Variable Speed of Light Cosmology, Primordial Fluctuations and Gravitational Waves

A variable speed of light (VSL) cosmology is described in which the causal mechanism of generating primordial perturbations is achieved by varying the speed of light in a primordial epoch. This yields an alternative to inflation for explaining the formation of the cosmic microwave background (CMB) and the large scale structure (LSS) of the universe. The initial value horizon and flatness problems in cosmology are solved. The model predicts primordial scalar and tensor fluctuation spectral indices $n_s=0.96$ and $n_t=- 0.04$, respectively. We make use of the $\delta{\cal N}$ formalism to identify signatures of primordial nonlinear fluctuations, and this allows the VSL model to be distinguished from inflationary models. In particular, we find that the parameter $f_{\rm NL}=5$ in the variable speed of light cosmology. The value of the parameter $g_{\rm NL}$ evolves during the primordial era and shows a running behavior.


Introduction
Although inflationary models have been successful in fitting cosmological data [1,2,3], there have been issues raised about the fundamental consequences of the models [4]. In particular, the need for chaotic and eternal inflation models has raised the specter of a multiverse cosmological scenario [5,6]. The question as to whether such a scenario can be falsified by observations and the lack of predictability of inflation models has been a cause for concern. Moreover, the standard single-field inflation models suffer from significant finetuning, such as the requirement of a slow-rolling potential and the fine-tuning needed to fit the magnitude of the CMB amplitude [7,8,9,10,11,12].
We shall consider a model based on the idea that the speed of light c can have a significantly larger value in the very early universe [13,14,15,16,17,18,19,20,21,22,23,24]. This assumption leads to a resolution of the horizon and flatness problems, thereby solving the initial value problem in early universe cosmology. A bimetric variable speed of light (VSL) model [25,26,27,28,29] has been proposed with two metric tensors. One metric and its light cone describe a varying speed of light and a constant speed of gravitational waves, while the other metric and its light cone describe a constant speed of light and a varying speed of gravitational waves. This model cannot produce an observed value for relic gravitational waves and a non-zero tensor mode spectral index n t .
In the following, we will formulate a version of VSL based on an earlier VSL model with one metric. In Section 2, we postulate a gravitational action in which the speed of light c = c(x) is a dynamical field. The total action also contains an action for a minimally coupled scalar "seed" field φ, which will produce quantum primordial fluctuations. It also contains an action inducing a spontaneous violation of Lorentz invariance [13,14,30,31,32,33] by means of a non-vanishing vacuum expectation value of a vector field ψ µ . The Lorentz group SO(3, 1) is spontaneously broken resulting in SO(3,1)→ O(3) × R, where R is the absolute preferred time corresponding to the comoving time t in the preferred frame associated with the Friedmann-Lemaître-Robertson-Walker (FLRW) metric. In Section 3, we develop the VSL cosmology and we show how the model can solve the horizon and flatness problems in initial value cosmology. Section 5 presents the calculations of the power spectra and spectral indices for primordial scalar density perturbations and tensor gravitational waves. In Section 6 we end with concluding comments.
The VSL cosmology can remove the fine-tuning of the initial values in the standard big-bang cosmology and fit the available observational data, such as an almost scale invariant, adiabatic and Gaussian scalar matter power spectrum and the potential observation of a gravitational wave power spectrum. As a viable alternative to standard inflationary models, it can relieve the need for fine-tuning present in these models and not require a multiverse scenario in the form of eternal inflation.

The action and field equations
We adopt the following action: where Here, g = det(g µν ), R = g µν R µν , Λ is the cosmological constant and Φ(x) = c 4 (x). The action S ψ is given by where ψ µ is a vector field, B µν = ∂ µ ψ ν − ∂ ν ψ µ , and W (ψ µ ) is a potential. The scalar field action S φ is defined by where φ is a dimensionless scalar field, V (φ) is a potential and c 0 is the measured speed of light today. The matter action is S M = S M (φ m , g µν ) where φ m denotes matter fields.
