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Is the cosmological constant an eigenvalue?


We propose to reinterpret Einstein’s field equations as a nonlinear eigenvalue problem, where the cosmological constant \(\Lambda \) plays the role of the (smallest) eigenvalue. This interpretation is fully worked out for a simple model of scalar gravity. The essential ingredient for the feasibility of this approach is that the classical field equations be nonlinear, i.e., that the gravitational field is itself a source of gravity. The cosmological consequences and implications of this approach are developed and discussed.

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  1. 1.

    An equation almost identical to Eq. (5) (apart from a factor of 2) can be derived directly from Einstein’s field equations by considering a metric where only the time-time component differs from its Minkowski value and requiring it to approach the Minkowski metric for large spatial distances [21, 22].

  2. 2.

    Equation (9) implies a modification of Newton’s acceleration in the gravitational field generated by a pointlike mass m, which becomes: \(g(r)=-Gm/r^2 + c^2 \Lambda \,r\). Such modification becomes sizeable only on cosmological scales. See [27] for further details.


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I wish to thank Gabriel Chardin for his thorough reading of the manuscript and many insightful comments. I also thank Omar Morandi and Raffaele Chiappinelli for helping with some mathematical issues. Needless to say, I am solely responsible for the errors or inaccuracies that may still remain in this paper.

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Appendix A: mathematical digressions

Appendix A: mathematical digressions

Appendix A.1: nonlinear eigenvalue problems

Nonlinear eigenvalue problems occur frequently in the mathematical literature. A very readable review was published recently [26]. More extensive discussions can be found in the two monographs [41, 42]. A nonlinear eigenvalue problem can be written generally as: \({{\mathcal {F}}}(\lambda ,u)=0\), where \(\lambda \) is the eigenvalue, \({{\mathcal {F}}}\) is a nonlinear function, and u usually belongs to a Banach space. In our case, the problem takes the special form: \({{\mathcal {F}}}(u)=\lambda u\). A typical example is the p-Laplacian eigenvalue problem with Dirichlet boundary conditions:

$$\begin{aligned}&\text {div} (|\nabla u|^{p-2} \nabla u) + \lambda |u|^{p-2} u = 0, \quad \mathrm{in}\,\, \Omega , \end{aligned}$$
$$\begin{aligned}&u = 0, \quad \mathrm{on }\,\, \partial \Omega \end{aligned}$$

where \(\Omega \subset {\mathbb {R}}^3\), and \(p>1\) is an integer [43]. Note that the above equation is homogeneous in u, just as Eq. (8) is homogeneous in \(\Phi \). This means that if u is a solution, then Cu is also a solution, for any real or complex number C. This is a property shared with linear equations.

Appendix A.2: relevant functional spaces

Let us consider the following linear eigenvalue problem, which corresponds to Eq. (10) in 1D, with \(2\pi G/c^2=1\) and \(\lambda =\Lambda /2\):

$$\begin{aligned}&-u_{xx} + \rho u = \lambda u, \quad x \in I \equiv [0,2\pi ], \end{aligned}$$
$$\begin{aligned}&u(0)=u(2\pi )=1, \end{aligned}$$
$$\begin{aligned}&u_x(0)=u_x(2\pi )=0, \end{aligned}$$

with \(\rho \ge 0\) and \(\int _I \rho (x)dx < \infty \), and the subscript stands for differentiation. Elliptic PDEs are usually defined in Sobolev spaces [44], because such spaces guarantee the existence of the derivatives, at least in a weak sense. Here, the appropriate space seems to be \(H^1\), which is also a Hilbert space equipped with the inner product \((u,v)=\int _I uv dx + \int _I u_x v_x dx\) and the related norm \(\Vert u\Vert =\sqrt{(u,u)}\).

We first consider the simple case where \(\rho =0\) (no matter density). Then, the eigenfunctions of Eq. (A3) are cosines: \(\varphi _n = \cos (nx)\), with eigenvalues \(\lambda _n = n^2\). Then, if a solution u(x) exists for the full Eq. (A3) with non-vanishing \(\rho \), it can be represented as:

$$\begin{aligned} u(x) = \frac{\sum _n a_n \varphi _n(x)}{\sum _n a_n}, \end{aligned}$$

which satisfies the required boundary conditions (\(a_n\) are real numbers). Of course, we have not proven that such a solution actually exists, which is a nontrivial mathematical problem.

