Detecting a Lorentz-Violating Field in Cosmology

Detecting a Lorentz-Violating Field in Cosmology

Baojiu Li B.Li@damtp.cam.ac.uk Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom    David F. Mota D.Mota@thphys.uni-heidelberg.de Institute of Theoretical Physics, University of Heidelberg, 69120 Heidelberg, Germany    John D. Barrow J.D.Barrow@damtp.cam.ac.uk Department of Applied Mathematics & Theoretical Physics, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WA, United Kingdom
July 14, 2019
Abstract

We consider cosmology in the Einstein-Æther theory (the generally covariant theory of gravitation coupled to a dynamical timelike Lorentz-violating vector field) with a linear Æ-Lagrangian. The spacetime splitting approach is used to derive covariant and gauge invariant perturbation equations which are valid for a general class of Lagrangians. Restricting attention to the parameter space of these theories which is consistent with local gravity experiments, we show that there are tracking behaviors for the Æ field, both in the background cosmology and at linear perturbation level. The primordial power-spectrum of scalar perturbations in this model is shown to be the same that predicted by standard general relativity. However, the power-spectrum of tensor perturbation is different from that in general relativity, but has a smaller amplitude and so cannot be detected at present. We also study the implications for late-time cosmology and find that the evolution of photon and neutrino anisotropic stresses can source the Æ field perturbation during the radiation and matter dominated epochs, and as a result the CMB and matter power spectra are modified. However these effects are degenerate with respect to other cosmological parameters, such as neutrino masses and the bias parameter in the observed galaxy spectrum.

pacs:
04.50.+h, 98.80.Jk, 04.80.Cc

I Introduction

For more than two decades Milgrom’s modified Newtonian dynamics (MOND) mond (1983) has been able to explain galaxy rotation curves which are conventionally considered as an evidence of cold dark matter (CDM) on galactic scales. MOND modifies Newton’s second law of motion to where and are the acceleration and Newtonian gravitational potential, respectively; is an effectively free function tending to unity in the limit , with being a new fundamental constant, which must have a numerical value of in order to match observations on a galactic scale. This theory looks like Newton’s when accelerations are large but is significantly different when accelerations are small. On galactic scales, , so the Newtonian dynamics is modified, but in a way that can fit spiral galaxy rotation curves. Subsequently, Bekenstein TeVeS (2004) built a relativistic theory which has MOND as a non-relativistic, weak-field limit, thus making the study of cosmology possible. In addition to the conventional tensor gravitational field, Bekenstein’s theory involves a vector and a scalar field, and is therefore dubbed TeVeS. Interestingly, it has been argued that TeVeS could also explain the large-scale structure formation of the Universe without recurring to CDM TeVeSperturbation (2006), thanks to the presence of the vector field TeVeSVector (2006).

Recently, the authors of Ref. TeVeSAether (2006) showed that TeVeS is equivalent to a vector-tensor theory of gravitation where the vector field has a non-fixed norm. They also showed that the correct MONDian limit could be realized with a single vector field having non-canonical kinetic terms NoncanonicalAether (2007). These results indicate that a vector field in the gravity sector might be an interesting component of the Universe (indeed the model in NoncanonicalAether (2007) and its generalized version we shall presented below in Eq. (1) could be used to explain dark energy and dark matter in background cosmology) and merits more detailed investigations.

The idea of a vector field coupled to gravity has a long worldline (see for example Aether (2001) for a review and others () for further references), but in this work we will focus on the model described in Aether (2001), which is the most well-studied one, and investigate its cosmological implications. This particular theory is based on a dynamical vector field coupled to gravitation that picks up a preferred frame and preserves general covariance. This vector field is unit-norm, timelike, and violates local Lorentz invariance. It is called the Æther field (or simply Æ-field) and we will refer to the associated Einstein-Æther theory as Æ-theory defined by the Æ-Lagrangian. The Æ-Lagrangian considered in Aether (2001) is a special case of our general model introduced in Eq. (1) below for a unit-norm vector field that includes terms up to second order in derivatives, and it has been extensively studied in various contexts AetherWF (2005); AetherWF2 (); AetherCompactStars (2007); AetherGravitationalWaves (2005); AetherBHs (2007).

