Month: February 2023

Take it where?

Take it where?

I had written most of the post below the line before an exchange with a senior colleague who accused me of asking us to abandon General Relativity (GR). Anyone who read the last post knows that this is the opposite of true. So how does this happen?

Much of the field is mired in bad ideas that seemed like good ideas in the 1980s. There has been some progress, but the idea that MOND is an abandonment of GR I recognize as a misconception from that time. It arose because the initial MOND hypothesis suggested modifying the law of inertia without showing a clear path to how this might be consistent with GR. GR was built on the Equivalence Principle (EP), the equivalence1 of gravitational charge with inertial mass. The original MOND hypothesis directly contradicted that, so it was a fair concern in 1983. It was not by 19842. I was still an undergraduate then, so I don’t know the sociology, but I get the impression that most of the community wrote MOND off at this point and never gave it further thought.

I guess this is why I still encounter people with this attitude, that someone is trying to rob them of GR. It’s feels like we’re always starting at square one, like there has been zero progress in forty years. I hope it isn’t that bad, but I admit my patience is wearing thin.

I’m trying to help you. Don’t waste you’re entire career chasing phantoms.

What MOND does ask us to abandon is the Strong Equivalence Principle. Not the Weak EP, nor even the Einstein EP. Just the Strong EP. That’s a much more limited ask that abandoning all of GR. Indeed, all flavors of EP are subject to experimental test. The Weak EP has been repeatedly validated, but there is nothing about MOND that implies platinum would fall differently from titanium. Experimental tests of the Strong EP are less favorable.

I understand that MOND seems impossible. It also keeps having its predictions come true. This combination is what makes it important. The history of science is chock full of ideas that were initially rejected as impossible or absurd, going all the way back to heliocentrism. The greater the cognitive dissonance, the more important the result.

Continuing the previous discussion of UT, where do we go from here? If we accept that maybe we have all these problems in cosmology because we’re piling on auxiliary hypotheses to continue to be able to approximate UT with FLRW, what now?

I don’t know.

It’s hard to accept that we don’t understand something we thought we understood. Scientists hate revisiting issues that seem settled. Feels like a waste of time. It also feels like a waste of time continuing to add epicycles to a zombie theory, be it LCDM or MOND or the phoenix universe or tired light or whatever fantasy reality you favor. So, painful as it may be, one has find a little humility to step back and take account of what we know empirically independent of the interpretive veneer of theory.

As I’ve said before, I think we do know that the universe is expanding and passed through an early hot phase that bequeathed us the primordial abundances of the light elements (BBN) and the relic radiation field that we observe as the cosmic microwave background (CMB). There’s a lot more to it than that, and I’m not going to attempt to recite it all here.

Still, to give one pertinent example, BBN only works if the expansion rate is as expected during the epoch of radiation domination. So whatever is going on has to converge to that early on. This is hardly surprising for UT since it was stipulated to contain GR in the relevant limit, but we don’t actually know how it does so until we work out what UT is – a tall order that we can’t expect to accomplish overnight, or even over the course of many decades without a critical mass of scientists thinking about it (and not being vilified by other scientists for doing so).

Another example is that the cosmological principle – that the universe is homogeneous and isotropic – is observed to be true in the CMB. The temperature is the same all over the sky to one part in 100,000. That’s isotropy. The temperature is tightly coupled to the density, so if the temperature is the same everywhere, so is the density. That’s homogeneity. So both of the assumptions made by the cosmological principle are corroborated by observations of the CMB.

The cosmological principle is extremely useful for solving the equations of GR as applied to the whole universe. If the universe has a uniform density on average, then the solution is straightforward (though it is rather tedious to work through to the Friedmann equation). If the universe is not homogeneous and isotropic, then it becomes a nightmare to solve the equations. One needs to know where everything was for all of time.

Starting from the uniform condition of the CMB, it is straightforward to show that the assumption of homogeneity and isotropy should persist on large scales up to the present day. “Small” things like galaxies go nonlinear and collapse, but huge volumes containing billions of galaxies should remain in the linear regime and these small-scale variations average out. One cubic Gigaparsec will have the same average density as the next as the next, so the cosmological principle continues to hold today.

Anyone spot the rub? I said homogeneity and isotropy should persist. This statement assumes GR. Perhaps it doesn’t hold in UT?

