The cosmological inflation is proposed as some resolutions for the important cosmological problems, e.g. the flatness, horizon and monopole problems. Most of successful models are based on classical scalar fields, although we have not observed such scalar bosons associated with the field.

The inflation can also be caused by other type of fields. The vector inflation has been proposed by Ford [1] and some authors [2-4]. It is emphasized that the massive vector field should non-minimally couple to gravity in such models [1-4].

The reason why the non-minimal coupling is important is as follows. Suppose the equation of motion for the vector field is given by

For the background field, we assume¹

,(1.2)

and depends only on t and. Then we define (1.1) becomes

which is very similar to the equation for a homogeneous scalar field in the Friedmann-Lemaître-Robertson-Walker universe. Moreover, the energy density is expressed as ~, which is also similar to the one for the scalar field. Thus, the slow evolution of the effective scalar field B_i can occur in the approximately isotropic inflating universe.

We have studied [5] Weyl invariant gravity [6-39] as a candidate for the theoretical model of the vector inflation. We found that the choice of the frame yields the mass of the Weyl gauge field, but the non-minimal coupling term is lost [5]. We come to the conclusion that we need further generalization of the gravitational theory.

In the different context, Deser and Gibbons considered Dirac-Born-Infeld (DBI)-Einstein theory [40] almost a decade ago, whose Lagrangian density is of the following type

where R_μν is the Ricci tensor and the α is a constant. Originally, electromagnetism of the DBI type has been considered as a candidate of the nonsingular theory of electric fields. Therefore the Dirac-Born-Infeld-Einstein theory as the highly-nonlinear theory is also expected as a theory of gravity suffered from no argument of singularity. The studies on the theory have been done by many authors [41-50]. Because of the nonlinearity in this theory, we expect the extension as the theory of gravity which realizes a successful vector inflation.

Consider the Weyl invariant D-dimensional extension of the Ricci curvature (see the next section) is

By simple replacement of the Ricci tensor by the Weyl invariant tensor in the action (1.4), the expansion

yields the terms RA_μA^μ and R_μνA^μA^ν and so on as well as R and F_μνF^μν. Other Weyl invariant terms are necessary, because the metric tensor must be combined with a scalar field which compensates the dimensionality. After the frame choice, the freedom of the scalar field is eaten by the vector field, then, the presence of the non-minimal terms mentioned above is still realized².

In the next section, we review the Weyl invariant gravity with the vector field [11-14,16-21,25-37]. The expression (1.4) is generalized to the Weyl invariant one. The Lagrangian for a Weyl-invariant DBI gravity is proposed in Section 3. In Section 4, the necessity condition for the vector inflation is investigated. In Section 5, another possible inflationary scenario is provided. The last section is devoted to the summary and prospects.

In this section, we review the Weyl’s gauge transformation to construct the gauge invariant Lagrangian.

Consider the transformation of metric (in D dimensions)

where is an arbitrary function of the coordinates x^μ.

We can define the field with weight which transforms as

In order to construct the locally invariant theory, we consider the covariant derivative of the scalar field

where A_μ is a Weyl’s gauge invariant vector field.

Under the Weyl gauge field transformation

we obtain the transformation of the covariant derivative of the scalar field as

The field strength of the vector field is given by

which is gauge invariant as

The modified Christoffel symbol is defined as

where. The modified curvature is given as follows:

The Ricci curvature in the Weyl invariant version is

where denotes the usual generally covariant derivative. Note that under the gauge transformation

Although we can use the Weyl invariant Ricci tensor in the DBI gravity, we should note that the metric tensor in the action is not Weyl invariant (which is shown in (2.1)). Thus, we use a combination instead of the metric tensor. The scalar Φ compensates the dimension of the metric. Now the use of and in the DBI type action leads to the theory of gravity, a vector field, and unexpectedly, a scalar field.

The introduction of the compensating scalar field tells us the action is far from general one. The monomial of the type of the kinetic term, in other words, two coordinate derivatives of the scalar field can be considered, while the curvature includes also two derivatives with no contraction. The possible monomials are

and (3.1)

Another notice is in order. The decomposition of a rank two tensor shows that there are three irreducible ones; an anti-symmetric tensor, a traceless symmetric tensor and a trace part.

Now, we must introduce the following independently Weyl invariant tensors into the determinant in the DBI theory:

where

and

We choose those as symmetric tensors are not traceless³.

