# Introduction

The LightCone tabular cosmological calculator is a versatile tool for understanding the expansion history of the universe: past, present and future. Stages in expansion history are designated by the corresponding scale factor **a**, which is the size of a generic distance compared with its size at present. For instance, **a**=1 denotes the present and **a**=0.5 the stage when cosmic distances were half their present size. In the same way, **a**=2 refers to a future stage when distances will be twice what they are now.

Each row of the table gives information (time, distances, expansion rates) about a given stage of expansion. If we are seeing light from a galaxy which was emitted back at **a**=0.5, the corresponding row of the table will tell us the galaxy's past and present distances, i.e. how far it was when it emitted the light and how far it is now. It also tells us the expansion speeds of those two distances, 'then' and 'now'. The same row in the table shows the time in years corresponding to the given scale factor: the year-count since the start of expansion. Also tabulated is the Hubble radius at that stage in history, i.e. the distance where cosmic distances increased at the speed of light. This gives a handle on the overall rate of expansion, as explained below.

When working with the scale factor a (the size of a generic distance relative to today) it is often convenient to use the reciprocal *S=1/a*. For instance the wavelengths of the Cosmic Microwave Background have been expanded (or "stretched") by a factor **S**=1090 since the time when it originated and this tells us what the scalefactor **a = 1/S** was at that time. Distances were then 1/1090 of their present size. In some cases it turns out to be easier to remember and use the "stretch" number 1090 instead of its reciprocal **1/S** = 0.000917. So don't be surprised to see the stretch S serving as an auxiliary or alternate handle on the scale factor. S is always one more than the corresponding redshift.

The calculator embodies the standard Lamda-Cold-Dark-Matter (LCDM) cosmological model derived from the Einstein field equation of General Relativity and employing the Friedmann Lemaitre Robertson Walker metric. The formulas used in its calculations are in a link at the bottom of the table.

The tooltips contained in the blue circles beside each parameter provide vital information about what each term means.

At the top left there are 7 inputs one can manipulate to generate different tables. The quantities listed on the right are derived from those inputs and can aid comparison with other source material. The default values of the inputs currently entered show a 10 step history of the universe growth from the origin of the microwave background to a future when distances are 100 times the size they are today.

# The 7 inputs (default values)

1) Hubble radius now (default **R _{0}** = 14.4 Gly). This is the size of any distance which at the present moment is expanding at the speed of light. Other cosmological distances are expanding in proportion to their size. A simple calculation shows that having the Hubble radius be 14.4 billion lightyears is equivalent to saying that cosmological distances are currently increasing at the rate of 1/144 percent per million years. This percentage growth rate applied to a Megaparsec distance (Mpc ≈ 3.26 million light years) amounts to a growth speed of 67.90 km/s, so a Hubble radius of 14.4 Gly is equivalent to the conventional Hubble rate being 67.90 km/s per Mpc.

It's important to note that the Hubble radius is not constant. If you refer to a typical table calculated by LightCone you will see that R has been increasing steadily in the past and is expected to continue doing so, approaching a final limit of 17.3 billion lightyears.

2) Future Hubble radius (default **R _{∞}** = 17.3 Gly). This is the size of distances which will be growing at the speed of light in the longterm future, again with other cosmological distances growing accordingly in proportion to their size. This is equivalent to saying that distances will eventually grow at the constant rate of 1/173 percent per million years.

3) The epoch of radiation-matter equality (default **S _{eq}** = 3400, in other words at scalefactor 1/3400). The early universe was dominated by radiation energy: a given volume held more energy in the form of radiation than in the form of matter. But expansion, by increasing both wavelengths and volume, reduces the energy density of radiation more rapidly than that of matter. So there had to come a time when they were equal, after which matter dominated. This epoch, given by the scale factor 1/3400 or its reciprocal, the corresponding

**S**= 3400, turns out to be a useful benchmark and serves as cosmic model parameter.

4) Total density parameter (default **Ω** = 1). This option allows one to adjust the curvature. The default value of **Ω**=1 represents a flat spatial geometry. A value of **Ω** >1 represents a positively curved spatial geometry, **Ω** < 1 represents one that is negatively curved.

5) Upper row stretch (default **S** =1/**a** = 1090). The range of the table history from start to finish can be specified by the user by putting in top and bottom row **S** values. These determine the beginning and ending scalefactors. In the default case, **S**=1090 denotes the epoch when the Cosmic Microwave Background light originated. Distances were then 1/1090 their present size.

6) Lower row stretch (default **S** =1/**a** = 0.01). This determines the final row scalefactor of the table. The default 0.01 means that the history will run until distances are 100 times their present size.

7) The minimum number of steps or rows (Default **S _{step}**=10) that the calculator will generate. There will sometimes be one row more, since when the

**S**-range includes

**S**=1 (the present), the calculator ensures that

**S**=1 (identically) is a separate row - sometimes by adding a row. It also allow

**S**=0, in which case it will only produce one row (

_{step}**S**).

_{upper}# The Display options

In the **Display As** block you will see 7 options. Consult the blue dot tooltips for additional information on these. To summarize:

**Standard** gives the normal table format.

**Office** gives a copy and paste-ready version for Microsoft office documents.

**Tex small** gives a copy and paste-ready LaTex version of the table which, despite its name, may be larger than you want.

**Tex script** gives a LaTex version which has so far been found to be a fitting size for many Forum applications (including PhysicsForums).

**Wikidot** gives a copy and paste-ready Wiki-HTML table to for sites that recognize that format.

**Chart** and **Chart Preview** allow one to graph the tabulated results. The scale factor is marked on the horizontal axis and the other columns selected for the table are represented as curves. You can select which curves to show in the chart by opening "**Column Definition and Selection**". Chart Preview gives you a quick look at a small version of the Chart to help while you are choosing curves and adjusting the scale factor range over which you want them plotted.

# The Column Definition and Selection menu

This lets you choose the quantities (time, distances, speeds etc.) you want presented in the columns of the table, and the desired degree of decimal-place precision. Each column listed on the menu has a place to check if you want it included, or to uncheck if you wish to exclude it. Each menu item also has a box where you can select the degree of precision you want shown. The blue dot tooltips explain how the numbers in a given column are defined and can help understand the table. The small round buttons in the top row allow one of them to be designated as the horizontal axis for the Chart options.

# More Guidance

Read the LightCone Tutorial for practical tips on how to use the calculator.

Read the Wiki http://cosmology101.wikidot.com/ for more explanations.

For more technical (mathematical) detail, visit the page: Advanced User