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Decoding the near-Planckian spectrum from CMB patterns

In March 2015, physics stars Juan Maldacena and Nima Arkani-Hamed (IAS Princeton) wrote their first paper together:

Cosmological Collider Physics
At that time, I didn't discuss it because it looked a bit technical for the blogosphere but things look a bit different now, partially because some semi-popular news outlets discussed it.



Cliff Moore's camera, Seiberg, Maldacena, Witten, Arkani-Hamed, sorted by distance

What they propose is to pretty much reverse-engineer very fine irregularities in the Cosmic Microwave Background – the non-Gaussianities – and decode them according to their high-tech method and write down the spectrum of elementary particles that are very heavy (comparably heavy to the Hubble scale during inflation) which may include Kaluza-Klein modes or excited strings.




Numerous people have said that "the Universe is a wonderful and powerful particle collider" because it allows us to study particle physics phenomena at very high energies – by looking into the telescope (because the expansion of the Universe has stretched these tiny length scales of particle physics to cosmological length scales). But Juan and Nima went further – approximately by 62 pages further. ;-)




What are the non-Gaussianities? If I simplify a little bit, we may measure the temperature of the cosmic microwave background in different directions. The temperature is about\[

T = 2.7255\pm 0.0006 \,{\rm K}

\] and the microwave leftover (remnant: we call it the Relict Radiation in Czech) from the Big Bang looks like a thermal, black-body radiation emitted by an object whose temperature is this \(T\). Such an object isn't exactly hot – it's about minus 270 Celsius degrees – but the absolute temperature \(T\) is nonzero, nevertheless, so the object thermally radiates. The typical frequency is in the microwave range – the kind of waves from your microwave oven. (And 1% of the noise on a classical TV comes from the CMB.) The object – the whole Universe – used to be much hotter but it has calmed down as it expanded, along with all the wavelengths.

The Universe was in the state of (near) thermal equilibrium throughout much of its early history. Up to the year 380,000 after the cosmic Christ (the Big Bang: what the cosmic non-Christians did with their Christ at that moment was so stunning that it still leaves cosmologists speechless) or so, the temperature was so high that the atoms were constantly ionized.

Only when the temperature gradually dropped beneath a critical temperature for atomic physics, it became a good idea for electrons to sit down in the orbits and help to create the atoms. Unlike the free particles in the plasma, atoms are electrically neutral, and therefore interact with the electromagnetic field much more weakly (at least with the modes of the electromagnetic field that have low enough frequencies).

OK, around 380,000 AD, the Universe became almost transparent for electromagnetic waves and light. The light – that was in equilibrium at that time – started to propagate freely. Its spectrum was the black-body curve and the only thing that has changed since that time was the simple cooling i.e. reduction of the frequency (by a factor) and the reduction of the intensity (by a similarly simple factor).



The CMB is the most accurate natural thermal black body radiation (the best Planck curve) we know in Nature. However, when we look at the CMB radiation more carefully, we see that the temperature isn't quite constant. It varies by 0.001% or 0.01% in different directions:\[

T(\theta,\phi) = 2.725\,{\rm K} + \Delta T (\theta,\phi)

\] The function \(\Delta T\) arises from some direction-dependent "delay" of the arrival of business-as-usual after the inflationary era. If the inflaton stabilized a bit later, we get a slightly higher (or lower?) temperature in that direction – which was also associated with a little bit different energy density in that region (region in some direction away from us, at the right distance so that the light it sent at 380,000 AD just hit our telescopes today).

The function \(\Delta T\) depends on the spherical angles and may be expanded in the spherical harmonics. To study the magnitude of the temperature fluctuations, you want to measure things like \[

\sum_{m=-\ell}^\ell\Delta T_{\ell, m} \Delta T_{\ell', m'}

\] The different spherical harmonic coefficients \(\Delta T_{\ell m}\) are basically uncorrelated with one another, so you expect to get close to zero, up to noise, unless \(\ell=\ell'\) and \(m=-m'\). In that case, you get a nonzero result and it's a function of \(\ell\) that you know as the "CMB power spectrum".



I wrote \(\Delta T\) as a function of the angles or the quantum numbers \(\ell,m\) but in the early cosmology, it's more natural to derive this \(\Delta T\) from the inflaton field and appreciate that this field is a function of \(\vec k\), the momentum 3-vector. By looking at the \(\Delta T\), we may only determine the dependence on two variables in a slice, not three.

