key: cord-0221208-44uh1lu9
authors: Abdullah, Muhammad; Bavor, Calvin; Chafamo, Biruk; Jiang, Xiaole; Kalim, Muhammad Hamza; Stiffler, Kory; Whiting, Catherine A.
title: Inflation from Dynamical Projective Connections
date: 2022-03-08
journal: nan
DOI: nan
sha: 36c58028de3b3df5675f8df2efb49818b978d063
doc_id: 221208
cord_uid: 44uh1lu9

We show how the recently developed string-inspired, projectively-invariant gravitational model Thomas-Whitehead gravity (TW gravity) naturally gives rise to a field acting as the inflaton. In the formulation of TW gravity, a field $mathcal{D}_{ab}$ is introduced into the projective connection components and is related to a rank-two tensor field $mathcal{P}_{ab}$. Through the dynamical action of TW gravity, in terms of projective curvature, the tensor field $mathcal{P}_{ab}$ acquires dynamics. By decomposing $mathcal{P}_{ab}$ into its trace and traceless degrees of freedom, and choosing the connection to be Levi-Civita, we demonstrate that TW gravity contains a non-minimally coupled scalar field with a specific potential. Considering only the trace degrees of freedom, we demonstrate that the scalar field acts as an inflaton in the slow roll approximation. We find a range of values for the parameters introduced by TW gravity that fit the experimental constraints of the most recent cosmological data.

Since the initial formulation of cosmological inflation in the late 1970's to early 1980's [1] [2] [3] [4] , dynamical scalar fields representing the so-called inflaton field have appeared in many unique forms. Despite the overwhelming evidence that the inclusion of scalar fields both alleviates long-standing cosmological problems and predicts the observed nearly scale-invariant spectrum of CMB perturbations, there are few proposals for a fundamental physical origin of the inflaton. Indeed, as explained by Kolb 

That paradigm [inflation], however, is still without a standard model for its implementation. Of course, that shortcoming should be viewed in light of the fact that our understanding of physics at energy scales well beyond that of the standard model of particle physics is still quite incomplete.

-E. Kolb and M. turner [5] Due to the expectation that physics beyond the standard model should have something to say about inflation, substantial effort has been focused on teasing out the emergence of inflation through string theory motivated models [6] . Attempts have also been made to realize the inflaton as the Standard Model Higgs field [7, 8] . Here, the Higgs is only a potentially viable inflaton if it couples non-minimally to the gravitational sector with a coupling of the form ξφ 2 R. More general models of non-minimally coupled inflation consider various other forms for the potential and coupling [9] [10] [11] [12] [13] .

Thomas-Whitehead gravity (TW gravity) [14] [15] [16] is a string-inspired model of gravity that emerges from the projective geometry of Thomas and Whitehead [17] [18] [19] . In this paper, we demonstrate this model could be the aformentioned raison d'etre for inflation, the foundational principle being projective symmetry. The importance of projective symmetry arises upon extending a coadjoint element of the Virasoro algebra to higher dimensions [20-26, 14, 16] . That the ultimate foundational piece of TW gravity is the Virasoro algebra is why TW gravity is said to be string-inspired. We find solutions to TW gravity that describe an early universe inflationary epoch fitting current cosmological data [27] . These solutions are parameterized by a set of three fundamental constants in TW gravity. One choice of these parameters would correspond to a certain combination of scalar field (non-minimally coupled) inflationary models investigated in [13] . This paper is structured as follows. In section 2 we review TW gravity. We demonstrate how projective symmetry is utilized in the construction of TW gravity and summarize the connection to the deeper underlying Virasoro algebra. In section 3 we demonstrate how a non-minimally coupled (NMC) model for inflation with a specific potential naturally emerges from TW gravity. This NMC model has three free parameters inherited from the full TW gravity model. In the slow roll approximation we constrain these three parameters within a range that matches the current observational data for the spectral index n s , tensor-to-scalar ratio r, and scalar amplitude A s for efoldings of N = 50, 60, and 70. In section 4 we make concluding remarks.

