the predicted reheating temperature is

\begin{equation}\label{eqn:ad122}
T_{R} = 2.11 \times 10^{13}\GeV,
\end{equation}

and using (<ref>) the predicted baryon-to-entropy ratio is

\begin{equation}\label{eqn:ad123}
\frac{n_{B}}{s} = 5.05 \times 10^{-5}.
\end{equation}

The numerical computation gives a baryon-to-entropy ratio for $\tau_{\Phi} = 5T_{asy}$ with $m_{\Phi} = 10^{16}\GeV$, $A^{\frac{1}{2}} = 10^{13}\GeV$, and $\sin 2\theta =1$ of

\begin{equation}\label{eqn:ad124}
\frac{n_{B}}{s} = 4.84 \times 10^{-5},
\end{equation}

which is in very good agreement with the analytical prediction.

Figures <ref> - <ref> show the evolution of the baryon-to-entropy ratio, the comoving condensate asymmetry, and the transferred asymmetry over time in the  $\tau_{\Phi} = 5T_{asy}$ scenario.

Numerically calculated comoving condensate asymmetry, $n_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.

Numerically calculated comoving transferred asymmetry, $\hat{n}_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.

Numerical baryon-to-entropy ratio vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.

Comparison of numerically calculated comoving condensate asymmetry with the analytical estimate from (<ref>), $n_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.

Comparison of numerically calculated comoving transferred asymmetry with the analytical estimate using (<ref>), $\hat{n}_{c}$, vs. $At/m_{\Phi}$ for $\tau_{\Phi} = 5T_{asy}$.

Figure <ref> shows the condensate asymmetry oscillating about zero with decreasing amplitude, until the condensate has completely decayed and the condensate asymmetry is zero. Since in this case we are considering $\tau_{\Phi} > T_{asy}$, this occurs over a large number of phase oscillations of the field, and therefore a large number of oscillations of the asymmetry. Similarly, Figure <ref> shows the transferred asymmetry increasing as an asymmetry is initially generated in the condensate and transferred through the initial inflaton decays. The transferred asymmetry then continues oscillating while the inflaton scalars continue decaying, transferring the asymmetry to the Standard Model until the inflaton condensate has decayed completely. The transferred asymmetry then settles at a small, positive, finite value, corresponding to the total asymmetry transferred to the Standard Model, and thus the baryon-to-entropy ratio shown in Figure <ref>. As in the case of $\tau_{\Phi} = T_{asy}$, we can see from Figures <ref> and <ref>, that the numerically calculated comoving condensate and transferred asymmetries are in perfect agreement with the comoving condensate and transferred asymmetries calculated analytically using the threshold approximation for $\tau_{\Phi} = 5T_{asy}$, demonstrating that the threshold approximation accurately models the behaviour of the condensate and transferred asymmetries, as well as accurately predicting the baryon-to-entropy ratio, both in the case of rapid decay of the inflaton condensate and decay over many oscillations of the asymmetry.

§.§ Validity of the Analytical Approximation of the Decay of the Condensate Asymmetry

In this section we examine the validity of the analytical approximation of the decay of the condensate asymmetry, and thus the validity of the analytical treatment of the threshold approximation. In our treatment thus far, we have assumed that the condensate asymmetry decays as

\begin{equation}\label{eqn:ad125}
n_{c}\left(t \right) = n_{0}\left(t \right)e^{-\Gamma_{\Phi}\left(t - t_{\ast}\right)},
\end{equation}

where $n_{0}\left(t \right)$ is the condensate asymmetry without decay. We will show that this is true if

\begin{equation}\label{eqn:ad126}
\Gamma_{\Phi} << m_{\Phi}^{2},
\end{equation}

\begin{equation}\label{eqn:ad127}
H\Gamma_{\Phi} << m_{\Phi}^{2}.
\end{equation}

We are considering rapid oscillations in the amplitude of the inflaton field, $\phi$, in this model, meaning that $m_{\Phi}^{2} >> H^{2}$, and up to the complete decay of the inflaton condensate we have that $\Gamma_{\Phi} \lesssim H$. From these two conditions, it follows that (<ref>), (<ref>) are satisfied.

We can prove (<ref>) is true under these conditions. As a starting point, we assume that the field solutions in the case of condensate decay are

\begin{equation}\label{eqn:ad128}
\phi_{i}\left(t \right) = \phi_{i0}\left(t \right)e^{-\frac{\Gamma_{\Phi}t}{2}}, \; \; \; i = 1,2
\end{equation}

where $\phi_{i0}\left(t \right)$ is the solution to the field equations without the decay terms. The full field equations with the decay terms are

\begin{equation}\label{eqn:ad129}
\ddot{\phi}_{1} + 3H\dot{\phi}_{1} + \Gamma_{\Phi}\dot{\phi}_{1} = -m_{1}^{2}\phi_{1},
\end{equation}

\begin{equation}\label{eqn:ad130}
\ddot{\phi}_{2} + 3H\dot{\phi}_{2} + \Gamma_{\Phi}\dot{\phi}_{2} = -m_{2}^{2}\phi_{2},
\end{equation}

and in order to verify the analytical approximation of the decay of the condensate asymmetry we need to substitute (<ref>) into the field equations with the decay terms. Using (<ref>) we have that

\begin{equation}\label{eqn:ad131}
\dot{\phi}_{i}\left(t \right) = \left[ \dot{\phi}_{i0}\left(t \right) - \frac{\Gamma_{\Phi}}{2}\phi_{i0}\left(t \right) \right]e^{-\frac{\Gamma_{\Phi}t}{2}},
\end{equation}

\begin{equation}\label{eqn:ad132}
\ddot{\phi}_{i}\left(t \right) = \left[\ddot{\phi}_{i0}\left(t\right) - \Gamma_{\Phi}\dot{\phi}_{i0}\left(t \right) + \frac{\Gamma_{\Phi}^{2}}{4} \right]e^{-\frac{\Gamma_{\Phi}t}{2}}.
\end{equation}

Substituting (<ref>), (<ref>) into the field equations (<ref>), (<ref>) we find that both sides are proportional to $e^{-\frac{\Gamma_{\Phi}t}{2}}$, and can therefore cancel the exponentials. The field equations with decay terms in terms of the field solution (<ref>) are then

\begin{equation}\label{eqn:ad133}
\ddot{\phi}_{i0}\left(t \right) + 3H\dot{\phi}_{i0}\left(t \right) = - \left[m_{i}^{2} - \frac{\Gamma_{\Phi}^{2}}{4} - \frac{3H\Gamma_{\Phi}}{2} \right]\phi_{i0}\left(t \right).
\end{equation}

These are the same as the field equations with no decay term if

\begin{equation}\label{eqn:ad134}
\frac{\Gamma_{\Phi}^{2}}{4} << m_{i}^{2}, \; \; \frac{3H\Gamma_{\Phi}}{2} << m_{i}^{2},
\end{equation}

and if these conditions are satisfied then we can infer that (<ref>) and (<ref>) are also satisfied, and are true for $\Gamma_{\Phi}^{2} << m_{i}^{2}$ and $H^{2} << m_{i}^{2}$. These conditions are generally satisfied for a rapidly oscillating field, which is a basic assumption of the dynamics of this model.

The condensate asymmetry is given by

\begin{equation}\label{eqn:ad135}
n\left(t \right) = \dot{\phi}_{1}\phi_{2} - \dot{\phi}_{2}\phi_{1}.
\end{equation}

Substituting (<ref>) and (<ref>) into the condensate asymmetry (<ref>) we find that

\begin{equation}\label{eqn:ad136}
n\left(t \right) = \left(\dot{\phi}_{10}\phi_{20} - \dot{\phi}_{20}\phi_{10}\right)e^{-\Gamma_{\Phi}t},
\end{equation}

which is simply

\begin{equation}\label{eqn:ad137}
n\left(t \right) = n_{0}\left(t \right)e^{-\Gamma_{\Phi}t}.
\end{equation}

The condensate asymmetry for a decaying condensate is therefore equal to the condensate asymmetry with no decay multiplied by $e^{-\Gamma_{\Phi}t}$, as put forward in (<ref>) in Section <ref>, and the analytical approximation of the decay of the condensate asymmetry is therefore valid in the limit of rapid oscillations, as we assume throughout this model.

§ BARYON ASYMMETRY WASHOUT BY INFLATON EXCHANGE

So far we have assumed that the condensate decays via $B$-conserving processes, and that the mean asymmetry formed in the inflaton condensate is transferred equally to the Standard Model particles. This means that we assume there are no additional processes happening which could remove the asymmetry from the particle plasma. In this section we will consider a $B$-violating process which follows the annihilation of two anti-fermions $\bar{\psi}$ to a virtual inflaton, which then decays to a pair of fermions $\psi$, which can subtract baryon number from the Universe and potentially wipe out the asymmetry. Such a process would arise from inflaton exchange combined with the interaction $\lambda_{\psi}\bar{\psi}^{c}\psi \Phi$, responsible for inflaton condensate decay. The rate of this process is given by

\begin{equation}\label{eqn:ad138}
\Gamma_{\Delta B} = n_{\Phi} \langle \sigma v \rangle,
\end{equation}

where we assume this process is relativistic, $v=1$. At reheating $t = t_{R}$, the number density of the inflaton particles is $n_{\Phi} \sim T_{R}^{3}$, and dimensionally the rate is given by

\begin{equation}\label{eqn:ad139}
\Gamma_{\Delta B} \sim \frac{\lambda_{\psi}^{4}A^{2}T_{R}^{5}}{m_{\Phi}^{8}},
\end{equation}

where inflaton exchange implies an amplitude proportional to $\lambda_{\psi}^{2}A$. This process will not be significant - and therefore washout due to inflaton exchange will not pose a problem for Affleck-Dine baryogenesis with quadratic symmetry-breaking terms - provided that $\Gamma_{\Delta B} < H\left(T_{R}\right)$. Using $H\left(T_{R}\right) \sim T_{R}^{2}/M_{pl}$, this requires that

\begin{equation}\label{eqn:ad140}
\frac{\lambda_{\psi}^{4}A^{2}T_{R}^{5}}{m_{\Phi}^{8}} < H\left(T_{R}\right) \sim \frac{T_{R}^{2}}{M_{pl}}.
\end{equation}

Rearranging (<ref>) gives the following constraint on the coupling $\lambda_{\psi}$

\begin{equation}\label{eqn:ad141}
\lambda_{\psi}^{4} \lesssim \frac{m_{\Phi}^{8}}{M_{pl}A^{2} T_{R}^{3}} \Rightarrow \lambda_{\psi}^{2} \lesssim \frac{m_{\Phi}^{4}}{M_{pl}^{\frac{1}{2}} A T_{R}^{\frac{3}{2}}}.
\end{equation}

Normalising the inflaton mass and the reheating temperature using $m_{\phi} = 10^{13}\GeV$ and $T_{R} = 10^{8}\GeV$, this can be expressed as

\begin{equation}\label{eqn:ad142}
\lambda_{\psi}^{2} \lesssim \left(6.5 \times 10^{4}\right)\left(\frac{m_{\Phi}^{2}}{A}\right)\left(\frac{m_{\Phi}}{10^{13}\GeV}\right)^{2}\left(\frac{10^{8} \GeV}{T_{R}}\right)^{\frac{3}{2}}.
\end{equation}

Provided that this condition is satisfied, the asymmetry will not be washed out by inflaton exchange. This can be rewritten as a constraint on the reheating temperature if we consider the fact that the inflaton decay rate is given by $\Gamma_{\Phi} = \lambda_{\psi}^{2}m_{\Phi}/4\pi $ [132], and $\Gamma_{\Phi} = H_{R}$. The reheating temperature is then

\begin{equation}\label{eqn:ad143}
T_{R} \approx \lambda_{\psi} \sqrt{m_{\Phi} M_{pl}},
\end{equation}

which allows the constraint (<ref>) to be expressed as

\begin{equation}\label{eqn:ad144}
T_{R} \lesssim \left(\frac{m_{\Phi}^{2}}{A}\right)^{\frac{2}{7}}\left(\frac{M_{pl}}{m_{\Phi}}\right)^{\frac{1}{7}}m_{\Phi}.
\end{equation}

Provided this is satisfied, there is little risk of the asymmetry being washed out by inflaton exchange. Since $A << m_{\Phi}^{2}$ and $m_{\Phi} < M_{pl}$, then this is generally satisfied for Affleck-Dine baryogenesis with quadratic symmetry-breaking terms if $T_{R} < m_{\Phi}$. A more in-depth examination of washout would require a specific model of the inflaton decay processes and the transfer of the baryon asymmetry.

§ CONSISTENCY WITH NON-MINIMALLY COUPLED INFLATION

In this section we consider the case of non-minimally coupled inflation with a quartic potential, which is naturally compatible with the potential of the model (<ref>). We explore the conditions needed for the analytic expressions of the baryon asymmetry we have derived in this work to be consistent with non-minimally coupled inflation. To achieve this we need that the inflaton dynamics stop being dominated by the non-minimal coupling before the Affleck-Dine dynamics become significant. This can be stated as requiring that the point of non-minimal coupling domination, $\phi_{c}$, must be larger than the threshold point $\phi_{\ast}$, $\phi_{\ast} < \phi_{c}$. From Section <ref>, we have that once $H^{2} = 4A$, at a point we denote as $\phi_{AD}$, the phase field $\theta$ becomes dynamical, and we need $\phi_{AD} < \phi_{\ast}$ in order for no asymmetry to be produced while the potential is in its $\left|\Phi\right|^{4}$ dominated regime and for the threshold approximation of the asymmetry to hold. We therefore require that $\phi_{AD} < \phi_{\ast} < \phi_{c}$ (see Figure <ref>) in order for this model of Affleck-Dine baryogenesis via quadratic $B$-violating terms as described by the threshold approximation to be compatible with the AD field being a non-minimally coupled inflaton.

Schematic of the inflaton potential illustrating the relationship between the different regimes of the potential as thresholds in the dominant dynamics of the inflaton field, assuming $\phi$ is canonically normalised, producing the characteristic plateau potential of non-minimally coupled models.

In the Palatini formalism, the point at which the non-minimal coupling ceases to be the dominant contribution to the dynamics of the inflaton field is when $\Omega^{2} = 1$. This represents the point at which the canonical inflaton in the Einstein frame becomes equivalent to the Jordan frame inflaton, and the potential is approximately given by the Jordan frame potential. We have that

\begin{equation}\label{eqn:ad145}
\Omega^{2} = 1 + \frac{\xi \phi_{c}^{2}}{M_{pl}^{2}} \approx \mathcal{O}\left(1\right).
\end{equation}

Since this corresponds to the edge of the plateau, we have

\begin{equation}\label{eqn:ad146}
\frac{\xi \phi_{c}^{2}}{M_{pl}^{2}} \sim 1 \Rightarrow \phi_{c} = \frac{M_{pl}}{\sqrt{\xi}},
\end{equation}

in the Palatini formalism.

In the metric formalism, the condition for equivalence of the inflaton between frames is (from the definition of the canonical field transformation in Section <ref>)

\begin{equation}\label{eqn:ad147}
\frac{\Omega^{2} + 6\xi^{2}\phi_{c}^{2}/M_{pl}^{2}}{\Omega^{4}} =1,
\end{equation}

for $\Omega^{2} = 1$. Using the slow-roll inflation condition on the non-minimal coupling, $\xi >> 1$, we have

\begin{equation}\label{eqn:ad148}
\frac{6\xi^{2}\phi_{c}^{2}}{M_{pl}^{2}} = 1 \Rightarrow \phi_{c} = \frac{M_{pl}}{\sqrt{6}\xi}.
\end{equation}

Using the condition $\phi_{\ast} = m_{\Phi}/\sqrt{\lambda_{\Phi}} < \phi_{c}$, in the Palatini formalism (<ref>) we therefore require

\begin{equation}\label{eqn:ad149}
\frac{m_{\Phi}}{\sqrt{\lambda_{\Phi}}} <  \frac{M_{pl}}{\sqrt{\xi}},
\end{equation}

which gives the condition on the inflaton mass for the model to be consistent with non-minimally coupled dynamics as

\begin{equation}\label{eqn:ad150}
m_{\Phi} < \frac{\sqrt{\lambda_{\Phi}}M_{pl}}{\sqrt{\xi}}.
\end{equation}

The comoving curvature power spectrum for a quartic potential in the Palatini formalism is (from (<ref>)[In Chapter 6 we derive the primordial curvature power spectrum and the inflation observables in the Palatini formalism generalised to include an inflaton mass term. We find that the power spectrum and observables are the same as those for conventional Palatini inflation with a quartic inflaton potential. See e.g. [62] for a recent review of the calculation of the observables in quartic Palatini inflation.])

\begin{equation}\label{eqn:ad151}
\mathcal{P}_{\mathcal{R}} = \frac{\lambda_{\Phi}N^{2}}{12\pi^{2}\xi },
\end{equation}

and using $N = 55$ as an estimate of the pivot scale and taking the amplitude of $\mathcal{P}_{\mathcal{R}}$ to be $A_{s} = 2.1 \times 10^{-9}$ [12], $\xi$ is

\begin{equation}\label{eqn:ad152}
\xi = 1.22\times 10^{10}\lambda_{\Phi}.
\end{equation}

Using $\xi$ normalised to the Palatini power spectrum $\mathcal{P}_{\mathcal{R}}$ given in (<ref>), the constraint on the inflaton mass (<ref>) becomes

\begin{equation}\label{eqn:ad153}
m_{\Phi} < 2.2 \times 10^{13}\GeV.
\end{equation}

For the case of the metric formalism (<ref>), for $\phi_{\ast} < \phi_{c}$, we have that

\begin{equation}\label{eqn:ad154}
\frac{m_{\Phi}}{\sqrt{\lambda_{\Phi}}} <  \frac{M_{pl}}{\sqrt{6}\xi},
\end{equation}

which gives

\begin{equation}\label{eqn:ad155}
m_{\Phi} < \frac{\sqrt{\lambda_{\Phi}}M_{pl}}{\sqrt{6}\xi}.
\end{equation}

The comoving curvature power spectrum in the metric formalism is given by (from (<ref>))

\begin{equation}\label{eqn:ad156}
\mathcal{P}_{\mathcal{R}} = \frac{\lambda_{\Phi}}{72\pi^{2}\xi^{2} }N^{2},
\end{equation}

and using $N =55$ and $A_{s} = 2.1 \times 10^{-9}$, this gives the value of the non-minimal coupling to be

\begin{equation}\label{eqn:ad157}
\xi = 4.5 \times 10^{4}\sqrt{\lambda_{\Phi}}.
\end{equation}

Using $\xi$ normalised to the metric power spectrum $\mathcal{P}_{\mathcal{R}}$ given in (<ref>) this gives

\begin{equation}\label{eqn:ad158}
m_{\Phi} < 2.2 \times 10^{13}\GeV,
\end{equation}

in order for the model to be consistent with non-minimally coupled inflaton dynamics in the metric formalism, which is the coincidentally the same numerical constraint for the inflaton mass for the Palatini formalism. This means that this model is compatible with an embedding into a non-minimally coupled inflation model within a well defined upper bound on the inflaton mass.

