Cosmologies in Horndeski’s second-order vector-tensor theory
Horndeski derived a most general vector-tensor theory in which the vector field respects the gauge symmetry and the resulting dynamical equations are of second order. The action contains only one free parameter, , that determines the strength of the non-minimal coupling between the gauge field and gravity. We investigate the cosmological consequences of this action and discuss observational constraints. For we identify singularities where the deceleration parameter diverges within a finite proper time. This effectively rules out any sensible cosmological application of the theory for a negative non-minimal coupling. We also find a range of parameter that gives a viable cosmology and study the phenomenology for this case. Observational constraints on the value of the coupling are rather weak since the interaction is higher-order in space-time curvature.
a]John D. Barrow,
a,b]and Kei Yamamoto
\affiliation[a]DAMTP, University of Cambridge,
Wilberforce Road, Cambridge, CB3 0WA, United Kingdom \affiliation[b]Institute of Theoretical Astrophysics, University of Oslo,
P.O. Box 1029 Blindern, N-0315 Oslo, Norway \emailAddJ.D.Barrow@damtp.cam.ac.uk \emailAddmikjel.email@example.com \emailAddK.Yamamoto@damtp.cam.ac.uk \keywordsCosmology of Theories beyond the SM, Classical Theories of Gravity, Spacetime Singularities
In the past few decades, modifying the Einstein’s theory of gravitation has been an active area of research , driven chiefly by the search for different varieties of inflation, the desire of some to explain flat galaxy rotation curves without dark matter , and the challenge of explaining why the expansion of the universe started to accelerate at late times. It is also natural to question the validity of general relativity, not least because of its ultra-violet behaviour which does not give a well-defined quantum field theory. There is a growing prospect of testing any such deviations from general relativity in very strong gravity fields by searching for the signatures of gravitational waves created by high-energy astrophysical phenomena, such as black hole mergers, or by scrutinising detailed observations of the microwave background anisotropy and statistics in the light of particular theories of inflation. While there are a plethora of inflationary models, it is difficult to modify general relativity without spoiling its appealing features and typical modifications end up introducing new scalar degrees of freedom, as was the case in the case of Brans-Dicke gravity  and its scalar-tensor generalisations [4, 5, 6], or lagrangian theories of gravity [7, 8, 9, 10, 11].
Maxwell’s classical theory of electromagnetism has also been extremely well tested and there are already strict constraints on potential modifications such as a non-zero photon mass [12, 13, 14, 15] or varying fine structure constant [16, 17, 18, 19, 20, 21, 22, 23, 24]. However, there has been renewed interest in modified electromagnetism in cosmology and there have been attempts to incorporate its effects into the dynamics of early universe, particularly during inflation, or to provide an explanation for cosmological magnetic fields [25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. Simple vector fields themselves are known to have difficulties in producing inflation. The time variation of the vector field is governed by the covariant time derivative of the field. Since the Christoffel symbols for an expanding cosmological model are of order the Hubble expansion rate it is not possible for the vector field to satisfy a slow-roll condition in the way that a scalar field can . However Einstein-aether theories offer an alternative that permits inflation [42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 24, 1]. In addition, although it was noted that a non-minimal coupling to the space-time curvature could drive accelerated expansion [52, 53, 54, 55], these scenarios suffer from various instabilities created by additional degrees of freedom arising from the lost gauge symmetry [56, 57, 58, 59, 60, 61]. A different type of extension of the Maxwell case is provided by the extension to Yang-Mills fields where there can be chaotic behaviour and arbitrarily low levels of anisotropy [62, 63, 64].