The appendices show our conventions, reviews of background material, and supporting calculations.

In this section we review TW gravity by defining projective connections, building from this projective curvature, and from this constructing projective invariants. We then show how the projectively invariant action of TW gravity is composed of these projective invariants.

Here we briefly review the necessary components of TW gravity, as generalized recently in [16] from the constant volume form of [14, 15] . TW gravity is a theory of dynamical projective connections. For a detailed review of the theory of projective connections, we refer the reader to [28] . A connection ∇ a with coefficients Γ a bc describes a geodesic path with coordinates x a and parameterization τ . These satisfy the geodesic equation

The same geodesic path with coordinates x a can be described by a different connection ∇ a with coefficientsΓ a bc and reparameterization σ = σ(τ ). These also satisfy the geodesic equation

so long as the connection coefficients and parameterizations are related as follows 

where Υ is the fundamental vector field of the Thomas-Cone N generating projective transformations. Explicitly, the projective connection coefficients are defined bỹ (2.4) . The one-form α is related to Υ by the conditions that α ρ Υ ρ = 1 and L Υ α ρ = 0, where L Υ denotes the Lie derivative with respect to Υ.

The symmetric field D ab , known as the diffeomorphism field, is not a tensor due to its appearance as part of the connection coefficientΓ λ ab . It transforms as follows

with | ∂x b ∂x c | the Jacobian of the transformation. For an infinitesimal coordinate trans- 11) to first order in ξ. The details of this dimensional reduction are shown in appendix B.

Equation (2.11) is the same transformation law as a coadjoint element D of the Virasoro algebra, up to rescalings of D. The diffeomorphism field D ab is usually seen as the dynamical extension of a coadjoint element of the Virasoro algebra to higher dimensions, rather than the dimensional reduction that was described here. From this vantage point, TW gravity as the resulting dynamical model for D ab is said to be string-inspired from the core foundational Virasoro algebra in one-dimension. More details can be found in the seminal works [20-26, 14, 16] .

Once in arbitrary d-dimensions, it is advantageous to form a tensor by adding various objects to D bc that remove the non-tensorial pieces from its transformation law. This is accomplished through the definition of the tensor field P bc

Note that P bc is symmetric only if the curl of α c vanishes, one possible solution being that the spacetime connection is Levi-Civita with respect to a metric g ab on M such that Γ m am = ∂ a ln |g|. This field P bc , related to the diffeomorphism field D bc via Eq. (2.12), transforms as a tensor on the spacetime M and will be shown to act as a source of cosmological inflation under certain assumptions.

Under a general coordinate transformation on M, the spacetime connection Γ a bc transforms as an affine connection

(2.13)

The projective connectionΓ α µν transforms as an affine connectioñ

under what we refer to as a Thomas-Cone transformation on N (TCN -transformation) [17, 18, 29, [14] [15] [16] x

The TCN -transformation is seen to be a general coordinate transformation on M with an additional Jacobian scaling of the λ-direction. We refer to objects transforming as a tensor with respect to TCN -transformations as TCN -tensors. Now we have the technology to write the manifestly TCN -covariant and manifestly projectively invariant geodesic equation

This equation is manifestly TCN -covariant in that dx µ /dτ and∇ µ are both TCNtensors. At the same time, it is also manifestly projectively invariant in thatΓ α µν is invariant with respect to projective transformations as in Eq. (2.3).

Since the projective connection coefficientsΓ α µν transforms as an affine connection under a TCN transformation, we can straightforwardly compute its curvature invariants. Explicitly, on a vector field κ α and co-vector κ α in N , we define the projective curvature tensor K γ ραβ through the usual relations

In terms of the connection coefficients, we can write the projective curvature as

The extended metric G αβ on the (d + 1)-dimensional manifold N is written succinctly as

where g α = (g a , 1/λ) and g a ≡ − 1 d+1 ∂ a ln |g|. This extension of a d-dimensional metric was detailed in [16] , where the previous restriction of constant-volume coordinates was lifted. The general volume coordinate metric G αβ can be written as the sum of the constant volume metric G (0) αβ and finite correction ∆G αβ :

Note again that if the spacetime connection Γ a bc is the Levi-Civita connection then g a = α a , since Γ a ab = ∂ b ln |g|. The determinate of the metric G αβ is the same as for G (0) αβ , [14, 15] G ≡ det(G αβ ) = det(G (0) αβ ) = − −2 g , g ≡ det(g ab ) .

