Continuum Field Theory
In Cosmology, the Universe is considered to be described by a homogeneous and isotropic spacetime background, characterized by the Friedmann–Lemaître–Robertson–Walker (FLRW) line element
\[
\begin{equation}
d s^2 \equiv g_{\mu\nu}d x^\mu d x^\nu = - d t^2 + a^2(t) \delta_{ij} d x^i d x^j \ ,
\label{eq_FLRWlineElem}
\end{equation}
\]
where \(t\) is the cosmic time, and \(a(t)\) is the scale factor. It is useful to define a new time variable \(\eta\) through the relation
\[
\begin{equation}
d \eta \equiv a^{-\alpha} (t) dt \ ,\label{eq_auto_001}
\end{equation}
\]
with \(\alpha\) a real number conveniently chosen for each problem at hand. We refer to \(\eta\) as the \(\alpha\)-time, and note that for \(\alpha=1\), it represents the conformal time, whereas for \(\alpha=0\), we recover cosmic time. In general, we will keep \(\alpha\) as an unspecified real number, writing all the relevant equations in terms of \(\alpha\)-time. Only when a physical problem is chosen, then one needs to make a concrete choice for \(\alpha\).
The evolution of the scale factor is dictated by the stress-energy tensor of the matter fields, which in order to be compatible with the statistical homogeneity and isotropy of the universe, must take the form of a perfect fluid as
\[
\begin{equation} {\bar T}_{\mu \nu} \equiv (\bar\rho + \bar p )u_{\mu} u_{\nu} + \bar p g_{\mu \nu} \ , \hspace{0.4cm} g_{\mu \nu} u^{\mu} u^{\nu} = -1 \hspace{0.4cm} \Longrightarrow \hspace{0.4cm} \begin{cases}
\bar\rho \equiv a^{-2 \alpha}\,{\bar T}_{00} \ , \\[0.5em]
\bar p \equiv {1\over 3a^2} \sum_j {\bar T}_{jj} \ ,
\end{cases} \label{eq_stresstensor} \end{equation}
\]
where \(\bar{p}\) and \(\bar{\rho}\) are the background values of the total pressure and energy densities of the field sectors, and \(u_{\mu} = (a^{\alpha},0,0,0)\)
is the four-velocity of a fluid at rest. The evolution of the scale factor is determined by the first and second Friedmann equations,
\[
\begin{eqnarray}
\mathcal{H}^2 \equiv \left({a'\over a}\right)^2 = a^{2 \alpha} \frac{\bar {\rho}}{3 m_p^2}
\,,~~~~~~~
{a''\over a} = \frac{a^{2 \alpha}}{6 m_p^2}\Big[ (2 \alpha - 1) \bar{\rho} - 3 \bar{ p} \Big]\,, \label{eq_Friedmann-full}
\end{eqnarray}
\]
where \({f}' \equiv \text{d} f/ \text{d} \eta\) and \(m_p \simeq 2.435\cdot10^{18}\) GeV is the reduced Planck mass.
Canonical Field Theory
We introduce now the equations of motion resulting in a scalar-gauge theory with three kinds of canonically-normalized scalar fields: a real scalar singlet \(\phi\), a U(1)-charged complex scalar field \(\varphi\), and a \([SU(N) \times U(1)]\)-charged doublet scalar field \(\Phi\). In the last two cases, there are also present, respectively, Abelian and non-Abelian gauge fields, \(A_{\mu}\) and \(C_{\mu} \equiv C_{\mu}^a T_a\), with \(T_a\) the \(N^2-1\) group generators, satisfying the \(SU(N)\) Lie algebra properties \([T_a, T_b] = i f_{abc} T_c\) , \({\rm Tr}(T_a) = 0\) , \({\rm Tr}(T_a T_b) = \frac{1}{2} \delta_{ab}\), and \(T^{\dagger}_a = T_a\), with \(f_{abc}\) the totally anti-symmetric structure constants of the group, and \([\cdot,\cdot]\) the usual matrix commutator. In the particular case of SU(2), \(T_a \equiv \sigma_a / 2\), \(a=1,2,3\), with \(\sigma_a\) the Pauli matrices, and \(f_{abc} = \epsilon_{abc}\) the totally anti-symmetric tensor. For further details see Chapter 2 of \(\mathtt{The~Art-I}\) (Ref. ). The scalar fields can be explicitly written in terms of real components as follows,
\[
\begin{eqnarray} \label{eq_ChargedScalars}
\begin{array}{ccccc}
\phi \in \mathcal{R}e & , & \varphi \equiv {1\over\sqrt{2}}(\varphi_1 +i\varphi_2) & , & \Phi = \left(
\begin{array}{c}
\varphi^{(1)} \\ \varphi^{(2)} \\ \vdots \\ \varphi^{(N)}
\end{array}
\right) =
{1\over\sqrt{2}}
\left(
\begin{array}{c}
\varphi_1 +i\varphi_2 \\ \varphi_3 +i\varphi_4 \\ \vdots \\ \varphi_{2N -1} +i\varphi_{2N}
\end{array}
\right) \,.