At any rate, the correlation functions\[

\langle \Delta T (\vec k_1) \Delta T (\vec k_2) \Delta T(\vec k_3) \rangle

\] averaged over all directions etc. seem to be zero according to our best observations so far. No non-Gaussianities have been observed. Again, why non-Gaussianities? Because the probability density for the \(\Delta T\) function to be something is given by the Ansatz\[

\rho[\Delta T(\theta,\phi)] = \exp \zav{ - \Delta T\cdot M \cdot \Delta T }

\] where \(M\) is a "matrix" that depends on two continuous indices – that take values on the two-sphere – and the dot product involves an integral over the two-sphere instead of a discrete summation over the indices. Fine. You see that probability density functional generalizes the function \(\exp(-x^2)\), a favorite function of Carl Friedrich Gauß, which is why this Ansatz is referred to as the Gaussian one.

The probability distribution is mathematically analogous to the wave function or functional of the multi-dimensional or infinite-dimensional harmonic oscillator or the wave functional for the ground state of a free (non-interacting, quadratic) quantum field theory (which is an infinite-dimensional harmonic oscillator, anyway). Or the integrand of the path integral in a free quantum field theory.

And this mathematical analogy may be exploited to calculate lots of things. In fact, it's not just a mathematical analogy. Within the inflationary framework, the \(n\)-point functions calculated from the CMB temperature are \(n\)-point functions of the inflaton field in a quantum field theory.

The \(n\) in the \(n\)-point function counts how many points on the two-sphere, or how many three-vectors \(\vec k\), the correlation function depends on. The correlation functions in QFT may be computed using the Feynman diagrams. In free QFTs, you have no vertices and just connect \(n\) external propagators. It's clear that in a free QFT, the 3-point functions vanish. All the odd-point functions vanish, in fact. And the 4-point and other even higher-point functions may be computed by Wick's theorem – the summation over different pairings of the propagator.

Back to 2015

No non-Gaussianities have been seen so far – all observations are compatible with the assumption that the probability density functional for \(\Delta T\) has the simple Gaussian form, a straightforward infinite-dimensional generalization of the normal distribution \(\exp(-x^2/2\sigma^2)\). However, cosmologists and cosmoparticle physicists have dreamed about the possible discovery of non-Gaussianities and what it could teach us.

It could be a signal of some inflaton (cubic or more complex) self-interactions, new particles, new effects of many kinds. But which of them? Almost all previous physicists wanted to barely see "one new physical effect around the corner" that is stored in the first non-Gaussianities that someone may discover.

Only Nima and Juan started to think big. Even though no one has seen any non-Gaussianity yet, they are already establishing a new computational industry to get tons of detailed information from lots of numbers describing the non-Gaussianity that will be observed sometime in the future. They don't want to discover just "one" new effect that modestly generalizes inflation by one step, like most other model builders.

They ambitiously intend to extract all the information about the particle spectrum and particle interactions (including all hypothetical new particle species) from the correlation functions of \(\Delta T\) and its detailed non-Gaussianities once they become available. Their theoretical calculations were the hardest step, of course. The other steps are easy. Once Yuri Milner finds the extraterrestrial aliens, he, Nima, and Juan will convince them to fund a project to measure the non-Gaussianities really accurately, assuming that the ETs are even richer than the Chinese.

OK, once it's done, you will have functions like\[

\langle \Delta T(\vec k_1) \,\Delta T (\vec k_2) \,\Delta T(\vec k_3) \rangle

\] By the translational symmetry (or momentum conservation), this three-point function is only nonzero for \[

\vec k_1+\vec k_2+ \vec k_3 = 0

\] which means if and only if the three vectors define sides of a triangle (oriented, in a closed loop). The three-point functions seem to be zero according to the observations so far. But once they will be seen to be nonzero, the value may be theoretically calculated as the effect of extra particle species (or the same inflaton, if it is self-interacting).

A new field of spin \(s\) and mass \(m\) will contribute a function of \(\vec k_1, \vec k_2,\vec k_3\) to the three-point function – a function of the size and shape of the triangle – whose dependence on the shape stores the information about \(s\) and \(m\). When you focus on triangles that are very "thin", they argue and show, the mass of the particle (naturally expressed in the units of the Hubble radius, if you wish) is stored in the exponent of a power law that says how much the correlation function drops (or increases?) when the triangle becomes even thinner.

Some dependence on the spin \(s\) is imprinted to the dependence on some angle defining the triangle.

And all the new particles' contributions add up. In fact, they "interfere" with each other and the relative phase has observable implications, too.

It's a big new calculational framework, basically mimicking the map between "Lagrangians of a QFT" and "its \(n\)-point functions" in a different context. They look at three-point functions as well as four-point functions. The contributions to these correlation functions seem to resemble correlation functions we know from the world sheet of string theory.

And they also show how these expressions have to simplify when the system is conformally (or slightly broken conformally or de Sitter) symmetric. Theirs is a very sophisticated toolkit that may serve as a dictionary between the patterns in the CMB and the particle spectrum and interactions near the inflationary Hubble scale.

