I’m making a fictional galaxy, and I want the inhabitants of this galaxy to be unable to calculate the age of the universe, I want them to be unaware of how much has already happened.

I’ve already decided that most other galaxies once visible from mine, has already passed beyond the cosmological horizon, and redshifted beyond detection; leaving only a few thousand local galaxies, close enough to be gravitationally bound, to be observed.

From what I can recall from school, the age of the universe is calculated by measuring the rate at which galaxies accelerate away from each other.

So would it still be possible to calculate the age of the universe using only these local galaxies?

If it is possible, when is the point (preferably before the heat death of the universe) where determining the age of the universe becomes impossible, or at least highly inaccurate?

Or does someone have a different idea of how I can achieve this without throwing too many laws of physics out the window?

The current 'measurements' of the age of the universe are actually measurements of some astronomical effects, that get mixed into calculations involving many constants. Most of what we know about the universe is (mostly implicitly) followed by '... if those values are indeed constant.'.

Some weird effects are actually easier to explain when we assume the constants to be not that, i.e. slightly changing over time (or space).

If your civilization realized that some important constant was actually variable, and a function of time that was not easily extrapolated (e.g. not a continuous function), their measurements might not be precise enough (maybe it would even be physically impossible to be precise enough) to pin it down, thus making the whole calculation impossible.

Your idea is sound. In the very far future when most galaxies are invisible and the cosmic background radiation has faded to the point of undetectability, evidence about the state of the early universe will be impossible to obtain at our current technology level, and some kind of steady-state theory may well be just as well supported by the available evidence. But remember that the visible universe will be very different from the one that we see around us today -- we're talking very far future, perhaps a trillion years from now. See this article.

You can just have them be primitive.

Alternatively, they did have the knowledge and tech to do it. They just lost of broke all their satellites and telescopes.

Edit: you edited the question to mention they do have technology. Alright, have them be surrounded by nebulae, such as the Eagle:

Or the Horsehead:

Those are thick enough that we cannot see past them, at least not in every frequency. If your inhabitants are in a bubble of dust, they won't be able to measure the doppler shift of distant galaxies.

First why not, and then how maybe ...

I'm not convinced this is possible for an advanced technological society capable of space travel.

One of the things they have to figure out on the way to getting that developed is the General Theory of Relativity. Although Einstein gets all the credit for this publicly, that's a gross simplification of a lot of investigation and discussion and ideas and theories that helped, so a society is going to get to it.

Almost as soon as you can do anything with this theory, people will inevitably start trying to develop a model of the universe. What we got (and this model is also pretty likely to be found by someone) is the FLRW metric. Expansion is what we found going on, but even in your scenario, they'll see effects in their "Local" galaxies (which won't be all that local any more !).

They'll also become aware of dark matter and dark energy, because this affects the motion of galaxies and even the motion of stars in galaxies.

Note even in the late universe when it's "dark" and they're all alone, even the fact that you are alone with nothing obvious to see will be useful data in terms of fitting it to a model of the FLRW type.

Quantum theory is also going to be found. Again inevitably people will seek explanations for the origin of the universe and look to it to provide explanations.

Like us they'll look for ways to combine quantum theories and general relativity (which we haven't quite managed yet :-) ). These will result in concepts that tell them to look at e.g. the relative distribution of different isotopes as evidence for their origin (this is one piece of evidence we use).

So they'll inevitably find clues and look for explanations, and go looking for more clues to test theories and find more data. They'll keep looking until they get an answer, because if there is one trait I suspect all intelligent life shares it's going to be "unrestrained nosiness". :-)

How to avoid this ...

One word : Desperation.

Give them a world that is in turmoil, always at war with itself, with an ecological nightmare that makes staying alive hard as blazes.

That pushes resources and all the inquisitive people into more practical work (if necessary in chains).

Also keep in mind that the main motivation for all that "going into space" stuff was not (in our case) pure scientific interest. We were building rockets to throw nukes at each other and science gave an excuse that didn't sound so insane. But in doing that we came extremely close to wiping ourselves out.

So we (and they) could just as easily have gotten to space travel and promptly almost wiped out most of the planet's life, including ourselves. (And the option is still there, kiddies - vote for sane people, please :-) ).

I don;t know about you but if e.g. the Cuban missile crisis had gone full scale nuclear exchange, none of us would care one iota about the age of the universe or studies to find out about it.

How do we calculate the age of the universe?

A very simple estimate of the age of the universe can be found via the Hubble constant:
$$tsimfrac1H_0$$
A more sophisticated (and exact) technique is to determine the standard cosmological parameters. By this, I mean the various relative densities of dark energy, matter, and radiation ($Omega_Lambda$, $Omega_m$, and $Omega_r$). We can then calculate the age of the universe by integrating the scale factor:
$$t=frac1H_0int_0^1 Big(Omega_m,0a^-1 + Omega_r,0a^-2 + Omega_Lambda,0a^2 + Omega_k,0Big)^-1/2 da$$
There are a number of ways to do determine the Hubble constant and the density parameters:

• Analyzing baryon acoustic oscillations
  


• Examining the sources of gravitational waves
  


• Looking at peaks and anisotropies in the cosmic microwave background
  


• Measuring gravitational lensing


• Studying the Sunyaev–Zeldovich effect
  

. . . and many others. We've recently seen discrepancies between some of the values derived by different methods, which implies that our standard cosmological model may be incomplete, but they're nonetheless all valid.

What your setup rules out

We can throw out all methods that require measurements of objects at high redshifts. Regrettably, this includes basically all the techniques I've described here. If you're looking for a time period at which it becomes difficult to determine the age of the universe - well, you picked a good one. All high-redshift sources aren't visible.

One possibility that might remain is looking for anisotropies in the cosmic microwave background (which I alluded to before), from which we can determine $H_0$. Unfortunately, it happens that $H_0$ is degenerate with the radiation density $Omega_Lambda$ and the equation of state parameter $w$; that is, you need two of those to determine the other one. This means you can only constrain the relationships between these three. Of course, if there was an independent way to determine $w$ and $Omega_Lambda$, you might be able to get somewhere.

I should note that this assumes that the CMB will still be detectable far in the future. Over time, it gets more and more redshifted; by this point, it may exist only at such long wavelengths that your civilization will be unable to observe it.

Make redshift the natural effect of a resistance caused by a currently unknown scalar field. Right now we assume we can track the universe's origins to a big bang because we believe it is expanding. We believe this because of redshift, but if redshift is just what happens to light over distance, then all of our theories about the universe's age, size, mass, and energy would just be wrong. Another reason we believe this is because of the universe's background radiation which we be believe to be caused by the big bang, might instead just be the result of light losing its energy as it travels through space. All of our models to date only work because the effect of this field is so predictable, but a predictable phenomenon and an accurate hypothesis to explain it are not mutually exclusive.

A more advanced civilization than our own could prove the existence a redshifting-scalar field by the inconsistencies it would create in travel times between stars (A thing we have not yet been able to test for ourselves). Once this field is proven, the new model of the universe would have to except that matter could go out indefinitely past our cosmological horizon, and observing the drift of galaxies would reveal nothing about the actual age of the universe if general expansion proves to just be an optical illusion.