Cosmology in a Globally U(1) Symmetric Scalar-Tensor Gravity

TL;DR

This paper proposes a scalar-tensor cosmological model where a complex scalar field drives cosmic evolution, avoiding traditional problems and generating primordial perturbations without tensor modes, challenging standard expansion-based cosmology.

Contribution

It introduces a novel scalar-tensor cosmology with a complex scalar field in Minkowski spacetime, eliminating the need for space expansion and addressing key cosmological issues.

Findings

Non-singular cosmological evolution with no horizon or flatness problems

Generation of a flat spectrum of scalar perturbations without tensor modes

Cosmology driven by mass growth rather than space expansion

Abstract

A cosmological model is formulated in the context of a scalar-tensor theory of gravity in which the entire cosmic background evolution is due to a complex scalar field evolving in Minkowski spacetime, such that its (dimensional) modulus is conformally coupled, and the (dimensionless) phase is only minimally coupled to gravitation. The former regulates the dynamics of masses; cosmological redshift reflects the growth of particle masses over cosmological time scales, not space expansion. An interplay between the energy density of radiation and that of the kinetic energy associated with the phase (which are of opposite relative signs) results in a non-singular cosmological model that encompasses the observed redshifting phase preceded by a turnaround that follows a blushifting phase. The model is essentially free of any horizon, flatness or anisotropy problems. Quantum excitations of the phase during the matter dominated blueshifting era generate a flat spectrum of adiabatic gaussian scalar perturbations on cosmological scales. No detectable primordial tensor modes are generated in this scenario, and cold dark matter must be fermionic. Other consequences are also discussed.

Equations

$$\mathcal{I}$$

$$F{G}_{\mu }^{\nu }$$

$$\frac{F^I}{2}R+\square {\varphi }^I$$

$$T_{M,\mu ;\nu }^{\nu }$$

$${\Theta }_{\mu }^{\nu }={\mathcal{G}}_{IJ}({\varphi }_{\mu }^I{\varphi }^{J,\nu }-\frac{1}{2}{\delta }_{\mu }^{\nu }{\varphi }^{I,\rho }{\varphi }_{\rho }^J)+F_{\mu }^{\nu }-{\delta }_{\mu }^{\nu }\square F.$$

$${\varphi }^I{\mathcal{L}}_{M,I}=T_M,$$

$${\rho }_M\propto {\rho }^{1-3w}.$$

$${\mathcal{I}}_{FRW}$$

$${\mathcal{Q}}^2+K$$

$$2{\mathcal{Q}}^{\prime }+{\mathcal{Q}}^2+K$$

$$\frac{x_e^2}{1-x_e}=\frac{45s}{4{\pi }^2}{\left(\frac{m_e}{2\pi T}\right)}^{\frac{3}{2}}\exp (-\Delta /T),$$

$${\theta }^{\prime \prime }+2\mathcal{Q}{\theta }^{\prime }=0,$$

$${\mathcal{Q}}^2=\lambda {\rho }^2+{\lambda }_p{n}_{NR}/\rho +{\rho }_{r,\ast }/{\rho }^2+{\rho }_{\theta ,\ast }/{\rho }^4,$$

$${\rho }^2={\rho }_{r,\ast }{\eta }^2-{\rho }_{\theta ,\ast }/{\rho }_{r,\ast }$$

$$\mathcal{Q}{\tilde{\phi }}^{\prime }+\left(\frac{k^2}{3}+{\mathcal{Q}}^2+{\theta }^{\prime 2}\right)\tilde{\phi }=-\frac{1}{2}({\mathcal{Q}}^2+{\theta }^{\prime 2}){\tilde{\delta }}_{{\rho }_M}-{\theta }^{\prime }\delta {\theta }^{\prime }$$

$${\tilde{\phi }}^{\prime }+\mathcal{Q}\tilde{\phi }=-\frac{3(1+w)}{2}({\mathcal{Q}}^2+{\theta }^{\prime 2})u+3{\theta }^{\prime }\delta \theta$$

$${\tilde{\phi }}^{\prime \prime }+2\mathcal{Q}{\tilde{\phi }}^{\prime }+\left[2{\mathcal{Q}}^{\prime }-4{\theta }^{\prime 2}-\frac{k^2}{3}\right]\tilde{\phi }=$$

$$\frac{(1+3w)}{2}({\mathcal{Q}}^2+{\theta }^{\prime 2}){\tilde{\delta }}_{{\rho }_M}+4{\theta }^{\prime }\delta {\theta }^{\prime }$$

$$\delta {\theta }^{\prime \prime }+2\mathcal{Q}\delta {\theta }^{\prime }+k^2\delta \theta =-4{\theta }^{\prime }{\phi }^{\prime }$$

$${\tilde{\delta }}_{{\rho }_M}^{\prime }+(1+w)(3{\tilde{\phi }}^{\prime }+k^{2}u)=0$$

$$u^{\prime }+(1-3w)\mathcal{Q}u+\frac{w^{\prime }u}{1+w}=-\tilde{\phi }+\frac{w{\tilde{\delta }}_{{\rho }_M}}{1+w},$$

$$[a^4{\rho }_M(1+w)v^{(v)}{]}^{\prime }$$

$$h^{\prime \prime }+2\mathcal{H}{h}^{\prime }+(k^2+2K)h$$

$$[{\rho }_M(1+w)v^{(v)}{]}^{\prime }$$

$$h^{\prime \prime }+2\mathcal{Q}{h}^{\prime }+(k^2+2K)h$$

$${\phi }^{\prime \prime }+3(1+w)\mathcal{Q}{\phi }^{\prime }+w{k}^2\phi =3(1-w){\theta }^{\prime }\delta {\theta }^{\prime }.$$

$$f_0$$

$$f_1$$

$$f_2$$

$$f_3$$

$$\delta \theta$$

$${\phi }^{\prime \prime }+\frac{6{\phi }^{\prime }}{\eta }=\frac{C}{{\eta }^8},$$

$${\Delta }^2=\frac{k^3|{\phi }^2|}{2{\pi }^2}=\frac{9\lambda |{\rho }_{\theta }|}{16{\pi }^2{\rho }_{DE}}$$

$$\phi =\frac{\beta }{{\eta }^5}+\frac{c_1^{\prime }}{{\eta }^3}+c_2^{\prime }$$

Full text

1 Introduction

The standard cosmological model with an early inflationary scenario has clearly been a very successful paradigm that provides a compelling interpretation of essentially all current cosmic microwave background (CMB), large scale structure (LSS) measurements, and the agreement between Big Bang nucleosynthesis (BBN) predictions and light element abundances. It is remarkable that the cosmological model provides a very good fit to extensive observational data, that sample phenomena over a vast dynamical range, using less than a dozen free parameters.

However, the essence of dark energy (DE) and cold dark matter (CDM) – two key ingredients in the model that determine the background evolution, LSS formation history, and gravitational potential on galactic scales – remain elusive. Additionally, what is considered by many as the most pristine fingerprint of cosmic inflation [1-3] – a major underpinning of the standard cosmological model – B-mode polarization of the CMB [4-6] induced by primordial gravitational waves (PGW), has not been detected. The latter is admittedly a very challenging measurement in the presence of e.g., polarized Galactic dust, nonlinear density perturbations, and instrumental systematics. In light of these hurdles and the theoretically viable broad window for the energy scale of inflation, it is not unlikely that this signal will never be measured at sufficient statistical significance. Yet, its non-detection will not rule out the inflationary paradigm, but would rather set an (arguably very weak) upper bound on the energy scale of inflation.

According to the standard cosmological model global evolution is driven by space expansion, namely the time-dependent Hubble scale provides the ‘clock’ for the evolving properties of radiation and matter, resulting in a sequence of cosmological epochs. This clock is only meaningful if other time scales, e.g., the Planck time, or characteristic Compton times, evolve differently, particularly if they are non-varying; thus, space expansion is a relative notion.

The main objective of the present work is to demonstrate the viability of an alternative, non-singular ‘bouncing’ cosmological model within a physical framework based on a $U(1)$ symmetric scalar-tensor theory of gravity. Spacetime, effectively Minkowski, is static in this model. Other non-scalar fields, e.g. Dirac, Weyl, and electromagnetic fields are similarly non-evolving on this static background space. Cosmic evolution is then achieved by evolving scalar fields, i.e. evolution of fundamental particle masses and the Planck mass, while their dimensionless ratios remain fixed to their standard values. The time-dependent scalar field which regulates the (dynamical) masses of particles starts infinitely large, monotonically decreases until it ‘bounces’, then grows again unboundedly. In other words, the Compton wavelength associated with fundamental particles starts infinitely small, increases until it peaks at the ‘bounce’, then decreases again. Described in terms of these length ‘units’ the universe is said to undergo a ‘contraction’ epoch, followed by a ‘bounce’ and ‘expansion’.

Since the universe does not actually contract and expand in the proposed model, it is perhaps more appropriate to refer to these epochs in a frame-independent fashion that is faithful to what is actually observed as blue-and red-shifting eras, respectively. For similar reasons it is more appropriate to refer to the ‘bounce’ by ‘turnaround’ or ‘turnaround point’ between the blue-and red-shifting eras. For the rest of this work we use these terms in reference to the proposed model instead of the more traditional ones.

We explore a wide range of the possible ramifications of the model (albeit not exhaustively) which a priori demotes the gravitational constant, particle masses, and all other dimensional constants, from their fundamental-physical-constants status, and replaces them with the conformally-coupled modulus of a single complex field.

