Has Planck closed the window on the Early Universe?

A combination of circumstances – including being a bit poorly – has made me rather late in getting around to reading the papers released by the Planck consortium a couple of weeks ago. I’ve had a bit of time this Sunday so I decided to have a look. Naturally I went straight for, er, paper No. 24, which you can find on the arXiv, here.

I picked this one to start with because it’s about primordial non-Gaussianity. This is an important topic because the simplest theories of cosmological inflation predict the generation of small-amplitude irregularities in the early Universe that form a statistically homogeneous and isotropic Gaussian random field. This means that the perturbations (usually defined in terms of departures of the metric from a pure Robertson-Walker form) are defined by probability distributions which are invariant under translations and rotations in 3D space.

In a nutshell, such perturbations arise quite simply in inflationary cosmology as zero-point oscillations of a scalar quantum field, in a very similar way the Gaussian distributions that arise from the quantized harmonic oscillator. Assuming the fluctuations are small in amplitude the scalar field evolves according to

which is similar to that describing a ball rolling down a potential , under the action of a force given by the derivative , opposed by a “frictional” force depending on the ball’s speed; in the inflationary context the frictional force depends on the expansion rate . If the slope of the potential is relatively shallow then there is a slow-rolling regime during which the kinetic energy of the field is negligible compared to its potential energy; the term in then becomes negligible in the above equation. The universe then enters a near-exponential phase of expansion, during which the small Gaussian quantum fluctuations in become Gaussian classical metric perturbations.

On the one hand, Gaussian fluctuations are great for a theorist because so many of their statistical properties can be calculated analytically: I played around a lot with them in my PhD thesis many moons ago, long before Planck, in fact long before any fluctuations in the cosmic microwave background were measured at all! The problem is that if we keep finding that everything is consistent with the Gaussian hypothesis then we have problems.

The point about this slow-rolling regime is that it is an attractor solution that resembles the physical description of a body falling through the air: eventually such a body reaches a terminal velocity defined by the balance between gravity and air resistance, but independent of how high and how fast it started. The problem is that if you want to know where a body moving at terminal velocity started falling from, you’re stumped (unless you have other evidence). All dynamical memory of the initial conditions is lost when you reach the attractor solution. The problem for early Universe cosmologists is similar. If everything we measure is consistent with having been generated during a simple slow-rolling inflationary regime, then there is no way of recovering any information about what happened beforehand because nothing we can observe remembers it. The early Universe will remain a closed book forever.

So what does all this have to do with Planck? Well, one of the most important things that the Planck collaboration has been looking for is evidence of non-Gaussianity that could be indicative of primordial physics more complicated than that included in the simplest inflationary models (e.g.  multiple scalar fields, more complicated dynamics, etc).  Departures from the standard model might just keep the window on the early Universe open.

A simple way of defining a parameter that describes the level of non-Gaussianity is as follows:

the parameter describes a quadratic contribution to the overall metric perturbation : you can think of this as being like a power series expansion of the total fluctuation in terms of a Gaussian component ; the term in angle brackets is just there to ensure the whole thing averages to zero. This definition of non-Gaussianity is not the only one possible, but it’s the simplest and it’s the one for which Planck has produced the most dramatic result:

which is clearly consistent with zero. If this doesn’t look impressive, bear in mind that the typical fluctuation in the metric inferred from cosmological measurements is of order . The quadratic terms are therefore of order , so the upper limit on the level of non-Gaussianity allowed by Planck really is minuscule. This is one of the reasons why some people have described the best-fitting model emerging from Planck as the Maximally Boring Universe…

So it looks like only very unwise investors will be buying shares in cosmological non-Gaussianity at least in the short-term. More fundamentally we may be approaching the limit of what we can learn about inflation in particular, or even the early Universe in general, using the traditional techniques of observational cosmology. But there remain very intriguing questions that may yet shed light on the pre-inflationary epoch. Among these are the large-scale anomalies seen in the very same Planck data that have put such stringent limits on non-Gaussianity. But that question, described in Planck Paper 23, will have to wait for another day…

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