Conformal Cosmology with a Positive
Effective Gravitational Constant

Mannheim [6] (henceforth M6) has developed a cosmological model (MM) based on conformal invariance. This model has a number of attractive features, but has not so far been accepted as a viable candidate for the correct theory of gravitation, in large part because it predicts, apparently unambiguously, a negative value for the effective gravitational constant, $G_{\rm eff}$. From this follows a thermal history markedly different from the usual one, and consequent difficulties in addressing questions such as primordial nucleosynthesis [4].

In this paper we point out a mathematical problem in the formulation of the model, and present a new approach that yields a positive $G_{\rm eff}$, so that the model closely resembles the conventional one.

The MM involves a scalar field, $S$, in a Friedmann-Robertson-Walker (FRW) background metric. It is this field $S$ that we will be concentrating on in this paper. The treatment of both $S$ and the gravitational field in M6 is classical. Important subsequent work has been done on the quantization of the fourth-order equations for the gravitational field that result from the conformal Lagrangian [7]. The scalar field, $S$, however, has always been treated classically, a procedure that we will question in this paper.

There are several things that are required of $S$ in a viable model:

• $\bullet$ 1

  An equation of motion that is conformally invariant.

• $\bullet$ 2

  An energy-momentum tensor whose 00 component (Hamiltonian) has an energy spectrum that is bounded below.

• $\bullet$ 3

  A constant vacuum expectation value, $S_{0}$, derived from the equation of motion. The $S_{0}^{2}$ in the action generates the effective gravitational constant, $G_{\rm eff}$.

• $\bullet$ 4

  A $S_{0}^{4}$ term in the action that gives the correct sign for the cosmological constant.

Let us see whether these criteria are met in the MM. Using (as Mannheim does) a metric signature $-,\;+,\;+,\;+$, and neglecting the coupling to the fermion field, the terms in Mannheim’s Lagrangian involving $S$ are (M6 (61)):

This results in an equation of motion (M6 (63)):

This equation, as required, is conformally invariant.

$S$ is assumed to acquire a non-zero vacuum expectation value, $S_{0}$, found by setting the derivative term in (2) equal to zero. This gives

as in [5] (13), with $h=0$.

Further development of the model (M6, section 10) shows that, in order to meet observational criteria at the present time, $R^{\mu}_{\;\;\mu}$ and $\lambda_{\rm M}$ are both negative, so that $S_{0}^{2}$ is real and positive. The effective gravitational constant, $G_{\rm eff}$, in the MM turns out to be negative (M6 (224)):

choosing units with $c=1$.

The stress tensor in the MM model is obtained by taking the variation of the action with respect to the metric, in the usual way (M6 (64); a similar expression is given in [3]). Retaining just the terms involving $S$ we get

The Lagrangian (1) is analogous to the Minkowski space Lagrangian for a $\phi^{4}$ field theory ([8] (7.2.14), with ${\cal H}(\phi)=\mu^{2}\phi^{4}$):

The $S^{2}$ term in (1) has the “right” sign for an $m^{2}$ term. But with $\lambda_{M}<0$, the $S^{4}$ term has the “wrong” sign, and, in a conventional treatment, will lead to a spectrum that is not bounded below. A theory of this sort has to be treated by methods appropriate to non-Hermitean Hamiltonians [1]. At the level of quantum mechanics, “wrong sign” $\phi^{4}$ theory is well developed. Nevertheless, even in Minkowski space, constructing a corresponding quantum field theory is a difficult problem, still incompletely understood [2]. Conformal cosmology will remain flawed until we can make progress in understanding the scalar field.

A conformally invariant theory with a single scalar field has a unique action, M6 (61), provided we use the familiar techniques appropriate for Hermitean Lagrangians. Once we recognize that our theory involves a non-Hermitean Lagrangian, however, a new approach is suggested, that we introduce in this section. We begin by working in Minkowski space, but retain $g_{\mu\nu}$ and $(-g)^{1/2}$ in formulae to simplify the transition to a FRW space.

A $\phi^{4}$ theory with a “wrong sign” $\phi^{4}$ term is non-Hermitean but is nevertheless ${\cal PT}$ symmetric, and can be treated by the methods outlined in [1]. The distinctive feature of this approach is the use of the ${\cal CPT}$ norm in place of the usual Dirac norm; for a quantized field, ${\mathsf{S}}$, we write this norm as

Here ${\cal P}$ and ${\cal T}$ represent the usual parity and time-reversal operations, while ${\cal C}$ represents a special operation designed to ensure the norm is real and positive definite and the theory is unitary. The ${\cal C}$ operator has to be specifically calculated for each Hamiltonian.

