Geometry of the Universe

Guide

Curvature of the Universe

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A geodesic is the shortest path between two points on a curved surface. For example, on a sphere, the geodesics are segments of great circles, while on a flat plane, they are straight lines. To measure the distance between two points in a curved geometry, one must take into account the curvature of the space. This is done by measuring the length of the geodesic between the two points.

The universe can be curved in various ways. We will consider three basic geometries:

Flat

In a flat universe, if a triangle is constructed with vertices joined by geodesics, the angles at the vertices $\alpha$ , $\beta$ , and $\gamma$ must satisfy

\alpha + \beta + \gamma = \pi

Such a universe is an infinite plane and has infinite area with no edges or boundaries.

Positively Curved (Closed)

In a spherical or positively curved geometry, the geodesics are part of great circles. The angles of a triangle satisfy

\alpha + \beta + \gamma = \pi + \frac{A}{R^2}

where $A$ is the area of the triangle and $R$ is the radius of the sphere. Such a geometry has finite area $4\pi R^2$ , but no edge or boundary.

Negatively Curved (Open)

A negatively curved geometry includes a hyperboloid. The angles of a triangle satisfy

\alpha + \beta + \gamma = \pi - \frac{A}{R^2}

where $A$ is the area of the triangle and $R$ is the radius of curvature. This geometry has infinite area and no edge or boundary.

We define $\kappa$ as the curvature constant and $R$ as the radius of curvature, such that

\tag{5.3.1} \kappa = \begin{cases} +1 &\text{positively curved} \\ 0 &\text{flat} \\ -1 &\text{negatively curved} \end{cases}

Hence, the sum of the angles of a triangle can be expressed as

\alpha + \beta + \gamma = \pi + \frac{\kappa A}{R^2}

Generally, $R (t)$ is a function of time. If $R_0$ is the radius of curvature at the present time, then the scale factor $a(t)$ is defined as

\tag{5.3.2} a(t) = \frac{R(t)}{R_0}

The value of the scale factor at the present time is $a(t_0) = 1$ .

FLRW Metric

An expanding, isotropic, and homogeneous universe can be described by the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, which is a solution to the Einstein field equations of general relativity. The FLRW metric is

\tag{5.3.3} ds^2 = -c^2 dt^2 + a(t)^2 \left[ \frac{dx^2}{1 - \kappa x^2 / R_0^2} + x^2 d \Omega^2 \right]

\tag{5.3.4} ds^2 = -c^2 dt^2 + a(t)^2 \left[ dr^2 + S_\kappa (r)^2 d \Omega^2 \right]

where $d\Omega^2 = d\theta^2 + \sin^2 \theta d\phi^2$ is the differential solid angle, $S_\kappa (r)$ is defined as

\tag{5.3.5} S_\kappa (r) = \begin{cases} R \, \sin \left( \frac{r}{R} \right) &\kappa = +1\\ r &\kappa = 0\\ R \, \sinh \left( \frac{r}{R} \right) &\kappa = -1\end{cases}

and $x = S_\kappa (r)$ .

Here, $ds$ is the infinitesimal proper distance. The spatial component of the FLRW metric is scaled by the scale factor $a(t)$ to account for the expansion. The time variable $t$ is the cosmic time, which is the time measured by an observer who sees the universe expanding uniformly around them. The spatial variables ( $r$ , $\theta$ , $\phi$ ) or ( $x$ , $\theta$ , $\phi$ ) are the comoving coordinates of a point in space. If the expansion of the universe is perfectly homogeneous and isotropic, the comoving coordinates of a point remain constant with time.

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In 1922 and 1924, the Soviet mathematician Alexander Friedmann and in 1927, Georges Lemaître, a Belgian astronomer, arrived independently at results that relied on the metric. Howard P. Robertson from the US and Arthur Geoffrey Walker from the UK explored the problem further during the 1930s. In 1935, Robertson and Walker rigorously proved that the FLRW metric is the only one on a spacetime that is spatially homogeneous and isotropic.

The FLRW metric assumes homogeneity and isotropy of space. It also assumes that the spatial component of the metric can be time-dependent. Thus, the validity of the FLRW metric is completely built upon the cosmological principle.

