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Mystery of the Cosmological Constant

The 2011 Nobel prize in physics has been awarded to S. Perlmutter, A. Riess and B. Schmidt for their path breaking discovery that the rate of expansion of the universe is increasing with time. The trio used Type Ia supernovae (SNe Ia) as standard candles to estimate their luminosity-distances. To appreciate some of the far reaching implications of their work, I have provided an elementary exposition of general theory of relativity, accelerated expansion of the universe, luminosity-distance, SNe Ia and the cosmological constant problem.


INTRODUCTION
Gravity is always attractive, right? Wrong, if we are to go by the research papers of S.
Perlmutter, A. Riess and B. Schmidt, for which they have been awarded this year's Nobel prize in physics [1][2][3][4][5]. The trio looked at Type Ia supernovae, located in far away galaxies, to measure their distances accurately, and landed up with a surprising discovery -cosmic anti-gravity at large scales. To appreciate some of the implications of their path breaking findings, and to have a glimpse of the repulsive side of gravitation, we need to look into gravity's fatal attraction, first.
According to the Newtonian theory, acceleration of a test body due to a massive object's gravity, while being proportional to latter's inertial mass and directed towards the massive body, is independent of the test particle's inertial mass. Newton's gravity also demands that the gravitational force be instantaneously transmitted by the source to the test particle, since it is inversely proportional to the square of the instantaneous separation between the two.
Instant transmission is unsatisfactory, as Einstein's special theory of relativity demands that no physical effect can propagate with a speed faster than c (speed of light in vacuum).
Einstein improved the situation by putting forward a relativistic theory of gravity in 1916 through his theory of general relativity (GR). GR is based on the observation that the trajectory of a test particle in any arbitrary gravitational field is independent of its inertial mass m (since the acceleration does not dependent on m), and therefore, it must be the geometry of the space-time that determines test particle trajectories. Note that for no other force, acceleration of a test particle is independent of its inertial mass (e.g. in classical electrodynamics, acceleration of a test charge is proportional to the ratio of its charge to mass).
Imagine that a small bundle of test bodies are freely falling in an arbitrary gravitational field. Since their accelerations due to gravity are nearly identical, if one were to sit on one such particle and observe the rest, one would find that the other test particles are freely floating as though gravity has simply disappeared! This is the principle of equivalence which states that no matter how strong or how time varying the gravity is, one can always choose a small enough frame of reference for a sufficiently small time interval such that gravity vanishes in this frame.
This small region is a locally inertial frame of reference, and laws of physics in this frame take the same form as they do in special theory of relativity. In special relativity, the proper distance ds between two nearby events with space-time coordinates x µ = (ct, x, y, z) and x µ + dx µ = (ct + cdt, x + dx, y + dy, z + dz) is given by, Note that in eq.(1), x i , i=1,2,3 are the Cartesian coordinates of the event, and η µν is the Minkowski metric with η 00 = 1 = −η ii , rest of the components of the metric being zero.
Einstein's summation convention has been used in eq.(1), so that repetition of Greek indices imply summation over 0,1,2 and 3.
In 3-dimensional Euclidean geometry, the line-element dl 2 = dx 2 + dy 2 + dz 2 has the same form whether you shift the Cartesian coordinate system by any constant vector or rotate the coordinate system about any axis by any constant angle. The line-element given by eq.(1) is similarly invariant under Lorentz transformations as well as constant space-time translations. According to the equivalence principle, whatever is the gravity around, in a locally inertial frame (i.e. freely falling frame), the line-element is given by eq.(1) and nongravitational laws of physics take the same form as in special relativity. But, what is the connection between this feature of gravitation and geometry?
Consider a generally curved two-dimensional surface (e.g. the surface of, say, a pear). No matter how greatly the surface is curved, one can always choose a tiny enough patch on it, such that it is sufficiently flat for Euclidean geometry to hold good over it. As one increases the size of the patch, the curvature of the pear's surface becomes apparent. This is so similar to the main characteristic of gravity that we discussed in the preceding paragraph.
The small patch on the pear over which the line-element is Euclidean ( is analogous to the local inertial frame in the case of 4-dimensional space-time where the line-element is described by eq.(1).
By going over to a small freely falling frame and choosing inertial (many a times called Minkowskian) coordinates, one manages to make gravity vanish so that special relativity is all that one needs to describe laws of physics, locally. What if one wants to study laws of physics over larger regions of space-time? In that case, one would have to employ other coordinates that are curvilinear in general. Then it ensues that, instead of the Minkowski metric, one would require a general metric tensor.
