynamic nature of the universe will be discussed based on General Relativity.

EXPANSION

German physicist Albert Einstein and Soviet physicist Alexander Friedmann can be considered the main pioneers of modern cosmology; without their works, we would not be able to describe the universe the way we do right now.

 

Albert Einstein’s General relativity is the main foundation on which many scientists’ works have been built, including Friedmann. Einstein believed that the universe was static; however, Friedmann demonstrated that the General theory of relativity could have non-static solutions [6]. His fundamental equations are what enable our current view of the universe, starting at a Big Bang and expanding.

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Fig. 1. Fundamental Einstein’s field equation (Amarashiki 2015)

 

Tμν: The energy-momentum tensor which determines the local geometry encoded in the metric tensor gμν.

Rμν: Ricci scalar –a contraction of the Ricci tensor – is the local curvature of space-time.

Λ: cosmological constant introduced in hopes of finding a static solution.

Finding cosmological solutions of field equations is a tough challenge. Before 1922, only two such solutions had been theorized. The first one by Einstein, whose solution enabled him to achieve his goal of finding a finite static universe with size determined by mass density. His solution represented a 3D spatial hyperspherical surface of constant radius r embedded in 4D space-time. The second solution was discovered by De Stitters. He, on the contrary, presented a different static universe with zero mass density and a negative spatial curve. It has a unique centre, and only light rays passing through it travel along geodesics[7].

While Albert Einstein and other scientists were looking for ways to avoid a dynamic nature predicted in general relativity, Friedmann sought to explain it. As he stated in the introduction of his paper ‘On the Curvature of Space,’ the goal of the paper was to bring to light the proof of the possibility of a universe whose spatial curvature is constant for the three spatial coordinates and dependent on time[8]. Based on his cosmological assumptions and General relativity, he asserted that the physical requirements of spatial homogeneity did not require a static universe. He found a new class of non-static solutions to the General relativity field’s equation. He featured the space as a 3D hypersphere with changing curvature in time with the hypersphere radius[9]. Albert Einstein claimed Friedmann’s idea of a non-static universe not to have any real meaning; however, Hubble later discovered expansion. Unfortunately, Alexander Friedmann – whose life was cut short at the young age of 37 – was not able to live long enough to see his prediction of an expanding universe be true.

In 1925, when the experimental data had begun to question the validity of Einstein’s static universe model, many astronomers started to favour De Sitter’s cosmological model. Vesto Silpher, an American astronomer, had measured and published 41 spectral shifts, out of which 36 were red-shifted. The Belgian priest George Lemaître, a student of the famous British astronomer Arthur Eddington, proved a linear relationship between distance and redshift in De Sitter’s solution. However, later he pointed out the weak spots in the De Sitter’s model.[10]

In 1927, Lemaître, having not heard of Alexander Friedmann’s work, discovered many new solutions to Einstein’s field equations.[11] In the most famous, he assumed a positive curvature of space, kept the cosmological constant non-zero, and allowed the radius of curvature and density of the universe to be a function of time. Thus, he obtained a model with perpetual accelerated expansion. However, his works remained unnoticed. [12]

When Hubble had begun to observe the red-shifted radiation from faraway galaxies, using the Doppler effect, he suggested that they were moving away from us[13]. However, he remained curious why the galaxies were moving away from us and could not link this with expansion. Lemaître was the one who derived the equation v=H.d where redshifts were treated as Doppler shifts corresponding to velocity v, d is the nebular distances, and H is the factor of proportionality. This was later named after Hubble after he experimentally calculated a much more precise value.[14]

Thus, using Hubble’s and Silpher’s observations, Lemaître postulated an expanding universe.

In 1930, Arthur Eddington discovered the flaws of Einstein’s model. He saw that the model was quite unstable. With the least disturbance in the model, the universe would begin expanding and contracting. At that time, Eddington discovered Lemaître’s work, found it remarkable, and promoted it.