(2.24)

The only non-vanishing components of K α βµν are

The projective curvature tensor satisfies the following

An important object to consider will be the projective Cotton-York tensor

which is a TCN -tensor. Its only non-vanishing components are

Notice on the Levi-Civita shell, ∆ a = 0 and the Cotton-York tensor becomes simply

and can be thought of as a gravitational analog of an electromagnetic field strength.

The existence of such an analogy is not surprising. 

We now detail the dynamical action describing TW gravity. This action is built from the projective curvature invariants described in the previous section and appears as

This first line of Eq. (2.36) includes a projective Ricci scalar and cosmological constant, mimicking the usual Einstein-Hilbert action. The second line is the projective Gauss-Bonnet action allowing for P ab to acquire dynamics. Specifically, dynamics is given to P ab through theK nab components of K α βµν as seen in Eq. (2.26). The trade off is the quadratic curvature terms over the manifold M which we will show how to manage.

Let us demonstrate these features by making the following expansions

These expansions can be easily derived via use of the succinct form of the metric in Eq. (2.19) and taking into account that the only non-vanishing λ components of either K αβ or K α βµν are K λ βµν . Notice the expansion Eq. (2.37b) contains the projective Cotton-York tensor, Eq. (2.33), which we see leads to quadratic derivatives on P ab in the action and thus provides dynamics to the field equations for P ab . Continuing our electromagnetic analogy, K nab is to F ab as P ab is to A a .

Expanding further the components of K a bcd , K ab , and |G| from the previous section, the action can be written as

with the following definitions in a slightly different convention from [15] 

Notice the term P [ab] = P ab − P ba in the action above. Generally, P ab and R ab are not independent as their antisymmetric parts are proportional to the curl of α b

Up until this point, the connection Γ a bc has been incompatible with the metric g ab . From here on in this paper, we set the connection Γ a bc to be a Levi-Civita connection,

compatible with the metric g ab . This forces α a = − 1 d+1 Γ e ea = − 1 d+1 ∂ a ln |g| = g a and P [ab] = R [ab] = 0, reducing the action to

where now we have from here on in this paper

We have performed the integrations over , absorbing the result into a redefinition of the constants

This integration along the projective direction allows for a natural scaling of both the gravitational coupling constant κ 0 and projective angular momentum parameter J 0 .

Notice if f and i are chosen to grow the angular momentum parameterJ 0 into a larger J 0 , the gravitational constantκ 0 necessarily shrinks to the smaller κ 0 . This could tie the weakness of the gravitational force to a large angular momentum of the Universe. The fact that the Gauss-Bonnet term is a topological invariant in four-dimensions removes any potential higher-order metric terms from the equations of motion. At this point it is also clear that the TW gravity action reduces to Einstein-Hilbert when P ab = 0.

Inflation via non-minimal coupling has been investigated for at least three decades. In a paper published in 1990 [9] , Fakir and Unruh sought to remedy the issue of generically large density perturbations inherent to chaotic inflation scenarios by removing the assumption of minimal coupling between the inflaton field and the Ricci scalar curvature. Indeed, inflation via this non-minimal coupling has been shown to result in acceptable values for the spectral index n s and tensor-to-scalar ratio r [7, 11, 12] .

Additionally, NMC inflation has been shown to provide a natural mechanism for the reheating phase occurring after inflation [10] . Initially, the form of the non-minimal coupling was commonly assumed to be φ 2 R, while more recent studies have investigated coupling of the more general form f (φ)R [11] . Inflationary scenarios based on the more general non-minimal coupling have since been shown to also result in viable experimental predictions for n s and r [11] . This section details the main result of this paper, which is how TW gravity naturally realizes inflation with a non-minimal coupling.

