Cosmological constant

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Author: Mr. Brendan Griffen, University of Queensland, AU
Author: Dr. Tamara Davis, The DARK Cosmology Centre

Mr. Brendan Griffen accepted the invitation on 14 May 2009 (self-imposed deadline: 14 December 2009).

Figure 1: The cosmological constant was originally introduced by Einstein in 1917 as a repulsive force required to keep the Universe in static equilibrium. He later called it the 'biggest blunder' of his career.

In the context of cosmology the cosmological constant is a homogeneous energy density that causes the expansion of the universe to accelerate. Originally proposed early in the development of general relativity in order to allow a static universe solution it was subsequently abandoned when the universe was found to be expanding. Now the cosmological constant is invoked to explain the observed acceleration of the expansion of the universe. The cosmological constant is the simplest realization of dark energy, which is the more generic name given to the unknown cause of the acceleration of the universe. Its existence has also been predicted by quantum theory, where it enters as a form of vacuum energy, although the magnitude predicted does not match that observed in cosmology.

1 History
2 The physics of the cosmological constant
3 Observational evidence
4 Unresolved issues
5 References
6 Further reading
7 External links
8 See also

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History

The cosmological constant first appeared in a 1917 paper by Einstein entitled "Cosmological Considerations in the General Theory of Relativity" (Einstein 1917), in which he motivates its introduction into the field theory of general relativity by the need to stabilize the universe against the attractive effect of gravity:

"The term is necessary only for the purpose of making possible a quasi-static distribution of matter, as required by the fact of the small velocities of the stars" (Einstein 1917).

At the time, observations of our universe were limited primarily to stars in our own galaxy, so there was indeed observational evidence justifying the assumption that the universe was static. The need to stabilize the equations of general relativity against expansion or collapse was pointed out by both Friedmann cite and Lemaître cite. After Einstein proposed the cosmological constant Lemaître cite and Eddington cite independently showed that this solution was not stable. These results could be considered a prediction that the universe must be expanding or contracting, a remarkable implication of general relativity that was later borne out by observation.

In the intervening years the cosmological constant came in and out of vogue as new observational results repeatedly seemed to require it, but then were explained in other ways. As of the early 1990's there were tantalising hints that the cosmological constant might again be needed. The universe appeared to be younger than the oldest stars it contained, a feature that was remedied if the universe was currently in an accelerating phase. Number counts of galaxies indicated that the volume contained within a solid angle at high redshift $\left(z = \frac{\lambda_{\mathrm{obsv}} - \lambda_{\mathrm{emit}}}{\lambda_{\mathrm{emit}}}\right)$ were larger than expected in a decelerating universe. Theoretical arguments from inflation and later observational results from the cosmic microwave background indicated that the universe should be flat but observations of large scale structure indicated that the matter density was inadequate to achieve this - vacuum energy could make up the shortfall.

This set the stage for the discovery of the accelerating universe by two teams in 1998/1999. The High-Z supernova team and the Supernova Cosmology project both discovered that high-redshift (from early times) supernovae were fainter than expected for a decelerating universe and that the difference could be explained if there was a cosmological constant of just the right magnitude needed for a flat universe.

Figure 2: The mass-energy composition of the Universe. The cosmological constant is believed to be behind the expansion of the Universe which is driven by dark energy.

This was a dramatic convergence of observation and theory. Since then increasingly accurate probes have confirmed to high precision the need for dark energy, but the nature of the dark energy is now the quality being discussed. To the current (2008) precision the measured properties of dark energy remain consistent with those of a cosmological constant. However, massive observing efforts are underway to test whether this is the correct explanation for the acceleration or whether some other sort of dark energy, perhaps one that changes with time or one that is motivated by some form of quantum gravity, is needed to explain the acceleration we see.

Greatest Blunder

Much later, when I was discussing cosmological problems with Einstein, he remarked that the introduction of the cosmological term was the biggest blunder of his life. (Gamow 1970).

This statement has become one of the most highly cited quotations in cosmology. Although Einstein introduced the cosmological constant in 1917 as a ‘patch-up’ to obtain a static Universe, he believed it ruined the elegant simplicity of the theory. Einstein himself was well aware that his model had non-static solutions, but because the stellar velocities of stars within the Milky Way were so small, he discarded these solutions as too implausible to be true. It may seem quite strange for us now to wonder how he could have missed that a global expansion would have negligible effect on the motions of the stars close to the observer, however, given the observational data available at the time, it is quite understandable how he thought his assumptions were reasonable. Not until he reviewed Lemaitre’s paper in the 1920s did he become convinced the expanding solutions were mathematically correct, even though he told Lemaitre in person that the physics was "tout à fait abominable!".