A dynamical vacuum expectation value (VEV) for the Higgs field is naturally accommodated by a Weyl-symmetric theory that essentially incorporates all the fundamental interactions into this framework [7]. In particular, the standard model (SM) of particle physics, where the Higgs VEV is a fixed constant, is just one convenient gauge choice. Applying a Weyl transformation to such field configurations that appropriately endows the Higgs VEV with exactly the same dynamics of the evolving scalar field in the background cosmological model guarantees that the dynamical particle masses are continuous everywhere, exactly as in the SM and general relativity (GR). Thus, a static field configuration, e.g. a planet or a galaxy, transforms into a stationary one, while observables in a planet or a star are the same as in standard physics.

While the intriguing possibility that the cosmological redshift could be explained by means of time-dependent fundamental ‘constants’ is nearly as old as (what has become) the standard expanding space interpretation [8], it has been waived off by big bang proponents as soon as it was proposed [9]. This basic idea has re-emerged later within the framework of e.g., scalar-tensor theories of gravity [10-14] and in the context of Weyl-geometry, e.g. [15].

Temporal variation of the gravitational constant, $G$, is a key feature in scalar-tensor theories, of which Brans-Dicke (BD) theory is archetypical. Standard interpretation of observational constraints, e.g. [16], usually renders this theory equivalent to GR due mainly to the convention that all other (dimensional) fundamental quantities are constant. Our approach is fundamentally different as it is based on the premise that all fundamental length scales have exactly the same dynamics which is regulated by the modulus of the complex scalar field. In other words, the often cited $\omega_{BD}>40000$ [16] that essentially fixes the BD scalar field (i.e. Newton’s $G$) to a constant thereby reducing the theory to GR, does not apply to the Bergmann-Wagoner type of theory [17, 18] studied here. In particular, our approach guarantees that dimensionless observables (in a sense that will be more clearly defined below) are by construction unchanged compared to their corresponding values in the standard cosmological model in the domain of the latter validity, i.e. down to the ‘bounce’.

Throughout, we adopt a mostly-positive signature for the spacetime metric $(-1,1,1,1)$. Our units convention is $\hbar=c=k_{B}=1$. We outline our theoretical approach in section 2, and the cosmological model is presented in section 3. In Section 4 we discuss and summarize our main results.

2 Theoretical Framework

Consider the following scalar-tensor theory of the Bergmann-Wagoner type [17, 18]

[TABLE]

where $\mathcal{G}_{IJ}$ is a two-dimensional metric in field space (that possibly depends on the fields), and capital Latin letters assume the values 1 & 2. A useful basis is $\phi^{(1)}=\rho e^{i\theta}$ & $\phi^{(2)}=(\phi^{(1)})^{*}$. The only non-vanishing components of $\mathcal{G}_{IJ}$ in this basis are the constant off-diagonal $\mathcal{G}_{12}=\mathcal{G}_{21}=-1$ components. In this complex fields frame $F\equiv\frac{|\phi|^{2}}{3}$ & $V=\lambda|\phi|^{4}$ where here $|\phi|\equiv|\phi^{(1)}|=|\phi^{(2)}|$. Alternatively, in a polar field frame with real $\phi^{(1)}=\rho$ & $\phi^{(2)}=\theta$ (which correspond to the modulus and phase, respectively, of the field $\phi^{(1)}$ of the complex field frame), the fields space metric is $\mathcal{G}_{IJ}=-2\cdot diag(1,\rho^{2})$, and $F\equiv\frac{\rho^{2}}{3}$ & $V=\lambda\rho^{4}$. In sections 2 & 3 we employ mainly the complex and real fields frames, respectively. Eq. (2.1) differs from the BD action in that $\rho$ appears not only in the curvature and kinetic terms but also in $\mathcal{L}_{M}$ and $V$. Eq. (2.1) is Weyl-symmetric in a sense that will be made clear below. Here and throughout, Greek indices run over spacetime coordinates, and $f_{\mu}\equiv\frac{\partial f}{\partial x^{\mu}}$ for any function $f$.

Employing $\mathcal{G}_{IJ}=diag(-2,-2\rho^{2})$ [or even $\mathcal{G}_{IJ}=diag(-2,2\rho^{2})$ for that matter] guarantees that gravitation is attractive rather than repulsive provided that the energy density $\rho_{M}$ satisfies $\rho_{M}=-\mathcal{L}_{M}$ in the non-relativistic (NR) limit. Since $\mathcal{L}_{M}$, $V$ & $F$ depend only on $\rho$ then $\theta$ is a free field that only minimally couples to gravity in this theory with a kinetic term of the canonical sign. The reason for it being only minimally coupled to gravity is that it is dimensionless and therefore blind to Weyl transformations. The fact that $\theta$ is massless simply reflects the assumed $U(1)$ symmetry. In case that this symmetry is only approximate then $\theta$ is in general massive. For the rest of this work we assume that $U(1)$ is an exact global symmetry of Eq. (2.1).

A necessary but insufficient requirement from a Weyl-symmetric action is that it contains no dimensional constants. The specific pre-factor $\frac{F}{2}=\frac{\rho^{2}}{6}$ in the curvature term combined with the canonical kinetic term of the scalar field guarantee that the action described by Eq. (2.1) is invariant under the Weyl transformation (or rather local field redefinition) $g_{\mu\nu}\rightarrow\Omega^{2}g_{\mu\nu}$, $\rho\rightarrow\rho/\Omega$ (or equivalently $|\phi|\rightarrow|\phi|/\Omega$) and $\mathcal{L}_{M}\rightarrow\mathcal{L}_{M}/\Omega^{4}$ where $\Omega(x)$ is an arbitrary function, e.g. [19]. Therefore, the proposed cosmological model is essentially described by a Weyl-symmetric theory – or as put by a few what only looks as such, e.g. [20-23] – through the entire cosmic history. Clearly, the single degree of freedom $\Omega$ is insufficient to simultaneously gauge-fix both $\rho$ & $\theta$. In particular, $\theta$ and its perturbations cannot be removed by Weyl transformations or by local gauge transformations.

Naively comparing the curvature term in Eq. (2.1) to the corresponding term in the Einstein-Hilbert (EH) action, while fixing $\phi$, implies the latter must be of order the Planck mass. However, $\mathcal{L}_{M}=\mathcal{L}_{M}(\rho)$ and therefore all particle masses are proportional to $\rho$ and although it is dynamical, dimensionless ratios of particle masses and of particle masses to the Planck mass are fixed to their SM values. This is a special property of the action described by Eq. (2.1) that significantly distinguishes it from other scalar-tensor theories in general, and BD in particular.

Variation of Eq. (2.1) with respect to $g_{\mu\nu}$ and $\phi^{I}$, results in the generalized Einstein equations, and scalar field equation, respectively, e.g. [24, 25]

[TABLE]

where $f^{I}\equiv\mathcal{G}^{IJ}\frac{\partial f}{\partial\phi^{J}}$ for any functional $f$. Here, $\Gamma^{I}_{JK}\equiv\frac{1}{2}\mathcal{G}^{IL}(\mathcal{G}_{LJ,K}+\mathcal{G}_{LK,J}-\mathcal{G}_{JK,L})$ is the ‘connection’ computed from the two-dimensional field space metric $\mathcal{G}_{IJ}$. Energy-momentum is not conserved,

[TABLE]

where semicolons stand for covariant derivatives. The effective energy-momentum tensor $\Theta_{\mu}^{\nu}$ associated with the scalar fields is

[TABLE]

Here and throughout, $f_{\mu}^{\nu}\equiv(f_{,\mu})^{;\nu}$, the covariant Laplacian is $\Box f$, and $(T_{M})_{\mu\nu}\equiv\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g}\mathcal{L}_{M})}{\delta g^{\mu\nu}}$ is the energy-momentum tensor. In the perfect fluid approximation the latter reads $(T_{M})_{\mu}^{\nu}\equiv\rho_{M}\cdot diag(-1,w,w,w)$ where $w$ is the equation of state (EOS) describing the fluid. From the combination of Eq. (2.3) and the trace of Eq. (2.2) we obtain the constraint

[TABLE]

in both the polar and complex field frames defined below Eq. (2.1). This constraint is indeed consistent with a traceless energy-momentum tensor but only in case that the matter Lagrangian density is independent of the scalar field, an often-made assumption (for example in GR or BD theory where all particle masses are fixed) that we relax in the present work. Therefore, traceless energy-momentum is a pre-requisite of Weyl-symmetric theories only when it is assumed that $\mathcal{L}_{M}$ does not depend on the scalar field.

It will be argued in section 3.1 that gravitation in the framework described by Eq. (2.1) is sourced only by the potential term of the $\phi$-dependent matter Lagrangian, while the kinetic term is in certain gauges the ‘curvature’ term itself. Since $\mathcal{L}_{M}=-\rho_{M}$ it immediately follows from Eq. (2.6) that

[TABLE]

Here, $\lambda\rho^{4}$ has been absorbed in $\mathcal{L}_{M}$ with an effective EOS parameter $w=-1$. As expected, $\rho_{M}$ is a quartic potential in the case $w=-1$, is independent of $\rho$, i.e. of masses, in the case $w=1/3$, and linear in masses in case of NR fermions, i.e. the $w=0$. For exactly the same reason CDM in the present model must be fermionic, in contrast with the standard cosmological model in which CDM could in principle be made up by either fermions or bosons. This, of course, is a direct result of Weyl invariance of the theory that does not allow any dimensional constants in Eq. (2.1). Therefore, CDM has to contribute a term $\propto\phi\bar{\psi}\psi$ to $\mathcal{L}_{M}$ in Eq. (2.1) where $\psi$ is some fermionic field. In the case of ‘stiff’ matter, $w_{stiff}=1$, $\rho_{M}\propto\rho^{-2}$. We assume that no contribution to the perfect fluid, described by $\mathcal{L}_{M}$, is characterized by a sound speed $c_{s}>1/\sqrt{3}$, i.e. $w>1/3$.