In Refs. AetherBackground (2004); AetherPerturbation (2005), the background cosmology and primordial power spectra of perturbations from inflation of a slight different model were also considered. Here, we investigate these for the model presented in Ref. Aether (2001), and also study the evolution of linear perturbation to the Æ-field during the radiation and matter-dominated epochs. As we will show below, if we restrict the parameter space of the underlying theory so as to satisfy the local experimental gravity constraints, AetherWF (2005); AetherGravitationalWaves (2005), this perturbation becomes sourceless and decays during the epoch of inflation and late-matter domination. However, it is sourced by the evolutions of the photon and neutrino anisotropic stresses during the radiation era and early matter era, which have some imprints on the cosmological observables.

Our presentation is organized as follows. In Sec. II, we briefly introduce the general Æ-theory and derive perturbation equations for the background Friedmann-like cosmology in the covariant and gauge invariant (CGI) formalism (see Zhao2007a (2007) for a derivation of the perturbation equations in conformal Newtonian gauge). In Sec. III we shall use these equations to discuss the perturbation dynamics for the cosmological models of Ref. Aether (2001). First, we summarize the existing constraints on the model in Sec. III.1; then, in Sec. III.2, we present the evolution equations for the perturbation variables and then use them to show how the primordial spectra of scalar and tensor perturbations in this theory are unmodified and modified, respectively, on comparing them with the predictions of general relativity (GR). The late-time evolution of the Æ-field perturbation and its effects on cosmological observables are also studied there. Finally, our discussion and conclusions are presented in Sec. IV.

Throughout this work our convention is , where are respectively the Riemann tensor and Ricci tensor; the metric signature is and the universe is assumed to be spatially flat, filled with photons, baryons, CDM, 3 species of neutrinos and a cosmological constant.

Ii Field Equations of Einstein-Æther Theory

In this section we briefly introduce the general form of the Æ -theory and derive the CGI perturbation equations which we will use to analyze the cosmological effects of the Æ -field. The equations presented here are for general Lagrangians. Later, we shall focus on a specific class of such theory characterized by a linear Lagrangian (with , see below).

ii.1 The General Einstein-Æ ther Theory

The model we consider here is a slight generalization of that presented in NoncanonicalAether (2007). It is characterized by a general gravitational action of the form

(1)

in which is the bare Newtonian gravitation constant, is a Lagrange multiplier ensuring that Æ -field has a unit norm, and is the Æ -Lagrangian expressed by

(2)

with

(3)

and being an arbitrary analytic function of its arguments. As long as the norm of is fixed (), the form of is the most general possibility we can have for our vector-field Lagrangian. Notice that Eq. (3) here differs from that given in NoncanonicalAether (2007) by the term and we shall refer to our model and that of NoncanonicalAether (2007) as the GEA (Generalized Einstein Æther) model to distinguish from the one considered in Aether (2001) (for which we instead called EA (Einstein Æther)). The matter Lagrangian is taken to be the same as in standard model.

We treat the Æ -field and inverse metric as the dynamical degrees of freedom and vary the action with respect to them to obtain the field equations. The former gives the Æ -field equation of motion (EOM):

(4)

where we have defined , and . The variation with respect to the metric leads to a modified Einstein equation. One could retain the form of Einstein equations in standard GR by treating the vector field as a new contribution (denoted by ) to the total energy-momentum tensor in the universe, in addition to that of the conventional fluid matter which is denoted by . Then, according to the definition

in which , we have

(5)

We also note that by varying the action with respect to the Lagrangian multiplier we simply get the normalization relation of the Æ -field, , as mentioned above.

ii.2 The Perturbation Equations in General Relativity

The CGI perturbation equations in general Æ -theories are derived in this section using the method of decomposition GR3+1 (1989) (see MG3+1 (1989) for applications of this method in modified-gravity models). First, we briefly review the main ingredients of decomposition and their application to standard general relativity GR3+1 (1989) for ease of later reference.

The main idea of decomposition is to make spacetime splits of physical quantities with respect to the 4-velocity of an observer. The projection tensor is defined as and can be used to obtain covariant tensors perpendicular to . For example, the covariant spatial derivative of a tensor field is defined as

(6)

The energy-momentum tensor and covariant derivative of the 4-velocity are decomposed respectively as

(7)
(8)

In the above, is the projected symmetric trace-free (PSTF) anisotropic stress, the heat flux vector, the isotropic pressure, the PSTF shear tensor, the vorticity, ( is the mean expansion scale factor) the expansion scalar, and the acceleration; the overdot denotes time derivative expressed as , brackets mean antisymmetrisation, and parentheses symmetrization. The 4-velocity normalization is chosen to be . The quantities are referred to as dynamical quantities and as kinematical quantities. Note that the dynamical quantities can be obtained from the energy-momentum tensor through the relations

(9)

Decomposing the Riemann tensor and making use the Einstein equations, we obtain, after linearization, five constraint equations GR3+1 (1989):

(10)
(11)
(12)
(13)
(14)

and five propagation equations,

(15)
(16)
(17)
(18)
(19)

Here, is the covariant permutation tensor, and are respectively the electric and magnetic parts of the Weyl tensor , defined by and . The angle bracket means taking the trace-free part of a quantity.