Friedmann’s Equations

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Fig.2. Friedmann’s Equations (The Friedmann Equation n.d)

Solving Einstein’s field equation for a homogeneous and isotropic universe has derived this equation.

ρ=inertial mass density of matter and radiation in the universe

k= curvature in the R-W metric

R is the curvature radius of space

G=gravitational constant

Friedmann assumed a positively curved space, a time variable matter density, and a vanishing cosmological constant. Therefore in 1922, he hypothesized his famous ‘closed universe model’ with dynamics of expansion-contraction.[15]

In 1924, Friedmann studied the possibility of a negative curvature of the universe. He thus created the open universe model with eternal expansion.

Friedmann- Lemaître’s Three Models of the Universe

A close up of a logo

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Fig.3 Rewriting the Friedmann Equation as a relation between expansion rate and density ratios. (Pela 2015)

H= Hubble parameter describing the rate of expansion (factor of proportionality)

H0=Hubble’s constant (expansion rate today). It is equal to 70kms−1Mpc−1

a= Expansion factor (1 today)

Ω=Density parameters

Friedmann’s first equation can be written in this way. This equation provides us with a relation between the expansion rate of the universe and the density of its components and its size. By measuring its densities, the evolution of the universe can be predicted.[16]

Ω0 is the ratio of the density of the universe to the critical density i.e., Ω0= ρ/ ρc

According to this relation, three major universe models can exist. These are:

Closed Universe Model

This is when Ω0 >1, i.e., the density of the universe is greater than the critical density. In this case, the expansion would slow down, and the universe would subsequently start collapsing into itself. This gives us a periodically occurring universe that expands and collapses into itself repeatedly.

Flat Universe Model

This is when Ω0 =1, i.e., the density of the universe, is equal to the critical density. Such a universe’s expansion has been decelerated to a minimal amount, and it grows asymptomatically. This is a flat 2D universe with no or negligible curvature.

Fig.4.

(i)Graphic representation of Friedmann Models (Gilson 2020)

(ii)Geometric shapes of Friedmann Models of the Universe (NASA 2014)

Open Universe Model

This is when Ω0 <1, i.e., the density of the universe is less than the critical density. Such a universe’s expansion, even though decelerated by gravity, has a sufficient expansion rate to escape gravity and expand forever. This universe has negative curvature and has a saddle-like shape.

 

For the ease of understanding, we can take the example of a projectile thrown vertically upward. It would probably decelerate by gravity and then fall back to earth. This depicts the closed universe model. However, suppose we have some exceptionally unrealistically strong cannon or perhaps, we find a way to transform someone into Hulk. In that case, the projectile will escape the earth’s gravity and will keep moving on the virtue of its inertia (as given by Newton’s First law). This depicts the open universe model. If we create a highly precise instrument that can throw the projectile with just enough force for it to escape gravity, it will depict the flat universe.

BIG BANG THEORY AND COSMIC BACKGROUND RADIATION

If the universe has been expanding, there must have been a time in the past in which it had been denser and hotter. All solutions of Friedmann predict that at some time in the past, the distance between the galaxies must have been zero. They also predict that there must have been a time in which the density of the Universe and the curvature of the universe must have been infinite, and from which it started expanding. This part of the universe’s life cycle is called the ‘Big Bang.’

Even though there is evidence that confirms this, cosmologists still do not know what triggered it.

This seemingly bizarre and outrageous but accurate idea was given by George Lemaitré in 1931. This idea was introduced in a world where other scientists, even Einstein, had finally accepted the idea of an expanding and dynamic universe. He called his idea ‘the primeval atom.’