As was noted already, a solution $g_{\mu\nu}$ & $\phi=(\rho,\theta)$ of the field equations (2.2) & (2.3) is symmetric under Weyl (local) rescaling $g_{\mu\nu}\rightarrow\Omega^{2}g_{\mu\nu}$ & $\rho\rightarrow\rho/\Omega$. Also as in GR, the metric ‘seen’ by a massless test particle is $g_{\mu\nu}$ and the metric governing the kinematics of a massive particle with mass $m$ is $m^{2}g_{\mu\nu}$. But in contrast with GR, in which masses are assumed to be constant, in the present model $m\propto\rho$, which implies that the effective metric ‘seen’ by a massive test particle is $\rho^{2}g_{\mu\nu}$. The latter is invariant under Weyl transformations, but $g_{\mu\nu}$ is not. However, null geodesics are blind to Weyl transformations, and the metric seen by massless particles is effectively $g_{\mu\nu}$.

As mentioned above, it has been argued that the Weyl-symmetric action, Eq. (2.1), is obtained in the case of a $\theta=0$ from the EH action by merely redefining the metric and scalar fields, that there are no conserved currents associated with the symmetry of Eq. (2.1), and that consequently this is a ‘sham’ or ‘fake’ Weyl-symmetry, e.g. [20-23]. Indeed, in the next section we illustrate this by going from the standard Friedmann-Robertson-Walker (FRW) action to Eq. (2.1) in the case $\theta=0$ by redefining fields. However, by adding another degree of freedom that is effectively only minimally-coupled to gravity (more specifically a free scalar field) to form a complex scalar field, the fields appearing in Eq. (2.1) cannot be generally redefined to recover the EH action; the two actions are clearly inequivalent in this case. Here we only point out that integrating both Eqs. (2.2) & (2.3) results in additional integration constants absent from GR. This will be addressed more concretely elsewhere [26].

3 Cosmological Model

In this section the background evolution, the evolution of linear perturbations, and a singularity-free early universe scenario are described. The modulus of the complex scalar field, $\rho$, has essentially the same dynamics that the scale factor $a(\eta)$ has in the standard cosmological model (except near the turnaround point, where the dynamics of $\theta$ significantly alters that of $\rho$). The field $\theta$ and its perturbations are responsible for the turnaround point and the flat power spectrum of density perturbations, respectively. Under certain plausible conditions, the entire observable cosmic evolution, from BBN onward, is identical to that of the standard cosmological model, but very early universe processes and scenarios can be much different, e.g. there is no initial curvature singularity in the model, and an early inflationary period of evolution.

3.1 Recasting FRW as a Scalar-Tensor Model and its Implications

To motivate the construction in the following sections of a cosmological model based on Eq. (2.1) we work out the equivalence between the FRW action and Eq. (2.1) in the homogeneous and isotropic case in this section (in the case of identically vanishing $\theta$).

The issue of ‘wrong’ relative sign of kinetic and potential energies as in Eq. (2.1) is characteristic of either general GR or other theories of gravitation ‘dressed’ with conformal symmetry, e.g. [21, 23]. This is usually interpreted as an indication for instability of the theory, e.g. [28-30], and was at the heart of an intense debate regarding the physical equivalence of theories in the Einstein frame and Jordan frame where the latter was considered by many as ‘unphysical’, in spite of the fact of being classically equivalent to the former, e.g. [31, 32], while others maintained that at least classically the theory with negative kinetic energy of the scalar field is stable. Indeed, the latter is the case with the classical FRW action, as we see below. Some even speculated that in a semi-classical context, when $\phi$ is quantized and the spacetime metric is treated as a classical field, stability may still be maintained, e.g. [33]. But, common lore holds it that if attempted to be quantized this ‘ghost’ field causes a catastrophic production of particles due to the unlimited phase space available for such processes to take place. We thus conjecture that $\rho$ of Eq. (2.1) is unquantized, i.e. that it is always found in its particle vacuum state (its energy states are unquantized) – a conjecture that could be better motivated by the following example. Again, we assume that $\rho$ of Eq. (2.1) is always classical and not only in the cosmological context [7].

Consider the EH action $\mathcal{I}_{EH}=(2\kappa)^{-1}\int(\tilde{R}-2\Lambda+\tilde{\mathcal{L}}_{M})\sqrt{-\tilde{g}}d^{4}x$ with a cosmological constant $\Lambda$ and a matter Lagrangian density $\tilde{\mathcal{L}}_{M}$. Here, $\kappa\equiv 8\pi G$ and $\tilde{R}$ is derived from the metric $\tilde{g}_{\mu\nu}$ in the usual way. The FRW action

[TABLE]

is obtained for the metric $\tilde{g}_{\mu\nu}=a^{2}\cdot diag(-1,\frac{1}{1-Kr^{2}},r^{2},r^{2}\sin^{2}\theta)$, and after the term proportional to the corresponding curvature scalar $\tilde{R}=6(a^{\prime\prime}/a+K)/a^{2}$ is integrated by parts. Here, $K$ is the spatial curvature, and a prime denotes derivatives with respect to conformal time, $d^{4}x=d^{3}xd\eta$.

One can readily verify that the Euler-Lagrange equation for the scale factor $a(\eta)$ that extremizes the action $\mathcal{I}_{FRW}$ is indeed the Friedmann equation $\mathcal{H}^{2}+K=a^{2}\tilde{\rho}_{M}+\frac{\Lambda a^{2}}{3}+const./a^{2}$, where $\mathcal{H}\equiv a^{\prime}/a$ is the conformal Hubble function, $\tilde{\rho}_{M}\equiv-\frac{1}{6}\tilde{\mathcal{L}}_{M}$, and it is assumed that $\mathcal{\tilde{L}}_{M}(a)$ is a power-law in $a$, as $\tilde{\rho}_{M}$ is a power-law in standard cosmology. Completing the derivation requires that the integration constant is related to the energy density of radiation $\tilde{\rho}_{r}=const./a^{4}$.

Defining $\rho^{2}\equiv 3a^{2}/\kappa$, the FRW action Eq. (3.1) is reformulated as a Weyl-symmetric scalar-tensor action with a non-positive kinetic term, defined on a static background

[TABLE]

Here, $g_{\mu\nu}$ is the static metric related to the FRW metric via a Weyl transformation, $g_{\mu\nu}\equiv\tilde{g}_{\mu\nu}/a^{2}=diag(-1,\frac{1}{1-Kr^{2}},r^{2},r^{2}\sin^{2}\theta)$, the 4D infinitesimal volume element is, e.g. $\sqrt{-g}d^{4}x=\sinh^{2}\chi\sin\theta d\eta d\chi d\theta d\varphi$, where $\chi$ is a ‘radial’ coordinate in the polar coordinate system (assuming $K<0$ for concreteness), $V(\rho)\equiv\lambda\rho^{4}$ is an effective potential, $\mathcal{L}_{M}=a^{4}\tilde{\mathcal{L}}_{M}$, $R=6K$ and $\lambda\equiv\kappa\Lambda/9$. The standard FRW spacetime is indeed described by a single degree of freedom, the scalar field $a(\eta)$, which is here replaced by $\rho(\eta)$, and therefore ignores the crucial role played by $\theta(\eta)$ as we will see below.

This example illustrates that a well-defined solution of the Einstein field equations is derived from an action which is essentially equivalent to Eq. (2.1), i.e. with a ‘wrong’ relative sign of the kinetic and potential terms. This wrong sign appearing in Eq. (3.1) is never considered a problem ($a$ is part of the metric field) and similarly the sign of $\rho^{\prime 2}$ should not alarm us, provided that $\rho$ is a genuinely classical field; same way that $a$ is a classical function in the standard formulation of the FRW solution so should be $\rho$. We emphasize that unlike in the standard cosmological model where $\tilde{g}_{\mu\nu}$ and $\tilde{R}$ are evolving, the latter being divergent at the initial singularity, the curvature scalar $R=6K$ is static. A Weyl transformation applied to Eq. (2.1) affects both the metric and scalar field to the extent that the dynamics of the metric (scale factor in our case) could be fully replaced by that of the scalar field in the FRW spacetime which is only described in terms of a single degree of freedom. In the standard cosmological model the kinetic and potential terms associated with the inflaton field appear with the canonical relative sign and thus could be quantized. In the present framework a different procedure is followed; whereas $\rho$, the analog of the scale factor $a$, is unquantized, the phase $\theta$ interacts only with the classical metric and $\rho$ fields and is therefore amenable to quantization.

Unlike $\rho(\eta)$ which is characterized by a vanishing current [23], the minimally-coupled field $\theta$ does have a conserved ‘charge’ associated with it which plays a central role in the present cosmological model as is shown below. We express the view in this work that the scale factor $a$ in the GR-based formulation of the FRW cosmology is not more genuine or fundamental degree of freedom than the field $\rho$. Indeed, the scale factor $a(\eta)$ is not an observable and it is conventionally fixed to unity at present $a(\eta_{0})=1$, but its logarithmic derivative, the conformal Hubble function $\mathcal{H}=a^{\prime}/a$, is. In analogy, $\mathcal{Q}\equiv\rho^{\prime}/\rho$ is an observable in the present reformulation of the FRW model, and can be viewed as the rate of variation of particle masses and the Planck mass, i.e. $m^{\prime}/m=O(10^{-18})\ {\rm sec^{-1}}$ at present, on a static background space as is described by Eq. (3.2).