Our cosmological model is written in terms of classical fields (expectation values, $S(x)$), which we take to be real. We assume $S^{\cal CPT}(x)$ can then be expressed as

(compare [1] (78)), so that

In an expression of this sort, $S(x)$ and $S^{*}(-y)$ describe the field $S$ at the same physical point, but use different coordinate systems to refer to that point.

Take the complex conjugate of (9), and let $x^{\mu}\rightarrow-y^{\mu}$ and $y^{\mu}\rightarrow-x^{\mu}$:

showing that $C(x^{\mu}-y^{\mu})$ must be real.

Mannheim points out (M6, section 10) that because the FRW space is conformally flat, the cosmological equation of motion (CEM) reduces to

$T^{\mu\nu}_{\rm total}$ is just the sum of (18) and $T^{\mu\nu}_{\rm kin}$, the contribution from ordinary matter (fermion fields, electromagnetic radiation, etc.). So our CEM, analogous to M6 (222), becomes

or, as in M6 (223)

with $\Lambda_{\rm M}\equiv\lambda_{\rm M}S_{0}^{4}<0$.

The $S_{0}^{2}$ term defines the effective gravitational constant in the theory. We find

analogous to M6 (224), but wth a positive sign.

Of particular interest, since it permits an analytic solution, is a model containing radiation only (M6, section 10.2). A corresponding solution exists for the present model also; let us see how it differs. The equation analogous to M6 (226) is (with $H\equiv\dot{R}/R$):

Since $k<0$ for the open geometry of the MM, all three terms in (27) are positive, in contrast to M6, where $\overline{\Omega}_{M}$ is negative.

Write the solution as in M6 (230), with $\rho_{M}(t)=A/R^{4}$:

The first term on the right of (28) is negative, not positive as in the MM. This means that $R$ must start from zero, as in the standard cosmology, not from some finite value, as in the MM. This is illustrated in figure 1. Both curves were drawn using the formula (28), but in the upper graph $\beta>1$, while in the lower graph $\beta<1$. The origin of $t$ is conventionally shifted in the lower graph so that $t=0$ is at point B, where $R=0$.

The present model has two features that are not present in conventional cosmology based on Einstein’s equations.

First, the model includes a scalar field that is essentially massless. The non-zero vacuum expectation value of this field is essential, but we have ignored any excitations. This may be permissible because the field couples very weakly to normal matter, and is difficult to observe, or it may undergo a spontaneous transition that renders it massive.

Second, Mannheim uses (19) for his basic cosmological equation, rather than the more complete one that results from the conformal action, M6 (188):

where $W^{\mu\nu}$ is the Weyl tensor, defined in M6 (107), (108) and (185). Mannheim justifies the neglect of $W^{\mu\nu}$ by observing that a FRW metric is conformally flat, and in such a space $W^{\mu\nu}=0$. This is true, but if we are to include perturbations to the metric (as, for example, in the study of anisotropies of the CMB) then this neglected term may become important.

We have pointed out two flaws in Mannheim’s conformal cosmological model.

• $\bullet$ 1

  The model predicts, apparently unambiguously, a negative value for the effective gravitational constant, $G_{\rm eff}$.

• $\bullet$ 2

  The model involves a scalar field, $S(x)$, that satisfies a conformally invariant equation of motion and develops a vacuum expectation value, $S_{0}$. The values of the parameters that are needed to satisfy observations lead to a “wrong sign $S^{4}$ theory”, with a Hamiltonian that has a spectrum that is unbounded below.

We have attempted to apply the techniques appropriate for such Hamiltonians to this cosmological problem, restricting ourselves to the classical limit of field equations that are still imperfectly understood. In this limit, using assumptions that appear reasonable, we find both a positive value for $G_{\rm eff}$ and a positive definite spectrum for the Hamiltonian.

Our derivation depends on one simple observation, the change of sign as we go from (41) to (42).

The derivation presented here will remain conjectural until progress is made in two main directions:

• $\bullet$ 1

  The techniques that have been successfully applied to $\phi^{4}$ quantum mechanics will have to be developed to cover the corresponding quantum field theory; this seems not to have been achieved at this time [2]. In particular, the form (8) for $S^{\cal CPT}(x)$ must be shown to be appropriate at the classical level.

• $\bullet$ 2

  The various manipulations we have employed are suitable for Minkowski space, but more detailed investigations are needed to show whether they can legitimately be extended to a FRW space.

If, on the other hand, we can accept the present model as viable, without first filling in these important gaps in our understanding, then we have to face the difficult question: how can we conclusively distinguish this model from the conventional one?