The name Big Bang was given later by Fred Hoyle in an attempt to mock his idea. Lemaître stated the universe to be so small and condensed in the past that it was a single entity. He considered this entity to be a ‘quantum of pure energy.’ He thus related cosmology to Quantum Mechanics, and this idea also introduced the branch of Quantum Cosmology. If the universe had been born as a single quantum, the notions of space and time would only have a sense after it had expanded[17]. Thus, the formation of the universe took place little before the beginning of space and time, according to him. He stated that there was no entropy at the time of this primaeval atom, and the time thus began after it started expanding, and entropy started building[18]. Afterwards, he assumed a positively curved space-time and pressure and density varying with time. This universe passes through a stagnation time during which it resembles Einstein’s static model before beginning an accelerated expansion again.[19]

His theory was refuted by all scientists. Albert Einstein and De Sitter had proposed a new Euclidean model, which due to their fame, was popular and more accepted. Other than that, the steady-state theory, given by Fred Hoyle, Thomas Gold, and Hermann Bondi, proposed a constant density of the universe, which was also much more acceptable. They emphasized on a field that allowed the matter to appear and compensate for dilution occurring due to expansion spontaneously. Their theory was disproved when a survey of radio waves from outer space was carried out at Cambridge by a group of astronomers led by Martin Ryle. They discovered that the radio source lay outside the galaxy, and the number of weaker ones was more than that of the strong sources. This meant that these sources were more numerous in the past.[20]

George Gamow, a student of Friedmann’s, gave the idea of a hot initial universe. He had predicted that the earlier light waves would not have been able to propagate just like light rays from the interior of the sun cannot reach us. He explained the genesis of the lighter nuclei and the formation of Hydrogen, Deuterium, and Helium. He also predicted the existence of Cosmic Radiation associated with the Big Bang.

Working on the Big Bang Theory, Alpen and Hermann calculated that one should today receive an echo of the Big Bang in the form of black body radiation at the temperature of 5K. In the 1960s, the theoretical physicists Robert Dicke and James Peebles studied oscillatory universe models (closed universe models undergoing expansion and contraction). They estimated the background radiation from the Big Bang to have a temperature of about 10K. They then learned that the engineers Arno Penzias and Robert Wilson had been receiving noise higher than expected. They discovered that the background noise was of a cosmic origin and was the same in all directions.[21]

That noise is now called CMB (cosmic microwave background). When the universe was young and extremely hot, it let out light that was stretched along with the expansion to such a long wavelength. At present, we find them in the form of microwave radiation. The CMB can be considered the echo of the big bang and thus is evidence for its occurrence. NASA has launched two missions to study the CMB and take pictures of a baby universe. The first one was COBE (Cosmic Background Explorer), launched in 1992. The satellite was able to map out the early universe and identify the primordial hot and cold spots in the cosmic background radiation. The second one was WMAP (Wilkinson Microwave Anisotropy Probe), which surveyed the skies at a greatly improved resolution[22].

Later, ESA launched the mission ‘Planck’, which has created the most accurate map of the CMB. Using theoretical models and observations from the CMB, theorists have been able to conclude that the universe is flat [23]. If the universe were curved, the photons reaching us from the early universe, which would have travelled the universe for billions of the years, would have caused the CMB map to appear distorted, which is not the case. The best measurements made indicate the universe is flat; however, some future equipment may indicate that the universe is curved. During the Big Bang, the universe had infinite curvature and was not flat[24]. Alan Guth, in the 1980s, gave his inflation theory in which he had proposed that the universe underwent a hyper expansion, which caused the universe to flatten out.[25]

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Fig.5.Planck CMB. (ESA 2013)

AN ACCELERATED EXPANSION

In 1998, two different teams of astronomers observed distant type Ia supernovae to measure the deceleration of the universe by gravity. They found the radiation from the supernovae to be much fainter than the model prediction[26]. This could only mean that the expansion of the universe, instead of being decelerated by the action of gravity, was accelerating[27].

Earlier, cosmologists had determined that the universe was flat. They then had added up the observed densities of the ordinary baryonic matter and dark matter. They found the energy density of the universe to be less than half the critical density required for the universe to be flat. Therefore, there was a simple proposal that the universe was filled with repulsive energy with negative pressure, which accounted for more than half the density of the universe [28]. Further observation also indicated that the density of dark energy was much higher today than in the past. Based on these observations, two major theoretical predictions of dark energy are possible.