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The physics of the cosmological constant

To explore more deeply the nature of the Universe, we must use the mathematical language in Einstein's general relativity to relate the geometry of space-time (expressed by the metric tensor, $g_{\mu\nu}(x_k)$ ) to the matter content of the universe, (expressed by the energy-momentum tensor, $T_{\mu\nu}(x_k)$ ).

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Einstein Field Equations

Universally, we consider concepts of space and time as two connect entities expressed as spacetime, with four coordinates (x,y,z,t) specifying an event. Arguably, Einstein's most significant discovery was creating an expression to connect the geometry of spacetime produced by a given distribution of mass and energy. From Lorentz invariance, we know that the energy-momentum tensor, $T_{\mu\nu}$ , should be proportional to the metric $\eta_{\mu\nu}$ ,

$T_{\mu\nu} = -\rho \eta_{\mu\nu}$

Generalising this for any coordinate system we have,

$T_{\mu\nu} = -\rho g_{\mu\nu}$

If we reduce the energy-momentum tensor into a matter component, $T_{\mu\nu}^{(vac)} = -\rho_{(vac)} \eta_{\mu\nu}$ and a vacuum component, $T_{\mu\nu}^{(M)}$ , Einstein's field equation is,

$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} = 8\pi G(T_{\mu\nu}^{(M)} -\rho_{(vac)}g_{\mu\nu})$

On the right is the stress-energy tensor, $T_{\mu\nu}$ which evaluates the effect of a given distribution of mass/energy on the degree of curvature of spacetime, as described via the Einstein tensor, $G_{\mu\nu}$ , on the left. It is interesting to note the inclusion of the speed of light, c, and Newton's gravitational constant, G, illustrating the extension of Einstein's earlier work on relativity theory.

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Implications

Solving Einstein's field equations for an isotropic, homogeneous universe leads to a description of the dynamical evolution of the universe in the form of a differential form for the scale factor, $R(t)$ . This new equation is known as the Friedmann equation, . Given in its most simplest form,

Figure 3: Aleksandr Friedmann (1888 - 1925) derived the famous equation which bares his name in 1922, shortly before his untimely death aged just 37 from Typhoid fever. George Gamow (1904 - 1968) was a student of his.

$\left[\left(\frac{1}{R}\frac{dR}{dt}\right)^2 - \frac{8}{3} \pi G \rho \right] R^2 = -kc^2$ .

As discussed, Einstein realised his original derivation could not produce a static universe and so introduced the infamous ad hoc term, the cosmological constant, $\Lambda$ . The modified form to produced a static universe (mathematically correct but this assumption was wrong) is now,

$\left[\left(\frac{1}{R}\frac{dR}{dt}\right)^2 - \frac{8}{3} \pi G \rho -\frac{1}{3}\Lambda c^2\right]R^2 = -kc^2$ ,

where k describes the overall geometry of the universe with three possible solutions (which one in particular represents our universe is still unknown though believed to be very close to k = 0. [REF]). These are: $k > 0$ , which means the total energy within the universe and is negative, and the universe is bound or closed (i.e. the universe will eventually halt and reverse in on itself), $k = 0$ which means the total energy is zero and the universe is flat (i.e. the universe will slow down its expansion and halt as $t \rightarrow \infty$ ) or $k < 0$ which means the total energy within the universe is positive and the universe is unbounded or open (i.e. the expansion will continue forever). Einstein's field equations can also be written to include the cosmological constant,

$R_{\mu\nu} - \frac{1}{2}Rg_{\mu\nu} + \Lambda g_{\mu\nu} = 8\pi G(T_{\mu\nu}^{(M)} -\rho_{(vac)}g_{\mu\nu})$

The new left hand term in the Friedmann equation comes from the addition of a new potential from Newtonian cosmology,

$U_\Lambda \equiv -\frac{1}{6}\Lambda m c^2 r^2$ ,

to the conservation of energy equation $E = K + U$ , where K and U are the total kinetic and potential energy within the universe, respectively. The force due to this new potential is thus,

$F_\Lambda = -\frac{\partial U}{\partial r}\hat{r} = \frac{1}{3}\Lambda c^2 \hat{r}$ ,

which is radially outward for $\Lambda > 0$ . This force outward allowed Einstein to include a repulsive force to keep the Universe static. As discussed is this introduction of the cosmological constant, $\Lambda$ , that he called the "biggest blunder of his life".

Another important point is that a nonzero cosmological constant implies that space should be curved in a universe devoid of matter, which goes against Einstein's tenant that matter-energy cause spacetime curvature. William de Sitter used Einstein's equations to describe an expanding, empty universe with the power of the expansion come from the cosmological constant.

