There are some basic assumptions that we can make about the universe. All agree with current observation, at least they do over a large enough scale, which for our purposes can be considered to be of the order of $100$'s of billions of *light-years*.

### The cosmological principle

The first two assumptions form the **cosmological principle**, which states that space, at any given instant of time, is both *homogeneous* and *isotropic* on large enough scales.

We say that an abstract space is **homogeneous** if there is no special place, or position, in that space. That is, the space is the same from point to point.

We say that an abstract space is **isotropic** if there is no special direction in that space. That is, the space is the same in every direction.

Notice that both of these assumptions don't hold for the *local* universe. The middle of a star is hardly the same as interstellar space, and looking at the sun is not the same as looking at the night sky, but, on large enough scales, they do hold, meaning that the universe is, on the whole, featureless and galaxies - even clusters of galaxies - can be disregarded.

Even though there is no mathematical or physical reason why the cosmological principle should hold, remarkably, it proves to be more and more accurate at larger and larger scales.

Nb. That might change, of course. We might discover, that at some even larger scale, the universe stops being homogeneous or isotropic.

### Space is flat

The third assumption about the universe is that, at any given instant of time, **space is flat**.

It is important to note this does not follow from the *cosmological principle*. For example, the curved surface of a sphere is both homogeneous and isotropic.

Nb. Here we are talking about all three dimensions of space, rather than the two-dimensional examples of the $2$-plane, the $2$-sphere and the $2$-saddle. An example of a curved space in this context would be a $3$-sphere, but, as mentioned before, the only $3$-space we can actually visualise is flat, Euclidean, space.

Observationally, on large enough scales, there is no detectable curvature of space. As mentioned in the last section, this only means that if space is curved, then its radius of curvature must be of an order larger than the observable universe (I think Professor Susskind refers to at least ten times the scale of the observable universe).

If the universe were *completely* flat, then either it would have to be infinite in space or be contained by some kind of ‘*wall in space*’, neither of which we really believe. But we can say it is flat to the same degree of accuracy with which we can say that a table is flat, or a football field is flat.

Without going into detail about how this is actually done, triangles are constructed in various *locations* in the observable universe and measurements are made of the sum of their angles. As far as we can tell, on large enough scales, this is always $180^{\circ}$.

Nb. By location, we are referring to location in space, rather than in space-time. This means that the method used to measure the triangles must take into account the passage of time.

### The universe is expanding

The fourth assumption is that the universe is expanding. This is a slightly more difficult notion to describe. Maybe a better way to say it is that, at large enough scales, distances between non-negligible objects in the universe increase over time.

Again, we note that this is not true locally. On solar scales, gravity determines that distances *decrease* over time, but on the galactic scale, say, distances increase.

Nb. Even on the galactic scale, it is possible that a pair of galaxies are not moving apart, but this will be the exception, rather than the rule.

The distance (and mass) scales at which distances decrease and increase have been determined to a very high degree of accuracy over the years, and recently led to the discovery that the observable mass in galaxies is not enough to hold those galaxies together gravitiationally according to Newton's Law, given their speed of rotation. Since the *required* missing matter was not observable, it was labelled **dark matter**.

This is not to be confused with **dark energy**, or **vacuum energy**, which is the energy associated with the expansion of the universe at large enough scales.

Up until recently, it was also assumed that the rate of expansion was slowing down. This corresponds either to the notion of the closed universe, which describes a universe that will eventually collapse back into a *big crunch*, or one which continues to expand, but at an ever-decreasing rate.

It's analogous to a cannon-ball fired vertically under the force of gravity. The ball will slow down and eventually *collapse* back to earth. Further, we could imagine that the ball was fired with exactly the escape velocity of the earth, meaning that the ball decreases over time but never reaches zero.

Neither of these are true. Not only is the universe expanding, but also the rate of expansion is increasing over time. Nobody really expected this, and we will discuss this again when we consider *energy conservation* in the next lecture.