An expanding universe did not mean that Einstein’s cosmological constant had to be zero. Wolfgang Pauli and Yakov Zel’dovich had realized the existence of quantum vacuum fluctuations due to Heisenberg’s uncertainty principle [29]. This led to the postulation of vacuum energy. This vacuum energy density would remain constant at all times and space, and thus the quantity would increase with expansion. The gravitational potential of the universe would be accountable for the increase in this energy. It would theoretically be spatially uniform and chemically inert[30]. However, when the vacuum energy had to be calculated, a huge cosmological constant was shown to be the value. This would have caused the universe to collapse in a short period if this energy had positive pressure or stretch and caused the big rip quickly if negative. This vacuum energy is the same as Einstein’s cosmological constant. Theorists believe in some kind of miscellaneous cancelling mechanism so that the amount of this energy is down to the observational value, i.e, two-third of the energy density. This energy, when it has negative pressure, could act as the driving force for the acceleration.[31]

The second possible cause is that Quintessence accounts for the missing mass and accelerated expansion of the universe. Quintessence is a quantum field, having a wavelength approximately the size of our (observable) universe. Since it has an extremely long wavelength, its particle description becomes impractical and kinetic energy also becomes negligible. Therefore, its behaviour is dominated by how it interacts with itself[1]. While cosmological constant is a very specific form of energy, Quintessence is a dynamic spatially dependent form of energy with negative pressure sufficient enough to induce cosmic expansion[32]. Its density, unlike its constant counterpart, decreases with time. However, it is still different from most of the other forms of energy like matter, radiation, or dark matter, which causes the expansion to slow down. For it to account for a large part of the density of the universe at present but a small fraction in the past, its density needs to decrease at a smaller rate than the density of matter, dark matter, and radiation with the expansion[33]. General relativity allows this as long as it exerts a negative pressure, causing the acceleration of the universe.

Even though having a cosmological constant is theoretically a viable solution, quintessence is also considered as it has different implications for fundamental physics, fits the observational data better, may explain the cosmic coincidence problem, and suggest a radically new picture of the overall universe.

Till now, dark energy has only been inferred using observational data however, several experiments have been planned to make searches for it. Other than these, a large number of variations of these two models exist, which have been created from the observational data of redshifts of supernovae, baryonic oscillations, and Hubble’s parameter. Different models can be differentiated based on the state parameter. State parameters or w is the ratio of pressure P to the energy density ρ i.e . The equation of the state parameter can be constant or be a function of time; however, for late-time acceleration, w must be less than -⅓. The lower the value of w, the faster cosmic acceleration is. For the vacuum state parameter, the value of w is -1, while for quintessence, the value varies between -1 and -⅓. [34]

THE FATE OF THE UNIVERSE

The answer to what will happen to this universe was simple before Hubble had discovered expansion; people believed that the universe was always there and would be there even after their deaths. Once the expansion was discovered, however, people realized the existence of a dynamic universe, and they became curious about the fate of the expanding cosmos.

Fig.6.The Big Crunch (Harris n.d)

The Big Crunch became a popular possibility for explaining the eventual fate of the universe. According to the big crunch model, the universe’s density is sufficient enough to ultimately stop the expansion. This will cause the universe to re-collapse into a singularity. This contracting universe will cause the formation of an increasing number of black holes, which keep integrating until the whole universe is inside one single black hole.[35]. This singularity could be the beginning of another cycle of the universe, and it would mean that the universe is eternal, and it keeps oscillating into the component stages.[36]

Though it has not been ruled out just yet, the Big Crunch does not agree with observational evidence and seems an unlikely scenario. Other possible explanations for the mechanism of the Big Crunch include the dark energy being a form of quintessence that could change its value over time. When dark energy turns negative, i.e exerts an attractive force, it can cause a big crunch.

The most likely scenario based on the observational evidence is the ‘Big Freeze.’