From here on out we write the characteristic projective length scale λ 0 in units of Planck length such that λ 0 = n λ √ 8πl p = n λ 8πG /c 3 , where n λ is a dimensionless scaling. We also write the projective angular momentum parameter in units of such

where n J is a dimensionless scaling. For the remainder of this paper, we take natural units = c = 1. The only remaining units will be written in terms of the reduced Planck mass

.

All physical constants will now be written as

Here we include arbitrary factors n λ , n κ , and n J noticing there is no naturalness argument to use to constrain n λ as it is the scale of the projective direction and we have no a priori notion of how large this scale should be. In the following, we seek to constrain n λ from experiment to give us a window into the size of the projective direction that gives rise to inflation. At the same time, a resulting constraint on n J gives us a window into the angular momentum scale involved in these projective directions as well. In a similar previous work [15] , we saw the angular momentum scale J 0 to be of the order of the observable Universe when the cosmological constant arising from the vacuum solution of TW gravity was used to constrain J 0 . It is possible that a constraint on n κ and the rescaling of the parametersκ 0 andJ 0 as shown in Eqs. (2.46) could be related to quantum gravitational effects. Further research into the quantization of this theory needs to be done to investigate this possibility.

Using the following tensor decomposition of P ab , we may cast the dynamics of TW gravity into its trace and traceless degrees of freedom

where the field dimensions are:

The constant w 0 is dimensionless while the factor of (λ 0 ) −1 = M p /n λ is included on the φ term to provide the correct units for a scalar field [φ] = L −1 = M and to cancel the λ 2 0 proportionality factor on the kinetic term in Eq. (2.44) as shown below. This leads to the following decomposition of the action,

The Lagrange multiplier 4λ is placed in by hand to enforce tracelessness of W ab at the equations of motion level but not at the Lagrangian level. This ensures that the equations of motion are the same whether performing the decomposition before or after they are derived from variation of the action. That is, varying the action Eq. (2.44) with respect to g ab , P ab and then performing the decomposition Eq. We notice a potential V (φ) that is quadratic in φ has developed from the decomposition. This is as expected as the action Eq. (2.44) was quadratic in P ab . The potential includes a linear term in φ which can be removed via field redefinition f (φ) =φ leading to a correction to the cosmological constant proportional to M 2 p /n J . In fact, the opposite was investigated in [15] where a field redefinition was used to generate an Einstein-Hilbert term with cosmological constant, setting all other dynamical fields to zero.

We notice that as the potential's dependence on f (φ) is purely second order the minimum of the potential occurs where f (φ min ) = 0

(3.7)

At this potential minimum, the Einstein-Hilbert term vanishes, as it is proportional to f (φ min ) which vanishes, and the theory is that of the the rank-two, symmetric traceless field W ab coupled to the metric. An interesting future work would be to investigate the theory at and around this potential minimum and possible connections to the cosmological constant. Our focus in this paper will instead be on making connections to slow roll inflationary cosmology, through non-minimally coupled inflation, to which we turn in the next sections.

We now consider Eq. (3.4), with the assumptions w 0 = 0 and Λ 0 = 0. in d = 4

dimensions. Taking w 0 = 0, we see P ab has only a trace degree of freedom held by the scalar field φ. The action in Eq. (3.4) now reduces to

where we have neglected S GB as in d = 4 dimensions the variation δS GB is a boundary term and thus adds nothing to the equations of motion [30] . This action is seen to be a particular case of a non-minimally coupled inflaton action in Jordan-frame with a potential of the form V (φ) = Aφ 2 + Bφ. We note that TW gravity has reduced in Eq. (3.8) to a composite of the linear and quadratic potential cases studied by [11] , [13] , after setting their dimensionless coupling parameter ξ = 8 n and our n κ = 1. Thus, our inflaton coupling parameter is formed from a combination of free parameters, as seen in Eq. (3.8), rather than being an additional free parameter of the model. Furthermore, TW gravity as reduced to Eq. (3.8) falls into the categorization of models in [13] as an F-dominant case.