§ BARYON ISOCURVATURE PERTURBATIONS IN NON-MINIMALLY COUPLED INFLATION

In this model, it is possible that the angular component of the inflaton field may undergo quantum fluctuations during inflation which result in density perturbations uncorrelated with the perturbations in the photon energy density. These are isocurvature fluctuations, as defined in Section <ref>. In Affleck-Dine baryogenesis models, the phase of the AD field can produce isocurvature fluctuations in the local baryon density, and therefore in baryon number density [133] - [137]. These isocurvature fluctuations in baryon number density can be observable, and this can lead to constraints on inflation models used in conjunction with the Affleck-Dine model in order for the model to be compatible with observations. In this section, we explore the baryon isocurvature perturbations generated in this model of Affleck-Dine inflation with quadratic $B$-violating terms, and examine whether this model is compatible with the bounds on the isocurvature fraction from the Planck experiment (2018).

We define the isocurvature perturbations as

\begin{equation}\label{eqn:ad159}
I = \left(\frac{\delta \rho_{i}}{\rho_{i}}\right)_{iso},
\end{equation}

for a given species $i$. In order to compare the isocurvature perturbations in baryon energy density in this model to the isocurvature bounds defined using Planck data [36], which are normalised to cold dark matter isocurvature perturbations, we begin with the isocurvature perturbations in cold dark matter density

\begin{equation}\label{eqn:ad160}
I = \left(\frac{\delta \rho_{CDM}}{\rho_{CDM}}\right)_{iso},
\end{equation}

and then use the fact that $\delta \rho_{CDM} \rightarrow \delta \rho_{B}$ to find the effective cold dark matter isocurvature perturbations due to baryon number perturbations

\begin{equation}\label{eqn:ad161}
I = \frac{\rho_{B}}{\rho_{CDM}}\left(\frac{\delta \rho_{B}}{\rho_{B}}\right)_{iso} = \frac{\Omega_{B}}{\Omega_{CDM}}\left(\frac{\delta \rho_{B}}{\rho_{B}}\right)_{iso},
\end{equation}

where the $\Omega$ here are the total energy densities as defined in Section <ref>. The isocurvature fraction $\beta_{iso}$ defined in [36] is

\begin{equation}\label{eqn:ad162}
\beta_{iso} = \frac{\mathcal{P}_{I}}{\mathcal{P}_{\mathcal{R}} + \mathcal{P}_{I}},
\end{equation}

assuming that the adiabatic and isocurvature fluctuations are uncorrelated. $\mathcal{P}_{\mathcal{R}} = A_{s}$ is the primordial curvature power spectrum, and to calculate $\mathcal{P}_{I}$ we need to calculate the power spectrum of the baryon isocurvature fluctuations $\delta \rho_{B}/\rho_{B}$ due to the fluctuations of the inflaton field, given by $\delta \theta$.

For non-minimally coupled inflation with a complex field, we have that

\begin{equation}\label{eqn:ad163}
\Phi = \frac{\phi}{\sqrt{2}}e^{i\theta} = \frac{1}{\sqrt{2}}\left(\phi_{1} + i\phi_{2}\right),
\end{equation}

where $\theta$ is an effectively massless field during inflation and the evolution of the inflaton field is purely along the radial direction, corresponding to $\theta = 0$ along the $\phi_{1}$ direction. Writing the inflaton as an effectively constant background plus a fluctuation in the field, we have that

\begin{equation}\label{eqn:ad164}
\Phi = \frac{\bar{\phi}}{\sqrt{2}}e^{i\delta\theta} \simeq  \frac{\bar{\phi}}{\sqrt{2}}\left( 1 + i\delta \theta \right) = \frac{1}{\sqrt{2}}\left(\bar{\phi}_{1} + i\delta \phi_{2}\right),
\end{equation}

for small $\delta \theta$, where

\begin{equation}\label{eqn:ad165}
\bar{\phi}_{1} = \bar{\phi}, \; \; \; \delta \phi_{2} = \bar{\phi}\delta \theta.
\end{equation}

We have that the inflaton kinetic term in both the metric and Palatini formalisms for a complex inflaton field is given by

\begin{equation}\label{eqn:ad166}
\frac{1}{\Omega^{2}}\partial_{\mu}\Phi^{\dagger}\partial^{\mu}\Phi = \frac{1}{2\left(1 + \frac{\xi \phi_{1}^{2}}{M_{pl}^{2}} + \frac{\xi \phi_{2}^{2}}{M_{pl}^{2}}\right)}\left( \partial_{\mu}\phi_{1}\partial^{\mu}\phi_{1} + \partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}\right),
\end{equation}

and during slow-roll inflation we have that

\begin{equation}\label{eqn:ad167}
\frac{\xi \bar{\phi}_{1}^{2}}{M_{pl}^{2}} >> 1, \frac{\xi \phi_{2}^{2}}{M_{pl}^{2}},
\end{equation}

and the kinetic term for the angular field can therefore be written

\begin{equation}\label{eqn:ad168}
\frac{M_{pl}^{2}}{2\xi \bar{\phi}_{1}^{2}}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2}.
\end{equation}

We define a canonical field $\chi_{2}$ such that

\begin{equation}\label{eqn:ad169}
\frac{d\chi_{2}}{d\phi_{2}} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}_{1}} \Rightarrow \chi_{2} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}_{1}} \phi_{2},
\end{equation}

and substituting this into (<ref>) we find the canonical kinetic term to be

\begin{equation}\label{eqn:ad170}
\frac{M_{pl}^{2}}{2\xi \bar{\phi}_{1}^{2}}\partial_{\mu}\phi_{2}\partial^{\mu}\phi_{2} \rightarrow \frac{1}{2}\partial_{\mu}\chi_{2}\partial^{\mu}\chi_{2}.
\end{equation}

In terms of fluctuations of the inflaton field we therefore have

\begin{equation}\label{eqn:ad171}
\delta \chi_{2} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}_{1}} \delta \phi_{2} = \frac{M_{pl}}{\sqrt{\xi}\bar{\phi}}\bar{\phi}\delta \theta,
\end{equation}

\begin{equation}\label{eqn:ad172}
\Rightarrow \delta \theta = \frac{\sqrt{\xi}}{M_{pl}}\delta \chi_{2}.
\end{equation}

Since $\chi_{2}$ is a canonically normalised field, its power spectrum is the standard expression given in Section <ref>

\begin{equation}\label{eqn:ad173}
\mathcal{P}_{\delta \chi_{2}} = \left(\frac{H}{2\pi}\right)^{2},
\end{equation}

and from (<ref>) we have that the corresponding power spectrum in $\delta \theta$ is

\begin{equation}\label{eqn:ad174}
\mathcal{P}_{\delta \theta} = \frac{\xi}{M_{pl}^{2}}\mathcal{P}_{\delta \chi_{2}} \Rightarrow \mathcal{P}_{\delta \theta} = \frac{\xi H^{2}}{4\pi^{2}M_{pl}^{2}}.
\end{equation}

The fluctuations in the baryon energy density relate to fluctuations in baryon number density through $\rho_{B} = m_{B}n_{B}$ for a fixed mass of the baryonic species, and we have

\begin{equation}\label{eqn:ad175}
\frac{\delta \rho_{B}}{\rho_{B}} = \frac{\delta n_{B}}{n_{B}}.
\end{equation}

In this model we have that

\begin{equation}\label{eqn:ad176}
n_{B} \propto \sin \left(2\theta \right),
\end{equation}

from (<ref>), (<ref>), (<ref>), for an initial phase angle $\theta$, and we can write the fluctuation in baryon number density as

\begin{equation}\label{eqn:ad177}
\frac{\delta n_{B}}{n_{B}} = \frac{d n_{B}}{d\theta}\frac{\delta \theta}{n_{B}}.
\end{equation}

Using (<ref>), we have that

\begin{equation}\label{eqn:ad178}
\frac{d n_{B}}{d\theta} = 2\cos \left(2\theta \right),
\end{equation}

and using (<ref>) and (<ref>) substituted into (<ref>), the fluctuation in baryon number density in terms of the fluctuation in $\theta$ is

\begin{equation}\label{eqn:ad179}
\frac{\delta n_{B}}{n_{B}} \propto \frac{2\delta \theta}{\tan \left(2\theta \right)}.
\end{equation}

The isocurvature fluctuation in baryon number density from (<ref>) is then

\begin{equation}\label{eqn:ad180}
I = \frac{\Omega_{B}}{\Omega_{CDM}}\frac{\delta n_{B}}{n_{B}} = \frac{\Omega_{B}}{\Omega_{CDM}}\frac{2\delta \theta}{\tan \left(2\theta \right)},
\end{equation}

the power spectrum of the baryon isocurvature fluctuations is thus

\begin{equation}\label{eqn:ad181}
\mathcal{P}_{I} =  \left(\frac{\Omega_{B}}{\Omega_{CDM}}\right)^{2}\frac{4}{\tan^{2} \left(2\theta \right)}\mathcal{P}_{\delta \theta},
\end{equation}

and using (<ref>), we can write (<ref>) as

\begin{equation}\label{eqn:ad182}
\mathcal{P}_{I} =  \left(\frac{\Omega_{B}}{\Omega_{CDM}}\right)^{2}\frac{\xi H^{2}}{\tan^{2} \left(2\theta \right)\pi^{2}M_{pl}^{2}}.
\end{equation}

The ratio $\mathcal{P}_{I}/\mathcal{P}_{\mathcal{R}}$ from (<ref>) can be written in terms of the isocurvature fraction

\begin{equation}\label{eqn:ad183}
\frac{\mathcal{P}_{I}}{\mathcal{P}_{\mathcal{R}}} = \frac{\beta_{iso}}{1 - \beta_{iso}},
\end{equation}

and using the observational limit on the isocurvature fraction, $\beta_{iso, lim}$, [36] we can constrain this ratio

\begin{equation}\label{eqn:ad184}
\frac{\mathcal{P}_{I}}{\mathcal{P}_{\mathcal{R}}} = \left(\frac{\Omega_{B}}{\Omega_{CDM}}\right)^{2}\frac{\xi H^{2}}{\tan^{2} \left(2\theta \right)\pi^{2}M_{pl}^{2}\mathcal{P}_{\mathcal{R}}} < \frac{\beta_{iso, lim}}{1 - \beta_{iso, lim}}.
\end{equation}

With some rearrangement this can be recast as a constraint on the Hubble parameter during inflation

\begin{equation}\label{eqn:ad185}
H_{inf} < \frac{\Omega_{CDM}}{\Omega_{B}}\frac{M_{pl}\pi A_{s}^{\frac{1}{2}}}{\sqrt{\xi}}\tan \left(2\theta \right) \left(\frac{\beta_{iso, lim}}{1 - \beta_{iso, lim}} \right)^{\frac{1}{2}}.
\end{equation}

Using $A_{s} = 2.1 \times 10^{-9}$, $\Omega_{CDM}/\Omega_{B} = 5.3$ and $\beta_{iso, lim} = 0.038$ from [36] this is

\begin{equation}\label{eqn:ad186}
H_{inf} < 3.6\times 10^{14}\left(\frac{\tan\left(2\theta \right)}{\sqrt{\xi}}\right)\GeV,
\end{equation}

which is the requirement in order to produce an isocurvature fraction consistent with observations.

The Einstein frame potential on the plateau in both the metric and Palatini formalisms for a quartic inflaton potential (see e.g. [62] for a review of the derivation of (<ref>) in both formalisms) is

\begin{equation}\label{eqn:ad187}
V_{E} = \frac{\lambda_{\Phi}M_{pl}^{4}}{4\xi^{2}},
\end{equation}

and from (<ref>) we have that

\begin{equation}\label{eqn:ad188}
H_{inf} = \left(\frac{\lambda_{\Phi}}{12}\right)^{\frac{1}{2}}\frac{M_{pl}}{\xi}.
\end{equation}

Using (<ref>) and (<ref>), we can write a constraint on the non-minimal coupling $\xi$ needed in order to produce an acceptable isocurvature fraction in this model,

\begin{equation}\label{eqn:ad189}
\left(\frac{\lambda_{\Phi}}{12}\right)^{\frac{1}{2}}\frac{M_{pl}}{\xi} < 3.6\times 10^{14}\left(\frac{\tan\left(2\theta \right)}{\sqrt{\xi}}\right)\GeV,
\end{equation}

\begin{equation}\label{eqn:ad190}
\xi > 3.7 \times 10^{6}\frac{\lambda_{\Phi}}{\tan^{2}\left(2\theta \right)}.
\end{equation}

We can now examine $\xi$ in the metric and Palatini formalisms and test whether the non-minimal coupling in each formalism satisfies the constraint (<ref>). The non-minimal coupling $\xi$ normalised from the comoving curvature power spectrum in the Palatini formalism is $\xi = 1.22\times 10^{10}\lambda_{\Phi}$ (<ref>) which easily satisfies the constraint (<ref>) provided that $\tan \left(2\theta \right)$ isn't exceptionally small, and also shows that the constraint on $\xi$ ultimately does not depend on the size of the inflaton self-coupling $\lambda_{\Phi}$. This shows that this model in the Palatini formalism produces an isocurvature perturbation in the baryon number density compatible with observations.

In the metric formalism we find $\xi = 4.5 \times 10^{4}\sqrt{\lambda_{\Phi}}$ (<ref>), and therefore

\begin{equation}\label{eqn:ad191}
\lambda_{\Phi} < 1.5\times 10^{-4}\tan^{4}\left(2\theta \right),
\end{equation}

is required in order for the constraint (<ref>) to be satisfied and isocurvature perturbations to be compatible with observations in the metric formalism.

Therefore in metric non-minimally coupled inflation, isocurvature perturbations place a significant constraint on the inflaton self-coupling. This also means that the baryon isocurvature fraction can be close to the present CMB bound if $\lambda_{\Phi} \sim 10^{-4}$, and so potentially observable as the bound improves, in contrast to the case of Palatini inflation where it will generally be much smaller than the CMB bound.

§ SEMI-CLASSICAL TREATMENT OF THE COHERENTLY OSCILLATING CONDENSATE

Throughout our derivation and analysis of the Affleck-Dine baryogenesis model with quadratic symmetry-breaking terms, we have assumed that $\phi_{1}$ and $\phi_{2}$ can be treated as classical fields. However, we must consider whether the classical treatment of the fields is valid in this context.

The inflaton field after inflation forms an oscillating coherent quantum condensate. In a second quantised theory, a coherent condensate can be treated as a classical field because the occupation number of the field states is much larger than one [138] [139]. In order for this to be satisfied, we require in the general sense that $X > m_{X}$ for the field $X$ which we are considering. However, in our model the condensate decays when $\phi < \phi_{\ast} = m_{\Phi}/\sqrt{\lambda_{\Phi}}$, and so $\phi < m_{\Phi}$ generally, meaning that the condensate can no longer be considered classically. Nevertheless, we will show that the classical calculation of the asymmetry is still correct.