Besides the above equations, it is useful to express the projected Ricci scalar into the hypersurfaces orthogonal to as

(20)

The spatial derivative of the projected Ricci scalar, , is then given as

(21)

and its propagation equation by

(22)

Finally, there are the conservation equations for the energy-momentum tensor:

(23)
(24)

As we are considering a spatially-flat universe, the spatial curvature must vanish on large scales and so . Thus, from Eq. (20), we obtain

(25)

This is the Friedmann equation in standard general relativity, and the other background equations (the Raychaudhuri equation and the energy-conservation equation) can be obtained by taking the zero-order parts of Eqs. (15, 23), yielding:

(26)
(27)

In what follows, we will only consider scalar perturbation modes, for which the vorticity and magnetic part of Weyl tensor are at most of second order GR3+1 (1989), and so will be neglected in our first-order analysis.

ii.3 The Perturbation Quantities in Æ -Theory

In the Einstein-Æther theories where we consider the Æ -field as a new species of matter, the gravitational field equations Eqs. (10 - 27) listed above preserve their forms, but the dynamical quantities appearing there should be replaced by the effective total quantities of the same type. For simplicity, we shall always use variables without superscripts to denote these effective total quantities, while for those of a specified matter species we shall add corresponding superscripts (e.g., denotes the energy density of the Æ -field ).

The vector Æ -field, , requires further discussion. As we mentioned above, it has the normalization relation . In the background Friedmann-Robertson-Walker (FRW) universe the requirements of homogeneity and isotropy require that is just equal to , which is unambiguously chosen as the 4-velocity of the fundamental observers. But in a perturbed, almost-FRW, universe this is no longer true and we can write where is another (first-order) vector field that vanishes in a FRW Universe: we call it the perturbation of the Æ -field. Furthermore, the relation implies that , i.e., is a spatial vector field which is perpendicular to , up to first order in perturbation. This fact is used extensively in deriving the perturbation equations (e.g., etc.).

With these preliminaries at hand, and after some lengthy manipulations, the Æ -field EOM Eq. (6) can be written as (up to first order)

(28)

and from the definitions Eq. (II.2) the Æ -field energy density, isotropic pressure, heat-flux vector and anisotropic stress can be identified from Eq. (7) (again up to first order) as

(29)
(30)
(31)
(32)
(33)

where, in Eq. (32), we have used Eq. (II.3). Here, we have defined the new parameters , and ; is given by . Including these Æ -contributions to Eqs. (10 - 27) one obtains the modified gravitational field equations for the general Æ -theory. It is also easy to check that the above results satisfy (separately) the conservation of the Æ -field’s energy-momentum tensor, Eqs. (23, 24). Note that Eqs. (29 - 33) are the general expressions of energy density, pressure, heat flux and anisotropic stress in the decomposition which include both zeroth order (background) and first order terms; to calculate the actually density contrast etc. (see Eqs. (51 - 54) below) one needs to take the covariant spatial derivatives of these equations GR3+1 (1989).

Iii A Specific Model: The linear Lagrangian

In above we presented the field equations for general Æ -theories, but in what follows we shall only analyze the cosmology of a specific edition of the theory which is defined by choice of a linear Lagrangian:

(34)

(note that the minus sign is because of our sign convention). This model is by far the most well known, in the sense that it has been investigated in the contexts of static weak-field limit AetherWF (2005) (see also AetherWF2 () for the weak-field limit of the Æ-model considered in NoncanonicalAether (2007)), background cosmology AetherBackground (2004), the radiation and propagation of the Æ -gravitational waves AetherPerturbation (2005); AetherGravitationalWaves (2005), compact stars AetherCompactStars (2007), and black holes AetherBHs (2007). Some of these studies have imposed stringent constraints on the viable parameter-space of s. In view of these restrictions we will confine our study to this constrained subset of possible theories. Note that the perturbation dynamics of the Æ -model has also been analyzed in AetherPerturbation (2005) in the absence of the term. Here, we shall include this term and perform a similar analysis but in slightly different manner and in more detail; we will also discuss on some detailed features which lead to our conclusions being different. The late-time evolution of the Æ -perturbation is also investigated. In particular, we shall find that within the locally-constrained parameter space the Æ -model will leave slightly different signatures on the perturbation evolutions from those left by the standard paradigm in general relativity, and so cosmological data on cosmic microwave background (CMB) and matter power spectra might place some constraint on the parameter space.

