**History of modern cosmology/ Rise of the Big Bang Model: **

Soon after Einstein had formulated the general theory of relativity, in 1917 he postulated that the universe is homogenous, isotropic, static, with a constant mean mass density and curvature of space. However, this model was not permitted by his field equations as they suggested that the universe would collapse due to the gravitational force acting on matter. In order for the universe to remain static, he added a cosmological constant, Λ to the field equation. In the same year, de Sitter also proposed an alternative model that satisfied the field equations. This however required the universe to be empty. Slipher had also published his findings that light from some of the observed nebulae are redshifted [1], receding from us with velocities from 290 to 1100 km s^{-1}.

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In 1922, Alexander Friedmann derived solutions to Einstein’s field equations that would allow for a dynamical universe. He assumed that matter density was time dependent. Five years later, George Lemaitre, a physicist, also a Belgian Roman Catholic priest, unaware of Friedmann’s work, independently came up with a dynamical model that included mass and pressure. By comparing this with the latest observations provided by Slipher [2] (whose data was published in Eddington’s book of 1923) and Hubble, he managed to theoretically derive a linear relation between receding velocities and distance to the observed cosmic objects(?), which is now known as the Hubble’s law. Lemaitre interpreted the observed redshifts as an effect of cosmic expansion. He published his findings that year in an obscure journal in French and also sent a copy to his former mentor, Eddington, but unfortunately it went unnoticed. During the 1927 Solvay congress, Lemaitre took the opportunity to give a copy of the reprint of his findings to Einstein, who found it abominable as it made no sense from a physical point of view. According to Lemaitre, it seemed like Einstein was unaware of the cosmological observations [3].

In 1929, Hubble published a paper which led to realisation of an expanding universe. Using data on Doppler redshifts mostly given by Slipher combined with the measured distances to the spiral nebulae, he found a linear correlation between distances and velocities v =Hr with H being a proportionality factor [4]. Hubble’s work was very similar to Lemaitre’s, however he merely interpreted them as Doppler effects and never considered them as a consequence of an expanding universe.

During the Royal Astronomical Society (RAS) meeting that took place in 1930, Eddington discussed the instability of Einstein’s model and called for the search of new explanations for the recession of velocities using dynamical models [5]. The proceedings of this meeting was published and read by Lemaitre, who then immediately wrote to Eddington to remind him of his 1927 paper. Eddington was impressed and arranged for it to be translated into English and was published in the monthly notices of the RAS in 1931. The paragraphs concerning Hubble’s law were deleted by Lemaitre himself and thus he was never credited for Hubble’s law and the discovery of an expanding universe [6]. In the same year, Lemaitre hypothesised that the universe came into being, governed by the laws of quantum mechanics, from an initial point known as the “Primeval Atom”. However, the idea was poorly received by the scientific community due to Lemaître’s personal religious beliefs.

In 1949, Thomas Gold, Hermann Bondi and Fred Hoyle came up with the “steady-state theory” which states that there was no beginning and no end in time, that the universe was unchanging and eternal. This opposed the idea of an expanding universe from a singularity. On a radio interview, Hoyle casually made fun of the idea that the universe was created in a “Big Bang”, a term that was popularised even though it was meant in a pejorative sense. Even though Hoyle found the big bang creation unacceptable both scientifically and philosophically, he was the one who coined the term “big bang”. (reference and summarise)

**Mathematical Basis of Big Bang cosmology:**

** **

In cosmology, the evolution of space-time is determined by the application of Einstein’s field equations based on his General Theory of Relativity. It is assumed that this is valid everywhere on a large-scale basis. The 10 field equations can be summarised by:

${\textcolor[rgb]{}{\mathrm{R}}}_{\textcolor[rgb]{}{\mathrm{\mu}}\textcolor[rgb]{}{\mathrm{\nu}}}\textcolor[rgb]{}{\u2013}{\textcolor[rgb]{}{\mathit{Rg}}}_{\textcolor[rgb]{}{\mathrm{\mu}}\textcolor[rgb]{}{\mathrm{\upsilon}}}\textcolor[rgb]{}{=}\textcolor[rgb]{}{\mathrm{}}\textcolor[rgb]{}{\u2013}\textcolor[rgb]{}{\mathrm{}}\frac{\textcolor[rgb]{}{8}\textcolor[rgb]{}{\mathit{\pi G}}}{{\textcolor[rgb]{}{c}}^{\textcolor[rgb]{}{2}}}{\textcolor[rgb]{}{T}}_{\textcolor[rgb]{}{\mathit{\mu \upsilon}}}\textcolor[rgb]{}{\mathrm{}}$