The oscillating classical field corresponds to a coherent state of the field $| \phi_{i}\left(t \right) \rangle$. In general, the expectation value of the field operator in the coherent state $| \phi_{i}\left(t \right) \rangle$ is given by its classical value,

\begin{equation}\label{eqn:ad192}
\langle \phi_{i}\left(t \right) | \hat{\phi}_{i}| \phi_{i}\left(t \right) \rangle = \phi_{i,cl}\left(t \right),
\end{equation}

where $\phi_{i, cl}$ is the classical value of the field $\phi_{i}$ which satisfies the classical equations of motion. Since we are treating the components of the fields $\phi_{1}$ and $\phi_{2}$ as independent scalar fields, the coherent state of the complex field is the product of the coherent states of $\phi_{1}$ and $\phi_{2}$

\begin{equation}\label{eqn:ad193}
| \Phi \left( t \right) \rangle = | \phi_{1}\left(t \right) \rangle | \phi_{2}\left(t \right) \rangle.
\end{equation}

The baryon number density operator is given by

\begin{equation}\label{eqn:ad194}
\hat{n} = \hat{\dot{\phi}}_{1}\hat{\phi}_{2} - \hat{\dot{\phi}}_{2}\hat{\phi}_{1},
\end{equation}

and the expectation value of the baryon number density in the coherent state  $| \phi_{1}\left(t \right) \rangle | \phi_{2}\left(t \right) \rangle$ is given by

\begin{multline}\label{eqn:ad195}
\langle \Phi \left( t \right) | \hat{n} | \Phi \left( t \right) \rangle = \langle \Phi \left( t \right) | \hat{\dot{\phi}}_{1}\hat{\phi}_{2} - \hat{\dot{\phi}}_{2}\hat{\phi}_{1} | \Phi \left( t \right) \rangle = \langle \phi_{1}\left(t \right) | \langle \phi_{2}\left(t \right) | \hat{\dot{\phi}}_{1}\hat{\phi}_{2} - \hat{\dot{\phi}}_{2}\hat{\phi}_{1}| \phi_{1}\left(t \right) \rangle | \phi_{2}\left(t \right) \rangle \\
= \langle \phi_{1} \left| \hat{\dot{\phi}}_{1} \right| \phi_{1} \rangle \langle \phi_{2} \left| \hat{\phi}_{2} \right| \phi_{2} \rangle  - \langle \phi_{1} \left| \hat{\phi}_{1} \right| \phi_{1} \rangle \langle \phi_{2} \left| \hat{\dot{\phi}}_{2} \right| \phi_{2} \rangle.
\end{multline}

Thus the expectation value of the baryon asymmetry operator in the coherent state is given by

\begin{equation}\label{eqn:ad196}
\langle \Phi \left( t \right) | \hat{n} | \Phi \left( t \right) \rangle  = \dot{\phi}_{1}\phi_{2} - \dot{\phi}_{2}\phi_{1} \equiv n_{cl}.
\end{equation}

Therefore the expectation value of the baryon asymmetry operator in the coherent state $i$ is equal to the baryon number density, $n_{cl}$, calculated using the classical fields $\phi_{1}$ and $\phi_{2}$. This is true even if the classical oscillating field is no longer a good approximation to the coherent state.

Since $\phi_{i} < m_{\Phi}$ in this model, the variance of the expectation value of the field becomes large compared to the squared classical field, which means that the quantum fluctuations of the fields around their expectation values will be large, and the fields cannot be considered classical. However, the length scale of the field fluctuations at the onset of inflaton decay must be smaller than the horizon. The present observed Universe evolves from a spacetime volume much larger than the size of the horizon at inflaton decay, therefore the observed baryon asymmetry today will be given by its spatial average, which corresponds to the classical value of the asymmetry. This means that the observed baryon-to-entropy ratio is equivalent to the baryon asymmetry calculated using the classical fields, $n_{cl}$, in this case, despite the fact that the fields themselves cannot be considered classical. The classical calculation of the baryon-to-entropy ratio is therefore still correct in the Affleck-Dine baryogenesis model with quadratic symmetry-breaking terms.

§ SUMMARY

In this chapter we have considered an Affleck-Dine baryogenesis scenario resulting from a model of non-minimally coupled inflation with a complex inflaton charged under a global $U(1)$ symmetry, with the complex inflaton taking the role of the AD field. We considered inflation from a renormalisable $U(1)$-symmetric $\left| \Phi \right|^{2} + \left| \Phi \right|^{4}$-type potential, with a quadratic $U(1)$-violating term $A\left( \Phi^{2} + \Phi^{\dagger^{2}}\right)$. This term introduces a phase dependence, altering the field's trajectory from being purely radial (along the $\phi_{1}$ direction) into a varying elliptical orbit in the complex plane, corresponding to an oscillating asymmetry.

At the end of slow-roll inflation, the inflaton is coherently oscillating. Initially these are $\left|\Phi\right|^{4}$ dominated oscillations, and then later becomes $\Phi^{2}$ oscillations as the oscillations become damped and the inflaton field oscillates about its minimum deep in the $\Phi^{2}$ region of the potential. At this stage the coherent condensate decays in a $B$-conserving process and subsequently reheats the universe. While the field undergoes coherent oscillations, it undergoes a periodic phase oscillation between its eigenstates $\Phi \leftrightarrow \Phi^{\dagger}$. This means that while the condensate is decaying, it is also periodically rotating between its $\Phi$ and $\Phi^{\dagger}$ states. So for each half-cycle in the phase oscillation, either $\Phi$ or $\Phi^{\dagger}$ scalars will be dominantly decaying, and will therefore produce either a baryon or anti-baryon number contribution to the particle plasma. Due to the fact that the condensate is decaying away while this is happening, it means that the produced baryon number and anti-baryon number of each oscillation doesn't exactly cancel with the charge produced in the previous half-cycle. This results in an overall asymmetry in baryon number being transferred to the particle content of the Standard Model, and this is how the observed small but finite baryon asymmetry is generated.

We were interested in studying the baryon asymmetry generated in this Affleck-Dine baryogenesis model from a quadratic symmetry-breaking term. In order to do this we derived the $U(1)$ asymmetry using a threshold approximation, wherein the asymmetry is generated when the potential is strongly dominated by its quadratic terms. We modelled the asymmetry of the condensate, and its evolution into the asymmetry which is transferred to the Standard Model, both in the case where the lifetime of the condensate is much larger than the oscillation period of the inflaton phase, $\tau_{\Phi} > T_{asy}$, and in the case where the condensate decays away before a phase rotation can occur, $\tau_{\Phi} < T_{asy}$. We find that in both cases the asymmetry transferred to the Standard Model derived using the threshold approximation could account for the observed baryon-to-entropy ratio. For $\tau_{\Phi} < T_{asy}$ it is typically much greater, and for $\tau_{\Phi} > T_{asy}$ it can be much less, due to the effect of suppression of the asymmetry due to averaging over a large number of oscillations. Given that the typical baryon asymmetry generated is much larger than that observed, the suppression due to oscillations is advantageous, requiring a smaller suppression of the quadratic symmetry-breaking terms.

We examine the dynamics of the angular component of the inflaton field $\theta$, and determine that the perturbations of the $\theta$ field do not significantly evolve before the potential becomes $\Phi^{2}$ dominated, provided that the constraint (<ref>) is satisfied, which is easily compatible with the constraints on $A^{\frac{1}{2}}/m_{\Phi}^{2}$, (<ref>) and (<ref>), derived in order to generate the observed baryon-to-entropy ratio from the transferred asymmetry. We numerically verified the constraint (<ref>) and showed that when this is violated there is an additional suppression of the asymmetry due to the damping of the phase during $\Phi^{4}$ oscillations.

We verify the validity of the threshold approximation in the derivation of the asymmetry, and the calculation of the baryon-to-entropy ratio using a numerical computation. In the numerical computation, we include decay terms in the field equations to account for the decay of the condensate, and we also model the dilution of the radiation energy density due to the expansion of the Universe. In the analytical calculation we assume that at time $t = t_{R}$, the condensate instantly decays to radiation and use this to calculate the reheating temperature. In the numerical computation of Section <ref>, we model the condensate as decaying away over a time period $\sim \Gamma_{\Phi}^{-1}$, and then calculate the reheating temperature directly from the energy density of radiation. We verify that the threshold approximation is reasonable for the analytical calculation for cases of the asymmetry generated with $\tau_{\Phi} = T_{asy}$ and $\tau_{\Phi} > T_{asy}$, with the predictions of the baryon-to-entropy ratio obtained numerically being within $10\%$ of the analytical predictions and the behaviour of the comoving condensate and transferred asymmetries in each case when modelled numerically is in perfect agreement with the behaviour predicted analytically using the threshold approximation. We also demonstrated that washout of the asymmetry through $B$-violating inflaton exchange is unlikely to pose a problem in this model for the range of reheating temperatures we would consider. This shows that this model provides a robust general framework for studying Affleck-Dine baryogenesis due to quadratic terms in models with a complex inflaton as the Affleck-Dine field.

In Section <ref> we discuss the compatibility of the Affleck-Dine baryogenesis via quadratic $B$-violating potential terms model with the dynamics of non-minimally coupled inflation. We find that in order for this to be realised, we require the non-minimally coupled dynamics of the inflaton to become insignificant before the threshold of $\Phi^{2}$ domination, and before the Affleck-Dine dynamics become significant. In both the metric and Palatini formalisms of non-minimally coupled inflation, we find that the model presented in this chapter can fulfill this requirement if the mass of the inflaton is smaller than $2.2 \times 10^{13}\GeV$.

We discuss baryon isocurvature perturbations in Section <ref> and we derive the constraints needed to ensure that the isocurvature fraction generated by a non-minimally coupled inflaton in this framework is not too large, and therefore consistent with the isocurvature bounds given in the Planck results [36]. We find that the isocurvature fraction predicted in this model can be consistent with the observational limit in both the metric and Palatini formalisms subject to constraints on the non-minimal coupling, $\xi$, (<ref>), (<ref>) and the inflaton self-coupling in the metric case (<ref>). In particular, in the metric case there is a significant upper bound on the size of the inflaton self-coupling, $\lambda_{\Phi} \sim 10^{-4}$. This also means that in the metric case, the baryon isocurvature perturbations could potentially become observable with an improved CMB bound on the isocurvature fraction in the future.

We also examine the validity of the classical treatment of the transfer of the asymmetry to the Standard Model. Since the inflaton field initially forms a coherently oscillating condensate with a large occupation number, the inflaton field can be treated as a classical field. However, when the inflaton decays, its amplitude $\phi$ is less than $m_{\Phi}$ and the coherent state is no longer in the classical limit. This raises the question of whether the classical treatment of the asymmetry in order to generate the observed baryon-to-entropy ratio is necessarily correct. We find that due to the spatial averaging of the baryon asymmetry, it is equal to the expectation value of the transferred asymmetry operator, which is equal to its classical value. Therefore, while it may not be true that the asymmetry can be considered strictly classical, the baryon asymmetry calculated using the classical fields still gives the correct value of the asymmetry.

While in this work we have considered a general Affleck-Dine scenario without studying a specific decay path of the inflaton or resulting transfer route of the $U(1)$-asymmetry to baryon number, it is possible that this work could be applied to a leptogenesis scenario from the decay of the inflaton to right-handed neutrinos, and then a subsequent transfer of the asymmetry to baryon number via sphaleron processes. This would allow a more in-depth analysis of the possibility of baryon asymmetry washout, and could also provide a framework for a model of inflation, baryogenesis and dark matter. This study is in progress.

CHAPTER: Q-BALLS FROM NON-MINIMALLY COUPLED INFLATION IN PALATINI GRAVITY

In this chapter we discuss the formulation of Q-balls in the framework of a non-minimally coupled Palatini inflation model. We present a study of the Palatini inflation model for a $\left|\Phi\right|^{2} + \left|\Phi\right|^{4}$-type potential in the Jordan frame (the first time an inflaton mass term has been included in Palatini inflation) and present the values for the inflationary observables. We derive the Q-ball field equation in a non-minimally coupled Palatini framework, discuss the existence conditions of these Q-balls and derive an inflaton mass range for which the model can both support inflation and produce Q-balls. In doing so, we will, for the first time, derive Q-ball solutions for the case of a complex scalar with a non-canonical kinetic term, corresponding to a new class of Q-ball. We derive the zeroes of the Q-ball equation and use these to solve the equation numerically. We verify the existence of Q-ball solutions over a range of field values and present some important properties of these Q-balls, including energy, charge and radius. We also present an analytical approximation of the Q-ball solution and obtain expressions for the energy and charge using this calculation. We show analytically that these Q-balls are stable and derive the energy-charge relation, both of which are then confirmed numerically. We consider the effects of curvature on these Q-balls and derive approximate relations between the inflaton self-coupling, $\lambda$, and the size and global energy of the Q-balls in order to predict, as a first approximation, the radius needed for one of these Q-balls to collapse to a black hole following formation in the presence of curvature. We discuss reheating in the case of Q-balls and Q-ball derived Black Holes, and speculate on the post-inflation cosmology which could result from an early matter-dominated era of Q-balls or their associated Black Holes. Finally the observability of the model is discussed, including the possibility of observable gravitational waves from Q-ball formation and/or decay and Q-ball dark matter. We also comment on the implications for the case of a real inflaton and oscillons.

§ Q-BALLS IN THE CONTEXT OF NON-MINIMALLY COUPLED PALATINI INFLATION

Q-balls are a subset of the class of field theory solutions known as non-topological solitons. Solitons are extended objects which can be visualised as droplets composed of many particles, held together by an attractive interaction between the particles. Their stability arises due to a conserved quantity of the theory, if the solitons are non-topological in nature.

In the case of Q-balls, the objects are composed of complex scalar particles charged under a conserved Noether symmetry of the theory, and the objects themselves carry a large charge. Charged scalar field configurations stable against small perturbations were originally theorised to exist by Rosen [140] in 1968. These ideas were then explored further and non-topological solitons were formally described by Friedberg, Lee et al  [141, 142] later in the 20th century. The formulation of Q-balls as we understand them today was developed by Coleman in 1985 [50].

Since then, Q-balls have been studied extensively in the context of supersymmetry (SUSY)  [105] - [113],[143, 144], generally forming from scalar condensates in the flat directions of supersymmetric potentials. They are often incorporated into supersymmetric theories in conjunction with Affleck-Dine baryogenesis [100] - [102]  as a dark matter candidate, either as complete Q-balls or as a source of particle dark matter when the Q-balls decay [103] - [108]. The globally charged nature of Q-balls makes them a good dark matter candidate [109] - [113], [143] - [145], [146] - [152] both as compact objects or as the seeds of particle dark matter from decays. The possibility of dark matter being accounted for by Primordial Black Holes seeded from the collapse of supersymmetric Q-balls has also recently been considered [153].

Many numerical simulations have been performed to model the formation of Q-balls from scalar condensates [154] - [160]. In [160], Hiramatsu et al showed that the fragmentation of a neutral condensate in a model based on a complex scalar first forms oscillons which then fragment to $\pm$ Q-ball pairs. Using Hiramatsu et al's results [160] as a benchmark, we will consider the possibility that the Q-balls in this model may also form from the fragmentation of the inflaton condensate into neutral lumps of scalars, which then decay into pairs of $\pm$ Q-balls.

The fragmentation of a scalar condensate is a complex process which depends heavily on the underlying particle physics and requires detailed numerical simulations to model. As such, a quantitative description of the process is beyond the scope of the research presented in this thesis. However, it is possible that the fragmentation of the inflaton condensate in this case could arise as a result of tachyonic preheating [161] - [165].

Tachyonic preheating is the process by which an inflaton condensate becomes unstable due to the amplified growth of specific wavelengths of perturbations arising due to a tachyonic instability of the inflaton potential. This can lead to the formation of overdensities within a previously homogeneous condensate, which - if soliton solutions are admissible in the underlying field theory - can lead to the condensate breaking apart to form solitons or other related objects [85, 86], the precise nature of which depends on the nature of the inflaton field itself.

In the work this chapter is based on we were interested in exploring Q-balls in a non-SUSY context. More specifically, we were interested in whether Q-balls composed of inflaton scalars could be produced in a model of non-minimally coupled Palatini inflation, possibly as a result of tachyonic preheating, whether the model could inflate and reheat successfully in conjunction with the production of Q-balls, what the properties of these Q-balls would be, how they would compare to the Q-balls of Coleman's solution, and what the broader implications for cosmology could be from the presence of these Q-balls.

§ THE MODEL

In this section we introduce the non-minimally coupled Palatini inflation model within which we want to produce inflatonic Q-balls. This model describes inflation driven by a complex scalar inflaton $\Phi$ charged under a global $U(1)$ symmetry and non-minimally coupled to gravity. In particular, it differs from existing analyses of Palatini inflation by the inclusion of an inflaton mass term, which has not been previously considered. We use the $\left( +, -, -, - \right)$ convention for the metric and $M_{pl}$ should be taken to be the reduced Planck mass.

