Within the framework of the expanding Universe models, we are going to answer the following question using a relatively simple calculation :
At this moment, what is the distance of the farthest object whose light has had the time to reach us since the beginning of the Universe ?
This imaginary spatial limit is called the cosmological horizon or particle horizon (not to confuse with event horizon). Any event that is now occurring or has already occurred at a point beyond this horizon cannot or cannot yet be observed by us. This is illustrated in the following figure :
Figure 1 : Representation of the cosmological horizon in an expanding Universe
This allows us to determine the radius of the observable universe, which is none other than the comoving distance to the cosmological horizon.
To calculate this distance, we must locate the points at which light is emitted and received by their co-moving spatial coordinates. We shall select our own position as the receiving point ($r=0$) and suppose that the angular coordinate of these two points is zero. By using the FLRW metric, one can obtain the equation for the trajectory of the ray of light emitted at $t_{1}$ from point $r_{1}$ and reaching us at $t_{0}$; this is a null geodesic.
\begin{equation}
c^{2}\text{d}t^{2}-R(t)^{2}\dfrac{\text{d}r^{2}}{1-kr^{2}}=0
\label{eq1}
\end{equation}
One thus obtains the source's $r_{1}$ coordinate:
Let us now examine the evolution of the Universe. Two primary types of model exist: those which consider the present age of the universe (or at least the age of its present phase of expansion) to be finite, and those that consider the universe to be infinitely old. In other words, for the first type (called Big-Bang models), the history of the universe can be described as beginning at an initial instant chosen as zero on the time scale. In contrast, history according to infinite models precludes any consideration of an initial instant in time. Let’s examine the case of Big-Bang models. Since their history of the Universe begins at $t=0$, they impose the following constraint: $t_{1} > t_{min}=0$. Let’s apply this constraint to the equation \eqref{eq2} : does this imply that a constraint is imposed on the spatial coordinate ?
This all depends on whether the integral converges or not :
\begin{equation}
{\large\int}_{0}^{t_{0}}\dfrac{c\text{d}t}{R(t)}
\label{eq4}
\end{equation}
If it converges, the cosmological horizon’s value can be defined. We are now going to express this integral in another form by replacing the time variable with redshift $z$ :
The term $H(z)$ representing the expansion rate as a function of redshift $z$ is determined by the Friedmann's equation (2). One can thus express it in this form :
In the following calculations, we use the $\Omega^{0}_{k}$ variable, which is determined by $\Omega^{0}_{m}$, $\Omega^{0}_{r}$ and $\Omega^{0}_{\Lambda}$ variables according to which :
We are going to calculate these three integrals numerically using Matlab’s integral function. We will also use the arrayfun function to perform a batch execution for 1D array redshift values :
% Function to integrate
function y = myfunc(x,Omega_m,Omega_r,Omega_l,Omega_k)
y=(Omega_m*(1+x).^(3)+Omega_r*(1+x).^(4)+Omega_l+Omega_k*(1+x).^(2)).^(-1/2);
In the program, we will set $z_{e}$ equal to $1100$, since this corresponds to the time when the decoupling of matter and radiation occurred.
Figure 2 : Cosmological horizon value with $z_{e}=1100$ for the 3 models
The red curve represents the cosmological parameters ($\Omega^{0}_{m}=0.3$, $\Omega^{0}_{\Lambda}\simeq 0.7$ and so $\Omega^{0}_{k} = 0$, i.e flat space) that correspond to the standard model ($\Lambda\text{CDM}$). By converting into light years ($1\,\text{pc} = 3.26\,\text{ly}$), we get, with $z_{e}=1100$ and $H_{0}=71\,\text{km/s/Mpc}$, a value for the radius of the observable Universe equal to: $R = 1.343\,10^4\,\text{Mpc} = 43.8\,\text{Gly}$, or roughly $\mathbf{44}$ billion light years.
Angular Diameter Distance is defined by the distance of a galaxy when it emitted a light ray at time $t_{1}$ that will be received today at $t_{0}$. In other words, its expression is :
Concerning the Angular Diameter Distance $D_{a}$, one can see a paradox : if we take 2 galaxies of redshifts $z_{1}$ and $z_{2}$ such that $z_{2} > z_{1}$, then the one that is the most distant today ($\text{galaxy}_{2}$) will appear larger in the sky than the one which is currently the least distant ($\text{galaxy}_{1}$). This is due to the fact that, starting from the values of $z_{max}$ given in the figure above, the Cosmological Horizon grows less quickly than the factor $(1+z)$, which implies a decreasing $D_{a}$ value for $z > z_{max}$ in each of the 3 models.
Lookback time is an estimation of the moment at which an astronomic object has emitted light that is received currently by the observer (i.e our galaxy). The origin time "0" is defined as the today cosmic time (redshift z=0). We also called this time as being the difference between age of Universe and age of Universe when considered object sent out light, or as the flying time of photon. Its general expression is :
For example, an object having a lookback time of 10 Giga years means that its light has travelled during 10 Giga years before reaching us, or the emission of its light has happened 10 Giga years ago. Given the figure below, with a lookback of 10 Gyr, redshift will be equal to 2, considering the ΛCDM model ($\Omega^{0}_{m}=0.3$ and $\Omega^{0}_{\Lambda}=0.7$).
Figure 4 : Lookback time as a function of Redshift
In practice, when observer receives light from an oject and compute its redshift, he's able to deduce, thanks to figure 4 above, the lookback time. Then, observer knows that he's seeing the object as it was at this value of time.
