I How do you get to the conclusion that there is accelerating expansion?

[Ben vdP]

TL;DR Summary

What is the reasoning behind the expansion of the universe is accelerating?

The observations could also be interpreted as a slowing down of the expansion since
the more distant you look the further you look into the past.

I'm trying to check the reasoning behind the "accelerating expansion of the universe",
but do not find anything sufficiently accurate or convincing.

For example in "https://en.wikipedia.org/wiki/Accelerating_universe" is put:

"Observations show that the expansion of the universe is accelerating, such that the velocity
at which a distant galaxy recedes from the observer is continuously increasing with time."

It is not clear what is exactly ment with "time" and what the context is.
My impression is that with "time" is ment something synonymous to "observed distance".
But they are two completely different parameters and they are not one on one.
Any relation between the two needs to be explained explicitly and how it relates to the measurements.

About "Supernova observations":

"The idea was that as type Ia supernovae have almost the same intrinsic brightness (a standard candle),
and since objects that are further away appear dimmer, the observed brightness of these supernovae can
be used to measure the distance to them. The distance can then be compared to the supernovae's
cosmological redshift, which measures how much the universe has expanded since the supernova occurred;
the Hubble law established that the further away an object is, the faster it is receding.
The unexpected result was that objects in the universe are moving away from one another at an accelerating rate."

It is a very suggestive picture, but the reasoning is full with large holes:

- Again there is the same confusion between "time" and "distance".
You are not measuring the light going from our position to the distant galaxy, and it is not instantly.
You are measuring the light that we receive now and was emitted long ago in the distant past in the distant galaxy.
You are looking at what happened in the past.

- Distance can be seen as the time it took for the light to travel from the emitting galaxy to us divided by the light speed.
But this is already implicitly dependant on the expansion of the universe and on relativistic effects.
That makes it a lot more complicated.

- The cosmological redshift does not directly measure how much the universe has expanded since the supernova occurred.
Also here relativistic effects and possibly other effects have to be taken into account and corrected for.

- A distant red-shifted galaxy cannot be used as a measure of the size of the universe, it is not on the edge of the universe.
It is tempting to see it as a kind of radius R of the universe or of the expansion. It is not however,
it is just a distant galaxy inbetween that is emitting light and that light appears red-shifted to us.
And that needs to be interpreted.

looking more closer at it:

If you see two galaxies g0 and g1 on the same line, with one galaxy g0 just a bit further than the other g1.
Use g0 as origine of your coordinate system. At time t0 = 0 light is emitted from g0.
At the later time t1, this light passes galaxy g1 and g1 is also emitting light.
At time T we receive the light from both galaxies.

From our perspective:

If the instrumentation measures more than expected red-shift from the light received from g0 and g1 and
if the light from galaxy g0 has more additional red-shift than the light from galaxy g1 (which is closer),
then light that was emitted later in time (from galaxy g1) shows less excess red-shift than light from g0.

I have to assume here that relativistic effects (and possible other mechanisms that can cause a red-shift effect),
have been taken into account and we are talking only about an additional red-shift unaccounted for.

So, moving forward in time (for subsequent emitted light), the amount of additional red-shift is diminishing over time,
i.e. it says expansion is slowing down and there is no accelerating expansion of the universe.

I do not see how you can get to the conclusion that "Observations show that the expansion of the universe is accelerating"
based on the measurements.
(In which the reasoning also needs to be correct obviously).

The question remains, how do you get at this conclusion from the observations?
What is the reasoning?

 

[Drakkith]

It is not clear what is exactly ment with "time" and what the context is.

The short, simple version is that time is what a clock measures. Start a clock in a reference frame. Measure the velocity of receding galaxies at time A, then measure again at a later time B. The measurements will show that galaxies are receding faster at time B than they should be if the expansion was purely 'inertial' without acceleration.

You are measuring the light that we receive now and was emitted long ago in the distant past in the distant galaxy.
You are looking at what happened in the past.

Of course. What is the problem here?

- Distance can be seen as the time it took for the light to travel from the emitting galaxy to us divided by the light speed.
But this is already implicitly dependant on the expansion of the universe and on relativistic effects.
That makes it a lot more complicated.