Our model of Weyl invariant DBI gravity is described by the Lagrangian density

with

where α₁, α₂, β, γ₁, γ₂, γ₃, γ₄ and λ are dimensionless constants⁴.

Furthermore the Lagrangian density can be expressed by the new metric conformally related to the original one and new variables. Here we choose

and

Note that a mass scale f was introduced here. By using the new metric and vector field, we rewrite the each term in the determinant of the Lagrangian as

We now can write M_μν as

where the “hat”s are dropped and dimensionless constants are

We can rewrite the Lagrangian as

This is the candidate Lagrangian for the vector inflation.

In this section, we apply our Weyl invariant DBI theory of gravity to cosmology in four dimensions (D = 4).

We take the metric for the homogeneous flat universe as

and, moreover, we assume the approximate isotropy .

We consider that only A₁(t) is homogeneously evolving, and A₂ = A₃ = A₀ = 0.

By these ansätze, we look for the condition that the vector field behaves much like a scalar field at classical homogeneous level. Substituting the ansätze, we find

After some calculations, we can subtract the part of the Lagrangian which includes bilinear and higher-order of the vector field A₁. We find that if the parameters are chosen as

and

the vector-field part becomes

where.

A simple case is realized when α₂ = γ₁ = γ₂ = γ₃ = γ₄ = 0, or these parameter take small values in comparison with α₁. Then the parameter is α₁ only. Equations (4.7) and (4.8) tell us, and. In this case, this is so simple that the effective mass for B₁ may be large. The tuning is possible; say, the choice of γ₄ does not affects (4.8) and makes the change in the effective mass.

An elaborate tuning may give the potential which induces the chaotic inflation [51]. In the next section, however, we show another simple inflation scenario.

The chaotic inflation in the model can occur by tuning of the parameters. We should remember that the model involves the higher-derivative gravity. Therefore another kind of inflation is worth to be considered.

First let us suppose the flat space. Then the potential, or the energy density for the constant B₁, can be easily written down as

Although other choices are possible, we consider here a simple choice as and ⁵. In this case, unfortunately, the previous conditions (4.7, 4.8) cannot be satisfied simultaneously, because for the positive coefficient of the Einstein-Hilbert term in the action. Then the potential is

This is the simplest potential. In the true vacuum, the vector field “condensates” and a “natural” choice λ = 1 leads to vanishing cosmological constants⁶!

This simplest version also has an inflationary phase. That is, for B₁ = 0, the scale factor behaves as a(t) ≈ e^Ht where.

Unfortunately, this phase is stabilized by the nonminimal coupling between curvatures and the vector field, because the effective potential in this phase becomes

The exit of the de Sitter phase is problematic, like the other higher-derivative models. Though the additional matter fields may play important roles, we will perform further study on them elsewhere.

The Weyl invariant DBI gravity is a candidate for a model which causes an inflationary universe. If the vector inflation can explain the possible anisotropy in the early universe, we may seriously investigate the Weyl invariant DBI gravity.

Here we examined slow development of the massive vector field. The inflation along with a fast evolution is shown to be possible in the DBI inflation, where the scalar degrees of freedom which originate from string (field) theory or D brane theory [52,53]. The similar scenario is feasible in our model, though the higher-derivatives make the detailed analysis difficult. Anyway, numerical calculations and large simulations will be needed to understand the minute meaning of the Weyl invariant DBI gravity, because the local inhomogenuity in the spatial directions as well as the strength of vector fields is important for thorough understanding in the early cosmology.

Finally, we think that some marginally related subjects are in order. The higher-dimensional cosmology in the Weyl invariant DBI gravity is worth studying because of its rich content. Incidentally, DBI gravity in three dimensions is eagerly studied [54-57], which is related to New Massive Gravity [58,59]. We think that the Weyl invariant extension of the lower-dimensional theory is also of much mathematical interest.

After completing this manuscript, we become aware of the paper “Higgs mechanism for New Massive gravity and Weyl invariant extensions of higher derivative theories” by Dengiz and Tekin [60]. They investigated a Weylinvariant DBI gravity in three dimensions.

We also become aware of two recent papers about the cosmology of Weyl invariant theory [61,62].

The authors would like to thank the organizers of JGRG20, where our partial result ([arXiv:1012.5375]) was presented. We also thank D. Comelli, S. Deser, T. Moon and B. Tekin for informing us about their important and interesting papers.

The present study is supported in part by the Grant-inAid of Nikaido Research Fund.