Select Questions-Answers From All Experts Astronomy Forum (1- General Relativity & Mercury's Perihelion Advance)

This series is intended to revisit some of the deeper, more thoughtful questions directed at me during my time serving on All Experts.   Each question will be given, followed by the answer.

Question: I am puzzled by Einstein’s conclusion that Mercury’s advance of perihelion is 43 arc seconds per year. How can that be obtained from the equation he used in The Principle of Relativity?

Diagram showing (exaggerated) advance of Mercury's perihelion.

 Ans.  What Einstein is really concerned with here is the amount of rotation of 'the planetary ellipse' due to the effects of gravitation in general relativity.  In this case the ellipse is that of the planet Mercury.

The equation used and purported to show the amount of this rotation (i.e. advance of perihelion) is given as:

e = [24 (p)³ a²]/ T² c² [1 - e²]


where e is the advance (or rotation) in seconds of arc, T is the period of revolution in seconds, c the velocity of light and e the eccentricity of the orbit.

Einstein, on page 164, states that for Mercury e= 43 seconds per century

The speed of light c = 3 x 10¹⁰ cm/s in the text.

This means if one uses cm/s all other units must be consistent.

In this context, 1 astronomical unit (AU) = 1.5 x 10¹³ cm. But from a Table of distances, Mercury's semi-major axis = 0.387 AU or:

a = (1.5 x 10¹³  cm) (0.387) = 5.8 x 10¹² cm

The period, T (in seconds) is just the length of Mercury's year (in days = 87.96, again from a Table) multiplied by the seconds-length of an Earth day, or 86,400 s, hence::

T = 7.6 x 10⁶ s

The eccentricity, e from a similar Table is e = 0.205.

Substituting all these values into the given equation yields:

e = 5.036 x 10^-7  radian

To get the equivalent seconds of arc (or arcsec) we use 1 rad (radian) = 57.3 degree where  one degree has 3600 seconds. Thus, 1 radian will have:

2.063 x 10⁵ arcsec

So, the associated arcsec for e will be:

(5.036 x 10^{- 7}   rad) x (2.063 x 10⁵ arcsec/ rad) =

0.104 arcsec

We are still not finished because the quantity is defined per CENTURY

At this point, you need to recall the PERIOD of Mercury is 0.2405 YRS.

So, the number of arcsec of perihelion advance per Earth years is:

0.104 arcsec/ 0.2405 years =  0.432

and over 100 years:

eta = 100 yr x (0.432 arcsec/ yr)   ~  43.2 arcsec or near to what Einstein noted on the page.

Miracles and Applying the Hume Test : Hint - Accept the Improbable Over the Impossible!

For the ardent supernaturalist, Philosopher David Hume’s test for accepting miracles often proves a “bridge too far.” According to Hume:

"No testimony is sufficient to establish a miracle, unless the testimony be of such a kind that its falsehood would be more miraculous than the fact which it endeavours to establish."

As one example of its application, evolutionary biologist Richard Dawkins, in his book Unweaving the Rainbow, chose the Fatima miracle of 1917, where 70,000 people reportedly saw the sun move. As Dawkins observed:

On the one hand, we are asked to believe in a mass hallucination, a trick of the light, or mass lie involving 70,000 people. This is admittedly improbable. But it is LESS improbable than the alternative: that the sun really did move...If the sun had moved in truth, but the event was seen only by the people of Fatima, an even greater miracle would have been perpetrated: an illusion of NON-movement had to be staged for all the millions of witnesses not in Fatima."

As an additional example, consider the miracle claim of Jesus “walking on water”. Prof. Hugh Schonfeld has a simple explanation for this: a mistranslation of the Hebrew word “al” which can mean “by” or “on”. So, when some scribe originally wrote: “walking by the water” it was later mistranslated as “walking on the water”, on account of the ambiguous meaning of the word 'al'.

Applying the Hume test: Is the Schonfeld claim of mistranslation MORE or LESS miraculous than a man actually violating the law of gravity and walking on water? It doesn’t require a lot of thought or effort to see that the supposed mistranslation of a passage of the New Testament is LESS miraculous (or if you prefer, less improbable) than that a man actually, literally walked on water.

