# Does a varying speed of light solve the cosmological problems?

###### Abstract

We propose a new generalisation of general relativity which incorporates a variation in both the speed of light in vacuum (c) and the gravitational constant (G) and which is both covariant and Lorentz invariant. We solve the generalised Einstein equations for Friedmann universes and show that arbitrary time-variations of c and G never lead to a solution to the flatness, horizon or problems for a theory satisfying the strong energy condition. In order to do so, one needs to construct a theory which does not reduce to the standard one for any choice of time, length and energy units. This can be achieved by breaking a number of invariance principles such as covariance and Lorentz invariance.

###### pacs:

PACS number(s): 98.80.Cq, 95.30.St^{†}

^{†}preprint: DAMTP-1999-31

## I Introduction

Inflationary models were originally proposed as a solution to some of the most fundamental problems of the standard cosmological model namely the horizon, flatness and monopole problems [1, 2, 3, 4]. In the context of inflation the solution to these problems is achieved through a period of very rapid expansion induced by a huge vacuum energy.

Despite the lack of a single, well motivated particle physics model for inflation, these models are highly successful in providing solutions to such cosmological puzzles. Still, it is crucial to investigate if other scenarios could also solve some of these cosmological problems, or even others which inflationary models do not address (such as the cosmological constant problem). In particular, it is important to establish what general conditions are required of a theory which is capable of providing solutions to such problems.

There have been recent claims for a time-varying fine structure constant [5] detected by comparing quasar spectral lines in different multiplets. These possible variations in the dimensionless parameter can be interpreted as variations in dimensional constants such as the electric charge, the Planck constant or the speed of light in vacuum [6, 7, 8]. Albrecht and Magueijo [9] have recently proposed a generalisation of General Relativity incorporating a possible change in the speed of light in vacuum (c) and the gravitational constant (G)—see also [7, 10, 11]. They have shown that their theory can solve many of the problems of the standard cosmological model including the horizon, flatness and cosmological constant problems, at the price of breaking covariance and Lorentz invariance. The also make the additional ex nihilio assumption of minimal coupling at the level of Einstein’s equations.

Here we take a pedagogical look at this problem by asking if one can restore some of the above principles and still obtain a theory which, prima facia, provides credible solutions to the standard cosmological enigmas. With this aim, we propose a new generalisation of General Relativity which also allows for arbitrary changes in the speed of light, , and the gravitational constant, . Our theory is both covariant and Lorentz invariant and for both mass and particle number are conserved. We solve the Einstein equations for Friedmann universes and show that the solution to the flatness, horizon or problems always requires similar conditions to the ones found in the context of the standard cosmological model.

We therefore argue that a theory that reduces to General Relativity in the appropriate limit and solves the horizon and flatness problems of the standard cosmological problems must either violate the strong energy condition (which is what inflation does), Lorentz invariance or covariance. Stronger requirements are needed in order to solve also the cosmological constant problem. In a subsequent publication [12] we shall show that our approach and that of Albrecht and Magueijo [9] can be further distinguished by an analysis of their corresponding structure formation scenarios.

## Ii A variable speed of light theory

Experiments can only measure dimensionless combinations of the fundamental parameters. This means that any evidence for variation in a dimensional parameter is dependent on the choice of units in which it is measured. Hence, before investigating the cosmological consequences of a variable speed of light theory we must specify our choice of units.

Here, we choose our unit of energy to be the Rydberg energy , our unit of length to be the Bohr radius () and our time unit to be . Using these units a measure of the velocity of light will be a measure of the dimensionless quantity

(1) |

where is the fine structure constant. Hence, by choosing appropriate units we are able to interpret a variation in the fine structure constant as being due to a change in the speed of light, c. It is possible to redefine our unit of time in such a way that and remain fixed while varies proportionally to by making

(2) |

In this article we shall not address the problem of which mechanism could induce a change in but we concentrate on the cosmological implications of such a variation.