, (1)

where ${\textcolor[rgb]{}{\mathrm{R}}}_{\textcolor[rgb]{}{\mathrm{\mu}}\textcolor[rgb]{}{\mathrm{\nu}}}$

is the Ricci tensor which depends on the metric tensor and its derivatives; R, the Ricci scalar is the contraction of the Ricci tensor; ${\textcolor[rgb]{}{T}}_{\textcolor[rgb]{}{\mathit{\mu \upsilon}}}\textcolor[rgb]{}{\mathrm{}}$

is the energy-momentum tensor; and G is the gravitational constant. The Robertson-Walker(RW) metric tensor ${\textcolor[rgb]{}{g}}_{\textcolor[rgb]{}{\mathrm{\mu}}\textcolor[rgb]{}{\mathrm{\upsilon}}}$

_{ }describes an expanding, perfectly isotropic and homogenous universe and is determined by *a*, the expansion scale factor as well as the curvature of spacetime. The metric can be described by [8]:

${\textcolor[rgb]{}{\mathit{ds}}}^{\textcolor[rgb]{}{2}}\textcolor[rgb]{}{=}\textcolor[rgb]{}{\mathrm{}}\textcolor[rgb]{}{\u2013}{{\textcolor[rgb]{}{c}}^{\textcolor[rgb]{}{2}}\textcolor[rgb]{}{\mathit{dt}}}^{\textcolor[rgb]{}{2}}\textcolor[rgb]{}{+}\textcolor[rgb]{}{\mathrm{}}{\textcolor[rgb]{}{a}}^{\textcolor[rgb]{}{2}}\left(\textcolor[rgb]{}{t}\right)\left[{\textcolor[rgb]{}{\mathit{dr}}}^{\textcolor[rgb]{}{2}}\textcolor[rgb]{}{+}\textcolor[rgb]{}{}{\textcolor[rgb]{}{S}}_{\textcolor[rgb]{}{k}}{\left(\textcolor[rgb]{}{r}\right)}^{\textcolor[rgb]{}{2}}\left({\textcolor[rgb]{}{\mathit{d\theta}}}^{\textcolor[rgb]{}{2}}\textcolor[rgb]{}{+}{\textcolor[rgb]{}{\mathit{sin}}}^{\textcolor[rgb]{}{2}}\textcolor[rgb]{}{\mathit{\theta}}{\textcolor[rgb]{}{\mathit{d\phi}}}^{\textcolor[rgb]{}{2}}\right)\right]$

,

where $\textcolor[rgb]{}{\mathit{ds}}$

is the spacetime interval between two events, a

${\textcolor[rgb]{}{S}}_{\textcolor[rgb]{}{k}}\left(\textcolor[rgb]{}{r}\right)\textcolor[rgb]{}{=}\textcolor[rgb]{}{\mathrm{sinh}}\left(\textcolor[rgb]{}{r}\right)$

if *k* = -1;
${\textcolor[rgb]{}{S}}_{\textcolor[rgb]{}{k}}\left(\textcolor[rgb]{}{r}\right)\textcolor[rgb]{}{=}\textcolor[rgb]{}{r}$

if *k* = 0;
${\textcolor[rgb]{}{S}}_{\textcolor[rgb]{}{k}}\left(\textcolor[rgb]{}{r}\right)\textcolor[rgb]{}{=}\textcolor[rgb]{}{\mathrm{sin}}\left(\textcolor[rgb]{}{r}\right)\textcolor[rgb]{}{}$

if *k* = +1

Einstein’s equations relates the geometry of the metric tensor which describes the geometry of spacetime, to ${\textcolor[rgb]{}{T}}_{\textcolor[rgb]{}{\mathit{\mu \upsilon}}}$

which is determined by mass and energy in that spacetime. This means that the paths in spacetime can be calculated.

