Alleviating the cosmological constant problem from particle production

Adopting equation (1), we do not know a priori the most suitable choice for the potential. Following Planck satellite results [26], there is no consensus about the most suitable scalar field potential that drives inflation. The corresponding experimental results provided several approaches that are still valid, ruling out other versions of $V(\phi)$. Among all the most promising possibilities, the hilltop potentials have not been excluded yet [27, 28] and may well-adapt to our scopes of healing the longstanding cosmological constant problem, producing de facto particles from quantum vacuum energy. Indeed, the choice

$$\begin{align} V(\phi)= \Lambda^4 (1-\phi^n/\mu_n^n+ \ldots), \end{align} \tag{ 3 }$$

with $n = 2;4$, involves a typically-large early cosmological constant, which may drive cosmological inflation. Even though assuming hilltop potentials is not the unique possibility, it appears as a remarkable toy approach that considers the presence of a potential driving inflation with vacuum energy and permits one to analytically integrate the subsequent equations related to particle production amount. Further, such potential has the advantage of driving inflation for small fields, $\phi\simeq0$, thus leaving the constant cosmological term to be responsible for a large scalar curvature and particle production. More complicated models can also be invoked, e.g. by assuming large field approaches, like the Starobinski potential, albeit in this case the amount of particles would be mostly due to the interaction between inflaton and curvature, thus complicating the overall treatment.

Hence, to guarantee the above prescriptions to hold, our strategy consists in the following two steps:

• We split the Universe into different epochs. The first is dominated by the inflaton field. The subsequent describes reheating and afterwards radiation dominated epoch until our era, i.e. late-time. During inflation, we write FRW perturbations within the de Sitter spacetime as generated by quantum fluctuations of the inflaton field;
• We evaluate geometric particle production [12, 29] during inflation, adopting the simplest choice for the coupling constant ξ, namely the conformal coupling ¹⁵ $\xi = 1/6$. To do so, we focus on geometry to fuel particle production, neglecting the quantum particle production related to the Bogoliubov coefficients [30, 31], as above remarked.

To work out our treatments, from equation (1), the free equation of motion for φ reads

$$\begin{align} \square_\eta \chi+ V^{\prime}(\phi \rightarrow \chi/a)\ a^3=0, \end{align} \tag{ 4 }$$

with $\square_\eta\equiv \partial_\mu\partial^\mu$ in conformal time and $V^{{\prime}}(\phi) \equiv \partial V/\partial \phi$. From equation (4), since we rescaled the field itself by the ansatz

$$\begin{align} \phi(x)= \chi(x)/a, \end{align} \tag{ 5 }$$

the friction term, namely ${\sim} \dot \phi H$, disappears as a natural consequence of our choice, as well-known in the literature, see e.g. [32].

Since our geometric particle production occurs at early stages of inflationary domain, namely as φ is small, the case n = 4 is disfavored to describe our prescription than n = 2, that also has the advantage to provide analytical dynamical solutions in strict analogy to the case of chaotic potential $V(\phi) = m^2 \phi^2/2$. Accordingly, we write the hilltop quadratic potential by

$$\begin{align} V(\phi)= \Lambda^4 (1-\phi^2/\mu_2^2+ \ldots), \end{align} \tag{ 6 }$$

where Λ⁴ corresponds to the vacuum energy density during inflation [33, 34].

The above potential is defined independently from the shift $V(\phi)\rightarrow V(\phi)+\mathcal C$, with $\mathcal C$ a generic constant, by simply rescaling the values of Λ⁴ and µ₂ ². This guarantees that, modifying the potential by adding a constant, the cosmological constant problem is not restored. The scale µ₂ is intimately related to the field width, i.e. to the field variation during inflation.

Inflation occurs as $\phi\simeq0$ and, by virtue of equation (1), we define the corresponding effective potential driving our inflationary phase as

$$\begin{align} V^{\,\mathrm{eff}}(\phi,R)\equiv \Lambda^4 (1-\phi^2/\mu_2^2)+\xi R\phi^2\,, \end{align} \tag{ 7 }$$

having constructed the sum of both hilltop potential and geometric coupling without any more complicated interactions. During inflation we can approximate it for small fields, having de facto that it can reduce to a slightly evolving vacuum energy contribution ${\sim} \Lambda^4$.

Clearly the dynamics of equation (7) is not fully-stable as due to the typology of coupling with scalar curvature, here the Yukawa-like one. In particular, once the original hilltop potential is modified through the geometric coupling, it is possible a priori not to get a graceful exit. In principle, we are here proposing a toy model where the coupling with curvature can play the role of producing particles, but further investigations on equation (7), to work out how inflation naturally ends, are essential. In other words, one has to investigate which kind of more complicated curvature coupling may be included into the above scenario, in order to exit inflation. Limiting to this toy model, we make some heuristic considerations on how inflation may end later in the manuscript.

At this stage, plugging equation (6) into (4), we get

$$\begin{align} \square_\eta \chi - 2 a^2 \frac{\Lambda^4}{\mu_2^2} \chi=0. \end{align} \tag{ 8 }$$

Here, equation (8) can be analytically solved to adapt throughout inflation occurs. We focus on two phases below, namely during and after inflation. We thus analyze how to produce particles and how to interpret them as dark matter, and then we discuss the consequences of our recipe immediately after inflation, up to our times.

Theoretically speaking, inflation lasts inside $-\infty \lt \tau \lt 0$. Around $\tau\simeq0$, i.e. as inflation ends, a de Sitter phase would naturally diverge and consequently unphysical divergences could occur. To avoid such singularities, the scale factor can be rewritten as prompted in [35]:

$$\begin{align} a(\tau)= \frac{1}{1-H_I\tau},\ \ \ \ \tau \leqslant 0, \end{align} \tag{ 9 }$$

where we conventionally baptize the Hubble constant with H_I , during inflation. It is evident that equation (9) does not provide any pathology within the range $-\infty \lt \tau \leqslant 0$, letting our model to work better during and immediately after inflation.

For the sake of clearness, it is well-established that during inflationary stage the Hubble rate is not exactly a constant. It may slightly change with time, leading to a quasi-de Sitter expansion. Assuming a de Sitter phase is therefore an approximation that, however, works well in describing the overall evolution of the scalar field during inflation. We will come back to analyze this issue later throughout the text.

2.3.1. Dynamical solutions.

Now, bearing the ansatz (9) in mind, equation (8) gives

$$\begin{align} \square_\eta \chi- \frac{2}{(1-H_I\tau)^2} \frac{\Lambda^4}{\mu_2^2}\chi=0, \end{align} \tag{ 10 }$$

whose general solution can be recast by

$$\begin{align} \chi({\mathbf{x}}, \tau)= \frac{f_k(\tau) e^{i {\mathbf{k}} \cdot {\mathbf{x}}}}{(2 \pi)^{3/2}}. \end{align} \tag{ 11 }$$

Here, the field modes f_k satisfy the differential equation

$$\begin{align} \ddot{f}_k+ \bigg[k^{\,2}- \frac{1}{(1-H_I\tau)^2} \left(\frac{2\Lambda^4}{\mu_2^2} \right) \bigg]f_k=0. \end{align} \tag{ 12 }$$

Hence, equation (12) can be more compactly written as

$$\begin{align} \ddot{f}_k+ \bigg[k^{\,2}- \frac{1}{\tilde{\tau}^2} \left(\frac{2\Lambda^4}{\mu_2^2H_I^2} \right) \bigg]f_k=0, \end{align} \tag{ 13 }$$

having introduced the new variable $\tilde{\tau} = \tau-1/H_I$. This equation has the form

$$\begin{align} \ddot{f}_k+ \left[ k^{\,2}- \frac{1}{\tilde{\tau}^2} \left(\nu^2-\frac{1}{4} \right) \right] f_k=0, \end{align} \tag{ 14 }$$

admitting general solutions given in terms of Hankel's functions [14]

$$\begin{align} f_k(\tau)= \sqrt{-\tilde{\tau}}\left[ c_1(k) H^{(1)}_\nu(-k\tilde{\tau})+ c_2(k) H^{(2)}_\nu(-k\tilde{\tau}) \right]. \end{align} \tag{ 15 }$$