Recently, it has been mentioned that a vector-tensor theory, first proposed by Horndeski in 1976 , could lead to an instability of conventional inflationary universe through the non-minimal coupling between the vector field and gravity . The action was derived by demanding second-order dynamical equations that reduce to Maxwell’s equations when evaluated on a Minkowski background and conservation of the current. These requirements result in only one additional term in the Lagrangian, and therefore a single free coupling parameter. Later, it was noticed that this theory falls into a special class of Kaluza-Klein reductions from higher-dimensional Lovelock invariants [67, 68, 69]. In contrast to the Horndeski scalar field theory , which has been discussed in attempts to construct the most general viable scalar-tensor theory recently, [71, 72, 73, 74, 75, 76], except for a brief examination of the static electromagnetism arising from this action , it appears to have escaped attention. Apart from being briefly mentioned in , its cosmological consequences have not been studied.
In this paper, we will investigate the simplest cosmological model, which is well understood in the minimally coupled case of a Maxwell electromagnetic field [78, 79], containing a perfect fluid and a vector field whose dynamics are described by the Horndeski Lagrangian. We find the following results;
The instability found in  for negative values of the coupling constant persists in the nonlinear regime and the universe eventually hits a singularity;
For a positive coupling constant, the electric field can still be amplified during the radiation-dominated era while giving a viable cosmology subject to some constraints on the allowed expansion rate changes at the epoch of primordial nucleosynthesis.
The first result effectively rules out any interesting cosmological application of this theory with a negative coupling. For a positive coupling constant, the modification is rather tame and an enormous value of the coupling in units of the Planck mass is allowed because of the higher-order nature of the modified term. However, the dynamics is of a phenomenological interest.
The article is organised as follows. In the next section, the theory is introduced and the modified Einstein-Maxwell equations are presented. Section 3 is the main part of the article where the dynamics of purely electric component are studied in an axisymmetric Bianchi type I universe. In section 4, we repeat the previous analysis for magnetic component. Section 5 discusses observational constraints and in section 6 we summarise our principal results.
2 Horndeski’s second-order vector-tensor theory
In 1976, Horndeski showed that the general Lagrangian that can be constructed from a metric and a vector field in four-dimensional space-time that satisfies the following conditions :
the field equations contain at most second-order derivatives of and (and do contain a second-order term);
the dynamical equations for respect charge conservation i.e. ;
the dynamical equations for reduce to Maxwell’s equations when evaluated on Minkowski space-time;
takes the following form:
where is the reduced Planck mass, is the Ricci scalar and is the Faraday tensor. The last term is Horndeski’s modification which can be expressed in several different ways as
The dimensionless non-minimal coupling constant is the only parameter of the theory. Our aim is to investigate the cosmological consequences with an arbitrary value of and to determine the parameter range yielding viable phenomenology. The other terms in the Lagrangian (1) are normalized so that it reduces to the Einstein-Maxwell theory when . Using the Levi-Civita tensor , we defined the generalised Kronecker’s delta by
and the double dual of Riemann by
was later identified with the Lagrangian obtained by Kaluza-Klein reduction from the five-dimensional Lovelock invariant
where is the Riemann tensor in five dimensions .
In this paper we use the sign conventions of  for the metric, Ricci and Riemann tensors which are different from those adopted in the previous studies of this model [65, 77]. The dynamical equations derived from this Lagrangian are given as follows:
- Variation with respect to
- Variation with respect to
We define the dual Faraday tensor as usual:
In ref., it was observed that (7) evaluated on a Friedmann-Lemaître-Robertson-Walker (FLRW) background could lead to an instability of the vector field. Ignoring the spatial gradient term, the solution for the comoving electric field strength evaluated on this background is given by
where is an integration constant, is the scale factor and is the Hubble expansion rate. When is negative and , the energy density of the electric field can rapidly increase and eventually diverge, even when the expansion of the universe is accelerated. Our first goal is to take into account the back reaction of this growing vector field and examine the fate of the inflationary universe.
3 Dynamics of electric fields in axisymmetric Bianchi type I universes
In this section and the next, we set .
