# Determining the acceleration of the Universe from a single star?

It Occurs to me we might be able to find an entirely independent method of determining the Universe's acceleration using a single source.

If one was to watch a single high source consistently one should be able to simply simply watch for the change in it's redshift with time. I know it would be a small effect (perhaps choosing a high Z source would help).

Maybe something like finding a source that could line up with the recoiless resonant absorption (Mossbauer effect) in some crystal would have enough sensitivity? (ie a cosmological scale Pound Rebka experiment)??

Anyway, I hadn't heard of it, but maybe someone can tell me why its a bad/good idea. Thanks!

NOTE: I get that ultimately one would use several sources for a proper analysis.

• The source and detector would have to have a completely fixed comoving distance. In practice, Andromeda is hurtling towards(!) us and other galaxies are moving away faster than the Hubble Law would predict. So on the individual level, it feels unlikely a star would be much use. Hence, we average over many galaxies. Oct 25, 2018 at 5:38
• @BigAL It doesn't matter their relative motion so long as it's constant, since it's a CHANGE in the redshift we're looking for. Oct 25, 2018 at 5:56
• @R.Rankin: but local gravitational fields can affect peculiar motions at least as much as the cosmological effects can -- for instance, the above cited example of andromeda moving toward us, which will be an advancing acceleration. Oct 25, 2018 at 6:39
• @JerrySchirmer Hence the note at the end of my question. Oct 25, 2018 at 6:50
• To use a single source repeatedly, you would need something other than the type Ia supernovas that are currently favored. Something that doesn't self-destruct. Oct 25, 2018 at 22:52

To answer this we need to find out how rapidly the red shift is changing, then decide if the change is large enough to measure on the sort of timescales we can use for the measurement.

We describe the expansion of the universe by a scale factor that we conventionally set to unity right now. Then if the current distance to a star is $$x_0$$ the variation of the distance with time is given by:

$$x(t) = a(t)x_0$$

And a quick differentiation gives the acceleration of the star as:

$$\ddot{x} = \ddot{a} x_0$$

The change in velocity, and hence the change in red shift, in some time $$t$$ is approximately $$\Delta v = \ddot x t = \ddot a x_0 t$$ if we approximate the acceleration as constant, which is a good approximation on human timescales.

The acceleration of the scale factor is given by the second Friedmann equation:

$$\frac{\ddot a}{a} = \frac{-4\pi G}{3}\left( \rho + \frac{3p}{c^2} \right) + \frac{\Lambda c^2}{3}$$

The pressure is approximately zero and currently $$a = 1$$ so this simplifies to:

$$\ddot a = -\tfrac{4}{3} \pi G\rho + \tfrac{1}{3}\Lambda c^2$$

The matter density (including dark matter) is around two hydrogen atoms per cubic metre or converting to SI:

$$\tfrac43\pi G\rho \approx 9 \times 10^{-37} \,\text{s}^{-2}$$

The current value of the cosmological constant is $$10^{−52}$$ m$$^{−2}$$, and this makes the second term:

$$\frac{\Lambda c^2}{3} \approx 3 \times 10^{-36} s^{-2}$$

Giving us the current value for $$\ddot a$$:

$$\ddot a \approx 2 \times 10^{-36} s^{-2}$$

It just remains to decide how far away we can reliably monitor a single star, and how long we want to do the experiment. Let's take a billion light years as the distance ($$10^{25}$$ metres) and ten years for the experiment duration ($$3 \times 10^8$$ seconds) and we get:

$$\Delta v \approx 0.006 \,\text{m/s}$$

We can measure the red shift due to velocities this small, for example using the Mossbauer effect, but only under very carefully controlled laboratory conditions. We would have absolutely no hope of doing it using a star a billion light years away. In any case the star will have some proper motion due to the gravitational fields it is travelling in and we could not be sure that a velocity change this small wasn't just due to local gravitational acceleration rather than the expansion of spacetime.

In summary, it's a nice idea but sadly infeasible.

• Wonderful answer! Thank you! As a random aside I had been thinking about optical lattices as in BEC's as a way to make a tunable reciever of the Mossbauer type (since you control all crystal properties of the lattice). maybe one day relevant here, hence the question. thanks again! Oct 25, 2018 at 11:02
• One note here, you computed the required sensitivity for the current rate of acceleration (your last equation), but wouldn't any source from the early universe (quasars as the called them come to mind) with sufficient absorbtion/ emission spectra, evidence a much larger acceleration than that figure (at least based upon what we know) Oct 25, 2018 at 12:06
• I mean penultimate equation (which sounds cooler anyway) Oct 25, 2018 at 12:21
• I'd been looking at using free electron lasers for the optical lattice source, whose input frequency was coupled via a second observation point to the light of the source itself. Diagrammatically it looks like an interferometer. Have been searching for a practical application of such a device Oct 25, 2018 at 12:34
• @R.Rankin you make a good point, if we're looking at a star a billion light years away we're seeing the effect of the acceleration a billion years ago. However unless you go back a very long way this actually reduces the velocity change because around 5 billion years age the acceleration was zero - this was the point that the matter density and dark energy exactly balanced each other making $\ddot a = 0$. Since then the acceleration has been increasing. Oct 25, 2018 at 13:14