According to the second law of thermodynamics, the total entropy is always increasing. This entropy will increase until it reaches a maximum point. The heat in the universe is evenly distributed; therefore the universe will be devoid of usable energy, and no mechanical motion will be able to take place in such a universe[37]. This scenario is also known as ‘the heat death’ and requires that the universe is flat or has negative curvature[38]. In this model, the dark energy could be vacuum energy with w=-1 or quintessence, which decreases with time. In the future under this scenario, stars will cease to form and run out of nuclear fuel; galaxies will disintegrate, dead black dwarf stars will drift away into the darkness along with the halts of the neutron stars and after trillions of years, black holes will follow the former celestial objects and disintegrate as well.[39]

The third possible fate of the universe is for it to end in a big rip. It takes place when the state parameter w is less than -1. This causes the cosmos to expand at an accelerated rate. The component of this dark energy is called phantom energy, which possesses negative kinetic energy. This phantom energy is often considered unstable[40]. According to this model, as the expansion rate keeps accelerating, it may exceed the speed of light and thus would disallow any being to observe the celestial bodies in the universe. Eventually, the expansion would tear apart the galaxies, stars, atoms, and even individual particles. This results in a universe with no matter and energy except the dark energy. This scenario is based on the assumption that the phantom energy will act like the cosmological constant and will not vary with time.[41]

Fig.7. From Big Bang to Big Rip (Teaford 2020)

CONCLUSION

In a hundred years, humankind went from being oblivious about the cosmos to discovering the expansion of the universe, determining its geometry and capturing the CMB to study the young universe. Based on all the aforementioned discoveries, cosmologists have hypothesised three models of the future of the universe: The Big Crunch, The Big Freeze and the Big Rip.

Even though we have theoretical models of how the universe is going to end, we know that this will not take place for at least a few billion years. It would not only be out of our lifespan, but also the span in which humans can exist. In fact, by that point, all the evidence of human existence will be erased.

Although it may not directly influence our lives, cosmology is still an integral part of astrophysics as it allows humans to explore the vastness of the cosmos. Looking at the Big Bang and the expansion of the universe fascinates curious individuals and also enables everyone to appreciate the universe. There is still the mystery of how and why the universe behaves the way it does. String theory and other unification theories may provide some insights, but currently, it is impossible to prove these theories.

Richard P. Feynman once said, “The present situation in physics is as if we know chess, but we don’t know one or two rules.” Adding to his quote, we also do not have the view of the entire chessboard, and our vision keeps diminishing due to expansion. However, by judging the moves played by the players, we can infer the position of all the pieces. Sometimes the moves played may seem quite bizarre at our first glance, however, they may be brilliant in reality. After all, physics, to be specific cosmology, is a science that keeps evolving.

There is always something left to discover in the universe; even though our understanding keeps increasing, it will never reach its peak. Just as Aristotle thought that Earth was the centre of the universe, what we currently “know” may be inaccurate, and the real nature may be much more far-fetched than we can even imagine. Still, that does not diminish Cosmology’s inherent value. Albert Einstein once stated, “the pursuit of knowledge is much more valuable than its acquisition.”

 