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Cosmology

In an homogeneous, isotropic universe the geometry is defined by the Friedamnn-Lemaître-Robertson-Walker metric (FLRW metric) and the dynamics of the universe are governed by the Friedmann equations (Friedmann equations). The dynamics are driven by the energy content of the universe and the equation of state of the components that make up the energy density. The equation of state relates density $\rho$ to pressure $p$ according to $w = p/\rho$ . The cosmological constant enters these equations in the following way, where $a$ is the scalefactor of the universe normalized to 1 at the present day, $H=\dot{a}/a$ is Hubble's constant (an overdot represents differentiation with respect to time), $G$ is the gravitational constant, and $k$ is the curvature of the universe given by +1, 0, and -1 for positive, flat, and negative curvature respectively.

(1)

$H^2 = \left(\frac{\dot{a}}{a}\right)^2 = \frac{8\pi G}{3}\rho - \frac{k}{a^2} +\frac{\Lambda}{3}$

(2)

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

These equations are more concisely written by considering the cosmological constant as a form of vacuum energy, so $\Lambda c^2=8\pi G \rho_\Lambda$ . Then Eqs. 1 and 2 become

(3)

$H^2 = \frac{8\pi G}{3}\sum_i \rho_i$

(4)

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho_i + 3p_i)$

The different components have different equations of state, which determines how their density changes with the expansion of the universe:

(5)

$\rho_i = \rho_{i0} a^{-3(1+w_i)}$

Pressureless matter has $w=0$ , radiation has $w=1/3$ , curvature has an effective $w=-1/3$ , cosmological constant has $w=-1$ . (We have used $w_\Lambda=-1$ , which implies $\rho_\Lambda + 3p_\Lambda = -2\rho_\Lambda$ in deriving Eq. 4.)

The current energy density of each component, $\rho_{i0}$ , is often represented as a fraction of the critical density, $\rho_{\rm c}=3H_0^2/8\pi G$ , which is the energy density required to close the universe (also calculated at the present day). Denoting this $\Omega_i = \rho_{i0}/\rho_{\rm c}$ and using Eq. 5 allows us to write

(6)

$H^2 = H_0^2\sum_i \frac{\rho_i}{\rho_{c}} = H_0^2\sum_i\Omega_i a^{-3(1+w_i)}$

In an homogeneous universe there are no pressure gradients, so a positive pressure does no work and has no expanding effect. On the contrary, in general relativity all forms of energy gravitate so pressure effectively pulls, strengthening the attractive force of gravity (thus the factor of $p$ in Eq. 2, which does not appear in Newtonian gravity). The cosmological constant has negative pressure, $w=-1$ , so its general relativistic contribution counteracts the normal force of gravity and provides an outwards acceleration.

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Quantum Mechanics and the Cosmological Constant

Vacuum Energy

Vacuum energy arises naturally in quantum mechanics due to the uncertainty principle. In particle physics the vacuum refers to the ground state of the theory --- the lowest energy configuration. The uncertainty principle does not allow states of exactly zero energy, even in vacuum (virtual particles are created). Since in general relativity all forms of energy gravitate, this ground state vacuum energy impacts the dynamics of the expansion of the universe.

Moreover, vacuum energy has $\rho_{\rm vac}=-p_{\rm vac}$ so the equation of state of vacuum energy is exactly that of a cosmological constant $w_{\rm vac}=w_\Lambda = -1$ .

Cosmological Constant From Quantum Field Theory

A relativistic field may be thought of as a collection of harmonic oscillators at all possible frequencies [REF]. For a scalar field $\phi$ in the form of a spinless boson and mass, $m$ , the vacuum energy is simply the sum over all possible modes (i.e. over all possible wavelengths, k),

$E_0 = \sum_i \frac{1}{2}\hbar\omega_i$ .

We carry out this sum by putting this system into a box of volume $L^3$ , and setting $L$ to infinity. Using periodic boundary conditions, $\lambda_i = \frac{L}{n_i}$ for some integer, ${n_i}$ , then since $k_i = \frac{2\pi}{\lambda_i}$ , there are $\frac{dk_iL_i}{2\pi}$ discrete values of $k_i$ in the range $(k_i, k_i + dk_i)$ . Hence the sum over all possible modes becomes,

$E_0 = \frac{1}{2}\hbar L^3 \int \frac{d^3k}{2\pi^3}\omega_k$ .

To evaluate this integral we require a cut-off for the maximum wavelength $k_{max} >> \frac{m}{\hbar}$ and set $\omega_k^2 = k^2 + \frac{m^2}{\hbar}$ . We then arrive at a vacuum density of,

$\rho_{vac} \equiv \lim_{L \to \infty} \frac{E_0}{L^3} = \hbar \frac{k_{max}^4}{16 \pi^2}$ .