The energy-momentum tensor is The T ψµν and T φµν are given by and Setting the cosmological constant Λ = 0, the variation of the action with respect to g µν yields the field equation: where G µν = R µν − 1 2 g µν R and where ∇ µ is the covariant derivative with respect to the metric g µν . We also obtain the field equations: and The energy-momentum tensor T µν satisfies the conservation law: In the action S we have made the speed of light c a dynamical degree of freedom, in addition to the dynamical degrees of freedom associated with the metric g µν and the fields ψ µ and φ.
We must now require that local Lorentz invariance and diffeomorphism invariance are violated, so that the speed of light c = c(t) cannot be made constant by a coordinate transformation as is the case in GR. Let us choose W (ψ µ ) to be of the form of a "Mexican hat" potential [13,14]: where λ > 0 and µ 2 > 0. If W has a minimum at then the spontaneously broken solution is given by We choose the ground state to be described by the timelike vector: The homogeneous Lorentz group SO(3, 1) is broken down to the spatial rotation group O(3). The three rotation generators J i (i = 1, 2, 3) leave the vacuum invariant, J i v i = 0, while the Lorentz boost generators K i break the vacuum symmetry K i v i = 0. The spontaneous breaking of the Lorentz and diffeomorphism symmetries produces massless Nambu-Goldstone modes and massive particle modes [33]. The spontaneous breaking of Lorentz invariance and diffeomorphism invariance has selected a preferred frame and direction of time.

Initial value conditions and cosmology
We will work in our application to cosmology in the preferred frame in which the metric is of the FLRW form: where K is the Gaussian curvature of space and K = 0, +1, −1 (in units of [length] −2 ) for flat, closed and open models, respectively. The metric has the group symmetry O(3) × R with a preferred comoving time t.
The energy-momentum tensor T µν will be described by a perfect fluid: where u µ = dx µ /ds is the fluid element four-velocity and p are the matter density and pressure, respectively. As in inflationary models, the horizon problem is solved in our VSL model [13]. Consider a locally flat patch with the line element: From the geodesic equation for light travel ds 2 = 0, we get In the limit when c → ∞ in an early phase of the universe, dt 2 → 0 and the Minkowski light cone is squashed. Now all points in an expanding bubble near the beginning of the universe will be in causal communication with one another. The horizon scale is determined by Let us assume that for t > t c we have c = c 0 . Then for t < t c we get d H → ∞ as c → ∞ and in the phase t < t c all points in the expanding spacetime will have been in causal communication.
The Friedmann equations are given in our model by where H =ȧ/a and ρ = ρ M + ρ ψ + ρ φ . In standard big-bang cosmology with c = c 0 : where Ω = ρ/ρ c , ρ c = 3H 2 /8πG and the Planck time t PL ∼ 10 −43 sec. This implies that the radius of curvature R curv at the Planck time was very large compared to the Hubble radius R H = c 0 /H: This means that the universe in standard big-bang cosmology was very special at the Planck time. The universe has survived some 10 60 Planck times without re-collapsing or becoming curvature dominated. We obtain from (22): where Ω = 8πGρ/3H 2 . In order to resolve the flatness problem, we separate out the initial value Ω i − 1 and the final value Ω 0 − 1 in (26): and Dividing (27) by (28) yields We have for c i = c 0 : Ifȧ i /ȧ 0 < 10 −5 corresponding to an early phase of inflationary acceleration and |Ω i − 1| ∼ O(1), then Ω 0 = 1 to a high accuracy [34]. This solves the flatness problem in inflationary models 3 . In our VSL model, the flatness problem can also be solved. When in an early phase: then from (29) Ω 0 = 1 is predicted to a high accuracy. During the radiation dominated phase with a(t) ∝ t 1/2 the radiation density is given by where σ is the the Stefan-Boltzmann constant, b = 8π 5 k 4 B /15h 3 and k B and h are Boltzmann's constant and Planck's constant, respectively. Thus, ρ r → 0 as c → ∞ and from this we can deduce that Ω i = 8πGρ r /3H 2 i 3 By implementing a measure Gibbons and Turok [35] find that the probability of Ne e-folds of inflation is of order exp(−3Ne).
in (29) is sufficiently diluted, so that for a large enough initial value of c i we have Ω 0 = 1 to a high degree of accuracy.