3.1 Electric fields in an anisotropic universe
In order to answer the question of back reaction, one needs to look at a fully non-linear system and solve both (5) and (7). The simplest generalisation of the FLRW universe that can accommodate a vector field is the axisymmetric Bianchi type I metric given by
This metric is spatially flat. Note that the (mean) Hubble and shear expansion rates are given by
We consider a homogeneous electric field along the -direction, which in the gauge corresponds to the following coordinate basis components for the vector potential:
The electric field strength seen by an observer moving with four-velocity is given by
where dots denote derivatives with respect to the comoving proper time . We also include a perfect fluid with the equation of state and constant . Hence, (5) yields the following:
The fluid obeys the usual adiabatic decay law:
To gain an insight into the effect of the Horndeski’s extra non-minimal coupling, let us assume that Horndeski’s modification term is dominant, that is
The Friedmann equation (10) becomes
which means that the universe must be strongly anisotropic when . Using the same approximation, (13) reduces to
Now, (11) becomes
which is equivalent to the usual Einstein equation in an empty Bianchi I universe dominated by the shear because the back-reaction of the Horndeski term exactly cancels out at leading order. The same is true in the shear evolution equation, as (12) yields
so that the universe isotropises in the same way as a (flat) universe containing only perfect fluids. We conclude that the Horndeski modification is fairly innocuous despite the formidable appearance of its energy-momentum tensor. It should only be able to affect the evolution of the universe when its contribution is comparable to the conventional Maxwell term or the matter.
Occurrence of a finite-time singularity
The evolution equation for electric field (13) is rather similar to the linearized equation yielding the solution (8), although it is fully non-linear in the present setup. In fact, we can integrate (13) analytically and obtain
This is essentially the same as the solution in FLRW background (8). Unless there is a mechanism within (10) - (14) that prevents from reaching , there should be a range of initial conditions for which the system eventually hits a singularity. Since the system comes close to this singularity precisely when Horndeski’s modified term becomes comparable to the Maxwell term
we may see unusual dynamical behaviours in this regime.
Let us see what happens to the evolution of the spatial geometry when the system approaches the singularity. For this purpose, it is useful to write down the evolution equation for , which we can cast into the following form;
It is immediately clear that the right-hand side is negative definite when and for initial conditions satisfying
Therefore, the singularity is inevitable regardless of the initial conditions for as long as is sufficiently large initially. Although it does not apply to inflationary universes with , we already know decreases monotonically while dominates the evolution of the universe. Thus, if the condition (18) holds initially, eventually grows and any matter domination, and hence inflation, comes to an end. Once the electric field starts to dominate the dynamics, the right-hand side of (17) is again negative definite and the singularity must be reached. While we are unable to eliminate the possibility that turns to positive and the universe manages to avoid the singularity during a brief period of matter-electric equality, numerical calculations suggest otherwise (see figure 1).
Since the instability condition (18) roughly corresponds to the one for Horndeski’s term to have a significant effect compared to the Maxwell term in the Lagrangian, it effectively rules out any sensible cosmological application of the theory with a negative . We have not discussed the case of negative since it describes either a collapsing universe or excessively anisotropic one. Close examination of (11) indicates recollapse or bounce right before hitting the singularity, both deriving from the violation of the weak energy condition. These behaviours are also observed in our numerical solutions of the equations (see figure 2).
3.2 Expansion-normalised autonomous system
In the previous subsection, we saw that the theory with a negative coupling, , should lead to pathological dynamics when the Horndeski modification has an appreciable effect. On the other hand, when is positive there is no danger of a singularity. The energy density is positive definite and we expect some viable cosmological dynamics. In order to carry out a more systematic investigation, it is always useful to rewrite the equations in terms of density parameters defined for each matter component, including the Horndeski contribution. It also enables us to apply the conventional methods of dynamical systems analysis.
We introduce the following normalised variables:
where correspond to respectively. The normalised Friedmann equation
will be used as the standard measure of the dynamical significance of each component. In particular, when , all the parameters are bounded by so that their values have a clear physical interpretation. Note that for , where and represent the minimal and non-minimal contribution to the vector field energy density, respectively. From (11), we define the deceleration parameter as
These four equations are not independent since they are related by the first integral (19).