This aspect of cosmology is so deeply embedded in everything that we do in the field that it was only recently that I realized it might not hold absolutely – and I’ve been actively contemplating such a possibility for a long time. Shouldn’t have taken me so long. Felten (1984) realized right away that a MONDian universe would depart from isotropy by late times. I read that paper long ago but didn’t grasp the significance of that statement. I did absorb that in the absence of a cosmological constant (which no one believed in at the time), the universe would inevitably recollapse, regardless of what the density was. This seems like an elegant solution to the flatness/coincidence problem that obsessed cosmologists at the time. There is no special value of the mass density that provides an over/under line demarcating eternal expansion from eventual recollapse, so there is no coincidence problem. All naive MOND cosmologies share the same ultimate fate, so it doesn’t matter what we observe for the mass density.

MOND departs from isotropy for the same reason it forms structure fast: it is inherently non-linear. As well as predicting that big galaxies would form by z=10, Sanders (1998) correctly anticipated the size of the largest structures collapsing today (things like the local supercluster Laniakea) and the scale of homogeneity (a few hundred Mpc if there is a cosmological constant). Pretty much everyone who looked into it came to similar conclusions.

But MOND and cosmology, as we know it in the absence of UT, are incompatible. Where LCDM encompasses both cosmology and the dynamics of bound systems (dark matter halos3), MOND addresses the dynamics of low acceleration systems (the most common examples being individual galaxies) but says nothing about cosmology. So how do we proceed?

For starters, we have to admit our ignorance. From there, one has to assume some expanding background – that much is well established – and ask what happens to particles responding to a MONDian force-law in this background, starting from the very nearly uniform initial condition indicated by the CMB. From that simple starting point, it turns out one can get a long way without knowing the details of the cosmic expansion history or the metric that so obsess cosmologists. These are interesting things, to be sure, but they are aspects of UT we don’t know and can manage without to some finite extent.

For one, the thermal history of the universe is pretty much the same with or without dark matter, with or without a cosmological constant. Without dark matter, structure can’t get going until after thermal decoupling (when the matter is free to diverge thermally from the temperature of the background radiation). After that happens, around z = 200, the baryons suddenly find themselves in the low acceleration regime, newly free to respond to the nonlinear force of MOND, and structure starts forming fast, with the consequences previously elaborated.

But what about the expansion history? The geometry? The big questions of cosmology?

Again, I don’t know. MOND is a dynamical theory that extends Newton. It doesn’t address these questions. Hence the need for UT.

I’ve encountered people who refuse to acknowledge4 that MOND gets predictions like z=10 galaxies right without a proper theory for cosmology. That attitude puts the cart before the horse. One doesn’t look for UT unless well motivated. That one is able to correctly predict 25 years in advance something that comes as a huge surprise to cosmologists today is the motivation. Indeed, the degree of surprise and the longevity of the prediction amplify the motivation: if this doesn’t get your attention, what possibly could?

There is no guarantee that our first attempt at UT (or our second or third or fourth) will work out. It is possible that in the search for UT, one comes up with a theory that fails to do what was successfully predicted by the more primitive theory. That just lets you know you’ve taken a wrong turn. It does not mean that a correct UT doesn’t exist, or that the initial prediction was some impossible fluke.

One candidate theory for UT is bimetric MOND. This appears to justify the assumptions made by Sanders’s early work, and provide a basis for a relativistic theory that leads to rapid structure formation. Whether it can also fit the acoustic power spectrum of the CMB as well as LCDM and AeST has yet to be seen. These things take time and effort. What they really need is a critical mass of people working on the problem – a community that enjoys the support of other scientists and funding institutions like NSF. Until we have that5, progress will remain grudgingly slow.

1The equivalence of gravitational charge and inertial mass means that the m in F=GMm/d2 is identically the same as the m in F=ma. Modified gravity changes the former; modified inertia the latter.

2Bekenstein & Milgrom (1984) showed how a modification of Newtonian gravity could avoid the non-conservation issues suffered by the original hypothesis of modified inertia. They also outlined a path towards a generally covariant theory that Bekenstein pursued for the rest of his life. That he never managed to obtain a completely satisfactory version is often cited as evidence that it can’t be done, since he was widely acknowledged as one of the smartest people in the field. One wonders why he persisted if, as these detractors would have us believe, the smart thing to do was not even try.

3The data for galaxies do not look like the dark matter halos predicted by LCDM.

4I have entirely lost patience with this attitude. If a phenomena is correctly predicted in advance in the literature, we are obliged as scientists to take it seriously+. Pretending that it is not meaningful in the absence of UT is just an avoidance strategy: an excuse to ignore inconvenient facts.