In an infinite space (\(I = {\mathbb {R}}^3\)), the norm generally diverges, because of the boundary condition on the first derivative (A5). However, this point should not be considered crucial: for instance, non-integrable wave functions are routinely used as solutions of the standard Schrödinger equation to describe propagating plane waves. In addition, as we have seen in the main text, these boundary conditions are to be applied on the Hubble sphere, i.e. in a finite volume, so this problem should not actually arise.

Appendix A.3: elliptic equations with Cauchy boundary conditions

An elliptic PDE, such as Eq. (A1), with Dirichlet boundary conditions, constitutes a well-posed problem. In contrast, if one takes Cauchy boundary conditions, the related problem is ill-posed [28]. Cauchy boundary conditions correspond to specifying both the function and its normal derivative on the boundary, e.g. \(u = \partial u/\partial n=0\) on \(\partial \Omega \). In that case, the problem does not always have a solution. Our Eqs. (10), (12), and (13) fall in this category.

However, as an eigenvalue problem, the problem makes perfect sense. The eigenvalue \(\lambda \) is determined precisely by the requirement that the problem does possess a solution for Cauchy boundary conditions. We can illustrate this on a simple problem in 1D. Let us consider the following first-order nonlinear differential equation:

$$\begin{aligned}&u^2 u_x + u = \lambda u, \quad x \in [0,1], \end{aligned}$$
$$\begin{aligned}&\quad u(0) = 0, \quad u_x(1) =\sqrt{2}/2 . \end{aligned}$$

The problem is obviously overdetermined, as a 1D differential equation admits only one boundary condition. Hence, for fixed \(\lambda \), Eq. (A7) does not generally admit a solution respecting both boundary conditions (A8). This is easily verified by checking that a solution of Eq. (A7) is \(u(x) = u_0 \sqrt{x}\), with \(u_0=\pm \sqrt{2(\lambda -1)}\), which satisfies the boundary condition in \(x=0\) but not (for arbitrary \(\lambda \)) in \(x=1\). But if one treats Eq. (A7) as an eigenvalue problem, the second boundary condition becomes a constraint that fixes the eigenvalue, in this case \(\lambda =2\). This is precisely what happens for our problem: adding the extra boundary condition on the gradient of \(\Psi \) on the boundary of the domain determines the value of \(\Lambda \).

As a further ‘physical’ example, let us consider the 1D heat equation with Cauchy boundary conditions:

$$\begin{aligned}&T_t = T_{xx} - S(x) T + \lambda T, \quad x \in [0,L], \end{aligned}$$
$$\begin{aligned}&T(x,0) = T_0, \end{aligned}$$
$$\begin{aligned}&T(0,t) = T(L,t) = T_0, \end{aligned}$$
$$\begin{aligned}&T_x(0,t) = T_x(L,t) = 0 , \end{aligned}$$

where T(xt) is the local temperature at an instant t, \(-S(x) T\) is a heat sink, and \(\lambda T\) is a heat source. The above problem corresponds to a system initially at temperature \(T_0\) everywhere, which evolves under the action of the sinks and sources of heat. The boundary conditions prescribe that the temperature must remain equal to \(T_0\) at \(x=0\) and \( x=L\) [Eq. (A11)] and that the heat flux must vanish at the boundaries [Eq. (A12)]. This is not physically realizable, of course: if no heat can escape the system, then the temperature at the boundaries cannot be fixed arbitrarily, but will be determined by the interplay of the sinks and sources. However, if \(\lambda \) is not fixed but rather considered as an eigenvalue, then the Cauchy boundary conditions can indeed be satisfied. Physically, this means that the source term \(\lambda T\) is tuned precisely so as to keep the temperature equal to \(T_0\) at the two boundaries. This determines the value of \(\lambda \).

We also note that a steady-state solution of Eq. (A9) corresponds to a solution of our model Eq. (10), in a 1D planar geometry. Indeed, the numerical results shown in this work were obtained by propagating the field \(\Psi (r,t)\) according to a heat-type equation like Eq. (A9), so that the solution relaxes naturally to the lowest-order eigenfunction.

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Manfredi, G. Is the cosmological constant an eigenvalue?. Gen Relativ Gravit 53, 31 (2021).

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  • Cosmological constant
  • Scalar gravity
  • Cosmology
  • Dark matter
  • Dark energy