Some more persistent cosmic tensions

I set out last time to discuss some of the tensions that persist in afflicting cosmic concordance, but didn’t get past the Hubble tension. Since then, I’ve come across more of that, e.g., Boubel et al (2024a), who use a variant of Tully-Fisher to obtain H₀ = 73.3 ± 2.1(stat) ± 3.5(sys) km/s/Mpc. Having done that sort of work, their systematic uncertainty term seemed large to me. I then came across Scolnic et al. (2024) who trace this issue back to one apparently erroneous calibration amongst many, and correct the results to H₀ = 76.3 ± 2.1(stat) ± 1.5(sys) km/s/Mpc. Boubel is an author of the latter paper, so apparently agrees with this revision. Fortunately they didn’t go all Sandage-de Vaucouleurs on us, but even so, this provides a good example of how fraught this field can get. It also demonstrates the opportunity for confirmation bias, as the revised numbers are almost exactly what we find ourselves. (New results coming soon!)

It’s a dang mess.

The Hubble tension is only the most prominent of many persistent tensions, so let’s wade into some of the rest.

The persistent tension in the amplitude of the power spectrum

The tension that cosmologists seem to stress about most after the Hubble tension is that in σ₈. σ₈ quantifies the amplitude of the power spectrum; it is a measure of the rms fluctuation in mass in spheres of 8h^-1 Mpc. Historically, this scale was chosen because early work by Peebles & Yu (1970) indicated that this was the scale on which the rms contrast in galaxy numbers* is unity. This is also a handy dividing line between linear and nonlinear regimes. On much larger scales, the fluctuations are smaller (a giant sphere is closer to the average for the whole universe) so can be treated in the limit of linear perturbation theory. Individual galaxies are “small” by this standard, so can’t be treated⁺ so simply, which is the excuse many cosmologists use to run shrieking from discussing them.

As we progressed from wrapping our heads around an expanding universe to quantifying the large scale structure (LSS) therein, the power spectrum statistically describing LSS became part of the canonical set of cosmological parameters. I don’t myself consider it to be on par with the Big Two, the Hubble constant H₀ and the density parameter Ω_m, but many cosmologists do seem partial to it despite the lack of phase information. Consequently, any tension in the amplitude σ₈ garners attention.

The tension in σ₈ has been persistent insofar as I recall debates in the previous century where some kinds of data indicated σ₈ ~ 0.5 while other data preferred σ₈ ~ 1. Some of that tension was in underlying assumptions (SCDM before LCDM). Today, the difference is [mostly] between the Planck best-fit amplitude σ₈ = 0.811 ± 0.006 and various local measurements that typically yield 0.7something. For example, Karim et al. (2024) find low σ₈ for emission line galaxies, even after specifically pursuing corrections in a necessary dust model that pushed things in the right direction:

As with so many cosmic parameters, there is degeneracy, in this case between σ₈ and Ω_m. Physically this happens because you get more power when you have more stuff (Ω_m), but the different tracers are sensitive to it in different ways. Indeed, if I put on a cosmology hat, I personally am not too worried about this tension – emission line galaxies are typically lower mass than luminous red galaxies, so one expects that there may be a difference in these populations. The Planck value is clearly offset from both, but doesn’t seem too far afield. We wouldn’t fret at all if it weren’t for Planck’s damnably small error bars.

This tension is also evident as a function of redshift. Here are measures of the combination of parameters fσ₈  =  Ω_m(z)^γσ₈ measured and compiled by Boubel et al (2024b):

The line representing the Planck value σ₈ = 0.81 overshoots most of the low redshift data, particularly those with the smallest uncertainties. The green line has σ₈ = 0.74, so is a tad lower than Planck in the same sense as other low redshift measures. Again, the offset is modest, but it does look significant. The tension is persistent but not a show-stopper, so we generally shrug our shoulders and proceed as if it will inevitably work out.