\end{array}
\end{eqnarray}
\]
More specifically, for canonical scalar-gauge theories we consider the action \(S = \int d^4x \sqrt{-g}\, \mathcal{L}\), with \(g \equiv {\rm det} (g_{\mu \nu})\), and the Lagrangian
\[
\begin{align}
-\mathcal{L} = \frac{1}{2}\partial_{\mu} \phi \partial ^{\mu}\phi + (D_{\mu}^A \varphi)^{*}(D_A^{\mu} \varphi) + (D_{\mu}\Phi )^{\dagger} (D^{\mu} \Phi) + \frac{1}{4} F_{\mu \nu} F^{\mu \nu} + \frac{1}{2}{\rm Tr}\{G_{\mu \nu}G^{\mu \nu}\} + V \ ,
\label{eq_lagrangian}
\end{align}
\]
with \(V \equiv V(\phi,|\varphi|, |\Phi|)\) the potential describing the interactions between the scalar fields. The covariant derivatives and field strength tensors associated to the gauge fields, are defined as
\[
\begin{eqnarray}
D_{\mu}^{\rm A} &\equiv & \partial _{\mu} - i g_A Q_AA_\mu \ , \hspace{4.5cm} F_{\mu \nu}\equiv \partial_{\mu} A_{\nu} - \partial_{\nu} A_{\mu} \ , \label{eq_AbCovDerivCont} \\
D_{\mu} & \equiv &
\mathcal{I}D^{\rm A}_\mu
- i g_C Q_C C_{\mu}^a \,T_a
\ , \hspace{2.76cm} G_{\mu \nu} \equiv \partial_{\mu} C_{\nu} - \partial_{\nu} C_{\mu} - i[C_\mu,C_\nu]\,, \label{eq_CovDerivCont}
\end{eqnarray}
\]
with \(g_{A}\) and \(g_C\) the Abelian and non-Abelian gauge couplings, \(Q_{A}\) and \(Q_C\) the Abelian and non-Abelian charges of the scalar fields, \(\mathcal{I}\) the \(N\times N\) identity matrix. The gauge-invariant electric and magnetic fields associated to the Abelian and non-Abelian fields can be written as
\[
\begin{equation}\label{eq_ElectricMagneticDefs}
E_i \equiv F_{0i} , \,\,\,\,\,\,\,\, B_i \equiv \frac{1}{2} \epsilon_{i j k} F^{j k} , \,\,\,\,\,\,\,\, E_i^a \equiv G_{0i}^a , \,\,\,\,\,\,\,\, B_i^a \equiv \frac{1}{2} \epsilon_{i j k} G^{j k}_a \ , \end{equation}
\]
where \(\epsilon_{ijk}\) is the Levi-Civita symbol in three dimensions with normalization \(\epsilon_{123}=+1\), and \(G_{\mu \nu}^a \equiv {\rm Tr}(2G_{\mu \nu} T_a) = {\rm Tr}(G_{\mu \nu} \sigma_a)\). Here it is important to note that the electric field definitions above depend on the \(\alpha\)-time \(\eta\), as \(F_{0i}\) and \(G_{0i}\) are defined with respect to \(\eta\), not \(t\).
The equations of motion for the matter fields and the scale factor have been derived in more detail in \(\mathtt{The~Art-I}\) (Ref. ). Here we simply quote their resulting form, which read
\[
\begin{eqnarray}
\phi'' - a^{-2(1 - \alpha)} {\vec\nabla}^{\,2} \hspace{-1mm}\phi + (3 - \alpha)\mathcal{H} {\phi'} &=& - a^{2 \alpha} V_{,\phi} \ , \label{eq_singlet-eomCONT} \\
\varphi'' - a^{-2(1 - \alpha)} {\vec D}_{\hspace{-0.5mm}A}^{\,2}\varphi + (3 - \alpha) \mathcal{H} {\varphi'} &=& - \frac{a^{2 \alpha}V_{,|\varphi|} }{2} \frac{\varphi}{|\varphi |} \ , \label{eq_higgsU1-eom}\\
\Phi'' - a^{-2(1 - \alpha)} {\vec D}^{\,2}\Phi + (3 - \alpha) \mathcal{H} {\Phi'} &=& - \frac{a^{2 \alpha} V_{,|\Phi|}}{2} \frac{\Phi}{|\Phi |} \ , \label{eq_higgsSU2-eom}
\\
\partial_0 F_{0i} - a^{-2(1 - \alpha )}\partial_j F_{ji} + (1 - \alpha) \mathcal{H} F_{0i} &=&
a^{2 \alpha}J^A_i \ , \label{eq_U1eom}
\\
(\mathcal{D}_0 )_{a b} (G_{0i})^b - a^{-2(1 - \alpha )} ( \mathcal{D}_j )_{a b} (G_{ji} )^b + (1 - \alpha) \mathcal{H} (G_{0i} )_a &=& a^{2 \alpha}(J_i)_a \ , \label{eq_SU2eom}
\\
\partial_i F_{0i} &=& a^2J^A_0 \ , \label{eq_GaussU1-eom}\\
(\mathcal{D}_i )_{a b} (G_{0i})^b &=& a^2(J_0)_a \ , \label{eq_GaussSU2-eom}
\end{eqnarray}
\]
where we have introduced the derivative operator \((\mathcal{D}_{\nu}O)_a = (\mathcal{D}_{\nu})_{a b}O_b \equiv ( \delta_{a b} \partial_{\nu} - f_{abc} C_{\nu}^c ) O_b\), and defined the matter currents
\[
\begin{eqnarray}
\label{eq_AbelianCurrent}
J_A^\mu & \equiv & 2g_A Q_A^{(\varphi)} \mathcal{I}m [ \varphi^{*} ( D_A^{\mu} \varphi )] + 2g_A Q_A^{(\Phi)} \mathcal{I}m [ \Phi^\dagger (D^{\mu} \Phi )]\,,\\
\label{eq_NonAbelianCurrent}
J_a^\mu & \equiv & 2g_C Q_C\mathcal{I}m [ \Phi^{\dagger} T_a( D^{\mu} \Phi )]\,.