Given that c is a constant in these units, we may specify our theory of gravity to be Einstein’s General Relativity with a variable gravitational constant . The Planck constant () and the gravitational constant () will be a function of the space-time position. However, we will assume, for the sake of simplicity, that to zeroth order and are functions of the cosmological time only. The theory specified in this way will clearly be Lorentz invariant. Our theory implies a modification of quantum mechanics in order to incorporate a variable (as indeed do the theories discussed in [7, 8, 9]). However, for the variation of the Planck constant is very small on an atomic timescale. This means that quantum mechanical results for atomic behaviour will hold to a very good approximation with only a simple modification . Nevertheless, we do expect that such changes will have observational consequences (eg. for black body curves). We will discuss this in more detail elsewhere, but here we simply point out that this (and many other constraints) will force any significant changes in the fundamental constants to happen very early in the history of the universe.

If we now switch to our original (and more natural) choice of units we will be left with a theory which has a variable but which is just analogous to General Relativity. The mass of an electron is a constant in these units, and it will be implicitly assumed that in the absence of any interactions (including gravity) the average distance between particles will remain a constant. From now on we will stick to our original choice of time unit . In passing, we note that it would also be possible, by making appropriate changes of units, to interpret the variation of the fine structure constant as a variation in the electric charge [6]. This different choice of units would lead to a different interpretation of the theory, but at the end of the day the physical consequences of the model would be the same.

In our model the Einstein equations take the usual form

(3) |

but now arbitrary variations in and will be allowed. Contrary to Albrecht and Magueijo we do not assume ab initio that variations in the speed of light do not introduce corrections to the curvature terms in the Einstein equations in the cosmological frame. In our model variations in the velocity of light, are always allowed to contribute to the curvature terms. These contributions are computed from the metric tensor in the usual way. Note that our only assumption is that both and (consequently) are a function of cosmological time only, to zeroth order. One can see that this is essentially similar to the much more familiar assumption that both density and pressure are functions of the cosmological time only, to zeroth order in the metric perturbations. Consequently, this does not break the covariance of the theory.

With the line element

(4) |

the Friedmann and Raychaudhuri equations in our theory are given by

(5) | |||||

(6) |

Here, and are the energy and pressure densities, and the curvature and the gravitational constants, and the dot denotes a derivative with respect to proper time. These can be combined into a conservation equation

(7) |

where is defined by . If the factor is a constant then the mass density () is conserved. In general, however, the conserved quantity will not be what one usually defines as ‘energy’. This is due to our particular choices of ‘fundamental’ units. Unlike in the theory proposed by Albrecht and Magueijo the curvature of the universe does not explicitly appear in the conservation equation. As we will show elsewhere [12], this difference is crucial for the ensuing structure formation scenarios.

Note that it is easy to transform between our general coordinate system as specified by the line element (4) and one in which (we use the subscript zero to denote quantities measured in these coordinates). The transformation rules are

(8) |

(9) |

(10) |

(11) |

It is then straightforward to check that, for example, the Friedmann and Raychaudhuri equations (5,6) transform in the correct way.

We note in passing that in our theory the Planck time given by

will be a variable for . This means that we may enter the quantum gravitational epoch sooner or later than in the standard cosmological scenario depending on the behaviour of both and at early times.

## Iii The flatness, horizon and Lambda problems

In order to solve the flatness problem the curvature term in equation (5) needs to be subdominant at late times. From the conservation equation we have that and so equation (5) can be re-written as

(12) |

where and are constants. We can see that the condition necessary for the the curvature term to be subdominant at large is

(13) |

This is just the condition necessary to solve the flatness problem in the standard cosmological model. Consequently, no solution to the flatness problem arrises naturally in this model.

On the other hand, a solution to the horizon problem can be achieved by having a period in the history of the universe in which the the scale factor can grow faster than the proper distance to the horizon,

(14) |

We can easily see that the scale factor can grow faster than only if

(15) |

Hence, the condition necessary to solve the horizon problem is again identical to the solution to the flatness problem and is no different from the one we obtain in the standard cosmological model: we must violate the strong energy condition.