2.3.2. Bunch–Davies state for vacuum.

The constants $c_1(k)$ and $c_2(k)$ are determined by selecting the vacuum state in the de Sitter space. As it is well-known from quantum field theory, a general curved spacetime does not admit a canonical, or even preferred, vacuum state [36]. So, a convenient choice in the de Sitter spacetime is the so-called Bunch–Davies state, which appears precisely thermal to a free-falling observer in such a space ¹⁶ [38]. In particular, imposing in our scheme the Bunch–Davies vacuum turns out to be equivalent to let our solution match the plane wave solution $e^{ik\tau}/\sqrt{2k}$ in the ultraviolet regime $k \gg aH_I$. Thus, we have

$$\begin{align} f_k(\tau)= \frac{\sqrt{\pi}}{2}e^{i(\nu +\frac{1}{2})\frac{\pi}{2}} \sqrt{-\tilde{\tau}} H_\nu^{(1)}(-k \tilde{\tau}), \end{align} \tag{ 16 }$$

where $H_\nu^{(1)}$ are first kind Hankel's functions and from equations (13) and (14) we get

$$\begin{align} \nu^2= \frac{1}{4}+ \frac{2 \Lambda^4}{\mu_2^2 H_I^2}. \end{align} \tag{ 17 }$$

2.3.3. Scalar field perturbations.

We assume now that scalar perturbations of the metric are generated by the quantum fluctuations of the inflaton field, as in the standard model of inflation [6, 14]. We neglect the effects of tensor modes (gravitational waves), which however are expected to produce similar outcomes ¹⁷ on super-Hubble scales. The most general line element for a perturbed spatially flat FRW Universe in case of scalar perturbations reads

$$\begin{align} \textrm ds^2=a^2(\tau) \left[(1+2\Phi)\textrm d\tau^2-2\ \partial_i B\ \textrm d\tau \textrm dx^i - \left((1-2\Psi) \delta_{ij}+ D_{ij} E \right) \textrm dx^i \textrm dx^j \right]. \end{align} \tag{ 18 }$$

In the longitudinal (or conformal Newtonian) gauge, we set $E = B = 0$. For a scalar field, one also obtains ¹⁸ $\Psi = \Phi$. Accordingly, the perturbation potential Ψ satisfies the differential equation [14, 40]

$$\begin{align} \dot{\Psi}+ \mathcal{H} \Psi=4\pi{G} \dot{\phi}_0 \delta \phi= \epsilon \mathcal{H}^2 \frac{\delta \phi}{\dot{\phi}_0}, \end{align} \tag{ 19 }$$

where we split the field as

$$\begin{align} \phi=\phi_0(\tau)+\delta \phi({\mathbf{x}}, \tau), \end{align} \tag{ 20 }$$

with φ₀ representing the 'classical' background field ¹⁹ and $\delta \phi({\mathbf{x}}, \tau)$ the quantum fluctuations around φ₀. In equation (19), G is the gravitational constant, $\mathcal{H} = \dot{a}/a$ and ε the usual slow roll parameter ²⁰ . From equation (9), we have

$$\begin{align} \mathcal{H}\equiv \frac{\dot{a}}{a}= \frac{H_I}{1-H_I\tau}, \end{align} \tag{ 21 }$$

with (always) vanishing slow-roll parameter ε, given as a consequence of adopting a quasi-de Sitter phase. Indubitably, a pure de Sitter spacetime implies ε = 0 at any time. To overcome this issue, we could easily modify the scale factor by including a slight correction, following the general idea of [14]. A plausible modified $a(\tau)$ then reads

$$\begin{align} a(\tau)= \frac{1}{(1-H_I\tau)^{1+m}}, \end{align} \tag{ 22 }$$

where m is a constant that weakly deviates from m = 0. For the sake of completeness, in principle we can assume $m = m(\tau)$ instead of a pure constant m, in order to properly solve equation (19). Both the possibilities, namely constant m and $m = m(\tau)$, as anticipated above, are related to the fact that inflation has to be described by a quasi de Sitter phase, first to avoid ε = 0 and second to guarantee that $a(\tau)$ is an approximate, but suitable, ansatz for our hilltop potential that, only asymptotically, evolves like a de Sitter phase.

With the ansatz (22), equation (21) is slightly modified by

$$\begin{align} \mathcal{H}= \frac{(1+m) H_I}{(1-H_I\tau)} \end{align} \tag{ 23 }$$

from which

$$\begin{align} \epsilon= 1- \frac{\dot{\mathcal{H}}}{\mathcal{H}^2}= 1-\frac{1}{1+m} \simeq m, \end{align} \tag{ 24 }$$

where the last equality on the r.h.s. is true for small m only. As stated above, by imposing a time-varying m term, the corresponding, more complicated, version of ε would weakly evolve during inflation to guarantee a graceful exit from it. We hereafter leave it fixed throughout the overall inflationary phase only to simplify our calculations and we will require $\epsilon \rightarrow 1$ in order to end inflation.

2.3.4. Potential solutions at super-Hubble scales.

From now on, we focus on super-Hubble scales, where the condition $k \ll a(\tau)H_I$ holds. This will provide a physically motivated cut-off for the momenta of particles produced, as we will see. On these scales, it can be shown [14] that $\dot{\phi}_0$ and δφ solve the same equation. The solutions are then related to each other by a function $c({\mathbf{x}})$ which depends upon space only:

$$\begin{align} \delta \phi= c({\mathbf{x}}) \dot{\phi}_0. \end{align} \tag{ 25 }$$

Setting $c({\mathbf{x}}) = e^{i {\mathbf{q}} \cdot {\mathbf{x}}}$, we can solve equation (19) for the scale factor (9) and obtain the general form of the potential

$$\begin{align} \Psi= \frac{\epsilon H_I-2 c_1(1-H_I\tau)^2}{2(1-H_I\tau)}\ e^{i {\mathbf{q}} \cdot {\mathbf{x}}}. \end{align} \tag{ 26 }$$

Assuming now that $\Psi(\tau \rightarrow - \infty) = 0$, we explicitly get

$$\begin{align} \Psi= \frac{\epsilon H_I}{2(1-H_I\tau)}\ e^{i {\mathbf{q}} \cdot {\mathbf{x}}}. \end{align} \tag{ 27 }$$

Having a functional form for Ψ, we can now compute the perturbation tensor from which our geometric particles arise.

An interesting point, from equation (27) is the following. As ε tends to one, namely as inflation ends, the perturbed potential does not vanish. This is a general feature of inflationary models, not only limited to our choice of $V(\phi)$. Consequently, a suppressing position-dependent exponent in the phase $e^{i {\mathbf{q}} \cdot {\mathbf{x}}}$ may be requested as cut-off scale in Ψ, physically motivated by the fact that once inflation ends the Universe is less inhomogeneous than during inflation and, gradually increasing the cosmic scale by cosmic expansion, one recovers the cosmological principle.

In other words, a generic form of Ψ for a unspecified potential that violates equation (25) may read

$$\begin{align} \Psi=\mathcal F(\epsilon,\tau,H_I)e^{i {\mathbf{q}}(\tau) \cdot {\mathbf{x}}}\,, \end{align} \tag{ 28 }$$

with ${\mathbf{q}}(\tau) = {\mathbf{q}}+i{\mathbf{K}}\tau$, and K a unconstrained positive-definite momentum.

A possible physical motivation to an ansatz of the form (28) may lie in the so-called back-reaction mechanism, which we now briefly discuss.

2.3.5. The issue of back-reaction.

2.3.6. Gauge transformations.

We now need to write the gravitational potential Ψ in the synchronous gauge ²¹ . In this gauge, the most general scalar perturbation takes the form $h_{ij} = h \delta_{ij}/3+h_{ij}^\parallel$.