It is notable that the time coordinate in Eq. (3.2) is conformal time $\eta$ rather than cosmic time $t$. The two are related via $d\eta=dt/a(t)$ and as mentioned $a(t)$ is arbitrarily set to unity at present. In this particular (and standard) convention the cosmic and conformal ‘clocks’ (that correspond to massive and massless test particles respectively) ‘tick’ at the same rate at present. At any given time in the past ($a<1$) in the expanding space the conformal clock was ticking relatively slower, thereby explaining cosmological redshift. In contrast, in the present reformulation of the FRW action there is only a single clock – the conformal clock – and cosmological redshift is now explained by the dynamics of growing masses in the redshifting era (i.e. growing Rydberg ‘constant’).

3.2 Evolution of the Cosmological Background

According to the alternative scenario proposed here a single complex scalar field does not only resolve the flatness and horizon puzzles, but also accounts for scale-free density perturbations. In this and the next section we derive the equations governing the background and linear perturbations, respectively.

The Einstein tensor components $G_{\mu}^{\nu}$, associated with the metric $g_{\mu\nu}=diag(-1,\frac{1}{1-Kr^{2}},r^{2},r^{2}\sin^{2}\theta)$, in conformal rather than cosmic time coordinates, are $G_{\eta}^{\eta}=-3K$ and $G_{i}^{j}=-K\delta_{i}^{j}$. In the following $f^{\prime}\equiv\frac{\partial f}{\partial\eta}$ denotes the derivative of a function $f$ with respect to conformal time $\eta$. Here, $i,j$ indices stand for the spatial coordinates. The energy-momentum tensor of a perfect fluid is $(T_{M})_{\mu}^{\nu}=\rho_{M}\cdot diag(-1,w,w,w)$, and employing Eq. (2.7) in Eqs. (2.1) & (2.3) we obtain that

[TABLE]

where $\mathcal{Q}\equiv\rho^{\prime}/\rho$ is the conformal Hubble-like function. In case that $\theta^{\prime}=0$ these reduce to the FRW equations but with $a$ & $\mathcal{H}$ replaced by $\rho$ & $\mathcal{Q}$. This implies that in the radiation-dominated (RD) or matter-dominated (MD) evolutionary eras, $\rho\propto\eta^{2/(1+3w)}$ is a monotonically increasing function of conformal time. This continues to be the case insofar $w>-1/3$. The evolution of the Rydberg ‘constant’ thus explains the observed cosmological redshift on this static spacetime. Since $\rho(\eta)$ is governed by a Friedmann-like equation it then follows that the effective background metric ‘seen’ by a massive test particle of mass $m$ is $m^{2}g_{\mu\nu}\propto\rho^{2}\eta_{\mu\nu}\propto a^{2}\eta_{\mu\nu}$ which is exactly the FRW metric. Massless test particles follow geodesics of a Minkowski background metric, as in the standard cosmological model (with $K=0$).

Before exploring the profound ramifications on the background evolution entailed by introducing the field $\theta$, we comment on the background evolution in the observable redshifting universe. We emphasize that exactly as in the standard cosmological model, $\rho_{M}$ is the sum of various contributions, i.e. $\rho_{M}=\sum_{i}\rho_{i}$ where each $\rho_{i}$ scales according to Eq. (2.7) with its respective $w_{i}$. In the same fashion that in the standard cosmological model the energy density associated with a species of fixed EOS $w_{i}$ is $\rho_{i}\propto a^{-3(1+w_{i})}$, i.e. the ratio of energy densities associated with two species which are characterized by $w_{i}$ & $w_{j}$ is $a^{-3(w_{i}-w_{j})}$, so it is the case here where $\rho_{i}\propto\rho^{1-3w_{i}}$ and the ratio becomes $\rho^{-3(w_{i}-w_{j})}$, consistent with the fact that $\rho(\eta)$ in the present model replaces $a(\eta)$ of the standard cosmological model. The sequence of RD, MD redshifting followed by the recent vacuum-like dominated redshifting is exactly as in the standard cosmological model. Clearly, since space is static (and assuming adiabatic evolution where particles are neither annihilated nor produced) particle number densities are fixed and the entire evolution is governed by the evolution of particle masses, which in turn is regulated by $\rho$. In other words, the monotonic growth of $\rho$, i.e. of particle masses, in a static background space (where the energy density of the CMB, i.e. its temperature, is fixed) implies that, e.g. an emitted atomic line at earlier epochs is determined by a relatively larger Rydberg ‘constant’ and is correspondingly characterized by a longer wavelength, i.e. the observed cosmological redshift simply reflects the monotonic growth of particle masses over cosmological time scales.

In addition, it is easy to see that the microphysics of the early universe is identical to that of the standard cosmological model. For example, the Saha equation

[TABLE]

determines the temperature $T$ at recombination, where $x_{e}$ is the ionization fraction, $s=(\frac{4\pi^{2}}{45})\frac{T^{3}}{n_{b}}$ and $n_{b}$ is the baryon number density, and $\Delta\equiv\alpha_{e}^{2}m_{e}/2$ with $\alpha_{e}$ & $m_{e}$ being the fine structure constant and the electron mass, respectively. In the standard cosmological model $m_{e}=constant$ & $T\propto a^{-1}$ and consequently the terms $\Delta/T$, $\left(\frac{m_{e}}{2\pi T}\right)^{\frac{3}{2}}$ & $s$ scale $\propto a$, $a^{3/2}$ & $a^{0}$, respectively. For comparison, in the present model $m_{e}\propto\rho(\eta)$ & $T\propto\rho^{0}$, and thus $\Delta/T$, $\left(\frac{m_{e}}{2\pi T}\right)^{\frac{3}{2}}$ & $s$ scale $\propto\rho$, $\rho^{3/2}$ & $\rho^{0}$, respectively. Consequently, the physics of recombination in the proposed model is identical to that of the standard cosmological model. Similar arguments apply to the physics of BBN which remains unchanged with respect to the standard cosmological model.

Another simple example is the Compton scattering cross-section in the Thomson limit, $\tau=\int n_{e}\sigma_{T}dl$, where $\sigma_{T}\propto m_{e}^{-2}$ is the cross section for Compton scattering in this limit. In the standard cosmological model $n_{e}\propto a^{-3}$, $\sigma_{T}\propto a^{0}$ & $dl\propto dt\propto a(\eta)d\eta$, i.e. $\tau=\int a^{-2}(\eta)d\eta$. In comparison, in the proposed model $n_{e}\propto a^{0}$, $\sigma_{T}\propto a^{-2}$ & $dl\propto d\eta$, i.e. $\tau=\int\rho^{-2}(\eta)d\eta$.

Back to the ‘Friedmann equations’, Eqs. (3.3) & (3.4). These are augmented with $\propto\theta^{\prime 2}$ terms, which come from $\Theta_{\mu}^{\nu}$, the effective energy-momentum tensor associated with the kinetic term of the scalar field (Eq. 2.5). These terms are comparable in size (but are of opposite sign) to other contributions to the energy budget of the universe only close to the turnaround time and are negligible at all other times as is shown below. Whereas $\rho(\eta)$ essentially replaces the scale factor $a(\eta)$ of the FRW model, $\theta$ is a new field. As is already mentioned and as shown in this section and in 3.4, inclusion of a non-vanishing $\theta^{\prime}$ in our cosmological model serves two purposes. First, it could be used to avoid the initial singularity by explicitly violating the weak, strong, and null energy conditions. Second, it couples excitations of $\theta$ to scalar metric perturbations, i.e. to the gravitational potential, ultimately resulting in a flat power spectrum. This mechanism has no parallel in the tensor perturbations sector and consequently no primordial generation of B-mode polarization on cosmological scales is expected (see also section 3.4).

Now, to close our system of background equations, the evolution of $\theta$ is governed by

[TABLE]

i.e. $\theta^{\prime}=c_{\theta}/\rho^{2}$, where $c_{\theta}$ is an integration constant. Eq. (3.6) is obtained for the second component of $\phi=(\rho,\theta)$ from Eq. (2.3) with $\mathcal{G}_{IJ}=2\cdot diag(-1,-\rho^{2})$, or is alternatively derived directly from variation of the kinetic term in Eq. (2.1) with respect to $\theta$ because all other terms are independent of $\theta$ and its derivatives. Another way to obtain Eq. (3.6) is to employ Eq. (2.3) to the complex field frame described below Eq. (2.1) while taking advantage of the fact that both $F$, $V$ & $\mathcal{L}_{M}$ depend purely on $|\phi|$ and that $\mathcal{G}_{IJ}=constant$ in this frame. The resulting equation is $\phi^{*}\frac{\partial\mathcal{I}}{\partial\phi^{*}}-\phi\frac{\partial\mathcal{I}}{\partial\phi}=2{\rm Im}(\phi^{*}\Box\phi)=0$ from which Eq. (3.6) readily follows.

