6th Week: The Early Universe and Big Bang Nucleosynthesis

The Big Bang Cosmology

Cosmological principle

The cosmological principle, which is the more general version of the Copernican principle, states that on large spatial scales, the Universe is homogeneous and isotropic. This means that there is no special point in the Universe. Homogeneity of the universe means that the universe has the same property at any regions from point to point. Isotropy of the universe means that the universe looks the same from all directions. We know that at small scales the universe is not homogenous and not isotropic otherwise any structures e.g. galaxies, stars, planets and humans would not even exist. However provided that we consider the universe on average on large scales, it looks approximately homogenous and isotropic. The observed cosmological scales are therefore approximately $300$ to $500$ $Mpc$ in which the cosmological principle works.

The expanding Universe

Before 1915, it was believed that the cosmos is static and infinite. But the infinite Universe (Newtonian Universe) was ruled out soon due to the Olbers paradox, that states for such a Universe the dark night should not exist. The Einsteins theory of gravitation suggested that the Universe is no more static. But however in order to get a static Universe solution Einstein added a so called cosmological constant, $\Lambda$ which later he called it his greatest blunder. In 1922 Friedmann solved the Einsteins equations for isotropic and homogeneous universe and found that the Universe is either expanding or collapsing. This was experimentally discovered by Hubble in 1928, that the Universe is expanding and the expansion law is the following

${\vec {v}}=H_{0}{\vec {R}}\ ,$

where $H_{0}$ is the so called Hubble constant, which is today believed to be

$H_{0}=70.1\pm 1.3\quad (km/s)/Mpc\ .$

The Hubble "constant" is not a constant, but is actually a time varying quantity. Defining a scale factor, $a$,

${\vec {R}}=a{\vec {x}}\ ,$

where ${\vec {x}}$ is the comoving coordinate, one can find the folowing relation

$H={\frac {\dot {a}}{a}}\ .$

Early Universe

Freeze -out time

weak reactions eventually end reacting slower than the cooling time of the

Cosmological Principle

According to the standard model of the cosmological principle, which only holds true on cosmological scales on approximately greater than 300 to 500 Mpc;

$k>0\Rightarrow$ the universe is viewed as closed $\Rightarrow$ end result Big Crunch

$k=0\Rightarrow$ the universe is viewed as flat $\Rightarrow$ end result forever expansion

$k<0\Rightarrow$ the universe is viewed as open $\Rightarrow$ end result Big Chill

Robertson-Walker Cosmology Now we come to the

From the large-scale distribution of galaxies and the near-uniformity of the CMB temperature, we have good evidence that the universe is nearly homogeneous and isotropic. Under this assumption, the space-time metric can be written in the FRW form

where r, θ, φ are comoving spatial coordinates and t is time, and where the expansion is described by the cosmic scale factor, a(t) (by convention, a = 1 today). The quantity k is the curvature of three-dimensional space: k = 0 corresponds to a spatially flat, Euclidean universe, k > 0 to positive curvature (three-sphere), and k < 0 to negative curvature (saddle). The wavelengths λ of photons moving through the universe scale with a(t), and the redshift of light emitted from a distant source at time tem, 1 + z = λobs/λem = 1/a(tem), directly reveals the relative size of the universe at that time. This means that time intervals are related to redshift intervals by dt = −dz/H(z)(1 + z), where H ≡ a˙/a is the Hubble parameter, and the overdot denotes a time derivative. The present value of the Hubble parameter is conventionally expressed as H0 = 100 h km/sec/Mpc, where h ≈ 0.7 is the dimensionless Hubble parameter. Here and below, a subscript 0 on a parameter denotes its value at the present epoch.

Density Evolution

Evolution of radiation, matter, and dark energy densities with redshift.For dark energy, the band represents w = −1 ± 0.2.