I bring this up to show that ascertaining whether a “miracle” occurred in such and such an event is not that difficult provided one adheres to a consistent principle. This came to the forefront recently in an encounter with a person (on the All Experts Atheist-Agnostic forum) who continually pressed me with assorted questions to accept that his odd citations of events, including: a baby found alive 11 days after an earthquake in Haiti, and another person surviving a plane crash, constituted “miracles”. As I pointed out to him or her:

Note the secular definition of "miracle" is:

“A claimed or perceived pathological event that overturns one or more laws of nature or physical principles.”

So any miracle report must be based on this,

Observe that none of your cited examples, or "incidents" do so, because none of them are technically in violation of a scientific law. A baby emerging from the rubble of an earthquake after a day or two (I've never seen any reports for 10 days - but I'd be interested to hear the exact event and source) may be improbable but it isn't scientifically impossible. Neither is surviving a plane crash when everyone else dies. The survivor may merely have had the good luck to be seated in the best place on the plane to survive!

So, no, none of the cited examples is truly an "extraordinary event" - say that overturns a law of nature- they are merely rare events, with improbable (but not impossible) outcomes,

Not satisfied with this, the respondent went on to cite an actual case of a Haitian infant surviving 11 days after the quake, and two incidents involving survivors of air crashes. S/he then asked: “Can you scientifically prove that a human can survive after falling from thousands of feet height. Also, scientists are convinced that they have understood less than 5% of physical laws governing the universe and the rest is still dark with concepts such as dark energy, dark matter and so on. How can we say with confidence that certain events/things are against physical laws which assumes that we have fully understood complete set of physical laws governing the universe which is wrong..

To which I replied the infant surviving 11 days after a quake still didn't qualify it for "sainthood" (given unknown plausible mitigating natural circumstances, i.e. hidden air flows, dripping water pipes, etc.) and I cited Milton Rothman’s ‘A Physicist’s Guide to Skepticism’ – Chapter Five, 'Laws of Permission and Laws of Denial'. Here, Rothman does an excellent job of laying out – based on existing physical laws (from which one need not extraordinarily extrapolate)- what constitutes impossible events and what does not. Also, merely because something “hasn’t happened yet” doesn’t mean it will or must at any future time. Once the basic laws are locked into place, and they are, those options are out. Above all, extraordinary claims demand extraordinary evidence at the time, not hypotheticals or unconfirmed reports, anecdotes.

I pointed out that all impossibility statements actually have far more generality and validity than possibility statements. But one must understand what this means. In science the actual, technical terms are "laws of permission" and "laws of denial". The first details all the actions that are allowed based on the existing laws of physics; the latter details all the actions disallowed.

According to Rothman (p. 129):

"The function of the laws of permission is to predict how objects will move through space or how a system will change with time under known forces. These laws are known with great precision. We can calculate with exquisite exactitude how a single charged particle will move in a known electromagnetic field. We know just how a pendulum will oscillate in a known gravitational field."

It also means that given a person falling freely with the acceleration of g = 9.8 m/s^2, and over some distance (height h) what the time will take him to fall, and the impacting resultant force when he lands. Also the extent to which that will break bones.

But one had to also note his cautionary point (p. 130) that: "it is usually impossible to make precise predictions of exactly what will happen in any real situation". This isn't because the laws are "wrong" but because extraneous, or previously unaccounted for random factors can enter including greater air resistance, say from an unforeseen updraft, or variable winds, or whatever.

Hence, a human body constructed as it is from fragile skeletal structure and organs that can rupture cannot normally fall freely thousands of feet and survive. That is, unless unusual mitigating circumstances are present, say that somehow the fall braked or the person lands on a massive hay stack or whatever.

I then went on to point out that the existence of dark energy doesn't mean a physical law has been violated, or that one emerged which we didn't already know. Hence, can't be placed in the same category as exceptional human events, or claimed "miracles". Dark energy is indeed an extrapolation of Einstein's general relativity law.

Still unsatisfied with my answers, the person provided more links to astounding cases of survivors purportedly dropping “from thousands of feet” up and living, while also dragging in other issues, such as when askling:

“Can't we assume that event which has a remote probability of occurring, say 1 in 100 million chances as miracle. How could a random big bang (disorderly event) lead to a orderly universe. Why are not there many meteorites, asteroid attacking our earth everyday if the universe evolved by chance? So many miracles I see everyday, how could I see with my eyes, how internal organs built in my body, external world and many things. I am surprised you don't consider any of these as miracles!”