Fixed points in the dynamical system
We first classify the fixed points. Since the subsystem specified by is identical to the magnetic Bianchi type I discussed in [79, 78], we know there must be at least four fixed points of physical interest:
- Flat Friedmann universe :
- Electric Bianchi type I :
The existence condition is .
- Kasner solutions :
In addition, there appears a fixed point describing a universe dominated by the Horndeski energy density:
- -dominated universe :
The deceleration parameter for this solution is , which is consistent with the analysis in section 3.1 where we showed that the back-reaction of the vector field exactly cancels out to leading order when the energy density is dominated by Horndeski’s modification term.
There are subtleties regarding these fixed points. First of all, and do not represent physical space-times when since they imply and respectively. It does not mean these fixed points are irrelevant in the dynamics, however, since they may be reached asymptotically from finite and in the far past or future. We shall see an example of this in figure 7. Secondly, and must be treated with care since some of the denominators appearing in the evolution equations (21)-(24) vanish on those fixed points. Nevertheless, it does not mean they are unphysical since they are well-behaved when appropriate limits are taken for the numerators. But the analysis requires evaluation of , which implies the stabilities may depend on the way the fixed point is approached. It is a consequence of the fact that is not decoupled from the normalised variables, and implicitly appears in the definition of . While a fixed point in the expansion-normalised equations usually represents a self-similar solution that is invariant under a scale transformation, the dynamical effect of the Horndeski modification depends on the scale of the curvature, or the size of the universe. Therefore, it is not surprising to see the stability change depending on each orbit with its specific value of . We will find this is indeed the case.
3.3 Dynamics around the matter-dominated solution
From physical point of view, by far the most interesting fixed point is because it can be regarded as a model of the late-time evolution for the universe when (dust) or (radiation), and also a model of inflation when . We have already mentioned the instability against perturbations of the electric field for . The condition for the occurrence of a singularity (18) translates into
in the new variables. Here, we shall see that this condition coincides with the instability condition for and otherwise the dynamics is trivial. We also study the stability for positive and show that it depends on the value of .
For the purpose of linearisation around , it turns out to be convenient to eliminate using the Friedmann equation (19) and rewrite the equations as
whose right-hand sides we avoid writing down explicitly as they are lengthy. While there are apparent s in those equations when they are evaluated on , they should be all well defined if an appropriate limit is taken along an arbitrary reference orbit. To proceed, we evaluate the functions for and then take the limit :
where we have introduced a notation
which will also be used later. When , or equivalently , we have and the fixed point is always well defined as it should be. The existence of the limit is inconclusive when () and the orbit satisfies
However, such an orbit merely represents one that ends up in the singularity . Since we have already discussed this case in detail, we exclude those orbits from our consideration here. As long as an orbit does not hit the singularity when approaching , should exist and be equal to zero.
Since the right-hand sides can be evaluated only as a limit associated with each reference orbit, the linearisation takes an extra step. We first expand the equations around an arbitrary point and then take the limit . The resultant linear equations are given as follows:
where the s preceding the variables denote their small perturbation. Unless the orbit is the singular one specified by (29), is finite and therefore these linearised equations are well defined. However, the asymptotic value of does depend on each orbit. Going back to its definition, one notices that
We already know its behaviour near since we have
Solving (14), we obtain
where is an orbit-specific constant. We also note that
and we can easily read off the eigenvalues of the linearization matrix
Notice that the eigenvalues are orbit and time dependent through . As it was necessary to take a non-standard approach to obtain the eigenvalues, we will later confirm the validity of the result by performing numerical calculations.