Similar to the discussion below Eq. (3.1), the field equation Eqs. (3.3), (3.4) & (3.6) could be directly derived from variation of $\mathcal{I}=\int\left(K\rho^{2}-\rho^{\prime 2}-\rho^{2}\theta^{\prime 2}-\rho_{M}(\rho)\right)\sqrt{-g}d^{4}x$ with respect to $\rho$ & $\theta$ while making use of Eq. (2.7). The metric $g_{\mu\nu}$ is the same static metric field used in Eq. (3.2). Substituting $\theta^{\prime}=c_{\theta}/\rho^{2}$ into Eqs. (3.3) & (3.4), and comparing with Eq. (2.7), shows that the $\propto\theta^{\prime 2}$ terms effectively play the role of a ‘stiff matter’ contribution with $w_{\theta}=1$. This contribution to the energy budget would have dominated the cosmic evolution at early epochs, i.e. in the small field limit $\rho/\rho_{0}\ll 1$, has it been positive. However, from Eq. (3.3) it is clear that $\rho_{\theta}\equiv-\rho^{2}\theta^{\prime 2}$ is negative. Assuming no turnaround takes place during the MD era, and if it ever takes place it does so deep into the RD era, then only when $|\rho_{\theta}|$ becomes comparable to the energy density of radiation does a turnaround take place, after which $|\rho_{\theta}|$ drops faster than any other form of energy density considered in this model in the redshifting era (we assume a theoretical upper limit of $w\leq 1/3$ for perfect fluids). Eq. (2.3) also gives a second-order evolution equation for $\rho$, but employing $R=6K$ and the solution of Eq. (3.6), $\theta^{\prime}=c_{\theta}/\rho^{2}$, this equation can be readily integrated once. The result of this integration is exactly the Friedmann-like equation, Eq. (3.3), and therefore the system of Eqs. (3.3), (3.4) & (3.6) fully determine the background evolution on cosmological scales. It should be mentioned that the relation $\theta^{\prime}=c_{\theta}/\rho^{2}$ has been similarly obtained in [34] in the context of another $U(1)$ symmetric cosmological model which is otherwise entirely different from the model put forward in the present work.

The proposed scenario is non-singular and the cosmic history comprises of blue- and redshifting evolution epochs. The former, blueshifting epoch, is nearly symmetric in this scenario to the redshifting era. It starts with a vacuum-like energy-, followed by MD, RD, and a brief era dominated by a mixture of radiation and ‘stiff’ energy density, a turnaround, and a redshifting epoch with these various eras occurring in reverse order.

Accounting for the vacuum-like, NR, radiation and effectively stiff energy densities, Eq. (3.3) implies that

[TABLE]

where $\rho_{i,*}$ is the energy density associated with the i’th species at $\eta_{*}$, $\rho(\eta_{*})\equiv 1$, $n_{NR}$ is the number density of NR particles (assuming a single species for illustration purposes), and $\lambda_{p}$ is a dimensionless parameter. In a static background, as the one considered here, both $n_{NR}$ and $\rho_{r}$ are fixed constants. The full analytic integration of this equation is not very illuminating, and therefore we treat two interesting limits separately that will suffice for our purposes. The first limit is obtained by neglecting the vacuum-like and NR terms near the turnaround point. In this case, the Friedmann-like equation integrates to

[TABLE]

where $\rho$ attains its minimum at $\eta=0$. It could be readily integrated to give the cosmic time around the turnaround point, $t\propto\int\rho(\eta)d\eta$, and is easily verified to be non-singular as well. In other words, the effective time coordinates of both massless and massive particles can be extended through the turnaround point, i.e. spacetime is geodesically-complete as is generically the case in non-singular bouncing cosmological models, [35]. In the absence of turnaround point (e.g. in case $\theta$ identically vanishes) the scalar field would have scaled $\rho\propto\eta$ as does the scale factor $a(\eta)$ in the RD era. The scalar field then vanishes at $\eta=0$. Whereas this is not a curvature singularity, it is a topological singularity and it is not entirely clear to us that the theory can be extended to $\eta<0$ as this would (at least naively) imply negative masses. Incoming null geodesics are described by $\eta_{0}-\eta=r$ in Minkowski spacetime. Therefore, since $\eta$ is bounded from below this would imply that observable $r$ is finite although underlying spacetime is Minkowski; this happens in that case because scalar fields hit an ‘initial (topological) singularity’ (in the scalar field $\rho$, not the metric or curvature scalar which are trivial) at a certain finite time in the past.

In the other extreme – where the background dynamics is dominated by the quartic potential – $\rho=\left(\sqrt{\lambda}(\eta_{c}-\eta)\right)^{-1}$, and $\rho$ scales according to its canonical dimension $length^{-1}$, i.e. $\propto\eta^{-1}$, not $\propto t^{-1}$. This again highlights the privileged role played by conformal time as compared to cosmic time, in contrast to the standard cosmological model where conformal time is only used for computational convenience, or as the natural time coordinate parameterizing null geodesics. The integration constant $\eta_{c}$ determines the lower and upper limit on the (conformal) time coordinate in the proposed nearly symmetric model, $\eta\in(-\eta_{c},\eta_{c})$. Again, since $\eta$ is bounded from below, this time due to the presence of the vacuum-like energy density component, then observed radial distances are bounded at $\eta=-\eta_{c}$ where the scalar field diverges and the model breaks down. Specifically, null geodesics in the case $K=0$ satisfy $r(\eta)=\eta_{0}-\eta$. Since $\eta>-\eta_{c}$ then the maximal observable distance is $r_{max}=\eta_{0}+\eta_{c}$. The latter can potentially be much larger than the Hubble scale if $\eta_{c}\gg\eta_{0}$. Although we focus here on DE-dominated asymptotics it is clear that a similar breakdown of the model is shared by any model asymptotically dominated by $w<-1/3$ as Eq. (3.3) integrates in this case to $\rho=c\left(\eta_{c}\pm\eta\right)^{\frac{2}{1+3w}}$ (in the cases $\eta<0$ & $\eta>0$ respectively) where $c$ and $\eta_{c}$ are positive integration constants. Note that this is also the solution of the (equivalent) Friedmann equation, the scale factor $a(\eta)$. Consequently, this scalar field singularity is present in the standard cosmological model as well. However, in the latter it is only a future singularity, whereas in the proposed model it is also a past singularity taking place during the blueshifting epoch, thereby rendering the observable universe finite although Minkowski space itself has no singularity and therefore no horizon associated with it.

One may argue that instead of applying Eq. (2.1) to a homogeneous and isotropic spacetime as done in the present section, we could have equally well used the EH action, Eq. (3.1), supplemented by $\mathcal{I}_{\theta}=-(3/\kappa)\int\theta^{\mu}\theta_{\mu}\sqrt{-g}d^{4}x=(3/\kappa)\int a^{2}\theta^{\prime 2}d^{4}x$, in formulating the cosmological model described in the present section on an expanding background space. However, this ad hoc procedure – while being mathematically equivalent to the approach taken here – seems somewhat less natural than the $U(1)$ symmetric model outlined here and formulated on a static background space.

The ‘flatness problem’ arises in the hot big bang model due to the monotonic expansion of space and the consequent faster dilution of the energy density of matter (either relativistic or NR) compared to the effective energy density dilution associated with curvature. It is thus hard to envisage how could space be nearly flat (as is indeed inferred from observations, e.g. [36]) if not for an enormous fine-tuning at the very early universe, or an early violent inflationary era. In the proposed scenario the matter content of the universe has always been the same, and in particular the present ratio of matter- to curvature-energy densities has been exactly the same when $\rho$ in the blueshifting era was equal to its present (redshifting era) value. However, in the blueshifting era matter domination over curvature is actually an attractor point as the blueshifting universe starts essentially from $\rho\rightarrow\infty$. In other words, had the universe been curvature-dominated (CD) at present (as is naively expected in the standard expanding hot big bang model but with no inflation), i.e. at $\rho_{0}$, it must have been CD at $\rho_{0}$ at the mirror blueshifting era, but since $\rho_{M}\propto\rho$ while $\rho_{K}\propto\rho^{2}$ then a curvature domination at $\rho_{0}$ in blueshifting era would amount to an extremely fine-tuned $\rho_{M}/\rho_{K}\rightarrow 0$ at $\rho\rightarrow\infty$ at the blueshifting era. As is well-known, entropy produced in the pre-bounce era could be processed at the bounce to thermal radiation, implying in effect that $\rho_{r}$ might somewhat change between pre- and post-bounce but the expectation in the proposed model is that the contracting and expanding epochs nearly mirror each other.

The kind of argument employed here in explaining away the flatness problem can be reversed to show that anisotropy actually does enormously grow relative to the other energy species in the blueshifting era. In terms of standard cosmology $\rho$ represents some (geometric) average scale factor and $\theta$ determines deviation from isotropy, such as in Bianchi-type cosmological models. In bouncing models this anisotropy is a measure of the variation of expansion or contraction rates between the principal axes of the homogeneous model, which change very fast near the bounce, the Belinskii-Khalatnikov-Lifshitz (BKL) instability, i.e. that anisotropy is a natural attractor in contracting cosmologies, and in order to avoid it an enormous fine-tuning of initial conditions has to be invoked, e.g. [37]. The analog of this in the proposed model would be the extreme variation over time of $\theta$ around the turnaround point. We will argue below that our model is free of these illnesses that generically afflict bouncing cosmological models. It should be emphasized that slightly anisotropic expansion or contraction could be mimicked by a stiff matter but the opposite is not true – an effective stiff EOS does not necessarily imply anisotropic evolution. For example, in the proposed model space is static and isotropic and the evolution is only in the scalar field, not space. An effective anisotropy, or in other words, an effective ‘stiff’ component ($w=1$) could arise, e.g. from terms in the Lagrangian Eq. (2.1) which are $\propto|\phi|^{-2}$. However, Weyl symmetry that prohibits the appearance of any dimensional quantity at the action level, severely limits the existence of such a term. It could appear in the form, e.g. $\Delta\mathcal{L}=(\bar{\psi}\psi)^{2}|\phi|^{-2}$, or higher-curvature terms of the form $\propto|\phi|^{-2}C_{\alpha\beta}^{\gamma\delta}C_{\gamma\delta}^{\rho\sigma}C_{\rho\sigma}^{\alpha\beta}$. The second term vanishes in the case of conformally-flat metric $g_{\mu\nu}$, as the one employed here. Perhaps more important, in the same fashion that no stiff matter component seems to be required in standard cosmology to explain observations we ignore such terms in the proposed model, and in general do not allow non-canonical negative powers of the scalar field to appear in the action (as discussed below Eq. 2.7) for a lack of an otherwise good reason to allow such terms.