To be able to see the acceleration of a star means that one should measure with an accuracy much better than the current measurement error in red shifts. By the way, the expansion of space is measured by the change in the spectrum of galaxies, not stars. Plot of distance (in giga light-years) vs. redshift according to the Lambda-CDM model. dH (in solid black) is the comoving distance from Earth to the location with the Hubble redshift z while ctLB (in dotted red) is the speed of light multiplied by the lookback time to Hubble redshift z. The comoving distance is the physical space-like distance between here and the distant location, asymptoting to the size of the observable universe at some 47 billion light years. The lookback time is the distance a photon traveled from the time it was emitted to now divided by the speed of light, with a maximum distance of 13.8 billion light years corresponding to the age of the universe.

The scale is in Giga light years, a error calculation on this curve would still be in fractions of Giga light years.To measure a change, over which a change in the red shift could register, so as to measure an acceleration, is not possible in human life times, which counts years, and so can see changes in light years.

In principle the experiment is doable (for a galaxy not a star), but not for humans.

• I appreciate your answer. I'm aware we use galaxies, after all that's how we can figure out how fast they're rotating (using the differential redshift from each side of it's rotation). Clearly the scale we look over is Giga light years. This merely places limits on how sensitive our spectometers are, it doesn't rule out the experiment in "a human lifetime" maybe someone can figure out how sensitive it would have to be? Oct 25, 2018 at 5:55
• giga is 10^9 .there are no such accuracies imo. in LIGO " change in distance between its mirrors 1/10,000th the width of a proton! This is equivalent to measuring the distance to the nearest star (some 4.2 light years) to an accuracy smaller than the width of a human hair!" ligo.caltech.edu/page/facts Oct 25, 2018 at 6:52
• " Mössbauer spectroscopy is a very sensitive technique in terms of energy (and hence frequency) resolution, capable of detecting changes in just a few parts per $10^{11}$ " en.wikipedia.org/wiki/M%C3%B6ssbauer_spectroscopy I get that that's for gamma rays, but it is proof of principle. I am familiar with giga Oct 25, 2018 at 7:01
• While I'm a huge (and I mean HUGE) fan of LIGO, I fail to see the relevance of that information here. Oct 25, 2018 at 7:06
• it shows the limits of accuracy that can be achieved in experiments of cosmological type. As I said, accuracy gives the size of a possible effect which will be statistically significant in measuring year after year the spectrum shift from a single galaxy. The acceleration determined from galaxy comparisons is very much smaller than the possible limits of a 100 year experiment .for direct detection from one source. Oct 25, 2018 at 8:27

You just need to measure the recession velocity of any astronomical source.

The proper distance to a given source $$d$$ is related to the comoving distance $$\chi$$ through: $$d(t) = a(t) \chi$$

where $$a(t)$$ is the scale factor for expansion of the universe. Then the recession velocity can be written as: $$\dot{d} = \dot{a} \chi = \frac{\dot{a}}{a} d$$

So the Hubble constant $$H\equiv \dot{a}/a$$ measures the rate of expansion of the universe in the above relation from the recession velocity of the source, and proper distance to it.

• You're not talking about accelerated expansion of the universe, thus it doesn't address my question. Oct 25, 2018 at 5:59

The idea of measuring the change in redshift over time of a distant galaxy has been around since at least the 1960s. Unfortunately, it remains far beyond our technical capabilities. In a previous post, I derived the equation for $$\dot{z}$$:

$$\dot{z} = (1+z)H_0 - H\!\left(\!\frac{1}{1+z}\!\right),$$ where $$H(a)$$ is the Hubble parameter, expressed in terms of the scale factor. In a $$\Lambda$$CDM model with $$H_0 = 68\ \text{km}\,\text{s}^{-1}\,\text{Mpc}^{-1}$$, this gives the following plot: As you can see, $$\dot{z}\sim 10^{-10}$$ per year. With current technology, quasar redshifts can be measured with an accuracy up to $$10^{-5}$$ (see Davis & Lineweaver (2003), section 4.3). In other words, with current technology it would still take ~100,000 years to measure any change in redshifts. It's a great idea, but we have a long way to go before it becomes feasible.