The Jordan frame inflaton action is given by

\begin{equation}\label{eqn:q1}
S_{J} = \int d^{4}x \sqrt{-g} \left[ -\frac{1}{2}M_{pl}^{2}\left( 1 + \frac{2\xi \mid \Phi \mid^{2}}{M_{pl}^{2}}\right)R + \partial_{\mu}\Phi^{ \dagger} \partial^{\mu}\Phi - V\left(\mid \Phi \mid \right) \right],
\end{equation}

where the Jordan frame potential is

\begin{equation}\label{eqn:q2}
V\left( \mid \Phi \mid \right) = m^{2} \mid \Phi \mid^{2} + \lambda \mid \Phi \mid^{4},
\end{equation}

where $m$ is the inflaton mass and $\lambda$ is the inflaton self-coupling. In order to recast the model in terms of conventional General Relativity, we perform a conformal transformation on the spacetime metric, given by

\begin{equation}\label{eqn:q3}
g_{\mu \nu} \longrightarrow \tilde{g}_{\mu \nu} = \Omega^{2}g_{\mu \nu},
\end{equation}

where the conformal factor $\Omega^{2}\left(\Phi \right)$ is

\begin{equation}\label{eqn:q4}
\Omega^{2} = 1 + \frac{2\xi \left| \Phi \right|^{2}}{M_{pl}^{2}}.
\end{equation}

In the Palatini formulation, the Ricci scalar transforms as (from (<ref>))

\begin{equation}\label{eqn:q5}
R \longrightarrow \tilde{R} = \frac{R}{\Omega^{2}},
\end{equation}

and the integration measure, $\sqrt{-g}$ in (<ref>), transforms as

\begin{equation}\label{eqn:q6}
\sqrt{-det \left(\frac{\tilde{g}_{\mu\nu}}{\Omega^{2}}\right)} = \sqrt{-\frac{1}{\Omega^{8}}det\left(\tilde{g}_{\mu\nu}\right)} = \frac{1}{\Omega^{4}}\sqrt{-det \tilde{g}_{\mu\nu}} = \frac{\sqrt{-\tilde{g}}}{\Omega^{4}}.
\end{equation}

The conformal transformation (<ref>) acts purely on the metric of the theory. This means that the transformation does not act on coordinates and the transformation on the derivative term in (<ref>) is due to the implicit factor of the metric used in the contraction of the indices

\begin{equation}\label{eqn:q7}
\partial_{\mu}\Phi^{\dagger} \partial^{\mu}\Phi = g_{\mu \nu}\partial^{\nu}\Phi^{\dagger} \partial^{\mu}\Phi,
\end{equation}

\begin{equation}\label{eqn:q8}
\Rightarrow \tilde{g}_{\mu \nu}\partial^{\nu}\Phi^{\dagger} \partial^{\mu}\Phi = g_{\mu \nu}\Omega^{2}\partial^{\nu}\Phi^{\dagger} \partial^{\mu}\Phi = \Omega^{2}\partial_{\mu}\Phi^{\dagger} \partial^{\mu}\Phi.
\end{equation}

The transformed action is then

\begin{equation}\label{eqn:q9}
S_{E} = \int d^{4}x \sqrt{-\tilde{g}} \left[ -\frac{1}{2}M_{pl}^{2}\tilde{R} + \frac{1}{\Omega^{2}}\partial_{\mu}\Phi^{ \dagger} \partial^{\mu}\Phi - V_{E}\left(\mid \Phi \mid \right) \right],
\end{equation}

where we are henceforth working in the Einstein frame (denoted by a subscript 'E' for the action or potential, or a tilde for geometric or other quantities). Our Einstein frame potential is then defined as

\begin{equation}\label{eqn:q10}
V_{E}\left(\mid \Phi \mid \right) = \frac{V\left(\mid \Phi \mid \right)}{\Omega^{4}}.
\end{equation}

As in Chapter 4, all calculations are performed in the Einstein frame in this chapter unless explicitly stated otherwise.

§ SLOW ROLL PARAMETERS AND INFLATIONARY OBSERVABLES

In this section we calculate the observables for non-minimally coupled Palatini inflation for the potential (<ref>) and check the compatibility of this model with the observations from the Planck satellite experiment. For ease of calculation we write the complex inflaton field in the form

\begin{equation}\label{eqn:q11}
\Phi = \frac{\phi}{\sqrt{2}}e^{i\theta},
\end{equation}

which means that the conformal factor (<ref>) becomes

\begin{equation}\label{eqn:q12}
\Omega^{2} = 1 + \frac{\xi \phi^{2}}{M_{pl}^{2}},
\end{equation}

and the Jordan frame potential is

\begin{equation} \label{eqn:q13}
V\left( \phi \right) = \frac{1}{2}m^{2}\phi^{2} + \frac{\lambda}{4}\phi^{4}.
\end{equation}

The Einstein frame potential in terms of (<ref>) is thus

\begin{equation}\label{eqn:q14}
V_{E}\left( \phi \right) = \frac{1}{2\Omega^{4}}m^{2}\phi^{2} + \frac{\lambda}{4\Omega^{4}}\phi^{4} = \frac{m^{2}\phi^{2}}{2\left( 1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}\right)^{2}} + \frac{\lambda \phi^{4}}{4\left( 1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}\right)^{2}},
\end{equation}

\begin{equation}\label{eqn:q15}
\Rightarrow V_{E} = \frac{\lambda M_{pl}^{4}}{4 \xi^{2}\left( 1 + \frac{M_{pl}^{2}}{\xi \phi^{2}}\right)^{2}}\left[ 1 + \frac{2 m^{2}}{\lambda \phi^{2}}\right].
\end{equation}

Since we are assuming that inflation takes place upon the plateau of the potential, and we expect the field to be very large in this regime, we define what we will refer to as "the plateau limit", where

\begin{equation}\label{eqn:q16}
\frac{\xi \phi^{2}}{M_{pl}^{2}} >> 1.
\end{equation}

It is assumed throughout this chapter that this approximation is a reflection of the dominant dynamics during and at the instantaneous end of slow-roll inflation, and more generally while the inflaton is otherwise on its plateau.

From (<ref>), we have that $M_{pl}^{2}/\xi \phi^{2} << 1$, and can therefore approximate

\begin{equation}\label{eqn:q17}
\left( 1 + \frac{M_{pl}^{2}}{\xi \phi^{2}}\right)^{-2} \thickapprox 1 - 2\frac{M_{pl}^{2}}{\xi \phi^{2}},
\end{equation}

\begin{equation}\label{eqn:q18}
\Rightarrow V_{E} = \frac{\lambda M_{pl}^{4}}{4 \xi^{2}}\left[ 1 + \frac{2 m^{2}}{\lambda \phi^{2}}\right] \left( 1 - \frac{2M_{pl}^{2}}{\xi \phi^{2}} \right).
\end{equation}

To leading order in small quantities, the Einstein frame potential is,

\begin{equation}\label{eqn:q19}
V_{E} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left[ 1 + \frac{2m^{2}}{\lambda \phi^{2}} - \frac{2M_{pl}^{2}}{\xi \phi^{2}} \right],
\end{equation}

which can be rewritten as

\begin{equation}\label{eqn:q20}
V_{E} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left[ 1 - \frac{2M_{pl}^{2}}{\xi \phi^{2}}\beta \right],
\end{equation}

where $\beta$ is defined as

\begin{equation}\label{eqn:q21}
\beta = 1 - \frac{\xi m^{2}}{\lambda M_{pl}^{2}}.
\end{equation}

To canonically normalise the inflaton field, we perform the field rescaling

\begin{equation}\label{eqn:q22}
\frac{d\sigma}{d\phi} = \frac{1}{\sqrt{1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}}},
\end{equation}

\begin{equation}\label{eqn:q23}
\Rightarrow \int d\sigma = \int  \frac{d\phi}{\sqrt{1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}}}.
\end{equation}

Writing $a = \xi/M_{pl}^{2}$ we can use

\begin{equation}\label{eqn:q24}
\int  \frac{dx}{\sqrt{1 + ax^{2}}} = \frac{1}{\sqrt{a}}\sinh^{-1} \left(\sqrt{a}x\right) + C,
\end{equation}

and the rescaled scalar field is therefore

\begin{equation}\label{eqn:q25}
\sigma \left(\phi \right) = \frac{M_{pl}}{\sqrt{\xi}}\sinh^{-1} \left(\frac{\sqrt{\xi}}{M_{pl}}\phi\right),
\end{equation}

where $\sigma \rightarrow \phi$ as $\phi \rightarrow 0$. It is important to note that this is a field rescaling and not a truly canonically normalised scalar field, due to the complex nature of the field $\Phi$. When discussing inflation in this model specifically, it can be regarded as a "canonical inflaton" since the inflation dynamics are determined by the radial component, $\phi$, of the inflaton field. Elsewhere in this chapter, $\sigma$ should be taken as a field redefinition, and the rescaling (<ref>) will be used multiple times in calculations throughout this chapter.

It follows from (<ref>) that

\begin{equation}\label{eqn:q26}
\phi \left(\sigma \right) = \frac{M_{pl}}{\sqrt{\xi}}\sinh \left(\frac{\sqrt{\xi}}{M_{pl}}\sigma \right).
\end{equation}

The hyperbolic sine function is

\begin{equation}\label{eqn:q27}
\sinh\left(x \right) = \frac{e^{x} - e^{-x}}{2},
\end{equation}

and we can therefore rewrite (<ref>) as

\begin{equation}\label{eqn:q28}
\phi \left(\sigma \right) = \frac{M_{pl}}{2\sqrt{\xi}}\left(e^{\frac{\sqrt{\xi}}{M_{pl}}\sigma} - e^{-\frac{\sqrt{\xi}}{M_{pl}}\sigma} \right).
\end{equation}

The rescaled field will be large in the same regime as the physical inflaton field. From (<ref>), we can say that the exponential arguments of (<ref>) will be large during inflation. This means that the decreasing exponential will be significantly smaller than the increasing exponential while on the plateau. During inflation, we can therefore say that the inflaton field in terms of the canonically normalised scalar is

\begin{equation}\label{eqn:q29}
\phi = \frac{M_{pl}}{2\sqrt{\xi}} e^{\frac{\sqrt{\xi}}{M_{pl}}\sigma},
\end{equation}

to a good approximation. Substituting (<ref>) into the Einstein frame potential (<ref>) gives

\begin{equation}\label{eqn:q30}
V_{E}\left(\sigma \right) = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left[ 1 - 8\beta e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma}  \right].
\end{equation}

This is the form of the inflaton potential in the Einstein frame which we will use in deriving the expressions for the inflationary observables (see Figure <ref>). In existing analyses of Palatini inflation, $m = 0$ and $\beta = 1$. Here we are generalising to $m \neq 0$ and $\beta \neq 1$.

Schematic of the inflaton potential in terms of the quasi-canonical field $\sigma$ indicating the edge of the plateau, as used in defining the plateau limit used in this research.

The slow roll parameters are given by

\begin{equation}\label{eqn:q31}
\epsilon = \frac{M_{pl}^{2}}{2}\left(\frac{V_{E}'}{V_{E}} \right)^{2},
\end{equation}

\begin{equation}\label{eqn:q32}
\eta = M_{pl}\frac{V_{E}''}{V_{E}},
\end{equation}

where a prime $'$ denotes a derivative with respect to a field, $\phi$, in this case. Working with the assumption that $\phi$ is large we approximate

\begin{equation}\label{eqn:q33}
V_{E} \thickapprox \frac{\lambda M_{pl}^{4}}{4\xi^{2}},
\end{equation}

while on the plateau. This is another approximation which will be used in this work when the plateau limit is invoked. The first and second derivatives of (<ref>) are

\begin{equation}\label{eqn:q34}
\frac{\partial V_{E}}{\partial \sigma} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\cdot \left(-8\beta \right) \cdot \left(-\frac{2\sqrt{\xi}}{M_{pl}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma} = \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(\frac{16\beta \sqrt{\xi}}{M_{pl}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma},
\end{equation}

\begin{equation}\label{eqn:q35}
\frac{\partial^{2}V_{E}}{\partial \sigma^{2}} =  \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(-\frac{32\beta \xi}{M_{pl}^{2}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma}.
\end{equation}

Using (<ref>), (<ref>) the slow roll parameters are thus

\begin{equation}\label{eqn:q36}
\epsilon = \frac{M_{pl}^{2}}{2}\left( \frac{4\xi^{2}}{\lambda M_{pl}^{4}} \cdot \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(\frac{16\beta \sqrt{\xi}}{M_{pl}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma}\right)^{2} = 128 \beta^{2} \xi e^{-4\frac{\sqrt{\xi}}{M_{pl}}\sigma},
\end{equation}

\begin{equation}\label{eqn:q37}
\eta = M_{pl}^{2} \frac{4\xi^{2}}{\lambda M_{pl}^{4}}\cdot \frac{\lambda M_{pl}^{4}}{4\xi^{2}}\left(-\frac{32\beta \xi}{M_{pl}^{2}}\right) e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma} = -32\beta \xi e^{-2\frac{\sqrt{\xi}}{M_{pl}}\sigma}.
\end{equation}

The number of e-folds of inflation is calculated using

\begin{equation}
\begin{split}\label{eqn:q38}
N = -\frac{1}{M_{pl}^{2}}\int^{\sigma_{end}}_{\sigma} & \frac{V_{E}}{V_{E}'} d\sigma  = -\frac{1}{M_{pl}^{2}}\int^{\sigma_{end}}_{\sigma} \frac{M_{pl}}{12\beta \sqrt{\xi}}\exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma \right) d\sigma \\
& = \frac{1}{32\xi \beta} \left[ \exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma \right) - \exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma_{end} \right)\right].
\end{split}
\end{equation}

Working with the assumption that $\sigma_{end} << \sigma $, the number of e-folds is given by

\begin{equation}\label{eqn:q39}
N = \frac{1}{32\xi \beta}\exp \left(\frac{2\sqrt{\xi}}{M_{pl}}\sigma \right).
\end{equation}

This can be rewritten to give an expression for $\sigma \left(N \right)$

\begin{equation}\label{eqn:q40}
\sigma \left(N \right) = \frac{M_{pl}}{2\sqrt{\xi}} \ln \left(32\xi \beta N \right).
\end{equation}

Substituting (<ref>) into (<ref>) and (<ref>) to find $\eta$ and $\epsilon$ in terms of the number of e-folds of inflation, we find

\begin{equation}\label{eqn:q41}
\epsilon = 128 \beta^{2} \xi \exp \left[-4\frac{\sqrt{\xi}}{M_{pl}} \left(\frac{M_{pl}}{2\sqrt{\xi}}\ln \left(32\xi \beta N \right)\right)\right] = \frac{1}{8\xi N^{2}},
\end{equation}

\begin{equation}\label{eqn:q42}
\eta = -32\beta \xi \exp \left[-2\frac{\sqrt{\xi}}{M_{pl}} \left(\frac{M_{pl}}{2\sqrt{\xi}} \ln \left(32\xi \beta N \right)\right)\right] = -\frac{1}{N}.
\end{equation}

These are the standard expressions for $\epsilon$ and $\eta$ in Palatini inflation, and are independent of the $\beta$ parameter. This means that $\eta$ and $\epsilon$ are not affected by the presence of the mass term during inflation, provided that $\beta > 0$.

It is instructive to examine the size of the $\epsilon$ parameter relative to the $\eta$ parameter in this model to check the consistency of the slow-roll approximation in this inflation model. From Section <ref> we have that slow-roll inflation ends when $\left| \eta \right| = 1$, and from (<ref>) we can see that the end of slow-roll corresponds to $N = 1$. Using (<ref>), we find that the value at the end of slow-roll inflation is $\epsilon = 1.03 \times 10^{-10}$, which is significantly smaller than the $\eta$ parameter. This shows that this model is consistent with slow-roll inflation, and that $\epsilon << \eta$ throughout.

The scalar spectral index, $n_{s}$ in terms of $\eta$ and $\epsilon$ is

\begin{equation}\label{eqn:q43}
n_{s} = 1 + 2\eta - 6\epsilon.
\end{equation}

Since $\epsilon << \eta$ in this case we approximate

\begin{equation}\label{eqn:q44}
n_{s} \approx 1 - \frac{2}{N}.
\end{equation}

The tensor-to-scalar ratio $r$ is

\begin{equation}\label{eqn:q45}
r \approx 16\epsilon = 16 \cdot \frac{1}{8\xi N^{2}} = \frac{2}{\xi N^{2}}.
\end{equation}

and the primordial curvature power spectrum is

\begin{equation}\label{eqn:q46}
\mathcal{P}_{\mathcal{R}} = \frac{V_{E}}{24\pi^{2} \epsilon M_{pl}^{4}} =  \frac{\lambda}{12\xi \pi^{2}} N^{2}.
\end{equation}

From the results of the Planck satellite experiment [12] (2018), the observed amplitude of the power spectrum is $A_{s} = 2.1 \times 10^{-9}$. Using $\lambda = 0.1$ and $N = 55$ as an estimate of the self-coupling and the pivot scale, (<ref>) gives a non-minimal coupling of  $\xi = 1.2163 \times 10^{9}$. Using the same estimate, the inflationary observables are $n_{s} = 0.9636$ and $r = 6.01 \times 10^{-13}$. The scalar spectral index is within the bounds of the 2018 results from the Planck satellite (assuming $\Lambda$CDM and no running of the spectral index), $n_{s} = 0.9649 \pm 0.0042$ (1-$\sigma$) [12], whilst the tensor-to-scalar ratio is heavily suppressed, as is typically the case in Palatini inflation models. These values of the observables are only an estimate of the predictions of the model since, as we will discuss further in Section <ref> - <ref>, the post-inflationary cosmology of this model may have an effect on the number of e-folds of inflation needed for the model to inflate successfully, and consequently the location of the pivot scale, leading to an adjustment in $n_{s}$ and $r$.