\end{eqnarray}
\]
Note that Eqs. \eqref{eq_GaussU1-eom} and \eqref{eq_GaussSU2-eom} are the Gauss constraint of the Abelian and non-Abelian sectors, respectively, which must be preserved at all times during the evolution.
The energy-momentum tensor of a system characterized by a lagrangian \(\mathcal{L}\), is given by
\[
\begin{equation}
T_{\mu \nu} \equiv -\frac{2}{\sqrt{-g}}\frac{\delta(\sqrt{-g} \mathcal{L})}{\delta g^{\mu \nu}}\,.
\label{eq_auto_002}
\end{equation}
\]
This definition leads, using Eqs. (\ref{eq_stresstensor}) and (\ref{eq_lagrangian}), to local expressions for the field's energy and pressure densities,
\[
\begin{eqnarray}
\rho &=& {K}_{\phi} + {K}_{\varphi} + {K}_{\Phi} + {G}_{\phi} + {G}_{\varphi} + {G}_{\Phi} + {K}_{U(1)} + {G}_{U(1)} + {K}_{SU(2)} + {G}_{SU(2)} + {V}, \label{eq_rhoLocal}\\
p &=& {K}_{\phi} + {K}_{\varphi} + {K}_{\Phi} -{1\over3}({G}_{\phi} + {G}_{\varphi} + {G}_{\Phi}) + {1\over3}({K}_{U(1)} + {G}_{U(1)}) + {1\over3}({K}_{SU(2)} + {G}_{SU(2)}) - {V}, \label{eq_pLocal}
\end{eqnarray}
\]
with the different energy density contributions given by
\[
\begin{align}
\label{eq_energy-contributions}
\hspace{-1cm}\left\lbrace
\begin{array}{rcl}
{K}_{\phi} &=& \frac{1}{2 a^{2\alpha} } \phi'^2 \\
{K}_{\varphi} &=& \frac{1}{a^{2\alpha} } (D_0^A \varphi)^*(D_0^A \varphi)
\\
{K}_{\Phi} &=& \frac{1}{a^{2\alpha} } (D_0 \Phi )^\dagger(D_0 \Phi)
\\
\end{array}\right.
\hspace{0.1cm};\hspace{0.75cm}
\left\lbrace
\begin{array}{rcl}
{G}_{\phi} &=& \frac{1}{2 a^2} \sum_i (\partial_i \phi)^2
\\
{G}_{\varphi} &=& \frac{1}{a^2} \sum_i (D_i^A \varphi)^*(D_i^A \varphi)
\\
{G}_{\Phi} &=& \frac{1}{a^2} \sum_i (D_i\Phi)^\dagger(D_i \Phi)
\\
\end{array}\right.
\hspace{0.1cm};\hspace{0.75cm}
\left\lbrace
\begin{array}{rcl}
{K}_{U(1)} &=& \frac{1}{2 a^{2 + 2 \alpha}} \sum_{i} F_{0i}^2
\\
{K}_{SU(2)} &=& \frac{1}{2 a^{2 + 2 \alpha}} \sum_{a,i} (G_{0i}^a)^2
\\
{G}_{U(1)} &=& \frac{1}{2 a^4} \sum_{i,j < i} F_{ij}^2
\\
{G}_{SU(2)} &=& \frac{1}{2 a^4} \sum_{a,i,j < i} (G_{ij}^a)^2 . \\
\end{array}\right.
\\\nonumber\\
\text{(Kinetic-Scalar)} \hspace{5cm} \text{(Gradient-Scalar)} \hspace{6.5cm} \text{(Electric & Magnetic)} \hspace{3.0cm}\nonumber\\
\end{align}
\]
If the fields dominate the energy budget of the Universe, the expansion rate can be determined through the Friedmann Eqs. \eqref{eq_Friedmann-full}, which in our case can be written as
\[
\begin{align}
\label{eq_FriedmannHub}
\mathcal{H}^2 \equiv \frac{a'^{ 2}}{a^2} &= \frac{a^{2 \alpha}}{3} \left( \frac{f_*}{m_p}\right)^2 \left[ E_K^{\phi} + E_K^{\varphi} + E_K^{\Phi} + E_G^{\phi} + E_G^{\varphi} + E_G^{\Phi} + E_K^A + E_K^B + E_G^A + E_G^B + E_V \right] ,
\\
\label{eq_FriedmannD2a}
{a''\over a} &= \frac{a^{2 \alpha}}{3} \left( \frac{f_*}{m_p}\right)^2 \left[ (\alpha-2)(E_K^{\phi} + E_K^{\varphi} + E_K^{\Phi}) + \alpha(E_G^{\phi} + E_G^{\varphi} + E_G^{\Phi}) + (\alpha + 1)E_V \right.\\
& \hspace{2cm} \left. + (\alpha-1)(E_K^A + E_K^B + E_G^A + E_G^B) \right] ,\nonumber
\end{align}
\]
where we have defined the volume-averaged energy contributions as \(E_{K}^{f} = \langle K_{f} \rangle\) and \(E_{G}^{f} = \langle G_{f} \rangle\) for the scalar fields \(f=\phi,\varphi,\Phi\), \(E_{K}^{A} = \langle K_{U(1)} \rangle\), \(E_{G}^{A} = \langle G_{U(1)} \rangle\), \(E_{K}^{B} = \langle K_{SU(2)} \rangle\), and \(E_{G}^{B} = \langle G_{SU(2)} \rangle\) for the gauge fields, and \({E}_V = \langle {V} \rangle\) for the potential energy, with \(\langle \dots \rangle\) denoting an average over sufficiently large volumes that encompass all relevant wavelengths of the fields. In \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\) we use Eq. (\ref{eq_FriedmannD2a}) to solve for the scale factor, while monitoring that the constraint equation (\ref{eq_FriedmannHub}) is verified throughout the evolution to some desired accuracy, see Section Evolution Algorithms.