+I’ve heard eminent scientists describe MOND’s predictive ability as “magic.” This also seems like an avoidance strategy. I, for one, do not believe in magic. That it works as well as it does – that it works at all – must be telling us something about the natural world, not the supernatural.

5There does exist a large and active community of astroparticle physicists trying to come up with theories for what the dark matter could be. That’s good: that’s what needs to happen, and we should exhaust all possibilities. We should do the same for new dynamical theories.

Imagine if you can

Imagine if you can

Imagine if you are able that General Relativity (GR) is correct yet incomplete. Just as GR contains Newtonian gravity in the appropriate limit, imagine that GR itself is a limit of some still more general theory that we don’t yet know about. Let’s call it Underlying Theory (UT) for short. This is essentially the working hypothesis of quantum gravity, but here I want to consider a more general case in which the effects of UT are not limited to the tiny netherworld of the Planck scale. Perhaps UT has observable consequences on very large scales, or a scale that is not length-based at all. What would that look like, given that we only know GR?

For starters, it might mean that the conventional Friedmann-Robertson-Walker (FRW) cosmology derived from GR is only a first approximation to the cosmology of the unknown deeper theory UT. In the first observational tests, FRW will look great, as the two are practically indistinguishable. As the data improve though, awkward problems might begin to crop up. What and where we don’t know, so our first inclination will not be to infer the existence of UT, but rather to patch up FRW with auxiliary hypotheses. Since the working presumption here is that GR is a correct limit, FRW will continue be a good approximation, and early departures will seem modest: they would not be interpreted as signs of UT.

What do we expect for cosmology anyway? A theory is only as good as its stated predictions. After Hubble established in the 1920s that galaxies external to the Milky Way existed and that the universe was expanding, it became clear that this was entirely natural in GR. Indeed, what was not natural was a static universe, the desire for which had led Einstein to introduce the cosmological constant (his “greatest blunder”).

A wide variety of geometries and expansion histories are possible with FRW. But there is one obvious case that stands out, that of Einstein-de Sitter (EdS, 1932). EdS has a matter density Ωm exactly equal to unity, balancing on the divide between a universe that expands forever (Ωm < 1) and one that eventually recollapses (Ωm > 1). The particular case Ωm = 1 is the only natural scale in the theory. It is also the only FRW model with a flat geometry, in the sense that initially parallel beams of light remain parallel indefinitely. These properties make it special in a way that obsessed cosmologists for many decades. (In retrospect, this obsession has the same flavor as the obsession the Ancients had with heavenly motions being perfect circles*.) A natural cosmology would therefor be one in which Ωm = 1 in normal matter (baryons).

By the 1970s, it was clear that there was no way you could have Ωm = 1 in baryons. There just wasn’t enough normal matter, either observed directly, or allowed by Big Bang Nucleosynthesis. Despite the appeal of Ωm = 1, it looked like we lived in an open universe with Ωm < 1.

This did not sit well with many theorists, who obsessed with the flatness problem. The mass density parameter evolves if it is not identically equal to one, so it was really strange that we should live anywhere close to Ωm = 1, even Ωm = 0.1, if the universe was going to spend eternity asymptoting to Ωm → 0. It was a compelling argument, enough to make most of us accept (in the early 1980s) the Inflationary model of the early universe, as Inflation gives a natural mechanism to drive Ωm → 1. The bulk of this mass could not be normal matter, but by then flat rotation curves had been discovered, along with a ton of other evidence that a lot of matter was dark. A third element that came in around the same time was another compelling idea, supersymmetry, which gave a natural mechanism by which the unseen mass could be non-baryonic. The confluence of these revelations gave us the standard cold dark matter (SCDM) cosmological model. It was EdS with Ωm = 1 mostly in dark matter. We didn’t know what the dark matter was, but we had a good idea (WIMPs), and it just seemed like a matter of tracking them down.

SCDM was absolutely Known for about a decade, pushing two depending on how you count. We were very reluctant to give it up. But over the course of the 1990s, it became clear [again] that Ωm < 1. What was different was a willingness, even a desperation, to accept and rehabilitate Einstein’s cosmological constant. This seemed to solve all cosmological problems, providing a viable concordance cosmology that satisfied all then-available data, salvaged Inflation and a flat geometry (Ωm + ΩΛ = 1, albeit at the expense of the coincidence problem, which is worse in LCDM than it is in open models), and made predictions that came true for the accelerated expansion rate and the location of the first peak of the acoustic power spectrum. This was a major revelation that led to Nobel prizes and still resonates today in the form of papers trying to suss out the nature of this so-called dark energy.