The general procedure to transform from the longitudinal to the synchronous gauge is the following [43]. Let us consider a general coordinate transformation from a system x^µ to another $\hat{x}^\mu$

$$\begin{align} x^\mu \rightarrow \hat{x}^\mu=x^\mu+d^\mu (x^\nu). \end{align} \tag{ 29 }$$

We write the time and the spatial parts separately as

$$\begin{align} &\hat{x}^0= x^0 + \alpha({\mathbf{x}},\tau) \end{align} \tag{ 30a }$$

$$\begin{align} &\hat{{\mathbf{x}}}= {\mathbf{x}}+ \nabla \beta({\mathbf{x}}, \tau)+ {\boldsymbol \epsilon}({\mathbf{x}}, \tau),\ \ \ \ \ \ \ \nabla \cdot {\boldsymbol \epsilon}=0, \end{align} \tag{ 30b }$$

where the vector d has been divided into a longitudinal component $\nabla \beta$ and a transverse component $\vec{\epsilon}$.

Let $\hat{x}^\mu$ denote the synchronous coordinates and x^µ the conformal Newtonian coordinates, with $\hat{x}^\mu = x^\mu+d^\mu$. We have

$$\begin{align} & \alpha({\mathbf{x}}, \tau)=\dot{\beta}({\mathbf{x}}, \tau), \end{align} \tag{ 31a }$$

$$\begin{align} & \epsilon_i({\mathbf{x}}, \tau)=\epsilon_i({\mathbf{x}}), \end{align} \tag{ 31b }$$

$$\begin{align} & h_{ij}^\parallel({\mathbf{x}}, \tau)=-2 \bigg( \partial_i \partial_j-\frac{1}{3} \delta_{ij} \nabla^2 \bigg) \beta({\mathbf{x}}, \tau), \end{align} \tag{ 31c }$$

$$\begin{align} & \partial_i \epsilon_j+ \partial_j \epsilon_i=0. \end{align} \tag{ 31d }$$

and

$$\begin{align} & \psi({\mathbf{x}}, \tau)= -\ddot{\beta}({\mathbf{x}}, \tau)- \frac{\dot{a}}{a} \dot{\beta}({\mathbf{x}}, \tau), \end{align} \tag{ 32a }$$

$$\begin{align} & \phi({\mathbf{x}}, \tau)= +\frac{1}{6} h({\mathbf{x}}, \tau)+ \frac{1}{3} \nabla^2 \beta({\mathbf{x}}, \tau)+ \frac{\dot{a}}{a} \dot{\beta}({\mathbf{x}}, \tau). \end{align} \tag{ 32b }$$

Now, setting $\Phi = \Psi$, as above stated, and recalling equation (27), we obtain

$$\begin{align} \ddot{\beta}({\mathbf{x}},\tau)+\frac{\dot{\beta}({\mathbf{x}}, \tau)}{1-H_I\tau}H_I+\frac{\epsilon H_I}{2(1-H_I\tau)}\ e^{i {\mathbf{q}} \cdot {\mathbf{x}}}=0, \end{align} \tag{ 33 }$$

whose general solution is

$$\begin{align} \beta({\mathbf{x}}, \tau)= \left( -\frac{\epsilon \tau}{2}+ \frac{c_1 \tau^2}{2}+c_2 \right) \ e^{i {\mathbf{q}} \cdot {\mathbf{x}}}.\end{align} \tag{ 34 }$$

2.3.7. Perturbation potential.

Let us now focus on the values of the integration constants c₁ and c₂. Concerning c₂ it is easy to see, from equation (32a ), that Ψ vanishes at $\tau\rightarrow0$ independently from the value of c₂ that, consequently, is fully-unconstrained. It is straightforward to set $c_2 = 0$ only to reduce our problem complexity. The situation mostly changes concerning c₁. There is no a priori reasons to fix it to a given value and apparently $\beta({\mathbf{x}},\tau)$ turns out to be quadratic in the conformal time. However, a conceptual caveat suggests how to get it. Indeed, subtracting then equation (32b ) from (32a ), we get

$$\begin{align} \ddot{\beta}({\mathbf{x}}, \tau)-\frac{2H_I}{1-H\tau} \dot{\beta}({\mathbf{x}}, \tau)+ \frac{h({\mathbf{x}}, \tau)}{6}=0 \end{align} \tag{ 35 }$$

leading to

$$\begin{align} h({\mathbf{x}}, \tau)= -6 \ddot{\beta}+ \frac{12 H_I}{1-H_I \tau} \dot{\beta}. \end{align} \tag{ 36 }$$

Here, $h({\mathbf{x}},\tau)$ would imply non-vanishing perturbations at $-\infty$ that actually diverge, as due to the first-order β time-derivative ²² . This fact appears clearly unphysical as we require perturbations to occur during and after inflation, rather than before. Plausibly we are thus forced to set $c_1 = 0$ to avoid any possible issue. Hence, we get from equation (37)

$$\begin{align} h({\mathbf{x}}, \tau)= -\frac{6 \epsilon H_I}{1-H_I\tau} e^{i {\mathbf{q}} \cdot {\mathbf{x}}} = -12 \Psi. \end{align} \tag{ 37 }$$

On super-Hubble scales, the term $h_{ij}^\parallel$ can be neglected. The perturbation tensor in synchronous gauge then reads

$$\begin{align} h_{\mu \nu}= \begin{pmatrix} 0 & 0 & 0 & 0\\ 0 & 4 \Psi & 0 & 0 \\ 0 & 0 & 4 \Psi &0 \\ 0 & 0 & 0 & 4 \Psi \end{pmatrix}, \end{align} \tag{ 38 }$$

from which the line element

$$\begin{align} \textrm ds^2=a^2(\tau)\left[\textrm d\tau^2-\delta_{ij}(1-4\Psi)\textrm dx^i \textrm dx^j \right]. \end{align} \tag{ 39 }$$

We are now ready to compute the corresponding geometric particle production.

In the external-field approximation, we can describe the interaction of the inflaton with spacetime geometry at first perturbative order via the Lagrangian [12]

$$\begin{align} \mathcal{L}_I= -\frac{1}{2} \sqrt{-g_{(0)}} H^{\mu \nu}T_{\mu \nu}^{(0)}\,, \end{align} \tag{ 40 }$$

where $g_{\mu \nu}^{(0)}\equiv a^2(\tau) \eta_{\mu \nu}$, $H_{\mu \nu} = a^2(\tau)h_{\mu \nu}$ and $T_{\mu \nu}^{(0)}$ is the zero-order energy-momentum tensor, namely

$$\begin{align} T_{\mu \nu}^{(0)}= &\partial_\mu \phi \partial_\nu \phi- \frac{1}{2}g_{\mu \nu}^{(0)} \left[ g^{\rho \sigma}_{(0)} \partial_\rho \phi \partial_\sigma \phi-2\Lambda^4(1-\phi^2/\mu_2^2) \right] \nonumber\\ &-\frac{1}{6}\left[\nabla_\mu \partial_\nu-g_{\mu \nu}^{(0)}\nabla^\rho \nabla_\rho+R_{\mu \nu}^{(0)}-\frac{1}{2} R^{(0)}g_{\mu \nu}^{(0)} \right]\phi^2. \end{align} \tag{ 41 }$$

The first-order $\hat S$-matrix can be obtained by Dyson's expansion formula (see e.g. [44])

$$\begin{align} \hat S=\hat T\exp^{-i\int_{-\infty}^{+\infty}d^4x \mathcal H_I}, \end{align} \tag{ 42 }$$

where $\mathcal{H}_I$ is the Hamiltonian density in interacting picture and $\hat T$ the time-ordering operator.

The exponential form of Dyson's expansion is not practical, since the integral in the exponent cannot be computed exactly. We may then expand out equation (42) at first order, recalling that the interaction Hamiltonian is smaller than the background one. As $\mathcal{H}_I = -\mathcal{L}_I$ in our model [45, 46], following the standard procedure in Dyson's expansion we get

$$\begin{align} \hat S \simeq 1+i \hat{T} \int \textrm d^4x\ \mathcal{L}_I. \end{align} \tag{ 43 }$$

Accordingly, the second order particle number density at time $\tau^*$ is

$$\begin{align} N^{(2)}(\tau^*)= \frac{a^{-3}(\tau^*)}{\left(2\pi \right)^{3}} \int \textrm d^3k\ \textrm d^3p\ \lvert \langle 0 \lvert \hat S \rvert k,p \rangle \rvert^2. \end{align} \tag{ 44 }$$

We remark that second order terms are not required in the $\hat S$-matrix expansion (43), since the interaction Lagrangian is still quadratic in the field at second geometric order, thus contributing at higher orders to the particle number density. Moreover, in equation (44) we have assumed that no 'quantum' particle production is involved, namely the Bogoliubov coefficients β_k and β_p obtained in [12] have been neglected. In appendix C we discuss the generalization of equation (44) to the case of non-zero Bogoliubov coefficients. Quantum particle production is also responsible for the generation of particle-antiparticle pairs at zero and first geometric order [12, 16, 29], as we will discuss in section 5.