Yes, it does make it more complicated. Cosmology is not an easy science.

Also here relativistic effects and possibly other effects have to be taken into account and corrected for.

All known effects have already been taken into account. Cosmologists and astronomers did not come to the conclusion that the universe is expanding and that the expansion is accelerating in a day, nor did they do so based on a single or small set of observations. Modern cosmology is built upon roughly a century's worth of observations and theoretical advancements done by thousands of professionals putting in every ounce of their skill and ability over the course of their lifetimes.

- A distant red-shifted galaxy cannot be used as a measure of the size of the universe, it is not on the edge of the universe.

A single redshifted galaxy, no. Thousands of galaxies at various redshifts, coupled with our best understanding of how that redshift occurred and what it means? Yes it can measure the size of the universe.

It is tempting to see it as a kind of radius R of the universe or of the expansion. It is not however,
it is just a distant galaxy inbetween that is emitting light and that light appears red-shifted to us.
And that needs to be interpreted.

It is being interpreted. And that interpretation is that the redshift is due to the galaxy receding from us, and the amount of redshift is directly related to distance, which, along with other measurements and observations, can help us determine the size of the observable universe.

From our perspective:

If the instrumentation measures more than expected red-shift from the light received from g0 and g1 and
if the light from galaxy g0 has more additional red-shift than the light from galaxy g1 (which is closer),
then light that was emitted later in time (from galaxy g1) shows less excess red-shift than light from g0.

I'm not quite sure what you're saying here. G0 will be more redshifted since it is further from us than G1 (assuming the two are not so close as to be bound together by gravity). This, by itself, means little. To determine the rate of expansion you have to look at many, many galaxies and combine redshifts with other factors like supernova brightness, baryon acoustics, CMB properties, etc.

I do not see how you can get to the conclusion that "Observations show that the expansion of the universe is accelerating"
based on the measurements.
(In which the reasoning also needs to be correct obviously).

I'm not sure where to start, to be honest. First, I'd say you need to have a good grasp on the Cosmic Distance Ladder. Have you looked into that yet? Many different observations and measurements are done to finally arrive at the Hubble parameter, which is where we get the basic redshift vs distance relationship.

 

[Ibix]

Roughly speaking, the point is that by looking far away you get a cross-section of the universe throughout its history. Nearby stuff looks as it did recently, and further away stuff looks as it did a long time ago. You can then think about how cosmological expansion affects the light in flight, and back an expansion history out of that data.

You need the details of the observations and the maths of the cosmological model to make actual numerical deductions like whether there is a non-zero $\ddot{a}$.

 

[Janus]

It works like this: As we look across the universe, we see recession speeds increasing with distance. This alone, only tells us that the universe is expanding. To work out how it expanded over time, we look at exactly how that recession speed changes with distance. If for example, if the rate increases linearly with distance(plotting distance vs. recession gives a straight line), than then the expansion rate has been constant through time. If however, it isn't linear (The plot line curves), then the expansion rate hasn't been constant over time. Which way the line curves tells us whether the rate has decreased or increased over time.

 

[LastScattered1090]

What is meant by time is "cosmic time" i.e. proper time measured by observers who are co-moving with the expansion, but who are otherwise not moving "through" space (in co-moving coordinates). We imagine their clocks all having been synchronized at cosmic time t = 0: the beginning of the expansion. Another way to think about it is that this is time as defined in reference frame in which (ignoring primordial fluctuations), the CMB appears isotropic (no kinematic dipole).

