Dead or Alive
|The Cosmological Constant|
Aka: Einstein's biggest blunder , or the vacuum energy
Committed Crimes: Being 'the most embarrassing observation in physics' ; being the 'the worst prediction in physics' ; being either too small, or too large, or too coincidental; being bad for astronomy, and being generally an annoyance
Last seen: In high redshift supernovae and the WMAP data
Watch out, here comes an equation!
Apologies if I scared any unprepared readers but I *really* can't do without. These are Einstein's field equations  of General Relativity, and aren't they just pretty? Here is in a nutshell what they say:
The quantities on the left side are the g, which is the metric of our spacetime. The metric tells us how we measure angles and distances. Then there are the R's with varying amount of indices. They describe the curvature of space time, and are build up of second order derivatives of the metric. Thus, the left side has dimensions Energy2 . If space-time is flat, the curvature is identically zero.
On the other side of the equation we have a T which is called the stress-energy tensor, and describes the matter and energy content in the space-time. It has dimension energy per volume, and contains energy density, as well as pressure and energy flux. In energy units it has dimension Energy4. The G is a coupling constant, and one now easily concludes it has dimension of 1/Energy2. If one investigates the Newtonian limit, one finds that G=1/MP2, where MP is the Planck Mass.
Thus, the equations say how matter and energy (right side) affects the curvature of the space-time we live in (left side). If space time is flat, there are no matter sources (Tμν = 0). An important point is that you can not just chose matter that moves as you like it, because it generally will not be consistent with what the equations say. You can only chose an initial configuration, and the equations will tell you how that system will evolve, matter and space-time together. Different matter types have different effects, and result in different time-evolutions.
That's the thing to keep in mind for the next section: different stuff causes different curvature. The details are what you need the PhD for.
Now cosmology is an extremely simplified case in which one describes the universe as if it was roughly speaking the same everywhere (homogeneous), and the same in all directions (isotropic). This is called the Cosmological Principle, and if you look around you, it is evidently complete nonsense. However, whether or not such a description is useful is a matter of scales.
Look e.g. at the desk in front of you. It looks like a plane surface with a certain roughness. If you look really close you'd find lots of structure, but if you are asking for some large scale effect - like how far your coffee cup will slide - the exact shape of a single tiny hills or dips in the surface doesn't matter. It's the same with the universe. If you look from far enough away, the finer details don't matter, galaxies are roughly equally distributed over the sky. With the cosmological principle, one neglects the details of the structures. One describes matter by an average density ρ and pressure p that does not depend on the position in spacetime. It has the same value everywhere, but can depend on time.
We have today extremely strong evidence that the universe is expanding, thus its volume grows. The ratio of this expansion is usually measured with the scale-factor a(t) , a dimensionless, increasing function of time. The universe's expansion is the same in all three spatial directions, so a given volume grows with ~ a(t)3. When the volume grows, stuff inside it thins out. The energy density of ordinary matter drops just inversely to the volume ~ 1/a(t)3.
The energy density of radiation drops even faster, because not only does the volume increase - in addition its wavelength gets also stretched, and therefore the frequency drops with an additional 1/a(t). Taken together, the energy density of radiation drops with 1/a(t)4. Thus, the density of all all kind of matter that we know, and have observed on earth should drop.
Because the expansion of the universe causes light to be stretched to higher wavelengths, and be shifted towards lower - 'redder' - frequencies, cosmologists like to date events not by the time t, but by the redshift, commonly denoted as z.
The Cosmological Constant and its Relatives
The Cosmological Constant (CC) is usually denoted with the Greek symbol Lambda, Λ. It is the constant in front of an additional term that can be added to Einstein's field equations. Depending on your taste, you can either interpret it as belonging on the left 'space-time' side, or the right 'matter' side of the equation. For the calculations this doesn't matter, so lets put it to the matter-side:
What have we done? Well, consider as before the case of 'empty' space, where Tμν is equal to zero. This empty space can no longer be flat: if it was, the curvature and thus the left side would vanish. But the right side doesn't. Thus, with the CC term, empty space is no longer flat. It is therefore very tempting to interpret this as the energy density of the vacuum which creates curvature even if 'nothing' is really there. As is appropriate for an energy density, the dimension of the CC is Energy4.