My reply, of course, didn’t diverge as I disclaimed the further examples, noted that even in the source skepticism was expressed, i.e. in the Chinese case, “further investigation after the fall of communism has cast doubt on the official story”. In the other case, the survivor suffered a fractured skull, a broken leg and collarbone and third-degree burns covering much of her body so she didn't end up unscathed. The point was that neither example trumped the use of the Hume test – to accept an improbable natural event over a supernatural “miracle”. I then rebutted the other arguments or issues raised:

- I did not "assume" objects cannot travel faster than light, only pointed out the existing evidence - including Einstein's special relativity - shows they do not. Many scientists being "hopeful" ( you don't say who) doesn't mean diddly. Hopeful doesn't cut it in terms of hard evidence. Show me the compelling data (and also if it's independently confirmed), then I might accept it.

- Assuming Weinberg did say what you claimed (you again give no source) that doesn't mean he's in favor of miracles. Obviously, on account of ever iimproving methods, technology 'gaps' can exist in assorted scientific domains, but that doesn't mean "anything goes". I will still be fairly sure (99.999999%) that my car won't stop working one day on gasoline and will use water instead.

- No, we can't "answer everything by science" (yet) but that doesn't mean such answers are perpetually foreclosed. It means we patiently allow the scientific process in any given area to play out before we try to trump it by balderdash or unsubstantiated nonsense.

- NO, we can't assume an event which has a "remote probability of occurring, say 1 in 100 million" is a "miracle". As per my previous extensive response. We adopt the HUME Miracle criterion and take an improbable natural cause before any supernatural one. If the supernaturalist doesn't like that, too bad. He needs then to come up with a coherent supernatural "theory" to account for what occurred. Naturalists- Materialists are under no obligation to do so.

- WHO says we have an "orderly" universe?

In cosmological terms, the whole concept of "order" has been relegated to a minor and tiny niche of the extant cosmos. For example, the recent balloon-borne Boomerang and MAXIMA UV measurements to do with Type Ia supernovae, have disclosed a cosmic content:

7% - ordinary visible matter

93% - dark component, of which:

- 70% is DARK (vacuum) energy and

- 23% is dark matter

In effect, 93% of the universe can’t even be assessed for “order” since it isn’t visible. In the case of dark matter, one can only discern its presence indirectly by the visible effects on neighboring matter. In the case of dark energy, the underlying physical basis isn’t even known – though we know the result is an increase in the acceleration of the universe (arising from a cosmic repulsion attributed to dark energy.

The actual "orderly" systems to which you refer (gravitationally bound clusters, galaxies etc.) are really only a tiny percent (0.001) of the 7% ordinary matter total, the rest is undifferentiated plasma. Also these systems' formation can be accounted for by ordinary physical processes, i.e. aggregation under the influence of electrostatic fores (binding dust particles together initially) then the larger scale action of gravity to accrete larger mass into orbiting dynamic systems.

- Many meteorites DO "attack" our Earth every day - up to 10 million kg worth. But most are so tiny they burn up in our atmosphere owing to friction with the atmosphere. Also, Earth hasn't been immune from large meteroid or asteroid strikes. Such as asteroid DA 14 2012, which made a close pass to Earth on Feb. 15 this year. On the same date, an 11 ton (2.2. x 10^ 4 kg) meteoroid exploded over the Russian Urals - caught on video, causing much damage.

-The point is that these major hits -strikes are rare,

-first, because our planet is usually not in the path of any large asteroids or meteroids, and second, because the ones most likely to clobber us are the smaller sort (such as the meteroid that struck Russia) but they only occur once ever hundred years.

Again, there's absolutely no reason to reach for extravagant or unusual explanations.

As per the "internal organs" of humans, only a person that doesn't grasp evolution would accord them any special significance.

In this sense, if you are open minded enough it would help to get Francis Crick’s book: ‘The Astonishing Hypothesis’, (Touchstone Books, 1995) in which he goes into exceptional detail to explain how natural selection forged the eye and the human vision system (Chapter 3, ‘Seeing’ ; Chapter 4, ‘The Psychology of Vision’; Chapter 10, ‘The Primate Visual System – Initial Stages’, Chapter 11, ‘The Visual Cortex of Primates’)

In the end, "miracles" only exist to those who don't fuilly understand the extent to which the natural world and its assorted objects, including people, can already be explained.