The answer to your question about how acceleration is measured is complicated. The short answer is that it's indirect, and model-dependent. How it's determined is also dataset-dependent. In one dataset (observations of Type Ia supernovae or SNe), the two physical observables are (1) distance to a galaxy (as measured at the present time using a standard candle like a Type Ia SNe) and (2) redshift of that galaxy, as measured using spectroscopy. Measure these two for enough SNe, and you can produce a distance-redshift curve. You compare this empirical curve to what different versions of cosmology theory (different models) predict. The models have different expansion histories due to the presence of different constituents driving the expansion. Models with no Lambda (no cosmological constant) predict a curve that differs from our data by many many standard deviations (many sigma). Which means the deviation between the no-Lambda theory and experiment is very unlikely to be due to random measurement error. The distance-vs-redshift curve that best fits our data comes from a model in which Lambda is not zero. Again, the evidence for this, statistically, is very strong (that's why Reiss and Perlmutter and whoever else won the Nobel prize for these SNe observations). So at no point did anyone directly measure the recession rate of a galaxy at many different times and note that it was increasing with time. They measured present-day distance and present-day redshift for a large sample of galaxies. But those observations can only be explained if Lambda is non-zero, and hence the expansion rate has been increasing with time exponentially over recent cosmological history (say, the last 6 billion years).

For other datasets, like measurements of the Cosmic Microwave Background (CMB) fluctuations, the physical observables are totally different, but the results agree. Again, the determination is indirect. In the CMB you look at the fluctuations and say, "how are they statistically distributed as a function of scale on the sky?". That produces another curve called the angular power spectrum. This curve is really telling you the size distribution of density fluctuations (sound waves) in the primordial plasma of the early universe. You again compare the shape of this curve to what what early-Universe theory predicts for universes with different compositions and expansion histories. The comparison again shows that the model that best fits that data is one in which Lambda (the cosmological constant) is non-zero, to exceedingly high statistical significance.

Now we're looking for evidence that Lambda may not be a constant. We're attributing its effects to some mystery quantum field permeating all of space (dubbed "dark energy") whose density may actually vary with time rather than being fixed. But the present-day value for the energy density of that field (Lambda, essentially) is not in doubt, given the data.

Hope that helps!

 

[weirdoguy]

I do not see how you can get to the conclusion that "Observations show that the expansion of the universe is accelerating" based on the measurements.

Well, to fully see what is going on you need more then a wikipedia article. You need to study a cosmology textbook, or even few, and also papers that are more recent then textbooks, since it takes time for a portion of new knowdledge to be put in a textbook form. Wiki may be good for a start, but cosmology is a vast portion of physics, that also needs background in general relativity and particle physics. And that in itself takes a lot of additional time to study.

 

[Jaime Rudas]

For example in "https://en.wikipedia.org/wiki/Accelerating_universe" is put:

"Observations show that the expansion of the universe is accelerating, such that the velocity
at which a distant galaxy recedes from the observer is continuously increasing with time."

Measure the velocity of receding galaxies at time A, then measure again at a later time B. The measurements will show that galaxies are receding faster at time B than they should be if the expansion was purely 'inertial' without acceleration.

No, that's not the case. When we talk about the acceleration of the expansion, we're not referring to the fact that the recession velocity of galaxies increases with time, but rather to the increase in the growth rate of the scale factor.

In fact, in the model that best fits observations, the recession velocity of galaxies currently decreases with time.

Edit: I just realized this isn't true. What's currently decreasing is the Hubble parameter, not the recession velocity of galaxies.

 

[Ben vdP]

The short, simple version is that time is what a clock measures. Start a clock in a reference frame. Measure the velocity of receding galaxies at time A, then measure again at a later time B. The measurements will show that galaxies are receding faster at time B than they should be if the expansion was purely 'inertial' without acceleration.


Of course. What is the problem here?


Yes, it does make it more complicated. Cosmology is not an easy science.


All known effects have already been taken into account. Cosmologists and astronomers did not come to the conclusion that the universe is expanding and that the expansion is accelerating in a day, nor did they do so based on a single or small set of observations. Modern cosmology is built upon roughly a century's worth of observations and theoretical advancements done by thousands of professionals putting in every ounce of their skill and ability over the course of their lifetimes.


A single redshifted galaxy, no. Thousands of galaxies at various redshifts, coupled with our best understanding of how that redshift occurred and what it means? Yes it can measure the size of the universe.


It is being interpreted. And that interpretation is that the redshift is due to the galaxy receding from us, and the amount of redshift is directly related to distance, which, along with other measurements and observations, can help us determine the size of the observable universe.