Proper Distance

The proper distance $d_\text{p} (t)$ between two points in space is defined as the length of the geodesic between them when the cosmic time is fixed at a value $t$ . It can be found using the FLRW metric:

ds^2 = a(t)^2 \left[ dr^2 + S_\kappa (r)^2 d\Omega^2 \right]

Along the spatial geodesic between the observer and the object, ( $\theta$ , $\phi$ ) are constant, therefore

ds = a(t) dr

\implies d_\text{p}(t) = a(t) r

d_\text{p}(t) = \begin{cases} a(t) \, R \, \sin^{-1} \left( \frac{x}{R} \right) &\kappa = +1\\ a(t) \, x &\kappa = 0\\ a(t) \, R \, \sinh^{-1} \left( \frac{x}{R} \right) &\kappa = -1\end{cases}

Differentiating the proper distance with respect to time gives

\dot{d_\text{p}(t)} = \frac{\dot{a}}{a} d_\text{p}(t)

\tag{5.3.6} v_\text{p}(t) = H(t) \, d_\text{p} (t)

where $v_\text{p}(t) = \dot{d_\text{p}(t)}$ is the object’s proper velocity and $H(t) = \frac{\dot{a}}{a}$ is the Hubble parameter at some time $t$ .

Since light travels along null geodesics, $ds = 0$ . Therefore, by the FLRW metric, we have the proper distance at the current time, or the comoving distance as

\tag{5.3.7} \boxed{d_\text{p} (t_0) = r = \int_{t_\text{e}}^{t_0} dr = c \int_{t_\text{e}}^{t_0} \frac{dt}{a (t)}}

If we observe a galaxy’s emission line $\lambda_0$ at time $t_0$ , and it was emitted with a shorter wavelength $\lambda_\text{e}$ at time $t_\text{e}$ , then

\tag{5.3.8} \frac{\lambda_0}{a(t_0)} = \frac{\lambda_\text{e}}{a(t_\text{e})} \implies \lambda(t) \propto a(t)

This change in wavelength is what leads to redshift. The redshift $z$ is given by

\tag{5.3.9} \boxed{1 + z = \frac{\lambda_0}{\lambda_\text{e}} = \frac{a(t_0)}{a(t_\text{e})} = \frac{1}{a(t_\text{e})}}

Distance Measures

Luminosity Distance

If the luminosity of an object is known to be $L$ , and the flux received from that body is $F$ , its luminosity distance $d_\text{L}$ is given by

\tag{5.3.10} d_\text{L} = \sqrt{\frac{L}{4 \pi F}}

From the FLRW metric, we get $d_\text{L} = S_\kappa (r) (1+z)$ . If $\kappa = 0$ ,

\tag{5.3.11} d_\text{L} = d_\text{p} (t_0) (1+z)

In a flat universe, the flux received from a source with redshift $z$ is

F = \frac{L}{4 \pi d_\text{p}^2} \cdot \frac{1}{1+z} \cdot \frac{1}{1+z}

The first factor of $1+z$ comes from the fact that as the photons are redshifted, their energy decreases. The second factor of $1+z$ comes from the fact that the interval of time between the emission and reception of the photons is $dt_\text{e} = dt_0 (1+z)$ .

Angular Diameter Distance

The angular diameter distance to an object is defined as

\tag{5.3.12} d_A = \frac{l}{\theta}

where $l$ is the length of the object and $\theta \ll 1$ is the angle subtended by the object at the observer.

From the FLRW metric, we get that in a flat universe

\tag{5.3.13} d_A = \frac{d_\text{p} (t_0)}{1+z}

The reason why a factor of $1+z$ comes in the angular diameter distance is because the object we are viewing is as it looked when the light was emitted. The object is not at the same distance as it was when the light was emitted. The object is at a distance $d_\text{p} (t_0)$ , but the light was emitted at a distance $d_\text{p} (t_\text{e}) = \frac{d_\text{p}(t_0)}{1+z}$ .

The angular diameter distance is related to the luminosity distance by

\tag{5.3.14} d_\text{L} = (1+z)^2 \, d_A

In an expanding universe, $d_\text{L} > d_A$ .

Cosmic Horizons

Hubble Distance

The Hubble time is defined as $t_\text{H} = H_0^{-1}$ .

The Hubble distance is defined as the distance beyond which $v_\text{p} > c$ .

\tag{5.3.15} d_\text{H} = \frac{c}{H_0}

Cosmic Event Horizon

The cosmic event horizon is the maximum comoving distance a light beam emitted now can travel:

\tag{5.3.16} d_\text{EH} = \int_{t_0}^{\infty} \frac{c}{a(t)} dt

Particle Horizon

The most distant objects one can see are those for which light emitted at $t = 0$ is just reaching us now at $t = t_0$ . The proper distance to such an object is called the particle horizon distance:

\tag{5.3.17} d_\text{hor} = \int_{0}^{t_0} \frac{c}{a(t)} dt

The portion of the universe lying within the horizon, called the visible universe or the observable universe, is causally connected to the observer.

At present time, $d_\text{H}(t_0) \: (\approx 4.3 \, \mathrm{Gpc}) < d_\text{EH}(t_0) \: (\approx 5 \, \mathrm{Gpc} ) < d_\text{hor}(t_0) \: (\approx 14.4 \, \mathrm{Gpc})$ .

Our Universe The Friedmann Equations