In the VSL model, traces of an initial inhomogeneity will be sufficiently smoothed out as the universe expands. However, in inflation models bubbles of inflation can be produced that do not inflate sufficiently and these will generate a non-uniformity problem as the universe expands to the present day [4].
We have adopted the scenario that in the very early universe the speed of light c(t) has a large value during a short time duration when t i < t < t c , and it has the value c = c 0 for t > t c when the Friedmann equations and the cosmology are described by GR.
Let us assume for our scalar fluctuation "seed" field φ that in (11) V (φ) = 0, yielding The solution to this equation isφ where B is a constant and a * is a reference value for a. For largeφ the kinetic contribution to ρ φ ∝ 1 2 (φ) 2 will dominate the matter densities ρ M and ρ ψ and the Friedmann equation (22) for c = c 0 becomes Substituting the solution (34), we get Because the field φ dominates in the early universe, we can neglect the spatial curvature for c = c 0 and we obtain the approximate solution given by The equation of state for a massless scalar field gives for the exponent n in a(t) ∝ t n the value n = 2/3(1+w), so for n = 1/3 we get w = 1. For w = 1 the kinetic term 1 2 (φ) 2 dominates the potential term V (φ). We observe that there is no explicit source for the field φ in Eq. (11), which implies that as the universe expands the field φ becomes increasingly diluted to the point where it has unobservable effects at present. By performing a post-Newtonian expansion of the gravitational field in the solar system, we have: g 00 ≈ 1 + 2GM ⊙ /c 2 0 1 AU. If the contribution of the field φ is to be significant in the solar system, then 2GM ⊙ /c 2 0 1 AU ∝ φ 2 . However, as the universe expands we have according to (34) thatφ ∼ 1/a 3 , so that the effects of the scalar field φ will become unobservable in the present universe.

Primordial fluctuations and gravitational waves
Let us consider in our VSL model the mechanism for the generation of cosmological fluctuations and the growth of large structures without inflation. In inflationary models, scalar quantum fluctuations oscillate until their wavelengths become equal to the Hubble radius R H = c 0 /H. When they pass beyond R H the oscillations are damped and the fluctuation modes are "frozen" as classical fluctuations with amplitude δφ ∼ H/2π [34]. In an inflationary spacetime the wavelengths of quantum field fluctuations δφ are stretched by rapid expansion: where H is approximately constant. Short-wavelength fluctuations are quickly redshifted by the inflationary expansion until their wavelengths are larger than the size of the horizon R H . In our VSL model, the wavelengths of the quantum field fluctuations δφ are stretched by the short duration large increase of c: where ν f is the frequency. The redshifted wavelengths of the short-wavelength fluctuations become larger than the horizon and are frozen in as classical fluctuations. The amplitudes of quantum modes are calculated at the horizon crossing, when the wavelength of a mode is equal to a/k = c 0 /H. For an inflationary epoch with 60 e-folds of inflation of the cosmic scale a, the initial matter fluctuations and gravitational waves with wavelengths λ i are stretched to their wavelengths today by the amount: where N is the number of e-folds of inflation, N ≥ 60. In the VSL model, the initial fluctuation wavelengths λ i are stretched by an equivalent amount: where Q 10 30 . The fluctuations are of two kinds: scalar matter quantum fluctuations and tensor gravitational wave fluctuations. Let us consider first the scalar fluctuations. We will consider a scenario in which we assume that V (φ) = 0. The fluctuations δφ(t, x) about a cosmological background will be considered in the comoving frame with the minimally-coupled Klein-Gordon field equation: where k = |k| and By substituting the solution (37) into (42), we get the equation of motion: This equation has the general Bessel function solution [27]: where A 1 and A 2 are constant coefficients and (46) Here, H ℓ = (3B) 2/3 /3a * and we define H ℓ = c 0 /ℓ 0 , where ℓ 0 is the length scale at which the normalized wave function is in its ground state. We can obtain from (45) the normalized plane wave solution: where κ = 16πG/c 4 0 and This solution is approximately equivalent to the one we get from adopting a flat Minkowski spacetime. We assume that after the fluctuation modes cross the horizon they are in a classical state in the Minkowski spacetime 4 .
