# Open inflationary universes in a brane world cosmology

###### Abstract

In this paper, we study a type of one-field model for open inflationary universe models in the context of the brane world models. In the scenario of a one-bubble universe model, we determine and characterize the existence of the Coleman-De Lucia instanton, together with the period of inflation after tunneling has occurred. Our results are compared to those found in the Einstein theory of Relativistic Models.

###### pacs:

98.80.Jk, 98.80.Bp

GACG-04/07

## I Introduction

Recent observations from the WMAP wmap are entirely consistent with a universe having a total energy density that is very close to its critical value, where the total density parameter has the value . Most people interpret this value as corresponding to a flat universe. But, according to this result, we might take the alternative point of view of having a marginally open or close universe model ellis-k , which early presents an inflationary period of expansion. This approach has already been considered in the literature, re6 re8 or shiba ramon , in the context of the Einstein theory of relativity or a Jordan-Brans-Dicke (JBD) type of gravitation theory, respectively. All these models have been worked out in a four dimensional spacetime. At this point, we should mention that Ratra and Peebles were the first to elaborate on the open inflation model Ratra .

The idea of considering an extra dimension has received great attention in the last few years, since it is believed that these models shed light on the solution to fundamental problems when the universe is traced back to a very early time. Specifically, the possibility of creating an open or closed universe from the context of a Brane World (BW) scenario MBPGD , has quite recently been considered.

BW cosmology offered a novel approach to our understanding of the evolution of the universe. The most spectacular consequence of this scenario is the modification of the Friedmann equation, in a particular case when a five dimensional model is considered, and where the matter described through a scalar field, is confined to a four dimensional Brane, while gravity can be propagated in the bulk. These kinds of models can be obtained from a higher superstring theory witten . For a comprehensible review of BW cosmology, see lecturer for example. Specifically, consequences of a chaotic inflationary universe scenario in a BW model was described maartens , where it was found that the slow-roll approximation is enhanced by the modification of the Friedmann equation. The purpose of the present paper is to study an open inflation universe model, where the scalar field is confined to the four dimensional Brane.

The plan of the paper is as follows: In Sec. II we specify the effective four dimensional cosmological equations from a five-AdS BW model. We write down the field equations in a Euclidean spacetime, and we solve them numerically. Here, the existence of the Coleman-De Lucia (CDL) instanton for two different models are described. In Sec. III we determine the characteristic of an open inflationary universe model that is produced after tunneling has occurred. In Sec. IV we determine the corresponding density perturbations for our models. In any case, our results are compared to those analogous results obtained by using Einstein’s theory of gravity. Finally, we conclude in Sec. V.

## Ii The Euclidean cosmological equations in Randall-Sundrum type II Scenario

We start with the action given by

(1) |

where

describes the matter confined in the brane is the Ricci scalar curvature of the metric of the five dimensional bulk; and denote the five-dimensional Planck mass and cosmological constant, respectively. The following relations are found to be valid in this case

(2) |

and

(3) |

where represents the effective cosmological constant on to brane and corresponds to the brane tension.

For this theory, Shiromizu et al.maeda have shown that the four-dimensional Einstein equations induced on the brane can be written as

(4) |

where is the energy-momentum tensor of the matter in the brane; is the local correction to standard Einstein equations due to the extrinsic curvature; and is the nonlocal effect corrections from a free gravitational field, which arises from the projection of the bulk Weyl tensor. These quantities are given by

(5) |

where and represent the energy density and pressure of a fluid, respectively, and

(6) |

where is a constant, and is the anisotropic stress. Since we are considering an AdS bulk and a Friedman-Robertson-Walker (FRW) brane, we should have On the other hand, an extended version of Birkhoff’s theorem tells us that if the bulk spacetime is AdS, then lecturer ; Bowcock:2000cq . Finally, we can put when a finned tuning is made over .

In order to write down the field equations, recall that, due to Bianchi identity, i.e. , we have

(7) |

The first term is automatically satisfied and thus the following constraint is thus obtained

(8) |

Now, since in our case we have , eq. (8) becomes maartens

(9) |

The invariant Euclidean spacetime metric is written as

(10) |

The scalar field equation becomes

(11) |

where is the scale factor, and the prime represents a derivative with respect to the Euclidean time () and .

