We can see from (<ref>) and (<ref>) that the energy and charge of the Q-balls are both proportional to $r_{Q}^{3}$, and therefore to the volume of the Q-ball. From (<ref>), we find that $r_{Q} \propto \phi_{0}$, and $E, Q \propto \phi_{0}^{3}$ in the analytical approximation. From the energy and charge calculated for the numerical solutions in Section <ref>, we find that there is a proportionality between the Q-ball energy and charge and $\phi_{0}^{3}$, as illustrated more clearly in Figures <ref> and <ref>, and that the relationship is truest and essentially exact for the large $\phi_{0}$ Q-balls. Figures <ref> and <ref> in Section <ref> show the analytical approximate solution for $\phi$, (<ref>), as a function of $r$ for large and small $\phi_{0}$. Comparing these with the numerical solutions for the same $\phi_{0}, \omega$ shown in Figures <ref> and <ref>, it is clear that the analytical approximation closely follows the numerical solution for small $r$. For $r < r_{Q}$ specifically, the analytical approximation (<ref>) is a good fit to the exact solution calculated by solving (<ref>) in the case of larger $\phi_{0}$ ($\phi_{0} >> M_{pl}/\sqrt{\xi}$), but the agreement is less apparent for smaller $\phi_{0} \rightarrow M_{Pl}/\sqrt{\xi}$. These results show that the analytical approximation of the Q-ball solution, and therefore the analytical approximations of the energy and charge, are a good approximation of the behaviour of the Q-ball solutions and the properties of the Q-balls themselves in the limit that $\phi_{0}$ is large and $r$ is small.

Taking the ratio of the analytical approximation of energy (<ref>) and charge (<ref>) gives

\begin{equation}\label{eqn:q139}
\frac{E}{Q} = \frac{1}{2}\left[ \omega + \frac{\omega_{c}^{2}}{2\omega}\right],
\end{equation}

as an estimate of the energy-charge ratio. In Section <ref> we found that the numerical results for the energy-charge ratio show that $E/Q \approx \omega$ when $\phi_{0}$ is large, with the deviation from $E/Q = \omega$ increasing for the smaller $\phi_{0}$ Q-balls. We see here that the analytical approximation of the energy-charge ratio does not produce this relation. It is likely that this is due to the fact that $E$ and $Q$ are only integrated to $r = r_{Q}/2$ in the analytical approximation. In Section <ref> we derive an exact expression for the energy-charge ratio in terms of $\phi \left(r \right)$, which will allow for a more accurate determination of the energy-charge ratio.

Table <ref> shows the estimates of the energy and charge calculated using the analytical approximation expressions (<ref>) and (<ref>), for each of the ten Q-ball solutions presented in Section <ref>.

$\omega / \omega_{c}$     $\phi_{0}/\GeV$     $r_{Q}/\GeV^{-1}$     $Q$      $E/\GeV$

$0.707155$     $3.2217991 \times 10^{17} $     $9.46 \times 10^{-10}$     $3.23 \times 10^{13}$     $4.97 \times 10^{26}$

$0.709$     $1.3464098 \times 10^{16}$     $3.94 \times 10^{-11}$     $2.34 \times 10^{9}$     $3.60 \times 10^{22}$

$0.71$     $8.855792 \times 10^{15}$     $2.58 \times 10^{-11}$     $6.58 \times 10^{8}$     $1.01 \times 10^{22}$

$0.72$     $1.960795\times 10^{15}$     $5.60 \times 10^{-12}$     $6.82 \times 10^{6}$     $1.05 \times 10^{20}$

$0.73$     $1.090258 \times 10^{15}$     $3.05 \times 10^{-12}$     $1.12 \times 10^{6}$     $1.72 \times 10^{19}$

$0.74$     $7.45339 \times 10^{14}$     $2.04 \times 10^{-12}$     $3.39 \times 10^{5}$     $5.22 \times 10^{18}$

$0.75$     $5.61953 \times 10^{14}$     $1.51 \times 10^{-12}$     $1.39 \times 10^{5}$     $2.15 \times 10^{18}$

$0.80$     $2.29632 \times 10^{14}$     $5.60 \times 10^{-13}$     $7.58 \times 10^{3}$     $1.18 \times 10^{17}$

$0.85$     $1.16877 \times 10^{14}$     $2.62 \times 10^{-13}$     $825$     $1.29 \times 10^{16}$

$0.89$     $4.7918 \times 10^{13}$     $1.01 \times 10^{-13}$     $50$     $7.82 \times 10^{14}$

Table showing $r_{Q}$, $E$ and $Q$ as a function of $\phi_{0}$ from the analytical approximation from (<ref>) and (<ref>) for the $m=0.9\omega_{c}$ Q-ball solutions presented in Section <ref>.

Comparing these results to the numerically calculated values of energy and charge in Tables <ref> and <ref> it is clear that the analytical estimates (<ref>) and (<ref>) consistently underestimate the energy and charge by about an order of magnitude for most of the Q-ball solutions. This makes sense when one considers the fact that the analytical expressions are only integrated to $r = r_{Q}/2$, as this is the limit at which the analytical approximation ceases to be a valid description of the behaviour of the Q-ball solution. The underestimate becomes greater as $\phi_{0}$ decreases, which is consistent with the analytical Q-ball profile $\phi(r)$ becoming a poorer fit to the numerical profile as $\phi_{0}$ approaches the edge of plateau, and therefore the limit of plateau description of the inflaton field dynamics. However, since the analytical approximation is a good approximation for the behaviour of the Q-ball solutions with larger $\phi_{0}$, it can be used as an estimate of the properties of these Q-balls and to examine the dependence of these quantities on the other parameters of the model.

§.§ Analytical Derivation of the Q-ball Chemical Potential Relation

In this section we derive the energy per unit charge for Palatini non-minimally coupled Q-balls and demonstrate that this is can be equated to the $\omega$ parameter in this model. This discussion follows very closely the derivation in Heeck et. al. [169] for the case of conventional minimally coupled Q-balls.

As a starting point we consider the Q-ball energy functional (<ref>)

\begin{equation}\label{eqn:q140}
E_{Q} = \int 4\pi r^{2} dr \left[ \frac{\omega^{2}\phi^{2}}{2\Omega^{2}} + \frac{1}{2\Omega^{2}}\left(\frac{\partial \phi}{\partial r}\right)^{2} + \frac{V}{\Omega^{4}}\right] + \omega Q - \omega \int 4\pi r^{2} dr   \rho_{Q},
\end{equation}

\begin{equation}\label{eqn:q141}
\rho_{Q} = \frac{\omega \phi^{2}}{\Omega^{2}}.
\end{equation}

Absorbing the $\rho_{Q}$ integral in (<ref>) into the first integral and rewriting the $\omega$ term, this becomes

\begin{equation}\label{eqn:q142}
E_{Q} = \int 4\pi r^{2} dr \left[\frac{1}{2\Omega^{2}}\left( \frac{\partial \phi}{\partial r}\right)^{2} + \frac{\omega^{2}\phi^{2}}{\Omega^{2}} - \frac{\omega^{2}\phi^{2}}{2\Omega^{2}} - \frac{\omega^{2}\phi^{2}}{\Omega^{2}} + \frac{V}{\Omega^{4}}\right] + \omega Q.
\end{equation}

We now differentiate (<ref>) with respect to $\omega$,

\begin{equation}\label{eqn:q143}
\frac{d E_{Q}}{d\omega} = \int 4\pi r^{2} dr \frac{\partial}{\partial \omega}\left[\frac{1}{2\Omega^{2}}\left( \frac{\partial \phi}{\partial r}\right)^{2} + \frac{\omega^{2}\phi^{2}}{\Omega^{2}} - \frac{\omega^{2}\phi^{2}}{2\Omega^{2}} - \frac{\omega^{2}\phi^{2}}{\Omega^{2}} + \frac{V}{\Omega^{4}}\right] + Q + \omega \frac{d Q}{d\omega},
\end{equation}

and rewrite the integral using the field rescaling outlined in Section <ref> to proceed with this part of the calculation in terms of $\sigma$

\begin{multline}\label{eqn:q144}
\frac{d E_{Q}}{d\omega} = \int 4\pi r^{2} dr \left[ \frac{1}{2}\frac{\partial}{\partial \omega}\left(\frac{\partial \sigma}{\partial r}\right)^{2} + \frac{\partial}{\partial \omega}\left(\frac{\omega^{2}\phi^{2}}{\Omega^{2}}\right) - \frac{\omega^{2}}{2}\frac{\partial}{\partial \omega}\left(\frac{\phi^{2}}{\Omega^{2}}\right) \right. \\
\left. - \frac{\partial}{\partial \omega}\left(\frac{\omega^{2} \phi^{2}}{\Omega^{2}}\right) + \frac{\partial}{\partial \omega}\left(\frac{V}{\Omega^{4}}\right)\right] + \omega \frac{d Q}{d\omega}.
\end{multline}

Rescaling the Q-ball equation in terms of $\sigma$ (<ref>),we find that the left-hand side can be written

\begin{equation}\label{eqn:q145}
\frac{d^{2}\sigma}{d r^{2}} + \frac{2}{r}\frac{d \sigma}{d r} = \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\sigma}{dr}\right),
\end{equation}

which can then be substituted back into the Q-ball equation (<ref>) and rearranged to give

\begin{equation}\label{eqn:q146}
\frac{\partial}{\partial \sigma}\left(\frac{V}{\Omega^{4}}\right) = \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\sigma}{dr}\right) + \frac{\partial}{\partial \sigma}\left(\frac{\omega^{2}\phi^{2}}{2\Omega^{2}}\right).
\end{equation}

We can write the left-hand side of (<ref>)

\begin{equation}\label{eqn:q147}
\frac{\partial}{\partial \omega}\left(\frac{V}{\Omega^{4}}\right) = \frac{\partial}{\partial \sigma}\left(\frac{V}{\Omega^{4}}\right)\frac{\partial \sigma}{\partial \omega},
\end{equation}

\begin{equation}\label{eqn:q148}
\Rightarrow \frac{\partial}{\partial \omega}\left(\frac{V}{\Omega^{4}}\right) = \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\sigma}{dr}\right)\frac{\partial \sigma}{\partial \omega} + \frac{\partial}{\partial \sigma}\left(\frac{\omega^{2}\phi^{2}}{2\Omega^{2}}\right)\frac{\partial \sigma}{\partial \omega}.
\end{equation}

Substituting (<ref>) back into (<ref>) gives

\begin{multline}\label{eqn:q149}
\frac{d E_{Q}}{d\omega} = \int 4\pi r^{2} dr \left[\frac{1}{2}\frac{\partial}{\partial \omega}\left(\frac{\partial \sigma}{\partial r}\right) \left(\frac{\partial \sigma}{\partial r}\right) -\frac{\omega^{2}}{2}\frac{\partial}{\partial \omega}\left(\frac{\phi^{2}}{\Omega^{2}}\right) + \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\sigma}{dr}\right)\frac{\partial \sigma}{\partial \omega} \right. \\
\left. + \frac{\partial}{\partial \sigma}\left(\frac{\omega^{2}\phi^{2}}{2\Omega^{2}}\right)\frac{\partial \sigma}{\partial \omega}\right] + \omega \frac{d Q}{d\omega},
\end{multline}

after canceling the $\pm \frac{\partial}{\partial \omega}\left(\frac{\omega^{2}\phi^{2}}{\Omega^{2}}\right)$ terms. Rewriting the second term in (<ref>)

\begin{equation}\label{eqn:q150}
\frac{\omega^{2}}{2}\frac{\partial}{\partial \omega}\left(\frac{\phi^{2}}{\Omega^{2}}\right) = \frac{\omega^{2}}{2}\frac{\partial}{\partial \sigma}\left(\frac{\phi^{2}}{\Omega^{2}}\right) \frac{\partial \sigma}{\partial \omega},
\end{equation}

and canceling with the fourth term in (<ref>) leaves

\begin{equation}\label{eqn:q151}
\frac{d E_{Q}}{d\omega} = \int 4\pi r^{2} dr \left[\frac{1}{2}\frac{\partial}{\partial \omega}\left(\frac{\partial \sigma}{\partial r}\right) \left(\frac{\partial \sigma}{\partial r}\right) + \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\sigma}{dr}\right)\frac{\partial \sigma}{\partial \omega}\right] + \omega \frac{dQ}{d\omega}.
\end{equation}

We then integrate the second term by parts

\begin{equation}\label{eqn:q152}
\int \frac{1}{r^{2}}\frac{d}{dr}\left(r^{2}\frac{d\sigma}{dr}\right)\frac{\partial \sigma}{\partial \omega} dr = \left. \frac{\partial \sigma}{\partial r}\frac{\partial \sigma}{\partial \omega}\right| - \int \frac{\partial \sigma}{\partial r} \frac{\partial^{2} \sigma}{\partial r \partial \omega},
\end{equation}

and obtain

\begin{equation}\label{eqn:q153}
\frac{d E_{Q}}{d\omega} = \int 4\pi r^{2} dr \left[\frac{1}{2}\frac{\partial}{\partial \omega}\left(\frac{\partial \sigma}{\partial r}\right) \left(\frac{\partial \sigma}{\partial r}\right) -  \frac{\partial \sigma}{\partial r} \frac{\partial^{2} \sigma}{\partial r \partial \omega} \right] + \omega \frac{dQ}{d\omega},
\end{equation}

where the boundary term from the integration is neglected due to the fact that the gradient of the scalar field goes to zero at infinity. The remaining terms in the integrand cancel and we are left with precisely

\begin{equation}\label{eqn:q154}
\frac{d E_{Q}}{d\omega} = \omega \frac{dQ}{d\omega}.
\end{equation}

This is the same result as derived in [169] for the case of a conventional canonical scalar, and shows that for Q-balls in non-minimally coupled Palatini gravity, the $\omega$ parameter can be equated to the chemical potential of the system

\begin{equation}\label{eqn:q155}
\frac{d E_{Q}}{dQ} = \omega,
\end{equation}

and that $E_{Q}\left(\omega \right)$ and $Q\left( \omega \right)$ increase and decrease together, as in the case of conventional Q-balls [50].

