Tag Archives: accelerating universe

Dark Energy and the Comological Constant

I am seeing a lot of confusion around dark energy and the cosmological constant. What are they? Is gravity always attractive? Or is there such a thing as negative gravity or anti-gravity?

First, what is gravity? Einstein taught us that it is the curvature of space. Or as famous relativist John Wheeler wrote “Matter tells space how to curve, and curved space tells matter how to move”.

Dark Energy has been recognized with the Nobel Prize for Physics, so its reality is accepted. There were two teams racing against one another and they found the same result in 1998: the expansion of the universe is accelerating!

Normally one would have thought it would be slowing down due to the matter within; both ordinary and dark matter would work to slow the expansion. But this is not observed for distant galaxies. One looks at a certain type of supernova that always has a certain mass and thus the same absolute luminosity. So the apparent brightness can be used to determine the luminosity distance. This is compared with the redshift that provides the velocity of recession or velocity-determined distance in accordance with Hubble’s law.

A comparison of the two types of distance measures, particularly for large distances, shows the unexpected acceleration. The most natural explanation is a dark energy component equal to twice the matter component, and that matter component would include any dark matter. Now do not confuse dark energy with dark matter. The latter contributes to gravity in the normal way in proportion to its mass. Like ordinary matter it appears to be non-relativistic and without pressure.

Einstein presaged dark energy when he added the cosmological constant term to his equations of general relativity in 1917. He was trying to build a static universe. It turns out that such a model is unstable, and he later called his insertion of the cosmological constant a blunder. A glorious blunder it was, as we learned eight decades later!

Here is the equation:

G_{ab}+\Lambda g_{ab} = {8\pi G \over c^{4}}T_{ab}

The cosmological constant is represented by the Λ term, and interestingly it is usually written on the left hand side with the metric terms, not on the right hand side with the stress-energy (and pressure and mass) tensor T.

If we move it to the right hand side and express as an energy density, the term looks like this:

\rho  = {\Lambda \over8\pi G }

with \rho  as the vacuum energy density or dark energy, and appearing on the right it also takes a negative sign. So this is a suggestion as to why it is repulsive.

The type of dark energy observed in our current universe can be fit with the simple cosmological constant model and it is found to be positive. So if you move \Lambda to the other side of the equation, it enters negatively.

Now let us look at dark energy more generally. It satisfies an equation of state defined by the relationship of pressure to density, with P as pressure and ρ denoting density:

P = w \cdot \rho \cdot c^2

Matter, whether ordinary or dark, is to first order pressureless for our purposes, quantified by its rest mass, and thus takes w = 0. Radiation it turns out has w = 1/3. The dark energy has a negative w, which is why you have heard the phrase ‘negative pressure’. The simplest case is w = -1, which the cosmological constant, a uniform energy density independent of location and age of the universe. Alternative models of dark energy known as quintessence can have a larger w, but it must be less than -1/3.



Why less than -1/3? Well the equations of general relativity as a set of nonlinear differential equations are usually notoriously difficult to solve, and do not admit of analytical solutions. But our universe appears to be highly homogeneous and isotropic, so one can use a simple FLRW spherical metric, and in this case one end up with the two Friedmann equations (simplified by setting c =1).

\ddot a/a  = - {4 \pi  G \over 3} ({\rho + 3 p}) + {\Lambda \over 3 }

This is for a (k = 0) flat on large scales universe as observed. Here \ddot a is the acceleration (second time derivative) of the scale factor a. So if \ddot a is positive, the expansion of the universe is speeding up.

The \Lambda term can be rewritten using the dark energy density relation above. Now the equation needs to account for both matter (which is pressureless, whether it is ordinary or dark matter) and dark energy. Again the radiation term is negligible at present, by four orders of magnitude. So we end up with:

\ddot a/a  = - {4 \pi  G \over 3} ({\rho_m + \rho_{de} + 3 p_{de}})

Now the magic here was in the 3 before the p. The pressure gets 3 times the weighting in the stress-energy tensor T. Why, because energy density is just there as a scalar, but pressure must be accounted for in each of the 3 spatial dimensions. And since p for dark energy is negative and equal to the dark energy density (times the square of the speed of light), then

\rho + 3 p is always negative for the dark energy terms, provided w < -1/3. That unusual behavior is why we call it ‘dark energy’.

Overall it is a battle between matter and dark energy density on the one side, and dark energy pressure (being negative and working oppositely to how we ordinarily think of gravity) on the other. The matter contribution gets weaker over time, since as the universe expands the matter becomes less dense by a relative factor of (1=z)^3 , that is the matter was on average denser in the past by the cube of one plus the redshift for that era.

Dark energy eventually wins out, because it, unlike matter does not thin out with the expansion. Every cubic centimeter of space, including newly created space with the expansion has its own dark energy, generally attributed to the vacuum. Due to the quantum uncertainty (Heisenberg) principle, even the vacuum has fields and non zero energy.

Now the actual observations at present for our universe show, in units of the critical density that

\rho_m \approx 1/3

\rho_{de} \approx 2/3

and thus

p_{de} \approx - 2

And the sum of them all is around -1, just coincidentally. Since there is a minus sign in front of the whole thing, the acceleration of the universe is positive. This is all gravity, it is just that some terms take the opposite side. The idea that gravity can only be attractive is not correct.

If we go back in time, say to the epoch when matter still dominated with \rho_m \approx 2/3 and  \rho_{de} \approx 1/3 , then the total including pressure would be 2/3 +1/3 – 1, or 0.

That would be the epoch when the universe changed from decelerating to accelerating, as dark energy came to dominate. With our present cosmological parameters, it corresponds to a redshift of z \approx 0.6, and almost 6 billion years ago.

Image: NASA/STScI, public domain


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.

Planck Mission Full Results Confirm Canonical Cosmology Model

Dark Matter, Dark Energy values refined

The Planck satellite, launched by the European Space Agency, made observations of the cosmic microwave background (CMB) for a little over 4 years, beginning in August, 2009 until October, 2013.

Preliminary results based on only the data obtained over the first year and a quarter of operation, and released in 2013, established high confidence in the canonical cosmological model. This ΛCDM (Lambda-Cold Dark Matter) model is of a topologically flat universe, initiated in an inflationary Big Bang some 13.8 billion years ago and dominated by dark energy (the Λ component), and secondarily by cold dark matter (CDM). Ordinary matter, of which stars, planets and human beings are composed, is the third most important component from a mass-energy standpoint. The amount of dark energy is over twice the mass-energy equivalent of all matter combined, and the dark matter is well in excess of the ordinary matter component.


This general model had been well-established by the Wilkinson Microwave Anisotropy Probe (WMAP), but the Planck results have provided much greater sensitivity and confidence in the results.

Now a series of 28 papers have been released by the Planck Consortium detailing results from the entire mission, with over three times as much data gathered. The first paper in the series, Planck 2015 Results I, provides an overview of these results. Papers XIII and XIV detail the cosmological parameters measured and the findings on dark energy, while several additional papers examine potential departures from a canonical cosmological model and constraints on inflationary models.

In particular they find that:

Ωb*h²  = .02226 to within 1%.

In this expression Ωb is the baryon (basically ordinary matter) mass-energy fraction (fraction of total-mass energy in ordinary matter) and h = H0/100. H0 is the Hubble constant which measures the expansion rate of the universe, and indirectly, its age. The best value for H0 is 67.8 kilometers/sec/Megaparsec  (millions of parsecs, where 1 parsec = 3.26 light-years). H0 has an uncertainty of about 1.3% (two standard deviations). In this case h = .678 and the expression above becomes:

Ωb = .048, with uncertainty around 3% of its value. Thus, just under 5% of the mass-energy density in the universe is in ordinary matter.

The cold matter density is measured to be:

Ωc*h²  = .1186 with uncertainty less than 2% and with the h value substituted we have Ωc = .258 with similar uncertainty.

Since the radiation density in the universe is known to be very low, the remainder of the mass-energy fraction is from dark energy,

Ωe = 1 – .048 – .258 = .694

So in approximate percentage terms the Planck 2015 results indicate 69% dark energy, 26% dark matter, and 5% ordinary matter as the mass-energy balance of the universe. These results are essentially the same as the ratios found from the preliminary results reported in 2013. It is to be emphasized that these are present-day values of the constituents. The components evolve differently as the universe expands. Dark energy is manifested with its current energy density in every new unit of volume as the universe continues to expand, while the average dark matter and ordinary matter densities decrease inversely as the volume grows. This implies that in the past, dark energy was less important, but it will dominate more and more as the universe continues to expand.

Why is dark energy produced as the universe expands? The simplest explanation is that it is the irreducible quantum energy of empty space, of the vacuum. Empty space – space with no particles whatsoever – still has fields (scalar fields, in particular) permeating it, and these fields have a minimum energy. It also has ‘virtual’ particles popping in and out of existence very briefly. This is the cosmological constant (Λ) model for the dark energy.