When the metric (10) is introduced into equation (4), we obtain the following field equations for scalar factor:

(12) |

where corresponds to Euclidean energy density associated with the scalar field, . From now on we will use units where and .

We consider the effective scalar potential, , to be of the form analogous to that described in Ref. ramon :

(14) |

where , and are arbitrary constants. We will take the particular values and from this potential. The second term controls the bubble nucleation. Its role is to create an appropriate shape in the inflaton potential, , with a maximum value near . The first term controls inflation after quantum tunneling has occurred, and its shape coincides with that used in the simplest chaotic inflationary universe model, . Following Ref. ramon we take and . Certainly, this is not the only choice, since other values for these parameters can also lead to a successful open inflationary scenario (with any value of , in the range ).

We have numerically solved the field equations (11) and (13) for the values and in the effective potential (14). However the instanton has the topology of a four-sphere, and there are two places at which . These are the points at which and . Then, the boundary conditions on arise from the requirement that be finite, i.e . From Eq.(12), we obtain that at the zeros of scalar factor, . Since we have used units where the Planck mass in four dimension is equal to one, then the Planck mass in five dimension becomes maartens and by means of Eq.(3) we arrive to .

On the other hand, we take the value of the constant in such a way that an appropriated amplitude for density perturbation is obtained. Thus, we take the values and . We choose and , since they provide the needed e-folds of inflation after tunneling has occurred.

At , the scalar field lies in the true vacuum, near the maximum of the potential, which (in Euclidean signature) correspond to . At , the same field is found close to the false vacuum, but now with a different value, . Specifically, for and and when the scalar field evolves from some initial value, i.e. to the final value , numerically we have found that the CDL instanton does exist, and the brane world open inflationary universe scenario can be realized. Table 1 summarizes our results, which are compared with those corresponding to Einstein’s Theory of Relativity.

Models | ||
---|---|---|

GR | ||

BW | ||

GR | ||

BW |

Note that the interval of tunneling, specified by , decreases when the , but its shapes remain practically similar. The evolution of the inflaton field as a function of the Euclidean time is shown in Fig. 1.

In Fig. 2 we show as a function of the Euclidean time for our model. From this plot we observe that, most of the time during the tunneling, we obtain , analogous to what occurs in Einstein’s GR theory. Note that, for , the peak becomes narrower and deeper, and thus the above inequality is better satisfied.

It is numerically possible to show that the CDL instanton exists for various values of the parameter. The principal difference shown is the deviation to general relativity, but this solution presents a similar behavior to that described by Linde re6 . The values coincide for large , and its values (after tunneling has occurred) coincide in the two theories, i.e. Einstein’s GR and BW theories.

The following expression represents the instanton action for the quantum tunneling between the false and the true vacuums in the BW theory. In order to reproduce the field equations of motions (11)-(13) and using the constraint (9), we write for the instanton action

(15) |

We should note that this action is not obtained from Eq.(1). At the moment, as far we know nobody has performed this task. Integrating by parts and using the Euclidean equations of motion, we find that the action may be written as

(16) |

Note that this action coincides with that corresponding to its analogous in the Einstein’s general relativity theory if we take the limit that re7 .

The inflaton field is initially trapped in its false vacuum, and a value specified by is obtained. After tunneling to the true vacuum, the instanton gets the value , and a single bubble is produced . Similar to the case in the GR theory, the instanton (or bounce) action is given by , i.e. the difference between the action associated with the bounce solution and the false vacuum. This action determines the probability of tunneling for the process. We have defined and as the false and true vacuum energies, respectively. Under the approximation that the bubble wall is infinitesimally thin, we obtain the reduced action for the thin-wall bubble:

(17) |

where we have taken into account the contributions from the wall (first term) and the interior of the bubble (the second and third terms). Here is the radius of the bubble, and

The surface tension of the wall becomes defined by

(18) |

or

The radius of curvature the bubble is one for which the bounce action (17) is an extremum. Then, the wall radius is determined by setting = 0, which gives

This could be solved for the radius of the bubble, and we found that

(19) |

or equivalently

(20) |

It is straightforward to check that when , a correct limit to GR is obtained. We can introduce a dimensionless quantity , which represents the strength of the wall tension in the thin-wall approximationn MSTTYY

(21) |

Following the values given in Table 1 and assuming that value , we find that the differences in the strength of the wall tension in the thin-wall approximation becomes , which can be compared to the corresponding value in Einstein’s GR theory, . For the first model, i.e, , becomes this difference becomes on the order of . Then, we can see that this difference in the strength of the wall tension for the two theories becomes insignificant. . In the second model, when

## Iii Inflation after tunneling

After the tunneling has occurred, we make an analytical continuation to the Lorentzian space - time, and we could see what is the time evolution of the scalar field and the scale factor . The field equations of motion for the fields and are given by

(22) |

and

(23) |

where the dots now denote derivatives with respect to the cosmological time.