This makes sense in terms of $\omega$ being the Lagrange multiplier, since for a given $Q$ there will be a value of $\omega$ which extremises the energy of the scalar field subject to the constraint of constant charge. Hence $Q$ is a function of $\omega$, and extremising the scalar field for a fixed $Q(\omega)$ will produce an energy configuration $E_{Q}(\omega)$ corresponding to a Q-ball.

It is also straightforward to show analytically that the $\omega$ parameter is approximately the ratio of the energy to the global charge. The following derivation closely follows that of the derivation of Eq.$12$ in Heeck et al. [169] for the case of conventional Q-balls.

From (<ref>), the effective Q-ball action can be written

\begin{equation}\label{eqn:q156}
\int dr \; \mathcal{L_{Q}} = \int 4\pi r^{2} \; dr \left[\frac{1}{2}\left(\frac{d\sigma}{dr}\right)^{2} -\frac{\omega^{2}\phi^{2}}{2\Omega^{2}} + \frac{V}{\Omega^{4}}\right],
\end{equation}

where the inflaton is understood as being a function of the rescaled field $\sigma$ in this case. We then perform a coordinate rescaling $r \rightarrow \chi \tilde{r}$, and write (<ref>) in terms of $\tilde{r}$

\begin{equation}\label{eqn:q157}
\int d\tilde{r} \; \mathcal{L_{Q}} = \int 4\pi \; d\tilde{r} \;  \tilde{r}^{2}  \;  \chi^{3} \, \left[\frac{1}{2\chi^{2}}\left(\frac{\partial \sigma}{\partial r}\right)^{2} - \frac{\omega^{2}\phi^{2}}{2\Omega^{2}} + \frac{V}{\Omega^{4}}\right].
\end{equation}

This (<ref>) contains an explicit dependence on $\chi$ from the factors present in the integrand, and an implicit dependence on $\chi$, since the quasi-canonical field is a function of $r$ and transforms as $\sigma \left( r\right) \rightarrow \sigma \left(\chi \tilde{r} \right)$, meaning that a small variation in $\chi$ will cause a small variation in $\sigma$. This means that we can write the variation of the Q-ball effective Lagrangian with respect to $\chi$ as

\begin{equation}\label{eqn:q158}
\frac{d\mathcal{L_{Q}}}{d\chi} = \left(\frac{d\mathcal{L_{Q}}}{d\chi}\right)_{explicit} +  \left(\frac{d\mathcal{L_{Q}}}{d\chi}\right)_{implicit},
\end{equation}

where $\chi = 1$ corresponds to a Q-ball solution which extremises the Q-ball effective action. This means that $\int d\tilde{r} \left(\frac{d \mathcal{L}_{Q}}{d\chi}\right)_{implicit} = 0$ at $\chi =1$. Since the value of (<ref>) is unchanged following the rescaling of the radial coordinate, we require that

\begin{equation}\label{eqn:q159}
\int d\tilde{r} \; \left(\frac{d \mathcal{L}_{Q}}{d\chi}\right)_{explicit} = 0,
\end{equation}

when $\chi = 1$. The left-hand side of this expression is

\begin{multline}\label{eqn:q160}
\int d\tilde{r} \; \left(\frac{d \mathcal{L}_{Q}}{d\chi}\right)_{explicit} = 4\pi \int d\tilde{r} \; \frac{\partial}{\partial \chi}\left[ \frac{\chi}{2}\left(\frac{d\sigma}{dr}\right)^{2} - \frac{\chi^{3}\omega^{2}\phi^{2}}{2\Omega^{2}} + \frac{\chi^{3}V}{\Omega^{4}}\right] \\
= 4\pi \int \; d\tilde{r} \; \tilde{r}^{2} \left[\frac{1}{2}\left(\frac{d\sigma}{d\tilde{r}}\right)^{2} + 3\chi^{2}\left(\frac{V}{\Omega^{4}} - \frac{\omega^{2}\phi^{2}}{2\Omega^{2}}\right)\right],
\end{multline}

which means that for $\chi = 1, \tilde{r} \rightarrow r$, and in order for (<ref>) to be satisfied we require that

\begin{equation}\label{eqn:q161}
4\pi \int dr \; r^{2} \left[\frac{1}{2}\left(\frac{d\sigma}{dr}\right)^{2} + 3\left(\frac{V}{\Omega^{4}} - \frac{\omega^{2} \phi^{2}}{2\Omega^{2}}\right)\right] = 0.
\end{equation}

This can be rewritten to obtain the condition

\begin{equation}\label{eqn:q162}
\begin{split}
4\pi \int dr \; r^{2} \left[ \frac{1}{2}\left( \frac{d\sigma}{dr} \right)^{2} + \frac{3V}{\Omega^{4} } \right] = 4\pi \int r^{2} & \; dr \frac{ 3\omega^{2} \phi^{2} }{ 2 \Omega^{2} } \\
& = 4\pi \cdot \frac{3}{2}\int r^{2} \; dr \frac{ \omega^{2} \phi^{2} }{ \Omega^{2} } = \frac{3}{2}\omega Q,
\end{split}
\end{equation}

\begin{equation}\label{eqn:q163}
\Rightarrow 4\pi \int dr \;  r^{2} \left[\frac{1}{2}\left(\frac{d\sigma}{dr}\right)^{2} + 3\left(\frac{V}{\Omega^{4} }\right) \right] = \frac{3}{2}\omega Q.
\end{equation}

The Q-ball energy functional (<ref>) can be written

\begin{equation}\label{eqn:q164}
E_{Q} = \omega Q + 4\pi \int dr \; r^{2} \left[\frac{1}{2}\left(\frac{d\sigma}{dr}\right)^{2} + \frac{V}{\Omega^{4}}\right] - 4\pi \int dr \; r^{2} \frac{\omega^{2}\phi^{2}}{2\Omega^{2}},
\end{equation}

where the last term is precisely

\begin{equation}\label{eqn:q165}
4\pi \int dr \; r^{2} \frac{\omega^{2}\phi^{2}}{2\Omega^{2}} = \frac{1}{2}\omega Q.
\end{equation}

Using (<ref>) we can write

\begin{equation}\label{eqn:q166}
\frac{1}{2}\omega Q = 4\pi \int dr \;  r^{2} \left[\frac{1}{6}\left(\frac{d\sigma}{dr}\right)^{2} + \frac{V}{\Omega^{4}}\right],
\end{equation}

which, substituted into the Q-ball energy functional (<ref>), gives

\begin{equation}\label{eqn:q167}
E_{Q} = \omega Q + 4\pi \int dr \; r^{2} \left[\frac{1}{2}\left(\frac{d\sigma}{dr}\right)^{2} + \frac{V}{\Omega^{4}}\right] - 4\pi \int dr \; r^{2} \left[\frac{1}{6}\left(\frac{d\sigma}{dr}\right)^{2} + \frac{V}{\Omega^{4}}\right],
\end{equation}

which is the expression for the energy of the field extremised with respect to the global charge $Q$

\begin{equation}\label{eqn:q168}
E_{Q} = \omega Q + \frac{4\pi}{3}\int dr \; r^{2} \left(\frac{d\sigma}{dr}\right)^{2}.
\end{equation}

Thus, the energy of the Q-ball is, in terms of $\phi$,

\begin{equation}\label{eqn:q169}
E_{Q} = \omega Q + \frac{4\pi}{3}\int dr \; r^{2} \frac{1}{\Omega^{2}}\left(\frac{d\phi}{dr}\right)^{2},
\end{equation}

where the integral can be interpreted as a surface energy term. There are some interesting things which can be said about this relation. Dividing through by the charge gives

\begin{equation}\label{eqn:q170}
\frac{E_{Q}}{Q} = \omega  + \frac{4\pi}{3Q}\int dr \; r^{2} \frac{1}{\Omega^{2}}\left(\frac{d\phi}{dr}\right)^{2},
\end{equation}

which shows that the $\omega$ parameter is equal to the energy-charge ratio of the Q-ball up to a small correction due to a contribution from the surface energy of the Q-ball. In the limit $\Omega \rightarrow 1$, this expression is equal to that obtained for conventional Q-balls in [169]. It also provides some analytical verification of the condition for absolute stability. If the Q-ball is absolutely stable, $E_{Q} < mQ$, then

\begin{equation}\label{eqn:q171}
\frac{E_{Q}}{Q}  = \omega  + \frac{4\pi}{3Q}\int dr \; r^{2} \frac{1}{\Omega^{2}}\left(\frac{d\phi}{dr}\right)^{2} < m.
\end{equation}

Provided that the surface term is significantly smaller than $\omega$, we find that

\begin{equation}\label{eqn:q172}
\omega < m,
\end{equation}

is an alternative formulation of the absolute stability condition for these Q-balls. Since $\omega < m$ in order for Q-ball solutions to exist, it follows that if (<ref>) holds then all Q-ball solutions will be stable.

§.§ Application of the Energy-Charge Relation to the Analytical Approximate Q-ball Solution

In this section we use the analytical Q-ball solution to show that the energy to charge ratio is close to $\omega$ for $\phi_{0} >> M_{pl}/\sqrt{\xi}$, confirming the numerical results in Figure <ref>. The analytical approximation for the scalar field is given by (<ref>) and the analytical expression for the Q-ball charge is given by (<ref>). The analytical approximation of $\phi$ is valid provided that the field does not deviate too far from its initial value, $\phi \approx \phi_{0}$, which is true for $r < r_{Q}/2$. The analytical expression for the charge is calculated for $r < r_{Q}/2$, so the integral in this expression must be taken as far as $r = r_{Q}/2$.

Differentiating the inflaton field in the analytical approximation (<ref>) gives

\begin{equation}\label{eqn:q173}
\frac{d\phi}{d r} = -\frac{2r \phi_{0}}{r_{Q}^{2}} \Rightarrow \left(\frac{d\phi}{d r}\right)^{2} = \frac{4r^{2}\phi_{0}^{2}}{r_{Q}^{4}}.
\end{equation}

Substituting this into (<ref>) in the plateau limit $\Omega^{2} \approx \xi \phi_{0}^{2}/M_{pl}^{2}$, and the charge $Q$ from (<ref>) gives

\begin{equation}\label{eqn:q174}
\frac{E}{Q} = \omega + \frac{4\pi \cdot 6\xi}{\pi \omega M_{pl}^{2}}\frac{1}{r_{Q}^{3}} \int^{r_{Q}/2}_{0} r^{2} \; dr \frac{M_{pl}^{2}}{3\xi \phi_{0}^{2}} \frac{4r^{2}\phi_{0}^{2}}{r_{Q}^{4}} = \omega + \frac{32}{\omega r_{Q}^{7}}\int^{r_{Q}/2}_{0} r^{4} dr.
\end{equation}

Performing the integration in (<ref>) we obtain

\begin{equation}\label{eqn:q175}
\frac{E}{Q} = \omega + \frac{1}{5\omega}\frac{1}{r_{Q}^{2}}.
\end{equation}

Substituting in the expression for $r_{Q}^{2}$ from (<ref>) gives

\begin{equation}\label{eqn:q176}
\frac{E}{Q} = \omega + \frac{1}{30\omega}\frac{M_{pl}^{2}}{\xi \phi_{0}^{2}}\left( m^{2} + \omega^{2} - \omega_{c}^{2}\right).
\end{equation}

For an inflaton mass within the Q-ball window (<ref>), $m^{2} + \omega^{2} - \omega_{c}^{2} \le m^{2}$, and we have

\begin{equation}\label{eqn:q177}
\frac{E}{Q} \le \omega + \frac{\omega}{30}\frac{M_{pl}^{2}}{\xi \phi_{0}^{2}}\left[ 1 + \frac{m^{2}}{\omega^{2}}\right].
\end{equation}

Taking $m^{2} \approx \omega^{2}$, we have that, at most

\begin{equation}\label{eqn:q178}
\frac{E}{Q} = \omega \left[ 1 + \mathcal{O}\left( \frac{M_{pl}^{2}}{\xi \phi_{0}^{2}}\right) \right],
\end{equation}

and therefore

\begin{equation}\label{eqn:q179}
\frac{E}{Q} \thickapprox \omega,
\end{equation}

for $\phi_{0} >> M_{pl}/\sqrt{\xi}$. This means that we can safely say to a very good approximation that the $\omega$ parameter in this model is the energy-charge ratio for $\phi_{0} >> M_{pl}/\sqrt{\xi}$.

§.§ Analytical Dependence of Q-ball Energy and Radius on the Inflaton Self-Coupling

The $\gamma$ parameter as defined (<ref>) is

\begin{equation}\label{eqn:q180}
\gamma = \frac{M_{pl}^{2}}{\xi}\left(m^{2} + \omega^{2} - \omega_{c}^{2} \right),
\end{equation}

and $\omega_{c}^{2} = \lambda M_{pl}^{2}/\xi$. If we approximate $m^{2} \approx \omega^{2} \approx \omega_{c}^{2}$, we have that

\begin{equation}\label{eqn:q181}
\gamma \approx \frac{M_{pl}^{2}}{\xi}\omega_{c}^{2} = \frac{\lambda M_{pl}^{4}}{\xi^{2}} \approx 4 V_{E},
\end{equation}

from (<ref>) in the plateau approximation.