The persistent tension in the cosmic mass density

A persistent tension that nobody seems to worry about is that in the density parameter Ω_m. Fits to the Planck CMB acoustic power spectrum currently peg Ω_m = 0.315±0.007, but as we’ve seen before, this covaries with the Hubble constant. Twenty years ago, WMAP indicated Ω_m = 0.24 and H₀ = 73, in good agreement with the concordance region of other measurements, both then and now. As with H₀, the tension is posed by the itty bitty uncertainties on the Planck fit.

Experienced cosmologists may be inclined to scoff at such tiny error bars. I was, so I’ve confirmed them myself. There is very little wiggle room to match the Planck data within the framework of the LCDM model. I emphasize that last bit because it is an assumption now so deeply ingrained that it is usually left unspoken. If we leave that part out, then the obvious interpretation is that Planck is correct and all measurements that disagree with it must suffer from some systematic error. This seems to be what most cosmologists believe at present. If we don’t leave that part out, perhaps because we’re aware of other possibilities so are not willing to grant this assumption, then the various tensions look like failures of a model that’s already broken. But let’s not go there today, and stay within the conventional framework.

There are lots of ways to estimate the gravitating mass density of the universe. Indeed, it was the persistent, early observation that the mass density Ω_m exceeded that in baryons, Ω_b, from big bang nucleosynthesis that got got the non-baryonic dark matter show on the road: there appears to be something out there gravitating that’s not normal matter. This was the key observation that launched non-baryonic cold dark matter: if Ω_m > Ω_b, there has^% to be some kind of particle that is non-baryonic.

So what is Ω_m? Most estimates have spanned the range 0.2 < Ω_m < 0.4. In the 1980s and into the 1990s, this seemed close enough to Ω_m = 1, by the standards of cosmology, that most Inflationary cosmologists presumed it would work out to what Inflation predicted, Ω_m = 1 exactly. Indeed, I remember that community directing some rather vicious tongue-lashings at observers, castigating them to look harder: you will surely get Ω_m = 1 if you do it right, you fools. But despite the occasional claim to get this “right” answer, the vast majority of the evidence never pointed that way. As I’ve related before, an important step on the path to LCDM – probably the most important step – was convincing everyone that really Ω_m < 1.

Discerning between Ω_m = 0.2 and 0.3 is a lot more challenging than determining that Ω_m < 1, so we tend to treat either as acceptable. That’s not really fair in this age of precision cosmology. There are far too many estimates of the mass density to review here, so I’ll just note a couple of discrepant examples while also acknowledging that it is easy to find dynamical estimates that agree with Planck.

To give a specific example, Mohayaee & Tully (2005) obtained Ω_m = 0.22 ± 0.02 by looking at peculiar velocities in the local universe. This was consistent with other constraints at the time, including WMAP, but is 4.5σ from the current Planck value. That’s not quite the 5σ we arbitrarily define to be an undeniable difference, but it’s plenty significant.

There have of course been other efforts to do this, and many of them lead to the same result, or sometimes even lower Ω_m. For example, Shaya et al. (2022) use the Numerical Action Method developed by Peebles to attempt to work out the motions of nearly 10,000 galaxies – not just their Hubble expansion, but their individual trajectories under the mutual influence of each other’s gravity and whatever else may be out there. The resulting deviations from a pure Hubble flow depend on how much mass is associated with each galaxy and whatever other density there is to perturb things.

This result is in even greater tension with Planck than the earlier work by Mohayaee & Tully (2005). I find the need to invoke interhalo matter disturbing, since it acts as a pedestal in their analysis: extra mass density that is uniform everywhere. This is necessary so that it contributes to the global mass density Ω_m but does not contribute to perturbing the Hubble flow.

One can imagine mass that is uniformly distributed easily enough, but what bugs me is that dark matter should not do this. There is no magic segregation between dark matter that forms into halos that contain galaxies and dark matter that just hangs out in the intergalactic medium and declines to participate in any gravitational dynamics. That’s not an option available to it: if it gravitates, it should clump. To pull this off, we’d need to live in a universe made of two distinct kinds of dark matter: cold dark matter that clumps and a fluid that gravitates globally but does not clump, sort of an anti-dark energy.