Stability of the matter-dominated solution
The first of the eigenvalues (38) is negative and represents the stability of against perturbation of . Since the third eigenvalue is always smaller than the second, the condition for the stability is
Let us first consider which corresponds to . In this case, we have
and consequently is definitely stable for , which means positive cannot be relevant in the context of inflation, and is unstable for . For , the stability is orbit-dependent. When an orbit satisfies
it runs away from . When is smaller than this threshold value, the orbit is attracted towards . Note that the value of is time-dependent so that the stability can change over the course of the evolution. In particular, from (33), is monotonically decreasing as long as the orbit stays close to the matter-dominated solution. We can immediately conclude that for orbits with , the stability does not change as the universe expands. For those satisfying (39), they typically become stable asymptotically in the future since can only increase when the universe is dominated by both Maxwell’s and Horndeski’s terms. For a physically interesting range of initial conditions, we shall later confirm that the instability of the electric field saturates before the orbit goes too far from and eventually comes back to it.
For , the dynamics is very different – depending on . As was already mentioned, the critical value corresponds to the singularity discussed in section 3.1.1. Firstly, implies and therefore the orbits in this range are stable as long as . For , we have and monotonically increases in the vicinity of . It approaches from below and therefore any orbit eventually becomes unstable. Since we already know is the singularity, we conclude that the orbits with this range of initial can never settle down at , regardless of the equation of state parameter ; see figure 3 for simulation of an inflationary universe with initial conditions satisfying . The solution approaches until is close to the critical value in which case the universe moves away from .
After leaving , we expect the system enters the regime where is dynamically negligible. Then the analysis in the section 3.1.1 applies and the singularity is inevitable.
Dynamics with in a radiation- or dust-dominated universe
We have found that the theory is quite innocuous for positive non-minimal coupling constant, in which case there is no singularity and the Friedmann solution is stable at late times. However, as shown above, there is a transient period where is unstable for fluids satisfying . In this period, and grow and, if the instability does not saturate before they become too large, the universe will eventually become strongly anisotropic. Since this introduces potential problems in the radiation or dust-dominated epoch, it is of interest to specify the range of initial conditions such that the universe is close to at all the subsequent times, i.e., . In this subsection, therefore, we investigate the dynamics close to in more detail for dust () and radiation () with a positive non-minimal coupling constant ().
where and are integration constants. At late times, when , the Maxwell and Horndeski densities decay as and . This is consistent with being an attractor at late times as shown above. When the Maxwell and Horndeski densities grow as and . We note that the maximum values of and occur when and , respectively. Taking into account the linear constraint (34), it follows that the maximum values of both and are roughly equal. Now we find that the initial conditions must satisfy
in a radiation-dominated universe and
in a dust dominated universe to ensure that at all the subsequent times. In figures 4 and 5, we show a numerical integration of the full non-linear equations (21)-(23) for radiation and dust, respectively. Since the initial conditions just barely satisfy the conditions (42)-(43), the peak values of and are at the one-percent level of the total energy budget. Note that, to good accuracy, we have when . This introduces the possibility of cancelling the effects of spatial anisotropy on the Cosmic Microwave Background (CMB), which will be discussed in section 5. Under the assumption that the theory describes a generalised electrodynamics, we show in section 5 that the amplification of the electric field must come to an end before the start of big bang nucleosynthesis, when the temperature is MeV. This still leaves the possibility open for a huge amplification of electric fields in the period between inflationary reheating and the nucleosynthesis. In the period of amplification, the electric field grows very quickly, . After the peak value is reached, rapidly decays and soon becomes negligible. At that stage, the dynamics becomes similar to the conventional electrodynamics; decays logarithmically (constant at the linear level) until the dust-dominated epoch when it decays as .
3.4 Stability of the other fixed points
Given the complexity of the dynamics, it is also helpful to analyse the stability of the other fixed points. Here we present the eigenvalues for , and . The linear stability analysis for is inconclusive since we have there, and so the Horndeski energy density is generically second order in perturbation. Our numerical simulations in the next subsection indicate that is a past attractor for .