The primordial curvature power spectrum is given by

\begin{equation}\label{eqn:q182}
\mathcal{P}_{\mathcal{R}} = \frac{\lambda N^{2}}{12\pi^{2}\xi},
\end{equation}

and this can be rearranged to give an expression for the inflaton self-coupling, $\lambda$,

\begin{equation}\label{eqn:q183}
\lambda = \left(\frac{12\pi^{2} P_{R}}{N^{2}}\right)\xi \Rightarrow \lambda \propto \xi.
\end{equation}

Using this, from (<ref>), we have that

\begin{equation}\label{eqn:q184}
\gamma \propto \frac{\lambda}{\lambda^{2}} \Rightarrow \gamma \sim \frac{1}{\lambda}.
\end{equation}

This means that, taking the expression for $r_{Q}$ from (<ref>), we can establish the dependence of $r_{Q}$ on the inflaton self-coupling

\begin{equation}\label{eqn:q185}
r_{Q} = \frac{\sqrt{6}\phi_{0}}{\sqrt{\gamma}} \Rightarrow r_{Q} \propto \sqrt{\lambda}\phi_{0}.
\end{equation}

The analytical expression for the Q-ball energy is given by (<ref>). Since in general $\omega^{2} \approx \omega_{c}^{2}$, which follows from $m^{2} \simeq \omega^{2}$ and (<ref>), the energy of the Q-ball becomes

\begin{equation}\label{eqn:q186}
E \approx \frac{3}{2} \frac{\pi M_{pl}^{2}}{12\xi}\omega_{c}^{2}r_{Q}^{3} \propto \frac{\omega_{c}^{2}r_{Q}^{3}}{\xi} \sim \frac{\lambda}{\xi}\frac{r_{Q}^{3}}{\xi},
\end{equation}

\begin{equation}\label{eqn:q187}
\Rightarrow E \propto \sqrt{\lambda}\phi_{0}^{3}.
\end{equation}

These relations will become important in Section <ref> when we consider the effects of curvature on non-minimally coupled Palatini Q-balls.

§ FORMATION OF NON-MINIMALLY COUPLED PALATINI Q-BALLS FOLLOWING TACHYONIC PREHEATING

As discussed in Section <ref>, we will consider the possibility, indicated by the results in [160], that the Q-balls in this model may arise as a result of the fragmentation of the inflationary condensate, where the inflationary condensate breaks into smaller compact lumps of scalars which subsequently decay into $\pm$ Q-ball pairs. If the inflationary potential is “flatter than $\phi^{2}$”, then this in general results in an attractive interaction between the scalars which creates a negative pressure within the inflaton condensate. This is one of the conditions which serves as a prerequisite for condensate fragmentation.

A requirement is that there is a tachyonic instability in the inflationary potential (second derivative of the effective potential/effective mass squared term is less than zero). This is satisfied on the plateau of the inflaton potential. When the field passes the tachyonic instability, quantum fluctuations of the field will be amplified as the field rolls quickly down its effective potential from the top of the local maximum. This tachyonic growth of certain modes during inflation can cause the condensate to fragment, provided that the perturbations in the energy density of the condensate are not diluted away by expansion before the tachyonic growth becomes significant. However, if there are no non-topological soliton solutions present in the scalar field theory of the inflaton then the condensate will not fragment and any large perturbations induced from tachyonic preheating will be smoothed out by expansion.

It has been shown in [168] for Palatini inflation with a real scalar field that the inflaton fluctuations grow rapidly due to tachyonic preheating at the end of slow-roll inflation, where this tachyonic growth occurs within a single inflaton oscillation for non-minimal couplings of the size relevant to this study. The non-linear fluctuations generated then become the dominant contribution to the dynamics of the inflaton field after growing very rapidly, and it is these amplified non-linear fluctuations which may cause the inflaton condensate to become unstable and fragment. In [168], the analysis is undertaken for a $\phi^{4}$ inflaton potential, which means that any amplified perturbations will be smoothed out somewhat more than in our case due to the repulsive interaction between the scalars dominating the dynamics of the inflaton within the condensate.

This analysis from [168] can therefore be applied to our model to provide a lower estimate on the range of perturbation wavelengths which undergo tachyonic growth and become non-linear. This provides a useful check on whether tachyonic preheating could be the underlying mechanism behind the formation of Q-balls in non-minimally coupled Palatini inflation, since if this is the case then the range of wavelengths of the amplified non-linear perturbations in this model should overlap with the radii of the relevant Q-balls generated in our numerical analysis (see Table <ref>). Namely, we consider the radius of the $\phi_{0} = 3.2217991 \times 10^{17} \GeV$ Q-ball, since this is within the range of estimated field values at the end of Palatini inflation, in order to estimate whether these Q-balls are likely to form from the fragmentation of the inflaton condensate following tachyonic preheating.

The inflaton begins to undergo oscillations when slow-roll inflation ends. We find that the value of the inflaton field at the onset of these oscillations in the model outlined in Section <ref> is

\begin{equation}\label{eqn:q188}
\phi_{end} = 2\sqrt{2} M_{pl} \sqrt{\beta},
\end{equation}

where $\beta = 0.1-1$, which gives $\phi_{end} \sim (1-3)M_{pl}$ for the inflaton masses relevant to this work. When the non-linear fragments form we can expect that the value of the field will actually be less than this due to expansion taking place while tachyonic preheating is occurring. Field values in the range $10^{17}-10^{18} \GeV$ are therefore a reasonable estimate.
For the purpose of this study we utilise a quantity introduced in [168] defined as the perturbation wavenumber

\begin{equation}\label{eqn:q189}
\kappa = \sqrt{\frac{\xi}{\lambda}}\frac{k}{M_{pl}},
\end{equation}

which is a rescaling of the physical wavenumber $k$. The range of perturbation wavelengths for which strong tachyonic growth of the perturbations occurs is given by $\kappa_{min} < \kappa < \kappa_{max}$, where the authors in [168] define $\kappa_{max} \thickapprox 0.4a$ [168], where $a$ is the scale factor. $\kappa_{min}$ corresponds to the size of the horizon. This range in perturbation wavenumber corresponds to a range of physical wavelengths, $\lambda_{min} < \lambda < \lambda_{max}$, for which the perturbations are expected to become non-linear. $\lambda_{min}$ is determined by $\kappa_{max}$ and $\lambda_{max} = 2 \pi H^{-1}$.

The maximum physical wavenumber $k/a$ from (<ref>) is

\begin{equation}\label{eqn:q190}
\frac{k_{max}}{a}  = \sqrt{\frac{\lambda}{\xi} } \frac{\kappa_{max}}{a} M_{Pl}  = \frac{2 \pi}{\lambda_{min}},
\end{equation}

and using $\kappa_{max} = 0.4 a$, we find that the minimum physical wavelength of the perturbations which will become non-linear due to tachyonic preheating is

\begin{equation}\label{eqn:q191}
\lambda_{min}  = \frac{2\pi}{0.4}\sqrt{\frac{\xi}{\lambda}}M_{pl}^{-1} \equiv 15.7 \sqrt{\frac{\xi}{\lambda}}M_{pl}^{-1}.
\end{equation}

Normalising the non-minimal coupling and the inflaton self-coupling using $\xi = 1.2 \times 10^{9}$ and $\lambda = 0.1$ respectively, we can write this as

\begin{equation}\label{eqn:q192}
\lambda_{min} = 6.5 \times 10^{-13}  \left(\frac{0.1}{\lambda}\right)^{1/2} \left(\frac{\xi}{10^{9}}\right)^{\frac{1}{2}} \GeV^{-1}.
\end{equation}

Taking $\lambda = 0.1$ and $\xi = 1.2 \times 10^{9}$ we find that the smallest physical wavelength in this model which we expect to produce non-linear perturbations in the inflaton condensate is $\lambda_{min} = 7.16 \times 10^{-13} \GeV^{-1}$.

The Hubble parameter at the end of slow roll inflation is

\begin{equation}\label{eqn:q193}
H^{2} = \frac{V_{E}}{3 M_{pl}^{2}},
\end{equation}

where we take $V_{E}$ to be the plateau approximation of the potential in the Einstein frame

\begin{equation}\label{eqn:q194}
V_{E} = \frac{\lambda M_{pl}^{4}}{4 \xi^{2}}.
\end{equation}

Substituting (<ref>) into (<ref>) we find that

\begin{equation}\label{eqn:q195}
H^{2} = \frac{\lambda}{12 \xi^{2}}M_{pl}^{2}.
\end{equation}

This means that

\begin{equation}\label{eqn:q196}
H^{-1} = \left(\frac{12}{\lambda}\right)^{\frac{1}{2}} \frac{\xi}{M_{pl}} = 4.56 \times 10^{-9} \left(\frac{0.1}{\lambda}\right)^{1/2} \left(\frac{\xi}{10^{9}} \right) \GeV^{-1},
\end{equation}

using the normalisation of (<ref>). Taking $\lambda = 0.1$ and $\xi = 1.2\times 10^{9}$, we find $H^{-1} = 5.47 \times 10^{-9} \GeV^{-1} $, and the maximum physical perturbation wavelength is then

\begin{equation}\label{eqn:q197}
\lambda_{max} = 2\pi H^{-1} = 2.87 \times 10^{-8}\left(\frac{0.1}{\lambda}\right)^{1/2}\left( \frac{\xi}{10^{9}}\right) \GeV^{-1}.
\end{equation}

For $\lambda = 0.1$ and $\xi = 1.2 \times 10^{9}$, this gives the maximum physical wavelength of the perturbations which will become non-linear in this model to be $\lambda_{max}= 3.4 \times 10^{-8} \GeV^{-1}$.

We find therefore that strong perturbation growth occurs for modes with wavelengths in the range $7.16 \times 10^{-13} \GeV^{-1} < \lambda  < 3.4 \times 10^{-8} \GeV^{-1}$, using $\lambda = 0.1$ and  $\xi = 1.2 \times 10^{9}$ as our parameter estimates for the non-minimally coupled Palatini inflation model. The radii of all of the Q-ball solutions presented in Section <ref> fall within this range. Most notably, the $\phi_{0} = 3.2217991 \times 10^{17}\GeV$ Q-ball has a radius $r_{X} = 8.89\times 10^{-10} \GeV^{-1}$, which is comfortably within the range of mode wavelengths we expect to see undergoing tachyonic growth. It is therefore possible that  $\phi_{0} \sim 10^{17}-10^{18} \GeV$ Q-balls could form in this model as a result of the fragmentation of the inflaton condensate due to tachyonic preheating, assuming that all other conditions for fragmentation are met, which means that non-minimally coupled Palatini Q-balls are physically feasible for the model parameters considered here, although a numerical simulation of the dynamics and fragmentation of the inflaton condensate in an expanding background would be needed in order to establish tachyonic preheating as a formation mechanism for these non-minimally coupled Palatini Q-balls properly.

§ THE EFFECTS OF CURVATURE ON NON-MINIMALLY COUPLED PALATINI Q-BALLS

In this analysis we have studied Q-balls in flat space, as they have been derived historically. In this section we consider the possible effects of curvature on these Q-ball solutions, in order to ascertain the validity of the flat space solutions. As a first pass approximation, we conduct this analysis as though the Q-balls form in flat space and then curvature is switched on. In this approximation we can consider the Q-ball solution to have a spherically symmetric energy density. This means that the spacetime outside of the Q-ball is described by the Schwarzschild metric

\begin{equation}\label{eqn:q198}
ds^{2} = \left(1 - \frac{2 G M(r)}{r} \right) dt^{2} -  \left(1 - \frac{2 G M(r)}{r} \right)^{-1} dr^{2} - r^{2} d \Omega^{2},
\end{equation}

where $M(r)$ is the mass-energy contained within radius $r$. Let the radius of the Q-ball in this scenario be $R$. If the radius of the Q-ball is smaller than its Schwarzschild radius, $r_{S} = 2G M\left(r \right)$, then the Q-ball will collapse to form a black hole. The condition to ensure that the Q-balls are not signifcantly affected by the curvature of the surrounding spacetime is therefore

\begin{equation}\label{eqn:q199}
R > 2 G M(R) \equiv r_{S},
\end{equation}

assuming that the mass-energy $M\left(r \right)$ increases faster with $r$ than $r$ increases. Essentially this means that the flat space Q-ball solutions are a good approximation of non-minimally coupled Palatini Q-balls provided that the Q-balls are sufficiently large compared to their Schwarzschild radii $r_{S}$.

Setting $R = r_{Z}$ and $M\left(r \right) = E$ from Table <ref>, we find that this condition is easily satisfied for all of the Q-balls which have been generated numerically in this model (see Table <ref>). This means that if curvature is switched on, all of the Q-ball solutions presented in Table <ref> remain as Q-balls and do not collapse to black holes. As a first approximation, this means that the flat space description is reasonable. It has been shown for conventional canonical scalar field theories that spatial curvature can affect the size of Q-balls [166], and the expression for the energy of the Q-balls will be different in curved space, since the energy density is derived from the energy-momentum tensor. In order to determine the full effects of gravity on these Q-balls, the Q-ball equation and the energy and charge of the Q-balls would need to be derived in a spherically symmetric spacetime. However, for the purposes of this thesis the flat space Q-ball solutions can be used to approximate the effects of curvature on Q-balls in a non-minimally coupled Palatini inflation framework.

max width=

$\omega/\omega_{c}$     $\phi_{0}/\GeV$     $r_{Z}/\GeV^{-1}$     $r_{S}/\GeV^{-1}$     $r_{S}/r_{Z}$

$0.707155$     $3.221799 \times 10^{17}$     $8.94 \times 10^{-10}$     $4.61 \times 10^{-11}$     $0.052$

$0.709$     $1.3464098 \times 10^{16}$     $3.79 \times 10^{-11}$     $3.31 \times 10^{-15}$     $8.74 \times 10^{-5}$

$0.71$     $8.855792 \times 10^{15}$     $2.5 \times 10^{-11}$     $9.33 \times 10^{-16}$     $3.73 \times 10^{-5}$

$0.72$     $1.960795 \times 10^{15}$     $6.09 \times 10^{-12}$     $9.65 \times 10^{-18}$     $1.58 \times 10^{-6}$

$0.73$     $1.090258 \times 10^{15}$     $3.73 \times 10^{-12}$     $1.57 \times 10^{-18}$     $4.22 \times 10^{-7}$

$0.74$     $7.45339 \times 10^{14}$     $2.72 \times 10^{-12}$     $4.89 \times 10^{-19}$     $1.80 \times 10^{-7}$

$0.75$     $5.61953 \times 10^{14}$     $2.28 \times 10^{-12}$     $2.04 \times 10^{-19}$     $8.96 \times 10^{-8}$

$0.80$     $2.29632 \times 10^{14}$     $1.47 \times 10^{-12}$     $1.39 \times 10^{-20}$     $9.48 \times 10^{-9}$

$0.85$     $1.16877 \times 10^{14}$     $1.49 \times 10^{-12}$     $2.66 \times 10^{-21}$     $1.79 \times 10^{-9}$

$0.89$     $4.7918 \times 10^{13}$     $2.70 \times 10^{-12}$     $1.08 \times 10^{-21}$     $4.00 \times 10^{-10}$

Table showing the radii $r_{Z}$, Schwarzschild radii $r_{S}$, and the ratio of the two, $r_{S}/r_{Z}$, for all of the Q-ball solutions presented.