Will the persistent questioner finally appreciate and accept that? Highly unlikely, but we will see!

Is Science Resistant to Novelty? Not Really!

In his recent WSJ column 'Mind and Matter' ('Is That Scientific Heretic a Genius or a Loon?'), p.C2, Nov 12-13, Matt Ridley ponders whether "scientific heretics" have been "persecuted" too often for their "radical ideas". He cites a number of examples including Daniel Schechtman who won a Nobel for the discovery of "quasi-crystals" when he was told earlier (by Linus Pauling) there was no such thing, "only quasi-scientists". Then there was Australian Barry Marshall, who ran a gauntlet of sorts when he hypothesized that bacterial infections caused stomach ulcers.

Then there was Charles Darwin, who postulated evolution by natural selection, and Albert Einstein, who proposed that planets orbit suns not because of a direct pulling action at long range (as Newton had proposed) but because a much larger mass object creates a curvature in space-time about which the smaller mass is constrained to move.

Using all these cases, Ridley then suggests that "science seems strangely resistant to novelty". Well, not really! You see, what an outsider or non-scientist may regard as "strangely resistant to novelty" in fact is the demand for rigor in showing that a new hypothesis is justified in effectively replacing or trumping hundreds of years of established science. Because a replacement of paradigm or model is indeed traumatic, overthrowing the life's work of thousands of dedicated minds, it is not taken lightly.

The issue then is not so much resistance to novelty, as insisting that whatever the "novelty" is it first pass a series of basic tests for falsification and also show how its own predictions enhance our understanding of the aspect of nature under scrutiny.

For example, Darwin early made a number of serious missteps before he was able to forge a coherent and correct theory! The problem was that in some cases, Darwin didn’t logically make the necessary connections that his observations actually disclosed, at the time he made them during his Beagle sojourn to the Galapagos.

Indeed, Darwin was so initially imbued with the creationist perspective that he failed to collect one single species of giant tortoise present in the islands. Fortunately, the occurrence of other species - and their recording- ultimately held away with the assistance of a consummate taxonomic researcher: John Gould - an ornithological expert at the London Zoological Society.

Indeed, had Darwin's mind already been "radicalized" to natural selection, and how adaptations operate within its purview, he'd have easily seen that the variety of beaks displayed among the Galapagos finches was directly traceable to the sort of foods they ate, in the respective islands. In point of fact, four of the fourteen finch species fed on seeds (as finches generally do), and another two species consumed fruits, flesh and flowers of cacti. Seven other finch species were primarily insectivorous, while one fed exclusively on leaves. Thus, from his creationist disadvantage point, it wasn't surprising that Darwin was fooled to the extent of believing some of the birds weren't finches at all.

According to Frank J. Sulloway, in his essay, ‘Why Darwin Rejected Intelligent Design’:

“Faced with an absence of critical information to resolve the finches issue, Darwin continued to give a nod to the prevailing creationist assumption that variation within immutable species can lead to new varieties or subspecies that are adapted to local environments”

Sulloway goes on to note that:

“Darwin returned to England, on October 2, 1836. Three months later, he deposited his Beagle collections of birds with John Gould, the ornithological expert at the London Zoological society.”

He goes on to emphasize it was none other than Gould who “immediately realized the extraordinary nature of Darwin’s Galapagos specimens and analyzed and described them first”

In other words, in a real sense, it was Gould who was the original radical thinker here and whose input then radicalized Darwin's!

Based on Gould’s analysis, contained in a full report he published March 1837, Darwin was finally informed that 3 of his 4 mockingbird species were distinct - new to science - and different from all other known mockingbirds. It was at that juncture Darwin found himself confronted by the problem of the origin of species that escaped him while actually in the Galapagos, imbued with his creationist mentality.

It would be no exaggeration to say Darwin was initially stunned by Gould's results. Indeed, if Gould was correct about the mockingbirds it meant that the supposed "barrier" between species had been broken by these birds on a set of isolated islands. Thus, gradual evolution through geographic isolation was the only plausible explanation that fit with the observational record (clarified by Gould, an expert ornithologist)

Darwin was later compelled to write in his Journal of Researches:"Seeing this gradation and diversity of structure in one small, intimately related group of birds, one might really fancy that from an original paucity of birds in this archipelago, one species had been taken and modified for different ends."