When the RW metric is applied to the Einstein equation, we obtain the Friedmann equation, which describes the expansion of the universe:

${\left(\frac{\stackrel{\u0307}{\textcolor[rgb]{}{a}}}{\textcolor[rgb]{}{a}}\right)}^{\textcolor[rgb]{}{2}}\textcolor[rgb]{}{=}\textcolor[rgb]{}{}\frac{\textcolor[rgb]{}{8}\textcolor[rgb]{}{\pi}}{\textcolor[rgb]{}{3}}\textcolor[rgb]{}{\mathit{G\rho}}\textcolor[rgb]{}{\u2013}\frac{\textcolor[rgb]{}{k}{\textcolor[rgb]{}{c}}^{\textcolor[rgb]{}{2}}}{{\textcolor[rgb]{}{a}}^{\textcolor[rgb]{}{2}}}$

. (2)

From the first law of thermodynamics, combined with the knowledge that the expansion of the universe is an adiabatic process, the fluid equation can be derived to give:

$\stackrel{\u0307}{\rho}+\frac{3\stackrel{\u0307}{a}}{a}\left(\rho +\frac{P}{{c}^{2}}\right)$

. (3)

This describes the evolution of energy density as the universe expands. Using the (2) and (3), the acceleration equation can be derived:

$\u20133\left(\frac{\stackrel{\u0308}{a}}{a}\right)=4\mathit{\pi G}\left(\rho +\frac{3P}{{c}^{2}}\right)$

. (4)

Since *a* is the scale factor representing the radius of the universe at time *t*,
$\stackrel{\u0307}{\textcolor[rgb]{}{a}}$

is the rate at which the universe expands (or contracts), and $\stackrel{\u0308}{\textcolor[rgb]{}{a}}$

is the acceleration of expansion (or deceleration of contraction). P is the pressure of matter and $\textcolor[rgb]{}{\mathit{\rho}}$

is its density; *k *is a constant that describes the spatial curvature and can take values of +1, 0 and -1, which represents a closed, flat and open universe respectively.

The FLWR universe is one that has a geometry with RW metric and dynamics governed by equations (2), (3), (4). The FLWR models represent a whole class of standard cosmological models that assume perfect isotropy and homogeneity in the universe. Since the universe is currently expanding and that density of matter and pressure are positive, it follows from equation (4) that there is a decrease in the acceleration of the expansion. This means that the further back we go into the past, the greater acceleration was, and the smaller the scale factor *a *of the radius of the universe. Hence, implying that there must be a time *t** _{0}* when

*a*= 0. Furthermore, as $\stackrel{\u0308}{\textcolor[rgb]{}{a}}$

increases and *a *decreases, P and
$\rho $

on the right-hand side of equation (4) will also increases until *t** _{0 }*when the values are infinite. Friedmann equations have predicted a spacetime singularity – point of infinite density, curvature and temperature.

**3 Pillars of Big Bang:**

- Hubble’s Law:

If the universe was static, galaxies should move about randomly. However, observations showed that most galaxies were redshifted and receeding from us. Hubble’s law implies that the universe is expanding and that it cannot be static.

When the wavelength of light or sound emitted by objects that are receeding from us, it appears stretched out and this expansion factor, *a*, can be defined as the redshift *z*:

$\frac{{\lambda}_{\mathit{obs}}}{{\lambda}_{\mathit{em}}}=1+z=\frac{1}{a}$

, (5)

It is often convenient to use a cosmic scale factor to express distances at different epochs, i.e at time *t* the scale was *a*(*t*) and the present value of *a*(*t** _{o}*) is set to 1.

If the speed of recession of an object, *v* is small compared to *c*, the standard Doppler effect applies and
$\mathit{z}\cong \frac{v}{c}$

.Therefore, measurement of the amount by which the absorption or emission lines have redshifted then gives us a direct measure of the recession velocity of the object.

In 1917, Slipher had already noted that most of the galaxies measured showed a redshift. Combining this available data with the distances to the 24 galaxies that Hubble measured by studying the Cepheid variables in them, in 1929, he proposed a relationship known as the Hubble’s law:

$v=\mathit{Hr}$

(7)

where *H* is the Hubble’s parameter with units km s^{-1} Mpc ^{-1 } and *r* is simply the distance. Hubble showed that the further away the galaxies, the faster they are receeding from us. This is also what we expected for the case in an expanding universe as it can be shown that the receeding velocity increases linearly with distance between two comoving objects:

$\textcolor[rgb]{}{v}\textcolor[rgb]{}{=}\frac{\textcolor[rgb]{}{\mathit{dr}}}{\textcolor[rgb]{}{\mathit{dt}}}\textcolor[rgb]{}{=}\frac{\textcolor[rgb]{}{d}}{\textcolor[rgb]{}{\mathit{dt}}}\left(\textcolor[rgb]{}{a}{\textcolor[rgb]{}{r}}_{\textcolor[rgb]{}{0}}\right)\textcolor[rgb]{}{=}\stackrel{\u0307}{\textcolor[rgb]{}{a}}{\textcolor[rgb]{}{r}}_{\textcolor[rgb]{}{0}}\textcolor[rgb]{}{=}\left(\frac{\stackrel{\u0307}{\textcolor[rgb]{}{a}}}{\textcolor[rgb]{}{a}}\right)\textcolor[rgb]{}{r}\textcolor[rgb]{}{=}\textcolor[rgb]{}{\mathit{Hr}}$