The scale at which the fluctuation mode exists is given by the condition: From this condition, we get from (46): where y k is the value of y evaluated for a = a k and γ = ℓ 3 0 /2a 3 * . If we assume that we should use (47) as initial data for the classical solution of (42), then we will match not only the initial perturbation, but also its time derivative. Keeping only the dominant contribution as y → 0 and ω k = c 0 k/a k gives From (48), (50) and the Planck length, ℓ PL = (G /c 3 0 ) 1/2 ∼ 10 −33 cm, we obtain the density perturbation power spectrum: Recalling thatφ 2 ∼ 12H 2 , we obtain the curvature power spectrum: Moreover, we have where A s is the scalar fluctuation amplitude and n s is the spectral index. The power spectrum (53) is scale invariant except for the factor ln 2 (y k ). The non-scale invariant contribution results from matching the initial state and its time derivative, and that the Bessel function Y 0 (y k ) is logarithmically divergent when y k → 0. This will lead to a slight deviation from a scale invariant spectrum. The scalar mode spectral index is given by From (50) and (53), we get n s = 1 + 6 ln(y k ) .
The running of the spectral index is calculated from which yields We have in the large scale limit the anisotropic amplitude: By adopting 2 5 we have δ H ∼ ℓ PL /ℓ 0 . Fixing the length scale ℓ to be ℓ 0 ∼ 10 5 ℓ PL ∼ 10 −28 cm, which is of the order of the grand unification scale, we can match the amplitude of the observed CMB fluctuations, δ H ∼ 10 −5 .
For the value ln(y k ) ∼ −150, we obtain the result for the spectral index: This is in good agreement with the result obtained by Planck Mission [1,2,3]: For the running of the spectral index n s , we get which is in approximate agreement with the Planck Mission result: In inflationary models the derivation of the power spectrum and the spectral index depend sensitively on the shape of the inflaton potential V (φ inflaton ) and its derivatives with respect to φ. The condition of a slow-roll potential is required to produce enough e-folds of inflation. This is not the case in our VSL derivation of the power spectrum and the spectral index. Our derivation of the scalar fluctuation power spectrum does not depend sensitively on the shape of the potential V (φ). This can reduce the VSL model dependence and associated fine-tuning problems.