My point here is that Charles Darwin, contrary to the mythology circulated by those like Ridley, didn't suddenly arise as some instant maverick - like Zeus from Mount Olympus- but he had to be led in that direction by others first, and then re-process his original data to clarify it for himself, that he in fact observed and doumented what he did!

Then there was Albert Einstein, another poor guy who was allegedly initially impeded by the lethargic starched shirts of academe, to hear Ridley tell it. But not so fast! Einstein couldn't even begin to make sense of his own notions until he was able to incorporate non-Euclidean geometry into the picture as well as apply a comprehensive form of advanced mathematics known as tensor analysis (see attached image of Einstein writing one form of a tensor equation, i.e. R_ik - ½Rg_ik= 0 which can be rewritten, R_ik= 0 for the vacuum state, also known as "Ricci flat"). It was only after Einstein expended vast energy in formulating his curvature-tensor equations that HE was able to understand what it was he had accomplished, and could then make it comprehensible to others - including the basis for empirical tests such as the angle of deflection of a star's light passing near the Sun.

Thereby, Newton's theory of gravitation was more superseded in the context of a host of new uses and tests, as opposed to being "overthrown" by a crazed radical. For example, Newton's theory couldn't account for the bending of starlight near gravitating masses, but Einstein's could. Hence, General Relativity was used in these instances.

Neither could Newton's theory account for the solar oblateness (i.e. difference between polar and equatorial diameters), but Einstein's could to within a very good degree (though the Brans-Dicke theory of gravitation is giving some stiff competition). Finally, Newton's theory could say nothing whatever about gravitational lenses and lensing - now being incorporated into more and more distant, cosmological investigation. Einstein's could account for such lensing on the basis of curvature of space-time near gravitating masses.

Again, we see that the alleged sudden appearance of a "heretic" is nonsense! There is no "sudden appearance" because by the time the final hypothesis or theory was presented, the alleged heretic already had to undergo much difficult inner work and mental testing to make it sensible to himself!

When that doesn't happen, then really absurd embarrassments ensue. For example, Pons and Fleischmann's claim of "cold fusion" some 15 years ago, which was merely the result of them not adequately taking into account systematic errors in their apparatus. No one else was ever able to replicate any actual evolution of heat or energy in a lab at room temperature, under those conditions. Again, this shows the importance of moving very slowly and methodically and testing every step of the way. As the CERN team which claimed superluminal neutrinos already is, and which I predict will uncover extraneous sources of error hitherto unconsidered.

Finally, Ridley brings up another alleged "Heretic" in the person of one Henrik Svensmark. According to Ridley:

"In 1997, he suggested that the Sun's magnetic field affects the Earth's climate by shielding the atmopshere against cosmic rays, which would otherwise create or thicken clouds and thereby cool the surface. So he reasoned, a large part of the natural fluctuations ni the climate over recent millennia might reflect variation in solar activity."

But I already skewered the basis for this nonsense in a previous blog:

http://brane-space.blogspot.com/2010/12/carbon-isotope-ratios-and-climate.html

Which also showed a critical graph (see attached) , bearing a 2000-year record of C14:C12 ratio deviations as been compiled by P.E. Damon ('The Solar Output and Its Variation', The University of Colorado Press, Boulder, 1977) . As solar physicist John Eddy has pointed out:

“The gradual fall from left to right (increasing C14/C12 ratio) is…probably not a solar effect but the result of the known, slow decrease in the strength of the Earth’s magnetic moment.[1] exposing the Earth to ever-increased cosmic ray fluxes and increased radiocarbon production.

The sharp upward spike at the modern end of the curve, representing a marked drop in relative radiocarbon, is generally attributed to anthropogenic causes—the mark of increased population and the Industrial Age"

But evidently the Über-radical Svensmark was in such a heated hurry to establish his heresy he didn't bother to examine the work already done on the solar-cosmic ray- climate nexus by Eddy and Damon!

Ridley then continues his pseudo-heretic promotion:

"Dr. Svensmark is treated as a heretic mainly because his theory is thought to hinder the effort to convince people that recent climate variation is largely manmade, not natural, so there is his bias toward resisting the idea"

Actually, that's largely bollocks and the reason is that one part of his theory ( a slowing magnetic cycle or cooler Sun) has already been proven wrong as the Sun ramps up its activity once again. Meanwhile, Svensmark was one of those who erroneously construed the data in a Nature paper by Dr. Noel Keenlyside et al, i..e. that cooling has been occurring since 1997,. and that doesn't put his statistical acumen in much stead.

