Curvature Corrections as the Source

of the Cosmological Acceleration

Dan N. Vollick

Department of Physics and Astronomy

and

Department of Mathematics and Statistics

Okanagan University College

3333 College Way

Kelowna, B.C.

V1V 1V7

Abstract

Corrections to Einstein’s equations that become important at small curvatures are considered. The field equations are derived using a Palatini variation in which the connection and metric are varied independently. In contrast to the Einstein-Hilbert variation, which yields fourth order equations, the Palatini approach produces second order equations in the metric. The Lagrangian is examined and it is shown that it leads to equations whose solutions approach a de Sitter universe at late times. Thus, the inclusion of curvature terms in the gravitational action offers an alternative explanation for the cosmological acceleration.

## Introduction

One of the most interesting aspects of modern cosmology concerns the acceleration of the cosmological expansion. Recent supernovae [1, 2, 3, 4] and CMBR [5, 6, 7, 8, 9, 10] observations indicate that the expansion of the universe is accelerating, contrary to previous expectations.

Most attempts to explain this acceleration involve the introduction of dark energy, as a source of the Einstein field equations. The nature of the dark energy is unknown but it behaves like a fluid with a large negative pressure. One possible candidate for the dark energy is a very small cosmological constant. Another, possibly related, problem involves the existence of dark matter. Observations of spiral galaxies, elliptical galaxies and galactic clusters indicates that these objects contain a large amount of dark matter. The difference between dark energy and dark matter is that dark matter clusters with the visible matter and dark energy is more or less uniformly spread throughout the universe.

Of course, one possibility is that we do not understand gravity on these large scales. Since dark energy and dark matter are needed to explain phenomena in regions of low curvature we can attempt to modify Einstein’s theory by adding corrections that become important when the curvature is small (see [13, 14, 15, 16] for other approaches that involve modifications of Einstein’s theory). Recently two attempts [11, 12] to explain the cosmic acceleration along these lines have been made. They involve adding a term proportional to to the Einstein-Hilbert action and varying the action with respect to the metric (see [17, 18] for other papers that consider non-polynomial terms in the action). This approach leads to complicated fourth order equations that can be simplified by performing a canonical transformation and introducing a fictitious scalar field. It was shown in these papers that the modified field equations can produce the observed cosmological acceleration without the need of dark energy.

In this paper I also consider a correction to the action that is proportional to , but I use the Palatini variational principle to derive the field equations. In this approach and are taken as independent variables. In Einstein gravity the variation with respect to gives the usual relationship between and the metric, i.e. is the Christoffel symbol associated with the metric . The variation with respect to gives , where . Thus, the Palatini variation is equivalent to the Einstein-Hilbert variation in Einstein’s theory. This is not the case however for other Lagrangians. In fact, it has been shown [19, 20] that the Palatini variation gives the usual vacuum Einstein equations for generic Lagrangians of the form . This is to be contrasted to the purely metric variation that produces fourth order equations. If matter is included the Palatini variation still produces second order equations, but the are no longer identical to Einstein’s equations. In this paper I show that the Palatini variation of the Lagrangian , where is a constant, leads to field equations that give an accelerating universe at late times.

## The Field Equations

The field equations follow from the variation of the action

(1) |

where

(2) |

, , and is the matter Lagrangian. Here we consider a Palatini variation of the action, which treats and as independent variables.

Varying the action with respect to gives

(3) |

where is the energy-momentum tensor and is given by

(4) |

Varying the action with respect to gives

(5) |

By contracting over and it is easy to see that this is equivalent to

(6) |

This equation can be solved for the connection using a similar approach to that used in general relativity. Alternatively, one can define a metric and it is easy to see that equation (6) implies that the connection is the Christoffel symbol with respect to the metric . A conformal transformation back to the metric gives (see [21] for the details on conformal transformations)

(7) |

where the first term is the Christoffel symbol with respect to the metric . At first sight it might appear that this does not really define the connection since contains derivatives of the connection. However, if we contract the field equation (3) we get

(8) |

If this equation can be solved for , as we will assume here, then the terms in (7) involving can be expressed as derivatives of . Since contains only the metric and not its derivatives the connection will involve only first derivatives of the metric and the field equations will then be second order in the metric .

The Ricci tensor and Ricci scalar are given by

(9) |

and

(10) |

where is the usual expression for in terms of and .