The inflaton field can be expressed in terms of the number of e-folds as

\begin{equation}\label{eqn:q47}
\phi \left(N \right) = \frac{M_{pl}}{2\sqrt{\xi}} \exp \left(\frac{\sqrt{\xi}}{M_{pl}} \left( \frac{M_{pl}}{2\sqrt{\xi}} \ln \left(32\xi \beta N \right)\right)\right) = 2\sqrt{2}M_{pl} \sqrt{\beta N},
\end{equation}

so the mass term directly influences the amplitude of the field throughout inflation. Defining the end of slow-roll inflation to be when $\left| \eta \right| \approx 1$, giving $N_{end} \approx 1$, the value of the field at the end of slow-roll inflation is then

\begin{equation}\label{eqn:q48}
\phi_{end} = 2\sqrt{2}M_{pl} \sqrt{\beta}.
\end{equation}

$\beta$ will be in the range $0.1-1$ for the values of inflaton mass relevant here. In the following section we consider limits on the inflaton mass in this model needed to produce successful inflation, and we consider how this governs the effect of the mass term on the value of the field at the end of slow-roll inflation.

§.§ The Upper Bound on the Inflaton Mass

For the purposes of the research presented here, we are primarily interested in Q-ball solutions which are compatible with non-minimally coupled Palatini inflation. Since the Q-balls will be formed from inflaton scalars, the effects of the inflaton mass from the presence of the $\beta$ parameter on the Einstein frame potential, and subsequently on the existence of Q-ball solutions, must be established, including the range of inflaton masses for which inflation is possible. It is important to ascertain the limits placed on the inflaton mass by the necessity that the potential be compatible with inflation first, in order to later establish the range of inflaton masses for which Q-ball solutions are compatible with Palatini inflation.

We start from the Einstein frame potential

\begin{equation}\label{eqn:q49}
V_{E}\left(\phi \right) = \frac{m^{2} \phi^{2}}{2\Omega^{4}} + \frac{\lambda \phi^{4}}{4\Omega^{4}},
\end{equation}

and differentiate to give

\begin{equation}\label{eqn:q50}
\frac{\partial V_{E}}{\partial \phi} = \frac{1}{\Omega^{6}}\left[m^{2}\phi + \left(\lambda - \frac{\xi m^{2}}{M_{pl}^{2}}\right) \phi^{3} \right].
\end{equation}

The inflaton potential must have a positive gradient with respect to the inflaton for inflation to occur, we therefore require

\begin{equation}\label{eqn:q51}
\frac{\partial V_{E}}{\partial \phi} > 0    \Rightarrow \lambda - \frac{\xi m^{2}}{M_{pl}^{2}} > 0.
\end{equation}

This means that, in order for inflation to occur in this non-minimally coupled Palatini model, the inflaton mass must obey the upper bound

\begin{equation} \label{eqn:q52}
m^{2} < \frac{\lambda M_{pl}^{2}}{\xi},
\end{equation}

where the upper limit on $m^{2}$ corresponds to $\beta =0$. For the purposes of studying these Q-balls numerically, we will consider masses which give a value for $\beta$ in the range $\sim 0.1 - 1$, which correspond to inflaton field values at the end of inflation (approximated as the end of slow-roll) in the range $\phi_{end} \sim \left(1 - 3 \right) M_{pl}$. The size of this mass term directly affects the shape of the potential, but will only have a significant effect on the value of the field at the end of inflation for inflation masses squared close to the upper limit (<ref>).

§ DERIVATION OF THE Q-BALL EQUATION

In this section we derive the Q-ball equation in non-minimally coupled Palatini gravity. We work in the Einstein frame in flat space with the action

\begin{equation} \label{eqn:q53}
S = \int d^{4}x \; \; \frac{1}{\Omega^{2}}\partial_{\mu}\Phi^{\dagger} \partial^{\mu}\Phi - \frac{1}{\Omega^{4}}V\left(\left| \Phi \right| \right).
\end{equation}

We will show that the flat space calculation is a valid approximation for the Q-ball solutions that we will obtain. While it is true that gravitational effects can affect the stability of Q-balls within certain limits, or alter the size of the Q-balls (see e.g. [166], [167]), the attractive interaction between scalars dominates over any gravitational effects for Q-balls of the size we are working with in this model. The effects of gravity, and indeed the presence of a non-minimal coupling of gravity to the scalar field in the Jordan frame are factors to consider in the post-inflationary cosmology of this model and may affect the survival of any relic Q-balls to the present day. We we will later consider a first-pass approximation of the effects of curvature on these Q-ball solutions in Section <ref> of this chapter, and proceed with the understanding that the purpose of this work is to ascertain the flat space properties of these Q-balls. Expansion could affect the formation of these Q-balls and their subsequent evolution, and this is something we touch upon in Sections <ref> - <ref>. For the purposes of establishing the existence of Q-balls made from scalars with a non-minimal coupling to gravity without the cosmological considerations of their existence we derive the Q-ball equation using the energy-momentum tensor for a scalar field in a non-expanding spacetime, and compare these Q-balls to the conventional flat space Q-balls as derived in [50].

In order to derive the Q-ball equation we begin by using the method of Lagrange multipliers in order to minimise the energy of the inflaton field with respect to the conserved $U(1)$ charge of the field

\begin{equation} \label{eqn:q54}
E_{Q} = E + \omega \left( Q - \int d^{3}x \rho_{Q} \right),
\end{equation}

which gives a "Q-ball action", $E_{Q}$, which we will refer to as the Q-ball energy functional. This can be extremised to obtain the field equations of the theory, the solutions of which - subject to a number of conditions which we discuss in Sections <ref> - <ref> - correspond to Q-balls. The global energy, $E$, and charge, $Q$, are

\begin{equation} \label{eqn:q55}
Q = \int d^{3}x \; j^{0} = \int d^{3}x \rho_{Q},
\end{equation}

\begin{equation} \label{eqn:q56}
E = \int d^{3}x \;  T^{00} = \int d^{3}x  \rho_{E},
\end{equation}

where $j^{0}$ is the temporal component of the conserved $U(1)$ Noether current, $j^{\mu}$, and $T^{00}$ is the temporal component of the energy-momentum tensor $T^{\mu \nu}$ for a complex scalar field, given by

\begin{equation} \label{eqn:q57}
T^{\mu \nu} = \frac{\partial \mathcal{L}}{\partial \left(\partial_{\mu}\phi_{a}\right)}\eta^{\nu \rho}\partial_{\rho}\phi_{a} - \delta^{\mu}_{\rho} \eta^{\nu \rho}\mathcal{L},
\end{equation}

where $\eta^{\nu \rho}$ is the Minkowski metric. The energy density of the $\Phi$ field is therefore

\begin{equation} \label{eqn:q58}
\rho_{E} = T^{00} = \frac{1}{\Omega^{2}}\partial_{t}\Phi^{\dagger} \partial_{t}\Phi + \frac{1}{\Omega^{2}}\partial_{i}\Phi^{\dagger} \partial^{i}\Phi + \frac{V(\mid \Phi \mid)}{\Omega^{4}}.
\end{equation}

This includes the effect of the non-canonical kinetic term of $\Phi$, which has not been previously studied in the context of Q-ball solutions.

The conserved Noether current $j^{\mu}$ of the $U(1)$ symmetry of the model is defined by

\begin{equation} \label{eqn:q59}
\partial_{\mu} j^{\mu} = 0, j^{\mu} = \frac{\partial \mathcal{L}}{\partial \left(\partial_{\mu}\phi_{a}\right)}\delta \phi_{a}.
\end{equation}

The temporal components of (<ref>) are

\begin{equation} \label{eqn:q60}
j^{0} = \frac{\partial \mathcal{L}}{\partial \left(\partial_{t}\Phi \right)} i\Phi - \frac{\partial \mathcal{L}}{\partial \left(\partial_{t} \Phi^{\dagger}\right)} i\Phi^{\dagger},
\end{equation}

which from (<ref>) gives the charge density of the inflaton field to be

\begin{equation} \label{eqn:q61}
\rho_{Q} = \frac{i}{\Omega^{2}}\left( \Phi \partial_{t}\Phi^{\dagger} - \Phi^{\dagger}\partial_{t}\Phi \right).
\end{equation}

Substituting (<ref>) and (<ref>) into the Q-ball energy functional (<ref>) gives

\begin{equation} \label{eqn:q62}
E_{Q} = \int d^{3}x \left[ \frac{1}{\Omega^{2}}\partial_{t}\Phi^{\dagger} \partial_{t}\Phi + \frac{1}{\Omega^{2}}\partial_{i}\Phi^{\dagger} \partial^{i}\Phi + \frac{V(\left|\Phi \right|)}{\Omega^{4}} - \frac{\omega i}{\Omega^{2}}\left( \Phi \partial_{t}\Phi^{\dagger} - \Phi^{\dagger}\partial_{t}\Phi \right) \right] + \omega Q.
\end{equation}

The temporal derivative terms in (<ref>) can be rewritten as

\begin{equation}
\begin{split} \label{eqn:q63}
\left(\partial_{t}\Phi - i\omega \Phi \right) \left(\partial_{t}\Phi^{\dagger} + i\omega \Phi^{\dagger} \right)
& = \partial_{t}\Phi \partial_{t}\Phi^{\dagger} + i\omega \Phi^{\dagger} \partial_{t}\Phi - i\omega \Phi \partial_{t} \Phi^{\dagger} - \omega^{2}\Phi^{\dagger}\Phi \\
& = \mid \partial_{t}\Phi - i\omega \Phi \mid^{2},
\end{split}
\end{equation}

in order to rewrite the Q-ball energy functional in a more insightful way. If we define the time derivatives in the integrand of (<ref>) as

\begin{equation} \label{eqn:q64}
I = \frac{1}{\Omega^{2}}\left[ \partial_{t}\Phi^{\dagger}\partial_{t}\Phi + i\omega \Phi^{\dagger}\partial_{t}\Phi - i\omega \Phi \partial_{t}\Phi^{\dagger} \right],
\end{equation}

these can be rewritten as in (<ref>)

\begin{equation} \label{eqn:q65}
I = \frac{1}{\Omega^{2}}\left[ \mid \partial_{t}\Phi - i\omega \Phi \mid^{2} - \omega^{2} \mid \Phi \mid^{2} \right].
\end{equation}

Substituting (<ref>) back into the Q-ball energy functional (<ref>) we have that

\begin{equation} \label{eqn:q66}
E_{Q} = \int d^{3}x \left[ \frac{1}{\Omega^{2}} \mid \partial_{t}\Phi - i \omega \Phi \mid^{2} - \frac{1}{\Omega^{2}} \omega^{2} \mid \Phi \mid^{2} + \frac{1}{\Omega^{2}}\partial_{i}\Phi ^{\dagger} \partial^{i} \Phi + \frac{1}{\Omega^{4}}V\left(\mid \Phi \mid \right) \right] + \omega Q,
\end{equation}

whereupon it is clear that a solution which extremises $E_{Q}$ is

\begin{equation} \label{eqn:q67}
\Phi \left(x,t \right) = \Phi \left( x \right) e^{i \omega t} \longrightarrow \partial_{t} \Phi = i\omega \Phi.
\end{equation}

Applying (<ref>), the Q-ball energy functional (<ref>) becomes

\begin{equation} \label{eqn:q68}
E_{Q} = \int d^{3}x \left[ \frac{1}{\Omega^{2}} \mid \overrightarrow{\nabla} \Phi \mid^{2} + \frac{1}{\Omega^{4}} V\left(\mid \Phi \mid \right) - \frac{1}{\Omega^{2}}\omega^{2} \mid \Phi \mid^{2} \right].
\end{equation}

\begin{equation} \label{eqn:q69}
V_{\omega} \left( \mid \Phi \mid \right) = \frac{1}{\Omega^{4}}V \left( \mid \Phi \mid \right) - \frac{1}{\Omega^{2}}\omega^{2}\mid \Phi \mid^{2},
\end{equation}

be defined as the Q-ball potential. Assuming that the Q-balls are spherically symmetric, we assume that the coordinate dependence of the inflaton field, when extremised with respect to its charge, is purely radial, which gives

\begin{equation} \label{eqn:q70}
\Phi \left(\textbf{x}\right) = \frac{\phi \left(\textbf{r}\right)}{\sqrt{2}} e^{i\omega t} = \frac{\phi\left(r\right) \hat{r}}{\sqrt{2}}e^{i\omega t},
\end{equation}

as the field solution corresponding to Q-balls.

Since the coordinate dependence of the field is purely radial, the gradient operator is reduced to

\begin{equation} \label{eqn:q71}
\overrightarrow{\nabla}\Phi = \frac{\partial \Phi}{\partial r}\hat{r} \\
\longrightarrow \frac{1}{\Omega^{2}} \left| \overrightarrow{\nabla}\Phi \right| = \frac{1}{\Omega^{2}}\left| \frac{1}{\sqrt{2}}\frac{\partial \phi}{\partial r}\hat{r} \right|^{2} = \frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2}.
\end{equation}

Rewriting in spherical polar coordinates, we have that the Q-ball energy functional is

\begin{equation} \label{eqn:q72}
E_{Q} = \int dr \; 4\pi r^{2} \left[ \frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2} + V_{\omega} \left(\phi \right) \right] + \omega Q.
\end{equation}

We define the function $\mathcal{L}_{Q}$ (which we refer to as the "Q-ball effective Lagrangian") to be extremised as

\begin{equation} \label{eqn:q73}
\mathcal{L}_{Q}= 4\pi r^{2} \left[ \frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2} + V_{\omega} \left(\phi \right) \right],
\end{equation}

and apply the Euler-Lagrange equations (derived for a complex scalar field in flat space in (<ref>))

\begin{equation} \label{eqn:q74}
\frac{\partial \mathcal{L}_{Q}}{\partial \phi} - \frac{d}{dr}\left(\frac{\partial \mathcal{L}_{Q}}{\partial \left(\partial_{r}\phi\right)}\right) = 0,
\end{equation}

in order to extremise (<ref>).

Using (<ref>), the second term of (<ref>) is

\begin{multline} \label{eqn:q75}
\frac{d}{dr}\left(\frac{\partial \mathcal{L}_{Q}}{\partial \left(\partial_{r}\phi\right)}\right) = \frac{4\pi r^{2}}{\Omega^{2}} \frac{\partial \phi}{\partial r}\frac{d}{dr}\left(\frac{\partial \mathcal{L}_{Q}}{\partial \left(\partial_{r}\phi\right)}\right) \\
= \frac{8\pi r}{\Omega^{2}}\frac{\partial \phi}{\partial r} + \frac{4 \pi r^{2}}{\Omega^{2}}\frac{\partial^{2}\phi}{\partial r^{2}} - \frac{8 \pi r^{2}}{M_{pl}^{2}}\frac{\xi \phi}{\Omega^{4}}\left(\frac{\partial \phi}{\partial r}\right)^{2},
\end{multline}

and the first term of (<ref>) is

\begin{equation} \label{eqn:q76}
\frac{\partial \mathcal{L}_{Q}}{\partial \phi} = 4\pi r^{2} \frac{\partial V_{\omega}}{\partial \phi} - \frac{4 \pi r^{2}}{M_{pl}^{2}}\frac{\xi \phi}{\Omega^{4}}\left(\frac{\partial \phi}{\partial r}\right)^{2}.
\end{equation}

Substituting (<ref>) and (<ref>) into (<ref>), dividing through by a factor of $4\pi r^{2}$ and multiplying by $\Omega^{2}$ gives the Q-ball equation

\begin{equation} \label{eqn:q77}
\frac{\partial^{2}\phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} - K\left(\phi \right) \left(\frac{\partial \phi}{\partial r}\right)^{2} = \Omega^{2}\frac{\partial V_{\omega}}{\partial \phi},
\end{equation}

\begin{equation} \label{eqn:q78}
K(\phi) = \frac{\xi \phi}{M_{pl}^{2} \Omega^{2}}.
\end{equation}

Equation (<ref>) corresponds to an entirely new and different form of the Q-ball equation than that for a scalar theory with canonical kinetic terms, and therefore to a new class of Q-ball. The result of the Q-ball equation for the minimally coupled case [50] can be recovered by setting $\Omega^{2} = 1 \left(\xi = 0 \right)$

\begin{equation}\label{eqn:q79}
\frac{\partial^{2} \phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} = \frac{\partial V_{\omega}}{\partial \phi},
\end{equation}

where in this instance

\begin{equation}\label{eqn:q80}
V_{\omega} \left( \phi \right) = \frac{1}{2}m^{2}\phi^{2} + \frac{\lambda}{4}\phi^{4} - \frac{1}{2}\omega^{2}\phi^{2}.
\end{equation}

Where we see that the difference the presence of the non-minimal coupling for (<ref>) makes as compared to (<ref>) is in the additional gradient squared term, and the dependence of the conformal factor in the Q-ball potential and on the right-hand side. This shows that, although the non-minimal coupling of the scalar field to gravity is recast in the Einstein frame using the conformal transformation, its effects are still manifest in the construction of the Q-balls in the mathematical sense, in that the attractive interaction binding the scalars into Q-balls seems to contain an additional component due to the non-minimal coupling. Physically, in the Jordan frame, this would be gravitational in nature, whereas as the Q-balls are studied here in the Einstein frame, it is merely a component of the regular attractive interaction between the inflaton scalars.

§.§ Existence Conditions of Q-balls

The existence of Q-ball solutions in a given theory is heavily dependent on the form of the potential, and the realisation of physical Q-balls relies on the $\omega$ parameter being within an acceptable range. The physical reasons for this will be discussed in later sections. In this section we explore the form of the potential in this model of non-minimally coupled Palatini Q-balls, derive the constraints on the existence of Q-balls in this model from the potential and compare the results to those derived in the case of conventional Q-balls.