iii.1 The Constrained Parameter Space

In this subsection we briefly summarize the constraints on the Æ -model described in Eq. (34). It has been well known that in the weak-field, slow-motion, limit and in the background cosmology the Æ -model displays tracking behavior. For the former environment, it can be shown that in the presence of the Æ -field the observed effective gravitational constant is a rescaling of the bare one AetherBackground (2004); AetherWF (2005) by (note that here our s have different signs from those in AetherWF (2005))

(35)

and in the latter environment the observed gravitational constant is also a rescaling of but with a different factor AetherBackground (2004)

(36)

Correspondingly, we define and to be used below. Note that the rescaled is generally not equal to and as such the background cosmic expansion rate will be different from that in standard GR. However, we note that the numerical value of gravitational constant we find in the textbooks and use in the numerical calculations is not but rather the locally-measured . It is possible to obtain limits on non-local values of by considering the primordial nucleosynthesis of light elements (see for example Ref. bs ()). If then we have , which indicates that the background cosmological dynamics will be exactly the same as assuming standard GR AetherBackground (2004); Comment (2007); otherwise we can use constraints from primordial nucleosynthesis on the value of gravitational constant to show that AetherWF (2005). However, note that the particle-horizon size at the epoch of neutron-proton freeze-out (), which is most sensitive to variations in the value of , is only and this causally linked region expands in size by about a factor of by the present-day to a size which is a sub-galactic scale but there has then been local gravitational collapse by a factor of . Such collapse may however also affect the local value of the gravitational constant tim (), but we will not consider this in the present work.

There are also constraints from the observations of parameterized post-Newtonian (PPN) parameters PPNGR (1993). It is shown in AetherWF (2005) that for all the PPN parameters to coincide with those in GR (otherwise the parameters may need to be fine-tuned) one reduces the full four-parameter space of the model to a two-parameter subspace characterized by

(37)

In addition, Æ -theories contain five gravitational and Æ -wave modes, which include the two usual spin-2 gravitational waves, and three additional modes: two spin-1 transverse Æ -gravity waves and one spin-0 longitudinal Æ -gravity wave. The speeds of the three additional modes are generally not equal to . It has been shown in Ref. AetherGravitationalWaves (2005) that if these speeds are less than 1 then the high-energy particles will produce erenkov radiation when passing through vacuum, which imposes stringent constraints on the model. However, as suggested in AetherWF (2005), these constraints do not apply if these modes propagate superluminally. The requirement that the additional Æ -gravity waves do not propagate subluminally further limits the parameter space to

(38)

Finally, when the above constraints Eq. (38) are satisfied, the positive energy requirement and the stability of additional wave modes AetherPerturbation (2005) also hold. In addition, we will have so that the Big Bang nucleosynthesis constraint does not apply. Thus, we can see that, even after using all the current constraints, there is still a large parameter space remaining for the model. In what follows we shall ask whether linear-order cosmological observations such as the CMB and the form of the matter power spectrum could reduce this parameter space further, and as we will show, the answer is positive. However, the modifications are small and depend weakly on the model parameters, which mean that the data on CMB and matter power spectra cannot give very stringent limits on the parameter space.

iii.2 Linearly Perturbed Equations

In this subsection we consider the perturbation evolutions of our Æ -model. For generality, we will derive the equations for arbitrary choices of parameters and only later confine ourselves to the parameter space described in Eq. (38). Besides, since the presence of the Æ -field in general will modify the cosmology at all times, we will also investigate its effects during the inflationary era as in AetherPerturbation (2005); after that we will turn to its effects on late time cosmology.

Following GR3+1 (1989), we shall make the following harmonic expansions of our perturbation variables

(39)

in which is the eigenfunction of the comoving spatial Laplacian satisfying

and are given by . Note that æ is dimensionless.