This analysis raises the question of whether a Q-ball which collapses to a black hole could be produced in this model. In order to determine this we consider the ratio of Schwarzschild radius to Q-ball radius, $r_{S}/r_{Z}$. The largest Q-ball presented in Table <ref> is the $\omega = 0.707155\omega_{c}$ Q-ball which has $\phi_{0} = 3.22 \times 10^{17}\GeV$, a radius of $r_{Z} = 8.94 \times 10^{-10}\GeV^{-1}$and a mass-energy $M = E_{Z} = 3.34 \times 10^{27}\GeV$. Using $G = 6.9 \times 10^{-39}\GeV^{-2}$, its Schwarzschild radius is $r_{S} = 4.64 \times 10^{-11}\GeV^{-1}$ and the curvature ratio is $r_{S}/r_{Z} = 0.052$. In order to form a black hole, the Q-ball would thus need to be of larger $\phi_{0}$ than this. From Section <ref> we know that $E \propto \sqrt{\lambda}\phi_{0}^{3}$ and $r_{Q} \propto \sqrt{\lambda} \phi_{0}$, and we have confirmed that these relations approximately hold for the numerically calculated values of $E_{Z}$ and $r_{Z}$. Since $r_{S} \sim M = E_{Z}$, $r_{S} \propto \sqrt{\lambda}\phi_{0}^{3}$, and we therefore have that $r_{S}/r_{Z} \propto \phi_{0}^{2}$, independent of the inflaton self-coupling, $\lambda$. We can normalise this proportionality using the $\phi_{0} = 3.22 \times 10^{17}\GeV$ Q-ball to obtain

\begin{equation}\label{eqn:q200}
\frac{r_{S}}{r_{Z}} = 0.052 \left(\frac{\phi_{0}}{3.22 \times 10^{17} \GeV} \right)^{2}.
\end{equation}

This is in good agreement with the results of Table <ref>.

From this, a Q-ball would form a black hole once $\frac{r_{S}}{r_{Z}} = 1$ if curvature is included. Setting (<ref>) equal to unity gives the value of $\phi_{0}$ which would result in a black hole, which we denote by  $\phi_{0, c}$, to be

\begin{equation}\label{eqn:q201}
\phi_{0, c} = 1.41 \times 10^{18}\GeV.
\end{equation}

This means that Q-balls of the $\phi_{0}$ we would expect from Q-balls formed from tachyonic preheating at the end of Palatini inflation could form black holes directly if the effects of curvature are taken into account.

We can use the proportionality of $E_{Z}$ and $r_{Z}$ to $\phi_{0}$ to estimate the mass-energy and radius of this Q-ball before it collapses to a black hole. We have that $E_{Z} \propto \phi_{0}^{3}$ and $r_{Z} \propto \phi_{0}$, and can write

\begin{equation}\label{eqn:q202}
\frac{E_{Z,c}}{E_{Z}} = \frac{\phi_{0,c}^{3}}{\phi_{0}^{3}},
\end{equation}

\begin{equation}\label{eqn:q203}
\frac{r_{Z,c}}{r_{Z}} = \frac{\phi_{0,c}}{\phi_{0}}.
\end{equation}

Normalising these using the $\phi_{0} = 3.22 \times 10^{17}\GeV$ Q-ball gives

\begin{equation}\label{eqn:q204}
E_{Z,c} = 3.34 \times 10^{27}\left(\frac{1.41 \times 10^{18}}{3.22 \times 10^{17}}\right)^{3} \GeV \Rightarrow E_{Z,c} = 2.80 \times 10^{29}\GeV,
\end{equation}

\begin{equation}\label{eqn:q205}
r_{Z,c} = 8.94 \times 10^{-10}\left(\frac{1.41 \times 10^{18}}{3.22 \times 10^{17}}\right) \GeV^{-1} \Rightarrow r_{Z,c} = 3.91 \times 10^{-9}\GeV,
\end{equation}

where if calculated using a curved space Q-ball solution, $\phi_{0,c}, E_{Z,c}$ and $r_{Z,c}$ would likely be slightly smaller.

We can therefore say that Q-balls of $\phi_{0} \sim 10^{18}\GeV$ could form black holes in curved space in this model, and that these Q-balls could feasibly be produced from the fragmentation of the inflaton condensate from tachyonic preheating at the end of Palatini inflation.

As aforementioned, this analysis considers the effects on flat space Q-balls if curvature is switched on once the Q-balls have formed, and the Q-ball equation would need to be derived and solved in curved space to obtain an accurate picture of how gravity affects the properties of non-minimally coupled Palatini Q-balls. We expect however that the overall effect of curvature on the properties of the Q-balls will be small since the deviation of the metric from the flat space metric is likely to be small, and the estimate of $\phi_{0,c}$ in the flat space case should therefore be reasonable.

To illustrate this consider the fact that the ratio of the Schwarzschild radius to the Q-ball radius is proportional to $\phi_{0}^{2}$ (<ref>). If we reparameterise the metric of the spacetime surrounding the Q-balls using this ratio we find that we can rewrite (<ref>) as

\begin{equation}\label{eqn:q206}
ds^{2} \sim \left(1 - \frac{\phi_{0}^{2}}{\phi_{0, c}^{2}} \right) dt^{2} -  \left(1 - \frac{\phi_{0}^{2}}{\phi_{0,c}^{2}} \right)^{-1} dr^{2} - r^{2} d \Omega^{2},
\end{equation}

which means that for a small patch of spacetime at the surface of the Q-ball, the local deviation of the metric from the flat space metric is proportional to $\phi_{0}^{2}$. For example, if $\phi_{0} = \phi_{0,c}/2$, then the radius of the Q-ball is $R = 4 r_{S}$, and the local deviation from the flat space metric is $25\%$. Since Q-balls are composed of a scalar field it is not strictly correct to model them as solid spheres in curved space, and we therefore cannot say that the metric inside the Q-balls automatically corresponds to the flat space metric. The mass-energy of the Q-balls in Schwarzschild space falls off as $M\left(r \right) \propto r^{3}$, and therefore the deviation of the metric from flat space inside the Q-ball will be more like $M\left(r \right)/r \propto r^{2}$, therefore

\begin{equation}\label{eqn:q207}
ds^{2} \sim \left(1 - \frac{2G r^{2}}{r_{S}^{2}} \right) dt^{2} -  \left(1 - \frac{2G r^{2}}{r_{S}^{2}} \right)^{-1} dr^{2} - r^{2} d \Omega^{2},
\end{equation}

which means that the deviation from the flat space metric inside the Q-balls will be small and will decrease further very rapidly as we get closer to the centre of the Q-ball. For $\phi_{0} < \phi_{0,c}/2$ we can therefore estimate that the presence of curvature will not significantly affect the properties of the Q-balls, and that the flat space approximation is a reasonable estimate for the properties of the Q-balls presented in Section <ref>. We also estimate that the prediction of the $\phi_{0}$ at which the Q-balls will start to collapse to black holes, $\phi_{0,c}$, will not be altered significantly when curvature is taken into account. It is likely that the presence of curvature may make the formation of black holes easier, since gravitational attraction may decrease the radius at which the gravitational force is ceases to be balanced by the gradient pressure of the scalar field inside the Q-ball. The formation of Q-ball derived black holes would also be affected by the effects of curvature, requiring treatment in curved spacetime, or Schwarzschild de-Sitter for considering the effects of expansion. We do not consider these effects in this thesis, in order to investigate the quality of the flat space Q-ball solutions as an approximation for the formation of Q-ball derived black holes, however a numerical simulation of the tachyonic preheating dynamics in this model of non-minimally coupled Palatini inflatonic Q-balls could be extended to explore the effects of curvature and expansion on the formation of Q-ball derived black holes.

§ REHEATING VIA Q-BALL EVAPORATION

In the following sections we discuss some cosmological consequences which could arise as a result of Q-ball formation at the end of inflation.

The formation of Q-balls at the end of non-minimally coupled Palatini inflation could theoretically lead to an early period of matter domination in the form of solitonic matter [84, 170]. This epoch of the energy density being largely composed of solitonic matter would lead to a different reheating mechanism and post-inflationary evolution in comparison to the usual decay of a homogeneous inflaton condensate. This is due to fact that the inflaton scalars are bound in compact objects. In general, the inflaton field decays throughout the volume of the inflationary condensate and can be modelled as a volume effect. When the scalars are bound in solitons, reheating occurs as a surface effect, with the scalars decaying perturbatively and the decay products evaporating away [171] - [175] from the surface of the Q-ball. In this scenario, the evaporation rate is initially suppressed by the surface area to volume ratio of the object, therefore the larger the Q-ball the larger the suppression of the evaporation rate from its surface. In the case of decay of the inflaton to fermion pairs, the decay rate is further suppressed due to the effects of Pauli blocking [172, 173]. This effect is due to the fact that as the scalars decay to fermion pairs, the energy levels of the fermions fill up. No more decays can then proceed until there are energy levels available for the decay products to occupy, or fermion pairs have evaporated away. This suppression of the decay rate leads to a lower reheating temperature. This slower reheating also means that there are a greater number of e-folds of expansion following inflation in these models, which in principle could lead to a shorter duration of inflation depending on the precise nature of the decay processes and the size of the Q-balls.

§ REHEATING FROM Q-BALL DERIVED BLACK HOLES AND ASSOCIATED COSMOLOGICAL IMPLICATIONS

The black holes which could result from the gravitational collapse of non-minimally coupled Palatini Q-balls in this model are generally light black holes. The $\phi_{0}$ of the Q-balls which could evolve to black holes start at $\phi_{0} = 1.41 \times 10^{18}\GeV$, and this Q-ball has a mass-energy of $M = E_{Z} = 2.80 \times 10^{29}\GeV$. Using the conversion of $1\GeV = 1.79 \times 10^{-27}$kg  gives a mass for this Q-ball of $M_{BH} = 501.2$kg. Black holes of mass around $500$kg have a very high temperature, which means that the energy of the particles emitted by the black hole is also very high and thus the black hole can energetically access a large range of evaporation channels. $M_{BH} \approx 500$kg black holes can produce all the particles of the Standard Model as they decay. It is therefore possible that the Universe could be reheated via the decay of these Q-ball black holes if a sufficient number could form such that they came to dominate the energy density of the Universe. In this section we explore the possibility of reheating through Q-ball black hole decay, and whether this is compatible with the observed scalar spectral index.

The time taken for a black hole of mass $M \lesssim 10^{8} \, \kg$, with a black hole temperature is greater than $\mathcal{O}\left(100\right) \GeV$, to evaporate, $t_{ev}$, is given by [176]

\begin{equation} \label{eqn:q208}
t_{ev} = \frac{40960 \pi}{27 g_{*\,BH}} G^{2} M^{3},
\end{equation}

where $g_{*\,BH} = 108.5$ is the number of spin degrees of freedom for the Standard Model. We can equate this to the age for a matter-dominated Universe at reheating, $t = (2/3)H(T_{R})^{-1}$ (where the expansion rate at reheating is $H(T_{R}) = k_{T_{R}} T_{R}^{2}/M_{Pl}$, $k_{T_{R}} \approx 3.3$), to obtain the expression for the reheating temperature if the decay of these black holes were to reheat the Universe. This is

\begin{equation} \label{eqn:q209}
T_{R} \approx \left(\frac{2}{3}\right)^{1/2} \left(\frac{27 g_{*\,BH}}{40960 \pi k_{T}}\right)^{1/2} \frac{8 \pi M_{Pl}^{5/2}}{M^{3/2}} =  1.145 \times 10^{6} \left(\frac{1 \, {\rm kg}}{M}\right)^{3/2} \GeV,
\end{equation}

where $M = M_{BH}$ is the mass-energy of the black hole.

In Section <ref> we showed that the Q-ball with the smallest $\phi_{0}$ which could form a black hole, once curvature is included, is a Q-ball with $\phi_{0} = 1.41 \times 10^{18} \GeV$ and  $M = E = 2.80\times 10^{29} \GeV$. Taking $\lambda = 0.1$, we have that, using $1\kg = 5.62 \times 10^{26}\GeV$, the mass of this black hole is $M = 498\kg$, and substituting this into (<ref>) we obtain the reheating temperature

\begin{equation}\label{eqn:q210}
T_{R} = 102.6 \GeV,
\end{equation}

which is a very low reheating temperature. We now want to establish whether reheating temperatures this low can produce an acceptable value of the scalar spectral index, in order to determine whether reheating via the decay of Q-ball black holes is a viable reheating mechanism.

In order to calculate the scalar spectral index we need to find the number of e-folds of inflation at the pivot scale, $N_{\ast}$. We use the pivot scale used by the Planck satellite, $k_{\ast} = 0.05\Mpc^{-1}$, and the wavelength of the pivot scale is then

\begin{equation}\label{eqn:q211}
\lambda_{\ast} = \frac{2\pi}{k_{\ast}} = 125.7\Mpc.
\end{equation}

We assume that there is sufficient formation of Q-ball black holes that they come to dominate the energy density of the Universe, and that the Universe enters into a period of early matter domination after inflation until $T = T_{R}$. At this point the black holes evaporate and reheat the Universe, transitioning the energy density into an era of radiation domination. During inflation, we have that (<ref>)

\begin{equation}\label{eqn:q212}
\lambda \left(t \right) = \frac{a}{a_{0}}\lambda_{\ast} = \frac{a}{a_{end}}\frac{a_{end}}{a_{R}}\frac{a_{R}}{a_{0}}\lambda_{\ast},
\end{equation}

where the $R$ subscripts correspond to the value of the quantity at reheating, "$0$" subscripts correspond to present-day values and "$end$" refers to the end of slow-roll inflation. We have that $a/a_{end} = e^{-N}$ and

\begin{equation}\label{eqn:q213}
\frac{a_{end}}{a_{R}} = \left(\frac{\rho_{R}}{\rho_{end}} \right)^{\frac{1}{3}},
\end{equation}

during matter domination where

\begin{equation}\label{eqn:q214}
\rho_{end} = V_{E} \approx \frac{\lambda M_{pl}^{4}}{4\xi^{2}},
\end{equation}

is the energy density at the end of inflation, dominated by the inflaton potential, and

\begin{equation}\label{eqn:q215}
\rho_{R} = \frac{\pi^{2}}{30}g\left(T_{R}\right) T_{R}^{4},
\end{equation}

is the energy density of radiation.