§.§.§ Rescaling the Non-Canonical Q-ball Equation in Terms of a Quasi-Canonical Scalar

The Palatini Q-ball equation in terms of the rescaled field $\sigma$ has a similar (but not the same) form as the general form of the Q-ball equation for the conventional case (<ref>), and the dynamics can therefore be understood as being similar in this form.

To illustrate this, we begin with the non-canonical Q-ball equation derived at the beginning of the section

\begin{equation}\label{eqn:q81}
\frac{\partial^{2} \phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} - \frac{\xi \phi}{M_{pl}^{2} \Omega^{2}} \left(\frac{\partial \phi}{\partial r}\right)^{2} = \Omega^{2}\frac{\partial V_{\omega}}{\partial \phi},
\end{equation}

and apply the rescaling

\begin{equation}\label{eqn:q82}
\frac{d\phi}{d r} \longrightarrow \Omega \frac{d\sigma}{d r},
\end{equation}

such that

\begin{equation}\label{eqn:q83}
\frac{d^{2}\phi}{d r^{2}} = \frac{d}{d r}\left(\frac{d\phi}{d r}\right) = \frac{d}{d r}\left(\Omega \frac{d\sigma}{d r}\right) = \Omega \frac{d^{2}\sigma}{d r^{2}} + \frac{d\Omega}{d r }\frac{d\sigma}{d r},
\end{equation}

\begin{equation}\label{eqn:q84}
\frac{d V_{\omega}}{d\phi} = \frac{d \sigma}{d \phi} \frac{d V_{\omega}}{d \phi} = \frac{1}{\Omega}\frac{d V_{\omega}}{d \sigma}.
\end{equation}

Substituting (<ref>) - (<ref>) into (<ref>) gives the rescaled Q-ball equation

\begin{equation}\label{eqn:q85}
\Omega \frac{d^{2}\sigma}{d r^{2}} + \frac{2\Omega}{r} \frac{d\sigma}{d r} + \frac{d\Omega}{d r} \frac{d \sigma}{d r} - \frac{\xi \phi \Omega^{2}}{M_{pl}^{2} \Omega^{2}}\left(\frac{d \sigma}{d r}\right)^{2} = \frac{\Omega^{2}}{\Omega}\frac{d V_{\omega}}{d \sigma}.
\end{equation}

We can write

\begin{equation}\label{eqn:q86}
\frac{d\Omega}{d r } = \frac{d\phi}{d r} \frac{d\Omega}{d\phi} = \frac{d\sigma}{d r} \frac{d\phi}{d\sigma} \frac{d\Omega}{d\phi},
\end{equation}

\begin{equation}\label{eqn:q87}
\frac{d\Omega}{d\phi} = \frac{d}{d\phi}\sqrt{1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}} = \frac{1}{2} \frac{2\xi \phi}{M_{pl}^{2}}\left(1 + \frac{\xi \phi^{2}}{M_{pl}^{2}}\right)^{-\frac{1}{2}} = \frac{\xi \phi}{M_{pl}^{2} \Omega}.
\end{equation}

\begin{equation}\label{eqn:q88}
\frac{d\Omega}{d r} = \frac{d\sigma}{d r} \frac{d\phi}{d\sigma} \frac{d\Omega}{d\phi} = \frac{d\sigma}{d r} \Omega \frac{\xi \phi}{M_{pl}^{2} \Omega} = \frac{d\sigma}{d r} \frac{\xi \phi}{M_{pl}^{2}}.
\end{equation}

Using (<ref>) on the third term of (<ref>), the rescaled Q-ball equation becomes

\begin{equation}\label{eqn:q89}
\Omega \frac{d^{2}\sigma}{d r^{2}} + \frac{2\Omega}{r} \frac{d\sigma}{d r} + \frac{\xi \phi}{M_{pl}^{2}}\left(\frac{d \sigma}{d r}\right)^{2} - \frac{\xi \phi}{M_{pl}^{2}}\left(\frac{d \sigma}{d r}\right)^{2} = \Omega\frac{d V_{\omega}}{d \sigma}.
\end{equation}

We can then cancel the gradient squared terms and divide through by $\Omega$ to obtain the Q-ball equation in terms of the $\sigma$ field

\begin{equation}\label{eqn:q90}
\frac{d^{2}\sigma}{d r^{2}} + \frac{2}{r}\frac{d\sigma}{d r} = \frac{d V_{\omega}}{d\sigma},
\end{equation}

which is exactly the same form as the Q-ball equation for $\phi$ in the $\Omega \rightarrow 1$ limit (<ref>), but with $V_{\omega}$ given by (<ref>). This shows that this is not the same as the equation for a canonically normalised field, as in that case we would have $\sigma^{2}$ rather than $\phi^{2}(\sigma)/\Omega^{2}(\sigma)$ in (<ref>). Therefore (<ref>) will produce significantly different Q-ball solutions compared to the conventional Q-ball solution, although it does suggest that there are some broad similarities between the two solutions.

It is at this point that we highlight the fact that it is not in general possible to transform a complex field such as $\Phi$ to a canonically normalised scalar in the strict sense, and that the transformation used in this case to compare the dynamics of the non-minimally coupled case to conventional Q-balls is quasi-canonical. The rescaled field $\sigma$ is therefore not interpreted as a physically meaningful field, but a function of the radial component $\phi$ of the physical complex inflaton used to illustrate some of the properties of the non-canonical Q-ball solution.

§.§.§ Q-ball Existence and Coleman's Mechanical Analogy in the Case of Non-Minimally Coupled Palatini Q-balls

In this section we discuss the application of Coleman's mechanical analogy [50], originally used as a tool for understanding the precise nature of Q-ball solutions and the dependence of their existence on the form of the scalar potential, to Q-balls in the non-minimally coupled Palatini case. This analogy was originally derived for Q-balls in the canonical scalar field case, and the underlying dynamics in this model therefore look a little different, as we will discuss later.

We first introduce the analogy in the context of the conventional case of a canonically normalised scalar, for which $\sigma = \phi$. If we interpret the field $\sigma$ $(\phi)$ as a position coordinate, and $r$ as a time coordinate, then (<ref>) ((<ref>)) looks like it is describing the damped motion of a particle moving in the potential $-V_{\omega}$. Figure <ref> schematically shows the potential $-V_{\omega}$ and the motion of the "particle" within it .

Schematic potential and particle motion as described by (<ref>) and (<ref>).

The "particle" begins its motion at the black circle on the right, at a starting point we will refer to as $\phi_{0}$. The local maximum at the origin of the potential $\phi = 0$ corresponds to the field in the vacuum at $r \rightarrow \infty$, and the idea of this analogy is that in order to obtain a Q-ball solution in a given potential $V_{\omega}$, the conditions must be precisely right such that the particle rolling down from $\phi_{0}$ on the right comes to rest exactly at the origin as illustrated in Figure <ref>. One such condition is the initial placement, or starting location of the particle, $\phi_{0}$. If the particle is initially placed too close to $\phi = 0$ - $\phi_{0}$ is too small - then the particle will not gain enough momentum to reach the top of the local maximum at $\phi = 0$, and it will roll backwards to oscillate about the local minimum to the right in Figure <ref> (clear circle on the right hand side). This is known as undershoot, and can also occur if $\phi_{0}$ is sufficiently high but the gradient of the potential is too shallow. Conversely, if $\phi_{0}$ is too large (starting placement is too far from $\phi = 0$, or the potential itself is too steep on the approach to the positive $\phi$ local minimum, the field will gather too much momentum and will roll clear of the local maximum at $\phi = 0$ and into the negative $\phi$ regime, where it will oscillate about the negative $\phi$ local minimum on the left in Figure <ref> (clear circle on the left hand side). This is known as overshoot.

Plots of $-V_{\omega}\left(\phi \right)$ for $\phi > 0$ are shown in Figure <ref> and Figure <ref>. Interestingly, the inverted potential before rescaling has the same form as the inverted potential in the conventional case (comparing to the $\phi > 0$ region of the schematic in Figure <ref>) corresponding to (<ref>). This demonstrates that the idea of Coleman's mechanical analogy can still be applied to explore the dependence of the existence of non-canonical Q-balls on the form of the scalar field potential in the underlying theory, although the underlying dynamics are different. As shown at the beginning of this section, in the non-minimally coupled Palatini case, the Q-ball equation in terms of $\phi$, (<ref>), has an additional gradient squared term with a negative sign and a dependence on the conformal factor on the right-hand side, both explicitly and in the form of the Q-ball potential, $V_{\omega}$, in the non-minimally coupled case. The gradient squared term can be interpreted as some external energy input, which makes the motion of the particle dynamically different from the simple Newtonian motion described in (<ref>), although the inverted potential looks very similar and the general idea as an analogy for the precise conditions on the potential and the field itself to produce Q-balls is still applicable. When considered in terms of the rescaled field $\sigma$, the Q-ball equation, has the same form as in the conventional case but with a modified potential $V_{\omega}$, (<ref>). Equation (<ref>) can therefore also be used as a means for comparison to the conventional case when considering the mechanical analogy with the understanding that, in our definition of the quasi-canonical field, $\phi$ is a function of $\sigma$ such that $V_{\omega}\left(\phi \right) = V_{\omega}\left(\phi \left(\sigma \right) \right)$.

$-V_{\omega}(\phi)$ for a non-minimally coupled Palatini Q-ball solution with large $\omega$ .

$-V_{\omega}(\phi)$ for a non-minimally coupled Palatini Q-ball solution with small $\omega$.

We can now use the rules derived from this analogy [50] for conventional minimally-coupled Q-balls to derive an analogous set of rules for the case of non-minimally coupled Q-balls to ensure that the Q-ball potential is compatible with producing Q-ball solutions. To avoid undershoot, the initial placement of the particle ($\phi_{0}, -V_{\omega}\left(\phi_{0}\right)$), must sit at least at, or higher than zero in order for the particle to reach the local maximum of the origin, otherwise the particle will not gather sufficient momentum. This condition can be expressed as

\begin{equation}\label{eqn:q91}
max\left(-V_{\omega}\right) \geq 0,
\end{equation}

as it is in the conventional model of Q-balls [50], and is significant because it implies that there is a lower bound on $\omega$ for which Q-balls can exist in a given potential. For the $V_{\omega}$ used in this model (<ref>), this result does not appear to produce anything meaningful analytically, which is not an unexpected result as the inflaton potential in the Einstein frame corresponds to a plateau. It is possible that there is another way to attain a lower bound on the value of $\omega$ for non-minimally coupled Palatini Q-balls, and we will return to this possibility in Section <ref> when numerical solutions of the Q-balls are explored, although a definitive lower bound on $\omega$ in this model is at the point of submitting this thesis unconfirmed.

To avoid overshoot we require that there is an extremum at $\phi = 0$ which is a local maximum, so that the particle can come to rest in the vacuum and not continue rolling into the $\phi < 0$ region

\begin{equation}\label{eqn:q92}
\left.\frac{d^{2}\left(-V_{\omega} \right)}{d\phi^{2}}\right|_{\phi = 0} < 0,
\end{equation}

this condition may be satisfied if and only if

\begin{equation}\label{eqn:q93}
\left.\omega^{2} < \frac{d^{2}V}{d\phi^{2}} \right|_{\phi = 0}.
\end{equation}

Calculating the left-hand side of (<ref>) we have

\begin{equation}\label{eqn:q94}
\frac{d^{2}\left(-V_{\omega} \right)}{d\phi^{2}} = \frac{\omega^{2}}{\Omega^{2}} - \frac{m^{2}}{\Omega^{4}} - \frac{3\lambda \phi^{2}}{\Omega^{4}}.
\end{equation}

At $\phi = 0$, $\Omega =1$ and (<ref>) therefore gives the result

\begin{equation}\label{eqn:q95}
\left.\frac{d^{2}\left(-V_{\omega} \right)}{d\phi^{2}}\right|_{\phi = 0} = \omega^{2} - m^{2} < 0 \Rightarrow \omega^{2} < m^{2},
\end{equation}

and from (<ref>) we find

\begin{equation}\label{eqn:q96}
\left.\omega^{2} < \frac{d^{2}V}{d\phi^{2}} \right|_{\phi = 0} = m^{2}.
\end{equation}

This means that

\begin{equation}\label{eqn:q97}
\omega < m,
\end{equation}

is a hard constraint on the existence of non-minimally coupled Palatini Q-balls, and is the same result as that on the existence of conventional minimally-coupled Q-balls derived from the mechanical analogy in [50], as the condition to avoid overshoot.

§.§ Q-ball Stability

Stability of Q-balls is an important property, particularly when discussing Q-balls as cosmological objects. A Q-ball is said to be absolutely stable if [50]

\begin{equation}\label{eqn:q98}
E < mQ,
\end{equation}

holds. $E$ is the energy of the Q-ball and $mQ$ can be interpreted as the energy of $Q$ free quanta of scalar particles of mass $m$ in the vacuum. This essentially means that in order to be absolutely stable the energy of a Q-ball must be less than the sum of the energies of the individual component scalars it is composed of when treated as free quanta. This difference in energy corresponds to the binding energy of the Q-ball. Classical stability implies a solution that is stable against small perturbations. As an approximation, the classical stability of a Q-ball can be gauged by the sign of the derivative of the charge with respect to $\omega$, i.e. if $\partial Q/\partial \omega <0$ then a Q-ball is classically stable [141] [142]. The absolute stability of the numerically generated non-minimally coupled Palatini Q-balls is discussed in Section <ref>.

§.§ Mass Range of the Inflaton from Inflation and the Existence of Q-balls

In Section <ref>, we demonstrated that the form of the inflaton potential - and the state of the inflaton field at the end of inflation - depends explicitly on the size of the mass term (<ref>), which means that the size of this mass term determines the initial conditions for Q-balls in non-minimally coupled Palatini inflation. It is therefore important that we establish the range of inflaton masses which are compatible with the existence condition (<ref>) of these Q-balls. In Section <ref>, we derived the constraint (<ref>) which gives an upper bound on the inflaton mass squared, from the condition that the inflaton potential must be of positive gradient in order for inflation to proceed. In this section, we use this, and the existence condition of non-minimally coupled Palatini Q-balls derived in the previous section, (<ref>), to derive a range of inflaton masses for which both inflation can occur and inflatonic Q-balls can exist. We will refer to this range of compatible inflaton masses which produce both inflation and Q-balls as the "Q-ball window".

We begin with the non-canonical Q-ball potential

\begin{equation}\label{eqn:q99}
V_{\omega} \left(\phi \right) =  \frac{1}{\Omega^{4}}\left(\frac{1}{2} m^{2} \phi^{2} + \frac{\lambda}{4} \phi^{4} \right)  - \frac{\omega^{2} \phi^{2}}{2\Omega^{2}},
\end{equation}

and expand to leading order in $M_{pl}^{2}/\xi \phi^{2}$ to give

\begin{equation}\label{eqn:q100}
V_{\omega} \left(\phi \right) = \frac{M_{pl}^{4}}{2\xi^{2}\phi^{2}}\left( m^{2} + \omega^{2} - \frac{2m^{2}M_{pl}^{2}}{\xi \phi^{2}} \right) - \frac{\lambda M_{pl}^{6}}{2 \xi^{3}\phi^{2}} + \frac{\lambda M_{pl}^{4}}{4\xi^{2}} - \frac{\omega^{2}M_{pl}^{2}}{2\xi}.
\end{equation}

In other words,

\begin{equation} \label{eqn:q101}
V_{\omega}\left(\phi \right) = \frac{M_{pl}^{4}}{2\xi^{2}\phi^{2}}\left[ m^{2} + \omega^{2} - \frac{\lambda M_{pl}^{2}}{\xi} \right] + \hspace{1mm}higher\hspace{1.0mm} order\hspace{1.0mm} and \hspace{1.0mm} constant ~\hspace{0.8mm} terms.
\end{equation}

For (<ref>) to yield a Q-ball solution, the solution $\phi \left(r \right)$ must decrease as $r$ increases from zero. This is true if the $\phi$ dependent term on the right hand side of (<ref>) is positive. The existence condition for a Q-ball from the Q-ball potential (<ref>) is therefore

\begin{equation}\label{eqn:q102}
m^{2} + \omega^{2} > \frac{\lambda M_{pl}^{2}}{\xi}.
\end{equation}

In Section <ref> we will analytically demonstrate that if $\omega < m$, as required in order for Q-balls to exist, then the condition (<ref>) must also be satisfied in order for the right-hand side of the Q-ball equation to have zeroes for some $\phi \neq 0$. Both interpretations of (<ref>) correspond to a necessary condition on the Q-ball potential for Q-ball solutions to exist in the model.