Glossary

  1. Cosmology: Branch of astronomy that studies the universe as a whole ( specifically the origin and the development of the universe).
  2. General Relativity: Einstein’s theory stating that space-time gets distorted under massive objects
  3. Spectrum: A continuum of electromagnetic radiation arranged in increasing wavelength.
  4. Red Shifted: Increase in wavelength of electromagnetic radiation.
  5. Isotropic: a physical property the same in all directions.
  6. Static Universe: A universe of a fixed size ( not expanding or contracting).
  7. Dynamic Universe: A universe that does not have a fixed size and changes its dimensions continuously or periodically.
  8. Fields Equation: partial differential equations that determine the dynamics of a physical field (specifically the time evolution and spatial distribution of the field).
  9. Hypersphere: A series of points equidistant from the centre in an n-dimensional space.
  10. 4D space-time: 3-dimensional space with 1 dimension of time.
  11. Doppler effect: The increase or decrease in the wavelength of a wave due to the movement of the source with respect to the observer.
  12. Heisenberg uncertainty principle: It is impossible to determine simultaneously the exact position and exact momentum of a particle.
  13. Quantum: A discrete quantity.
  14. Quantum Mechanics: Physics mainly dealing with subatomic particles on which classical physics is not applicable.
  15. Black Body Radiation: Radiation emitted by a blackbody ( a theoretical body that absorbs all electromagnetic radiation falling on it without reflecting it).
  16. Supernovae: Powerful stellar explosions at the end of the life of a star that has a mass 8 to 15 times that of the Sun’s.
  17. Baryonic Matter: Ordinary matter, made of protons and neutrons.
  18. Entropy: The measure of disorder in a body.
  19. Black Holes: Astronomical objects with deep gravitational wells where even light cannot escape.
  20. Quantum Vacuum fluctuations: Temporary appearance and existence of subatomic particles. The uncertainty principle allows it, however, energy is required for these subatomic particles to pop into existence.
  21. Vacuum Energy: Underlying Background energy of space
  22. De Broglie relation: De Broglie states that matter, like radiation, also shows dual behaviour- particle and wave. De Broglie also gave the equation where is the wavelength, h is Planck’s constant and mv is the linear momentum.

ENDNOTES

REFERENCES:

[1] Einstein, Albert. “The Foundation of the General Theory of Relativity.” Annalen der Physik 354, no. 7 (1916). doi:10.1002/andp.19163540702.

[2]Hubble, Edwin. “A relation between distance and radial velocity among extra-galactic nebulae.” PNAS 15, no. 3 (January 1929). https://doi.org/10.1073/pnas.15.3.168.

[3]Friedmann, Alexander A. “On the curvature of space.” Z.phys 10 (1922). doi:10.1007/BF01332580.

[4]Luminet, Jean-Pierre. “Lemaitre’s Big Bang.” Proceedings of Frontiers of Fundamental Physics 14 — PoS(FFP14), 2016.

[5] Perlmutter, Saul. “Supernovae, Dark energy, and the Accelerated expansion.” Physics Today 56, no. 4 (April 2003). doi:10.1063/1.1580050.

[6]Belenkiy, Ari. “Alexander Friedmann and the origins of modern cosmology.” Physics Today 65, no. 10 (2012)

[7] Ibid

[8] Friedmann, curvature of space

[9] Belenkiy, Alexander Friedmann

[10Luminet, Lemaitre’s Big Bang

[11] Lemaître, Abbé G. “The Expanding Universe.” General Relativity and Gravitation 29, no.5 (1997)

[12]Luminet, Lemaitre’s Big Bang

[13] Hubble, Relation between distance and radial velocity

[14] Nussbaumer, Harry. “Einstein’s conversion from his static to an expanding universe.” The European Physical Journal H 39, no. 1 (2014).

[15]Luminet, Lemaitre’s Big Bang

[16] Pela. “What is a Friedmann Model?” Astronomy Stack Exchange. Last modified April 2015. https://astronomy.stackexchange.com/questions/10344/what-is-a-friedmann-model.

[17] Luminet, Lemaitre’s Big Bang

[18]Frengel, V., and A. Grib. “Einstein, Friedmann, Lemaitre: Discovery of the Big Bang.” Inspirehep, 1994.

[19] Luminet, Lemaitre’s Big Bang.

[20]Hawking, Stephen. “Second Lecture: The Expanding Universe.” In The Theory Of Everything. Mumbai: Jaico Publishing House, 2006.

[21]Luminet, Lemaitre’s Big Bang

[22]NASA.”The Big Bang.” Science Mission Directorate | Science. Last modified June 19, 2020. https://science.nasa.gov/astrophysics/focus-areas/what-powered-the-big-bang.

[23]Cain, Fraser. “How Do We Know the Universe is Flat? Discovering the Topology of the Universe.” Phys.org – News and Articles on Science and Technology. Last modified June 7, 2017. https://phys.org/news/2017-06-universe-flat-topology.html.

[24] Cadwell, Robert R. “Quintessence – Physics World.” Physics World. Last modified October 24, 20. https://physicsworld.com/a/quintessence/.