Coming back to equation (44), the probability amplitude for particle pair creation can be derived from equations (40)–(43), namely

$$\begin{align} \langle 0 \lvert \hat S \rvert k,p \rangle=& -\frac{i}{2} \int \textrm d^4x\ 2a^4 H^{ij} \bigg[ \partial_{(i} \phi_k^* \partial_{j)}\phi_p^*-\frac{1}{2}\eta_{ij} \eta^{ab} \partial_{(a} \phi_k^* \partial_{b)}\phi_p^*+g_{ij}^{(0)}\Lambda^4\left(-\frac{\phi^*_k \phi^*_p}{\mu_2^2}\right) \nonumber\\ & -\frac{1}{6} \left( \nabla_i \partial_j-g_{ij}^{(0)}\nabla^a \nabla_a+R_{ij}^{(0)}-\frac{1}{2} R^{(0)}g_{ij}^{(0)} \right)\phi_k^* \phi_p^* \bigg]\,, \end{align} \tag{ 45 }$$

with $i,j = 1,2,3$ as consequence of working in the synchronous gauge. We have also defined the field modes

$$\begin{align} \phi_k ({\mathbf{x}}, \tau)= \frac{f_k(\tau)}{(2 \pi)^{3/2} a(\tau)} e^{i {\mathbf{k}}\cdot {\mathbf{x}}}, \end{align} \tag{ 46 }$$

which can be derived from equation (11) together with the solutions equation (16).

On super-Hubble scales, these modes can be written as [14]

$$\begin{align} \phi_k= \frac{1}{(2 \pi)^{3/2}} e^{i\left(\nu-\frac{1}{2}\right)\frac{\pi}{2}} 2^{\nu-\frac{3}{2}} \frac{\Gamma(\nu)}{\Gamma(3/2)} \frac{H_I}{\sqrt{2k^3}} \left(\frac{k}{aH_I} \right)^{\frac{3}{2}-\nu} e^{i {\mathbf{k}}\cdot {\mathbf{x}}}. \end{align} \tag{ 47 }$$

Exploiting now the fact that the perturbation tensor is diagonal and writing explicitly all the curvatures, equation (45) can be recast in the compact form

$$\begin{align} \langle 0 \lvert \hat S \rvert k,p \rangle= -\frac{i}{2} \int \textrm d^4x\ 2a^4\left(A_1({\mathbf{x}},\tau)+A_2({\mathbf{x}},\tau)+A_3({\mathbf{x}},\tau) \right), \end{align} \tag{ 48 }$$

where $A_i({\mathbf{x}}, \tau)$ are the only non-zero contributions to the probability amplitude, namely

$$\begin{align} A_i({\mathbf{x}},\tau)=H^{ii}\cdot \bigg[&\partial_i \phi_k^* \partial_i \phi_p^*+\frac{1}{2} \eta^{ab} \partial_a \phi_k^* \partial_b \phi_p^*+a^2 \Lambda^4 \left( \frac{\phi_k^* \phi_p^*}{\mu_2^2}\right) \nonumber\\ & -\frac{1}{6} \left(\partial_i \partial_i+\frac{\dot{a}}{a}\partial_0+\eta^{ab}\partial_a \partial_b-\left(\frac{\ddot{a}}{a}+\left( \frac{\dot{a}}{a} \right)^2 \right)+3\frac{\ddot{a}}{a} \right) \phi_k^* \phi_p^* \bigg]. \end{align} \tag{ 49 }$$

2.4.1. Dark matter from 'geometric particles'?.

With all the above ingredients, we can now compute the final number density of geometric particles produced, namely $N^{(2)}(\tau)$ at τ = 0. As anticipated, these are interpreted in terms of dark matter quasiparticles. Dark matter seems the most plausible candidate in our model, since it only interacts gravitationally with ordinary matter and, in fact, the way of obtaining it derives from the Yukawa-like potential only. We expect that any particle pair creation, got from purely quantum processes, becomes subdominant over quasiparticles obtained directly from vacuum fluctuations [47], as above discussed.

Hence, to determine dark matter microphysics and properties, we first need to specify initial inflationary settings, i.e. to properly define super-Hubble scales, introducing a cut-off scale to have enough e-foldings, say N, that are needful to speed up the Universe during inflation [6], having

$$\begin{align} N= \int\ \textrm dt H(t)= \int\ \textrm d\tau\ \frac{H_I}{1-H_I\tau} \simeq 60, \end{align} \tag{ 50 }$$

where conventionally we took 60 as minimal number of e-foldings. We thus obtain $\log(1-H_I \tau)\lvert_0^{t_I} = 60$, where $t_I\lt0$ is assumed to be the initial time for inflation, and it can be inferred once the fixed values are imposed on our free parameters.

Lying on super-Hubble scales, namely

$$\begin{align} \frac{k}{a(\tau)H_I} \ll 1 \implies k \ll a(\tau)H_I, \end{align} \tag{ 51 }$$

quantum fluctuations of the inflaton field become classical ²³ , i.e. they no longer oscillate in time (cfr equation (47)). In this respect, we can properly get particles only after horizon exit ²⁴ .

Easily, equation (47) is valid throughout all the inflationary epoch, as we take the minimum of aH_I , say $a(\tau_I)H_I$, as required cut-off. This ensures that the field modes are described by equation (47) as $\tau \gt \tau_I$. However, since $a(\tau_I) \propto \exp(-60)$, this choice would result in a very small cut-off for particle momenta. Consequently, from a genuine physical perspective, this issue is healed by assuming that geometric particle production started at $t_i \gt t_I$, i.e. not exactly at the beginning of the inflationary era. In our computation we can show that, in view of our effective potential parameters, realistic values for the dark matter number density may be obtained within the range $t_i \in [-\exp(45)/H_I, -\exp(40)/H_I]$. For completeness, however, we remark that our choice of time-independent cut-off inevitably leads to underestimating the total number density. This happens because we essentially neglect all the momenta whose horizon crossing is subsequent the time τ_i .

2.4.2. Constraints on the effective potential.

Concerning the requirements of our effective potential, we invoke the following basic demands.

• Since inflation is thought to follow a quantum gravity regime, we expect vacuum energy scales to lie on Planck mass scales, namely
  $$\begin{align} \Lambda^4 \simeq M_{\mathrm{pl}}^4, \end{align} \tag{ 52 }$$
  where $M_{\textrm{pl}} = 1.22 \times 10^{19}$ GeV is the Planck mass. This ansatz agrees with current understanding about the value of the cosmological constant as predicted by quantum field theory fluctuations [1].
• The corresponding slightly evolving Hubble rate during inflation is therefore
  $$\begin{align} H^2(t)\equiv H^2_I\simeq \frac{8 \pi{G}}{3} \Lambda^4, \end{align} \tag{ 53 }$$
  and Planck satellite data [26] impose the following constraint (at a 95% confidence level):
  $$\begin{align} \frac{H_I}{M_{\textrm{pl}}} < 2.5 \times 10^{-5}, \end{align} \tag{ 54 }$$
  which accordingly would give $\Lambda^4 \lesssim 10^{65}$ GeV⁴. In particular, this energy scale for vacuum energy is the typical regime of spontaneous symmetry breaking in grand unified theories [48, 49].
• The minimally coupled hilltop quadratic potential requires [26]
  $$\begin{align} 0.3 < \log_{10} (\mu_2/M_{\textrm {pl}}) < 4.85, \end{align} \tag{ 55 }$$
  namely $2\ M_{\mathrm{pl}} \lesssim \mu_2 \lesssim 10^5\ M_{\mathrm{pl}}$. Our effective potential, instead, includes additional field-curvature coupling contribution, that provide relevant consequences on inflation. Nevertheless, in case of conformal coupling, large Λ⁴ values would result in vacuum energy domination. Hence, it appears licit to consider the prescription of equation (55) in our computation as prior for equation (7).