Non-Canonical Field Theories
There are many possibilities to consider when dealing when field theories with non-canonical interactions. Below we consider few representative examples, such as scalar fields non-minimally coupled (NMC) to gravity through a term of the form \(\propto \phi^{2}R\), with \(R\) the Ricci scalar; scalar fields with non-minimal kinetic (NMK) terms of the form \(f(\phi)X\), with \(X \equiv \partial^{\mu}\phi\partial_{\mu}\phi\); and axion-like-particles (ALPs) interacting with gauge fields through a coupling of the form \(\phi F\tilde{F}\). More details on these cases can be found on Chapters 3 and 4 of \(\mathtt{The~Art-II}\) (Ref. ).
Scalar Non-Minimally Coupled to Gravity
We begin with an action containing a curvature interaction proportional to \(\xi R\phi^2\) and a generic potential \(V(\phi,\{\varphi_{\rm m}\})\) for \(\phi\) and the remaining matter fields \(\{\varphi_{\rm m}\}\), represented by \(\mathcal{L}_{\rm m}\). In the Jordan-frame, this reads
\[
S =
\int d^{4}x\,\sqrt{-g}
\left[
\frac{1}{2}m_p^2R
-\frac{1}{2}\xi R\phi^{2}
-\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi
-V(\phi,\{\varphi_{\rm m}\})
+\mathcal{L}_{\rm m}
\right]\,,
\label{eq_action}
\]
where \(R\) is the Ricci scalar and \(\xi\) the non-minimal coupling parameter. Neglecting gravitational perturbations and restricting the metric to a spatially flat FLRW background, the equation of motion for \(\phi\) becomes
\[
\begin{eqnarray}
\phi''
+(3-\alpha)\frac{a'}{a}\phi'
-a^{-2(1-\alpha)}\nabla^2\phi
+a^{2\alpha}
\left(
\xi\bar R\phi+\frac{\partial V}{\partial\phi}
\right)
=0\,,
\label{eq_eom}\\
{\rm where} ~~~~~~\bar R =
\frac{6}{a^{2\alpha}}
\left[
\frac{a''}{a}
+(1-\alpha)\left(\frac{a'}{a}\right)^2
\right]\,.~~~~~~~~~~~~~~
\label{eq_cosmic_R}
\end{eqnarray}
\]
Assuming homogeneity and isotropy on large scales, the energy-momentum tensor takes the perfect-fluid form \(T^\mu{}_\nu=\operatorname{diag}\{-\bar\rho(\eta),\bar p(\eta),\bar p(\eta),\bar p(\eta)\}\). The background pressure and energy density are decomposed into contributions from the non-minimally coupled scalar and the remaining matter sectors, \(\bar p=\bar p_\phi+\bar p_{\rm m}\) and \(\bar\rho=\bar\rho_\phi+\bar\rho_{\rm m}\). The Einstein equations then reduce to the Friedmann equations in \(\alpha\)-time,
\[
\begin{eqnarray}
\mathcal{H}^{2}
\equiv
\left(\frac{a'}{a}\right)^2
=
\frac{a^{2\alpha}}{3m_p^2}
\left(\bar\rho_\phi+\bar\rho_{\rm m}\right)
\label{eq_Hu}\,~~;~~~~~~~~~~
\frac{a''}{a}
=
-\frac{a^{2\alpha}}{6m_p^2}
\left[
(1-2\alpha)
\left(\bar\rho_\phi+\bar\rho_{\rm m}\right)
+3\left(\bar p_\phi+\bar p_{\rm m}\right)
\right]\,,
\end{eqnarray}
\]
where the energy density and pressure of the non-minimally coupled scalar field are
\[
\bar\rho_\phi(\eta)
=
\frac{1}{2a^{2\alpha}}\left\langle\phi'^2\right\rangle
+\frac{1}{2a^2}\left\langle(\nabla\phi)^2\right\rangle
+\left\langle V(\phi)\right\rangle
+\frac{3\xi}{a^{2\alpha}}\mathcal{H}^2
\left\langle\phi^2\right\rangle
+\frac{6\xi}{a^{2\alpha}}\mathcal{H}
\left\langle\phi\phi'\right\rangle
-\frac{\xi}{a^2}\left\langle\nabla^2\phi^2\right\rangle\,.