To manipulate this action into a form where we can easily apply the usual slow-roll analysis, we transform from Jordan to Einstein frame via the conformal transformation

where g ab andg ab are Jordan and Einstein frame metrics, respectively. We demonstrate the details in switching from Jordan to Einstein frame in Appendix C, which follows closely [31, 32] . For d = 4 we have

Under a conformal transformation with this ω, the two parts of the Lagrangian in

Substituting into the action Eq. (3.8) and combining like terms we find

with

Integrating by parts the B(φ) term and neglecting boundary terms yields

Defining the canonical field as 5

leads to the canonical scalar field action

(3.20)

The differential equation Eq. (3.19) defining the canonical field h can be solved piecewise exactly by separation of variables

(3.21)

The potential minimumṼ = −∞ corresponds to the point h = −∞. In the large φ limit, this relationship becomes 

Carrying out the derivatives and simplifying yields the following forms of the slow-roll parameters = 32

(3. 26) which, in the large field limit become

We see that the necessary conditions 1 and |η| 1, for the slow-roll approximation to hold, are satisfied in the large field limit of Eq. (3.27). Once we have solved for these slow-roll parameters, we will use them to calculate the scalar-mode spectral index n s , scalar-mode amplitude A s , and tensor-to-scalar amplitude ratio r via the equations

where φ * is the field-value corresponding to the number of e-foldings during inflation.

In order to evaluate these expressions we need to obtain φ * as a function of e-foldings N . We calculate this using the standard slow-roll expression

as we do not have closed form expressions forṼ in terms of h, we must change variables back to φ using dh = C(φ)dφ from Eq. (3.19 )

Integrating this equation yields

, f end = f (φ end ) . 

TW gravity has three free parameters (n J , n λ , n κ ) that we will fit using data from the Planck, BICEP2, and Keck Array [27] . Our solution proceeds as follows 1. Solve Eq. 4. Fit these solutions to the Planck, BICEP2, and Keck Array data [27] . For the range of solutions for f * that fits the r and n s data, solve Eq. (3.28) for the corresponding range for n J that also fits the A s data.

5. This will give a range of values for the three parameters (n J , n λ , n κ ) of the TW gravity model of slow roll inflation that fits the current inflatiory data [27] . V >>ḣ 2 /2 is valid. An asymptotenn λ → ∞ occurs at this lower limit. The upper limit f end < 1 + 2/ √ 3 is enforced by the fact thatnn λ must be positive (the free TW gravity parameters all must be positive).

Constraining f end withnn λ Next, we insert this range of solutions fornn λ into Eq. (3.31) and solve for f end as a function of the initial condition f * = f (φ * ). Solutions are plotted in Figure 2 for the number of efolds N = 50, 60, and 70. These solutions all satisfy the necessary condition f * > f end which ensures that inflation occurs prior to the condition φ * ≈ 1 is realized, which shuts off inflation. These solutions also satisfy f end > 1 necessary for the validity of the slow roll approximation as previously described.

Non-minimal coupling f (φ) f * > f end > 1 that is a necessary condition for inflation in the slow roll approximation.

The range of f * corresponding to positivenn λ is to three significant figures as follows 1 .00 < f * 11.8 N = 50 (3.34a)

1.00 < f * 12.9 N = 60 (3.34b)

1.00 < f * 13.9 N = 70 (3.34c)

Each upper limit on f * results in the upper limit for f end = 1 + 2/ √ 3 for whicĥ nn λ → 0 as previously explained.

Now that we have valid ranges for f * for various efolds N , using Eq. (3.28) we can calculate a range of n s and r predicted by our TW gravitational model. Using the most recent combination of Planck, BICEP2, and Keck Array data presented in [27] to constrain n s and r, we are able to place constraints on the product of dimensionless constantsnn λ . In Figure 3 we display the 68% and 95% confidence intervals from [27] with predictions from our TW gravitational model overlayed. Figure 3 is in agreement with the predictions of [13] for the case of chaotic inflation with power-law F in the metric formulation for (n, p) = (1, 1).