In order to solve this set of Equations numerically, we use the following boundary conditions: , and . As in re6 shiba and ramon , the scalar field slowly rolls down to its minimum of effective potential, and its field starts to oscillate near this minimum. During this stage the e-folds parameter presents different values for our models under study; those results are summarized in Figure 3. Clearly, the e-folds parameter increases in the BW scenarios.

## Iv Spectrum of scalar perturbation

Even though the study of scalar density perturbations in open universes is quite complicated re8 , it is interesting to give an estimation of the standard quantum scalar field fluctuations inside the open bubble. The corresponding density perturbation in the brane world cosmological model becomes maartens

(24) |

where is taken. Certainly, other contributions must be added in order to get an exact expression re6 ; re8 , but those contributions do not change expression (24) significantly if we use it for . . Note that the latter equation coincides with its analogous Einstein Eqs., when the limit

Figure 4 shows the magnitude of the scalar perturbations for our models as a function of the e-folds of inflation, after the open universe was formed. Even though the shape of the graph is similar to that of Einstein’s GR theory , the maximun value of has become bigger in the BW then in Einstein’s GR models. For instance, in Einstein’s GR they it becomes maximum for , while for BW model its maximum is found at . Also in the the model with , the values of where vanishes, becomes bigger in BW than in Einstein’s GR theory. On the other hand, we should mention that there is relationship between the values of and the scale where is measured. For , where gets it maximum value, it is found that the scale where the scalar perturbation is measured corresponds to the . However, for it decreases to , and for this practically comes to zero. Something similar happened with for the case . There, the corresponding values of were smaller.

Also, it is interesting to give an estimation of the tensor spectral index in the brane world cosmological model. Using Ref.maartens , this index for a flat universe is given by

(25) |

By numerically solving the field equation associated with field , we obtain for the cases and in the GR theory the following values: which is evaluated in the value the , where presents a maximum, i.e, and for . In BW cosmological models for the cases and , we obtain for and for .

## V conclusion

In this paper we have studied one-field open inflationary universe models in which the gravitational effects are described by BW cosmology. In this kind of theory, the Friedmann equation gets modified by an additional term: . We have solutions to an effective potentials in which the CDL instantons exist ramon . The existence of these instantons becomes guaranteed since the inequality is satisfied, and thus, with the slow-roll approximation, inflationary universes models are realized for different values of the parameter . For the two values ( and ) that we considered, remains greater than during the first e-folds of inflation.

Also, we have generalized the CDL instanton action to brane world cosmology. This action is described by expression (15).

On the other hand, it seems that, according to equations (17) and (18), the result for the probability of nucleation of a bubble is the same as in Einstein gravity. However, this is superficial, since there is a modified relationship between the Hubble rate and the potential , given by the well known modified Friedmann equation and thus a modified expression between the wall tension and the potential occurs. In the thin-wall limit, we have found that the strength of the wall tension, -- minimum increase, when compared with their analogous results obtained in Einstein’s GR theory.

We have also found that the inclusion of the additional term () in the Friedmann’s equation improves some of the characteristic parameters of inflation. For instance, this is accentuated in the number the e-folds (see Fig.[3]).

Finally, in the graphs the maximum presents a displacement when compared with that obtained in Einstein’s GR theory, this would change the value of the fundamental parameter that appears in the scalar potentials.

###### Acknowledgements.

S.d.C.and J.S was supported from COMISION NACIONAL DE CIENCIAS Y TECNOLOGIA through FONDECYT Grant Nos. 1030469; 1010485 and 1040624 and Postdoctoral Grant 3030025. Also, it was partially supported by UCV Grant No. 123.752. R. H. is supported from PUCV through Proyecto de Investigadores Jóvenes ao 2004. The authors wish to thank CECS for its kind hospitality.## References

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