Concerning the choice of the slow-roll parameter, we have previously discussed that small deviations from a pure de Sitter evolution are required in order to have a non-zero ε. Since we are dealing with inhomogeneities at a perturbative level, we also have to satisfy [12, 29]

$$\begin{align} \lvert h_{ij}(x) \rvert \ll 1. \end{align} \tag{ 56 }$$

Hence, by virtue of equation (27) we see that in order to preserve the perturbative treatment, we further need $\epsilon H_I \ll 1$. In this respect, we draw in figure 1 the number density of geometric particles, namely $N^{(2)}(0)$, for given values of the hilltop parameter µ₂. In figure 2 we show the dependence of the number density on the vacuum energy term driving inflation, Λ⁴.

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Zoom In Zoom Out Reset image size

2.4.3. Cut-off scales and vacuum energy.

A further inspection of equations (17) and (47) reveals that the modes of the field become exactly 'frozen' on super-Hubble scales if $\nu \simeq 3/2$, namely

$$\begin{align} \mu_2 \simeq 0.42 \times 10^{19}\ {\mathrm{GeV}}= 0.34\ M_{\mathrm{pl}}. \end{align} \tag{ 57 }$$

This value is well outside the range provided in equation (55) that, however, we know to be valid only in case of minimally-coupled inflaton. Smaller µ₂ values generally lead to a larger number density. Accordingly, if we require equation (57) to hold in general, then two possibilities arise:

• The cut-off scale on particle momenta is forced to be much smaller, in order to preserve realistic values for the number density $N^{(2)}(0)$;
• Alternatively, vacuum energy should be several orders of magnitude below Planck energy scale, i.e. closer to the scales of the particle physics standard model.

The latter possibility is still an open and promising scenario. Constraining $N^{(2)}(0)$ requires to know the cut-off scale that a priori cannot be known. Disclosing how to constrain the vacuum energy cut-off scales deserves further investigation and will be object of future works.

We featured the inflationary epoch by using the de Sitter solution of equation (9). This represented a suitable approximation that, however, fails to be predictive during the reheating transition, i.e. as inflation ends. In particular, at the end of inflation equation (53) no longer holds and the behavior of the scale factor is not determined by the sole vacuum energy, Λ⁴. Instead, it depends on the full effective potential V^eff that, consequently, should be evaluated in toto.

In particular, the potential V^eff of equation (7) also includes the coupling to the Ricci scalar curvature, which is crucial to interpret geometric particles as dark matter quasiparticles. However, this interacting term grows as φ increases. By construction, it grows when the hilltop potential evolves towards its minimum. Accordingly, the presence of such coupling would not allow a graceful exit from inflation, since the full potential never reaches its minimum. In principle, this may suggest that a more complicated version the single-field inflationary potential would be required in order to properly address the transition from inflation to reheating. For instance, in [7] a Morse potential reducing to the Starobinski one is investigating, unifying de facto inflation with dark energy.

2.5.1. Approximating the reheating phase.

Here, as naive estimation we can assume that during reheating the background geometry behaves in average as a matter dominated Universe [35, 50] and so, accordingly to this hypothesis, we select a scale factor that fulfills an Einstein–de Sitter (EdS) Universe dominated by matter through

$$\begin{align} a(\tau)= \left( 1+ \frac{H_I \,\tau}{2} \right)^2,\ \ \ \ \ 0 < \tau < \tau_{\textrm r} \end{align} \tag{ 58 }$$

where we denote with $\tau_{\textrm r}$ the time at which reheating is expected to end.

We now need continuity of each epoch, without passing through any transition. To do so, we notice that equation (58) ensures the validity of the matching conditions [51] at time τ = 0, essentially implying the continuity of the scale factor and Hubble parameter on the junction hypersurface. For τ > 0, the zero-order scalar curvature takes the form:

$$\begin{align} R^{(0)}= \frac{3 H_I^2}{(1+\frac{H_I\tau}{2})^6},\ \ \ \ \ 0 \lt \tau \lt \tau_\textrm r \end{align} \tag{ 59 }$$

and introducing the usual slow-roll parameter

$$\begin{align} \epsilon= \frac{1}{16 \pi{G}}\left(\frac{V^{\prime}}{V} \right)^2, \end{align} \tag{ 60 }$$

we can exploit equation (7) and set ε = 1 to obtain a realistic value $\phi_{\mathrm{end}}$ for the field at the end of inflation. For $R^{(0)}(\tau \gt10^{-12})$ the hilltop contribution dominates over the field-curvature coupling in $V^{\,\mathrm{eff}}$, and we expect a result close to the one obtained for minimally coupled hilltop models [52], namely

$$\begin{align} 2q^2 \frac{(\phi_{\mathrm{end}}/\mu_2)^2}{(1-(\phi_{\mathrm{end}}/\mu_2)^2)^2}=1\,, \end{align} \tag{ 61 }$$

having defined $q \equiv M_{\mathrm{pl}}/\mu_2$. We obtain the following expansion for small q,

$$\begin{align} \phi_{\mathrm{end}}= \mu_2\left(1-\frac{1}{\sqrt{2}}q + \frac{1}{4}q^2+ \mathcal{O}(q^3) \right)\,, \end{align} \tag{ 62 }$$

so that we have a nonzero remaining contribution to the hilltop component of the potential. We expect this remaining contribution to the potential to be responsible for baryonic particle creation, as usually discussed in preheating and reheating models (see e.g. [50]).

2.5.2. Approximating the radiation dominated phase.

As the reheating stops, the subsequent phase of radiation domination can be modeled by a scale factor of a EdS Universe of the form

$$\begin{align} a(\tau)=b\tau+c,\ \ \ \ \ \tau_{\textrm r} < \tau < \tau_\textrm m \end{align} \tag{ 63 }$$

where the constants b and c are determined by again imposing the matching conditions on the matching hypersurface from reheating to radiation phase, at time $\tau = \tau_\textrm r$. The quantity $\tau_\textrm m$ denotes the instant of time at which transition to the matter-dominated era is expected to happen. During radiation domination, the corresponding EdS Hubble parameter and temperature satisfy [48]

$$\begin{align} H &\sim \sqrt{G}\,T^2, \end{align} \tag{ 64 }$$

where T is the corresponding temperature of the Universe. The expected dark matter energy density at $\tau_\textrm r$ is then ²⁵

$$\begin{align} \rho^{\mathrm{DM}}\left (\tau_{\textrm r}\right) &\simeq (1+z_{\textrm r})^3 \rho^{\mathrm{DM}}_0, \end{align} \tag{ 65 }$$

where $\rho_0^{\mathrm{DM}}$ is the current value got at redshift z = 0, namely $\rho^{\mathrm{DM}}_0\simeq 0.25\ \rho_{\mathrm{cr}}$, with $\rho_{\mathrm{cr}}\equiv 3H_0^2/ 8\pi{G}$ and H₀ is the Hubble constant. Further, we introduced the redshift $z_\textrm r$ that certifies the beginning of the radiation phase. Since we are dealing with the radiation-dominated epoch, $z_\textrm r$ can be obtained within a EdS Universe dominated by radiation only, i.e.

$$\begin{align} H(z) \simeq H_0^2 \Omega_{0\textrm r} (1+z)^4. \end{align} \tag{ 66 }$$

Here, $\Omega_{0\textrm r}$ is the today radiation density, say $\Omega_{0\textrm r}\equiv \rho_0^\textrm r/\rho_{\mathrm{cr}} \simeq 9.29 \times 10^{-5}$ [53]. Using now the ansatz made in equation (64), we get

$$\begin{align} z_{\textrm r}\equiv z\left(T_{\textrm r}\right) \simeq \sqrt[4]{\frac{G T^4_{\textrm r} }{H_0^2\ \Omega_{0\textrm r}}}, \end{align} \tag{ 67 }$$

where $T_{\textrm r}$ is the temperature corresponding to $z_\textrm r$.