If then the solutions to equation (8) will be constants and will be equal to times a (positive) constant. This implies that , and . Thus, in a vacuum the field equations will reduce to the Einstein field equations with a cosmological constant for a generic (see [19, 20] for the vacuum case). The field equations (3) can be written in the Einstein form

(11) |

with a modified source given by

(12) |

Now consider the Lagrangian

(13) |

where is a positive constant with the same dimensions as and the factor of 3 is introduced to simplify future equations. The field equations for this Lagrangian are

(14) |

Contracting the indices gives

(15) |

and the solution to this algebraic equation is

(16) |

For large we expect the above to reduce to , which follows from the Einstein field equations. Thus, if we need to select the positive sign and if we need to select the negative sign. In a universe filled with an ideal fluid , so that if the dominant energy condition holds (i.e. and ). Thus we have

(17) |

The vacuum solution is

(18) |

so that at late times, as , the universe will approach a de Sitter spacetime and the expansion of the universe will accelerate.

From equations (14) and (16) we see that the field equations reduce to the Einstein equations if . Thus, in a dust filled universe the evolution will be governed by the Einstein field equations at early times. Eventually the corrections to the equations of motion will become important and the universe will make a transition to a de Sitter universe at late times. To match the observations of the cosmological acceleration we must take . Note that in a vacuum we get the Einstein field equations plus a small cosmological constant, so that this theory will pass all the solar system tests that general relativity has passed. It is also interesting to note that in a radiation dominated universe so that the dynamics is not governed by the Einstein’s equations even at large curvature. As we will see below (see equation (24)) the equations of motion are the Einstein field equations with a cosmological constant and a modified Newton’s constant.

Now consider the evolution of a universe with metric

(19) |

at late times, when . In this regime

(20) |

(21) |

(22) |

(23) |

and the field equations are

(24) |

The matter in the present universe can be approximated by dust with , where is a constant. The nonvanishing components of the Ricci tensor are

(25) |

and

(26) |

At late times the universe will almost be in a de Sitter phase and we can take

(27) |

where and . To lowest order in the field equations are

(28) |

and

(29) |

Subtracting these two equations gives the first order equation

(30) |

and the particular solution to this equation is

(31) |

Thus, at late times the universe approaches a de Sitter spacetime exponentially fast. This behaviour is analogous to the cosmic no hair theorem for fourth order gravity discussed by Kluske and Schmidt [22].

## Conclusion

Using a Palatini variation the field equations for a nonlinear gravitational Lagrangian coupled to matter were found. The vacuum field equations are the Einstein equations with a cosmological constant. Thus, at late times as our universe will approach a de Sitter spacetime. The inclusion of matter gives field equations that differ Einstein’s equations. Using these equations it was shown that the approach to de Sitter space is exponentially fast when the term dominates. Thus, the inclusion of non-polynomial curvature terms in the gravitational action offers an alternative explanation for the cosmological acceleration.

## References

- [1] J.L.Tonry et al., astro-ph/0305008
- [2] S. Perlmutter et al., Ap.J. 517, 565 (1999), astro-ph/9812133
- [3] A.G. Riess et al., Astron. J. 116, 1009 (1998), astro-ph/9805201
- [4] S. Perlmutter et al., Bull. Am. Astron. Soc. 29, 1351 (1997), astro-ph/9812473
- [5] C.L. Bennett et al., astro-ph/0302207
- [6] C.B. Netterfield et al., astro-ph/0104460
- [7] N.W. Halverson et al., Ap.J. 568, 38 (2002), astro-ph/0104489
- [8] A.H. Jaffe et al., Phys. Rev. Lett. 86, 3475 (2001), astro-ph/0007333
- [9] A.E. Lange et al., Phys. Rev. D 63, 042001 (2001)
- [10] A. Melchiorri et al., Ap.J. 563, L63 (2000), astro-ph/9911445
- [11] S.M. Carroll, V. Duvvuri, M. Trodden and M. Turner, astro-ph/0306438
- [12] S. Capozziello, S. Carloni and A. Troisi, astro-ph/0303041
- [13] C. Deffayet, G. Dvali and G. Gabadadze, Phys. Rev. D65, 044023 (2002), astro-ph/0105068
- [14] G. Dvali, G. Gabadadze and M. Shifman, Phys. Rev. D67, 044020 (2003), hep-th/0202174 and hep-th/0208096
- [15] G. Dvali, A. Gruzinov and M. Zaldarriaga, hep-ph/0212069
- [16] G. Dvali and M. Turner astro-ph/0301510
- [17] H. Kleinert, H.-J. Schmidt, Gen. Rel. Grav. 34, 1295 (2002)
- [18] R. Brandenberger, V. Mukhanov and A. Sornborger, Phys. Rev. D48, 1629 (1993)
- [19] M. Ferraris, M. Francaviglia and I. Volovich, Nouvo Cim. B108,1313 (1993), gr-qc/9303007
- [20] M. Ferraris, M. Francaviglia and I. Volovich, Class. Quant. Grav. 11, 1505 (1994)
- [21] S.W. Hawking and G.F.R. Ellis, The large scale structure of spacetime, Cambridge University Press (1973), pg 42
- [22] S. Kluske and H.-J. Schmidt, Astron. Nachr. 317, 337 (1996), gr-qc/9503021