using C=3x108 m/s
[3x108 /68x103] [1x106]=4.4x109 light yrs.
this assumes constant acceleration (as quoted; 68k m/s for every 1x106 light years distance), and the universe can expand at the speed of light.
where do they get 14.7?
Hi sinx. The answer to this is not short, but ok [deep breath], here goes:
sinx, your calculation doesn't work because you have treated the 68k number as being an acceleration when it clearly is not. Notice that it actually has dimensions of speed/distance = 1/time. To understand what this quoted parameter actually is, we need a toy model. Consider a 1D universe that is just an infinite ruler with markings at regular intervals (and a galaxy at each marking, if you like!). Suppose that initially these markings are 1 "unit" apart, but after one timestep (let's say one second) the amount of space between all the markings has doubled:
Code:
(t = 0) | | | | | | | |
(t = 1) | | | | | | | |
Let's say we, the observer, happen to be located at the leftmost marking in the above drawing. We notice something interesting. In one timestep, our nearest neighbour has receded from us by a distance of 2 units, so its apparent recessional speed is 2 units/second. However, the rightmost marking that has been drawn has appeared to move away from us by 14 units in the same interval of time. Its recessional velocity is 14 units/second! For this type of uniform expansion of space, then, it seems clear that
the farther away an object is, the faster it appears to be receding away from you. Its recessional speed \(\displaystyle v\) is proportional to its distance \(\displaystyle d\). In fact the relation is given by
\(\displaystyle \displaystyle v = H_0 d\)
with the constant of proportionality being called
Hubble's constant after the famous astronomer Edwin Hubble who discovered this relationship for distant galaxies in the 1920s. The typical speed/distance units used to express \(\displaystyle H_0\) (see below) seem weird: in SI it would just be 1/seconds. But they are chosen to be convenient given the characteristic recession speeds and distances of astronomical objects. Currently we think that:
\(\displaystyle \displaystyle H_0 \approx 70~\frac{\mathrm{km}/\mathrm{s}}{\mathrm{Mpc}}\)
So this is the change in recession speed in
kilometres per second that occurs
per megaparsec of distance. The original poster (OP) didn't quite get the units right. It's per megaparsec (Mpc) rather than per megalightyear (Mly). For those of you who don't know your SI prefixes, a megaparsec is one million parsecs. And a parsec is about 3.26 light years. Professional astronomers tend to use parsecs rather than light years to state distances, for reasons I won't get into right now. As another comment: the exact value of \(\displaystyle H_0\) depends on who you ask. Results from the
Planck satellite (which observed the Cosmic Microwave Background radiation from 2009-2013) say 68 (km/s)/Mpc. Other results from observations of "standard candles" (distant astronomical sources of a known intrinsic brightness) like Cepheid variable stars and Type Ia supernovae, give a result that is more like 72 or 74 (km/s)/Mpc, and there is currently a roughly 3.6\(\displaystyle \sigma\) tension in the values of \(\displaystyle H_0\) obtained by these different cosmological probes. This is a mildly interesting/annoying problem in observational cosmology right now.
You might notice something very peculiar. If \(\displaystyle H_0\) really is constant, then the rate of change of the distance is proportional to the distance itself. This is a recipe for exponential growth! In other words, if the farther away you are, the faster you are receding, then obviously this will be a runaway effect. But do we really observe that? If so, why was it supposedly such a shock/surprise when scientists discovered that the expansion of the Universe was accelerating? By all media accounts, didn't they expect the expansion to be slowing, due to the mutual gravitational attraction of all the matter in the Universe? Yes. How can this be reconciled? By realizing that the Hubble constant is not actually a constant at all. It varies with time. It's only
called a constant because, at any given moment, it is the same
throughout space (i.e. the same for all observers). The actual equation involves the Hubble
parameter \(\displaystyle H\):
\(\displaystyle v = H(t)d\)
which is a function of time, and which has always been expected to be decreasing with time. But, in accordance with Einstein's General Relativity (GR), things like the geometry and expansion rate of the Universe depend on its
mass-energy content. If the mysterious "dark energy" (in the form of a cosmological constant in the Einstein field equations) were the only constituent of the Universe i.e. if there were no dark matter, no atomic matter, and no radiation (photons), then you can show from GR that \(\displaystyle H\)
would be constant and there would be exponential growth (see
https://en.wikipedia.org/wiki/De_Sitter_universe). In any case, \(\displaystyle H_0\) is just the present-day value \(\displaystyle H(t=t_0)\), where \(\displaystyle t_0\) is "now."