What if the issue is even more fundamental? Taking a long view, subsuming many essential details, we’ve gone from a natural cosmology (EdS) to a less natural one (an open universe with a low density in baryons) to SCDM (EdS with lots of non-baryonic dark matter) to LCDM. Maybe these are just successive approximations we’ve been obliged to make in order for FLRW** to mimic UT? How would we know?

One clue might be if the concordance region closed. Here is a comparison of a compilation of constraints assembled by students in my graduate cosmology course in 2002 (plus 2003 WMAP) with 2018 Planck parameters:

The shaded regions were excluded by the sum of the data available in 2003. The question I wondered then was whether the small remaining white space was indeed the correct answer, or merely the least improbable region left before the whole picture was ruled out. Had we painted ourselves into a corner?

If we take these results and the more recent Planck fits at face value, yes: nothing is left, the window has closed. However, other things change over time as well. For example, I’d grant a higher upper limit to Ωm than is illustrated above. The rotation curve line represents an upper limit that no longer pertains if dark matter halos are greatly modified by feedback. We were trying to avoid invoking that deus ex machina then, but there’s no helping it now.

Still, you can see in this diagram what we now call the Hubble tension. To solve that within the conventional FLRW framework, we have to come up with some new free parameter. There are lots of ideas that invoke new physics.

Maybe the new physics is UT? Maybe we have to keep tweaking FLRW because cosmology has reached a precision such that FLRW is no longer completely adequate as an approximation to UT? But if we are willing to add new parameters via “new physics” made up to address each new problem (dark matter, dark energy, something new and extra for the Hubble tension) so we can keep tweaking it indefinitely, how would we ever recognize that all we’re doing is approximating UT? If only there were different data that suggested new physics in an independent way.

Attitude matters. If we think both LCDM and the existence of dark matter is proven beyond a reasonable doubt, as clearly many physicists do, then any problem that arises is just a bit of trivia to sort out. Despite the current attention being given to the Hubble tension, I’d wager that most of the people not writing papers about it are presuming that the problem will go away: traditional measures of the Hubble constant will converge towards the Planck value. That might happen (or appear to happen through the magic of confirmation bias), and I would expect that myself if I hadn’t worked on H0 directly. It’s a lot easier to dismiss such things when you haven’t been involved enough to know how hard they are to dismiss***.

That last sentence pretty much sums up the community’s attitude towards MOND. That led me to pose the question of the year earlier. I have not heard any answers, just excuses to not have to answer. Still, these issues are presumably not unrelated. That MOND has so many predictions – even in cosmology – come true is itself an indication of UT. From that perspective, it is not surprising that we have to keep tweaking FLRW. Indeed, from this perspective, parameters like ΩCDM are chimeras lacking in physical meaning. They’re just whatever they need to be to fit whatever subset of the data is under consideration. That independent observations pretty much point to the same value is far compelling evidence in favor of LCDM than the accuracy of a fit to any single piece of information (like the CMB) where ΩCDM can be tuned to fit pretty much any plausible power spectrum. But is the stuff real? I make no apologies for holding science to a higher standard than those who consider a fit to the CMB data to be a detection.

It has taken a long time for cosmology to get this far. One should take a comparably long view of these developments, but we generally do not. Dark matter was already received wisdom when I was new to the field, unquestionably so. Dark energy was new in the ’90s but has long since been established as received wisdom. So if we now have to tweak it a little to fix this seemingly tiny tension in the Hubble constant, that seems incremental, not threatening to the pre-existing received wisdom. From the longer view, it looks like just another derailment in an excruciatingly slow-moving train wreck.

So I ask again: what would falsify FLRW cosmology? How do we know when to think outside this box, and not just garnish its edges?

*The obsession with circular motion continued through Copernicus, who placed the sun at the center of motion rather than the earth, but continued to employ epicycles. It wasn’t until over a half century later that Kepler finally broke with this particular obsession. In retrospect, we recognize circular motion as a very special case of the many possibilities available with elliptical orbits, just as EdS is only one possible cosmology with a flat geometry once we admit the possibility of a cosmological constant.

