Simple Math

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The problems of this section need basic understanding of Friedman equations, definitions of proper, comoving and conformal coordinates, the cosmological redshift formula and simple cosmological models (see Chapters 2 and 3).

Let us make our definitions a little more strict.

A particle horizon, for a given observer $A$ and cosmic instant $t_0$ is a surface in the instantaneous three-dimensional section $t=t_0$ of space-time, which divides all comoving particles [1] into two classes: those that have already been observable by $A$ up to time $t_0$ and those that have not.

An event horizon, for a given observer $A$, is a hyper-surface in space-time, which divides all events into two non-empty classes: those that have been, are, or will be observable by $A$, and those that are forever outside of $A$'s possible powers of observation. It follows from definition that event horizon, and its existence, depend crucially on the observer's (and the whole Universe's) future as well as the past: thus it is said to be an essentially global concept. It is formed by null geodesics.

The following notation is used hereafter: $L_p (t_0)$ is the proper distance from observer $A$ to its particle horizon, measured along the slice $t=t_0$. For brevity, we will also call this distance simply "the particle horizon in proper coordinates", or just "particle horizon". The corresponding comoving distance $l_p$ is the particle horizon in comoving coordinates.

Likewise, $L_e$ is the proper distance from an observer to its event horizon (or, rather, its section with the hypersurface $t=t_0$), measured also along the slice $t=t_0$. It is called "space event horizon at time $t_0$", or just the event horizon, for brevity. The respective comoving distance is denoted $l_e$.

Problem 1

The proper distance $D_p (t_0)$ between two comoving observers is the distance measured between them at some given moment of cosmological time $t=t_0$. It is the quantity that would be obtained if all the comoving observers between the given two measure the distances between each other at one moment $t=t_0$ and then sum all of them up. Suppose one observer detects at time $t_{0}$ the light signal that was emitted by the other observer at time $t_e$. Find the proper distance between the two observers at $t_0$ in terms of $a(t)$.

Problem 2

Show, that the proper distance $L_p$ to the particle horizon at time $t_0$ is \begin{equation} L_p (t_0)=\lim\limits_{t_e \to 0}D_p (t_e ,t_0). \label{LpDp} \end{equation}

Problem 3

The past light cone of an observer at some time $t_0$ consists of events, such that light emitted in each of them reaches the selected observer at $t_0$. Find the past light cone's equation in terms of proper distance vs. emission time $D_{plc} (t_e)$. What is its relation to the particle horizon?

Problem 4

The simplest cosmological model is the one of \emph{Einstein-de Sitter}, in which the Universe is spatially flat and filled with only dust, with $a(t)\sim t^{2/3}$. Find the past light cone distance $D_{plc}$ (\ref{Dplc}) for Einstein-de Sitter.

Problem 5

Demonstrate that in general $D_{plc}$ can be non-monotonic. For the case of Einstein-de Sitter show that its maximum -- the maximum emission distance -- is equal to $8/27 L_H$, while the corresponding redshift is $z=1.25$.

Problem 6

In a matter dominated Universe we see now, at time $t_0$, some galaxy, which is now on the Hubble sphere. At what time in the past was the photon we are registering emitted?

Problem 7

Show that the particle horizon in the Einstein-de Sitter model recedes at three times the speed of light.

Problem 8

Does the number of observed galaxies in an open Universe filled with dust increase or decrease with time?

Problem 9

Draw the past light cones $D_{plc}(t_e)$ for Einstein-de Sitter and a Universe with dominating radiation on one figure; explain their relative position.

Problem 10

Find the maximum emission distance and the corresponding redshift for power law expansion $a(t)\sim (t/t_0 )^n$.

Problem 11

Show that the most distant point on the past light cone was exactly at the Hubble sphere at the moment of emission of the light signal that is presently registered.

Problem 12

Show that the comoving particle horizon is the age of the Universe in conformal time

Problem 13

Show that \begin{align} \frac{dL_p}{dt}&=L_p (z)H(z)+1;\\ \frac{dL_e}{dt}&=L_e (z)H(z)-1. \end{align}

Problem 14

Find $\ddot{L_p}$ and $\ddot{L_e}$.

Problem 15

Show that observable part of the Universe expands faster than the Universe itself. In other words, the observed fraction of the Universe always increases.

Problem 16

Show that the Milne Universe has no particle horizon.

Problem 17

Consider a universe which started with the Big Bang, filled with one matter component. How fast must $\rho(a)$ decrease with $a$ for the particle horizon to exist in this universe?

Problem 18

Calculate the particle horizon for a universe with dominating

  • radiation;
  • matter.

Problem 19

Consider a flat universe with one component with state equation $p =w \rho$. Find the particle horizon at present time $t_0$.

Problem 20

Show that in a flat universe in case of domination of one matter component with equation of state $p=w\rho$, $w>-1/3$ \begin{equation} L_{p} (z)=\frac{2}{H(z) (1+3w)},\qquad \dot{L_p}(z)=\frac{3(1+w)}{(1+3w)}. \end{equation}

Problem 21

Show that in a flat universe in case of domination of one matter component with equation of state $p=w\rho$, $w<-1/3$ \begin{equation} L_{e} (z)=-\frac{2}{H(z) (1+3w)},\qquad \dot{L_p}(z)=-\frac{3(1+w)}{(1+3w)}. \end{equation}

Problem 22

Estimate the particle horizon size at matter-radiation equality.

  1. Rindler uses the term "fundamental observers".