Using the fact that the entropy remains constant during slow roll inflation we have that (from Section <ref>),

\begin{equation}\label{eqn:q216}
a^{3}g_{s}\left(T \right) T^{3} = constant,
\end{equation}

we can write

\begin{equation}\label{eqn:q217}
\frac{a_{R}}{a_{0}} =  \left(\frac{g_{s}\left(T_{0}\right)}{g_{s}\left(T_{R}\right)}\right)^{\frac{1}{3}} \frac{T_{0}}{T_{R}}.
\end{equation}

Substituting (<ref>) into (<ref>), the wavelength as a function of time is

\begin{equation}\label{eqn:q218}
\lambda \left(t \right) = \left(\frac{g_{s}\left(T_{0}\right)}{g_{s}\left(T_{R}\right)}\right)^{\frac{1}{3}} \frac{T_{0}}{T_{R}} \left(\frac{\rho_{R}}{\rho_{end}} \right)^{\frac{1}{3}}e^{-N }\lambda_{\ast}.
\end{equation}

At horizon exit during inflation, we have that

\begin{equation}\label{eqn:q219}
\lambda = H^{-1} = \left(\frac{V_{E}}{3M_{pl}^{2}}\right)^{-\frac{1}{2}}.
\end{equation}

At the pivot scale we therefore have

\begin{equation}\label{eqn:q220}
\left(\frac{V_{E}}{3M_{pl}^{2}}\right)^{-\frac{1}{2}} = \left(\frac{g_{s}\left(T_{0}\right)}{g_{s}\left(T_{R}\right)}\right)^{\frac{1}{3}} \frac{T_{0}}{T_{R}} \left(\frac{\rho_{R}}{\rho_{end}} \right)^{\frac{1}{3}}e^{-N_{\ast} }\lambda_{\ast},
\end{equation}

equating (<ref>) and (<ref>). Rearranging (<ref>) gives the expression for the number of e-folds at the pivot scale

\begin{equation}\label{eqn:q221}
N_{\ast} = ln\left[ \left(\frac{g_{s}\left(T_{0}\right)}{g_{s}\left(T_{R}\right)}\right)^{\frac{1}{3}} \frac{T_{0}}{T_{R}} \left(\frac{\pi^{2}g\left(T_{R}\right)T_{R}^{4}}{30V_{E}} \right)^{\frac{1}{3}}\left(\frac{V_{E}}{3M_{pl}^{2}}\right)^{\frac{1}{2}}\lambda_{\ast}\right].
\end{equation}

Here $T_{R} = 102.6\GeV$ is the reheating temperature, $T_{0} = 2.4 \times 10^{-13}\GeV$ is the temperature of the CMB, $g_{s}\left(T_{R}\right) = g\left(T_{R}\right) \approx 100$ and $g_{s}\left(T_{0}\right) = 3.91$. For $\lambda = 0.1$ and $\xi = 1.2163 \times 10^{9}$, $V_{E} = 5.61 \times 10^{53}\GeV^{4}$, and using the conversion $1\Mpc = 1.56 \times 10^{38}\GeV^{-1}$ we have $\lambda_{\ast} = 1.96 \times 10^{40}\GeV^{-1}$.

Substituting these numbers into (<ref>) gives the number of e-folds at the pivot scale to be

\begin{equation}\label{eqn:q222}
N_{\ast} = 43.1,
\end{equation}

which by (<ref>) gives a value for the scalar spectral index of $n_{s} = 0.9536$, which is excluded by the $2-\sigma$ lower bound from the Planck satellite $n_{s} > 0.9565$ [36]. This means that reheating via the decay of Q-ball black holes in the case where the Q-ball derived black holes dominate the energy density at early times is not a viable reheating mechanism for non-minimally coupled Palatini inflation because it predicts a scalar spectral index which is too small.

An alternative scenario for a Q-ball black hole post-inflation cosmology can be realised if only a small fraction of the Q-balls collapse to black holes, leaving a small initial fraction of energy density as Q-ball derived black holes while the rest is converted to radiation as the remaining Q-balls decay. This results in a long period of radiation domination before the Q-ball black holes come to dominate at a later time. This means that there will be a greater number of e-folds of inflation at the time the pivot scale exits the horizon, which would give a larger value for the scalar spectral index, and could potentially bring the prediction for $n_{s}$ into the acceptable bounds on the scalar spectral index from the Planck results.

In this scenario we use the assumption of instantaneous reheating, whereby the energy density of the inflaton is all immediately converted to radiation at the end of slow-roll inflation, such that

\begin{equation}\label{eqn:q223}
\rho_{end} = 3M_{pl}^{2}H^{2} = V_{E} = \rho_{R} = \frac{\pi^{2}}{30}g\left(T_{R}\right) T_{R}^{4}.
\end{equation}

Using $V_{E} = 5.61 \times 10^{53}\GeV^{4}$, for $g\left(T_{R}\right) \approx 100$, this equality gives a reheating temperature of

\begin{equation}\label{eqn:q224}
T_{R} = 1.14 \times 10^{13}\GeV,
\end{equation}

which is significantly higher than the reheating temperature in the case of reheating from Q-ball derived black hole evaporation after a period of matter domination from these black holes.

The number of e-folds at the pivot scale $k_{\ast} = 0.05Mpc^{-1}$, $\lambda_{\ast} = 1.96 \times 10^{40}\GeV^{-1}$ is derived similarly to the case of Q-ball black hole domination. The wavelength of the inflaton fluctuations as a function of time is (<ref>)

\begin{equation}\label{eqn:q225}
\lambda \left( t \right) = \frac{a}{a_{0}}\lambda_{\ast} = \frac{a}{a_{end}} \frac{a_{end}}{a_{0}}\lambda_{\ast},
\end{equation}

with $a/a_{end} = e^{-N}$ and

\begin{equation}\label{eqn:q226}
\frac{a_{end}}{a_{0}} = \left(\frac{\rho_{0}}{\rho_{end}} \right)^{\frac{1}{3}},
\end{equation}

we have that

\begin{equation}\label{eqn:q227}
\lambda\left(t \right) = \left(\frac{\rho_{0}}{\rho_{end}} \right)^{\frac{1}{3}}e^{-N} \lambda_{\ast}.
\end{equation}

Using the fact that $\lambda = H^{-1}$ at horizon exit, and (<ref>) we have that

\begin{equation}\label{eqn:q228}
e^{N_{\ast}} = \left( \frac{g_{s}\left(T_{0}\right)}{g_{s}\left(T_{end}\right)}\right)^{\frac{1}{3}}\frac{T_{0}}{T_{end}}\left(\frac{V_{E}}{3M_{pl}^{2}}\right)^{\frac{1}{2}}\lambda_{\ast}.
\end{equation}

Since we are assuming instantaneous reheating, this means that the Q-balls which do not collapse to black holes decay instantly to radiation. So we can say $T_{end} = T_{R} = 1.14 \times 10^{13}\GeV$, and using $T_{0} = 2.4 \times 10^{-13}\GeV$, $g_{s}\left(T_{R}\right) \approx 100$, $g_{s}\left(T_{0}\right) = 3.91$ and $V_{E} = 5.61 \times 10^{53}\GeV^{4}$ we find that

\begin{equation}\label{eqn:q229}
N_{\ast} = 51.6,
\end{equation}

which using (<ref>) gives a value for the scalar spectral index of $n_{s} = 0.9612$, which is within the $1-\sigma$ bound of the Planck value of $n_{s}$ [36]. Thus in general, $43.1 < N_{\ast} < 51.6$, when a density of black holes from Q-balls is included. This means that although reheating via the evaporation of Q-ball black holes is not viable in the case of complete black hole matter domination at early times, an acceptable value of the scalar spectral index can be generated in the case where a small fraction of Q-balls collapse to black holes and come to dominate the energy density of the Universe at a later time once reheating has proceeded through the instant decay of the remaining Q-balls to radiation, and the radiation energy density has decayed away with expansion. This provides a small insight into some of the rich implications that non-minimally coupled Palatini Q-balls could have for cosmology, since we have demonstrated that for a certain mass range of the inflaton, (<ref>), the model can inflate successfully, may produce Q-balls, and could also seed Primordial Black Holes from the collapse of Q-balls for sufficiently large $\phi_{0}$.

§.§ Q-ball Black Hole Mass Dependence on the Inflaton Self-Coupling, and Contraints from Baryogenesis

Thus far we have explored a fairly minimal model of inflation and Q-balls which has the potential to be a self-sufficient basis for a cosmological model. We are, however, yet to discuss a possible mechanism for baryogenesis in the case of Q-balls from non-minimally coupled Palatini inflation. A complete study of the generation and transfer of the baryon asymmetry is beyond the scope of this thesis, however there are important aspects of the model in relation to baryogenesis which can be discussed.

Since these Q-ball black holes start at $M \sim 500$kg, as discussed in Section <ref> these Q-balls can produce all the particles of the Standard Model upon evaporation. It is therefore reasonable to think that there might be some decay channel accessible for these black holes which could seed a lepton number asymmetry, this could then be the basis for baryogenesis via thermal leptogenesis in this model if the asymmetry in lepton number could be transferred to a baryon number asymmetry through sphaleron processes. Alternatively, electroweak baryogenesis could occur if there is a first-order electroweak symmetry breaking transition. The issue with this is that the temperature required for sphalerons to instigate baryogenesis, or for a first-order phase transition to occur, must be at least equal to the temperature of electroweak symmetry breaking $T_{EW} = 159.5\GeV$ [177]. Thus the case which gives a reheating temperature of around $T_{R} \approx 100\GeV$ is too low for electroweak symmetry breaking, and therefore insufficient for the precipitation of the baryon asymmetry by sphalerons. If there were a way to raise the reheating temperature in the case of reheating via Q-ball derived black hole decay, then it would be more feasible to embed non-minimally coupled Palatini Q-balls into a cosmological model which includes the generation of the baryon asymmetry.

In order to explore this idea, we first note that, as found in Section <ref>, that the mass-energy of the Q-ball derived black holes depends on the square root of the inflaton self-coupling, $M = E_{Z} \propto \sqrt{\lambda}\phi_{0}^{3}$, and we find that for the $\phi_{0} = 1.41 \times 10^{18}\GeV$ Q-ball derived black hole the reheating temperature is insufficient for baryogenesis for $\lambda = 0.1$. From (<ref>), we can write

\begin{equation}\label{eqn:q230}
M \propto \sqrt{\lambda}\phi_{0}^{3},
\end{equation}

and for the same $\phi_{0} = 1.41 \times 10^{18}\GeV$, for smaller inflaton self-coupling, say $\lambda = 0.01$, we therefore have

\begin{equation}\label{eqn:q231}
M_{new} = \sqrt{0.01}\frac{M}{\sqrt{0.1}} = \sqrt{0.1}M.
\end{equation}

For $M = E = 2.8 \times 10^{29}\GeV$, this gives a mass energy for the $\phi_{0} = 1.41 \times 10^{18}\GeV$ Q-ball derived black hole of

\begin{equation}\label{eqn:q232}
M_{new} = 8.85 \times 10^{28}\GeV,
\end{equation}

for an inflaton self-coupling of $\lambda = 0.01$. Using the conversion $1\kg = 5.62 \times 10^{26}\GeV$, this gives a mass in kilograms of

\begin{equation}\label{eqn:q233}
M_{new} = 158\kg.
\end{equation}

Using (<ref>), we find that the reheating temperature in this case is

\begin{equation}\label{eqn:q234}
T_{R} = 574\GeV.
\end{equation}

This shows that lowering the strength of the inflaton self-coupling results in a smaller mass for the black holes and subsequently a higher reheating temperature. For $\lambda = 0.01$, the reheating temperature is sufficiently high that a lepton number asymmetry generated by interactions of the decay products of these black holes could be transferred to the Standard Model particle content via sphalerons, or a first-order electroweak transition and electroweak baryogenesis could occur. This shows that the reheating temperature from Q-ball derived black hole evaporation can be adjusted using the inflaton self-coupling, and that if $\lambda$ is small enough the model can reheat to a $T_{R} > T_{EW}$ sufficient for baryogenesis. This allows more scope to embed this model of non-minimally coupled Palatini Q-balls with an early era of  Q-ball derived black hole domination and reheating into a more complete cosmological model which can account for the observed baryon asymmetry today.

There is also the possibility of baryogenesis through the evaporation of the primordial black holes formed from collapsing Q-balls themselves, with a non-zero baryon number being produced through Hawking radiation. The possibility of a baryon asymmetry being produced through the evaporation of black holes was originally proposed by Hawking in 1974 [178] and has been explored in a number of particle physics models in conjunction with other aspects of early Universe cosmology such as inflation, electroweak symmetry breaking and dark matter generation (see for example [179] - [186]) through either an electroweak or gravitational [187] baryogenesis mechanism. Baryogenesis from evaporation of Q-ball derived black holes also allows for the possibility of extending the non-minimally coupled Palatini inflation model with Q-balls into a more complete model of cosmology, with the potential for embedding into Beyond the Standard Model particle models.

§.§ Gravitational Waves from Black Holes via the Poltergeist Mechanism

It may also be possible to produce gravitational waves from the decay of primordial black holes through a phenomenon known as the Poltergeist mechanism [188]. If there is an early era of matter domination in the form of primordial black holes, these primordial black holes can rapidly decay, converting sub-horizon density perturbations into sub-horizon perturbations in radiation density. These perturbations in the radiation density can generate pressure waves within the thermal bath, which in turn generate gravitational waves. These gravitational waves may be observable in the next generation of detection experiments (DECIGO and LISA [189] - [193]), for black holes in the mass range $2-400$kg, provided that the spread of mass is in the range $\sigma\left(\delta M/M\right) \lesssim 0.01$ [188] .

