Cosmological models

1. For a universe with only non-relativistic matter, $P_m=0$. If it is static, then $\ddot{a}=\dot{a}=0$ and Friedmann’s second equation is written :

Injecting this equation into the first Friedmann equation gives :

So, to obtain a static universe, Einstein had to introduce a non-zero cosmological constant. Moreover, we deduce that $c^2 k / a^2 > 0$ so $k=+1$: the static universe must be spherical. Einstein originally argued that the universe should be bounded using Mach’s principle^[2], so the solution of a spherical universe seemed satisfactory from this point of view.

The scale factor can be associated with the radius of the spherical universe. Using the second Friedmann equation, we find that the value of the radius is :

2. Combining the two Friedmann equations for a universe with cold matter only, we find:

Let’s consider a perturbation of the radius, $a=\Lambda^{-1/2} + \delta a$, which induces a perturbation of the density of matter on the sphere $\rho_m = \rho_0 + \delta \rho = c^2 \Lambda/4 \pi G + \delta \rho$. This last equation becomes

Now the quantity of matter is conserved, so we also have :

and :

i.e. the perturbation $\delta a$ grows exponentially with time ($\Lambda>0$). The universe is unstable.

1. In a flat universe $k=0$ and the first Friedmann equation gives in Sitter’s model :

The solution of this differential equation is

The de Sitter universe grows exponentially with time.

2. To show that a de Sitter universe can be considered a static universe, let’s write $T_0 = \sqrt{c^2 \Lambda / 3}$ and change the coordinates $r'(t) = a_0 e^{t/T_0}\sigma$. The FLRW metric then becomes

Using the transformation :

we find :

Historically, Sitter’s model was discovered as a static universe with this coordinate system, whose shape closely resembles Schwarzschild’s solution. As soon as the idea of an expanding universe was accepted by the scientific community, Sitter’s universe was considered in its dynamical form, as an exponentially expanding universe dominated by the cosmological constant.

1. In the single-matter Lemaître model, the first Friedmann equation is written :

At $t\approx 0$, the Universe was extremely small so the matter term dominates:

2. Then, after some time, $a$ becomes large and the cosmological constant term dominates:

3. Deriving equation (24), we find that :

When $a$ is small, we see that $\ddot{a}$ is negative and expansion decelerates. However, when $a$ is large, $\ddot{a}>0$ and the expansion of the universe accelerates. The transition occurs at :

For the ΛCDM model with $\Omega_m^0\approx 0.3$ and $\Omega_\Lambda^0\approx 0.7$, we have $a_*/a_0 \approx 0.6$ hence a transition redshift to $z\approx 0.67$.

1. In terms of $\Omega_i^0$, the first Friedmann equation is written :

which gives

This last equation resembles the equation for the conservation of mechanical energy for a massive body in one-dimensional motion. Let’s make the analogy:

• $\frac{1}{2}\Omega_k^0$ is constant with $a$ can be identified as the conserved mechanical energy of the massive body.
• $\frac{1}{2}\frac{\dot{a}^2}{H_0^2}$ represents the kinetic energy of the solid body.
• $\frac{1}{2}\frac{\Omega_m^0}{a}$ resembles a gravitational potential centered around $a=0$.
• $\frac{1}{2}\frac{\Omega_r^0}{a^2}$ is another type of attractive potential.
• $\frac{1}{2}\Omega_\Lambda^0 a^2$ is an inverted harmonic potential (repulsive) centered around $a=0$.

2. $$\frac{\ddot{a}}{H_0^2} = - \frac{1}{2}\frac{\Omega_m^0}{a^2 } -\frac{\Omega_r^0}{a^3 } + \Omega_\Lambda^0 a$$

   (33)

This equation resembles Newton’s law applied to a massive body in one-dimensional motion. Let’s make the analogy:

• $\frac{\ddot{a}}{H_0^2}$ acceleration of the solid body
• $\frac{1}{2}\frac{\Omega_m^0}{a^2 }$ gravitational force (attractive)
• $\Omega_\Lambda^0 a$ elastic repulsive force

Let’s define :

3. In this universe model, we have $\Omega_\Lambda^0 = \Omega_m^0 / 2$ and :

At $a=1$ or $t=0$, the first Friedmann equation gives :

The model is spherical so $\Omega_k^0 = -k c^2 / H_0^2 < 0$ with $k=+1$ which implies that $\Omega_m^0 > 2/3$. In Einstein’s Universe, $\Omega_m^0=1$.

Source:Mechanical analogy

According to figure Figure 1, Einstein’s solution at $a=a_0$ is unstable.

$\Omega_m^0=1\Rightarrow k=0$	$\Omega_m^0=1.5\Rightarrow k=+1$	$\Omega_m^0=0.5\Rightarrow k=-1$
Source:Mechanical analogy	Source:Mechanical analogy	Source:Mechanical analogy

In these models, curvature is again given by :

By analyzing the three plots in figure Table 1, we can say that a spherical universe composed solely of matter will necessarily collapse at some point, whatever its initial conditions (Einstein’s need to add the cosmological constant). An expanding flat universe expands indefinitely and asymptotically stops expanding at $t\rightarrow \infty$. An expanding hyperbolic universe also expands to infinity.

5. The transition scale factor is given by :

$\Omega_m^0=0.3, \Omega_\Lambda^0=0.7$	$\Omega_m^0=0.3, \Omega_\Lambda^0=1.5$
Source:Mechanical analogy	Source:Mechanical analogy
$\Omega_m^0=0.3, \Omega_\Lambda^0=0.5$	$\Omega_m^0=0.3, \Omega_\Lambda^0=-0.7$
Source:Mechanical analogy	Source:Mechanical analogy

Depending on the parameter values, the transition scale occurs in the future or in the past. If the cosmological constant is positive, expanding universes have a decelerated expansion and, after the transition scale, an accelerated expansion. If the cosmological constant is negative, the universe must collapse after a certain time.

So why is the Universe expanding today? That depends entirely on the initial conditions, and in particular on whether the universe was born of a Big Bang. And why was there a Big Bang? We can let our imaginations run wild: brane collisions, God, pan-dimensional mice... but the answer is not (yet) given by the physical sciences.

1. We use the first Friedmann equation :

because at $a(t_0)=1$ we still have $1=\Omega_k^0 + \Omega_m^0 + \Omega_\Lambda^0$.

• We have $\Omega_m^0=1, \Omega_\Lambda^0=0, \Omega_r^0 \approx 0$ then :

With $t=0$ at the Big Bang,

We thus obtain :

with :

• $\Omega_m^0=\Omega_\Lambda^0\approx 0,\Omega_r^0 \approx 0$ then :

With $t=0$ at the Big Bang,

• $\Omega_m^0=\Omega_r^0\approx 0, \Omega_\Lambda^0 = 1$ then:

With $t=t_0$ now,

We have $a(t)\Rightarrow 0$ when $t\rightarrow -\infty$ so the age of the universe is infinite in Sitter’s model.

2. Einstein de-Sitter’s value is incompatible with measuring the age of the first stars with such a value of $H_0$. However, the ΛCDM model agrees with the “empty universe” model. However, in the “empty universe”, the assumption that matter plays no role is very strong.

The first measurement of the Hubble constant was incompatible with measurements of the Earth’s age. Early cosmological models were therefore affected by this fact until the measurement of the constant $H_0$ was corrected.