Misconception about flat universe

What does it mean by flat universe? Does it mean it is two dimensional –flat universe?  Or if universe is flat then what is curvature of space-time?

According to Einstein theory of general relativity, matter-energy curves space-time around it. On the other hand, based on WMAP measurements, universe is observed to be flat with only a 0.4% margin of error. These two statements seem to contradict each other as presence of matter should have curved space-time around it. What does it mean by flat universe when the presence of so much matter should curve the space-time around it?

A flat universe is a universe which looks same from any point of observation in the universe in any direction of the observation. The word flat is referred to describe it as equal in any direction and at any point. As human we can only perceive three dimensions of space and one dimension of time. So, to us a flat universe is a sphere of observable universe which looks similar in any direction.

But, what led to this conclusion that universe is flat and why is it important. How do we measure universe flatness?

Well, before understanding about flat-universe we need to learn few things

1.     Cosmological Principle- Universe is homogenous and isotropic

2.     Friedmann Cosmological models

3.     WMAP measurements- Why we need dark energy

Cosmological Principle- Universe is homogenous and isotropic

The cosmological principle is an axiom that implies that the distribution of matter in the universe is homogenous and isotropic when viewed on a large scale because forces are expected to act uniformly throughout the universe. Cosmological principle is based on the fact that we don’t hold a special place in this universe.

            Homogenous universe: A universe is homogenous if the position of observation doest change the distribution of energy-density in the universe. Example of homogenous object is sphere and an observation made from the surface of the sphere (Note: Any observation apart from surface will not be homogenous).

            Isotropic universe: A universe is isotropic if the angle of observation doesn’t change the distribution of energy-density in the universe. Example of isotropic universe is a sphere and an observation made from center of the sphere (Note: Any observation apart from the center will not be isotropic)

Note: In either of the universe, if there is increase/decrease in energy-density from the observation point, the increase/decrease must be in proper order/pattern.

Now, we know that we can’t observe the entire universe and we can’t find the center and surface of the universe. But, from any point in the universe, we can only observe certain portion of the universe, with that point as the center of the sphere. We call this as observable universe. So, according to cosmological principle, universe must be homogenous and isotropic from any point in the universe or distribution of energy-density must be same anywhere in that observable universe.

Friedmann Cosmological models

Friedmann is Russian physicist. Based on the Einstein field equations and cosmological principle he derived a set of equations, called as Friedmann equations that govern the expansion of universe in the homogenous and isotropic models of the universe. Based on these equations he defined three types of cosmological models called Friedmann cosmological models in which universe can be flat, Closed or Open shaped. The shape of the universe depends on density parameter which is the ratio of actual density of the universe to the critical density. It can also be known by applying pythogorous theorem to spatial coordinates or by finding the sum of angle of a triangle formed by any three points in space-time. Curvature of space-time is basically is a mathematical description of whether or not Pythagorean Theorem is valid for spatial co-ordinates.

Closed Universe: A Closed universe is a universe with positive curvature. The ratio of actual density to critical density is greater than 1. It means that the density of the universe at any point will be more than its critical density. Also, the sum of the three angles of a triangle will always be greater than 180.  The fate of the Closed universe will be big crunch- Because of gravity universe should undergo deceleration and all the mass in universe should come down to one single point.

Saddle-shape universe or Open universe: A saddle-shaped universe is a universe with negative curvature. The ratio of actual density to critical density is less than 1. It means that the density of the universe at any point will be less than its critical density. Also, the sum of the three angles of a triangle will always be less than 180. The fate of the open universe will be ever forever expanding.

Flat- Universe: A flat universe is a universe with zero-curvature. The ratio of actual density of the universe and critical density is exactly 1. It means if we measure the density of the universe at any point in the universe the value should be same and exactly equal to critical density. Also, the sums of the three angles of a triangle will always exactly 180. The fate of the universe depends on present state of universe that is either accelerating/decelerating but acceleration/deceleration should slow-down in time.(However, we later found there is also repulsive gravity in this universe: Dark energy. If Dark energy is true fate of flat-universe and open universe will be similar with different rates of expansion)

Summary of above two things

The above discussed points are purely theoretical and obviously universe would take any one of 3 shapes. So, the summary of the above two points is that the universe is homogeneous and isotropic and can take any of the above said shapes.

WMAP measurements- Why we need dark energy

Wilkinson Microwave Anisotropy probe (WMAP) is a NASA explorer mission, launched in 2001 to make some fundamental measurements of cosmology. The mission is so successful that it helped in producing a new standard model of cosmology.

Here are the top 10 list of measurements made my WMAP- WMAP's Top Ten.

Of these measurements that two observations that I want discuss are

1.     It measured the universe is flat within 0.4% difference.

Flatness measurement is done through CMB triangle test. Cosmological microwave background radiation (CMBR) provides the evidence of flat universe. CMBR is the left over thermal radiation from big bang. It is the oldest radiation of the universe, radiation when the age of the universe was just 380,000 years old that can be observed because universe existed in the state of plasma before that. This radiation is observed to be isotropic to 1 part in 100,000.

“The CMB triangle test involves using a standard ruler on the last scattering surface and then attempting to measure its angular size. The ruler here is the sound horizon at last scattering, corresponding to the size of the fundamental acoustic mode at that epoch. The lengths of the two longest sides of the triangle are the distance to the last scattering surface, which is essentially fixed. With all 3 sides of the triangle having their lengths established, all that remains is to measure the angle subtended by the shortest side, the standard ruler on the last scattering surface, and we have determined whether or not the Universe is flat”(Source: http://arxiv.org/pdf/astro-ph/0004003.pdf”)

2.     Measured the matter and dark matter make up only about 28.6% critical density.

But from the above discussed points we understood that a flat universe must have same energy density as that of critical density. So, the missing 71.4% energy went unnoticed. Even before this observation, two teams: High-Z supernova search team and Supernova cosmology project observed that the universe is expanding and this expansion is accelerating. But, these teams were expecting a decelerating universe as the effect of gravity must cause them to pull each other. They found out that there is some kind of repulsive gravity that is causing this accelerating expansion. According to their observations, this repulsive gravity energy must be at least 71.3% of total energy.  

Hence they introduced a term called Dark-energy which is causing this repulsive gravity and that makes up this missing energy-density.

So, a flat universe is a homogenous and isotropic universe (is roughly same everywhere) with the average energy density equal to its critical density.