Testability

I was encouraged to write this blog post by this text in the Symmetry Magazine,
Looking for strings inside inflation
Troy Rummler wrote about a very interesting topic and he has included some useful and poetic remarks. For example, Edward Witten called Juan's and Nima's work "the most innovative one" he heard about at Strings 2015. Juan's slides are here and the 29-minute YouTube talk is here. And Witten has also said that science doesn't repeat itself but it "rhymes" because Nature's clever tricks are recycled at many levels.

Well, I still feel some dissatisfaction with that article.

First, it doesn't really make it clear that Arkani-Hamed, Maldacena, and Witten are not just three of the random physicists or even would-be physicists that are the heroes of most of the hype in the popular science news outlets. All of them are undoubtedly among the top ten physicists who live on Earth right now.

Second, I just hate this usual post-2006 framing of the story in terms of the slogan that "string theory would be nearly untestable which is why all the theoretical physicists have to work hard on doable tests of string theory".

What's wrong with that slogan in the present context?
  1. String theory is testable in principle, it has been known to be testable for decades, and that's what matters for its being a 100% sensible topic of deep scientific research.
  2. String theory seems hard to test by realistic experiments that will be performed in several years and almost all sane people have always thought so already when they started to work on strings.
  3. The work by Arkani-Hamed and Maldacena hasn't changed that: it will surely take a lot of time to observe non-Gaussianities and observe them accurately enough for their dictionary and technology to become truly relevant. So even though theirs is a method to look into esoteric short-distance physics via ordinary telescopes, it's still a very futuristic project.
  4. The Nima-Juan work doesn't depend on detailed features of string theory much.
It is meant to recover the spectrum and interactions of an effective field theory at the Hubble scale, whether this effective field theory is an approximation to string theory or not. In fact, the term "string" appears in one paragraph of their paper only (in the introduction).

The paragraph talks about some characteristic particles predicted by string theory whose discovery (through the CMB) could "almost settle" string theory. For example, they believe that a weakly coupled (but not decoupled) spin-4 particle would make string theory unavoidable because non-string theories are incompatible with weakly coupled particles of spin \(s\gt 2\). This is a part of the lore, somewhat ambitious lore. I think it's morally correct but it only applies to "elementary" particles and the definition of "elementary" is guaranteed to become problematic as we approach the Planck scale. For example, the lightest black hole microstates – the heavier cousins of elementary particles but with Planckian masses – are guaranteed to be "in between" composite and elementary objects. Quantum gravity provides us with "bootstrap" constraints that basically say that the high-mass behavior of the spectrum must be a reshuffling of the low-mass spectrum (UV-IR correspondence, something that is seen both in perturbative string theory as well as the quantum physics of black holes).

The scale of inflation is almost certainly "at least several orders of magnitude" beneath the Planck scale so this problem may be absent in their picture. But maybe it's not absent. Theorists want to be sure that they have the right wisdom about all these things – but truth to be told, we haven't seen a spin-4 particle in the CMB yet. ;-)

It's a very interesting piece of work that is almost guaranteed to remain in the domain of theorists for a very long time. And it's unfortunate that the media – including "media published by professional institutions such as the Fermilab and SLAC" – keep on repeating this ideology that the theorists are "obliged" to work on practical tests of theories and they are surely doing so. They are not "obliged" and they are mostly not doing these things.

The Planckian physics has always seemed to be far from practically doable experiments. The Juan-Nima paper is an example of the efforts that have the chance to reduce this distance. But I think that they would agree that this distance remains extremely large and the small chance that the distance will shrink down to zero isn't the only and necessary motivation of their research. Theorists just want to know – they are immensely curious about – the relationships between pairs of groups of ideas and data even if both sides of the link remain unobservable in practice!



The most likely shape of a newborn galaxy, extracted from the spectrum of our Calabi-Yau compactification through the Nima-Juan algorithm

I am a bit confused about the actual chances that the sufficient number of non-Gaussianities and their patterns may ever be extracted from the CMB data. The low enough \(\ell\) modes of the CMB simply seem to be Gaussian and we won't get any new numbers or new patterns stored in them, will we? The cosmic variance – the unavoidable noise resulting from the "finiteness of the two-sphere and or the visible Universe" i.e. from the finite number of the relevant spherical harmonics – seems to constrain the accuracy of the data we may extract "permanently". So maybe such patterns could be encoded in the very high values of \(\ell\) i.e. small angular distances on the sky?

For example, there could be patterns in the shape of the galaxies (inside the galaxies), and not just the broad intergalactic space. For example, if it turned out that the most likely shape of the newborn galaxy is given by the blue picture above (one gets the spiraling mess resembling the Milky Way once the shape evolves for a long enough time), it could prove that God and His preferred compactification of string/M-theory is neither Argentinian nor Persian. I am not quite sure whether such a discovery would please Witten, however. ;-)

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