The fundamental entity in GR that describes space-time geometry is the space-time dependent metric tensor g µν (x α ), which determines the invariant proper distance ds between any two nearby events with coordinates x µ and x µ + dx µ , Here, x µ , µ, ν=0,1,2,3, now represents a general curvilinear coordinate, specifying the location of an event. The metric g µν is a generalization of η µν , the Minkowski metric tensor. If the space-time geometry was not curved, one could choose a coordinate system such that everywhere the metric tensor is just the Minkowski metric tensor. But GR states that energy and momentum associated with matter warp the space-time geometry, entailing that in general it is not possible to choose inertial coordinates everywhere so that the metric is globally Minkowskian.
However, according to the principle of equivalence, by choosing an appropriate coordinate system, even in an arbitrarily curved space-time, the metric tensor can be made to take the form of η µν in a sufficiently small space-time region (physically, this corresponds to choosing a sufficiently small freely falling frame). In GR, the mathematical forms of physical laws remain the same even when one makes an arbitrary coordinate transformation.
Although in a local inertial frame, gravitational force disappears, tidal force does not.
For instance, earth is freely falling towards the sun because of latter's pull. But we do not feel sun's gravity since the freely falling earth constitutes a local inertial frame. However, as sun's gravity is non-uniform, portions of earth closer to the sun feel a greater tug than those located farther. This differential pull is the source of tidal force which causes the commonly observed ocean tides. In GR, the tidal acceleration is due to a fourth rank tensor called Riemann tensor that is constructed out of the metric and its first as well as second derivatives. Therefore, the ocean tides owe their existence to the non-zero Riemann tensor describing the curvature of space-time geometry around the sun (as well as the moon).
GR tells us that matter distorts the space-time from an Minkowskian geometry to a non-Minkowskian one, and test bodies just move along straightest possible paths in such a curved space-time. As to, how the matter warps the space-time geometry, is given by the so called Einstein equations which relate tensors created out of the metric and the Riemann tensor to the matter energy-momentum tensor multiplied by a combination of Newton's constant G and light speed c. Einstein equations possess a pristine beauty, with space-time geometry on one side, and the energy and momentum of matter on the other. When the geometrical curvature of space-time is small and the motion within the source is slow enough, GR leads automatically to Newton's laws of gravitation.
GR as a theory of gravitation gained immediate acceptance among the physics community as soon as its prediction of bending of light was actually seen during the solar eclipse of 1919.
Of course, GR had already correctly explained the anomalous precession of the perihelion of Mercury. Later, existence of gravitational waves (ripples in space-time geometry) predicted by Einstein was also verified with the discovery of inspiralling Hulse-Taylor binary pulsar 1913 + 16. Indeed, gravitational effects too propagate as gravitational waves with finite speed c, consistent with the demands of special relativity.

GEOMETRY OF THE UNIVERSE
Armed with these successes, Einstein in 1917 turned his attention towards building a general relativistic model of the whole universe. He found that GR is unable to produce a static universe. Astronomers, those days, believed that stars and nebulae do not exhibit any large scale ordered motion. The prevalent view in the past was that universe on large scales is static. GR, being a theory of attractive gravity, predicted that a large mass (like our universe) will either keep collapsing under its own weight or exhibit deceleration in its expansion rate if it was growing in size to begin with. In either case, a static universe from GR was out of question. Einstein was in a fix.
His next step was to modify GR by introducing a term representing a kind of universal repulsion. This extra feature Λ g µν , called the cosmological constant term, on the left hand side of Einstein equations helps in preventing gravitational collapse of the universe and, hence, leads to a static solution provided that value of the constant is positive (corresponding to cosmic repulsion).
In 1929, the renowned American astronomer Edwin Hubble while studying the spectra of radiation from galaxies, discovered that spectroscopic lines were shifted to the red end as though there was some kind of a Doppler redshift. The so called Hubble's law stated that the redshift z of a galaxy satisfies z = H 0 c d, where d and H 0 = 72 km/s/Mpc are the distance of the galaxy from us and Hubble constant, respectively. Therefore, it appears that galaxies recede from each other with a speed proportional to their distance of separation.
According to Hubble's law, if the distance between two galaxies is 100 Mpc (i.e. about 326 million light years), the separation between the galaxies would be increasing at the rate of about 7200 km/sec. The universe actually is not static at all! Instead, the universe is expanding. Einstein called the action of introducing a cosmological constant term in GR to be his greatest blunder. Why? Because, he could have predicted that universe is not static, purely from theoretical calculations using his original theory.
In fact, way back in 1922, Friedmann had discovered (without using the Λ -term) general relativistic solutions representing homogeneous, isotropic and expanding (or, contracting) universe. From the point of view of observations, although matter is clumpy and inhomogeneous at small scales, if one considers distribution of galaxies and clusters of galaxies on scales larger than about 500 Mpc, the distribution of matter is remarkably homogeneous and isotropic. Exploiting these symmetries, relativists like Friedmann, Lemaitre, Robertson and Walker could write down the line-element describing the geometry of such an isotropic and homogeneous universe, where the constant k can take either the value 0 (spatially flat universe, infinite in extent), +1 (closed universe without a boundary, having a finite spatial volume) or -1 (open universe, infinite in extent).
It is evident from eq.(3), that the time dependent function a(t) (called the scale factor) determines the spatial distance between a galaxy at (r, θ, φ) and another at (r+dr, θ+dθ, φ+ dφ). An increasing a(t) with time describes an expanding universe. Hubble's law can be explained as follows: Light from a galaxy starts at time t and after travelling an enormous distance, reaches us at time t 0 . During this period, the scale factor increases from a(t) to a(t 0 ), stretching the wavelength of light, and hence the redshift. The observed redshift is mainly due to the expansion of the universe and not due to any Doppler shift. One can show that the redshift z is given by, Farther the galaxy, earlier is the time t of emission, smaller is a(t), and therefore, larger is the redshift z. Distant galaxies and quasars have been observed with redshifts as high as 6, implying that radiation from such distant objects started when the universe was smaller by a factor of 7. The Hubble constant is nothing but, The geometry described by eq. (3) is applicable only at cosmic scales (since homogeneity and isotropy is valid only at such large scales). The sizes of planets, stars or galaxies do not get affected by the cosmological expansion.
If one takes the Friedmann-Robertson-Walker metric from eq.(3) and uses it in the Einstein equations corresponding to a homogeneously and isotropically distributed matter, then one finds that the scale factor a(t) satisfies, and,ä where ǫ and p are the total energy density and total pressure, respectively, of the contents of the universe.
When k = 0 (flat universe) and matter is non-relativistic so that p ≪ ǫ, one finds from eqs. (6) and (7) that a(t) is proportional to t 2/3 . From eq.(4), one has the redshift-age relation for k=0 matter-dominated model, For such a model, the Hubble constant (eq.(5)) is simply, whereby the knowledge of H 0 = 72 km/s/Mpc implies that the universe right now is about 13 billion years old.
From Friedmann-Robertson-Walker equations (FRWE) (6) and (7), it follows that if the universe is just made up of standard matter like protons, neutrons, electrons, photons, neutrinos etc., the rate of expansion of the universeȧ slows down steadily with time (ä < 0), as energy density and pressure are positive quantities. This is natural since gravity is attractive. Just to give you an analogy, when one throws a ball up with a high velocity, although the distance between earth and the ball increases with time, the latter's velocity keeps decreasing because of earth's gravity.
Had there been an anti-gravity between the ball and earth, the velocity would have increased with time. Fortunately, inertial masses are always positive (otherwise, kinetic energy will be negative, leading to runaway situations, e.g. extraction of energy from a moving negative mass object would make the object travel faster!), entailing an attractive gravitation that keeps us grounded to the earth. But, is there no possibility of gravitational repulsion?

AND DARK ENERGY
In GR, even pressure associated with matter causes space-time geometry to be curved (e.g. eq. (7)). This is expected as pressure is related to the density of random kinetic energy of particles making up normal matter. Energy density is usually taken to be positive, otherwise matter would not be stable (as matter would keep decaying to more and more negative energy density states). For standard matter, thermodynamic pressure is positive.
Under such conditions, gravity can only be attractive.
On the other hand, if a weird form of matter with a sufficiently large negative pressure permeates space, it will result in anti-gravity, since from eq.(7) the expansion rate would increase with time (i.e.ä > 0). From the first law of thermodynamics one knows that when ordinary gas expands adiabatically, its internal energy decreases since it performs work against the surroundings while growing in volume. Now imagine that one has expanding exotic matter (having negative pressure) instead. Its internal energy would then grow while expanding! It is such counter-intuitive properties that lead to cosmic repulsion when GR is combined with presence of exotic matter.
For the cosmological constant, one can show from the first law of thermodynamics that while expanding, its internal energy grows in such way as to maintain a constant energy density (see eq.(14) below). It is of course consistent with the fact that cosmological constant is actually a constant .
As Einstein and Lemaitre had envisaged, a sufficiently large and positive cosmological constant would lead to repulsive gravity on cosmic scales, since pressure p Λ in the case of cosmological constant Λ is exactly negative of its energy density ǫ Λ . But how would one determine observationally whether in our universe the second derivative of a(t) is positive or negative? This is where the investigations of Perlmutter, Riess and Schmidt assume importance.
In the framework of Euclidean geometry, consider a source placed at a distance d from us, emitting radiation isotropically with a luminosity L. The flux of radiation F received by us is given by, since in unit time, L amount of energy crosses a spherical surface of area 4πd 2 . But when the space-time geometry is curved as given by eq. (3), it can be shown that, where d L (defined to be the luminosity-distance) is given by, where r and z are the radial coordinate and the redshift of the source, respectively, while t is the time when the radiation left the source to reach us (i.e. r = 0) at time t 0 .
In eq.(12), presence of a(t 0 )r is normal since it is the physical distance of the source at the present epoch t 0 . The factor 1 + z arises because of the stretching of time intervals due to cosmological expansion. Photons radiated from the source in a time interval dt is received by us over a period of (1 + z)dt.
Now, particles with zero rest mass like photons (light quanta) travel with speed c in local inertial frames, so that from eq.(1) one has ds 2 = 0 for light. From equivalence principle, therefore, ds 2 = g µν dx µ dx ν = 0 for zero rest mass particles in any coordinate system. In particular, in the cosmological setting, when light from a distant source moves along a radial ds 2 = 0 trajectory to reach us, one has from eq.(3), after taking the necessary square roots, One can solve the FRWE (eqs. (6) and (7)) to obtain a(t) and thereafter, eq.(13) can be integrated to express the radial coordinate of the source r as a function of its redshift z, so that the luminosity-distance d L (eq.(12)) becomes a function of z, H 0 and the value ofä at the present epoch t 0 [8].
Let us derive the expression for luminosity-distance when k = 0 and the cosmological constant is positive, in an otherwise empty universe. The energy density ǫ Λ associated with a positive cosmological constant Λ is given by with pressure, When eqs. (14) and (15) are substituted in the FRWE for k = 0 case, one obtains, implying that indeed there is cosmic repulsion, sinceä > 0. In the flat and empty universe with a positive cosmological constant,˙a a does not change with time (eq.(16)), so that the Hubble constant (eq. (5)) is independent of the epoch.
Solving the simple differential eq.(16) for an expanding universe, one obtains the scale factor, Substituting eq.(18) in eq.(13) with k = 0, one gets after integrating, where use has been made of the redshift-scale factor relation given by eq.(4).
From eqs. (12) and (19), the luminosity-distance in this model is simply, Using a similar procedure, and making use of eqs. (8) and (13) with k = 0, it is easy to work out the following expression for luminosity-distance in the case of flat matter-dominated model, It is evident from a comparison of eqs.(20) and (21) that the luminosity-distance in the case of an accelerating universe increases with redshift much more rapidly than that in a decelerating universe. In general, when there is matter as well as a non-zero cosmological constant, one can derive the expression for d L (z) following the above procedure. Also, for  A white dwarf, with a companion star, both going round their common centre of mass due to their mutual gravitational attraction, is likely to accrete matter steadily from its companion. At some point of time, the infalling matter raises the white dwarf's mass to above the threshold of Chandrasekhar limit. The core then becomes gravitationally unstable and implodes, giving birth to a SNe Ia.
The sudden decrease in the gravitational potential energy as the core collapses rapidly to a smaller radius results in the release of a huge amount of energy that blows apart the star.
A core with mass M c , shrinking from a large size R to a compact radius R c , has to give up an energy, since its gravitational potential energy decreases from ∼ − GM 2 c R to ∼ −E given by eq.(22). For a 1.4 M ⊙ core collapsing to form a neutron star of radius R c ≈ 10 km, the explosion energy E could be as high as ∼ 10 53 ergs.
The light curve of a typical supernova exhibits radiation luminosity rising with time rapidly to reach a peak value, followed by a gradual decay over a period of 30 to 40 days. According to quantum mechanics, the ground state energy of a harmonic oscillator is nonzero. It is also a common knowledge that every physical field is associated with a particle.