The upper bound of this mass range is slightly smaller than the mass of the lowest $\phi_{0}$ Q-ball black hole predicted in this model, which starts around $M = 500$kg. It should be noted however that these mass predictions come from approximating the effects of curvature on flat space Q-balls, rather than from the direct formation of black holes - either directly from the collapse of overdensities within the inflaton condensate or from mergers of Q-balls post-formation [194] - in a curved spacetime, so it is likely that if the effects of gravity are taken into account, smaller Q-balls of a lower mass could be formed, and the Schwarzschild limit for these Palatini Q-balls may be lowered.

We also demonstrated in Section <ref>, that Q-ball black holes of a lower mass can be generated for a smaller inflaton self-coupling, $\lambda$, in a regime where the Q-ball black holes dominate the energy density at early times and then rapidly decay to radiation later on, so it is possible in this scenario that the Poltergeist mechanism could produce observable gravitational waves, although a numerical simulation of Q-ball formation in this model would be necessary to determine this formally.

§ SUMMARY AND DISCUSSION

In this chapter we have presented a study of Q-balls within the framework of a non-minimally coupled Palatini inflation model. The underlying scalar field theory is comprised of a complex scalar field charged under a global $U(1)$ symmetry with a coupling to the Ricci scalar in a simple $\phi^{2} + \phi^{4}$ potential in the Jordan frame which fulfills the role of the inflaton. In the Einstein frame the non-minimal coupling of the inflaton field to gravity is recast as a non-canonical kinetic term in the inflaton action. The predictions for the inflationary observables match the standard predictions for Palatini inflation, and the non-minimal coupling is calculated to be $\xi = 1.2163 \times 10^{9}$. We find that the presence of the inflaton mass term during inflation does not affect the form of the slow-roll parameters, or the observables, but it does directly affect the value of the inflaton field at the end of slow-roll inflation, where the inflaton at the end of inflation in this model is predicted to be $\phi \sim \left(1-3 \right)M_{pl}$. We also use the fact that a positive gradient is a necessary condition on the inflaton potential in order for inflation to proceed successfully to establish an upper bound on the inflaton mass squared.

The Q-balls in this model correspond to a complex scalar with a non-canonical kinetic term, which constitute a new type of Q-ball not previously studied. We derived the Q-ball equation analytically in the Einstein frame, by extremising the Q-ball action in flat space with respect to the conserved $U\left(1 \right)$ charge, and found that the form of the Q-ball equation in non-minimally coupled Palatini gravity is different from the conventional case in that there is an additional gradient squared term and a dependence on the conformal factor on the right hand side, and we find that this Q-ball equation (<ref>) is in exact agreement with the equation for non-canonical Q-balls presented in [195], which came out shortly after the article detailing the research this chapter is based on was released [196]. This demonstrates that non-minimally coupled Palatini Q-balls are a new class of Q-ball with different underlying dynamics to that of conventional Q-balls.

Despite this, we find that the existence condition for these Q-balls is the same as in the conventional case, $\omega < m$, and that Coleman's mechanical analogy for Q-ball existence can still be applied, although the underlying dynamics are not the same. We then used this to derive a range of inflaton masses squared for which the inflaton potential is both compatible with the existence of Q-balls and can inflate successfully, which we refer to as the Q-ball window. This is significant because it can allow for suitable candidates for the inflaton field in order to produce Q-balls to be distinguished.

We solved the Q-ball equation numerically for a fixed inflaton mass within the Q-ball window for a range of $\omega < m$. We then used the location of the zeroes of the Q-ball equation, corresponding to the minima of the effective scalar potential, in order to narrow the range of field values for which Q-ball solutions can form for each $\omega$, these values being denoted by $\phi_{0}$. A code was then run between the upper and lower bounds of $\phi_{0}$ for each $\omega$ until a Q-ball solution was found. We presented ten Q-ball solutions in the range $\omega = \left(0.707155 - 0.89 \right)\omega_{c}$, and have calculated the important properties of these Q-balls including the radii, the charge of the Q-balls and the global energy of the Q-balls. We established that the radius of this type of Q-ball is proportional to $\phi_{0}$, the energy of the Q-balls is proportional to $\phi_{0}^{3}$, and that the $\omega$ parameter is approximately equal to the ratio of the Q-ball energy to the Q-ball charge $E/Q$. We find that these relations hold very strongly for the larger $\phi_{0}$ Q-balls and the relations become less well-defined for the smaller $\phi_{0}$ Q-balls. We also confirm that the Q-balls are absolutely stable, $E < mQ$, with the exception of the $\omega = 0.89\omega_{c}$ Q-ball, which is very close to the existence limit of $\omega < m$. This is a particularly significant result for the $\phi_{0} \sim 10^{17}\GeV$ Q-ball, since this shows that Q-balls of the size we expect to form from the inflaton condensate at the end of slow-roll inflation in non-minimally coupled Palatini gravity exist, and could therefore survive long enough to affect the post-inflationary cosmology of this model.

We derived an analytical approximation to the Q-ball equation for $\phi >> M_{pl}/\sqrt{\xi} \approx \phi_{0}$ in the limit that $r < r_{Q}/2$. We find that the field profiles generated from the analytical approximations of the Q-ball solutions for large $\phi_{0}$ closely match the Q-ball profiles from the numerical solutions up to about $r = r_{Q}/2$, and become a decreasingly good approximation as the $\phi_{0}$ of the Q-balls decreases. We also use this approximation to derive analytical expressions for the energy and charge of the Q-balls in the small $r$ limit, and we use these as a test on the numerical calculation of the Q-ball properties, including a confirmation using the analytical solution that $E \propto \sqrt{\lambda}\phi_{0}^{3}$ and $r_{Q} \propto \sqrt{\lambda}\phi_{0}$, as predicted by the approximate proportionality of the numerical radii and energies. We closely followed a derivation presented in [169] applied to the case of non-minimally coupled Palatini Q-balls to verify that the energy and charge vary with $\omega$ together, as is the case for conventional Q-balls. We calculated that the energy-charge ratio is equal to $\omega$ plus an integral term which can be interpreted as a contribution from the surface energy of the Q-ball, as opposed to the exact prediction of $E/Q = \omega$ as the chemical potential of the theory in the conventional case, although we later confirm that for $\phi >> M_{pl}/\sqrt{\xi}$ the contribution from this term is very small and that $E/Q \approx \omega$, as confirmed by the numerical calculations of the energy and charge. We also find that this provides an alternative formulation of the absolute stability condition, $\omega < m$.

We speculate that these Q-balls could form from the fragmentation of the inflaton condensate at the end of slow roll inflation, possibly as a result of tachyonic preheating. We confirm that this is possible by following a calculation presented in [168] in order to determine the range of perturbation wavelengths which undergo tachyonic growth, $7.16 \times 10^{-13} \GeV^{-1} < \lambda < 3.4 \times 10^{-8} \GeV^{-1}$. We find that the largest perturbation length scale to undergo tachyonic amplification in this model is within the horizon during inflation, and that the range of perturbation wavelengths encapsulates the radii of all the Q-balls produced numerically in this model. This means that all of the Q-balls presented here could form from the growth of the inflaton field perturbations due to tachyonic preheating at the end of slow-roll inflation.

Due to the fact that at the end of inflation in this scenario the inflaton scalars are bound into compact objects, reheating will proceed differently than in conventional inflation when non-topological soliton solutions are not present. The production of stable Q-balls at the end of inflation leads to an early era of matter domination, where the energy density of the Universe is dominated by solitons for a period after inflation before the Universe reheats via the evaporation of the Q-balls to radiation. This can proceed very rapidly unless the decay of the Q-balls to fermions is suppressed by Pauli blocking. This can mean that more e-folds of expansion are required to reheat the model than in conventional inflation, which can alter the prediction of the reheating temperature, and subsequently the scalar spectral index.

We explored the effects of curvature on these Q-balls by considering the flat space Q-ball solutions in a Schwarzschild spacetime, and we find that these Q-balls will collapse to black holes of mass $\sim 500$kg for $\phi_{0} \geq \phi_{0,c} = 1.41 \times 10^{18}\GeV$ when $\lambda = 0.1$. This shows that Primordial Black Holes can form from the collapse of Q-balls of the field values expected from tachyonic preheating after Palatini inflation, making the proposed formation mechanism of these Q-balls and the resultant black holes to be physically feasible. It also raises the question of whether, in curved space, sufficiently large perturbations from tachyonic growth could directly seed primordial black holes even if the underlying scalar field theory does not admit non-topological soliton solutions. We also find that the flat space approximation of the curved space properties of these Q-balls/Q-ball derived black holes are reasonable for $\phi_{0} < \phi_{0,c}/2$, since below this limit the local deviation of the metric is less than $25\%$ from the flat space metric. We also confirm that the mass of these Q-ball black holes scales with $\sqrt{\lambda}$, and the ratio of the Schwarzschild and Q-ball radius is independent of the inflaton self-coupling.

We examined the viability of reheating via the evaporation of Q-ball derived black holes in the event that they come to dominate the energy density after inflation and then decay to radiation at later times. We find that this reheats the Universe at a very low temperature $T_{R} \approx 100\GeV$, which produces an insufficient value for the scalar spectral index of $n_{s} = 0.9536$, which is excluded by the $2-\sigma$ Planck bound of $n_{s} = 0.9565$ [36]. An alternative scenario is where only a small fraction of the Q-balls produced collapse into black holes, and the remaining Q-balls rapidly decay to radiation, instantly reheating the Universe. This means that the Universe enters immediately into an era of radiation domination following inflation, and that the Q-ball black holes come to dominate the energy density at a later time. By this mechanism, reheating completes within a greater number of e-folds of expansion than the case of early Q-ball derived black hole matter domination. It can give a much large reheating temperature, up to $T_{R} = 1.14 \times 10^{13}\GeV$, and a scalar spectral index of $n_{s} = 0.9612$ which is within the $1-\sigma$ Planck bounds of $n_{s} = 0.9649 \pm 0.0042$ [36]. This shows that although reheating purely via the evaporation of Q-ball derived black holes is not a viable reheating mechanism, the model can still reheat successfully if a only small fraction of the Q-balls produced collapse to black holes.

Adjusting the value of the inflaton self-coupling can also adjust the reheating temperature. We found that, using the dependence of the black hole mass-energy on $\sqrt{\lambda}$, if the inflaton self-coupling is $\lambda = 0.01$, the mass of the black hole resulting from the $\phi_{0} = 1.41 \times 10^{18}\GeV$ Q-ball is $M = 158$kg, and this has a reheating temperature of $T_{R} = 574 \GeV$. This is significant because it shows that altering the value of the inflaton self-coupling for a given Q-ball of $\phi_{0} \gtrsim \phi_{0,c}$ can raise the reheating temperature, bringing it to a scale $T_{R} > T_{EW} = 159 \GeV$, greater than the electroweak phase transition temperature. This is the temperature at which sphalerons can exist, and can therefore be considered the temperature at which baryogenesis becomes viable in the model, which opens up the possibility of being able to generate the baryon asymmetry in this model in addition to inflation, Q-balls and Primordial Black Holes, forming a more complete cosmological model.

Altering the value of the inflaton self-coupling for a Q-ball of $\phi_{0} \gtrsim \phi_{0,c}$ also lowers the mass of the black hole produced by the collapse of the Q-ball. This can be significant because it has been shown that black holes in the mass range $2-400$kg may produce gravitational waves which are observable in the next generation of detection experiments via the Poltergeist mechanism. We have shown for $\lambda = 0.1$, the black holes produced from the Q-balls presented here start at $M \approx 500$kg which is slightly higher than the masses which can produce observable gravitational waves through the Poltergeist mechanism. It may be therefore that these Q-balls could produce lighter black holes in the $2-400$kg range for smaller $\lambda$, which is an exciting possibility for the observability of a model of non-minimally coupled Palatini inflation with Q-balls.

It should be noted that the post-inflationary cosmology of this model is rich with other potential gravitational wave sources. It has previously been shown that non-linear eras in the early Universe can generate their own stochastic gravitational wave background ([91] - [93]); and that the rescattering of amplified modes during preheating ([94] - [97]); the inhomogeneous, anisotropic motions of the condensate associated with Q-ball formation ([197, 198]); the fragmentation of an inflaton condensate [199]; and fluctuations in the numbers of $\pm$Q-balls [200] - [203], can also source gravitational waves. A sudden transition from an early matter dominated era to a radiation dominated era can also cause an enhancement of the gravitational waves induced at second order in the primordial curvature perturbations [98], [204], [205]. This may be especially relevant in our case as a rapid decay of Q-balls can cause such a sudden change in the equation of state of the Universe [A recent work [206] has demonstrated that the decay of Q-balls from the Affleck-Dine mechanism may proceed in such a fashion and may result in an observable gravitational wave signature.]. Since the frequency and energy density of these gravitational wave signatures are largely model dependent and will be affected by subtleties in model evolution, it is difficult to speculate at present as to the potential observability of any gravitational wave signatures generated in this framework of Palatini Q-balls, however it is possible that the signals produced could be observable in future experiments able to detect higher frequencies of gravitational waves [99].

As mentioned in the Section <ref>, Q-balls and possibly their decay products can be dark matter candidates. Although we do not address this possibility in depth in this thesis, if the Q-balls form only a very small component of the energy density at the end of inflation, then assuming that the Q-balls did not dissociate or dissipate due to thermal effects, they could decay at a later time or survive to the present day and contribute to the Dark Matter content of the Universe. Depending on the interactions of the inflaton with the particle content of the Standard Model, these objects could be observable with direct dark matter searches.