I'm not quite sure what you're saying here. G0 will be more redshifted since it is further from us than G1 (assuming the two are not so close as to be bound together by gravity). This, by itself, means little. To determine the rate of expansion you have to look at many, many galaxies and combine redshifts with other factors like supernova brightness, baryon acoustics, CMB properties, etc.


I'm not sure where to start, to be honest. First, I'd say you need to have a good grasp on the Cosmic Distance Ladder. Have you looked into that yet? Many different observations and measurements are done to finally arrive at the Hubble parameter, which is where we get the basic redshift vs distance relationship.

There is a very crucial point and is exactly what the question is about.
There is a large gap between "the measurements" and "show that galaxies are receding faster ..."

It was not clear at all that the same galaxy was measured several times and the data compared at these different times.
Also what these measurements actually are or show always stays rather vague and gets to quickly interpreted into galaxies receding.


There are a lot of non trivial and possible tricky issues when galaxies are very far away and relativistic effects become important.

The situation at time A and that at time B are not identical.
For the remote galaxy time has passed and the distance is also different.
It can affect how to interpretate redshift with respect to distance.
The Hubble parameter is different as it is time dependant.
The light has taken a longer time to reach us, that affects the timing.


I tried to make clear that you do not observe how objects in the universe are moving away from one another now, but how they did in the past as this information is reaching us only now after having travelled billions of years through the universe.


With the simple model on g0 and g1 I tried to illustrate that an excess in decelerating expansion can appear to us as an excess in accelerating expansion, a kind of optical illusion, due to that it takes time for the light to reach us.

If the galaxy g0 is showing more unaccounted red shift than the closer galaxy g1 then that effect goes away for galaxies closer to us emitting later in time. So later in time the extra red-shift effect becomes smaller and not larger.

Or larger and larger distances, in this case from g1 to g0: Distance delta (D) is positive; but for time of emitted light delta (T) is negative.


It is maybe over dramatic but how can you rule out that a decelerating expansion in the early universe
has not been mistaken as an accelerating expansion in the current universe.
This in the context where the Hubble parameter H differs significantly from H0.

 

[phinds]

I tried to make clear that you do not observe how objects in the universe are moving away from one another now, but how they did in the past as this information is reaching us only now after having travelled billions of years through the universe.

Exactly. And this is part of what leads to the conclusion that the expansion of the universe is accelerating (because, unlike you, scientists correctly account for exactly that). You are drawing the completely wrong conclusion from a correct fact.

 

[Ibix]

Also what these measurements actually are or show always stays rather vague and gets to quickly interpreted into galaxies receding.

Vague? What sources are you reading? Because from the quoted passage I suspect you are reading non-technical sources and complaining that they don't give technical details.

With the simple model on g0 and g1 I tried to illustrate that an excess in decelerating expansion can appear to us as an excess in accelerating expansion, a kind of optical illusion, due to that it takes time for the light to reach us.

The problem is that you gave no mathematical description of your thinking. It's just hand-waving about some mental picture you have of what's going on that may or may not bear any resemblance to actual cosmological models. So your "can appear" is a very strong claim from very little evidence that you actually understand the physical model.

An example of something you could do is calculate redshift as a function of the $r$ coordinate where light was emitted (which is a textbook calculation requiring assumptions of the matter, radiation and dark energy densities now) and the amount of matter in a thin shell at that radius from us at that time. That gives you a simple estimate of the form of a graph showing the number of objects you should see as a function of their redshift. Play around with the three parameters and you will see the form of the graph change.

In an ideal world you would simply find the form of the graph that best matched observation, and the input parameters are your best estimates of those parameters in the universe. The acceleration or lack thereof is then a two line calculation. In reality you would need to correct the graph for telescope sensitivity and the fraction of the sky blocked from view by nearer objects, but then you would match your corrected graph to observation just as the ideal case.

The key point here is the numbers and maths. Anybody can say, as you are doing, that they don't believe the models can give the results they do. But it's an empty claim until you've generated two identical graphs using different input parameters that more or less match reality.

 

[Jaime Rudas]

It is maybe over dramatic but how can you rule out that a decelerating expansion in the early universe
has not been mistaken as an accelerating expansion in the current universe.