If one plugs the matter content and the CC term into Einstein's field equations, one obtains the Friedmann equations that relate the behavior of the scale factor to the density and pressure
The appearing constant κ is either 0, +1 or -1, depending on the spatial curvature. The first equation has a square on the left side, meaning that the right side needs to be positively valued. The second equation determines the acceleration of the universe. Note that for usual matter, energy density and pressure are positively valued. Thus, the only factor that can make a positive contribution to the acceleration is the CC.
The most stunning fact about the Cosmological Constant is that it is constant. No kidding: remember we've seen before that all kind of matter that we know dilutes when the universe expands.
But the Cosmological Constant is constant.
The corollary of this insight is that if you start with an arbitrary amount of usual matter, sooner or later it will have dropped to the value of the CC term. And if you wait even longer, the CC term will eventually dominate, causing an eternally accelerating universe. Who could possibly want that?
A term with a CC is not the only way to get a positive contribution to the universe's acceleration. Unusual equations of state that relate ρ with p in a way no standard matter could do would have a similar effect. The family of stuff with such behavior is the mafia of the 'dark energy'. A lot of creativity has gone into its investigation. The suspects in the family include quintessence, k-essence, h-essence, phantom fields, tachyon fields, Chaplygin gas, ghost condensates, and probably a couple more aunts and uncles that I haven't been introduced to.
The CC appears in the Einstein's field equation and can be treated as a source term. For this analysis it is irrelevant whether the term might actually be of geometrical origin. In this context, the constant Λ just is a parameter to describe a specific behaviour.
- Supernovae redshift
Supernovae of type Ia show a very uniform and reliable dependence of luminosity on time. This makes them ideal candidates for observation, as they allow (to a certain extent) to disentangle the source effects from the effects occurring during light propagation. The emitted light travels towards us, and while it does so it has to follow the curvature of space-time. The frequency and luminosity that we receive then depends on the way the universe has evolved while the light was traveling. From the observation on can then extract knowledge about the curvature, and thus about the matter content of the universe.
As it turns out, distant supernova (z > .5) are fainter than would be expected for a decelerating universe with vanishing CC. If one explains the data with the dynamics of the scale factor, one is lead to the conclusion that the universe presently undergoes accelerated expansion. As we have seen above, this can only be caused by a positive cosmological constant. In addition, the data also shows that the transition from deceleration to acceleration happened rather recently, i.e. around z ~ 0.3. For more detail's see Ned Wright's excellent tutorial.
Last November, the observations of high redshift supernovae could be extended above z~1, which allows us to conclude that the dark energy component at these times didn't do anything too spectacular. For more, see Sean's post Dark Energy has long been Dark-Energy-Like.
- The Age of the Universe
A recurring argument for a CC has come from the age and present expansion rate of the universe. The presence of the CC influences the expansion of the universe. If it exists, the age of the universe one can extract from today's value of the Hubble parameter would be larger than without a CC. The age of the oldest stars that have been observed seems to indicate the necessity of the CC. However, this analysis is not without difficulties since determining the age of these stars, as well as the present value of the Hubble parameter, is still subject to uncertainties that affect the conclusion .
- The Cosmic Microwave Background
WMAP measures the temperature inhomogeneities imprinted in the Cosmic Microwave Background (CMB). These very low temperature photons have been traveling freely since the time the universe became transparent for them, called the 'surface of last scattering'. The photon's distribution shows small fluctuations on specific scales, a snapshot of the universe as it was only 300,000 years old. Commonly depicted are the temperature fluctuations as a function of the multipole moment, roughly speaking the angular size of the spots (for a brief intro, see here). Recall from above that different stuff causes different curvature. Thus, from these structures in the CMB one can draw conclusions about the evolution, and thus about the matter content, of the universe.
Since the CC only became important recently, its dominant effect is to change the distance to the last scattering surface, which determines the angular scale of the observed CMB anisotropies. Most prominently, it affects the position of the first peak in this figure. Based on current measurements of the Hubble scale, the WMAP data is best fitted by a spatially flat universe in which 70% of the matter is described by the CC term.
The value of the CC that can be inferred from the presently available data is approximately Λ1/4 ~ 10-12GeV.