We now turn to the spectrum of relic gravitational waves. The conformally flat background metric is In our model the wavelength of gravity waves is given by λ g = c g /ν g , where ν g is the frequency of a gravitational wave. During the phase when c g ≫ c 0 gravitational waves are generated and their wavelengths are stretched and cross the horizon. As the universe expands, the amplitude of the gravitational wave spectrum passes back into the observable universe and we can observe a B-polarization with a non-zero spectral index n t and ratio r=tensor/scalar. The tensor perturbations can be expressed as where and where e ij (k, λ) is a polarization tensor satisfying e ij = e ji , e ii = 0, k i e ij = 0 and e ij (k, λ)e * ij (k, µ) = δ λµ . Moreover, we have ψ k,λ , ψ * l,λ = 2π 2 P δψ δ(k − l), where P δψ is the gravitational wave spectrum. The gravitational tensor component can be expressed as the superposition of two scalar polarization wave modes: h tensor where +, × refer to the longitudinal and transverse polarization modes, respectively. The modes obey the scalar equation of motion at the horizon and super-horizon scales when c = c 0 : The sizes of the second and third terms depend on the magnitude of c 0 k/aH. If c 0 k/aH → 0, then the gravitational wave mode is outside the horizon and we can neglect the third term and the solution to the wave equation approaches a constant. If c 0 k/aH is large, the mode is inside the horizon and the second term becomes sub-leading. The mode then undergoes a damped oscillation and decays as 1/a. The tensor modes of interest are outside the horizon with constant values determined by the primordial distribution, generated at a sub-horizon scale during the phase when c g has a large value c g ≫ c 0 . During the radiation and matter domination epochs the modes gradually re-enter the horizon and damp away. Only the tensor modes which entered the horizon just before the surface of last scattering lead to important effects in the CMB. The tensor mode power spectrum is where k p = 0.004 Mpc −1 is the pivotal scale. By solving the scalar wave equation (70) in the same manner as was done for the scalar matter fluctuations, we obtain the power spectrum: where N is a constant. As in the case of the scalar matter fluctuations, the gravitational wave mode fluctuations are scale invariant up to the slight scale breaking factor ln 2 (y k ). The tensor mode spectral index is given by This yields the result n t = 6 ln(y k ) .
Choosing, as before for the scalar perturbations, the value ln(y k ) ∼ −150 we obtain This spectral index (tilt) is red which agrees with the standard inflationary model result when the Null Energy Condition is violated,Ḣ < 0.
If we now adopt the BICEP2 B-polarization result r = 0.2 +0.07 −0.05 [38] 5 , then we get r/n t = −5, which is close to the single-field inflationary model consistency condition determined by the slow-roll parameter, ǫ = −Ḣ/H 2 , related to the equation of state p = ǫ − ρ. We have r = 16ǫ and n t = −2ǫ giving r/n t = −8, which is satisfied irrespective of the form of the single-field inflationary potential.

Conclusions
We have formulated a VSL model in which the homogeneous Lorentz group SO(3, 1) is spontaneously broken to the rotation group O(3) by the non-zero vacuum expectation value 0|ψ µ |0 . This determines a preferred time t in the cosmological model corresponding to the comoving time in an FLRW spacetime.
In contrast to the inflationary scenario, our VSL model prediction of the almost scale invariant, Gaussian fluctuation spectra for matter and relic gravitational waves does not rely on determining the shape of a potential and its derivatives. The model can relieve the fine-tuning that is inevitably a consequence of inflationary models. Once the short duration phase of c ≫ c 0 and c g ≫ c g0 has taken place and the wavelengths of the initial primordial quantum fluctuations and gravitational waves have been stretched through the horizon, then classical solutions of the wave equations for the quantum and gravitational wave fluctuations can be employed to generate the power spectra and the spectral indices of the scalar and tensor fluctuation modes. Our VSL model can produce the non-zero scalar and tensor spectral indices n s = 0.96 and n t = −0.04. Combined with the tensor/scalar ratio potentially observed by BICEP2, r = 0.2 +0.07 −0.05 , we obtain the ratio r/n t = −5, which is close to the single inflation field consistency condition r/n t = −8.
Since the ordered phase in the spontaneous symmetry breaking of Lorentz invariance is at a much lower entropy than the restored, disordered symmetry phase and due to the existence of a domain determined by the direction of the vev, 0|ψ µ |0 , a natural explanation is given for the cosmological arrow of time and the origin of the second law of thermodynamics [13,14,40]. The ordered state of low entropy in the symmetry broken phase with c ≫ c 0 , becomes a state of high entropy in the symmetry restored disordered phase with c = c 0 . The spontaneous symmetry breaking of the gravitational vacuum leads to a manifold with the structure O(3) × R, in which time appears as an absolute external time parameter. The vev 0|ψ µ |0 points in a chosen direction of time to break the symmetry of the vacuum creating an arrow of time.