Beyond the case of a complex inflaton and Q-balls, an additional motivation for this study of Q-ball solutions is the possible insights it may give into the case of a real inflaton and oscillons. A non-minimally coupled Palatini model of this type could also inflate successfully for the case of a real inflaton, and neutral oscillons could be formed from the fragmentation of such a condensate. There have been numerous works which explore the formation and existence of oscillons and related objects from condensate fragmentation [74], [84] - [88]; the evolution of a subsequent period of oscillon domination [170], and the observational signatures these objects could leave [89],[90], and as such this is a many-faceted and active area of research in cosmology. However, unlike the case of Q-balls, there are no analytical solutions for oscillons. Given the similarity of the underlying physics of real and complex scalars, it is possible that the Q-ball properties will be similar to those of the corresponding oscillons and that an oscillon window for oscillon formation will exist, analogous to the Q-ball window.

We have demonstrated in this chapter that a model of non-minimally coupled Palatini inflation with non-canonical Q-balls can produce a post-inflation cosmology potentially very different from the standard post-inflation evolution in conventional Palatini inflation, with possible consequences for observability and the arrival at today's cosmology which are broad and far-reaching. In order to determine the precise implications, a numerical simulation detailing the tachyonic growth of the inflaton perturbations, the fragmentation of the inflaton condensate, the formation and possible decay of the Q-balls, and the formation and evaporation of any of the Q-ball black holes in curved spacetime on a subsequent expanding background would need to be performed.

CHAPTER: CONCLUSION

The main focus of the research presented in this thesis has been the application of non-minimally coupled, and $R^{2}$ term, inflation models in the Palatini formalism to other problems and phenomena in cosmology, as well as the compatibility of these models with Big Bang cosmology and particle physics in the broader sense. The non-minimal coupling of the inflaton to gravity, and any higher order curvature terms in the gravitational action, modify the dynamics of the scalar field, and as such the non-minimal coupling or higher order term is typically recast by performing a conformal transformation on the metric of the theory, such that the gravitational dynamics beyond the Einstein-Hilbert action manifest as non-canonical kinetic terms and a rescaled inflaton potential when examined in the Einstein frame - a frame defined such that the transformed gravitational action is Einstein-Hilbert, and the gravitational dynamics can be considered as equivalent to General Relativity. Non-minimally coupled and $R^{2}$ inflation models are therefore typically considered in the Einstein frame and we follow this approach in this thesis. The effect of these terms is that the inflaton potential becomes a plateau in the Einstein frame, and they therefore provide a very natural mechanism for producing a flat inflaton potential, as is favoured by observations of the inflationary observables. Non-minimally coupled and $R^{2}$ inflation models are therefore very favourable observationally, and we find that the results of the calculations of the predicted inflationary observables in the inflation models considered in this thesis are in good agreement with observations.

We primarily consider inflation in the Palatini formalism in this thesis. In the Palatini formalism, the conformal transformation of the Ricci scalar is much more straightforward - owing to the treatment of the spacetime metric $g_{\mu \nu}$ and the affine connection $\Gamma^{\rho}_{\mu \nu}$ as independent quantities - in comparison to the transformation in the metric formalism, and no additional non-canonical inflaton kinetic terms arise in the Einstein frame action as a result. This means that the canonical normalisation of the inflaton field is different depending on the formalism used, which results in the Einstein frame potential in terms of the canonically normalised inflaton being different, even if the Jordan frame inflaton potential is the same in both formalisms. Since the slow-roll parameters - and hence the inflationary observables built from them - depend explicitly on the inflaton potential, the expressions and the predictions for these quantities will therefore be different depending on the formalism used. Most notably, the Palatini formalism typically produces a heavily suppressed tensor-to-scalar ratio compared to the metric formalism, and also typically requires a larger value of the non-minimal coupling $\xi$ in order to produce the observed primordial curvature power spectrum $\mathcal{P}_{\mathcal{R}}$.

In Chapter 4, we consider a $\phi^{2}$ Jordan frame inflaton potential in the Palatini formalism in a framework where there is an $R^{2}$ term in the gravitational action, setting the non-minimal coupling to zero in order to exclusively study the effects of the $R^{2}$ term on the $\phi^{2}$ model in the Palatini formalism. We were interested in whether the $\phi^{2}$ inflation model could be brought into observational favour - the original $\phi^{2}$ chaotic inflation model having been ruled out observationally on the basis of the tensor-to-scalar ratio being too large - by the addition of the $R^{2}$ term and the additional suppression of the tensor-to-scalar ratio by the Palatini formalism treatment. Another issue with the original $\phi^{2}$ chaotic inflation model is the fact that in order to complete inflation the inflaton must be super-Planckian, and we wanted to examine the constraints which would be placed on the model from the requirement that the inflaton be sub-Planckian for the duration of inflation.

The transformation of the inflaton action to the Einstein frame if there is an $R^{2}$ term present is more involved that the case of a non-minimal coupling, requiring a transformation of the action to include an auxiliary field $\chi$, which must then be eliminated via its equations of motion to define the conformal transformation and recast the $R^{2}$ term into non-canonical inflaton kinetic terms. We find that the quartic kinetic term $\sim \left(\partial_{\mu}\phi\partial^{\mu}\phi\right)^{2}$ makes a negligible contribution to the dynamics of the inflaton during slow-roll, and so we neglect this term and proceed in the same way as for non-minimally coupled inflation. Canonically rescaling the inflaton field, we find that the inflaton dynamics are well-described by a threshold approximation utilising the dominant terms of the potential in the given regime of the field. Above the threshold, the field is on the plateau of the potential and below we find that the canonical scalar field is approximately the untransformed inflaton field, such that the dynamics of the potential are equivalent to the quadratic dependence in the Jordan frame, and we regard the frames to be dynamically equivalent below this point.

Imposing the constraint that the canonically normalised inflaton remains sub-Planckian during inflation places a constraint on the coefficient of the $R^{2}$ term to be $\alpha \gtrsim 10^{12}$. We therefore find that the issue of the too-large inflaton in $\phi^{2}$ inflation can be remedied in the Palatini formalism by adding an $R^{2}$ term to the gravitational part of the action with a coefficient of $\alpha \gtrsim 10^{12}$.

In addition to the constraint of a sub-Planckian inflaton, we also examined the possible effects of embedding this inflation model into a UV completion of gravity. An issue of inflation as an effective theory in quantum gravity is that higher-order corrections scaled by the scale of quantum gravity - the Planck mass in the Einstein frame - are introduced into the inflaton potential. As aforementioned, the inflationary observables depend explicitly on the inflaton potential, therefore any additional corrections to the inflaton potential will modify the predicted values of the inflationary observables and could possibly move a theory out of observational favour. If we want to embed inflation in a more complete cosmological model, and embed Big Bang cosmology into a more complete theory of physics, it is important to test the compatibility of inflation with a UV completed gravity framework. We therefore tested the robustness of the model with respect to observations when Planck-suppressed effective theory corrections are added to the inflaton potential. We find that the model remains consistent with observations provided that the shift on the $\eta$-parameter due to the introduction of the Planck-suppressed corrections from the UV completion of gravity does not exceed $\left| \Delta \eta \right| \leq 0.001$.

Placing this constraint on the potential shift due to these Planck-suppressed corrections places the constraint on the coefficient of the $R^{2}$ term that $\alpha \gtrsim 10^{32}$ in order for the model to remain in agreement with observations when Planck-suppressed potential corrections from a UV completion of gravity are included. A similar constraint on the $\alpha$ parameter can be derived for the case of Planck-suppressed potential corrections with a broken shift symmetry, and in this case we find that the requirement for the model to remain consistent with observations is that $\alpha \gtrsim 10^{20}$. We note from these results that the $\phi^{2}$ model of Palatini inflation with an $R^{2}$ term is compatible with observations when embedded as an effective theory in quantum gravity framework provided that the contribution of the $R^{2}$ term to the inflaton action in the Jordan frame is very large. In the Einstein frame this can be interpreted as the conformal factor being very large, and the potential therefore being very flat and the inflaton kinetic term being heavily suppressed.

Using the threshold approximation of the potential, we find that the number of e-folds of inflation at the pivot scale in this model can be expressed as a function of $\alpha$, and that for larger $\alpha$, the Planck pivot scale exits the horizon after fewer e-folds of inflation. For each of the derived lower bounds on $\alpha$ we calculate the scalar spectral index, the tensor-to-scalar ratio, the Hubble parameter, the number of e-folds at the pivot scale, the reheating temperature, the value of the inflaton field at the end of inflation, the unitarity cutoff in the Einstein frame, and the unitarity cutoff in the present vacuum. For all of the regimes of $\alpha$ we find that unitarity is conserved in the model, and that the scalar spectral index in each case is within the $1-\sigma$ bound on the scalar spectral index from the Planck observations (2018), with the exception of the case of Planck-suppressed potential corrections ($\alpha \gtrsim 10^{32}$), whereby the scalar spectral index is slightly outside of the $2-\sigma$ bounds on $n_{s}$ from the Planck results. It is possible that this result could be modified by the addition of other corrections, or by the combinatorial suppression factor in the $\sigma^{6}$ term from the UV completion of gravity.

In all cases the tensor-to-scalar ratio is heavily suppressed, with $r \sim 10^{-25} - 10^{-6}$, as is typical for Palatini inflation. This shows that the presence of the $R^{2}$ term in the Palatini formalism can bring the tensor-to-scalar ratio of the $\phi^{2}$ inflation model into observational favour in conjunction with a sub-Planckian inflaton, and possibly with the case of Planck-suppressed corrections for a sufficiently large $\alpha$. However, the predicted values of $r$ are below the observable limit of $r \sim 10^{-3}$ in the next generation of CMB experiments.

We consider two possible reheating mechanisms in this model: reheating via the decay to right-handed neutrinos and reheating via Higgs portal annihilation. In both cases we use the Coleman-Weinberg corrections to the inflaton potential in order to estimate the shift of the potential due to the couplings between the inflaton and right-handed neutrinos, and the inflaton and the Higgs.  For each case we choose the mass scale of the decay channel as the renormalisation scale, and use the constraint on the $\eta$-shift due to the potential corrections, $\left| \Delta \eta \right| \leq 0.001$, in order to constrain the size of the couplings. We find that for Higgs portal annihilation, $\lambda_{\phi H} < 8.2 \times 10^{-7}$, and for decay to right-handed neutrinos $\lambda_{\phi N} < 1.1 \times 10^{-3}$.

In the case of reheating via Higgs portal annihilation, there are two different possibilities for how reheating can proceed; through rapid preheating or through fragmentation of the inflaton condensate. We find that Higgs portal annihilation is not a viable reheating mechanism for $\phi^{2}$ Palatini inflation with an $R^{2}$ term unless the annihilation rate exceeds the Hubble rate quickly enough that the model reheats immediately at the end of inflation, or the condensate fragments to oscillons. We find that the condition for the inflaton condensate to fragment is that $\alpha > 2.9 \times 10^{13}$ for the $\phi^{2}$ Palatini inflation model with an $R^{2}$ term, and this is therefore compatible with the inclusion of Planck-suppressed potential corrections but may not be satisfied for the minimum requirement of a sub-Planckian inflaton, $\alpha \gtrsim 10^{12}$, if $\alpha$ is close to its lower bound. Considering the fact that the $\alpha$ parameter is generally very large, we conclude that the condensate is likely to fragment, and that reheating via the Higgs portal then proceeds through the decay of inflatonic oscillons, in which case reheating is not instantaneous. Given that the condensate fragments, we find that the model can reheat successfully through the annihilation of oscillons for $\alpha \gtrsim 10^{20}, 10^{32}$, although given that the scalar spectral index $n_{s}$ is in tension with the $2-\sigma$ lower bound for the case of general Planck-suppressed corrections, $\alpha \gtrsim 10^{32}$, even for the case of instantaneous reheating, this suggests that reheating via Higgs portal annihilation is observationally disfavoured for this case.

For the case of reheating via the decay to right-handed neutrinos we find this can only reheat the model instantaneously for the case of general Planck-suppressed potential corrections, $\alpha \gtrsim 10^{32}$. This is because the minimum value of $\lambda_{\phi N}$ needed in order for the decay to proceed is only compatible with the upper bound on $\alpha$ for the case of general Planck-suppressed potential corrections. If the reheating is not instantaneous, then this reheating would proceed for a lower reheating temperature and consequently a lower $n_{s}$, which rules out this reheating mechanism for the case of general Planck-suppressed potential corrections, since the scalar spectral index is already in tension with the $2-\sigma$ Planck lower bound, but can allow the cases with smaller values of $\alpha$.

In Chapter 5 we consider an application of the Affleck-Dine mechanism to a scalar field in the early Universe with a $\left|\Phi\right|^{2} + \left|\Phi\right|^{4}$-type potential with $U(1)$ symmetry-breaking terms for a complex field. This form of the potential is naturally compatible with non-minimally coupled inflation models. In this work we consider quadratic $U(1)$-violating terms and set the cubic and quartic terms to zero in order to exclusively study the effects of the quadratic term. We use a threshold approximation for the potential and treat the model such that the Affleck-Dine dynamics of the field do not become significant until the potential is deep into the quadratic regime. We use this to derive an analytical approximation of the $U(1)$ asymmetry generated in the model. We then use this to derive expressions for the asymmetry generated within the coherently oscillating condensate in the $\Phi^{2}$ part of the potential - the condensate asymmetry - and the asymmetry transferred to the Standard Model by the decay of the condensate - the transferred asymmetry. We check the transferred asymmetry for both the case of no decay of the condensate and the condensate decaying. These two scenarios can be treated as the limit of the condensate decaying on timescales small and large compared to the scalar lifetime.

Dynamically the asymmetry is generated within the condensate as the inflaton scalars oscillate between their $\Phi$ and $\Phi^{\dagger}$ states and the condensate decays. This means that there is not an exact cancellation of the number density of $\Phi$ and $\Phi^{\dagger}$ scalars within the condensate, and allows for the generation of an asymmetry in $B$ within the inflaton condensate. As the condensate decays this also means that there isn't an exact cancellation of the decay products with the antiparticles when the scalars decay via a $B$-conserving interaction, resulting in an overall asymmetry in the $B$ number of the decay products of the inflaton field, which is then transferred to the Standard Model.

