Planck 2015 Constraints on Dark Energy and Inflation

The European Space Agency’s Planck satellite gathered data for over 4 years, and a series of 28 papers releasing the results and evaluating constraints on cosmological models have been recently released. In general, the Planck mission’s complete results confirm the canonical cosmological model, known as Lambda Cold Dark Matter, or ΛCDM. In approximate percentage terms the Planck 2015 results indicate 69% dark energy, 26% dark matter, and 5% ordinary matter as the mass-energy components of the universe (see this earlier blog:

Dark Energy

We know that dark energy is the dominant force in the universe, comprising 69% of the total energy content. And it exerts a negative pressure causing the expansion to continuously speed up. The universe is not only expanding, but the expansion is even accelerating! What dark energy is we do not know, but the simplest explanation is that it is the energy of empty space, of the vacuum. Significant departures from this simple model are not supported by observations.

The dark energy equation of state is the relation between the pressure exerted by dark energy and its energy density. Planck satellite measurements are able to constrain the dark energy equation of state significantly. Consistent with earlier measurements of this parameter, which is usually denoted as w, the Planck Consortium has determined that w = -1 to within 4 or 5 percent (95% confidence).

According to the Planck Consortium, “By combining the Planck TT+lowP+lensing data with other astrophysical data, including the JLA supernovae, the equation of state for dark energy is constrained to w = −1.006 ± 0.045 and is therefore compatible with a cosmological constant, assumed in the base ΛCDM cosmology.”

A value of -1 for w corresponds to a simple Cosmological constant model with a single parameter Λ  that is the present-day energy density of empty space, the vacuum. The Λ value measured to be 0.69 is normalized to the critical mass-energy density. Since the vacuum is permeated by various fields, its energy density is non-zero. (The critical mass-energy density is that which results in a topologically flat space-time for the universe; it is the equivalent of 5.2 proton masses per cubic meter.)

Such a model has a negative pressure, which leads to the accelerated expansion that has been observed for the universe; this acceleration was first discovered in 1998 by two teams using certain supernova as standard candle distance indicators, and measuring their luminosity as a function of redshift distance.

Modified gravity

The phrase modified gravity refers to models that depart from general relativity. To date, general relativity has passed every test thrown at it, on scales from the Earth to the universe as a whole. The Planck Consortium has also explored a number of modified gravity models with extensions to general relativity. They are able to tighten the restrictions on such models, and find that overall there is no need for modifications to general relativity to explain the data from the Planck satellite.

Primordial density fluctuations

The Planck data are consistent with a model of primordial density fluctuations that is close to, but not precisely, scale invariant. These are the fluctuations which gave rise to overdensities in dark matter and ordinary matter that eventually collapsed to form galaxies and the observed large scale structure of the universe.

The concept is that the spectrum of density fluctuations is a simple power law of the form

P(k) ∝ k**(ns−1),

where k is the wave number (the inverse of the wavelength scale). The Planck observations are well fit by such a power law assumption. The measured spectral index of the perturbations has a slight tilt away from 1, with the existence of the tilt being valid to more than 5 standard deviations of accuracy.

ns = 0.9677 ± 0.0060

The existence and amount of this tilt in the spectral index has implications for inflationary models.


The Planck Consortium authors have evaluated a wide range of potential inflationary models against the data products, including the following categories:

  • Power law
  • Hilltop
  • Natural
  • D-brane
  • Exponential
  • Spontaneously broken supersymmetry
  • Alpha attractors
  • Non-minimally coupled

Figure 12 from Constraints on InflationFigure 12 from Planck 2015 results XX Constraints on Inflation. The Planck 2015 data constraints are shown with the red and blue contours. Steeper models with  V ~ φ³ or V ~ φ² appear ruled out, whereas R² inflation looks quite attractive.

Their results appear to rule out some of these, although many models remain consistent with the data. Power law models with indices greater or equal to 2 appear to be ruled out. Simple slow roll models such as R² inflation, which is actually the first inflationary model proposed 35 years ago, appears more favored than others. Brane inflation and exponential inflation are also good fits to the data. Again, many other models still remain statistically consistent with the data.

Simple models with a few parameters characterizing the inflation suffice:

“Firstly, under the assumption that the inflaton* potential is smooth over the observable range, we showed that the simplest parametric forms (involving only three free parameters including the amplitude V (φ∗ ), no deviation from slow roll, and nearly power-law primordial spectra) are sufficient to explain the data. No high-order derivatives or deviations from slow roll are required.”

* The inflaton is the name cosmologists give to the inflation field

“Among the models considered using this approach, the R2 inflationary model proposed by Starobinsky (1980) is the most preferred. Due to its high tensor- to-scalar ratio, the quadratic model is now strongly disfavoured with respect to R² inflation for Planck TT+lowP in combination with BAO data. By combining with the BKP likelihood, this trend is confirmed, and natural inflation is also disfavoured.”

Isocurvature and tensor components

They also evaluate whether the cosmological perturbations are purely adiabatic, or include an additional isocurvature component as well. They find that an isocurvature component would be small, less than 2% of the overall perturbation strength. A single scalar inflaton field with adiabatic perturbations is sufficient to explain the Planck data.

They find that the tensor-to-scalar ratio is less than 9%, which again rules out or constrains certain models of inflation.


The simplest LambdaCDM model continues to be quite robust, with the dark energy taking the form of a simple cosmological constant. It’s interesting that one of the oldest and simplest models for inflation, characterized by a power law relating the potential to the inflaton amplitude, and dating from 35 years ago, is favored by the latest Planck results. A value for the power law index of less than 2 is favored. All things being equal, Occam’s razor should lead us to prefer this sort of simple model for the universe’s early history. Models with slow-roll evolution throughout the inflationary epoch appear to be sufficient.

The universe started simply, but has become highly structured and complex through various evolutionary processes.


Planck Consortium 2015 papers are at – This site links to the 28 papers for the 2015 results, as well as earlier publications. Especially relevant are these – XIII Cosmological parameters, XIV Dark energy and modified gravity, and XX Constraints on inflation.


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