\label{eq_nmcrho}\\
\]
\[
\begin{aligned}
\bar p_\phi(\eta)
={}&
\frac{1-4\xi}{2a^{2\alpha}}\left\langle\phi'^2\right\rangle
-\frac{1-12\xi}{6a^2}\left\langle(\nabla\phi)^2\right\rangle
-\left\langle V(\phi)\right\rangle
+\frac{2\xi}{a^{2\alpha}}\mathcal{H}
\left\langle\phi\phi'\right\rangle
-\frac{\xi}{3a^2}\left\langle\nabla^2\phi^2\right\rangle
\\
&+
2\xi\left\langle\phi V_{,\phi}\right\rangle
+\frac{\xi}{a^{2\alpha}}
\left[
\mathcal{H}^2
+12\left(\xi-\frac{1}{6}\right)
\left(
\frac{a''}{a}
+(1-\alpha)\mathcal{H}^2
\right)
\right]
\left\langle\phi^2\right\rangle\,,
\end{aligned}
\label{eq_nmcp}
\]
with \(V_{,\phi}\equiv\partial V/\partial\phi\), and \(\langle \dots \rangle\) denoting volume-averaging over sufficiently large scales that encompass all relevant wavelengths of the fields. The scale factor may, in principle, be evolved using the Friedmann equations in \(~\)\eqref{eq_Hu}. We note, however, that contrary to canonical scenarios, the r.h.s. of the Friedmann equations depend on time derivatives of \(a(\eta)\), given the expression of \(\bar\rho_\phi(\eta), \bar p_\phi(\eta)\).
As proposed in Ref. , the scale factor evolution can be obtained alternatively from the trace of the energy-momentum tensor of the non-minimally coupled field,
\[
T_\phi
=
(6\xi-1)
\left(
\partial^\mu\phi\,\partial_\mu\phi
+\xi R\phi^2
\right)
+6\xi\phi\frac{\partial V}{\partial\phi}
-4V\,.
\label{eq_4dtrT}
\]
Given the traced Einstein equations \(R = -\frac{1}{m_p^2}g^{\mu\nu}\left(T^\phi_{\mu\nu}+T^{\rm m}_{\mu\nu}\right)\) = \(-\frac{1}{m_p^2}\left(T_\phi+T_{\rm m}\right)\), it follows that the background curvature satisfies
\[
m_p^2\bar R =
(1-6\xi)
\left[
\left\langle
\partial^\mu\phi\,\partial_\mu\phi
\right\rangle
+\xi\bar R\left\langle\phi^2\right\rangle
\right]
-6\xi\left\langle\phi V_{,\phi}\right\rangle
+4\left\langle V\right\rangle
-\left\langle T_{\rm m}\right\rangle\,,
\label{eq_EFEtrBack}
\]
where \(\langle\cdots\rangle\) denotes again volume-averaging of the corresponding local spatial inhomogeneities, over length scales much larger than the inverse gradient-scales of the problem. Solving the above expression for \(\bar R\), then gives
\[
\begin{eqnarray}
\bar R
=
\frac{F(\phi)}{m_p^2}
\left[
(1-6\xi)
\left\langle
\partial^\mu\phi\,\partial_\mu\phi
\right\rangle
+4\left\langle V\right\rangle
-6\xi\left\langle\phi V_{,\phi}\right\rangle
-\left\langle T_{\rm m}\right\rangle
\right]\,.
\label{eq_eomR}\\
~~~~{\rm where}~~~
F(\phi)
\equiv
\frac{1}{
1+(6\xi-1)\xi\left\langle\phi^2\right\rangle/m_p^2
}\,.~~~~~~~~~~~~~~~
\label{eq_Fphi}
\end{eqnarray}
\]
Using Eq.\(~\)\eqref{eq_cosmic_R}, we then obtain the following differential equation for the scale factor,
\[
\frac{a''}{a}
+(1-\alpha)\left(\frac{a'}{a}\right)^2
=
\frac{a^{2\alpha}F(\phi)}{6m_p^2}
\left[
(1-6\xi)
\left\langle
\partial^\mu\phi\,\partial_\mu\phi
\right\rangle
+4\left\langle V\right\rangle
-6\xi\left\langle\phi V_{,\phi}\right\rangle
-\left\langle T_{\rm m}\right\rangle
\right]\,.
\label{eq_piadot}
\]
This equation can be evolved simultaneously with the equations of motion of the non-minimally coupled scalar and the remaining matter fields. In \(\mathcal{C}\mathtt{osmo}\mathcal{L}\mathtt{attice}\) we actually follow this method.