Alternatively, we might live in an underdense region such that the local Ω_m is less than the global Ω_m. This is an idea that comes and goes for one reason or another, but it has always been hard to sustain. The convergence to low Ω_m looks pretty steady out to ~100 Mpc in the plot above; that’s a pretty big hole. Recall the non-linearity scale discussed above; this scale is a factor of ten larger so over/under-densities should typical be ±10%. This one is -60%, so I guess we’d have to accept that we’re not Copernican observers after all.

The persistent tension in bulk flows

Once we get past the basic Hubble expansion, individual galaxies each have their own peculiar motion, and beyond that we have bulk flows. These have been around a long time. We obsessed a lot about them for a while with discoveries like the Great Attractor. It was weird; I remember some pundits talking about “plate tectonics” in the universe, like there were giant continents of galaxy superclusters wandering around in random directions relative to the frame of the microwave background. Many of us, including me, couldn’t grok this, so we chose not to sweat it.

There is no single problem posed by bulk flows^, and of course you can find those that argue they pose no problem at all. We are in motion relative to the cosmic (CMB) frame^$, but that’s just our Milky Way’s peculiar motion. The strange fact is that it’s not just us; the entirety of the local universe seems to have a unexpected peculiar motion. There are lots of ways to quantify this; here’s a summary table from Courtois et al (2025):

As we look to large scales, we expect the universe to converge to homogeneity – that’s the Cosmological Principle, which is one of those assumptions that is so fundamental that we forget we made it. The same holds for dynamics – as we look to large scales, we expect the peculiar motions to average out, and converge to a pure Hubble flow. The table above summarizes our efforts to measure the scale on which this happens – or doesn’t. It also shows what we expect on the second line, “predicted LCDM,” where you can see the expected convergence in the declining bulk velocities as the scale probed increases. The third line is for “cosmic variance;” when you see these words it usually means something is amiss so in addition to the usual uncertainties we’re going to entertain the possibility that we live in an abnormal universe.

Like most people, I was comfortably ignoring this issue until recently, when we had a visit and a talk from one of the protagonists listed above, Richard Watkins (W23). One of the problems that challenge this sort of work is the need for a large sample of galaxies with complete sky coverage. That’s observationally challenging to obtain. Real data are heterogeneous; treating this properly demands a more sophisticated treatment than the usual top-hat or Gaussian approaches. Watkins described in detail what a better way could be, and patiently endured the many questions my colleagues and I peppered him with. This is hard to do right, which gives aid and comfort to the inclination to ignore it. After hearing his talk, I don’t think we should do that.

The data do not converge with increasing scale as expected. It isn’t just the local space density Ω_m that’s weird, it’s also the way in which things move. And “local” isn’t at all small here, with the effect persisting out beyond 300 Mpc for any plausible h = H₀/100.

This is formally a highly significant result, with the authors noting that “the probability of observing a bulk flow [this] large … is small, only about 0.015 per cent.” Looking at the figure above, I’d say that’s a fairly conservative statement. A more colloquial way of putting it would be “no way we gonna reconcile this!” That said, one always has to worry about systematics. They’ve made every effort to account for these, but there can always be unknown unknowns.

Mapping the Universe

It is only possible to talk about these things thanks to decades of effort to map the universe. One has to survey a large area of sky to identify galaxies in the first place, then do follow-up work to obtain redshifts from spectra. This has become big business, but to do what we’ve just been talking about, it is further necessary to separate peculiar velocities from the Hubble flow. To do that, we need to estimate distances by some redshift-independent method, like Tully-Fisher. Tully has been doing this his entire career, with the largest and most recent data product being Cosmicflows-4. Such data reveal not only large bulk flows, but extensive structure in velocity space:

We have a long way to go to wrap our heads around all of this.