The last item, essential for our purposes, occurs because at the end of inflation the energy of the initial inhomogeneous Universe is much smaller than vacuum energy. The energy lost, and transformed into geometric particles, is responsible for $\rho^{(1)}$ magnitude. This fact can naturally fix the fine-tuning issue today. Indeed, if $\rho^{(2)}$ is equal to the density at the end of inflation, i.e. without the degrees of freedom of quantum vacuum energy, the fine-tuning issue is not a well-posed problem, but rather it turns out to be naturally overcome. Accordingly, since $\rho^{(2)} = \rho^{(1)}$ if we get $P^{(2)}-P^{(1)}\leqslant \rho_{\textrm {cr}}$ then the two magnitudes of pressure would be comparable and so $P^{(2)}$, i.e. the pressure today, will be proportional to the matter density at late-times alleviating the coincidence problem [55].

Following the nomenclature of section 1, we can schematically sketch the corresponding net values reached by Λ as:

$$\begin{align} \Lambda\simeq\begin{cases} \rho_{\textrm {vac}},\quad & \mathrm{Before}\ \mathrm{inflation}, \\ \Lambda_B,\quad &\mathrm{After}\ \mathrm{inflation}. \end{cases} \end{align} \tag{ 69 }$$

In this picture, $\Lambda_B$ is the magnitude inferred once we evaluate $P^{(1)}-P^{(2)}$. Thus, the bare cosmological constant arises since not all the vacuum energy is cancelled. In fact, the pressure jump, namely $P^{(1)}-P^{(2)}$ is due to the fact that vacuum energy is not completely fine-tuned to give geometric particles, but rather a (large) fraction of it provides particles. Hence, if we indicate with $\Lambda_{\textrm {geom}}\leqslant \rho_{\textrm {vac}}$ the amount used for getting particles, we conclude

$$\begin{align} \Lambda_B=\rho_{\textrm {vac}}-\Lambda_{\textrm {geom}}. \end{align} \tag{ 70 }$$

In the above equation, we neglected the fact that, at the end of inflation, there is also a remaining contribution due to the inflaton potential, since $\phi_{\mathrm{end}}/\mu_2$ may have small deviations from unity, as reported in equation (62). However, as already explained, we expect this contribution to be responsible for ordinary baryonic matter production during reheating and, for this reason, it is not involved in our argument here.

Equation (70) is then true if inflation ends before cancelling completely the overall vacuum energy pressure, leaving a residual constant pressure to contribute the spatial part of the energy momentum tensor after inflation, being proportional to $\Lambda_B$. In such a picture, $\Lambda_B$ is therefore reinterpreted as the difference of pressures before and after the transition that is associated to the particle production. This mechanism fully-agrees with the one presented in [13], but the here-adopted hilltop potential differs from the one prompted in [7].

It is finally useful to remark that we use the end of inflation, assuming that $\epsilon\rightarrow1$, to compute the contribution of $\Lambda_{\textrm {geom}}$. This furnishes the remaining contribution to the potential that leads to baryonic particle creation. The corresponding value for $\Lambda_{\textrm {geom}}$ is therefore not fine-tuned but determined by when the inflationary time ends.

We now consider the transition from the inhomogeneous inflationary scenario to the matter-dominated reheating previously discussed. We write for the two spacetimes the corresponding line elements to hold ²⁶

$$\begin{align} g_1&=a_1^2(\tau_1)\left[\textrm d\tau_1^2-\delta_{ij}\Gamma^2\,\textrm dx^i \textrm dx^j \right], \end{align} \tag{ 71a }$$

$$\begin{align} g_2&=a_2^2(\tau_2)\left[\textrm d\tau_2^2-\delta_{ij}\textrm dX^i \textrm dX^j \right], \end{align} \tag{ 71b }$$

with $\Gamma\equiv \sqrt{1-4\Psi}$ and a priori $\tau_1\neq\tau_2$, $x_i\neq X_i$. Here the perturbed metric g₁ is associated to equation (39) whereas g₂ is the current spatially-flat homogeneous and isotropic FRW spacetime, still valid up to our time by simply fulfilling the cosmological principle. We evaluate the Israel–Darmois junction conditions and we assume the following recipe:

• We measure time regardless the cosmological epoch, leading to $\tau_1 = \tau_2$;
• The angular part of both the spacetimes remains unaltered before and after the matching.

The equivalence of the metric tensors (71a ) and (71b ) induced on τ = 0 (chosen as the junction time between the two phases) gives that both spatial line elements must coincide, $x_i = X_j$, up to radial rescaling. This in turn implies a condition that the metric coefficient Γ containing the potential Ψ must satisfy on the matching hypersurface τ = 0:

$$\begin{align} \Gamma=\frac{a_2(0)}{a_1(0)}. \end{align} \tag{ 72 }$$

Moreover, equivalence of the two extrinsic curvatures gives the further condition

$$\begin{align} \Gamma_{,\tau}=H_2(0)-H_1(0), \end{align} \tag{ 73 }$$

where $\Gamma_{,\tau}\equiv\frac{\partial\Gamma}{\partial\tau}$, evaluated at τ = 0 again.

When we discussed about reheating time, we required matching continuity of our functions. So, in analogy, assuming the Universe not to pass through any transition and/or discontinuity, we take its size and radius to be continuous. Consequently, from (72) and (73), it is licit to write down

$$\begin{align} a_2(0)&=a_1(0)\,, \end{align} \tag{ 74a }$$

$$\begin{align} H_2(0)&=H_1(0)\,. \end{align} \tag{ 74b }$$

From equations (72) and (73), by virtue of the above relations, we get the intriguing fact that Ψ must be constant on the junction hypersurface, in order to permit the matching between the two spacetimes to occur.

By construction from Einstein's equations, one expects the pressure term to be proportional to the second derivative of Γ with respect to τ, namely $\Gamma_{,\tau\tau}$. If the pressure difference between the first and second stage of our spacetimes is proportional to ρ_cr, then the coincidence problem would be alleviated.

Bearing in mind equations (72) and (73) with the recipe of equations (74a ) and (74b ), we therefore obtain the $G^{1}_{1}$ component of Einstein's tensor, $G^{\mu}_{\nu}\equiv R^{\mu}_{\nu}-\frac{1}{2}\delta^{\mu}_{\nu}R$, namely the pressure, as follows

$$\begin{align} P=-\frac{1}{8\pi{G}}\left[\frac{(a^{\prime}(\tau))^2}{a(\tau)^4}-2\frac{a^{^{\prime\prime}}(\tau)}{a(\tau)^3}-2\frac{\Gamma_{,\tau\tau}}{a(\tau)^2}\right]\,. \end{align} \tag{ 75 }$$

Immediately, assuming a matter dominated EdS Universe after inflation, we find

$$\begin{align} \Delta \rho&=0\,, \end{align} \tag{ 76a }$$

$$\begin{align} \Delta P&=-3H_I^2-2\Gamma_{,\tau\tau}\,, \end{align} \tag{ 76b }$$

where $\rho^{(1;2)}\equiv G_{0}^{0,(1;2)}$ and $P^{(1;2)}\equiv-\frac{1}{8\pi{G}} G_{1}^{1,(1;2)}$, i.e. density and pressure respectively whereas the density and pressure shifts are, by definition, $\Delta \rho\equiv\rho^{(1)}-\rho^{(2)}$ and $\Delta P\equiv P^{(1)}-P^{(2)}$.

As a consequence, requiring $\Delta P\sim \rho_{\textrm {cr}}$ provides

$$\begin{align} \Gamma_{,\tau\tau}\simeq \frac{\rho_{\textrm {cr}}-3H_I^2}{2}\,. \end{align} \tag{ 77 }$$

Forcing $\Gamma_{,\tau\tau}$ to vanish suggests that $H_I^2\simeq \rho_{\textrm {cr}}$. Since this value is the today critical density previously introduced, it is quite likely that $H_I^2$ should be close to this value immediately after inflation. This heuristic proof is supported by the fact that, although we denoted with H_I the inflationary Hubble rate, it is not exactly constant throughout inflation and in particular its value is much smaller than the one in equation (53) as inflation is ending. Hence, its value, once the process of geometric particle production ends, is proportional to the current critical density as a consequence of our cancellation mechanism.