Non-Minimal Kinetic Scalar Theories
Another example of non-canonical interactions is given by models with an internal field-space metric (that may, in principle, depend on both the field amplitudes and their conjugate momenta), \(\mathcal{G}_{ab}\equiv\mathcal{G}_{ab}[\{\phi_a\},\{\pi_{\phi_a}\}]\). These theories are characterized by an action of the form
\[
S_{\rm NMK}
=
-\int d^4x\,\sqrt{-g}
\left[
\frac{1}{2}g^{\mu\nu}\mathcal{G}_{ab}
\partial_\mu\phi_a\partial_\nu\phi_b
+V(\{\phi\})
\right]\,,
\label{eq_ScalarActionNonCanonicalCont}
\]
where the field-space metric is symmetric, \(\mathcal{G}_{ab}=\mathcal{G}_{ba}\). The case of canonically normalized scalar fields is trivially recovered by \(\mathcal{G}_{ab}=\delta_{ab}\). Likewise, if \(\mathcal{G}_{ab}=\beta\delta_{ab}\), with \(\beta>0\) constant, the kinetic term can be brought to canonical form through the field redefinition \(\phi_a\rightarrow\sqrt{\beta}\phi_a\). Non-canonically normalized fields correspond therefore only to \(\mathcal{G}_{ab}\) depending explicitly on the field amplitudes and/or their derivatives. In such a case, one further assumes \(\det(\mathcal{G}_{ab})\neq 0\), so that an inverse metric \(\mathcal{G}^{-1}_{ab}\) exists and satisfies \(\mathcal{G}_{ac}\mathcal{G}^{-1}_{cb}=\delta_{ab}\). For simplicity, we restrict below to field-space metrics that depend only on the field amplitudes, \(\mathcal{G}_{ab}=\mathcal{G}_{ab}(\{\phi_a\})\). Varying the above action and considering the metric given by the FLRW background, we obtain field equations of motion as
\[
\begin{eqnarray}
\mathcal{G}_{ab}\phi_b''
+
(3-\alpha)\frac{a'}{a}\mathcal{G}_{ab}\phi_b'
-a^{-2(1-\alpha)}
\mathcal{G}_{ab}\nabla^2\phi_b
+ \gamma_{abc}
\left(
\phi_b'\phi_c'
-a^{-2(1-\alpha)}
\vec{\nabla}\phi_b\cdot\vec{\nabla}\phi_c
\right)
+a^{2\alpha}\frac{\partial V}{\partial\phi_a}
=0\,,
\label{eqn_FLRWeqnforNMK}\\
{\rm where}~~~~~~~~ \gamma_{abc} \equiv \mathcal{G}_{ab,c}
-\frac{1}{2}\mathcal{G}_{bc,a}\,,~~~~{\rm and}~~~~ \mathcal{G}_{bc,a}\equiv\partial\mathcal{G}_{bc}/\partial\phi_a\,. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
\label{eq_gammaNMK}
\end{eqnarray}
\]
Introducing the conjugate momenta \(\pi_a\equiv\pi_{\phi_a}\), Eq.\(~\)\eqref{eqn_FLRWeqnforNMK} can be recast as the first-order system
\[
\begin{eqnarray}
\left\lbrace
\begin{array}{l}
\phi_a'
\equiv
\mathcal{G}^{-1}_{ab}\pi_b\,,
\\[3mm]
\pi_a'
+(3-\alpha)\frac{a'}{a}\pi_a
= (\mathcal{F}_a)_{bc}\pi_b\pi_c +
a^{-2(1-\alpha)}
\left[
\mathcal{G}_{ab}\nabla^2\phi_b
+\gamma_{abc}
\vec{\nabla}\phi_b\cdot\vec{\nabla}\phi_c
\right]
-a^{2\alpha}\frac{\partial V}{\partial\phi_a}\,,
\end{array}\right.
\label{eqn_NMKcontinuum}\\
{\rm where}~~~~~~~~ (\mathcal{F}_a)_{bc} \equiv \mathcal{G}_{ae}\mathcal{G}^{-1}_{ec,d}\mathcal{G}^{-1}_{db} + \left(\mathcal{G}_{ae,d}-\frac{1}{2}\mathcal{G}_{ed,a}\right)\mathcal{G}^{-1}_{ec}\mathcal{G}^{-1}_{db}\,.~~~~~~~~~~~~
\label{eq_FaNMK}
\end{eqnarray}
\]
To solve these equations, one must also specify the equation of motion for the scale factor, whose evolution is governed by Eq.\(~\)\eqref{eq_FriedmannD2a}. For scalar fields with non-canonical kinetic terms, the kinetic and gradient energy-density contributions, compared with Eq.\(~\)\eqref{eq_energy-contributions}, become
\[
K_{\rm NMK}
=
\frac{1}{2a^{2\alpha}}
\mathcal{G}_{ab}\phi_a'\phi_b'\,,
\qquad
G_{\rm NMK}
=
\frac{1}{2a^2}
\mathcal{G}_{ab}
\vec{\nabla}\phi_a\cdot\vec{\nabla}\phi_b\,.
\label{eq_energy-contrib-NMK}
\]
The corresponding Friedmann equations are
\[
\begin{eqnarray}
\left(\frac{a'}{a}\right)^2
&=&
\frac{a^{2\alpha}}{3m_p^2}
\left\langle
K_{\rm NMK}+G_{\rm NMK}+V
\right\rangle\,,
\label{eq_FriedmannHubble-NMK}\\
{\rm and} ~~~~~~~~~~~~~~~~~~~~~~~~~~~ && \nonumber \\
\frac{a''}{a}
&=&
\frac{a^{2\alpha}}{3m_p^2}
\left\langle
(\alpha-2)K_{\rm NMK}
+\alpha G_{\rm NMK}
+(\alpha+1)V
\right\rangle\,.
\label{eq_FriedmannDDa-NMK}
\end{eqnarray}
\]
where, as usual, \(\langle\cdots\rangle\) denotes volume averaging over regions sufficiently large to encompass all relevant field wavelengths.