- First Crime: Diverging
- Second Crime: Being large
Infinity is not a particularly good result. Now you might argue that we can't trust quantum field theory up to arbitrarily high energy scales, because quantum gravity should become important there. If you drop virtual particles with energy higher than the Planck scale, you find that Λ should be of the order MP4, a huge value.
Of one neglects gravity, one can argue that the absolute value of the energy density can't be observed, and one should only be interested in energy differences. One thus can set 'by hand' the vacuum energy to zero, a process that is called renormalization. Unfortunately, one can't do this with gravity, because all kind of energy creates curvature, and it is not only energy differences that are observable. However, one can't take the above huge value seriously. If the CC was really that large, we wouldn't exist. In fact, this value for the CC is 120 orders of magnitude larger than the observed one. Thus the reason why it has been called 'The worst prediction in physics'.
- Third Crime: Being Small
- Forth Crime: Why now?!
- Fifth Crime: Other Coincidences
- The CC is the fourth power of an energy scale, which happens to be close by scale in which the (differences of the squared) neutrino masses have been measured, and the absolute masses of the lightest neutrinos are believed to fall. Coincidence?
- It further turns out that the ratio of that mass scale Λ1/4 to the vacuum expectation value (VEV) of the Higgs, is about the same as the ratio of the Higgs VEV to the Planck mass. Coincidence?
- And then the CC is about the geometric mean of the (forth power of) the Hubble scale and the Planck mass . Coincidence?
This data analysis is of course not completely watertight. To begin with, one has to notice that all of the above mentioned rests on General Relativity. If instead the equations of motions were modified, if might be possible to do without dark energy. However, studies in this direction so far have not been particularly convincing, as consistent modifications of GR are generally hard to do.
But besides this, the data interpretation is still subject of discussion. An input in all the analysis is the value of the Hubble constant. It has been argued for various reasons that the presently used value should be corrected by taking into account local peculiar motions, or a possible spatial variations. Measuring the Hubble value through time delays in gravitationally lensed systems e.g. yielded a significantly lower result.
Likewise, the supernovae data could be biased through the sample, or the effect could stem from other effects during propagation, like dust with unusual properties. For an recent summary of all such uncertainties, see .
Generally, on can say that there remains the possibility that the data can be fitted by other models. But to date the the CC term is the simplest, and most widely accepted explanation we have. One should keep in mind however that the desire to come up with a theory that produces a kind of dark energy uses GR as relay station. Instead it might be possible that a satisfactory explanation reproduces the observational facts, yet is not cast into the form of GR with dark energy because it eludes the parametrization implied in writing down the Friedmann equations.
Our observations can currently be best described by to so-called ΛCDM model. It has a Cosmological Constant, and a significant amount of cold 'dark matter' (explained in our earlier post). The parameters in this model have been well constrained by experiments, but the theoretical understanding is still very unsatisfactory. ΛCDM is a macroscopic description, but we have so far no good theory explaining the microscopic nature of dark energy or dark mater.
Investigations will be continued.
- Sean Carroll's Living Review: The Cosmological Constant.
- T. Padmanabhan, Dark Energy and Gravity
- Ned Wright, Vacuum Energy Density, or How Can Nothing Weigh Something?
- Subir Sarkar: Is the evidence for dark energy secure?
- Wikipedia on The Cosmological Constant
 George Gamow, My World Line (Viking, New York). p 44, 1970
 Ed Witten, quoted from Renata Kallosh's talk "Towards String Cosmology", slide 5.
 Lawrence Krauss, quoted from "Physicists debate the nature of space-time", NewScientist Blog, 02/20/07
 It's a plural because there is one for every choice of indices μν, and each index runs over 3 space and one time dimension, from 0 to 3. This would make for 16 equations, but the set is symmetric under exchange of the indices, so actually there are only ten different equations.
 This is a theoretical physics blog, so Planck's constant and the speed of light is equal to one. This then means the dimension of length is that of time, and both is an inverse of energy. If that doesn't make sense to you, don't worry it's not really relevant. Just accept it as a useful way to check the order of coupling constants.
 For details, see arxiv: 0710.5307
 See T. Padmanabhan, hep-th/0406060
TAGS: PHYSICS, DARK ENERGY, VACUUM ENERGY, COSMOLOGY, COSMOLOGICAL CONSTANT