Perhaps one way to rule out this possible confusion is the fact that the standard model contemplates both a decelerating expansion in the early universe, and an accelerating expansion in the current universe.

 

[Drakkith]

It was not clear at all that the same galaxy was measured several times and the data compared at these different times.

It's not. Or, rather, two measurement taken even several years apart will not show any change in redshift, as the change is so small over that time period that it is not measurable to our instruments. I used that as a simple example of what would happen over time, not as an example of what you would actually measure. In reality you would measure many different galaxies at the same time and, using various methods, construct a model that gives you the distance to each one.

Also what these measurements actually are or show always stays rather vague and gets to quickly interpreted into galaxies receding.

First, the measurements are not vague, and second, that's the only known way to cause a redshift that preserves emission line spacing that could occur throughout the large-scale structure of the universe. That is, there is no known way to redshift the spectrum of light emitted by an atom or molecule while keeping the spacing of each line proportional. So at z=2, the wavelength and spacing between of each emission line is doubled. Two emission lines at, say, 440 nm and 420 nm would change to 880 nm and 840 nm and the spacing between them would go from 20 nm to 40 nm. This holds true for all wavelengths of all emission lines of all atoms and molecules at all z values.

There are a lot of non trivial and possible tricky issues when galaxies are very far away and relativistic effects become important.

What effects? Why are they important? What do they do?

The situation at time A and that at time B are not identical.
For the remote galaxy time has passed and the distance is also different.

So?

It can affect how to interpretate redshift with respect to distance.

How so?

The Hubble parameter is different as it is time dependant.
The light has taken a longer time to reach us, that affects the timing.

Okay. How?

I tried to make clear that you do not observe how objects in the universe are moving away from one another now, but how they did in the past as this information is reaching us only now after having travelled billions of years through the universe.

That's true everywhere, as light has a finite speed.

With the simple model on g0 and g1 I tried to illustrate that an excess in decelerating expansion can appear to us as an excess in accelerating expansion, a kind of optical illusion, due to that it takes time for the light to reach us.

I think you're saying that a greater 'excess' of redshift for galaxies further away could mean that the universe is in a decelerating expansion? If so, then you would be partially correct. In fact, the universe was in a decelerating expansion up until recently (in cosmological timescales). The Hubble parameter is smaller now than it was in the past and is actually continuing to decrease.

I hope the following math is correct. Someone correct me if I've made a mistake. I used a post on this page as a reference.

In other words, the Hubble parameter is $H = \frac{\dot a(t)}{a(t)}$, where $H$ is the Hubble parameter and $a(t)$ is the scale factor. An expanding universe has $\dot a(t) > 0$ while an accelerating expansion has $\ddot a(t) > 0$ (first and second derivatives).

What's happening is that the expansion is slowing but since $\ddot a(t)$ is greater than zero, the slowing itself is decreasing (becoming more positive).

This is why it is important to use math. Just saying 'excess redshift' isn't specific enough since the amount of 'excess' and how it appears to change over time is important.

 

[Ben vdP]

OK, Thanks, I think we can close this discussion.
I was hoping to get the reasoning steps in the interpretation of the measured data and its patterns (before it is getting interpreted) explicit to a level that you can start some logical and physical analysis.
It might only be in hundreds of pages of articles.

 

[Ibix]

I was hoping to get the reasoning steps in the interpretation of the measured data and its patterns (before it is getting interpreted)

I think if you want data without interpretation you're sunk. I don't think you can even answer the question of how much you weigh without interpretation.

Depending on how much detail you want, though, a basic cosmology book might be enough. Something like Hartle would at least explain how redshift works and introduce you to things you could measure, as well as the maths of FLRW universes. Depending on your physics background you might need other texts first.

 

[Drakkith]

OK, Thanks, I think we can close this discussion.
I was hoping to get the reasoning steps in the interpretation of the measured data and its patterns (before it is getting interpreted) explicit to a level that you can start some logical and physical analysis.
It might only be in hundreds of pages of articles.

You're probably looking for something that is beyond the scope of an internet forum.

 

[PeterDonis]

I was hoping to get the reasoning steps in the interpretation of the measured data and its patterns (before it is getting interpreted) explicit to a level that you can start some logical and physical analysis.