Observational Comparison (Plank+BICEP2/Keck Array) efolds. The corresponding confidence interval is also shown, using the boundary value of f * from Table 1 . As f * decreases the range of n J decreases but does not vanish.

For N = 60 efolds, the range of values that fits completely within the 95% confidence interval (including that part that is also within the 68% confidence interval) is to three 

In [15] it was shown how TW-gravity (in the n κ → ∞ limit of this paper) could give rise to corrections to the cosmological constant on the order of today's measured value, Λ ∼ 10 −120 M 2 p , if n J ∼ 10 120 , corresponding to approximately the angular momentum scale for the observable universe. In the general n κ case in Eq. (3.36)

The potentialṼ is plotted in Figure 5 for the case that fits the 68% confidence level boundary for N = 60 efolds as shown in Figure 3 .

Potential that Fits the 68% Confidence Level Boundary Data In Figure 5 , we haveṼ ∼ 10 −11 M 4 p at the end of inflation, thus predicting that Λ ∼ 10 −11 M 2 p according to Eq. 3.36. This would be the case if the inflaton field is completely frozen at the end of inflation. The steepness of the slope at the end of inflation in Figure 5 suggests the possibility that h is not completely frozen at the end of inflation and thatṼ might yet decrease substantially for a small change in h. Thus perhaps the prediction of Λ ∼ 10 −11 M 2 p at the end of inflation is not the end of the story for TW-gravity born inflatons, but Λ is actually much smaller.

Assuming inflation occurs at time scale of 10 −36 s, we can approximate the time derivative squared of the canonical scalar field h asḣ 2 ≈ ((h end − h * )/10 −36 s) 2 , with h end and h * calculated from Eq. (3.21) evaluated at f end and f * , respectively. In this approximation, the following 3D plots are ofḣ 2 /2 versus n λ and n κ for the upper and lower bounds of n J associated with the A s constraint Eq. (3.35) at N = 60 efolds.

The potentialṼ (φ end ) = constant for given n J , as calculated from Eq. (3.16) , is used to demonstrate the validity of the slow roll approximation as it is the lowest value of the potential over the slow roll inflationary epoch. These plots clearly show that the slow roll approximation is valid for N = 60 as the solution n λ = n λ (n κ ) for fixed n J , marked by the red line, is clearly in the range whereṼ >>ḣ 2 /2. Figure 6 uses the maximum values f * = 12.9 and f end = 1 + 2/ √ 3 for N = 60 efolds as described in Changing the scale of Figure 8a as in Fig. 9 allows for some analysis of LHC measurable effects. Figure 9 demonstrates that n κ ∼ 10 20 for an n λ ∼ 10 16 that is associated with projective length scale effects that hypothetically could be probed at the LHC. As seen in Eq. n λ (n κ ) for fixed n J . This solution has precisely n κ = 1.10 × 10 20 for n λ = 10 16 . This is the smallest value of n κ that corresponds to an LHC-scale n λ = 10 16 , for all values of f * within the 95% confidence level for N = 60. This plot is the same as Fig. 8a with a modified n κ and n λ range that is within scales that can be probed at the LHC.