When either or is nonzero, the stability analysis is straightforward since there is no orbit-dependence. We obtain the following eigenvalues:
- Fixed point
- Fixed point
Whenever exists (), it is a future attractor. The new fixed point is a saddle regardless of , and the orbit is temporarily attracted towards it for initial conditions close to that fail to satisfy the conditions (42)-(43). We did not find the dynamics around this solution to be of any phenomenological interest.
The eigenvalues of is dependent on each reference orbit and the linearisation can be carried out in a similar way as for in section 3.1.1. In the end, we derive the following equations:
and read off the eigenvalues:
The first eigenvalue is associated with the perturbation of and positive so that cannot be a future attractor. The time-dependence of around is given by evaluating (17) with as
For , the situation is analogous to except that is irrelevant to future asymptotic behaviour. An orbit with initially reaches the singularity while is a saddle point for the others. For , the stability changes over the course of the evolution.
3.5 Numerical Analysis
We conclude this section by showing two visualizations that confirm the analysis of this section, and suggest that is a past attractor (which is the case in the minimally coupled theory) for . Figure 6 shows a phase portrait of the invariant subset (a universe without the fluid). This subset is effectively two dimensional by the Friedmann equation (19) and the phase space is completely characterised by the variables and . The green shaded region corresponds to . It clearly shows that is a past attractor for on this subspace. Note the existence of stream lines directed towards in the region indicating orbit dependence of the stability for a negative non-minimal coupling constant. In the region , there are two distinct flows inside and outside the bold red lines that denote the location of the singularity. The outer region is marked with a red mesh and corresponds to the pathological region (25) where the singularity is inevitable. Note that (25) can be written which explains why the entire region is non-pathological on the subspace . When a fluid is included, however, the singularity can be reached even from a small initial shear (for example figure 3). Figure 7 shows the full three-dimensional phase flow for a few orbits with , which demonstrates the past-stability of and confirms the time-dependent stability of .
4 Dynamics of a magnetic field in axisymmetric Bianchi type I
In the usual electromagnetism, Maxwell’s equations treat electric and magnetic fields in a symmetric manner. In the context of Bianchi cosmologies, source-free pure electric and magnetic fields are mathematically indistinguishable. The modification breaks this duality. Let us consider a homogeneous magnetic field along -axis in the spacetime (9). In parallel to the electric field of the previous section, we define as the magnetic field seen by a comoving observer:
Contrary to the electric case, equation (7) is trivially satisfied for this Faraday tensor. Instead the evolution of the comoving magnetic field is given by the Bianchi identity which leads to
In contrast to the peculiar dynamics of the electric field, there is no modification to the evolution equation for the magnetic field since it comes from the closedness of the field-strength 2-form. Then, we can solve it easily to obtain
This simply means the magnetic field is adiabatically decaying due to the expansion of the universe.
The Einstein equations can be written in the following convenient from:
Again, there appears to be a problem for negative when . This time, the singularity stems from the Einstein equations instead of the evolution equation for as in the electric case. From (49), it is clear that is positive definite in an expanding universe () for initial conditions satisfying
According to (46), decays in this regime and consequently the system has to reach the singularity . As one can see in figure 8, the shear becomes negative before reaching the singularity. The behaviour is insensitive to the initial conditions or value of whenever initially. This condition coincides with the one for the Horndeski modified term to have a significant contribution to the dynamics. We conclude that is pathological for the magnetic case too.
While there is no divergent behaviour for positive , the energy density of the magnetic field is not necessarily positive, in contrast to the electric field case. This will not cause any problem for ordinary expanding universes with since the solution (46) ensures monotonic decrease of . However, it may result in unusual behaviours when we go backwards in time. We repeat the dynamical system analysis of the previous section and show that is cosmologically viable.
4.1 Expansion-normalised autonomous system
We introduce the following normalised variables:
The normalised Friedmann equation takes the canonical form
In contrast to the electric field, is not positive definite regardless of the sign of . Following the steps of the previous section, we rewrite (48) as a defining equation for the deceleration parameter:
We derive the following evolution equations for the normalised variables: