1.1. The many faces of the cosmological constant
Einstein's equations, which determine the dynamics of the spacetime, can be derived from the action (see, eg. [7]):
(1) |
where L_{matter} is the Lagrangian for matter depending on some dynamical variables generically denoted as . (We are using units with c = 1.) The variation of this action with respect to will lead to the equation of motion for matter (L_{matter} / ) = 0, in a given background geometry, while the variation of the action with respect to the metric tensor g_{ik} leads to the Einstein's equation
(2) |
where the last equation defines the energy momentum tensor of matter to be T_{ik} 2(L_{matter} / g^{ik}).
Let us now consider a new matter action L'_{matter} = L_{matter} - ( / 8G) where is a real constant. Equation of motion for the matter (L_{matter} / ) = 0, does not change under this transformation since is a constant; but the action now picks up an extra term proportional to
(3) |
and equation (2) gets modified. This innocuous looking addition of a constant to the matter Lagrangian leads to one of the most fundamental and fascinating problems of theoretical physics. The nature of this problem and its theoretical backdrop acquires different shades of meaning depending which of the two forms of equations in (3) is used.
The first interpretation, based on the first line of equation (3), treats as the shift in the matter Lagrangian which, in turn, will lead to a shift in the matter Hamiltonian. This could be thought of as a shift in the zero point energy of the matter system. Such a constant shift in the energy does not affect the dynamics of matter while gravity - which couples to the total energy of the system - picks up an extra contribution in the form of a new term Q_{ik} in the energy-momentum tensor, leading to:
(4) |
The second line in equation (3) can be interpreted as gravitational field, described by the Lagrangian of the form L_{grav} (1/G)(R - 2), interacting with matter described by the Lagrangian L_{matter}. In this interpretation, gravity is described by two constants, the Newton's constant G and the cosmological constant . It is then natural to modify the left hand side of Einstein's equations and write (4) as:
(5) |
In this interpretation, the spacetime is treated as curved even in the absence of matter (T_{ik} = 0) since the equation R_{ik} - (1/2)g_{ik}R - g_{ik} = 0 does not admit flat spacetime as a solution. (This situation is rather unusual and is related to the fact that symmetries of the theory with and without a cosmological constant are drastically different; the original symmetry of general covariance cannot be naturally broken in such a way as to preserve the sub group of spacetime translations.)
In fact, it is possible to consider a situation in which both effects can occur. If the gravitational interaction is actually described by the Lagrangian of the form (R - 2), then there is an intrinsic cosmological constant in nature just as there is a Newtonian gravitational constant in nature. If the matter Lagrangian contains energy densities which change due to dynamics, then L_{matter} can pick up constant shifts during dynamical evolution. For example, consider a scalar field with the Lagrangian L_{matter} = (1/2)_{i} ^{i} - V() which has the energy momentum tensor
(6) |
For field configurations which are constant [occurring, for example, at the minima of the potential V()], this contributes an energy momentum tensor T_{b}^{a} = ^{a}_{b} V(_{min}) which has exactly the same form as a cosmological constant. As far as gravity is concerned, it is the combination of these two effects - of very different nature - which is relevant and the source will be T_{ab}^{eff} = [V(_{min}) + ( / 8 G)] g_{ab}, corresponding to an effective gravitational constant
(7) |
If _{min} and hence V(_{min}) changes during dynamical evolution, the value of _{eff} can also change in course of time. More generally, any field configuration which is varying slowly in time will lead to a slowly varying _{eff}.
The extra term Q_{ik} in Einstein's equation behaves in a manner which is very peculiar compared to the energy momentum tensor of normal matter. The term Q^{i}_{k} = _{} ^{i}_{k} is in the form of the energy momentum tensor of an ideal fluid with energy density _{} and pressure P_{} = - _{}; obviously, either the pressure or the energy density of this "fluid" must be negative, which is unlike conventional laboratory systems. (See, however, reference [8].)
Such an equation of state, = - P also has another important implication in general relativity. The spatial part g of the geodesic acceleration (which measures the relative acceleration of two geodesics in the spacetime) satisfies the following exact equation in general relativity (see e.g., page 332 of [9]):
(8) |
showing that the source of geodesic acceleration is ( + 3P) and not . As long as ( + 3P) > 0, gravity remains attractive while ( + 3P) < 0 can lead to repulsive gravitational effects. Since the cosmological constant has (_{} + 3P_{}) = - 2_{}, a positive cosmological constant (with > 0) can lead to repulsive gravity. For example, if the energy density of normal, non-relativistic matter with zero pressure is _{NR}, then equation (8) shows that the geodesics will accelerate away from each other due to the repulsion of cosmological constant when _{NR} < 2_{}. A related feature, which makes the above conclusion practically relevant is the fact that, in an expanding universe, _{} remains constant while _{NR} decreases. (More formally, the equation of motion, d (_{} V) = - P_{} dV for the cosmological constant, treated as an ideal fluid, is identically satisfied with constant _{}, P_{}.) Therefore, _{} will eventually dominate over _{NR} if the universe expands sufficiently. Since ||^{1/2} has the dimensions of inverse length, it will set the scale for the universe when cosmological constant dominates.
It follows that the most stringent bounds on will arise from cosmology when the expansion of the universe has diluted the matter energy density sufficiently. The rate of expansion of the universe today is usually expressed in terms of the Hubble constant: H_{0} = 100h km s^{-1} Mpc^{-1} where 1 Mpc 3 × 10^{24} cm and h is a dimensionless parameter in the range 0.62 h 0.82 (see section 3.2). From H_{0} we can form the time scale t_{univ} H_{0}^{-1} 10^{10}h^{-1} yr and the length scale cH_{0}^{-1} 3000h^{-1} Mpc; t_{univ} characterizes the evolutionary time scale of the universe and H_{0}^{-1} is of the order of the largest length scales currently accessible in cosmological observations. From the observation that the universe is at least of the size H_{0}^{-1}, we can set a bound on to be || < 10^{-56} cm^{-2}. This stringent bound leads to several issues which have been debated for decades without satisfactory solutions.
At present we have no clue as to what the above questions mean and how they need to be addressed. This review summarizes different attempts to understand the above questions from various perspectives.