A good cosmology textbook will go into this. Liddle's Introduction to Modern Cosmology, for example.

The basic reasoning was given in post #4.

 

[DAH]

TL;DR Summary: What is the reasoning behind the expansion of the universe is accelerating?

The observations could also be interpreted as a slowing down of the expansion since
the more distant you look the further you look into the past.

You are not measuring the light going from our position to the distant galaxy, and it is not instantly.
You are measuring the light that we receive now and was emitted long ago in the distant past in the distant galaxy.
You are looking at what happened in the past.

I once asked the same question, how do we know if the distant galaxies are accelerating now since the light we observe here and now is from billions of years ago?
Then I found that cosmologists used methods such as measuring the apparent size of galaxies v redshift, otherwise known as the angular size redshift relation.

The figure above shows two types of expansion (a) is accelerating and (b) a decelerating expansion. Time goes from bottom to top, the scale factor $R_1$ is when the universe was much smaller and $R_0$ is the scale factor today.
The light rays from a distant galaxy that we observe today were emitted when the universe was much smaller and these galaxies can appear larger to us and its this apparent size v redshift that can determine the history of the expansion of the universe.
I don't know the exact numbers but, initially at low redshifts ( up to around z = 1.3)the angular size of galaxies decreases, to then increase at higher redshifts, and finally at very high z they decrease again.

The measured change in the angular size at specific redshifts is exactly what we would expect from a universe that transitions from deceleration to accelerated (figure (a)), and cosmologist can use it to estimate the acceleration started around 4 to 5 billion years ago.
If the universe was currently decelerating as in figure (b), then the observed angular size redshift relation would be much different, pretty much the opposite to what we observe. These methods have been used since the 90's and are still used today using different sources at many wavelengths over a wide range of redshifts, and the results always point toward a Robertson Walker universe with accelerated expansion.

Although the universe is accelerating in its expansion, the Hubble parameter actually decreases with time, and will continue to do so until the dark energy constant totally dominates and then it will finally be truly a constant giving
$$H=\sqrt{\frac{\Lambda c^2}{3}}$$
Since the scale factor is related to Hubble constant by $H=\frac{1}{R}\frac{dR}{dt}$ and substitute this into the above we have the differential equation:
$$\frac{1}{R}\frac{dR}{dt}=\sqrt{\frac{\Lambda c^2}{3}}$$
which has the solution $R\propto e^{ct\sqrt{\Lambda /3}}$
So in the remote future, the scale factor $R(t)$ will increase exponentially with time forever, according to our best models anyway.

 

[Bandersnatch]

@DAH angular diameter distance is actually hard to use, as finding a standard ruler is harder than finding a standard candle.
Both teams that made the discovery used luminosity distance instead (that is, brightness).

 

[DAH]

@DAH angular diameter distance is actually hard to use, as finding a standard ruler is harder than finding a standard candle.
Both teams that made the discovery used luminosity distance instead (that is, brightness).

You're probably right and I would think all methods are challenging especially at very high redshift because the galaxies were much younger and smaller than typical nearby ones.

 

[Jaime Rudas]

I once asked the same question, how do we know if the distant galaxies are accelerating now since the light we observe here and now is from billions of years ago?
Then I found that cosmologists used methods such as measuring the apparent size of galaxies v redshift, otherwise known as the angular size redshift relation.

@DAH angular diameter distance is actually hard to use, as finding a standard ruler is harder than finding a standard candle.
Both teams that made the discovery used luminosity distance instead (that is, brightness).

You're probably right and I would think all methods are challenging especially at very high redshift because the galaxies were much younger and smaller than typical nearby ones.

I don't know if I'm misunderstanding the approach, but both teams that discovered the acceleration of the expansion used the luminosity of Ia Supernovae as standard candle, and it is the method most commonly used by cosmologists today to measure distances. I don't think the method of measuring the apparent size of galaxies vs redshift has ever been used to determine the acceleration of the expansion.

 

[Jaime Rudas]

Then I found that cosmologists used methods such as measuring the apparent size of galaxies v redshift, otherwise known as the angular size redshift relation.