[25]Guth, Alan. “Inflationary universe: A possible solution to the horizon and flatness problems.” PHYSICAL REVIEW D 23, no. 2 (January 15, 1981).

[26] “Dark Energy | COSMOS.” Centre for Astrophysics and Supercomputing. Accessed July 31, 2020. https://astronomy.swin.edu.au/cosmos/D/Dark+Energy.

[27] Elizalde, Emilio. “Cosmological Constant and Dark Energy: Historical Insights.” Open Questions in Cosmology, 2012

[28] Caldwell, Quintessence

[29] Elizalde, Dark Energy

[30] Steinhardt, Paul J. “A Quintessential Introduction to Dark Energy.” Philosophical Transactions: Mathematical, Physical and Engineering Sciences 361, no. 1812 (2003).

[31]Bousso, Raphael. “Cosmological Constant, or Vacuum Energy.” Edge.org. Last modified 2017. https://www.edge.org/response-detail/27194.

[32]Steinhardt, Quintessence

[33] Caldwell, Quintessence

[34]Steinhardt, Quintessence

[35]Hillyard, William. “Astronomy & Cosmology.” William (Bill) Hillyard. n.d. https://www.whillyard.com/science-pages/cosmology.html.

[36]Harris, William. “How the Big Crunch Theory Works.” HowStuffWorks. Last modified March 2, 2009. https://science.howstuffworks.com/dictionary/astronomy-terms/big-crunch.htm.

[37]Villaneuva, John C. “What is the Big Freeze?” Universe Today. Last modified August 8, 2009. https://universetoday.com/36917/big-freeze.

[38]Hillyard, Astronomy and Cosmology

[39] Kaku, Michio. “The Big Freeze.” Big Think. Last modified October 6, 2018. https://bigthink.com/dr-kakus-universe/the-big-freeze.

[40]Hillyard, Astronomy and Cosmology

[41]Harris, Big Crunch

 

 

FIGURE REFERENCES:

Fig.1. Amarashiki. 2015. Einstein’s Field Equation. Image. https://thespectrumofriemannium.files.wordpress.com/2015/06/einsteinfieldeq.jpg.

Fig.2. The Friedmann Equation. Image. Accessed July 30. http://burro.cwru.edu/Academics/Astr330/Lect03/friedmann.html.

Fig.3. Pela. 2018. What Is A Friedmannn Model. Image. https://astronomy.stackexchange.com/questions/10344/what-is-a-friedmann-model.

Fig.4.

(i) Gilson, David. The Ever Expanding Universe In Modern Cosmology. Image. Accessed July 30. http://www.laidback.org/~daveg/academic/expandinguniverse/index2.html.

(ii) NASA. 2014. Geometry Of The Universe. Image. Accessed July 30. https://map.gsfc.nasa.gov/media/990006/index.html.

Fig.5. Planck CMB. 2013. Image. https://www.esa.int/var/esa/storage/images/esa_multimedia/images/2013/03/planck_cmb/12583930-4-eng-GB/Planck_CMB_pillars.jpg.

Fig.6.Harris, William. 2020. How The Big Crunch Theory Works. Image. Accessed September 4. https://cdn.hswstatic.com/gif/big-crunch-theory-big-bounce.jpg.

Fig.7. Teaford, Jeremy. 2020. From Big Bang To Big Rip. Image. Accessed September 4. https://i.guim.co.uk/img/static/sys-images/Guardian/Pix/pictures/2015/7/2/1435853967107/7aefb94d-ee19-4f0b-ae10-ff2cbdbfe880-1020×788.jpeg?width=700&quality=85&auto=format&fit=max&s=ac376161e03f21a10767b4d649dc72b5.

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  1. Based on De-Broglie relation.

Biography

Biography: Sahil Khandelwal is an enthusiastic high-school student. He is passionate about science, especially physics, and aspires to work in the field of scientific research. Besides that, his hobbies include Chess, programming and guitar.

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