Developing the metric

For understanding this vast universe of ours, how it came into existence and what is its future, we need mathematical constructs or models. Through these models we expect to explain the observed phenomenon, explain the current state of the universe its start and its end.
The first assumption that we make while constructing models for the universe is that it is homogenous and isotropic. Well you will say where can you see the uniformity, the Sun is obviously different from the earth on which we live, the stars look different in two different directions of the sky. What I ask you to see is not these local differences. Let me modify the previous statement a bit. We expect the universe to be highly homogenous and isotropic at the largest of scales. The observations of the universe at intergalactic scales shows these properties. If you scan any part of the sky and have a count of the number of galaxies in the direction you will roughly see the same number of galaxies. That is why we say that the universe is isotropic. Also no point is special. And as the Universe exhibits isotropy at every point and it is homegenous around us so it must be homogenous everywhere.
We have been using the words Isotropy and homogeneity very freely. But what exactly do we mean by them. Homogeneity is the property that makes every point indistinguishable from every other point and isotropy is the property that makes every direction to be indistinguishable from every other. Homogeneity is invariance under translations while isotropy is invariance under rotations. We can have homogenous manifolds without being isotropic, e.g. the surface of a cylinder. We also have the example of a paraboloid. If you sit at the vertex we have isotropy around that point but we do not have the paraboloid to be homogenous. But if we have a manifold to be isotropic and homogenous around one point then it is homogenous around every point.
Also we note that our universe is far from static but at every point of time we have homogeneity and isotropy. So how does this help? This observation helps us know that our Universe can be described as a manifold of the type $R\times\Sigma$. $R$ represents a real variable while $\Sigma$ represents a 3 dimensional isotropic and homogenous manifold which is also called a maximally symmetric space. This $R$ can be taken as representing cosmic time.
The metric measures distances between two points in spacetime and characterizes the spacetime. For such a spacetime we should have the metric as

$$ds^{2}=-dt^{2}+a^{2}(t)\gamma_{ij}(u)du^{i}du^{j}$$ (1.1)

Here $\gamma_{ij}$ is a maximally symmetric metric on $\Sigma$ and a(t) represents how big it is at any point of time. Also we note that we have no cross terms involving space and time in the metric above. And we have spacelike coordinates scaled by a function of time. Such coordinates are called comoving and observers at constant $u_{i}$ and $u_{j}$ are called comoving. The function a(t) is called the scale factor.