Accordingly, we infer that

• On the surface τ = 0, vacuum energy cancellation is associated to a minimum of the Γ function, as $\rho_{\textrm {cr}}\gt3H_I^2$;
• The total energy density is constant on τ = 0, i.e. $\rho^{(1)} = \rho^{(2)}$, implying energy conservation;
• The pressure shift suggests that the corresponding fluid evolves as a dark fluid [56], mimicking the predictions presented in [13].

As a consequence of our recipe, one argues that the standard cosmological model, i.e. the ΛCDM paradigm, is modified because, at the end of our process, we can model the corresponding total fluid as a single fluid of matter whose pressure is not exactly zero, but is constrained to current value, called before $\Lambda_B$. Thus, the fine-tuning issue is no longer a real problem because quantum fluctuations associated to Λ are removed by virtue of our cancellation mechanism.

The value of $\Delta P$, however, is fixed at τ = 0. It is natural to wonder whether it remains constant throughout the evolution of the Universe at late times, namely $\tau\rightarrow \infty$ or not. For the sake of simplicity, we may assume it to be constant without any time evolution for pressure, albeit we cannot exclude the pressure to vary at late-times. In the case of non-varying pressure, then the model reduces to the one presented in [13] with the great advantage to physically-explain how density, cancelled out by the mechanism, transforms to new species of particles.

To evaluate $\Delta \rho$ and $\Delta P$ we made the ansatz of having a matter dominated EdS Universe, characterized therefore by $P^{(2)} = 0$. However, shifting to a radiation dominated EdS Universe we again would get $\Gamma_{,\tau\tau}\simeq H_I^2$, implying that, at τ = 0, our model is not particularly influenced by choosing either matter or radiation. Then, by virtue of the continuity equation one computes a constant density that resembles the ΛCDM model, exhibiting a very different physical interpretation over the constant that fuels the Universe to speed up today.

This theoretical scheme works if Γ is constant on the hypersurface τ = 0. Since inflation ends, requiring a perfect homogeneous and isotropic Universe, one argues negligible Ψ at the end of inflation, say $\Psi\rightarrow0$ as $\tau\rightarrow0$. From equation (28), assuming $\rho_{\textrm {cr}}-3H_I^2 = \varepsilon\rho_{\textrm {cr}}$, we thus have

$$\begin{align} \ddot{\mathcal{F}}(\epsilon,0,H_I)=-\frac{\varepsilon}{2}\rho_{\textrm {cr}}\ e^{-i{\mathbf{q}}\cdot {\mathbf{x}}}\,, \end{align} \tag{ 78 }$$

where ε is a unknown constant that quantifies the deviation between H_I and ρ_cr. Afterwards, involving equation (27) and assuming the slow roll parameter to vanish after inflation in order to fulfill $\mathcal F = \dot{\mathcal{F}} = 0$, we get $\ddot{\mathcal{F}} = 0$, implying $H_I^2 = \rho_{\textrm {cr}}/3$, again addressing the coincidence problem ²⁷ . Clearly, a more suitable choice of $\mathcal F$ is required to guarantee that $\mathcal F = \dot{\mathcal F} = 0$ and $\ddot{\mathcal F} \neq0$ in general and this implies to select a more suitable version of the effective potential, instead of our hilltop quadratic one corrected by a Yukawa-like term involving a coupling with curvature.

In general, however, addressing this issue is a central problem related to any inflationary scenarios [6], whereas the here-employed potential only represents a first proposal to work out our model of dark matter production. The search for a more suitable version of the effective potential, however, requires to exit from inflation. This may be jeopardized by the coupling with curvature that, albeit it becomes negligibly small, is assumed to be small enough to guarantee $\epsilon\rightarrow1$ immediately before the jump to $\epsilon\rightarrow0$. Hence, a more suitable choice of the underlying potential would give new insights toward a graceful exit from inflation and at the same time the geometric production of dark matter particles. On the other side, we believe the need of curvature coupling is essential to interpret the corresponding dark matter fluid. In fact, if no coupling with curvature occurs, then the interpretation of particle production cannot be geometrical and only baryons can form during reheating as byproduct of the scalar field alone. We stressed such considerations throughout the text previously and we here underline that the more particles are produced from geometry the more coupling with R is clearly needful, i.e. to fix the current dark matter abundance one has to invoke a further coupling.

In view of the aforementioned prescriptions, our corresponding dark energy scenario can be modeled using a dark fluid [57, 58], effectively compatible with the one presented in [7, 13, 59, 60]. This fluid can be interpreted as a single fluid of matter with pressure, where the pressure is furnished by the additional bare cosmological constant. Indeed, if zero-point fluctuations are cancelled, leaving a $\Lambda_B\neq 0$, the remaining Universe density would be associated to matter (and radiation, clearly), but the corresponding pressure would be given by the sum of $\Lambda_B$ and the pressure of dust and radiation. As the Universe expands, radiation dominates over matter and $\Lambda_B$. But, since $\Lambda_B$ magnitude is comparable with matter, once the matter epoch finishes then $\Lambda_B$ tends to dominate over dark matter and baryons, reproducing de facto the behavior of current cosmological model. Accordingly, we can quantify the pressure throughout the Universe evolution as follows:

• During inflation, our choice of the hilltop potential leads to vacuum energy (ρ_vac) domination, resulting in a large and negative pressure. This is a common trait to all inflationary models, and of course implies the violation of the strong energy condition [14]. In this phase, the corresponding Universe dynamics is then described by a quasi de Sitter solution, whose deviations from the pure de Sitter are due to the inflaton fluctuations.
• At the end of inflation, a large part of vacuum energy has been transformed into geometric particles, while the remaining contribution is responsible for ordinary baryonic production. The pressure associated to geometric particles ($-\Lambda_B$) has been computed in section 4.2 and corresponds to the transition from the inhomogeneous quasi-de Sitter phase to a matter-dominated one, which represents a well-known simplified scheme for reheating. Clearly, the strong energy condition is here restored.
• After reheating, we find the usual radiation and then matter eras. In such phases the total pressure is due to dust, radiation and the $\Lambda_B$ contribution. However, the latter is expected to be small compared with radiation at early stages, being compatible with the standard Big Bang model.
• At late times, matter and radiation becomes subdominant with respect to the bare cosmological contribution, whose negative pressure is expected to drive the current Universe expansion. This implies a new violation of the strong energy condition, which however overcomes the coincidence and fine-tuning problems due to the geometric origin of such pressure, as previously discussed.

In table 2 we summarize the phases described above, specifying the corresponding pressure due to the dominant fluid in each phase.

Table 2. Summary of the phases of the Universe evolution and corresponding value of the pressure in our model.

	Phase	Pressure
$\tau_I \lt \tau \lt 0\ \ \$	Inflation$\ \$	$P \sim - \rho_{\mathrm{vac}} = - \Lambda^4$
$0 \lt \tau \lt \tau_\textrm r\ \ \$	Reheating$\ \$	$P \sim 0$
$\tau_\textrm r \lt \tau \lt \tau_\textrm m\ \ \$	Radiation era$\ \$	$P \sim \rho_{\mathrm{rad}}(\tau)/3$
$\tau \gt \tau_\textrm m\ \ \$	Matter era$\ \$	$P \sim 0$
$\tau \rightarrow +\infty\ \ \$	Bare CC domination$\ \$	$P \sim -\Lambda_B$

Hence, the here-depicted overall paradigm fully-degenerates with the ΛCDM model, being however physically highly-different from it.

Another feature of our model is to assume geometric particles to dominate over any quantum mechanisms of particle creations, as discussed in section 2. This contribution is usually computed assuming the Universe to expand from an asymptotically flat region (in) to another one (out), and computing the corresponding Bogoliubov coefficients for ladder operators. Details of such calculations are described in appendix C.