We have shown that TW gravity provides a raison d'etre for inflation, the underlying foundational principle being projective symmetry. This symmetry itself arises from the dynamical extension of a coadjoint element of the Virasoro algebra to higher dimensions. By decomposing the tensor field arising from TW gravity into trace and traceless degrees of freedom and setting the traceless components to zero, we have found that the resulting action reproduces that of (generalized) non-minimally coupled inflation with a novel inflaton potential. After performing a conformal transformation to Einstein frame, we recover the canonical scalar field inflaton action, where the free parameters of TW gravity n J , n κ , and n λ (TW-parameters) become embedded in the canonical field h and its potentialṼ . Calculating the slow roll parameters and applying conditions about the end of inflation allowed us to constrain relationships to the TW-parameters. Finally, using recent observational data for the scalar-mode spectral index, scalar-mode amplitude, and tensor-to-scalar amplitude ratio, we determined the ranges for the TW-parameters that fit recent data for N = 50, 60, and 70 efoldings. For N = 70, however, the model does not fit within the most constraining confidence interval of the Planck+BICEP2/Keck Array data. Generally, a lower number of efoldings paired with a larger value for the inflaton field at the start of inflation better matches the data. Furthermore, we confirmed that the range of TW-parameters that fits the data is consistent with the slow-roll approximation of a potential dominated expansion. The range of parameters that fits within the 95% confidence level is 1.15 × 10 8 n J 4.37 × 10 10 and 0 < n 2 λ /(n J n κ ) 1030 for N = 60. The TW-parameter n J corresponds to an angular momentum scale for TW gravity, associated with the TW coupling constant J 0 = n J . We find n J ∼ 10 10 fits the slow roll cosmological data and leads to a cosmological constant contribution Λ 10 −11 M 2 p from the scalar field settling down at the end of inflation. This correction is still much larger than the measured result Λ ∼ 10 −120 M 2 p , however, the shape of the TW-potential V at the end of inflation indicates the possibility that Λ could continue to decrease lower than 10 −11 M 2 p after inflation was over. We would like to return to this in future works.

The inflationary solutions of TW gravity presented include a regime where a large Jordan frame gravitational coupling constant might lead to LHC sensitive TW effects.

Such effects may arise as dark matter portals through the spin connection of the higherdimensional projective space [14] [15] [16] . Specifically, the spin connection gives rise to an axial coupling between the trace of P ab , that is the scalar field inflaton described here, and fermions [15, 16] . The inflationary solutions of TW gravity indicate that if such a dark matter portal exists at LHC scales, it could shed light on inflationary cosmology as well. Furthermore, TW gravity will contribute to the reheating process after inflation, through direct decay of TW inflatons and possibly through facilitating decays as portals. Investigations in these directions will be pursued in the future.

In this paper, we assumed the connection was Levi-Civita. An interesting future work would be to relax this constraint and consider the Palatini approach of [16] that leads to a model with more tensorial degrees of freedom along with more equations of motion associated with the connection. We also plan to investigate how adding the traceless component of the tensor field P ab in Eq. (3.3) will affect the analysis. Due to the addition of the Lagrangian Eq. (3.5), we will need to solve the field equations directly and make an Ansatz for the form of the components of W ab . This could possibly provide a theoretical origin to the anisotropic seeds needed for galaxy formation in the early universe. 

The units of the various constants used throughout this paper for d = 4 are

We may at times set c = 1 but expose factors of c when calculating numerical values.

Latin indices take values a, b, · · · = 0, 1, 2, . . . , d − 1 and Greek indices take values µ, ν, · · · = 0, 1, 2, . . . , d, with the exception of the Greek letter λ, which refers to the projective coordinate x d = λ = λ 0 . The covariant derivative acts on contravariant and covariant vectors as

A rank m-contravariant, n-covariant tensor, which we refer to as an (m, n) tensor, will have m-terms involving the connection Γ a bc as for contravariant vectors and n-terms involving the connection as for covariant vectors. The d + 1-dimensional covariant derivative is defined analogously withΓ α µν . At places in this paper where we have assumed compatibility between Γ a bc and the metric g ab we have 

Considering an infinitesimal coordinate transformation

the diffeomorphism field's transformation law in d = 1-dimensions becomes

The last step is to write D (x ) in terms of x via a Taylor series

Plugging this back into the transformation Eq. (B.5) and rearranging we have

, we can simply drop the prime in the third term on the right hand side, leaving us with

To first order in ξ, this is the same as Eq. (2.11).