**FLRW = Friedmann-Lemaître-Robertson-Walker. I intentionally excluded Lemaître from the early historical discussion because he (and the cosmological constant) were mostly excluded from considerations at that time. Mostly.

Someone with a longer memory than my own is Jim Peebles. I happened to bump into him while walking across campus while in Princeton for a meeting in early 2019. (He was finally awarded a Nobel prize later that year; it should have been in association with the original discovery of the CMB). On that occasion, he (unprompted) noted an analogy between the negative attitude towards the cosmological constant that was prevalent in the community pre-1990s to that for MOND now. NOT that he was in any way endorsing MOND; he was just noting that the sociology had the same texture, and could conceivably change on a similar timescale.

***Note that I am not dismissing the Planck results or any other data; I am suggesting the opposite: the data have become so good that it is impossible to continue to approximate UT with tweaks to FLRW (hence “new physics”). I’m additionally pointing out that important new physics has been staring us in the face for a long time.

Early Galaxy Formation and the Hubble Constant Tension

Early Galaxy Formation and the Hubble Constant Tension

Cosmology is challenged at present by two apparently unrelated problems: the apparent formation of large galaxies at unexpectedly high redshift observed by JWST, and the tension between the value of the Hubble constant obtained by traditional methods and that found in multi-parameter fits to the acoustic power spectrum of the cosmic microwave background (CMB).

Maybe they’re not unrelated?

The Hubble Tension

Early results in precision cosmology from WMAP obtained estimates of the Hubble constant h = 0.73 ± 0.03 [I adopt the convention h = H0/(100 km s-1 Mpc-1) so as not to have to have to write the units every time.] This was in good agreement with contemporaneous local estimates from the Hubble Space Telescope Key Project to Measure the Hubble Constant: h = 0.72 ± 0.08. This is what Hubble was built to do. It did it, and the vast majority of us were satisfied* at the time that it had succeeded in doing so.

Since that time, a tension has emerged as accuracy has improved. Precise local measures** give h = 0.73 ± 0.01 while fits to the Planck CMB data give h = 0.6736 ± 0.0054. This is around the 5 sigma threshold for believing there is a real difference. Our own results exclude h < 0.705 at 95% confidence. A value as low as 67 is right out.

Given the history of the distance scale, it is tempting to suppose that local measures are at fault. This seems to be the prevailing presumption, and it is just a matter of figuring out what went wrong this time. Of course, things can go wrong with the CMB too, so this way of thinking raises the ever-present danger of confirmation bias, ever a scourge in cosmology. Looking at the history of H0 determinations, it is not local estimates of H0 but rather those from CMB fits that have diverged from the concordance region.

The cosmic mass density parameter and Hubble constant. These covary in CMB fits along the line Ωmh3 = 0.09633 ± 0.00029 (red). Also shown are best-fit values from CMB experiments over time, as labeled (WMAP3 is the earliest shown; Planck2018 the most recent). These all fall along the line of constant Ωmh3, but have diverged over time from concordance with local data. There are many examples of local constraints; for illustration I show examples from Cole et al. (2005), Mohayaee & Tully (2005), Tully et al. (2016), and Riess et al. (2001). The divergence has occurred as finer angular scales have been observed in the CMB power spectrum and correspondingly higher multiples ℓ have been incorporated into fits.


The divergence between local and CMB-determined H0 has occurred as finer angular scales have been observed in the CMB power spectrum and correspondingly higher multiples ℓ have been incorporated into fits. That suggests that the issue resides in the high-ℓ part of the CMB data*** rather than in some systematic in the local determinations. Indeed, if one restricts the analysis of the Planck (“TT”) data to ℓ < 801, one obtains h = 0.70 ± 0.02 (see their Fig. 22), consistent with earlier CMB estimates as well as with local ones.

Photons must traverse the entire universe to reach us from the surface of last scattering. Along the way, they are subject to 21 cm absorption by neutral hydrogen, Thomson scattering by free electrons after reionization, blue and redshifting from traversing gravitational potentials in an expanding universe (the late ISW effect, aka the Rees-Sciama effect), and deflection by gravitational lensing. Lensing is a subtle effect that blurs the surface of last scattering and adds a source of fluctuations not intrinsic to it. The amount of lensing can be calculated from the growth rate of structure; anomalously fast galaxy formation would induce extra power at high ℓ.