Where did you find that? I ask because I think the first time this method was used to test the expansion of the universe was in 1924 by Carl Wirtz in this paper, as mentioned in this other one. It was also used by W. de Sitter for the same purpose in this 1930 paper, which is perhaps the last relevant paper where this method was used.

 

[Bandersnatch]

I don't know if I'm misunderstanding the approach

You pretty much rephrased the exchange you quoted, so I guess you understood it well enough.

which is perhaps the last relevant paper where this method was used.

If you look for papers on the Tolman surface brightness test on arxiv, they are all about using the angular diameter, together with brightness, to test for expansion (but not acceleration). There's a few recent, decent ones. But also a number of those by Lerner and his ilk, using the fuzziness of the method to argue for a static universe. So, reader beware.

 

[Jaime Rudas]

If you look for papers on the Tolman surface brightness test on arxiv, they are all about using the angular diameter, together with brightness, to test for expansion (but not acceleration). There's a few recent, decent ones. But also a number of those by Lerner and his ilk, using the fuzziness of the method to argue for a static universe. So, reader beware.

Ah, Tolman, yes, yes. His work is relevant.

As for Lerner... I'd rather not comment.

 

[ohwilleke]

The comparison again shows that the model that best fits that data is one in which Lambda (the cosmological constant) is non-zero, to exceedingly high statistical significance.

Now we're looking for evidence that Lambda may not be a constant. We're attributing its effects to some mystery quantum field permeating all of space (dubbed "dark energy") whose density may actually vary with time rather than being fixed. But the present-day value for the energy density of that field (Lambda, essentially) is not in doubt, given the data.

We can say with great confidence that Lambda (whether it is a true constant or a function of time) is non-zero, which implies that there is an accelerating expansion.

But, this isn't a physical constant that we know particularly precisely. The uncertainty in our measurement of it is on the order of 3% (according to the Particle Data Group's list of astrophysical constants).

In terms of relative uncertainty, the only other fundamental physical constants that we know less precisely are: the up quark mass, the down quark mass, and five neutrino physics related constants (the CP violating parameter of neutrino oscillation, the θ23 parameter of neutrino oscillation, and the three absolute neutrino masses - although they aren't independent and there is only one degree of freedom in those three physical constants with a relative uncertainty of 3% or more).

All of the other physical constants of the Standard Model of Particle Physics and General Relativity (as well as physical constants like the speed of light and Planck's constant that aren't normally considered free parameters of either of those theories) are known to greater precision.

And, there is increasing doubt that Lambda really is constant, with models with a non-constant Lambda that is a function of time preferred over a constant value, by something on the order of 3 to 5 sigma depending on the paper analyzing the question. See, e.g., Mateus Scherer, et al., "Challenging ΛCDM: 5σ Evidence for a Dynamical Dark Energy Late-Time Transition" (April 29, 2025).

It's also important to recognize that the uncertainty over whether Lambda is really constant or not means that the model upon which prior measurements of this physical constant are dependent is itself in doubt. We've recently gone from a situation where we have large measurement errors in measuring a conceptually and theoretically very simple parameter in a widely accepted model, to one in which we're trying to reconcile lots of different data points that aren't collectively consistent with the old model to find a new model.

Again, the evidence of an an accelerating expansion is still very strong, but the exact extent to which it is accelerating is more unclear than it has been at any time in many decades. And, to answer this question, we're going to have to puzzle over our grab bag of astronomy data, that isn't quite consistent with a constant cosmological constant, in a way that focuses in on a lot of details of those observations that hadn't been necessary as long as the constant cosmological constant model was still unshaken. We probably need, at a minimum, two physical constants to describe something that we'd previously thought that we could describe with just one physical constant.

This is only coming up now for a reason that is nothing to be sad about. We have torrents of new data coming in from multiple independent astronomy collaborations with data of more different kinds, at greater precision, than every before in the history of the world.

Until now, we didn't have the volume and precision of data to distinguish between a simple cosmological constant model and the alternatives in a statistically significant way. Now, we finally do, and it turns out that the universe seems to be more complicated than we'd thought.