That brings me to the thorny issues of time and distance in an expanding Universe. You have to be very careful what you mean when you talk about time, because, if you've ever studied Special Relativity, you know that in 4D spacetime,
which of the four spacetime coordinates is considered to be "time", is something that is different for different observers. So there is no universal notion of "all of space at a given moment in time" (i.e. there no single correct direction in which to take 3D spatial slices of the 4D spacetime). In cosmology, one convention, when we talk about
time is to mean "cosmic time" which is proper time \(\displaystyle t\) as measured by "comoving observers" (observers who are simply being carried along with the expansion). Since the
worldlines of these observers (i.e. their paths through spacetime) all intersect in only one spacetime location --- the initial singularity of the Big Bang --- there can be one spatial slicing that applies to all of them. They will experience time the same way.
Similarly, one has to be very careful what one means when talking about distance in the expanding Universe, because there are several ways to define it. A point brought up by the OP illustrates this nicely: the age of the Universe has been measured to be 13.8 billion years, so naively one would expect the longest distance out to which we are able to see to be 13.8 billion light years. Light from more distant objects would not yet have had time to reach us, so this radius would define the size of our
Observable Universe. However, the distance to this boundary is actually 46 billion light years, because although the most-distant photons reaching us now have been travelling for ~13.8 billion years, the places they came from have moved away from us in the intervening time interval due to the expansion of the Universe. The physical distance to the boundary now is
not the distance that you would infer from the
light travel time.
The distance of 46 billion light years is an example of what cosmologists refer to as the
proper (radial) distance \(\displaystyle r\). Loosely speaking, it is the distance you would measure if you could magically freeze the expansion, run a string out to the point in question, and then measure the length of the string. At this point I will introduce the very important concept of the
scale factor \(\displaystyle a(t)\). This is the ratio of the distance between any two observers at time \(\displaystyle t\) to their distance now:
\(\displaystyle \displaystyle a(t) \equiv \frac{r(t)}{r(t_0)}\)
So if two galaxies are separated by a distance of 500 Mpc today (\(\displaystyle t = t_0\)), and if you go back to a time \(\displaystyle t\) in the Universe's history at which the scale factor was \(\displaystyle a(t) = 0.5\), then those galaxies would have been separated by only 250 Mpc at that time. If they are separated by only 100 Mpc today, then they would have been separated by 50 Mpc at that time. Thus the function \(\displaystyle a(t)\) encapsulates the dynamics of the expansion of the universe. In the future, the scale factor will be greater than unity. And although people say informally that \(\displaystyle H(t)\) is the "expansion rate" of the Universe, it's actually \(\displaystyle da/dt\) that is the expansion rate, while
\(\displaystyle \displaystyle H = \frac{1}{a}\frac{da}{dt}\)
I'll "prove this" in an Appendix below, but you can see from this relation why \(\displaystyle H\) was expected to be decreasing with time. The only way it wouldn't be was if the rate of change of scale factor was growing faster than (or at the same rate as) the scale factor itself (which can only be true in an accelerating Universe).
Another arguably much more convenient way to measure distance is called "comoving" distance \(\displaystyle \chi\), which is the distance between observers as measured using a coordinate grid
that is expanding along with the expansion of space. Hence comoving distances between objects are fixed with time (assuming the objects' only relative motion is due to Hubble expansion). The definition of comoving distance is:
\(\displaystyle \displaystyle \chi \equiv \frac{r(t)}{a(t)} \)
In other words, by convention, the comoving distance between two objects is just scaled to be equal to what their proper distance is
today. So if two galaxies (with no relative motion in comoving coordinates) are separated by a proper distance of 500 Mpc today, their comoving distance has always been and will always be 500 Mpc.