We define

\begin{equation}\label{eqn:q103}
\omega_{c}^{2} = \frac{\lambda M_{pl}^{2}}{\xi},
\end{equation}

and use this to rewrite (<ref>)

\begin{equation} \label{eqn:q104}
m^{2} + \omega^{2} > \omega_{c}^{2}.
\end{equation}

If we combine (<ref>)  with the condition for the inflaton potential to be compatible with inflation (<ref>), which we can write as $m^{2} < \omega_{c}^{2}$, we find the condition for inflation and the existence of Q-balls in terms of the $\omega$ parameter and the inflaton mass to be

\begin{equation}\label{eqn:q105}
m^{2} < \omega_{c}^{2} < m^{2} + \omega^{2}.
\end{equation}

The existence of Q-balls also requires that $\omega < m$ (<ref>), and the right-hand side of the inequality (<ref>) can be constrained

\begin{equation}\label{eqn:q106}
\omega^{2} + m^{2} < 2m^{2}.
\end{equation}

(<ref>) can therefore be stated as

\begin{equation}\label{eqn:q107}
m^{2} < \omega_{c}^{2} < 2m^{2},
\end{equation}

which may be equivalently written as an range of inflaton masses squared

\begin{equation} \label{eqn:q108}
\frac{\omega_{c}^{2}}{2} < m^{2} < \omega_{c}^{2}.
\end{equation}

This gives the range in inflaton mass squared (the Q-ball window) for which the inflaton potential is compatible with both inflation and with the existence of Q-balls. It also provides a mass window within which we can numerically search for Q-ball solutions which are compatible with inflation, the topic of the next section. From (<ref>), the Q-ball window seems to favour inflaton masses close to the upper bound derived from inflation (<ref>), and while Q-balls may form for inflaton masses larger than the upper bound (<ref>) in this model, inflation would not be possible in conjunction with these Q-balls.

§ NUMERICAL SOLUTIONS OF THE Q-BALL EQUATION

In this section we discuss and present the results of the numerical analysis of non-minimally coupled Palatini Q-balls. In this project we solved the non-canonical Q-ball equation (<ref>) numerically for a fixed mass within the range specified in (<ref>), over a range of $\omega$ for the following boundary conditions at $r=0$

\begin{equation} \label{eqn:q109}
\phi \left(r =0\right) = \phi_{0} \;\;\;\; ; \;\;\;
\frac{\partial \phi}{\partial r} \left(r=0\right) = 0.
\end{equation}

For each $\omega$ we ran the code for trial values of the inflaton field, corresponding to $\phi_{0}$, and scanned for Q-ball solutions. The details of this methodology are outlined in the following subsection. When discussing numerical results, the inflaton mass $m$ and the parameter $\omega$ are listed in units of $\omega_{c}$.

§.§ Zeroes of the Q-ball Equation

In this section we discuss the zeroes of the right-hand side Q-ball equation (<ref>), and how they relate to the existence of Q-ball solutions. These zeroes are shown in Figure <ref>, and correspond to fixed points of the Q-ball equation, which in turn correspond to the extrema of the Q-ball effective potential $V_{\omega}\left(\phi\right)$, shown schematically in Figure <ref>. The fixed points of the Q-ball equation can be related to the mechanical analogy outlined in Section <ref> and more importantly, used to constrain $\phi_{0}$ when searching for Q-ball solutions numerically.

The non-minimally coupled Q-ball equation derived in this work (<ref>) has three zeroes (shown in Figure <ref>). These correspond to two stable fixed points, which are symmetric attractors about $\Omega^{2}\partial V_{\omega}/\partial \phi = 0$, and one unstable fixed point.

Plot illustrating the zeroes of the Q-ball equation.

Figure <ref> shows an example of the right hand side of the Q-ball equation as a function of $\phi$. The points where the function crosses zero are the zeroes or fixed points of the Q-ball equation. The attractor fixed points are the furthest to the left (denoted $\phi_{-}$) in the negative region of $\phi$ and the furthest to the right (denoted $\phi_{+}$) in the positive regime of $\phi$. These provide a reference for determining the value of $\phi$ at $r = 0$, $\phi_{0}$, for which a Q-ball solution exists. If the field gets caught by - and starts to oscillate around -  the positive fixed point, $\phi_{+}$, as $r$ increases, then $\phi_{0}$ is too low to produce a Q-ball. The lowest $\phi_{0}$ which produces this result can then be used as a lower bound for the range of $\phi_{0}$ which could generate a Q-ball solution for a given $\omega$. In the opposite limit, if $\phi_{0}$ is too high then the field can drop below zero as $r$ increases and begin to oscillate around the negative fixed point, $\phi_{-}$. The lowest $\phi_{0}$ which produces this result can then act as an upper bound on the range of $\phi_{0}$ for which there could exist a Q-ball solution for the chosen $\omega$. The fixed points of the Q-ball equation therefore produce behaviour in the function of the field over distance, $\phi \left(r \right)$, which can be used to restrict the parameter space of $\phi_{0}$ used when searching for Q-ball solutions numerically.

This can be understood in terms of the mechanical analogy of Q-balls outlined in Section <ref> by noting that the fixed points of the Q-ball equation correspond to the extrema of the inverted Q-ball effective potential $-V_{\omega}$, in the analogy. The local minimum in the $\phi >0$ region of Figure <ref> corresponds to the positive attractor fixed point $\phi_{+}$ (first zero from the right in Figure <ref>). The unstable fixed point corresponds to the local maximum at $\phi =0$ in Figure <ref>, and is the central fixed point in Figure <ref>. The local minimum in the $\phi < 0$ region of the potential in Figure <ref> corresponds to the symmetric negative fixed point $\phi_{-}$ (furthest zero from the right in Figure <ref>).

This illustrates how precise the values of the parameters need to be to obtain a Q-ball solution numerically, as well as how strict the conditions on the potential in order to produce Q-balls are. The field profile for a Q-ball solution is represented by $\phi\left(r\right)$ asymptoting to zero as $r \rightarrow \infty$, corresponding to the field coming to rest exactly at the top of the local maximum at $\phi = 0$ in Figure <ref> from its starting point $\phi_{0}$ in the language of Coleman's analogy. It is also important to emphasise that the existence of the zeroes of the Q-ball equation determine the existence of the Q-ball solutions themselves, without the fixed points there cannot be Q-ball solutions.

We now derive the condition for the existence of zeroes of the Q-ball equation. The requirement for extrema to exist is that $\partial V_{\omega}/\partial \phi = 0$ for some $\phi \neq 0$. Using (<ref>) and setting the first derivative to zero gives the location of the symmetric $\phi \neq 0$ zeroes to be

\begin{equation} \label{eqn:q110}
\phi = \pm \frac{M_{Pl}}{\sqrt{\xi}} \frac{\sqrt{m^{2} - \omega^{2}}}{\sqrt{m^{2} + \omega^{2} - \omega_{c}^{2} }  }.
\end{equation}

In order for the Q-ball solution to be stable we require that $\omega^{2} < m^{2}$. This will be verified both numerically later in this section and analytically in Section <ref>. The range of values of $m^{2}$ for which relevant zeros of the Q-ball equation exist is therefore

\begin{equation}\label{eqn:q111}
m^{2} + \omega^{2} > \omega_{c}^{2},
\end{equation}

which is exactly the condition (<ref>) derived from the Q-ball equation. We confirm numerically in Section <ref> that zeroes with $\phi \neq 0$ exist over the range of $\omega$ satisfying (<ref>) and (<ref>).

§.§ Numerical Results

In this section we present the results of solving the non-minimally coupled Palatini Q-ball equation numerically, and discuss the properties of these Q-balls. We present ten Q-ball solutions for the inflaton field of mass $m = 0.9\omega_{c}$, which is within the Q-ball window (<ref>), close to the upper bound of inflaton masses which inflate the model (<ref>), for a range of $\omega$ from $0.89\omega_{c}$ to $0.707155\omega_{c}$. This produces Q-balls with initial field values in the range $\phi_{0} \sim 10^{13} - 10^{17} \GeV$, illustrating that Q-balls could be produced from the kind of field values predicted at the end of non-minimally coupled Palatini inflation in this model. We present the radii, charge and energy of the Q-balls, in addition to some other parameters in relation to stability of the Q-balls which we define later in this section. The bounds on $\omega$ are chosen because they lie close to the bounds for which a Q-ball solution can be obtained for $m = 0.9\omega_{c}$. The upper bound, $\omega = 0.89\omega_{c}$, lies close to the stability limit of $\omega = m$ for Q-balls of the chosen mass, whereas the lower limit was discovered numerically while searching for Q-ball solutions for $m = 0.9\omega_{c}$. It may be expected that there are Q-balls with $\phi_{0}$ larger than this - of particular interest being a $\phi_{0} \sim 10^{18}\GeV$ Q-ball, discussed in depth in Section <ref> - but the precision on $\omega$ would need to be even greater to find it. It is interesting to note that the lower limit is close to being $\omega \sim \omega_{c}/\sqrt{2}$, which may have some kind of theoretical relevance to the existence of Q-balls in this framework.

Since there is no fixed definition of the radius of the Q-ball, two different possibilities are considered in this work for the purposes of evaluating the numerical Q-ball solutions. One possibility considered is a definition we will call $r_{X}$ - referred to as "the $X$ point" when discussing the numerical procedure - and a second definition, $r_{Z}$ - similarly "the $Z$ point" - as two different possibilities for understanding the concept of the Q-ball's "edge" when evaluating the properties numerically. The $X$ point is defined as being at a distance $r = r_{X}$ from the centre of the Q-ball $r = 0$ at which the field has decreased to $1 \%$ of its initial value $\phi_{0}$. This definition allows the properties of the Q-ball solutions to be examined using a fixed definition of what the radius is for all $\phi_{0}$. The $Z$ point is defined as being the point at which the code used to find a Q-ball solution for a given $\phi_{0}$ and $\omega$ cuts off having found a Q-ball solution. This distance from the centre of the Q-ball appears in the Q-ball profile, $\phi \left(r \right)$, as the point at which the field stops asymptoting towards zero and begins to increase again, moving away from the expected behaviour of a Q-ball solution. This occurs at $r = r_{Z}$ and this point corresponds to the maximum value of $r$ for which the Q-ball solution is valid for a given $\phi_{0}$ and $\omega$. Although less well defined than the $X$ point, since it will be different for every Q-ball solution, the $Z$ point cutoff is the more useful definition for calculating the energy $E$ and charge $Q$ of the Q-balls since the $Z$ point corresponds to the upper value of $r$ to which the Q-ball equation is integrated.

Two tables of values for the Q-ball properties are therefore presented, corresponding to those calculated at the $X$ point and the $Z$ point, as an exploration into how the radius of the Q-ball should be interpreted numerically and whether it makes any significant difference to the predicted properties of the Q-balls. Quantities with a subscript $X$ are calculated at the $X$ point, and those with a $Z$ are calculated at the $Z$ point.

Figures <ref> - <ref> show two Q-ball solutions, calculated numerically and using an analytical approximation derived in Section <ref>, (<ref>). Figures <ref> and <ref> show a Q-ball with $\phi_{0} \sim 10^{16}\GeV$, and the Figures <ref> and <ref> show a Q-ball with $\phi_{0} \sim 10^{13}\GeV$, with $\omega$ close to the upper bound of compatibility with the existence of Q-balls, $\omega = m$. The inflaton self-coupling is $\lambda = 0.1$ throughout and the corresponding non-minimal coupling is $\xi = 1.2163 \times 10^{9}$, using the estimate from (<ref>) derived in Section <ref>.

Q-ball solution for $\omega = 0.709\omega_{c} $, $m=0.9\omega_{c}$ and $\phi_{0} = 1.3464098 \times 10^{16} \GeV$ obtained by solving (<ref>) numerically.

Q-ball solution for  $\omega = 0.709\omega_{c} $, $m=0.9\omega_{c}$ and $\phi_{0} = 1.3464098 \times 10^{16} \GeV$ obtained using the analytical solution (<ref>).

Q-ball solution for $\omega = 0.89\omega_{c}$, $m=0.9\omega_{c}$ and $\phi_{0} = 4.7918\times 10^{13} \GeV$ obtained by solving (<ref>) numerically.

Q-ball solution for $\omega = 0.89\omega_{c}$, $m=0.9\omega_{c}$ and $\phi_{0} = 4.7918\times 10^{13} \GeV$ obtained using the analytical approximation (<ref>).

In Tables <ref> and <ref> we present the numerical properties of the Q-ball solutions using the $r_{Z}$ and $r_{X}$ definitions of the Q-ball radius, respectively. Quantities recorded are the $\omega$ parameter; the value of the field at $r = 0$, $\phi_{0}$; the radius of the Q-ball, $r_{Z}$ or $r_{X}$; the value of the field at the defined edge of the Q-ball, and the total energy, $E$, charge, $Q$, and the energy-charge ratio, $E/Q$ of the Q-ball.

In order to assess whether the Q-balls found are stable, we introduce the quantities $\Delta_{\omega}$ and $\Delta_{m}$. $\Delta_{\omega}$ is defined by

\begin{equation}\label{eqn:q112}
\Delta_{\omega} = \frac{1}{\omega}\left(\frac{E}{Q} - \omega \right).
\end{equation}

This quantity measures the extent to which the $\omega$ parameter for each Q-ball solution is equal to the energy-charge ratio. The relationship between $\omega$ and the energy-charge ratio is established analytically for this model in Section <ref>.

$\Delta_{m}$ is defined by

\begin{equation}\label{eqn:q113}
\Delta_{m} = \frac{1}{m}\left(\frac{E}{Q} - m \right).
\end{equation}

This quantity provides a measure of absolute stability. As discussed in Section <ref>, absolute stability is defined by the energy of a Q-ball in relation to the energy of its component scalars if they were free particles, $E < mQ$. $\Delta_{m} < 0$ is therefore the condition for a Q-ball is to be absolutely stable.

max width=

$\omega / \omega_{c}$     $\phi_{0} / \GeV$     $\phi_{Z}/\GeV$     $r_{Z}/\GeV^{-1}$     $Q_{Z}$     $E_{Z}/\GeV$     $(E/Q)_{Z}/\GeV$     $\Delta_{\omega, Z}$     $ \Delta_{m, Z}$

$0.707155$     $3.2217991 \times 10^{17} $     $1.61 \times 10^{10}$     $8.94 \times 10^{-10}$     $2.17 \times 10^{14}$     $3.34 \times 10^{27}$     $1.54 \times 10^{13}$     $-2.79 \times 10^{-4}$     $-0.21$

$0.709$     $1.3464098 \times 10^{16}$     $1.63 \times 10^{10}$     $3.79 \times 10^{-11}$     $1.56 \times 10^{10}$     $2.40 \times 10^{23}$     $1.54 \times 10^{13}$     $9.71 \times 10^{-4}$     $-0.21$

$0.71$     $8.855792 \times 10^{15}$     $2.05 \times 10^{10}$     $2.50 \times 10^{-11}$     $4.37 \times 10^{9}$     $6.76 \times 10^{22}$     $1.55 \times 10^{13}$     $1.64 \times 10^{-3}$     $-0.21$

$0.72$     $1.960795 \times 10^{15}$     $9.76 \times 10^{9}$     $6.09 \times 10^{-12}$      $4.43 \times 10^{7}$     $6.99 \times 10^{20}$     $1.58 \times 10^{13}$     $7.71 \times 10^{-3}$     $-0.19$

$0.73$     $1.090258 \times 10^{15}$     $6.60 \times 10^{9}$     $3.73 \times 10^{-12}$      $7.09 \times 10^{6}$     $1.14 \times 10^{20}$     $1.61 \times 10^{13}$     $1.30 \times 10^{-2}$     $-0.18$

$0.74$     $7.45339 \times 10^{14}$     $1.61 \times 10^{10}$     $2.72 \times 10^{-12}$      $2.16 \times 10^{6}$     $3.54 \times 10^{19}$     $1.64 \times 10^{13}$     $1.76 \times 10^{-2}$     $-0.16$

$0.75$     $5.61953 \times 10^{14}$     $1.14 \times 10^{10}$     $2.28 \times 10^{-12}$     $8.85 \times 10^{5}$     $1.48 \times 10^{19}$     $1.67 \times 10^{13}$     $2.17 \times 10^{-2}$     $-0.15$

$0.80$     $2.29632 \times 10^{14}$     $1.96 \times 10^{10}$     $1.47 \times 10^{-12}$     $5.60 \times 10^{4}$     $1.01 \times 10^{18}$     $1.80 \times 10^{13}$     $3.51 \times 10^{-2}$     $-8.00 \times 10^{-2}$

$0.85$     $1.16877 \times 10^{14}$     $1.57 \times 10^{10}$     $1.49 \times 10^{-12}$     $1.01 \times 10^{4}$     $1.93 \times 10^{17}$     $1.91 \times 10^{13}$     $3.51 \times 10^{-2}$     $-2.24 \times 10^{-2}$

$0.89$     $4.7918 \times 10^{13}$     $8.82 \times 10^{9}$     $2.70 \times 10^{-12}$      $3.98 \times 10^{3}$     $7.82 \times 10^{16}$     $1.97 \times 10^{13}$     $1.50 \times 10^{-2}$     $3.74 \times 10^{-3}$

Table listing important properties of the $m=0.9\omega_{c}$ Q-balls, calculated using the $Z$ point definition of the Q-ball radius. $\phi_{0}$ denotes the value of the field at $r = 0$. $\phi_{Z}$, $E_{Z}$, $Q_{Z}$, $(E/Q)_{Z}$, $\Delta_{\omega, Z}$ and $\Delta_{m, Z}$ are calculated numerically at the point $r_{Z}$.

max width=

$\omega / \omega_{c}$     $\phi_{0}/\GeV$     $\phi_{X}/\GeV$     $r_{X}/\GeV^{-1}$     $Q_{X}$     $E_{X}/\GeV$     $(E/Q)_{X}/\GeV$     $\Delta_{\omega, X}$     $\Delta_{m, X}$