Axion-Gauge interactions
Axion-like particles (ALPs) enjoy a shift-symmetry \(\phi=\phi+C\) that allows them to couple derivatively to gauge fields through Chern–Simons terms, as \(\phi F\tilde{F}\) (Abelian) or \(\phi G\tilde{G}\) (non-Abelian). Here we consider an axion–\(U(1)\) sector described by the action
\[
S =
\int d^4x\,\sqrt{-g}
\left[
\frac{1}{2}m_p^2R
-\frac{1}{2}\partial_\mu\phi\,\partial^\mu\phi
-V(\phi)
-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}
+\frac{1}{4}\frac{\phi}{\Lambda}
F_{\mu\nu}\tilde F^{\mu\nu}
\right]\,.
\label{eq_AxionAction}
\]
where \(\phi\) is a real pseudo-scalar field, the axion-like particle (ALP), \(V(\phi)\) is its interaction potential (which might break the shift symmetry explicitly in some scenarios), and the field-strength tensor of the Abelian gauge field \(A_\mu\) is defined as in standard canonical theories, \(F_{\mu\nu}\equiv\partial_\mu A_\nu-\partial_\nu A_\mu\).
To characterize the ALP-gauge coupling strength, we define a dimensionless parameter \(\alpha_\Lambda\equiv m_p/\Lambda\) associated to the scale \(\Lambda\). The dual field-strength tensor is defined as \(\tilde F_{\mu\nu}\equiv\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}F^{\rho\sigma}\), where \(\epsilon_{\mu\nu\rho\sigma}\) is the four-dimensional Levi-Civita pseudotensor in curved spacetime, normalized as \(\epsilon_{0123}=1/\sqrt{-g}\). Varying the action with respect to \(\phi\) and \(A_i\) in a FLRW background, and considering the temporal gauge, \(A_0=0\), yields the equations of motion in \(\alpha\)-time as
\[
\begin{eqnarray}
\phi''
+(3-\alpha)\mathcal{H}\phi'
-a^{2(\alpha-1)}\vec{\nabla}^{\,2}\phi
+a^{2\alpha}V_{,\phi}
&=&
\frac{\alpha_\Lambda}{m_p}
a^{\alpha-3}\vec E\cdot\vec B\,,
\label{eq_axion_eom}
\\
E_i'
+(1-\alpha)\mathcal{H}E_i
+a^{2(\alpha-1)}\epsilon_{ijk}\partial_jB_k
&=&
-\frac{\alpha_\Lambda}{m_p}a^{\alpha-1}
\left(
\phi'B_i-\epsilon_{ijk}\partial_j\phi\,E_k
\right)\,.
\label{eq_axion_gauge_eom}
\end{eqnarray}
\]
where primes denote derivatives with respect to \(\alpha\)-time, \(\eta\), and \(E_i \equiv F_{0i} = A_i'\). The system is subject to the Gauss constraint, which, in the absence of external charged currents, reduces to
\[
\partial_iE_i
=
-\frac{\alpha_\Lambda}{m_p}
a^{\alpha-1}\partial_i\phi\,B_i\,.
\label{eq_axion_gauss}
\]
The local energy density and pressure of the combined axion–gauge sector are
\[
\begin{eqnarray}
\rho
&=&
K_\phi+G_\phi+V(\phi)+K_{U(1)}+G_{U(1)}\,,
\label{eq_axion_rho}
\\
p
&=&
K_\phi-\frac{1}{3}G_\phi-V(\phi)
+\frac{1}{3}
\left[
K_{U(1)}+G_{U(1)}
\right]\,.
\label{eq_axion_p}
\end{eqnarray}
\]
The individual kinetic and gradient contributions of the scalar and vector fields coincide with those defined in Eq.\(~\)\eqref{eq_energy-contributions}. If the axion–\(U(1)\) gauge sector dominates the energy budget of the Universe, the expansion is governed by the Friedmann equations, Eqs.\(~\)\eqref{eq_FriedmannHub} and \eqref{eq_FriedmannD2a}, retaining only the contributions from this sector.
Scalar-Gauge-Fluid Dynamics
Coming soon ...
Gravitational Waves
Gravitational waves (GWs) are transverse and traceless tensor perturbations, \(h_{ij}\), of the background metric. Considering the FLRW solution as the background metric, the perturbed line element (in \(\alpha\)-time) is
\[
\begin{align}
\label{eq_GWmetric}
ds^2
=
-a^{2\alpha}(\eta)d\eta^2
+a^2(\eta)\left(\delta_{ij}+h_{ij}\right)dx^idx^j\,,
\end{align}
\]
which are transverse, \(\partial_i h_{ij}=0\), and traceless, \(h_{ii}=0\). Linearizing the Einstein equations in \(h_{ij}\), the equation of motion of GWs reads
\[
\begin{align}
\label{eq_GWEOMcontinuum}
h_{ij}''
+(3-\alpha){a'\over a}h_{ij}'
-a^{-2(1-\alpha)}\nabla^2h_{ij}
=
{2\over m_p^2a^{2(1-\alpha)}}\Pi_{ij}^{\rm TT} \,.