,

where ${\textcolor[rgb]{}{r}}_{\textcolor[rgb]{}{0}}$

is the comoving distance and *a* is the expansion factor. This is called the Hubble flow which does not take into account the peculiar velocities of the objects.

*H* represents the constant rate of cosmic expansion due to the stretching of spacetime itself andchanges with time throughout the life of universe. Determining the Hubble’s parameter involves measuring distances and redshifts of distant objects. The main challenge is finding reliable indicators of distance. (methods of determining distances?)

Since the Cosmological Principle states that the universe is isotropic and homogenous, the observed recession of distant objects can be interpreted as a general expansion of the universe. When the expansion is traced backwards in time towards the Big Bang, we find that the universe must be denser in the distant past until a single mathematical point of infinite density is reached.

- Cosmic Microwave Background Radiation(CMBR):

The first prediction for the residual radiation from the Big Bang was made by Alpha (doctoral student of Gamow) and Hermann in 1948 [9]. Not only was this often wrongly credited to Gamow, the prediction of the black-body radiation at around 5 K did not cause much excitement in the scientific community. Even though calculations were refined several times, no attempts were made to detect this CMBR. It was only in 1964 that Arno Penzias and Robert Wilson accidentally discovered this CMB in attempt to calibrate the antenna of the radio telescope in a radiation free environment. They found a constant background noise corresponding to an antenna temperature of around 3.5 K and was interpreted by Dicke et al as the left-over CMB from the hot past [11].

The CMBR saturates space after about 380,000 years since the Big Bang occurred due to the two main events that took place: recombination and photon decoupling. Before recombination, the universe was opaque and the plasma was in thermodynamic equilibrium. This is because the universe was ionised and photons cannot travel far without scattering efficiently off protons and electrons. Therefore, photons could not travel freely and no light escaped. When the universe cooled, through expansion, to around 3000 K, the electrons could finally combine with protons to form neutral hydrogen atoms. Hence, the photons decouple and can travel freely through space since then. The collision of photons with electrons before last scattering must ensure that the photons were in equilibrium which entails a blackbody spectrum.

In order to verify this prediction, observations were made by the FIRAS instrument aboard the COBE spacecraft. The data taken by the FIRAS instrument showed that it was very well fitted by a blackbody radiation curve with a temperature of $2.735\mathrm{}\pm 0.06K$

[10]. Thus, this is

[1] Slipher, V.M. 1917.Nebulae. Proceedings of the American Philosophical Society, vol 56, 403-409.

[2] Eddington, A.: The Mathematical Theory of Relativity, Cambridge University Press (1923), p. 162

[3] Lemaître, G. 1958. Rencontres avec A. Einstein. Revue des Questions Scientifiques, 129, 129-132

[4] Hubble, E.: A Relation between distance and radial velocitiy among extra-galactic nebulae. In: Pro-ceedings of the National Academy of Sciences15, March 15. Number 3, 168 (1929)

[5] A.S. Eddington, Proceedings at the Meeting of the Royal Astronomical Society, The Observatory 53, 39 (1930).

[6] G. Lemaître, A homogeneousuniverse of constant mass and increasing radius accounting for the radial velocity of extra-galactic nebulae, M.N.R.A.S. 41, 483 (1931).

[7] Lemaître, G. (1931b). The beginning of the world from the point of view of quantum theory. Nature, 127, 706.

[8] Robertson, H. P. 1929. “ON THE FOUNDATIONS OF RELATIVISTIC COSMOLOGY” *Proceedings of the National Academy of Sciences of the United States of America* vol. 15,11, p 822-9.

[9] Alpher, R., Hermann, “Evolution of the Universe”, Nature, Nov. 13, 1948, Vol. 162, p774-775

[10] Mather, J *et al*, The Astrophysical Journal, May 10, 1990, Vol. 354, L37-L40

[11] Dicke, R. at al. “Cosmic Black-Body Radiation”, *Astrophysical Journal*, May 7, 1965, Vol. 142, p414–419.

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