# What is the structure of the Universe? Geometry of spacetime

The Einstein equation predicts several possible ways for the Universe to evolve in time and space. What are these predictions, and how do they equate themselves to observation?

COSMOS AND UNIVERSES: The Final Theory of the Cosmos and the Universes

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The starting point of a theoretical exploration of cosmology is the Einstein equation

with a metric of the form

It is possible to divide the spacetime modelled by this equation conveniently into time and space, so we can speak of this spacetime as describing the evolution of space in time.

The space part of this spacetime is homogeneous (looks the same at any point in a given direction) and isotropic (looks the same in any order from a given point). This is an abstract ideal approach to the universe, but as will be seen below, it is one that has performed exceptionally well from an empirical point of view.

There are three options for the spatial geometry of spacetime with the above metric, represented by three choices for the value of the parameter k: k=1, k=0 or k=-1. The condition of being spatially homogeneous and isotropic means that the surfaces of constant time t have constant curvature, which can be either positive, zero or negative.

A sphere has constant positive curvature, whereas a hyperboloid has constant negative curvature.

To solve the Einstein equation, such as matter, radiation or vacuum energy, with the energy-momentum tensor Tmn, we need to postulate those’ stuff’ in spacetime. The energy density r and pressure p of the “stuff” in question are their components. The equations for the scale factor a(t) are

Here we have truncated Newton’s constant GN to plain G. The top equation contains the condition for the closure density of the Universe explained below.

There are many different kinds of “stuff” that can be a part of the energy density r, with different equations of the state relating r to the pressure p. Let’s mark each different r by index i; thus, ri refers to the energy density from the ith type of “stuff” in this spacetime. Let’s also set something called the critical density and then divide all by the critical density to make the above equation dimensionless:

where W is the density parameter. The equation that will tell us the curvature of space from the stuff content of the spacetime becomes

The three possibilities for the value of the parameter k correspond the three different possibilities for the curvature of space in this spacetime. A value of k=1 corresponds to constant positive curvature, k=0 to zero curvature and k=-1 to constant negative curvature.

The time evolution of space is more complex because it relies on the equation of state of the things in spacetime. The state equation is the relation in the material between pressure and density. The conservation of energy and the state equation determine how the energy density changes as space evolve in time.

Read more: What is Antimatter?

This is where vacuum energy becomes important. The energy densities for matter, radiation and vacuum energy changes with the change in the size of space (the scale factor a(t)) like

The energy density of matter and radiation will become smaller as the Universe gets bigger, but the density of vacuum energy will remain the same. Another name for vacuum energy is the cosmological constant. A cosmological constant eventually controls the time evolution of an expanding universe, because its energy density stays the same while those of matter and radiation are getting smaller.

In a spacetime with all three forms of energy present, the radiation part of the mix will dominate the dynamics when the scale factor a(t)<<1. In the Big Bang model, this is called the radiation dominated era and accounted for the first 10,000-100,000 years of the evolution of our Universe. Right now, the dominant forms of energy in our Universe are matter and vacuum energy.

That being said, we will avoid dealing with any vacuum energy right now and consider a spacetime with the only matter, with no radiation or cosmological constant. In this case, the time evolution of space is related to the curvature of space as follows:

If the amount of energy density in the spacetime is over the critical density, so that W > 1, then the fate of the Universe is to expand in a Big Bang but then eventually contract back into a Big Crunch. Despite the fact that this would take place on a time scale of billions of years, humans today find this possibility philosophically undesirable. More importantly, the data do not support it.

The visible matter in the Universe observed by humans today has barely a fraction of closure density. In fact, as seen today, the Universe seems to have barely a fraction of the mass required to prevent galaxies from flying apart, based on the rotations of the stars in the galaxy around the centre of the galaxy.

Contents

#### Dark matter

Something becomes visible when it interacts with light in such a way that we can see it. Astronomers studying the motions of stars in spiral galaxies noticed that the mean star velocity did not drop off with radius from the galactic centre as rapidly as the falloff in luminous mass in the galaxy dictated according to Newtonian gravity. The stars far from the centre were rotating too fast to be balanced by the gravitational force from the luminous mass contained within that radius. This led to the fact that the majority of the mass in a galaxy was some sort of low luminosity mass, and this unseen mass was called dark matter.

Dark matter is probably not baryonic matter, because the abundance of primordial elements such as hydrogen, helium and deuterium would be much higher if the Big Bang had produced enough baryon density to account for the dark matter in galaxies.

The amount of dark matter present in the Universe has been estimated using various techniques, including observing the velocities of galaxies in clusters and calculating the gravitational mass of galactic clusters by their gravitational lensing effects on surrounding spacetime. The end result is that the baryonic density WB is about 5% and the dark matter density WD is about 30%

The leading candidate for dark matter right now comes from supersymmetry. Supersymmetric versions of the Standard Model of elementary particle physics contain heavy supersymmetric partners of the electroweak gauge bosons and the Higgs field that are electrically neutral and hence don’t interact with electromagnetic radiation, aka light. These neutralinos, as they are called, are fermionic partners of the neutral gauge bosons and the Higgs field. They would have a high mass, yet interact very weakly, and those two qualities make them a good candidate for dark matter.

#### The Cosmological Constant

The observational evidence that the Universe was expanding didn’t come around until 1929, which was 14 years after Einstein’s General Theory of Relativity was first published. The Einstein equations predicted an expanding Universe for any kind of ordinary matter or radiation in existence.

There is no evidence yet to make people believe that the expanding solutions to the Einstein equations represented observed physics, Einstein postulated a new kind of energy density that could balance the matter density in the Universe and prevent the Universe from expanding. This new theoretical energy density is called the cosmological constant, known by the symbol L. The energy density and pressure for L are

The Einstein equations with a matter density rm and cosmological constant L become

A static solution has a(t) = constant = a0, which means that k=+1 and the matter density, cosmological constant L0 and scale factor are related by

A cosmological constant alters the time evolution that is associated with a given spatial curvature. The k=+1 spacetime with only matter expands and then recollapses, but the k=+1 spacetime with matter and a cosmological constant can either expand forever (for L > L0), stay the same forever (L = L0) or expand and recontracts (0 < L < L0). If L > 0 and k= 0 or -1, then space expands forever. If L < 0, then k=-1. When k=-1 with matter and no cosmological constant, the Universe is open and expands forever. But for L < 0, even though k=1 and the topology of space are open, this spacetime expands and then recontracts like the k=+1 model with matter and no cosmological constant.

#### What’s the final answer?

The Large-Scale Structure of the Universe

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1. Our Universe is fairly flat: The Big Bang thermal radiation relic cooled to the temperature of 2.73 ° Kelvin, is the cosmic microwave background. But it didn’t cool completely easily, and there were some lumps left over after the radiation cooled down. The angular size of those lumps as observed from our present location in spacetime depends on the spatial curvature of the Universe. The currently observed lumpiness in the temperature of the cosmic microwave background is just right for a flat Universe that expands forever.

2. There is a cosmological constant: There is vacuum energy or something that acts just like one, to make the Universe accelerate in time. The acceleration of the Universe can be seen in the redshifts of distant supernovae.

3. Most of the matter in the Universe is dark matter: Galactic motion experiments indicate that ordinary visible matter only makes up a tiny fraction of the Universe’s overall energy, i.e., matter in stars, galaxies, planets, and interstellar gas.

The Universe at our current epoch has (approximately)

So right now, the vacuum energy density in our Universe is just around twice as high as the dark matter energy’s density, with almost negligible contribution from visible, baryonic matter. The total adds up to a flat universe which can forever expand.