There is such a thing as *Hubble time*, which is simply the inverse
of the *Hubble parameter* (*H*). The inverse of the Hubble constant (*H*_{0})
is the current Hubble time (because the Hubble constant is the current value of
the Hubble parameter).

As I noted back in 2014, in *Is the Universe Expanding at the Speed of Light?*, the current value of Hubble time is interesting because it’s basically
the same as the age of the universe.
Now, because the (current) value of Hubble time is the inverse of the
Hubble constant, it varies with measurements of the Hubble constant.

- 2011 (Hubble) ~71.5 to ~76 km/s/Mpc
- 2012 (Spitzer) ~72 to ~76.5 km/s/Mpc
- 2012 (WMAP – after 9 years) 68.52-70.12 km/s/Mpc
- 2013 (Planck – after four years) 67.03 to 68.57 km/s/Mpc

By happy coincidence, Skydive Phil released a video on “*The Hubble Tension*” at about the same time as my retrospective podcast
listening of the *Infinite Monkey Cage* got me thinking
about the Hubble constant and the age of the universe all over again. What was bothering me was that there were
constant references to the acceleration of the expansion of the universe,
together with the assertion (claim, reminder, stating) of the fact that the
universe is 13 point something (7 or 8) billion years old (might have been
raised in *this specific episode* or *this one*).

If the rate at which the universe is expanding is accelerating,
then surely the age of the universe comes into question. The value given for age of the universe has not
changed significantly since 2014, when I reported it as 13.8 billion years –
the current values range from 13.772 (WMAP) to 13.813 (Planck 2015 data) km/s/Mpc. The value of the Hubble constant on the other
hand …

- 2018 (Planck) 67.66 (67.24 to 68.08) km/s/Mpc
- 2018 (Hubble and Gaia) 73.52 (71.90 to 75.14) km/s/Mpc
- 2017 (LIGO and Virgo) 70.0 (62.0 to 82.0) km/s/Mpc
- 2016 (eBOSS – after two years) 67.6 (67.0 to 68.3) km/s/Mpc

Skydive Phil’s video focusses on the difference between the
eBOSS (baryonic acoustic oscillation) and Planck measurements and the measurement
from Hubble and Gaia collaboration (also known as SH_{0}ES, sometimes
miswritten as SH_{o}ES, or SHoES).
The problem here is that the error bars no longer overlap, which
indicates some sort of problem – either they are measuring different things or
at least one of them is measuring the wrong way.

You occasionally read that the Hubble time is a useful estimate
of the age of the universe. In that case,
ignoring the ridiculously large error bars on the LIGO/Virgo result, the age of
the universe is between 13.01 and 14.60 billion years. In some cases, it is suggested that the Hubble
time indicates how long the universe has been expanding – but to all intents
and purposes this is what is meant by the age of the universe (much in the same
way as a baby is not strictly 0 days old at birth, usually having gestated in a
womb for about 9 months, we just pick a nice convenient reference point and count
from there).

However, we are now being told that, about 5 billion years
ago, the expansion of the universe started accelerating. If the Hubble time is a useful estimate of
the age of the universe *and* the age of the universe is what we
are being told (13.8 billion years, or near enough), then don’t we have a
problem? We can work out the value of
the Hubble parameter at a Hubble time of approximately 8.8 billion years (let’s
call it *H*_{-5}, meaning *H* at now minus 5
billion years), and it works out to be about 1.5 times that of today – ie about
111.1 km/s/Mpc.
(Perhaps the acceleration started only 4 billion years ago, at the end
of the matter-dominated era and the beginning of the *dark-energy-dominated era*. The value of the Hubble parameter
corresponding with a Hubble time of 9.8 billion years is a bit lower at *H*_{-4}=99.8
km/s/Mpc, but still about 40% higher than today.)

What is going on here?

Is the Hubble time only coincidentally a good estimate of
the age of the universe at the current time (but won’t be in the future and
wasn’t in the past)? This sounds like it
would be a fair addition to the list of fine tunings, and that surely can’t be
good.

If the Hubble time isn’t generally a good approximate of the
age of the universe, then there’s no reason to suggest that it ought to be a
good approximate today and maybe the age of the universe is not 13.8 billion
years after all. Not really, there are a
multitude of ways in which the age of the universe is measured, so cosmologists
don’t simply rely on inversing the Hubble constant (for example, they look at *cosmic background radiation fluctuations*).

Another possibility is that the Hubble parameter has been tracking
the age of the universe faithfully and 5 billion years ago it actually was *H*_{-5}=111.1
km/s/Mpc. Would that mean that, since that
time, Hubble expansion of the universe has decreased and some other expansion
of the universe (due to dark-energy, apparently) has got involved? If so, it still doesn’t really add up. We have the Hubble constant because that’s what
we measure as the current expansion rate of the universe. If the Hubble component of that is lower than
is required to account for that expansion rate, then *H*_{0}
< ~70, which increases the estimate for the age of the universe. Thinking about the notion that the universe
has been accelerating in its rate of expansion for the past 5 billion years,
let’s say at best case that *H*_{-5} was just a bit lower than
today, say just under the error bar for LIGO/Virgo at 61 km/s/Mpc. That would mean that at that time the Hubble
time was 16 billion years, and today the universe would be 21 billion years
old. That’s a big error in measurement. It just doesn’t seem right.

So, a question for the people in the know … if we were
around 5 billion years ago and were measuring the Hubble parameter using the
rate at which distant galaxies were receding, approximately what result would
we have come up with?

Do I have a solution?
Yes, I think I do. I just don’t quite
understand (yet) why most cosmologists would likely tell me I am wrong.