Finally, a cosmological constant can be accounted for by including the cosmological constant mass density () in equations (5) and (6). In this case equation (5) becomes

(16) |

The condition necessary for the cosmological constant term in equation (16) to become negligible at late times is just

(17) |

Again this is exactly the same condition we get in the standard cosmological model. We therefore conclude that our theory does not provide a solution to the standard cosmological problems. Note that our results do not depend on any assumptions about the specific behaviour of and . In particular, they hold whether or not mass and particle number are conserved.

## Iv Discussion and conclusions

In this article we have explicitly constructed a a generalisation of General Relativity which is both covariant and Lorentz invariant and obeys the strong energy condition, and shown that in such a theory any arbitrary time-like variations in and will not lead to a solution to some of the most important problems of the standard cosmological model. The main drawback of the theory we have constructed is that it is incomplete in the sense that a model for the dynamics of and is not presented. Also, another outstanding issue which needs further discussion is that of the possible observational consequences of the required modifications to quantum mechanics.

From our above discussion, it is easy to see that the reason why such a theory cannot solve the standard cosmological enigmas is that one can always find a choice of time unit in which this theory will be identical (roughly speaking) to the standard cosmological model. Furthermore, this should be true of any theory which is (a) covariant and Lorentz invariant, (b) reduces to GR in the appropriate limit, and (c) obeys the strong energy condition. The first two conditions ensure that there will be a choice of units such that the theory in question will reduce to the standard one, and then the condition required to solve the cosmological problems should be that (c) is violated. Note that this argument is not strictly a proof, since we have not provided a general way to find the required choice of units. However, we believe that it is physically clear from the discussion above (and in section II) that such a choice of units should exist.

On the other hand, the postulates of the theory of Albrecht and Magueijo [9], when translated in the above language, correspond to the assumption that one can not find any choice of time unit in which the theory reduces to the standard one. In their theory this is achieved by breaking covariance and Lorentz invariance.

Hence the above discussion leads us to conclude that to solve the standard cosmological problems in a theory which reduces to General Relativity (possibly after some appropriate changes in the ‘fundamental’ units) one must either violate the strong energy condition, Lorentz invariance or covariance. Of course the above condition is necessary but not sufficient. Inflation is an obvious example of a theory which violates the first of the above principles.

A theory such as the one proposed by Albrecht and Magueijo [9], on the other hand, can work because it violates the latter two. In this context, the fact that such a theory has a variable speed of light is, we think, only a minor ‘side effect’ of their postulates, and other possibilities would do just as well (eg, a varying electric charge [8], etc). Furthermore, we anticipate that it should be possible to construct theories which break covariance and Lorentz invariance, have ‘constant constants’ and still can solve the horizon and flatness problems, although one might also expect such theories to be even more contrived than the ones discussed here.

In a subsequent paper [12], we will show that the approach of Albrecht and Magueijo [9] has some other difficulties, in particular at the level of structure formation scenarios. Some of these are due to the fact that there are varying constants (and will therefore also appear in the theory we have proposed in the present paper), but others are specifically due to the way in which Lorentz invariance and covariance are broken in their approach. This implies that it is not clear that this approach can be a viable alternative to inflation. It is therefore interesting to ask if there is any other paradigm, apart from these two, that can provide analogous solutions to the cosmological enigmas.

###### Acknowledgements.

We would like to thank Paulo Carvalho and Paulo Macedo for enlightening discussions. P.P.A. is funded by JNICT (Portugal) under ‘Programa PRAXIS XXI’ (grant no. PRAXIS XXI/BPD/9901/96). C.M. is funded by JNICT (Portugal) under ‘Programa PRAXIS XXI’ (grant no. PRAXIS XXI/BPD/11769/97).## References

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