Persistent tensions persist

I’ve discussed a few of the tensions that persist in cosmic data. Whether these are mere puzzles or a mounting pile of anomalies is a matter of judgement. They’ve been around for a while, so it isn’t fair to suggest that all of the data are consistent with LCDM. Nevertheless, I hear exactly this asserted with considerable frequency. It’s as if the definition of all is perpetually shrinking to include only the data that meet the consistency criterion. Yet it’s the discrepant bits that are interesting for containing new information; we need to grapple with them if the field is to progress.

*This was well before my time, so I am probably getting some aspect of the history wrong or oversimplifying it in some gross way. Crudely speaking, if you randomly plop down spheres of this size, some will be found to contain the cosmic average number of galaxies, some twice that, some half that. That the modern value of σ₈ is close to unity means that Peebles got it basically right with the data that were available back then and that galaxy light very nearly traces mass, which is not guaranteed in a universe dominated by dark matter.

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⁺It amazes me how pervasively “galaxies are complicated” is used as an excuse⁺⁺ to ignore all small scale evidence.

Not all of us are limited to working on the simplest systems. In this case, it doesn’t matter. The LCDM prediction here is that galaxies should be complicated because they are nonlinear. But the observation is that they are simple – so simple that they obey a single effective force law. That’s the contradiction right there, regardless of what flavor of complicated might come out of some high resolution simulation.

⁺⁺At one KITP conference I attended, a particle-cosmologist said during a discussion session, in all seriousness and with a straight face, “We should stop talking about rotation curves.” Because scientific truth is best revealed by ignoring the inconvenient bits. David Merritt remarked on this in his book A Philosophical Approach to MOND. He surveyed the available cosmology textbooks, and found that not a single one of them mentioned the acceleration scale in the data. I guess that would go some way to explaining why statements of basic observational facts are often met with stunned silence. What’s obvious and well-established to me is a wellspring of fresh if incredible news to them. I’d probably give them the stink-eye about the cosmological constant if I hadn’t been paying the slightest attention to cosmology for the past thirty years.

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^&There is an elegant approach to parameterizing the growth of structure in theories that deviate modestly from GR. In this context, such theories are usually invoked as an alternative to dark energy, because it is socially acceptable to modify GR to explain dark energy but not dark matter. The curious hysteresis of that strange and seemingly self-contradictory attitude aside, this approach cannot be adapted to MOND because it assumes linearity while MOND is inherently nonlinear. My very crude, back-of-the-envelope expectation for MOND is very nearly constant γ ~ 0.4 (depending on the scale probed) out to high redshift. The bend we see in the conventional models around z ~ 0.6 will occur at z > 2 (and probably much higher) because structure forms fast in MOND. It is annoyingly difficult to put a more precise redshift on this prediction because it also depends on the unknown metric. So this is a more of a hunch than a quantitative prediction. Still, it will be interesting to see if roughly constant fσ₈ persists to higher redshift.

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^%The inference that non-baryonic dark matter has to exist assumes that gravity is normal in the sense taught to us by Newton and Einstein. If some other theory of gravity applies, then one has to reassess the data in that context. This is one of the first considerations I made of MOND in the cosmological context, finding Ω_m ≈ Ω_b.

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^MOND is effective at generating large bulk flows.

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^$Fun fact: you can type the name of a galaxy into NED (the NASA Extragalactic Database) and it will give you lots of information, including its recession velocity referenced to a variety of frames of reference and the corresponding distance from the Hubble law V = H₀D. Naively, you might think that the obvious choice of reference from is the CMB. You’d be wrong. If you use this, you will get the wrong distance to the galaxy. Of all the choices available there, it consistently performs the worst as adjudicated by direct distance measurements (e.g., Cepheids).

NED used to provide a menu of choices for the value of H₀ to use. It says something about the social-tyranny of precision cosmology that it now defaults to the Planck value. If you use this, you will get the wrong distance to the galaxy. Even if the Planck H₀ turns out to be correct in some global sense, it does not work for real galaxies that are relatively near to us. That’s what it means to have all the “local” measurements based on direct distance measurements (e.g., Cepheids) consistently give a larger H₀.