Here we follow closely [31, 32] to demonstrate how the Riemann tensor, Ricci tensor, and Ricci scalar transform under a conformal transformation in d-dimensions:

This transforms the square root of the determinate as

The connectionΓ c ab for the metric g ab and Γ c ab for the metric g ab differ by a tensor C c ab that has the same form for both metrics

Here ∇ a is the covariant derivative associated with the connection Γ c ab and∇ a is the covariant derivative associated with the connectionΓ c ab . Indices for the covariant derivatives are raised and lowered with their associated metrics:

(C.5)

The relationship between the Riemann curvature tensors R c adb constructed from Γ c ab andR c adb constructed fromΓ c ab is

(C.6)

Contracting first and third indices gives the relationship between the Ricci tensors

Contracting with the metric and simplifying gives the relationship between the Ricci scalars

Here the Ricci scalars are defined as the Ricci tensor contracted with the associated metric and the Laplacian is defined as contracted with is associated metric:

R =g ab R ab ,R =g abR ab =g abRc acb (C.9)

ω =∇ a ∇ a ω ,˜ ω =∇ a∇ a ω . (C.10)

In proving the above, the following calculations are useful

For a Lagrangian in d-dimensions of the form n −1 κ |g|f (φ)R to transform to the Einstein frame

we must have f (φ)e (2−d)ω = n κ , the solution for which is

This results in the following for the Laplacian and square of the divergence of ω

where f = df /dφ and f = d 2 f /dφ 2 . Substituting these results into Eq. (C.14) while multiplying the entire equation by −M 2 p /2 gives in terms of φ now

Under this same conformal transformation, a scalar field Lagrangian transforms as follows

The Friedmann-Robertson-Walker (FRW) metric is given by:

The non-vanishing Christoffel symbols are

Γ˘i 1j = 1 r δ˘ij , Γ 1ȋj = − 1 r g 11 g˘ij = 1−kr 2 ra 2 g˘ij , (D.3)

where the spatial indices are i, j, k, . . . 1, 2, 3, the spherical indices areȋ,j, · · · = 2, 3, and the the Hubble parameter is H(t) =ȧ/a, with an overhead dot indicating a time derivative. The non-vanishing Ricci tensor components and Ricci scalar are

(D.6)

The action for a scalar field is given by 7

whereṼ is the potential for the scalar field, 8 noting that in flat space the kinetic term becomes ∇ m h∇ m h = g mn ∇ m h∇ n h =ḣ 2 − (∇h) 2 . A massive scalar field has V = 1 2 m 2 h 2 . We can find the equations of motion for the scalar field through δS = 0 or equivalently through the Euler-Lagrange Equations, The stress-energy tensor for a perfect fluid in general coordinates is given by where g mn u m u n = 1 is satisfied for our conventions.

If we treat the field to be homogeneous, i.e. h(t), then the Lagrangian reduces to The defining condition of inflation is that it was a period of accelerated expansion of the universe, corresponding toä > 0. From the acceleration equation Eq. (D.9a),ä > 0 implies that H 2 > −Ḣ and thus H < 1.

As the universe expands, a increases and therefore the k a 2 term becomes negligible compared to the energy density of the scalar field. Then we have thaṫ There must be approximately 60 e-foldings (though values of 50-70 can be found without the inflation literature) and the model must incorporate an end to inflation in accordance with observation. For further information see [34, 35] .

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Einstein's equations are 6 R ab − 1 2 g ab R − g ab Λ 0 = κ T ab (D. 7) where for an FRW metric, the stress-energy tensor plus cosmological constant takes the form of a perfect fluid in co-moving coordinatesEinstein's equations for an FRW metric, known as the Friedmann equations, arëThe first equation relates to the acceleration of the scale factor, the second relates the energy density to the critical density 3H 2 /κ. Simultaneously solving Eqs. (D.11) is also equivalent to solving the Friedmann equations Eqs. (D.9). 6 Here κ = 8πG/c 4 which is exactly equivalent to M −2 p in natural units used throughout the body of the paper. This is not to be confused with κ 0 which appears throughout the paper and is merely proportional to M −2 p in natural units as shown in Eq. 3.2.