This is the ultimate free lunch in nature. The dark energy works as a negative gravity; it enters into the equations of general relativity as a negative pressure which causes space to expand. And as space expands, more dark energy is created! A wonderful self-reinforcing process is in place. Since the dark energy dominates over matter, the expansion of the universe is accelerating, and has been for the last 5 billion years or so. Why wonderful? Because it adds billions upon billions of years of life to our universe.

The Planck Consortium also find the universe is topologically flat to a very high degree, with an upper limit of 1/2 of 1% deviation from flatness at large scales. This is an impressive observational result.

One of the most interesting results is Planck’s ability to constrain inflationary models. While a massive inflation almost certainly happened during the first billionth of a trillionth of a trillionth of a second as the Universe began, as indicated by the very uniformity of the CMB signal, there are many possible models of the inflationary field’s energy potential.

We’ll take a look at this in a future blog entry.

Dark Energy Drives Runaway Universe

Accelerating universe

Accelerating universe graphic. Credit: NASA/STSci/Ann Field

Dark energy was first introduced as a possibility as a result of the formulation of Einstein’s equations of general relativity. When he considered how the universe as a whole would behave under the general relativity description of gravity, he added a term to his equations, known as the cosmological constant. At the time the prevailing view was that the universe was static, and neither expanding nor contracting. The term was intended to balance the self-gravitational energy of the universe, and it thus acts as a repulsive force, rather than an attractive one. His basis for introduction of the cosmological constant was erroneous in two respects. The first problem is that the static solution was unstable, as if balanced on a knife edge. If you nudged it a little bit by increasing the matter density in some region slightly, that region would collapse, or if you lowered the density ever so slightly, that region would expand indefinitely. The second problem is that by 1929 Edwin Hubble had demonstrated the universe is actually expanding at a significant rate overall.

Subsequently, Einstein called the introduction of the cosmological constant his “greatest blunder”. After the realization that we live in an expanding universe, while the possibility of the cosmological constant having a non-zero value was sometimes entertained in cosmological theory, it was mostly ignored (set to zero). Over the next several decades, attention turned to better measuring the expansion rate of the universe and the inventory of matter, both ordinary matter and the dark matter, with the amount of the latter implied by long range gravitational effects seen both within galaxies and between galaxies. Was there enough matter of both types to halt the expansion? It seemed not, rather that there was only about 1/4 of the required density of matter, and that was mostly in the form of dark, not ordinary matter. Matter of either type would slow down the expansion of the universe due to its gravitational effects.

After 1980, the inflationary version of the Big Bang gained acceptance due to its ability to explain the flat topology of the universe and the homogeneity of the cosmic microwave background radiation, the relic light from the Big Bang itself. The inflationary model strongly indicated that the total energy density should be about 4 times greater than seen from the matter components alone. It is the total of energy and matter (the energy content of matter) which determines the universe’s fate, since E = mc^2.

In 1998 the astounding discovery was made that the universe’s expansion rate is accelerating! This was determined by two different teams, each of which were making measurements of distant supernovae (exploding stars). And it was confirmed by measurements of tiny fluctuations in the intensity of the microwave background radiation. The two techniques are consistent, and a third technique based on X-ray emission from clusters of galaxies, as well as a fourth technique based on very large scale measurements of relative galaxy positions, also give results consistent with the previous two techniques. The inflationary predictions are satisfied with dark energy presently three times more dominant than the rest mass energy equivalent from dark matter plus ordinary matter. Further measurements have refined our understanding of the relative strength of dark energy in comparison to dark matter and ordinary matter. The best estimates are that, today, dark energy is 74% of the universe’s total mass-energy balance.

In the cosmological constant formulation, dark energy is constant in time, while the matter density drops as the universe expands, in proportion to the cube of the scale factor. So if we consider the universe in its early days the energy contained in the dark matter would have dominated over dark energy, as the mass density would have been much greater than today. The crossover from matter dominated to dark energy dominated came after the universe was about 9 billion years old, or about 5 billion years ago. This emergence of dark energy as the dominant force, due to its nature as a repulsive property of “empty” space-time, results in an accelerating expansion of the universe, which has been called the “runaway universe”. Our universe is apparently slated to become hugely larger than its current enormous size.

Why is dark energy important then? Since five billion years ago, and on into the indefinite future, it has dominated the mass-energy content of the universe. It drives a re-acceleration of the universe. It inhibits the re-collapse (“Big Crunch”) of our entire universe or even substantial portions of the universe. Thus it naturally extends the life of the entire universe to trillions of years or much more – far beyond what would occur were the universe to be dominated by matter only and with density at the critical value or above. Dark energy thus works to maximize the available time and space for life to develop and to evolve on planets found throughout the universe.