3.3 Linear Perturbation Theory

The standard cosmological model has successfully passed numerous tests and has been quite effective in explaining the formation and linear growth of density perturbations over the background spacetime, predicting the CMB acoustic peaks, polarization spectrum, and damping features on small scales. It also correctly describes the linear and nonlinear evolution phases of the LSS (on sufficiently large scales) and abundance of galaxy and galaxy cluster halos. Therefore, it would seem essential to establish equivalence of linear perturbation theory between our model and the standard cosmological model.

As in section 3.2, we assume an effective energy density $\rho_{M}$ characterized by a (generally time-dependent) EOS $w=w(\eta)$ that encapsulates NR and relativistic baryons, CDM, radiation, and a vacuum-like energy density. Two perturbation variables are the scalar metric perturbations $\varphi$ & $\alpha$ that appear in the rescaled perturbed FRW line element $ds^{2}=-(1+2\alpha)d\eta^{2}+(1+2\varphi)\gamma_{ij}dx^{i}dx^{j}$, where $\gamma_{ij}\equiv diag[1/(1-Kr^{2}),r^{2},r^{2}\sin^{2}\theta]$. We define the fractional energy density and pressure perturbations (in energy density units) $\delta_{\rho_{M}}\equiv\delta\rho_{M}/\rho_{M}$ and $\delta_{P_{M}}\equiv\delta P_{M}/\rho_{M}$, respectively. The matter velocity is $v$.

In the shear-free gauge, and in the case of vanishing curvature $K$ and stress anisotropy, i.e. $\varphi=-\alpha$, the Arnowitt-Deser-Misner (ADM) energy & momentum constraints, Raychaudhuri equation, $\delta\theta$ equation, and the perturbed continuity & Euler equations (Eqs. 39, 40, 42, 43, 48 & 49 of [24]) reduce to

[TABLE]

where $u\equiv v/k$ and we used Eq. (3.3) to eliminate $\rho_{M}$, and the ‘shifted’ perturbation quantities are defined as $\tilde{\varphi}\equiv\varphi+\delta_{\rho}$ & $\tilde{\delta}_{\rho_{M}}\equiv\delta_{\rho_{M}}-4\delta_{\rho}$, where $\delta_{\rho}\equiv\frac{\delta\rho}{\rho}$. Applying the conformal transformation $g_{\mu\nu}\rightarrow g_{\mu\nu}/\rho^{2}$ & $\rho_{M}\rightarrow\rho_{M}\rho^{4}$ takes us back to $\varphi$ and $\delta_{\rho_{M}}$ in the limit of linear perturbations $\delta_{\rho}\ll 1$. We note that the difference of the second-order perturbation equation for $\delta\rho$ and the perturbed trace of the generalized Einstein equation (Eqs. (43) & (44) of [24], respectively) results in the consistency relation $\delta_{P_{M}}=w\delta_{\rho_{M}}$ and therefore provides no additional information to Eqs. (3.9)-(3.14). The latter are exactly the linear perturbation equations over an FRW background if we make the replacement $\mathcal{Q}\rightarrow\mathcal{H}$ and recalling the conclusion from section 3.2 that $\mathcal{Q}$ is exactly $\mathcal{H}$ except of very near the turnaround point.

Vector and tensor perturbations, which are described by Eqs. (53)-(55) and (58) of [24] respectively, similarly satisfy the same equations they do in GR, provided that $a\rightarrow\rho$, i.e. $\mathcal{H}\rightarrow\mathcal{Q}$ in going from the GR formulation to the one adopted here, and the sources, e.g. anisotropic stress $\pi^{(t)}$, are correspondingly rescaled by a multiplicative factor $\rho^{4}$, e.g. $\pi^{(t)}\rightarrow\pi^{(t)}\rho^{4}$. For example, the vector and tensor perturbation modes, $v^{(v)}$ & $h$ respectively, satisfy the equations

[TABLE]

in the standard cosmological model, where $\pi^{(v)}$ & $\pi^{(t)}$ are the vector and tensor anisotropic stresses, respectively. For comparison, the corresponding Eqs. (54) & (58) of [24] are

[TABLE]

respectively. For example, the decay of superhorizon tensor modes, $h\propto a^{-2}$, in the standard cosmological model is replaced by the identical scaling $h\propto\rho^{-2}$ in the alternative model discussed here. This is not surprising; the two cosmological models are related via Weyl symmetry (ignoring the effect of the dynamical field $\theta$ near the turnaround point for the purpose of the present discussion) and thus the dynamics of dimensionless metric perturbations such as $v^{(v)}$ & $h$ should be identical in both models.

The full kinetic theory, pertaining to the theory described by Eq. (2.1) applied to homogeneous and isotropic background cosmology, involving collisional photons and collisionless neutrinos, e.g. [24], where the corresponding perturbed energy-momentum tensor components are given in terms of integrals over the respective distribution functions $f$, can be easily incorporated in our scheme with minor adjustments; neutrino masses are $\propto\rho(\eta)$, the dynamical Higgs VEV, and this has to be accounted for in the collisional Boltzmann equation.

The Newtonian limit of the gravitational interaction in the framework of Eq. (2.1) is obtained from Eqs. (3.9)-(3.14) by setting $\mathcal{Q}$, $K$, $w$, and $\theta^{\prime}$ to zero. In particular, Eqs. (3.9), (3.13) & (3.14) are the relativistic Poisson, continuity, and Euler equations, respectively.

In the standard cosmological model, a gravitational potential consistent with Eqs. (3.9)-(3.14) in the RD era, i.e. $w=1/3$, is $\varphi=A(k)j_{1}(q)/q+B(k)y_{1}(q)/q$ where $q\equiv\frac{x}{\sqrt{3}}$, with $x\equiv k\eta$, and $j_{l}$ & $y_{l}$ are the spherical Bessel and Neumann functions, respectively. Assuming the two modes have been generated at approximately equal amplitudes at some $q\ll 1$ during the inflationary period then the mode that diverges at $q=0$ is negligible at later times. Consequently, the initial condition $B(k)=0$ is selected in standard cosmology and $\varphi$ is essentially constant at $q\rightarrow 0$. In the present model the argument is different; since the model is extended through the turnaround point to $\eta<0$ the diverging mode which is an odd function of $\eta$ near the turnaround point is set to zero by the mere requirement that $\varphi$ is a continuous function of time, at $\eta=0$ in particular. Hence, the adiabatic initial condition merely follows from the very existence of a non-singular turnaround and required continuity of the gravitational potential. A similar argument applies to the tensor modes; the mode singular at $\eta=0$ is an odd function of $\eta$ and is consequently discontinuous at $\eta=0$. Although the simplest models of inflation guarantee that the initial conditions are adiabatic in the standard cosmological model, a mixture of adiabatic and isocurvature modes can also be accommodated by certain inflationary models. The latter are constrained at the few percent level but are not entirely ruled out [36].

It has been recently proposed that the universe has a CPT ‘anti-universe’ counterpart [38]. The two universes according to this picture share a common topological singularity at $\eta=0$. In contrast, as is shown in section 3.4 below, our model is characterized by a non-singular turnaround. In [38] it is shown that the adiabatic initial conditions could naturally result from time-reversal symmetry which is inherent to their model, a conjecture that is obviously not satisfied by the perturbed FRW universe in the scenario proposed here (but is clearly satisfied at the background level of the present model). Thus, rather than conjecturing a global time-reversal symmetry we make a more modest and natural requirement from our model – pointwise continuity. This implies in particular that the integration constants multiplying perturbation modes which are divergent odd functions of $\eta$ must vanish.

3.4 Primordial Flat Spectrum and Turnover Point

Although inflation provides a mechanism for generating scalar and tensor perturbations which are characterized by nearly-flat power spectra, it is not a prediction of the inflationary scenario; it has been known for nearly a decade before the advent of inflation that at least the density perturbations are described by a nearly flat spectrum [39-41], and as of yet there is no evidence for the existence of primordial tensor modes anyway. Other early universe scenarios, e.g. the varying speed of light cosmology [42], the ekpyrotic [43] and new ekpyrotic [44] scenarios, the cyclic universe [45, 46], string gas cosmology [47], Anamorphic cosmology [48], and pseudo-conformal universe [49, 50], are capable of explaining the observed flat spectrum as well.

It is well known that flat spectra are equally well generated from fluctuations of scalar fields during the MD contraction era in bouncing scenarios [51]. Alternatively, here we consider the massless transversal perturbation of the scalar field, $i\rho\delta\theta$ (accounting for its minimal coupling to scalar curvature perturbations) as a viable source for scale-invariant density perturbations.