\end{align}
\]
GWs are sourced by the transverse-traceless (TT) part of the anisotropic stress tensor, \(\Pi_{ij}^{\rm TT}\). For a generic background fluid, this anisotropic stress tensor takes the form
\[
\begin{align}
\label{eq_GWAnisotropicStress}
\Pi_{ij}
\equiv
T_{ij}-\bar p\,g_{ij},
\end{align}
\]
In practice, however, it is more convenient to define an effective anisotropic stress tensor containing only those contributions to \(\Pi_{ij}\) that have a non-zero TT projection. For example, for a generic model consisting of canonically normalized scalars and Abelian gauge fields, this takes the form (Ref. )
\[
\begin{align}
\label{eq_GWEffectiveAnisotropicStress}
\Pi_{ij}^\mathrm{eff}=\sum_a \nabla_i\phi_a \nabla_j\phi_a+2\sum_b \text{Re}\left[ \left(D_i^A\varphi_b\right)^*D_j^A\varphi_b \right]-a^{-2\alpha}E_iE_j-a^{-2}B_i B_j \,,
\end{align}
\]
from which the source of GWs is obtained after TT projection. This operation is non-local in real space, but corresponds to an algebraic relation in Fourier space,
\[
\begin{align}
\label{eq_GWTTsourceProjection}
\Pi_{ij}^{\rm TT}({\bf k},\eta)
=
\Lambda_{ij,lm}(\hat{\bf k})\Pi_{lm}^{\rm eff}({\bf k},\eta)\,,
\end{align}
\]
where the Fourier-space TT projector takes the form,
\[
\begin{align}
\label{eq_GWTTprojector}
\Lambda_{ij,lm}(\hat{\bf k})
\equiv
P_{il}(\hat{\bf k})P_{jm}(\hat{\bf k})
-{1\over2}P_{ij}(\hat{\bf k})P_{lm}(\hat{\bf k})\,,\quad\quad \text{with}\quad\quad P_{ij}(\hat{\bf k})
\equiv
\delta_{ij}-\hat k_i\hat k_j\,,
\end{align}
\]
and \(\hat k_i\equiv {k_i\over k}\). This guarantees the projected tensor is both transverse, \(k_i\Pi_{ij}^{\rm TT}=0\), and traceless, \(\Pi_{ii}^{\rm TT}=0\).
Finally, it is worth mentioning GW observables. The most relevant quantity related to GWs is the energy density of the GW background,
\[
\begin{align}
\label{eq_GWrhoContinuum}
\rho_{\rm GW}(\eta)
&=
{m_p^2\over 4a^{2\alpha}V}
\int_V d^3{\bf x}\,
h'_{ij}({\bf x},\eta)h'_{ij}({\bf x},\eta) \nonumber
\\
&\simeq
{m_p^2\over 4a^{2\alpha}V}
\int_V {d^3{\bf k}\over(2\pi)^3}
h'_{ij}({\bf k},\eta)h_{ij}^{\prime *}({\bf k},\eta) \nonumber
\\
&\equiv
\int {d\rho_{\rm GW}\over d\log k}d\log k \,,
\end{align}
\]
from which the spectral density can be defined as
\[
\begin{align}
\label{eq_GWrhoSpectrumContinuum}
\left({d\rho_{\rm GW}\over d\log k}\right)(k,\eta)
=
{m_p^2 k^3\over 8\pi^2a^{2\alpha}V}
\int {d\Omega_k\over4\pi}\,
h'_{ij}(\hat{\bf k},k,\eta)h_{ij}^{\prime *}(\hat{\bf k},k,\eta)\,.
\end{align}
\]
For stochastic sources the volume average can be replaced by an ensemble average \(\langle...\rangle\) over the independent realizations of the tensor fluctuations,
\[
\begin{align}
\rho_{\rm GW}(\eta)
&= \dfrac{m_p^2}{4a^{2\alpha}}
\left\langle h'_{ij}({\bf x},\eta) h_{ij}^{\prime *}({\bf x},\eta)\right\rangle
\nonumber \\
&= \dfrac{m_p^2}{4a^{2\alpha}}
\int \dfrac{\text{d}^3{\bf k}}{(2\pi)^3}
\dfrac{\text{d}^3{\bf k'}}{(2\pi)^3}
e^{-i {\bf x}\cdot({\bf k} - {\bf k'})}
\left\langle h'_{ij}({\bf k},\eta) h_{ij}^{\prime *}({\bf k'},\eta)\right\rangle
\nonumber \\
&\equiv
\dfrac{m_p^2}{8\pi^2a^{2\alpha}}
\int\dfrac{\text{d}k}{k} k^3 P_{h'}(k,\eta)\,,
\label{eq_stochasticPS}
\end{align}
\]
where we have introduced the power spectrum of the time derivative of \(h_{ij}\),
\[
\begin{align}
\left\langle h'_{ij}({\bf k},\eta)h_{ij}^{\prime *}({\bf k'},\eta) \right\rangle
=
(2\pi)^3 P_{h'}(k,\eta)\delta^{(3)}({\bf k} - {\bf k'})\,.
\label{eq_stochasticPS_2}
\end{align}
\]
In addition to the energy density, one usually also defines the fractional GW energy density power spectrum as
\[
\begin{align}
\label{eq_GWOmegaContinuum}
\Omega_{\rm GW}(k,\eta)
\equiv
{1\over\rho_\text{c}}{d\rho_{\rm GW}\over d\log k}
=
{k^3\over 24\pi^2\mathcal H^2}P_{h'}(k,\eta)
=
{k^3\over 24\pi^2a^{2\alpha}H^2}P_{h'}(k,\eta)\,.
\end{align}
\]
where \(\rho_\text{c}=3m_p^2 H^2\) is the critical energy density.