In terms of the above harmonic expansion coefficients, Eqs. (11, 13, 16, II.2, 21, 22) can be rewritten as GR3+1 (1989)

(40)
(41)
(42)
(43)
(44)
(45)

where and a prime denotes the derivative with respect to the conformal time (). Also, Eq. (24) and the spatial derivative of Eq. (23) become

(46)
(47)

Recall that we shall always neglect the superscript for the total dynamical quantities and add appropriate superscripts for individual matter species. Note that

(48)

and that the appearing above is the bare (not necessarily the measured) one. Furthermore, for convenience we define the frame-independent (FI) variables GR3+1 (1989)

(49)
(50)

is FI according to Eq. (33) because we know that the anisotropic pressure tensor is frame invariant. Hence, it follows from Eq. (13), that is also FI up to first order in perturbation. In the zero-shear frame (the Newtonian gauge), we have simply and .

Before presenting the evolution equations, we first write the dynamical quantities of the Æ -field in terms of the harmonic coefficients introduced above:

(51)
(52)
(53)
(54)

In these expressions we have used both æ and because not all these quantities are FI. However it can be shown that the two quantities

(55)

and

(56)

which will be used in the derivations, are FI, as they are expressed in terms of FI variables only. Note that in the above we have defined for convenience, where is the Newtonian gravitational potential, and

We now investigate in detail Eqs. (55, 56). On large scales, where , the terms involving (these include the term in Eq. (55)) can be safely disregarded, and as a result we have

(57)
(58)

where the superscript means the fluid matter. We see that on large scales these attributes of the Æ -field track those for other matter species in the universe. As the combination determines the type of perturbations (for example, the perturbation is adiabatic if the combination is equal to zero), this indicates that the Æ -field will not alter the type of the scalar perturbation produced in the inflationary era. Note that the above tracking behaviors are the same as that in the background cosmology, i.e., rescaling the gravitational constant by a same factor.

Now we can proceed to derive the evolution equations for our Æ -model. The propagation equation for the Æ -field Eq. (II.3), in terms of the FI variables, becomes

(59)

in which we have changed the variable to for simplicity. Taking the time derivative of Eq. (43), adding to it times the same equation, and using Eq. (46) to eliminate the term, we obtain the following equation second-order differential equation for :

(60)

Eqs. (59, 60) are the evolution equations for the coupled system that we are looking for (note that where is the fluid matter anisotropic stress and can also be expressed in terms of and its time derivatives by virtue of Eq. (54)). They are not closed if and are unknown and to know these quantitie we would need to know the matter content of the universe.

iii.3 The Primordial Power Spectra

In the analysis above, we have mentioned that the presence of the Æ -field does not affect the form of the scalar perturbation produced during inflation. But we also need to know whether other features of the inflationary power spectrum, such as the spectral index and the amplitude, are modified by the Æ -field, as compared with the predictions in standard GR. Here, we will investigated this issue (see AetherPerturbation (2005) for a study in the absence of the term in the Æ -Lagrangian) by considering a single-field model of inflation InflationBook (2000) in the presence of the Æ -field.

During the inflationary epoch a scalar inflaton field slowly rolls along its potential and has an almost constant energy density which drives an almost exponential expansion of the universal scale factor. The comoving Hubble length (the horizon) decreases with time so that in this process the quantum vacuum fluctuations of the inflaton field on the scales of interest to us leave the horizon (their scales become larger than the horizon). The curvature perturbations they generate remain constant during their subsequent super-horizon evolution, until these scales eventually reenter the horizon long after the inflationary period has ended. During the radiation-dominated era when these modes stay outside the horizon, the metric perturbation becomes a constant, which drives the density perturbations of different matter species, and leads to the observed CMB and matter power spectra after horizon re-entry.

There are no couplings between the scalar field and gravitational or Æ -fields, so we can write down its dynamical quantities for the scalar field as

(61)

Making the following harmonic expansion for

(62)

it is easy to get

(63)
(64)
(65)

where . Then parallel to Eqs. (55, 56) we have, following the standard procedure,

(66)

and

(67)

where we have defined the FI variable

(68)

Substituting Eqs. (55, 67) into Eq. (60), and using Eq. (41) to eliminate the term proportional to , we arrive at the following equation

(69)

Eqs. (59, 69) that form a closed set of evolution equations for the coupled Æ -inflaton system; they are a generalization of the results presented in AetherPerturbation (2005), and from them we can perform our analysis of the observational effects.

Let us look first at the large-scale evolution of . In this limit so the last term on the left-hand side, and the first two terms on the right hand side, of Eq. (69) can be dropped. Notice that during the inflationary era and Eq. (59) become