For example, electromagnetic field is associated with photons, Dirac field with electrons or quarks, and so on. There is some kind of a field-particle duality in quantum theory which is a generalization of de Broglie's wave-particle duality.
Any quantum field can be decomposed into infinitely many Fourier modes. It can be shown that the dynamics of each Fourier mode is analogous to that of a simple harmonic oscillator. Hence, for each Fourier component there is a quantum zero point energy. This implies that the ground state energy (i.e. the lowest energy) of the quantum field, which is a sum of all these zero point energies, is infinite! This is not so much of a problem in non-gravitational quantum theory. The reason is as follows.
The ground state of the total Hamiltonian encompassing all the quantum fields of nature is called the vacuum state (because, this state corresponds to 'no particle' state, i.e. literally a vacuum like condition). A single particle state is analogous to the first excited state, and so on. The energy of a single particle state is also infinite (obviously, as the vacuum or the lowest energy itself is infinite). Normally in experiments one deals with particle states, and hence, one can say that one will deal only with the difference of energy between the particle state and the vacuum. This is of course finite. So, by redefining the vacuum energy to be the 'zero level' one can consistently talk about the observed finite energy of particles. After all, the zero of a potential energy has no observable significance -one can add or subtract any constant quantity. It is only the change in the potential energy that is measurable. So, why should throwing away of vacuum energy be any different, right? Wrong.
The ground state actually has non-trivial significance. Firstly, vacuum state gets influenced when the boundary conditions are altered. For example, if one has two large, thin and parallel conducting plates, the electric field has to vanish on the plates. Because of this boundary condition, the Fourier modes corresponding to the electromagnetic (EM) field within the plates are discrete. That is, the EM wavelengths are quantized (just like it happens in the case of a stretched wire kept clamped at two points). The physicist Casimir showed that the difference between the vacuum energy outside the plates and between them is finite and increases with the plate separation. That means, if one takes two extremely smooth conducting plates, there should be a tiny force of attraction between them. This Casimir effect has been experimentally seen. Note that the force of attraction is not due to EM force. Rather, it is because of the non-trivial vacuum or the ground state configuration of the vacuum EM field.
Secondly, throwing away the vacuum energy is strictly illegal, as all forms of energy warp the space-time geometry (i.e. even vacuum energy is a source of gravity). An infinite vacuum energy is of course a bigger worry when gravitational interaction is brought in. So, how does one include vacuum energy's contribution to gravity?
The celebrated Russian physicist Zeldovich had drawn attention towards a very interesting point. We know that the Minkowski metric tensor is invariant under Lorentz transformations and space-time translations. That is, for any observer in any arbitrary inertial frame, the metric tensor is η µν . Also, the vacuum state must be invariant under Lorentz transformations as well as space-time translations. In other words, a 'no particle' state in an inertial frame S must be a 'no particle' state in any other inertial frame S ′ . Otherwise, an observer in an inertial frame can find out her/his absolute velocity using a particle detector in vacuum, which is absurd. Therefore, the energy-momentum tensor of the vacuum must look the same in all inertial frames. This means that the energy-momentum tensor corresponding to the vacuum state must be locally of the form V η µν (where V is a constant vacuum energy density). Why?
Because it is the only second rank tensor invariant under both Lorentz transformations as well as space-time translations, so that vacuum or the 'no particle' state in any inertial frame is still the'no particle' state in any other inertial frame.
In a local inertial frame, the metric tensor is identical to the Minkowski tensor. Hence, in an arbitrary coordinate system, vacuum energy-momentum tensor must be of the form V g µν .
This implies that both vacuum energy-momentum tensor and the cosmological constant term have similar forms. Einstein's cosmological constant term appears on the L.H.S. of Einstein equations in GR while the vacuum energy-momentum tensor sits on the right, along with the matter energy-momentum tensor.
There arises an interesting possibility. Suppose, Einstein's cosmological constant Λ and the vacuum energy density V are both infinite but their difference is finite and small. This tiny remnant can act as an effective cosmological constant. Now, that can explain all the current cosmological observations concerning accelerated expansion of the universe. But there is a catch! Why should the two infinite quantities be so adjusted as to leave a tiny difference? This is the so called cosmological constant fine tuning problem [9,10].
The riddle of cosmological constant persists even though the original motivation for its introduction has long vanished after the discovery of Hubble's law. It is like the smile of the Chesire cat -the cat disappeared but its smile remained! Up there, Einstein would now be smiling with the kind of havoc that his 'blunder' has been causing.