There, asymptotic flatness is required in order to properly define the notion of particle (and vacuum), which is not unique in curved spacetime. Including quantum particle production in our framework would imply:

• The production of particle-antiparticle pairs at zero and first geometric order [12, 16]. So, as already noted in section 2.4, these particles can annihilate, without having enough time to significantly contribute to the net dark matter budget of the Universe;
• An additional contribution to the second-order number density, equation (44), depending on the Bogoliubov coefficients β_k and β_p , would enter dark matter production. This contribution is always positive (cfr equation (C8)) and for this reason it would affect dark matter production increasing the total number density $N^{(2)}(0)$.

Concerning the last item above, we underline that when dealing with de Sitter spacetimes, the main conceptual problem of equation (9) is that it can describe an asymptotically flat Universe only in remote past, $\tau \rightarrow - \infty$ but not around τ = 0.

In other words, we do not have an asymptotically flat out region. This issue is discussed in detail in [36], where the authors show that if de Sitter spacetime is extended also to $0 \lt \tau \lt \infty$, asymptotic flatness is recovered at $\tau \rightarrow + \infty$ and no quantum particle production occurs ²⁸ . Of course, this approach cannot be employed in realistic models of Universe evolution, since it would neglect the EdS phases subsequent to inflation.

Alternatively, one could imagine that after the usual transition from inflation to radiation/matter domination, the Universe finally reaches an adiabatic regime at late times, where the notion of particle becomes meaningful again [64, 65]. In [65] the authors show that in this framework 'quantum' particle production from vacuum is non-negligible. This is true in particular for super-Hubble modes $k \ll a_{\mathrm{end}} H_I$, where $a_{\mathrm{end}}$ is the scale factor at the end of inflation. They also notice that the particle abundance is larger if the inflationary energy scale is of the order of 10¹⁶ GeV, which fully agrees with the here-considered scales.

As discussed above, we thus expect that non-zero Bogoliubov coefficients will increase the total number density of particles produced at second geometric order. The possible inclusion of such 'quantum' particle production in our framework will be subject of future works, albeit it is expected, in view of our above considerations, as a fraction of dark matter, rather than the main constituent.

A key assumption in our model is that dark matter arises as a geometric quasi-particle as a consequence of the coupling between inflaton field and spacetime curvature, without any further quantum couplings. So, our scenario does not involve the concept of weakly interacting massive particles (WIMPs) as derived from effective extensions of the particle physics standard model. However, the physical meaning of our geometric particles shows very stable configurations directly induced by gravity and, by construction, interacting with other objects only under the action of gravity/geometry. Better saying, we can figuring out sorts of weakly interacting geometric particles (WIGEPs) that, differently from WIMPs [66], may have a collective behavior in making structures to form sharply in the very early Universe, without extending the standard model of quantum field theory. These particles cannot form at more recent epochs, by virtue of the cosmological principle, i.e. when no perturbations are involved the WIGEP mechanism is suppressed. So, summing up, our WIGEP would

• Be stable immediately after the Big Bang, as a priori they do not exhibit charges and thus they do not interact electromagnetically;
• Have been created in a very large amount as consequence of deleting out vacuum energy that transforms into geometric particles, due to inhomogeneities;
• Behave as collective particles in order to create enough overdensities capable of having galaxies as today we observe ²⁹ by virtue of the perturbed spacetime involved into computation.

So, since our particles behave in a very similar way than previous expectations, despite non being WIMPs, it is possible to confront the kinds of interactions developed in previous literature, namely cosmologically-stable dark matter particles and standard model (SM) particles. These models mainly consider spin-0, spin-1/2 and spin-1 dark matter candidates interacting with SM fields (mostly fermions) through spin-0 or spin-1 mediator fields, usually dubbed portals ³⁰ .

Among the plethora of plausible portals, the simplest scenario is represented by the so-called standard model portals, in which dark matter interacts with the standard model of particle physics through the Higgs or the Z-boson. These portals have only two free parameters, i.e. the dark matter and portal mediator masses. For their simplicity, they are highly-predictive, albeit disfavored in some experimental limits [67]. The scheme which more closely resembles our approach includes the scalar Higgs boson. So, without assuming any charge conjugation and parity (CP) symmetry violation and taking into account a scalar field χ that describes dark matter, the corresponding interacting Lagrangian becomes

$$\begin{align} \mathcal{L}_I=\xi_0 \lambda_\chi^H \chi^* \chi H^\dagger H, \end{align} \tag{ 79 }$$

where $H = \left( 0\ \frac{v_h+h}{\sqrt{2}} \right)$ is the Higgs boson doublet, not to be confused with the Hubble rate, in unitary gauge [44], h the excited field, v_h the ground state and $\xi_0 = 1/2\ (1)$ in case dark matter is (not) its own antiparticle.

Confronting the two expectation limits for WIMPs and WIGEPs would therefore be extremely instructive to exclude mass ranges dropped out by observations. Hence, to do so we draw an excluding plot, see figure 6, where we show the mass value $m^*$ of our geometric dark matter candidate as function of the hilltop parameter µ₂. There, a blue region that delimits the mass values that have been excluded by recent LUX limits [66, 68] is prompted, as predicted by the above Higgs portal mechanism. Conventionally, for the sake of simplicity we have set $\xi_0 = 1$ and $\lambda_\chi^H = 1/6$ in analogy with our scheme of field-curvature coupling ³¹ . In case of large dark matter masses, we notice excluded regions lying inside

$$\begin{align} 1\ \mathrm{GeV} \lesssim m^{*,\textrm {excluded}} \lesssim 400\ \mathrm{GeV}\,, \end{align} \tag{ 80 }$$

where the superscript indicates the blue zone in figure 6. Consequently, by looking at our model, equation (80) would exclude hilltop µ₂ values approximately inside the range

$$\begin{align} \mu_{2}^{\textrm {excluded}} \in [4,8]\,. \end{align} \tag{ 81 }$$

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We may therefore easily conclude that the WIGEPs may lie either in very large intervals of masses that are, somehow, similar to those predicted by the so-called WIMPzillas [69] or in smaller intervals, namely $\lesssim$1 GeV, being compatible with recent axion search [70].

It is remarkable to stress an important point as follows. Slow mass WIGEPs that we predict, or better to say that we cannot exclude, may be less likely since axions and/or in general ultralight fields may arise for other different processes, being dominant over geometric quasiparticles, see e.g. [71–73] where examples of virtual gravitons associated to the gravity sector can be found. These predictions look extremely similar to our outcomes, since we adopted a Yukawa-like interaction that couples the gravity and scalar field sectors adopting a fast interaction, similar to that discussed in the previous references. Toward our model, since vacuum energy magnitude is huge, it is more likely that massive particles are produced, instead of light fields that would give a number of particles too large, i.e. far from expectations. Moreover, the more massive fields are predicted the more weakly interacting particles are expected, guaranteeing structures to form. We also stress that since dark matter is produced from the degrees of freedom coming from vacuum energy, but nothing has argued about the dynamics of the Universe before and after, such vacuum energy is associated to a symmetry breaking mechanism [2]. In this respect, it has been shown in [13, 24] that the solution of the cosmological constant problem would imply highly massive particles.

In other words, from the one side we do not exclude ultralight fields to contribute to dark matter [74]. On the other hand, however, we propose that the dominant contribution is geometrical, but with higher and more probable masses associated to it.

In all the above treatment, particles have been produced assuming the free parameters to lie in suitable intervals that are compatible with theoretical expectations on vacuum energy and non-excluded regions provided by experiments. However, departures from these bounds may lead to different values of mass candidates, albeit the physical expectations about ultralight and highly-massive fields remain unaltered.

Phrasing it differently, we emphasize that, as characteristics of our proposed quasi-particles are better understood, it is conceivable that stricter allowable mass limits could be found, leaving the possibility that the mass range here discussed will be modified accordingly.

In addition to our analysis, it would be crucial to stress that the quantum cosmological constant problem is not fully-addressed in our treatment. Indeed, even if we solve the classical cosmological constant problem, finding a convincing reason to put the minimum of the potential to zero, contributions from the zero-point fluctuations of all the quantum fields present in the Universe cannot be ignored. They correspond to $\langle 0|T_{\mu\nu}|0\rangle$, with $T_{\mu\nu}$ the full energy momentum tensor, providing then a net contribution to the energy density that might be equated to the dark matter constituent. In our results, however, we did not assume the density provided by such a contribution. Clearly, further developments will focus on this central point.