We find that our analytical approximation of the transferred asymmetry can generate the observed baryon-to-entropy ratio today, subject to certain constraints on $A^{\frac{1}{2}}/m_{\Phi}$, where $A$ is the $U(1)$ violating mass squared term. In the case where the lifetime of the scalars $\tau_{\Phi}$ is much larger than the oscillation time of the asymmetry $T_{asy}$, the generated asymmetry can easily be larger than the observed baryon-to-entropy ratio, and the suppression of the asymmetry over many oscillations is important in this case, whereas in the case where the condensate decays before the asymmetry can undergo many oscillations, the suppression of $A^{\frac{1}{2}}/m_{\Phi}$ must be much stronger to generate the observed baryon-to-entropy ratio.

We derive a constraint on $A^{\frac{1}{2}}/m_{\Phi}$ for which the threshold approximation of the asymmetry is valid from the fact that we require the quartic part of the potential to not make a significant contribution to the dynamics. This suppresses the dynamics of the angular component of the inflaton, and allows us to treat the Affleck-Dine dynamics of the field as being purely in the quadratic regime of the potential. This constraint on the validity of the threshold approximation is easily compatible with the constraints on $A^{\frac{1}{2}}/m_{\Phi}$ derived in order to produce the observed baryon-to-entropy ratio.

We analyse the behaviour of the condensate and transferred asymmetries from a numerical calculation, and we find that the analytical threshold approximation of the condensate and transferred asymmetries are in exact agreement with the numerically calculated asymmetries for $A^{\frac{1}{2}}/m_{\Phi} = 0.001$.

We find that the baryon-to-entropy ratio calculated numerically is the same order of magnitude as the baryon-to-entropy ratio calculated using the analytical expressions, for both $\tau_{\Phi} = T_{asy}$ and $\tau_{\Phi} > T_{asy}$, and we demonstrate that the analytical approximation of the decay of the condensate asymmetry is a valid approximation of the behaviour of the asymmetry.

For a general inflaton decay process to a pair of fermions, we show that the asymmetry is unlikely to be washed out by inflaton exchange, and express this as a constraint on the reheating temperature. This is an estimate since a proper calculation would require a specific decay model of the inflaton, however we find that the constraint we have should be easily satisfied.

We verify that this model of Affleck-Dine baryogenesis with quadratic $B$-violating potential terms is compatible with the dynamics of non-minimally coupled inflation provided that the Affleck-Dine dynamics become dominant below the threshold, well after the non-minimally coupled dynamics have ceased to dominate the inflaton potential. We find that this requirement in both the metric and Palatini formalisms leads to the same constraint on the inflaton mass of $m_{\Phi} < 2.2 \times 10^{13}\GeV$, and we therefore find that this mechanism with quadratic $B$-violating terms is compatible with a non-minimally coupled inflation model where the inflaton also acts as the Affleck-Dine field.

We consider the baryon isocurvature perturbations generated in this model and use the limits on the observable baryon isocurvature fraction from the Planck satellite in order to constrain the size of the non-minimal coupling $\xi$ in both the metric and the Palatini formalisms. We find that baryon isocurvature perturbations are generally much smaller than the observational limit in Palatini inflation, whereas in metric inflation they require that $\lambda_{\Phi} \lesssim 10^{-4}$. This also means that baryon isocurvature perturbations are close to the present CMB bound on the isocurvature fraction in metric non-minimally coupled inflation if $\lambda_{\Phi} \sim 10^{-4}$.

We also consider the validity of the classical treatment of the inflaton for the calculation of the baryon asymmetry. We find that although the inflaton field is inherently quantum in nature, the baryon asymmetry is equivalent to its classical value due to spatial averaging, and therefore treating the inflaton field as classical gives the correct expression for the asymmetry.

In Chapter 6 we present a model of non-minimally coupled Palatini inflation with a $\left|\Phi\right|^{2} + \left|\Phi\right|^{4}$-type potential with Q-balls. We begin by deriving the slow-roll parameters and the inflationary observables in this model in the Einstein frame and find that they are the same as the typical expressions for $\phi^{4}$ Palatini inflation. We then derive an upper bound on the inflaton mass from the requirement that the gradient of the inflaton potential must be positive in this case in order for inflation to take place. This has not been previously considered.

We then derive the Q-ball equation in the Einstein frame for a scalar non-minimally coupled to gravity, and we find that the Q-ball equation in this case has an additional gradient squared term and an additional conformal factor dependence on the right-hand side as compared to the equation for conventional Q-balls. This shows that these non-minimally coupled Palatini Q-balls are dynamically different to the conventional case of Q-balls and correspond to a new class of Q-ball.

We find that we can still apply Coleman's mechanical analogy for the existence of Q-balls to these non-minimally coupled Palatini Q-balls, due to the fact that the inverted Q-ball potential is still compatible with the analogy despite the fact that the underlying dynamics are different. We find that the resulting existence condition for Q-balls in the non-minimally coupled Palatini case is the same as for conventional Q-balls, $\omega < m$.

From the form of the Q-ball potential, we derive a mass range for the inflaton field within which the potential can produce Q-balls compatible with the upper bound on the inflaton mass needed for inflation to occur. We refer to this as the Q-ball window, and we then use this in order to narrow the parameter space within which we search for Q-ball solutions numerically.

We obtain ten Q-ball solutions for an inflaton mass of $m = 0.9\omega_{c}$ in the range of $\omega = (0.707155 - 0.89)\omega_{c}$, corresponding to $\phi_{0}$ in the range $10^{13}-10^{17}\GeV$ numerically. From here we numerically derive the key properties of these Q-balls, energy, charge and radius. We also derive an analytical solution of the Q-ball equation, and compare the properties of these Q-balls as calculated using the analytical approximation to the results calculated numerically. We find that these Q-balls are absolutely stable, with the exception of the $\omega = 0.89\omega_{c}$ Q-ball, and we also find that $E/Q \approx \omega$, in particular for the larger $\phi_{0}$ Q-balls.

The analytical approximation of the behaviour of the Q-ball solutions is a good fit for the Q-ball solutions for larger $\phi_{0}$, corresponding to the limit $\phi_{0} >> M_{pl}/\sqrt{\xi}$. Since the analytical approximation is calculated in the plateau limit, the poor fit for the smaller $\phi_{0}$ solutions makes sense as these Q-balls form close to the point where the plateau approximation starts to break down. We also find that the approximation is a good estimate of the Q-ball properties for $r < r_{Q}/2$, since this is the limit the analytical approximations of energy and charge are taken to.

The analytical approximation predicts that $E, Q \propto \phi_{0}^{3}$ and $r \propto \phi_{0}$, which are both confirmed numerically. Using the analytical approximation we confirm that the $\omega$ parameter corresponds to the chemical potential of the system, as it does in the case of conventional Q-balls, and we show that $E/Q = \omega$ up to a small correction from the surface energy density of the Q-ball. Subsequently, we find that the existence condition for Q-balls, $\omega < m$, is also, to a good approximation, an expression of the absolute stability requirement for Q-balls.

We discuss the possibility that these Q-balls could form as a result of the fragmentation of the inflaton condensate from tachyonic preheating. We apply the analysis used in [168] to derive a range of perturbation wavelengths which would experience tachyonic growth in this model during tachyonic preheating, and we find that the radii of all the Q-ball solutions calculated in this model are contained within this range, $7.16 \times 10^{-13}\GeV^{-1} < \lambda < 3.4 \times 10^{-8}\GeV^{-1}$. This means that it is possible that these non-minimally coupled Palatini Q-balls could be formed from the fragmentation of the inflaton condensate as a result of tachyonic preheating.

It is possible that formation of Q-balls through this channel could result in an early epoch of matter domination in the form of Q-balls, in which case reheating in this model would occur via the evaporation of Q-balls. This perturbative decay of the Q-balls could potentially result in a slower reheating than the usual decay of a coherent inflaton condensate, due to the effects of Pauli blocking and also the suppression of the evaporation rate of the scalars from the size of the surface area of the Q-balls. This slower reheating could result in a lower reheating temperature, and therefore a shorter duration of inflation owing to the fact that a greater number of e-folds of expansion would be needed to complete reheating.

We also estimate the effects of curvature on these Q-balls. In order to do this, we consider the flat space Q-ball solutions used in this work in a Schwarzschild metric. We find that the radii of all of the Q-balls generated in this work are greater than their Schwarzschild radii, and that therefore all the Q-balls generated here are consistent flat space Q-balls and will not collapse to black holes. We find that the smallest $\phi_{0}$ which would be needed to generate a Q-ball which would collapse to a black hole is $\phi_{0, c} = 1.41 \times 10^{18}\GeV$, which is typically of the size we would expect the inflaton field to be at the end of non-minimally coupled Palatini inflation. This Q-ball has a predicted radius of $r = 3.91 \times 10^{-9}\GeV^{-1}$, which is within the range of perturbation wavelengths in this model which could experience tachyonic growth. We therefore have a model of non-minimally coupled Palatini inflation which could feasibly produce Q-balls which could seed primordial black holes, and that these Q-balls could form from the fragmentation of the inflaton condensate following tachyonic preheating. Calculating the mass-energy contained within the Schwarzschild radius of this Q-ball gives a black hole of mass $M \sim 500\kg$.

We find that for $\phi_{0} < \phi_{0,c}/2$, the effects of curvature in this approximation (on the flat space solutions) are small, with a maximum of $25\%$ local deviation from flat space within the metric at the surface of the Q-balls. Therefore the estimated value of $\phi_{0,c}$ should be close to the true value. A derivation of the Q-ball equation in curved space would be required in order to verify the effects of curvature on non-minimally coupled Palatini Q-balls and their dynamics.

We also speculate on the possibility of reheating the model via the decay of primordial black holes. Since the black holes which could form in this framework start at a mass of $M \sim 500\kg$, the temperature of these black holes is high enough that the decay of these black holes could produce all of the particle content of the Standard Model, and could provide a mechanism for reheating the Universe. We consider this possibility both in the event of an early period of matter domination of primordial black holes formed from collapsed Q-balls, and in the event that only a few of the Q-balls in the early period of Q-ball matter domination collapse to form black holes, with the primordial black holes making up a small fraction of the energy density.

In the case of reheating from primordial black holes dominating the energy density after inflation, we find that the value of the scalar spectral index predicted in this scenario is excluded by the $2-\sigma$ lower bound on $n_{s}$ from the Planck results, and so is not a viable reheating mechanism for non-minimally coupled Palatini inflation with Q-balls.

For the case where the dominant component of the energy density is still Q-ball matter after inflation, and only a small fraction of the Q-balls collapse to black holes, we expect that the reheating temperature can be much higher, with only a short period of black hole domination at late times, and that the predicted scalar spectral index therefore can be within the $1-\sigma$ bounds of the Planck results, $n_{s} = 0.9612$. This means that an early period of radiation domination due to Q-ball decay with a small fraction of Q-ball derived black holes may be a viable reheating model for non-minimally coupled Palatini inflation with Q-ball derived black holes. This is one example of the potential broader cosmological implications which could result from the presence of Q-balls in a non-minimally coupled Palatini inflation model.

We speculate on a possible channel of baryogenesis from this model, which could result from the generation of an asymmetry through the interactions of the particles created from the decay of these Q-ball derived black holes. Black hole decay in the case of an early era of primordial black hole domination gives a reheating temperature of around $T_{R} \approx 100\GeV$, which is too low for either electroweak baryogenesis or the transfer of an asymmetry to baryon number by sphalerons, both of which require $T > 159.5\GeV$. Using the proportionality of the Q-ball mass-energy to $\phi_{0}$, $M \propto \sqrt{\lambda}\phi_{0}^{3}$, we find that for a smaller inflaton self-coupling, $\lambda = 0.01$, the $\phi_{0,c} = 1.41 \times 10^{18}\GeV$ Q-ball produces a black hole of mass $M = 158\kg$, which has a reheating temperature of $T_{R} = 574\GeV$ and is sufficiently high for electroweak baryogenesis or the generation of sphalerons. This shows that the reheating temperature of this non-minimally coupled Palatini inflation with Q-balls can be adjusted by adjusting the inflaton self-coupling. Therefore, reheating through the decay of Q-ball derived black holes in the case where these black holes dominate the energy density of the Universe after inflation can be consistent with baryogenesis for a smaller inflaton self-coupling.

Finally, we discuss the possibility of observable gravitational wave signatures from the model. Specifically, we find that the generation of gravitational waves through the decay of primordial black holes via the Poltergeist mechanism [188] is a possibility in this model. These gravitational waves would be observable in the next generation of gravitational wave detection experiments, DECIGO and LISA, for black holes in the mass range $2 - 400\kg$. Although the upper bound here is slightly lower than the black holes formed from the collapse of Q-balls in this model when $\lambda = 0.1$, we have demonstrated that the mass of these black holes can be within the observable range for a smaller inflaton self-coupling, so there is a possibility of observable gravitational waves from black hole decay in this model for different parameter values. There is also the fact that these estimates have been made using the flat space Q-ball solutions, and it is therefore possible that the curved space Q-ball solution could produce black holes of smaller masses, and the analysis would need to be performed in curved space to assess this.

There is also the possibility that future-observable gravitational waves could be generated from the decay of Q-balls themselves, mergers and interactions of Q-balls and possibly from the formation of the Q-balls, although a full numerical simulation of the formation and evolution of these Q-balls would need to be performed to ascertain the precise dynamics of these Q-balls and the subsequent possibilities for gravitational wave production.

In this thesis, we have presented three pieces of original research which propose solutions to problems in inflation or cosmology, or provide a basis for important phenomena in cosmology, within the framework of non-minimally coupled or $R^{2}$ inflation, primarily in the Palatini formulation of gravity. In every case we find that the conjunction of inflation with these other phenomena is at least feasible within a set of constraints on the model parameters. In addition to observational compatibility, in each case we have discussed the observability of the model, the compatibility of the model with the subsequent transition from the end of inflation to the standard cosmological evolution, and some aspect of the implications of the model or its place within cosmology in the broader sense. It is the hope of the author that the research presented has shown that inflation beyond a minimal coupling of the inflaton to gravity in the Palatini formalism is a strong candidate not just for inflation itself, and the observationally favourable status of the inflation model, but also as a basis for embedding inflation into the rest of cosmology, and indeed physics as a whole, in order to search for a more complete description of the origin of the Universe as we observe and understand it today.

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