Combining Eqs. (3.9) & (3.11) we obtain for an arbitrary $w(\eta)$

[TABLE]

Using Eq. (3.12) to express $\varphi^{\prime}$ in terms of $\delta\theta$ and its derivatives, taking the time-derivative of Eq. (3.17), and employing the relation $\theta^{\prime}\propto\rho^{-2}$, i.e. $\theta^{\prime\prime}/\theta^{\prime}=-2\mathcal{Q}$, and $\mathcal{Q}=\frac{2}{(1+3w)\eta}$ (by virtue of Eq. 3.3 and assuming the dynamics is dominated by a species characterized by $w=constant$, a very good approximation throughout the cosmic history except at very brief transitions between the various epochs), we obtain a fourth-order equation for $\delta\theta$ of the form $\delta\theta_{xxxx}+f_{3}\delta\theta_{xxx}+f_{2}\delta\theta_{xx}+f_{1}\delta\theta_{x}+f_{0}\delta\theta=0$ with

[TABLE]

where again $x\equiv k\eta$, and small $\propto\theta_{x}^{2}$ terms have been neglected. The latter is a very good approximation except for the immediate vicinity of the turnaround point. We have not been able to analytically solve Eq. (3.18) for arbitrary $w$. Specializing to the case $w=0$ results in

[TABLE]

where $c_{1}$, $c_{2}$, $c_{3}$ & $c_{4}$ are (possibly $k$-dependent) integration constants, and Ei is the Exponential Integral. As usual, we impose the Bunch-Davies condition in the limit $x\gg 1$ on the perturbed transversal component of the doublet, i.e. $i\rho\delta\theta\rightarrow\frac{1}{\sqrt{2k}}e^{-ix}$. Employing $\rho=\rho_{0}(\eta/\eta_{0})^{2}$ in the MD era then implies that $c_{3}=-\frac{k^{3/2}\eta_{0}^{2}}{\sqrt{2}\rho_{0}}$ and all other integration constants vanish. Using Eq. (3.6), $\theta^{\prime}=c_{\theta}/\rho^{2}$, in Eq. (3.17) we obtain in the limit $x\ll 1$

[TABLE]

where $C\equiv(-9c_{\theta}\eta_{0}^{6})/(\sqrt{2}k^{3/2}\rho_{0}^{3})$. Its solution is $\varphi=\tilde{c}_{1}+\tilde{c}_{2}\eta^{-5}+(C/6)\eta^{-6}$. In the blueshifting era the fastest growing mode is the $\propto\eta^{-6}$ term (assuming all modes are generated at approximately similar amplitudes, and neglecting the $\propto\eta^{-5}$ term that is discontinuous at $\eta=0$) and thus $\varphi=-9c_{\theta}/(6k\sqrt{2k}\rho^{3})$, which implies that

[TABLE]

is flat. Perturbations generated earlier on during the MD blueshifting era are relatively smaller at the time of production than those which are generated latter since $\rho_{\theta}$ is much smaller than $\rho_{DE}$ at large $\rho$ values. However, they have a longer time to grow than the perturbations generated closer to the turnaround point. Overall, the perturbations observed today are equally contributed at all times during blueshifting MD era.

It should be stressed that unlike the mechanism proposed in [51] which reflects the fact that any field fluctuates on an evolving background (and with rms fluctuation determined by the Hubble scale), the curvature perturbations discussed here are a direct result of the coupling between $\theta$ and $\varphi$ (and the amplitude of scalar perturbations is therefore $\propto\rho_{\theta}$). There is no tensorial analog in our scenario to this coupling.

Repeating the same procedure in the RD era ($w=1/3$), and imposing the Bunch-Davies vacuum on $i\rho\delta\theta$ in the $x\gg 1$ limit eliminates one of the four integration constants. In the other extreme, $x\ll 1$, we obtain $\delta\theta\propto\eta^{-4}$. Since $\theta^{\prime}=c_{\theta}{\rho^{2}}$ and since in the RD era $\rho=\rho_{0}\eta/\eta_{0}$, then neglecting the $wk^{2}\varphi$ term in Eq. (3.17) we obtain

[TABLE]

in the $x\ll 1$ limit, where $c^{\prime}_{1}$, $c^{\prime}_{2}$ & $\beta$ could possibly depend on $k$. The constant $\beta$ is also proportional to $c_{\theta}$. Again, the mere requirement of continuity at all times, and at $\eta=0$ in particular, implies that $\varphi=c^{\prime}_{2}(k)$, and consequently Eq. (3.21) is not modulated during the RD era.

Inflation generically predicts a slightly red power spectrum. The Harrison-Zeldovich spectrum $\Delta^{2}$ in Eq. (3.21) is ruled out by recent observations at the $\sim 7\sigma$ confidence level [36] assuming the vanilla $\Lambda$CDM cosmological model and provides yet another evidence for the inflationary scenario. However, some doubts concerning the robustness of this conclusion have been raised in light of the recently claimed tension in inference of Hubble constant from cosmological data and local universe measurements , e.g. [52] and references within. More specifically to our model, taking the results of [36] at face value (although the standard cosmological model is in tension with local measurements) the model proposed here would have to be modified, probably via introducing a cosmological scale that breaks the Weyl invariance of Eq. (2.1). This possibility will be explored elsewhere.

In generic bouncing scenarios it is implicitly assumed that the bounce takes place at sufficiently large redshifts, specifically prior to BBN [$z_{BBN}\lesssim O(10^{10})$]. In that case the observed abundance of light elements, thermalization of the CMB, and latter processes, are explained exactly as in the standard cosmological model. This requirement is naturally satisfied by bouncing scenarios that rely on modifications of the gravitational field at the Planck scale which naturally take place in quantum gravity-inspired models. In the classical model proposed here turnaround takes place for entirely different reasons; $\rho_{\theta}$ is always negative and is characterized by $w_{\theta}=1$, i.e. $|\rho_{\theta}|$ grows with decreasing $\rho$ during blueshifting even faster than does the energy density of radiation, $\rho_{r}$. This implies that there is a sufficiently large $z_{b}>z_{BBN}$ at which $\rho_{r}(z_{b})+\rho_{\theta}(z_{b})=0$, and the Hubble function momentarily vanishes $\mathcal{H}(z_{b})=0$ while changing sign from negative to positive. Unlike in the quantum gravity scenarios, the requirement that $z_{b}>z_{BBN}$ sets a constraint on the parameter space of our model. In the following this constraint is derived.

From Eq. (3.21), the power spectrum of scalar metric perturbations generated at a time $\eta$ (during the blueshifting epoch $\eta<0$) is $\Delta^{2}(\eta)=\frac{9\lambda}{16\pi^{2}}\left(\frac{|\rho_{\theta}|}{\rho_{DE}}\right)_{0}\left(\frac{\rho_{0}}{\rho(\eta)}\right)^{6}$, where as usual $F_{0}$ stands for the value of a function $F$ at present, $\eta_{0}$. Since the model is symmetric around the turnaround point then $\left(\frac{|\rho_{\theta}|}{\rho_{DE}}\right)_{0}$ stands for the ratio $\frac{|\rho_{\theta}|}{\rho_{DE}}$ at both $\eta_{0}$ and $-\eta_{0}$. The discussion following Eq. (3.22) implies that dynamical metric perturbations are generated only during the MD era. As implied in the discussion following Eq. (3.21), perturbations generated at $-\eta_{0}$ grow by a latter time $\eta$ [during which $\rho_{0}$ changed to $\rho(\eta)$] to $\Delta^{2}(\eta)=\frac{9\lambda}{16\pi^{2}}\left(\frac{|\rho_{\theta}|}{\rho_{DE}}\right)_{0}\left(\frac{\rho_{0}}{\rho(\eta)}\right)^{6}$. But, this is also the amplitude of perturbations generated at this $\eta$. Thus, perturbations generated at all times in the range $-\eta_{0}<\eta<-\eta_{eq}$, where $\eta_{eq}$ is the time at matter-radiation equality, grow by the time $-\eta_{eq}$ to $\Delta^{2}(-\eta_{eq})=\frac{9\lambda}{16\pi^{2}}\left(\frac{|\rho_{\theta}|}{\rho_{DE}}\right)_{0}\left(1+z_{eq}\right)^{6}$. Growth is halted during (blue- and redshifting) RD era. During the time periods $-\eta_{rec}<\eta<-\eta_{eq}$ & $\eta_{eq}<\eta<\eta_{rec}$ the gravitational potential is constant as in the standard cosmological model. All this implies that the observationally inferred normalization of curvature perturbations on super-horizon scales $\Delta^{2}=2.4\times 10^{-9}$ [36], which corresponds to $z_{rec}$, equals $\Delta^{2}(-\eta_{eq})=\frac{9\lambda}{16\pi^{2}}\left(\frac{|\rho_{\theta}|}{\rho_{DE}}\right)_{0}\left(1+z_{eq}\right)^{6}$. In the following consideration we make the idealized assumption that the model is perfectly symmetric around the turnaround point, in particular we assume that $\rho^{4}(\eta)\rho_{r}$ is fixed throughout cosmic history. Adopting $z_{eq}\sim 3400$ [36], and the observationally inferred $\left(\frac{\rho_{r}}{\rho_{DE}}\right)_{0}=1.3\times 10^{-4}$ we readily obtain $\left(\frac{|\rho_{\theta}|}{\rho_{r}}\right)_{0}=2.11\times 10^{-25}\lambda^{-1}$. Now, the condition $\mathcal{H}(z_{b})=0$, i.e. $|\rho_{\theta}|_{0}(1+z_{b})^{6}=\rho_{r,0}(1+z_{b})^{4}$ along with the requirement that $z_{b}>10^{10}$ implies that $\lambda>O(10^{-5})$. Therefore, the scenario proposed here is entirely consistent with the standard cosmological model at all epochs from BBN onwards insofar the dimensionless self-coupling $\lambda$ is sufficiently (yet reasonably) large.

Since $\varphi$ is linearly related to $\delta\theta$ via e.g. Eq. (3.17), and assuming all perturbations are linear, the former automatically inherits the statistical gaussian properties of the latter; scalar perturbations are thus expected to be gaussian and adiabatic (the latter property has been discussed in section 3.3).