In addition, as I showed above, the much longer term studies of Damon and Eddy totally refute Svensmark's hypothesis of a cosmic ray-cloud based natural variant of climate change, in the sense of controverting the anthropogenic warming hypothesis, via CO2 - accepted now by more than 97% of the climate scientists of the planet (Eos Transactions of the American Geophysical Union, Vol. 90, No. 3, p. 22) .

Ridley then refers to the recent CERN (CLOUD) studies on cosmic rays and climate variation, and implies that those will finally bear out the great rebel Svensmark's theories. Even Svensmark appears to agree as he said in one recent online interview:

"I welcome the CLOUD results. They basically confirm our own experimental results since 2006, and does so within a larger variation of parameters. It seems to say that ions are fundamental for the nucleation of new aerosols."

But again, I shot this crap down in another blog:

http://brane-space.blogspot.com/2011/09/cosmic-ray-blarney-will-flat-earther.html

in which I noted, regarding one of the CLOUD team's results :

"Kirkby and his colleagues were nevertheless cautious on the finding as they also reported that the presence of certain atmospheric pollutants, such as H2So4 (which can incept "acid rain"- which appears when water molecules in the atmosphere react with sulphur dioxide or SO2). In that case, seed formation is repressed by up to 1/1000 of those needed to account for cloud seeding. Worse, it's already known that clouds are very poorly parameterized in climate models as a whole. This has led to an ongoing debate over the past 8 years on whether in fact the sign of albedo change is positive or negative. (See e.g. ‘Can Earth’s Albedo and Surface Temperature Increase Together’ in EOS, Vol. 87, No. 4, Jan. 24, 2006, p. 37).

Ridley wages a column bet that based on his work, Svensmark "may yet prove to be a Schechtman".

I'd say, based on what I've seen, it's more likely he'll prove to be a Pons or Fleischmann!

DARK ENERGY – A NEW LAW OF PHYSICS? (II)

In the previous article I stated that the dark energy - vacuum equation of state:

w = (p / rho) = -1

is consistent with Einstein's general theory of relativity - which one could say approaches the status of a 'basic law of physics'.

I now want to delve into more detail. Take the equation that defines cosmic expansion:

R’’/R = - 4pi/ 3 G rho (1 + 3 w)

And if we let w = (-1/3):

the whole right side becomes zero, and

-1/3 = p/rho or -rho = 3p

If we set: 0 = (rho + 3p) then:

p = - rho /3

and if: p < (- rho /3) we have gravity that repels

Looking back to the earlier equation for w, one finds:

p = - rho (e.g. pressure = - energy density)

and - rho < (- p /3)

Specifically the term (rho + 3 p) acts as a source of gravity in general relativity, (where rho = energy density).


In this case, a negative pressure dovetails with general relativity's allowance for a "repulsive gravity" - since any negative pressure has associated with it gravity that repels rather than attracts.

This being the case, we may assert that dark energy represents no "new law" of physics, but rather an extrapolation of an existing one. The core issue that still must be addressed is whether this relationship implies the need for a "cosmological constant" and if so, what magnitude it might be.

This brings us to the question: What’s to become of the cosmos if the acceleration is ongoing?

All this in tandem supports the prediction by many dark energy theorists that the cosmos will ultimately expand forever and yield to an ultimate heat death. All objects in the cosmos will be so far apart that no exchange of energy can occur and they will simply die out. Or, exhaust all heat sources and become cold, dead cinders. There simply isn't any agency to counter the accelerating force of dark energy to prevent it.

Will the discovery of the Higgs particle (boson) by the large hadron collider cause us to reassess this end? Not really! Not unless there is some corelative data that also show there is no basis to posit an accelerating expansion.

In the absence of that, the best tack to try to overturn the dark energy thesis is to try to find an alternative explanation for the two close peaks in the power spectrum - indicating that the plasma is subject to the dark energy equation of state. Thus, the current primacy of the dark energy thesis rests on a particular interpretation for those two power spectrum peaks based on using the Legendre functions.

If anyone has any evidence to the contrary, and can PROVE it (using the same power spectrum and the Legendre functions) points to a collapsing universe instead, then go for it.