Early Galaxy Formation

JWST observations evince the early emergence of massive galaxies at z ≈ 10. This came as a great surprise theoretically, but the empirical result extends previous observations that galaxies grew too big too fast. Taking the data at face value, more structure appears to exist in the early universe than anticipated in the standard calculation. This would cause excess lensing and an anomalous source of power on fine scales. This would be a real, physical anomaly (new physics), not some mistake in the processing of CMB data (which may of course happen, just as with any other sort of data). Here are the Planck data:

Unbinned Planck data with the best-fit power spectrum (red line) and a model (blue line) with h=0.73 and Ωm adjusted to maintain constant Ωmh3. The ratio of the models is shown at bottom, that with = 0.67 divided by the model with h = 0.73. The difference is real; h = 0.67 gives the better fit****. The ratio illustrates the subtle need for slightly greater power with increasing ℓ than provided by the model with h = 0.73. Perhaps this high-ℓ power has a contribution from anomalous gravitational lensing that skews the fit and drives the Hubble tension.

If excess lensing by early massive galaxies occurs but goes unrecognized, fits to the CMB data would be subtly skewed. There would be more power at high ℓ than there should be. Fitting this extra power would drive up Ωm and other relevant parameters*****. In response, it would be necessary to reduce h to maintain a constant Ωmh3. This would explain the temporal evolution of the best fit values, so I posit that this effect may be driving the Hubble tension.

The early formation of massive galaxies would represent a real, physical anomaly. This is unexpected in ΛCDM but not unanticipated. Sanders (1998) explicitly predicted the formation of massive galaxies by z = 10. Excess gravitational lensing by these early galaxies is a natural consequence of his prediction. Other things follow as well: early reionization, an enhanced ISW/Rees-Sciama effect, and high redshift 21 cm absorption. In short, everything that is puzzling about the early universe from the ΛCDM perspective was anticipated and often explicitly predicted in advance.

The new physics driving the prediction of Sanders (1998) is MOND. This is the same driver of anomalies in galaxy dynamics, and perhaps now also of the Hubble tension. These predictive successes must be telling us something, and highlight the need for a deeper theory. Whether this finally breaks ΛCDM or we find yet another unsatisfactory out is up to others to decide.

*Indeed, the ± 0.08 rather undersells the accuracy of the result. I quote that because the Key Project team gave it as their bottom line. However, if you read the paper, you see statements like h = 0.71 ± 0.02 (random) ± 0.06 (systematic). The first is the statistical error of the experiment, while the latter is an estimate of how badly it might go wrong (e.g., susceptibility to a recalibration of the Cepheid scale). With the benefit of hindsight, we can say now that the Cepheid calibration has not changed that much: they did indeed get it right to something more like ± 0.02 than ± 0.08.

**An intermediate value is given by Freedman (2021): h = 0.698 ± 0.006, which gives the appearance of a tension between Cepheid and TRGB calibrations. However, no such tension is seen between Cepheid and TRGB calibrators of the baryonic Tully-Fisher relation, which gives h = 0.751 ± 0.023. This suggests that the tension is not between the Cepheid and TRGB method so much as it is between applications of the TRGB method by different groups.

***I recall being at a conference when the Planck data were fresh where people were visibly puzzled at the divergence of their fit from the local concordance region. It was obvious to everyone that this had come about when the high ℓ data were incorporated. We had no idea why, and people were reluctant to contradict the Authority of the CMB fit, but it didn’t sit right. Since that time, the Planck result has been normalized to the point where I hear its specific determination of cosmic parameters used interchangeably with ΛCDM. And indeed, the best fit is best for good reason; determinations that are in conflict with Planck are either wrong or indicate new physics.

****The sharp eye will also notice a slight offset in the absolute scale. This is fungible with the optical depth due to reionization, which acts as a light fog covering the whole sky: higher optical depth τ depresses the observed amplitude of the CMB. The need to fit the absolute scale as well as the tip in the shape of the power spectrum would explain another temporal evolution in the best-fit CMB parameters, that of declining optical depth from WMAP and early (2013) Planck (τ = 0.09) to 2018 Planck (τ = 0.0544).

*****The amplitude of the power spectrum σ8 would also be affected. Perhaps unsurprisingly, there is also a tension between local and CMB determinations of this parameter. All parameters must be fit simultaneously, so how it comes out in the wash depends on the details of the history of the nonlinear growth of structure. Such a calculation is beyond the scope of this note. Indeed, I hope someone else takes up the challenge, as I tire of solving all the problems only to have them ignored. Better if everyone else comes to grip with this for themselves.