4 Summary

While the standard cosmological model has no doubt been very successful in phenomenologically interpreting a wide spectrum of observations, it is fair to say that it still lacks a microphysical explanation of several key features, primarily the nature of CDM and DE. Direct spectral information on the CMB is unavailable (due to opacity) in the early RD era ($z\gg 3000$). From the observed cosmic abundance of light elements, BBN at redshifts $O(10^{9})$ could be indirectly probed. Earlier on, at $z=O(10^{12})$ and $z=O(10^{15})$ [energy scales of $O(200)$ MeV and $O(100)$ GeV, respectively], the quantum chromodynamics (QCD) and electroweak phase transitions had presumably occurred, although their (indeed weak) signatures in the CMB and LSS have not been found. In addition, inflation, a cornerstone in the standard cosmological model, is clearly beyond the realm of well-established physics; its detection via the B-mode polarization it induces in the CMB could be achieved only if it took place at energy scales $\sim 13$ orders of magnitude larger than achievable at present. Although the inflationary scenario is very flexible it is also plagued with certain undesirable problems, such as the $\eta$-problem, trans-Planckian problem, and the ‘measure problem’ in the multiverse. The latter is unavoidable in the currently favorite ‘eternal inflation’ scenario.

Ideally, an alternative cosmological model that agrees well with the standard cosmological model at BBN energies and lower, i.e. $z<10^{10}$, while still addressing the classical problems of the hot big bang model that inflation was designed to solve, and all this in the $\lesssim$ TeV range of energies, will definitely be an appealing alternative. This could be in principle achieved with a (relatively late) non-singular ‘bounce’ that also removes the technically and conceptually undesirable initial (curvature) singularity problem of GR-based cosmological models. In order to achieve such a bounce within GR, or a conformally-related theory, certain ‘energy conditions’ have to be violated. One specific realization of this program has been the focus of the present work.

Symmetries play a key role in our theories of fundamental interactions. For example, the SM of particle physics is based on a local $U(1)\times SU(2)\times SU(3)$ gauge group with quantizable gauge fields. In addition, our favorite theory of gravitation, GR, is diffeomorphism-invariant. In this work we entertained the possibility that in addition to diffeomorphism-invariance, the fundamental scalar-tensor theory of gravitation, Eq. (2.1), also respects a global $U(1)$ symmetry; the modulus of the scalar field is conformally coupled to gravity whereas its (dimensionless) phase is only minimally coupled. The cosmological model based on this alternative theory of gravitation has some appealing properties, only a few of them have been discussed in the present work.

We have shown here that the standard cosmic scale factor might be replaced by the conformally coupled modulus of the scalar field. Its kinetic and potential terms appear with the ‘wrong’ relative sign in the action (thereby guarantying that gravitation is an attractive force), and is therefore inherently classical, exactly as the scale factor of the standard cosmological model is. Ignoring the phase (as is effectively done in the standard cosmological model), which is a free quantizable field minimally coupled to gravitation, results in ignoring its possible perturbations as well, which are described by scale-invariant gaussian and adiabatic perturbations. In the standard cosmological model we then obtain scalar perturbations with these desired properties from the fluctuations of another scalar field – the inflaton. Perhaps even more important, the existence of this phase $\theta$ guarantees that cosmic history goes through a non-singular ‘bounce’ rather than an initial singularity.

A tantalizing alternative scenario explored in this work starts with a vacuum-like- followed by MD and RD deflationary evolution which culminates at a ‘bounce’ (essentially turnaround) when the (absolute value of the negative) energy density associated with the effective ‘stiff matter’ (provided by the kinetic term of $\theta$) momentarily equals that of radiation. We reiterate that the various cosmological epochs according to this scenario only result from the universal evolution of masses, not space expansion – the latter is static. In the vacuum-like-dominated epoch the energy density of the universe is dominated by the quartic potential of the scalar field which is genuinely classical with no quantum fluctuations. Therefore, DE according to the present scenario is not zero-point energy but rather a manifestation of the self-coupling of the scalar field, with all other fields (e.g. Dirac, electromagnetic, etc.) playing a subdominant role in the dynamically evolving background at the DE era. This DE contribution is characterized by a non-dynamical equation of state with no recourse to fine-tuning of the potential and with no need to introduce a new, e.g. quintessence, field.

During the MD blueshifting epoch, gaussian adiabatic scalar perturbations characterized by a flat spectrum, which are sourced by the (quantum) fluctuating minimally coupled field $\theta$, are generated. The observed power spectrum is efficiently produced during the entire MD deflationary epoch. As in the case of inflation, the observed gaussianity is explained by the correspondence between the (quantum) vacuum state of an essentially free scalar field and the ground state of an harmonic oscillator. Adiabatic ‘initial’ conditions, generically predicted by inflation, are instead a natural outcome of the very existence of a turnaround point in the present model instead of a big bang; the mere requirement of continuity then selects the ‘adiabatic’ initial conditions. Normalizing primordial scalar perturbations by their observed value and requiring that the turnaround takes place safely remote from standard BBN or any other lower-energy standard cosmological epochs sets a lower limit on the self coupling parameter of the scalar field, $\lambda\gtrsim 10^{-5}$.

The scalar field does not ‘slow-roll’ along its potential (as it does in the standard inflationary scenario) but rather its kinetic energy is comparable to its potential energy at all times. It is therefore free from the fine-tuning problem of the inflaton potential shape generically required in the standard cosmological model due to radiative corrections. From the present work perspective ‘slow-roll’ is an artifact of the standard units convention, in which all scalar fields (e.g. particle masses, the cosmological constant, the inflaton itself, etc.) are effectively set to constants.

Linear perturbations generated during the blueshifting era generically survive the turnaround due to continuity of metric perturbations and do not undermine the underlying homogeneity and isotropy of the cosmological model in the redshifting era. Matter is not created in the (non-singular and adiabatic) cosmological scenario layed out here, nor is it destroyed. The cosmological scenario from the BBN era onward is exactly as in the standard cosmological model (assuming that $\lambda\gtrsim 10^{-5}$). In addition, the ‘anisotropy problem’ that generally plagues bouncing scenarios does not exist in our construction. Weyl symmetry and the consequent absence of any dimensional parameter in the action severely limit the possibility that such an effective term is present in the action. The existence of inverse powers of the scalar field at the action level is not allowed, barring the existence of any energy contribution characterized by $w>1/3$ in the form of perfect fluid. In addition, although the energy density of $\theta$ behaves effectively as a perfect fluid with $w=1$ it does not cause any anisotropy problem simply because turnaround takes place exactly once the anisotropy-like density starts taking over the cosmic dynamics, and this only happens since this ‘stiff’ energy density is negative, a consequence of the assumed $U(1)$ symmetry – otherwise turnaround would not have taken place.

While conformal time is both past- and future-bounded in this scenario, i.e. $\eta\in(-\eta_{c},\eta_{c})$, the (effective) cosmic time in not. There is no ‘horizon problem’ associated with the model – not for radiation, and not even for, e.g. light (but still massive) neutrinos. Specifically, cosmic history starts with very large (and in principle infinite) particle masses and therefore the causal horizon is much larger than would be naively expected from monotonically growing masses (that corresponds to the redshifting era), i.e. essentially $\eta_{0}\ll\eta_{c}+\eta_{0}$ if $\eta_{c}\gg\eta_{0}$. Likewise, the ‘flatness problem’ afflicting the hot big bang scenario stems from the slower decay of the energy density associated with curvature as compared to that of matter in a monotonically expanding universe. In ‘bouncing’ scenarios the situation is reversed in the pre-bounce era; starting at infinitely large $\rho$ (particle masses) one typically expects to find that the energy density in the form of NR matter largely exceeds that of curvature at any finite $\rho$ value in the blueshifting era. Since this adiabatic model is very nearly symmetric in $\rho$ around the turnaround point (barring entropy processing effects at around the RD era), one generally expects the universe to look spatially flat at any finite $\rho$ after the would-be singularity (actually a non-singular ‘bounce’). From this perspective flatness is an attractor-, rather than an unstable-point that requires fine-tuning. The ‘monopole’ and ‘relic defects’ problems do not arise (in the proposed scenario) for any reasonable $\lambda$ value (unless the latter is exceptionally large).

The proposed model is falsifiable in several respects: First, $w_{DE}=-1$ due to Weyl invariance and any observationally inferred $w_{DE}\neq-1$ would rule out the model. Likewise, CDM is made of fermionic particles for the same reason and thus any credible evidence in favor of bosonic DM would similarly rule out the model. Gaussianity of CMB anisotropy – conventionally induced by single-field inflation models – is yet another tenet of the present (effectively single-field) model. If non-gaussianity ultimately turns out to be small but non-vanishing, then it could be explained by more elaborate inflationary models, whereas it will rule out the model proposed in this work. In addition, unlike the inflationary paradigm that generically predicts adiabatic initial conditions but allows a certain level of admixture of isocurvature modes, the proposed ‘bouncing’ model predicts that the initial conditions are purely adiabatic. Finally, within the framework adopted here, if B-mode polarization is ultimately measured, it would have to be of non-quantum origin, since gravitation in genuinely classical, the metric is unquantized, and its perturbations are not subject to the Bunch-Davies vacuum condition. Consequently, unlike inflationary-induced B-mode polarization, it does not have to be characterized by a flat spectrum. In any case, the phase fluctuations that induce density perturbations in the present model do not source tensorial metric modes.

We believe that, in addition to addressing the cosmological horizon, flatness and cosmological relic problems, the framework proposed here provides important insight on the nature of CDM, DE, initial singularity, cosmological ‘expansion’, the flatness of the matter power spectrum on cosmological scales, and primordial tensor modes. Even so, the work presented here is by no means exhaustive, and indeed many of its basic aspects will be further elucidated in future papers.

Acknowledgments

The author is indebted to Yoel Rephaeli for numerous constructive, critical, and thought-provoking discussions which were invaluable for this work. This work has been supported by the Joan and Irwin Jacobs donor-advised fund at the JCF (San Diego, CA).