$0.707155$     $3.2217991 \times 10^{17} $     $3.22 \times 10^{15}$     $8.89 \times 10^{-10}$     $2.15 \times 10^{14}$     $3.30 \times 10^{27}$     $1.54 \times 10^{13}$     $-4.47 \times 10^{-4}$     $-0.22$

$0.709$     $1.3464098 \times 10^{16}$     $1.35 \times 10^{14}$     $3.70 \times 10^{-11}$     $1.54 \times 10^{10}$     $2.38 \times 10^{23}$     $1.54 \times 10^{13}$     $ -1.27 \times 10^{-3}$     $-0.21$

$0.71$     $8.855792 \times 10^{15}$     $8.86 \times 10^{13}$     $2.43 \times 10^{-11}$     $4.34 \times 10^{9}$     $6.70 \times 10^{22}$     $1.54 \times 10^{13}$     $-7.78 \times 10^{-4}$     $-0.21$

$0.72$     $1.960795 \times 10^{15}$     $1.96 \times 10^{13}$     $5.38 \times 10^{-12}$     $4.42 \times 10^{7}$     $6.97 \times 10^{20}$     $1.58 \times 10^{13}$     $6.58 \times 10^{-3}$     $-0.19$

$0.73$     $1.090258 \times 10^{15}$     $1.09 \times 10^{13}$     $3.03 \times 10^{-12}$     $7.08 \times 10^{6}$     $1.14 \times 10^{20}$     $1.61 \times 10^{13}$     $1.23 \times 10^{-2}$     $-0.18$

$0.74$     $7.45339 \times 10^{14}$     $7.45 \times 10^{12}$     $2.13 \times 10^{-12}$     $2.16 \times 10^{6}$     $3.54 \times 10^{19}$     $1.64 \times 10^{13}$     $1.71 \times 10^{-2}$     $-0.16$

$0.75$     $5.61953 \times 10^{14}$     $5.62 \times 10^{12}$     $1.66 \times 10^{-12}$     $8.85 \times 10^{5}$     $1.48 \times 10^{19}$     $1.67 \times 10^{13}$     $2.13 \times 10^{-2}$     $-0.15$

$0.80$     $2.29632 \times 10^{14}$     $2.30 \times 10^{12}$     $8.97 \times 10^{-13}$     $5.60 \times 10^{4}$     $1.01 \times 10^{18}$     $1.80 \times 10^{13}$     $3.48 \times 10^{-2}$     $-8.02 \times 10^{-2}$

$0.85$     $1.16877 \times 10^{14}$     $1.17 \times 10^{12}$     $7.84 \times 10^{-13}$     $1.01 \times 10^{4}$     $1.93 \times 10^{17}$     $1.91 \times 10^{13}$     $3.49 \times 10^{-2}$     $-2.26 \times 10^{-2}$

$0.89$     $4.7918 \times 10^{13}$     $4.79 \times 10^{11}$     $1.21 \times 10^{-12}$     $3.96 \times 10^{3}$     $7.79 \times 10^{16}$     $1.97 \times 10^{13}$     $1.50 \times 10^{-2}$     $3.69 \times 10^{-3}$

Table listing important properties for the $m=0.9\omega_{c}$ Q-balls calculated using the $X$ point definition of the Q-ball radius. $\phi_{0}$ is the value of the scalar field at $r = 0$. $\phi_{X}$, $E_{X}$, $Q_{X}$, $(E/Q)_{X}$, $\Delta_{\omega, X}$ and $\Delta_{m, X}$ are calculated numerically at the point $r_{X}$.

The energy-charge ratio increases by less than an order of magnitude from the largest $\phi_{0}$ Q-ball to the smallest at $\omega = 0.89\omega_{c}$. Comparing the data in Tables <ref> and <ref> also shows that the calculated energy, $E$, charge $Q$, and energy-charge ratio, $E/Q$, using the $X$ point definition of radius are comparable to the values obtained for these quantities calculated using the $Z$ point definition for each Q-ball. The radii for each Q-ball are similarly close in magnitude for each definition, with the difference between $r\left(X\right)$ and $r\left(Z\right)$ becoming slightly more pronounced in the smaller $\phi_{0}$ Q-balls.

The larger $\phi_{0}$ Q-balls have generally larger radii and the increase in radius is linear with $\phi_{0}$ to a good approximation for $\phi_{0} > M_{Pl}/\sqrt{\xi}  \; ( \sim 6.9 \times 10^{13} \GeV)$. Increasing $\phi_{0}$ also increases the Q-ball energy and charge, as expected since a physically larger Q-ball (larger radius) will be composed of a larger number of scalars and will therefore carry a greater charge and energy.  Figures <ref> and <ref> show the relationships between the logarithm of the energy of the Q-balls and the logarithm of $\phi_{0}$, and the logarithm of the charge of the Q-balls and the logarithm of $\phi_{0}$ respectively. We can see from these plots that $E$ and $Q$ are proportional to $\phi_{0}^{3}$ to a good approximation for the larger $\phi_{0}$ Q-balls, and that the approximate proportionality becomes less representative for the smaller $\phi_{0}$ Q-balls. This proportionality is predicted by the analytical approximation derived in Section <ref>, and is therefore a useful test of the validity of the numerical Q-ball solutions.

Log-log plot of $\ln Q_{z}$, vs. $\ln \phi_{0}$ for the $m=0.9\omega_{c}$ Q-balls.

Log-log plot of $\ln E_{z}$, vs. $\ln \phi_{0}$ for the $m=0.9\omega_{c}$ Q-balls.

As we can see from Tables <ref> and <ref>, $\left| \Delta_{\omega} \right| \sim 10^{-4} - 10^{-2}$ in general. This shows that $\omega$ is close to the value of the energy-charge ratio, and therefore to a good approximation can be equated to the chemical potential for the Q-ball solutions considered in this work. $ \left| \Delta_{\omega} \right| $ increases as $\phi_{0}$ decreases, and from Figure <ref> it is clear that that $ \Delta_{\omega} $ increases as Q-ball radius decreases in general. This is consistent with the patterns seen in the other quantities considered, in that the behaviour patterns of the properties for the smaller $\phi_{0}$ Q-balls are less well-defined than for the larger $\phi_{0}$ Q-balls. It is possible that this is a reflection of the Q-balls becoming unstable as the potential begins to deviate from the plateau ($\phi_{0} \rightarrow M_{Pl}/\sqrt{\xi}$).

$\Delta_{\omega,X}$ vs $r_{X}$ for the $m=0.9\omega_{c}$ Q-balls.

We can see from Tables <ref> and <ref>, and clearly in Figure <ref>, that $\Delta_{m}$ is negative for all Q-ball solutions considered except the $\omega = 0.89\omega_{c}$ Q-ball, meaning that all of the Q-balls considered are absolutely stable except for the $\omega = 0.89\omega_{c}$ case. Since the magnitude of $\Delta_{m}$ for $\omega = 0.89\omega_{c}$ is still small although it is positive, it is possible that $\Delta_{m}$ is actually negative but smaller than the level of numerical errors in the Q-ball calculation.

$\Delta_{m,X}$ vs $r_{X}$ for the $m=0.9\omega_{c}$ Q-balls.

The most significant result of the numerical analysis is that absolutely stable Q-balls of $\phi_{0} \geq 10^{17} \GeV$ can be generated in this model. This shows that the Q-ball mass window allows for the formation of Q-balls from field values we expect at the end of non-minimally coupled Palatini inflation (see Section <ref>). It is therefore possible that these Q-balls could form from the fragmentation of the inflaton condensate following tachyonic preheating at the end of non-minimally coupled Palatini inflation [168]. The presence of these Q-balls could significantly affect the post-inflation cosmology of the model, with consequences such as faster reheating and the production of gravitational waves from the decay of these Q-balls, leading to measurable effects. This is discussed further in Sections <ref> - <ref>.

§ ANALYTICAL APPROXIMATION OF THE Q-BALL SOLUTION AND ESTIMATE OF Q-BALL PROPERTIES

In this section we derive an analytical approximation of the Q-ball solution, and calculate the energy and charge in this approximation as an estimate of the Q-ball properties. We compare these to the results obtained from the numerical solutions in Section <ref> and derive some important relations between energy and charge.

In this model we are assuming that inflation ends while the inflaton field is on the inflationary plateau. Using the plateau limit (<ref>), we can approximate the right hand side of the Q-ball equation (<ref>) to be

\begin{equation}\label{eqn:q114}
\Omega^{2}\frac{\partial V_{\omega}}{\partial \phi} = - \frac{\gamma}{\phi},
\end{equation}

to leading order in $1/\phi^{2}$, where $\gamma$ is defined as

\begin{equation}\label{eqn:q115}
\gamma = \frac{M_{pl}^{2}}{\xi}\left( m^{2} + \omega^{2} - \frac{\lambda M_{pl}^{2}}{\xi}\right).
\end{equation}

In the plateau limit we have that $\phi >> M_{pl}/\sqrt{\xi}$, and we also assume that

\begin{equation}\label{eqn:q116}
\left| \frac{\partial^{2}\phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} \right| >> K\left(\phi \right) \left(\frac{\partial \phi}{\partial r}\right)^{2},
\end{equation}

where $K\left(\phi \right) \sim 1/\phi$ on the plateau (we will confirm that (<ref>) is consistent below). Using (<ref>) and (<ref>), the Q-ball equation can be approximated as

\begin{equation}\label{eqn:q117}
\frac{\partial^{2}\phi}{\partial r^{2}} + \frac{2}{r}\frac{\partial \phi}{\partial r} = - \frac{\gamma}{\phi},
\end{equation}

where $\gamma > 0$ for Q-ball solutions to exist.

We assume that $\phi$ does not deviate from its initial value $\phi_{0}$ too much, and also that

\begin{equation}\label{eqn:q118}
\frac{\partial \phi}{\partial r} \rightarrow 0,
\end{equation}

as $r \rightarrow 0$.

Setting $\phi = \phi_{0}$ on the right hand side of (<ref>), we make the following ansatz

\begin{equation}\label{eqn:q119}
\phi \left( r \right) = \phi_{0} - A r^{2},
\end{equation}

where $A$ is a coefficient to be determined. In order for the field profile to have the typical Q-ball shape, we require that the right hand side of the equation is negative. This means that

\begin{equation}\label{eqn:q120}
\gamma > 0 \longrightarrow m^{2} + \omega^{2} > \frac{\lambda M_{pl}^{2}}{\xi}.
\end{equation}

Substituting (<ref>) into (<ref>), we find that the left hand side becomes a constant, $-6A$, and that the approximation (<ref>) holds because

\begin{equation}\label{eqn:q121}
K\left(\phi \right) \left(\frac{\partial \phi}{\partial r}\right)^{2}  \approx \frac{4A^{2}r^{2}}{\phi_{0}},
\end{equation}

is a very small number for $\phi >> M_{pl}/\sqrt{\xi}$, and goes to zero as $r \rightarrow 0$. This leaves

\begin{equation}\label{eqn:q122}
-6A = -\frac{\gamma}{\phi_{0}} \Rightarrow A = \frac{\gamma}{6 \phi_{0}},
\end{equation}

and approximation of the field profile (<ref>) is therefore

\begin{equation}\label{eqn:q123}
\phi \left( r \right) = \phi_{0} -  \frac{\gamma}{6 \phi_{0}}r^{2}.
\end{equation}

The assumption that $\phi \approx \phi_{0}$ holds to a good approximation if

\begin{equation}\label{eqn:q124}
r^{2} < \frac{r_{Q}^{2}}{4} = \frac{6 \phi_{0}^{2}}{4\gamma},
\end{equation}

since applying (<ref>) at $r_{Q}/2$ gives $\phi \left(r_{Q}/2\right) = \frac{3}{4}\phi_{0}$.

To a good approximation, we can therefore say that the Q-ball solution follows

\begin{equation}\label{eqn:q125}
\phi \left( r \right) = \phi_{0}\left[ 1 - \left(\frac{r}{r_{Q}}\right)^{2} \right],
\end{equation}

for values of $r < r_{Q}/2$, where the assumption (<ref>) is consistent with this solution. We can use this to estimate the Q-ball properties, such as energy and charge, which can be used as a check on the numerical solution and also to determine the parameter dependence of the numerical Q-ball solutions.

§.§ Analytical Estimate of Q-ball Energy and Charge

In this section we derive analytical expressions for the energy, $E$, and the global charge, $Q$, of the non-minimally coupled Palatini Q-balls. Assuming $\phi \approx \phi_{0}$, and working in the plateau limit, we have that the extremised Q-ball energy $E$ is approximately (from (<ref>))

\begin{equation}\label{eqn:q126}
E = \int^{r_{Q}/2}_{0} 4\pi r^{2} \; dr \left[\frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2} + \frac{V\left(\phi \right)}{\Omega^{4}}\right].
\end{equation}

For $r < r_{Q}/2$, and assuming $\phi \approx \phi_{0}$ we approximate the Jordan frame potential to be

\begin{equation}\label{eqn:q127}
V\left(\phi \right) \approx \frac{\lambda}{4}\phi_{0}^{4},
\end{equation}

since $\lambda\phi_{0}^{4} >> m^{2}\phi_{0}^{2}$ in the Jordan frame in this case,  and the conformal factor is $\Omega^{2} \approx \xi \phi_{0}^{2}/M_{pl}^{2}$. Using (<ref>), we have that

\begin{equation}\label{eqn:q128}
\frac{\partial \phi}{\partial r} = -\frac{2r \phi_{0}}{r_{Q}^{2}}.
\end{equation}

Using (<ref>) and (<ref>), we find that the Q-ball energy (<ref>) becomes

\begin{equation}\label{eqn:q129}
E = \left( \frac{\omega^{2}\phi_{0}^{2}}{2\Omega^{2}} + \frac{\lambda \phi_{0}^{4}}{4\Omega^{4}}\right) \int^{r_{Q}/2}_{0} 4\pi r^{2} \; dr + \frac{2\phi_{0}^{2}}{r_{Q}^{4}}\int^{r_{Q}/2}_{0} 4\pi r^{4} \; dr.
\end{equation}

Performing the integration gives

\begin{equation}\label{eqn:q130}
E = \left( \frac{\omega^{2}\phi_{0}^{2}}{2\Omega^{2}} + \frac{\lambda \phi_{0}^{4}}{4\Omega^{4}}\right)\frac{4\pi}{3}\left(\frac{r_{Q}}{2}\right)^{3} + \frac{2\phi_{0}^{2}}{r_{Q}^{4}}\frac{4\pi}{5}\left(\frac{r_{Q}}{2}\right)^{5},
\end{equation}

and with some rearrangement this is

\begin{equation}\label{eqn:q131}
E = \frac{\pi M_{pl}^{2}}{2 \xi}\left[ \frac{1}{6}\left(\omega^{2} + \frac{\lambda M_{pl}^{2}}{2\xi}\right) + \frac{1}{10 r_{Q}^{2}}\right]r_{Q}^{3}.
\end{equation}

Using (<ref>) and (<ref>), we have that

\begin{equation}\label{eqn:q132}
\frac{1}{r_{Q}^{2}} = \frac{\gamma}{6\phi_{0}^{2}} = \frac{M_{pl}^{2}}{6\xi \phi_{0}^{2}}\left(m^{2} + \omega^{2} - \omega_{c}^{2} \right),
\end{equation}

and since we are working in the plateau limit, $M_{pl}^{2}/\xi \phi_{0}^{2} << 1$, we can say that

\begin{equation}\label{eqn:q133}
\frac{M_{pl}^{2}}{6\xi \phi_{0}^{2}}\left(m^{2} + \omega^{2} - \omega_{c}^{2} \right) << \frac{1}{6}\left(m^{2} + \omega^{2} - \omega_{c}^{2} \right).
\end{equation}

This means that the $1/r_{Q}^{2}$ term is small compared to the $1/6$ term in (<ref>), and the Q-ball energy can be approximated as

\begin{equation}\label{eqn:q134}
E = \frac{\pi M_{pl}^{2}}{12 \xi}\left[ \omega^{2} + \frac{\lambda M_{pl}^{2}}{2\xi}\right] r_{Q}^{3}.
\end{equation}

Note that this expression holds for $r < r_{Q}/2$ and is therefore a lower bound estimate on the Q-ball energies calculated numerically. Similarly, the Q-ball charge can be approximated analytically. The charge density of the Q-balls from (<ref>) using the Q-ball ansatz (<ref>) is

\begin{equation}\label{eqn:q135}
\rho_{Q} = \frac{\omega \phi^{2}}{\Omega^{2}},
\end{equation}

where for $\phi \approx \phi_{0}$ and $\Omega^{2} \approx \xi \phi_{0}^{2}/M_{pl}^{2}$ this is

\begin{equation}\label{eqn:q136}
\rho_{Q} = \frac{\omega M_{pl}^{2}}{\xi}.
\end{equation}

The global charge from (<ref>) using (<ref>) is thus

\begin{equation}\label{eqn:q137}
Q = \int^{r_{Q}/2}_{0} 4\pi r^{2} \; dr \frac{\omega M_{pl}^{2}}{\xi} = \frac{4\pi \omega M_{pl}^{2}}{\xi}\left( \frac{r_{Q}}{2}\right)^{3},
\end{equation}

and integrating for $r < r_{Q}/2$ gives the Q-ball charge to be

\begin{equation}\label{eqn:q138}
Q = \frac{\pi \omega M_{pl}^{2}}{6 \xi} r_{Q}^{3}.
\end{equation}