Choosing $k_{max} = \frac{E}{\hbar}$ we obtain a numeric estimate of the vacuum energy as $\rho_{vac} = 10^{74} GeV^4\hbar^{-3}$ . However, this naïve prediction of the magnitude of vacuum energy is about 120 orders of magnitude from the measured cosmological value. This is known as the cosmological constant problem. From this one could postulate an additional cosmological constant is exactly $8\pi G\rho_{vac}$ to make the net cosmological constant zero.

For an elaboration on these topics see Carroll (1992).

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Observational evidence

Evidence for the accelerating universe includes (see review by Frieman, Turner and Huterer 2008):

Number counts of galaxies.

Galaxy number counts have been used widely in cosmology to estimate the deceleration parameter q0 (which is largely independent of the scale factor). Essentially, the number of galaxies per unit area and magnitude bin (N(m)) are observed from some very deep wide-field image at any place in the sky. The predictions for N(m) for a given range of deceleration parameters are compared to the observation of N(m) found in the image. Two assumptions are made; (1) the luminosity of galaxies is constant through time and (2) the number of galaxies per unit volume is the same as when they formed.

The age of the universe.
The observed flatness of the universe despite insufficient matter.

Using measurements of temperature fluctuations in the CMB from when the universe was ~380,000 years old one can conclude that the Universe is spatially flat to within a few percent. By combining CMB measurements of the large scale structure and accurate H_0 measurements, one finds that the matter in the Universe only contributes approximately 23% to the critical density. To account for the missing energy density, a smoothly distributed energy is required, i.e. dark energy.

The faintness of distant supernovae (0<z<0.5)
The relative brightness of extremely distant supernovae (z>0.5).

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Unresolved issues

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Cosmological Constant Problem

The cosmological constant problem arises because quantum field theory predicts a value of the vacuum energy far in excess of the cosmological constant value measured in cosmology. Quantum mechanical calculations that sum the contributions from all vacuum modes below an ultraviolet cutoff at the Planck scale give a vacuum energy density of $\rho_\Lambda\sim10^{112} {\rm erg/cm}^3$ . This exceeds the cosmologically observed value of $\rho_\Lambda\sim10^{-8} {\rm erg/cm}^3$ by about 120 orders of magnitude. See Carroll (2004), Section 4.5.

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Coincidence Problem

The cosmological constant is not diluted as the universe expands, whereas the density of matter drops in inverse proportion to the volume. This means that there is only a fleeting moment of cosmological time during which the matter density will be of comparable magnitude to the vacuum energy density. The fact that we appear to be living in that moment seems too unlikely to be coincidence. This has been called the coincidence problem, and has motivated theories beyond the cosmological constant with more general forms of dark energy that may change with time.

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Dark energy or cosmological constant

These unresolved issues have motivated the current observational effort to test whether the cosmological constant is a valid cause of the acceleration of the universe. Other theories, such as fledgling theories of quantum gravity (e.g. brane-motivated cosmologies), naturally produce dark energy candidates with properties different from the standard cosmological constant. Phenomenological theories such as quintessence have also been proposed, which have a time-varying value of dark energy that traces the energy density and thus naturally solves the coincidence problem.

Dark energy also encompasses the possibility that there is no additional energy density component to the universe, but rather that the equations of general relativity need revision. In this sense general relativity might be a limit of a more complete theory of gravity in the same way that Newtonian gravity is a low-energy limit of general relativity. This possibility is also known as dark gravity.

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References

Carroll, Press (1992). The cosmological constant Annual Review of Astronomy and Astrophysics 30: 499-542.
Carroll, Sean (2004). Spacetime and Geometry. Addison Wesley, San Francisco, CA. 171-174
Albert, {{{7}}} (142–152). Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie (Cosmological Considerations in the General Theory of Relativity) Koniglich Preußische Akademie der Wissenschaften, Sitzungsberichte 1917: Einstein.
- For an English translation see Einstein, Albert (1997). The collected papers of Albert Einstein (Alfred Engel, translator) Princeton University Press, Princeton, New Jersey.
Frieman, Josh; Turner, Michael and Huterer, Dragan (2008). Dark Energy and the Accelerating Universe Annual Review of Astronomy and Astrophysics 46: 385-432. arXiv:0803.0982
Gamow, George (1970). My World Line. Viking, New York. 44

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External links

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Invited by:	Dr. Søren Bertil F. Dorch, The Niels Bohr Institute and Center for Scholarly Communication (The Royal Library / Copenhagen University Library and Information Services), Denmark
Invited by:	Dr. Riccardo Guida, Institut de Physique Théorique; CEA, IPhT; CNRS; Gif-sur-Yvette, France

Cosmological constant

From Scholarpedia

Contents

History

The physics of the cosmological constant

Einstein Field Equations

Implications

Cosmology

Quantum Mechanics and the Cosmological Constant

Observational evidence

Unresolved issues

Cosmological Constant Problem

Coincidence Problem

Dark energy or cosmological constant

References

Further reading

External links

See also

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