• Spontaneous Quantum-to-Classical Cosmological Collapse Dynamics

    The cosmological primordial perturbations of the universe, implicitly defined by the Wheeler–DeWitt equation:

        \[\begin{array}{l}\tilde H\Psi = \left( {\frac{{2\pi G{\hbar ^2}}}{3}} \right.\frac{{{\partial ^2}}}{{\partial {\alpha ^2}}} - \frac{{{\hbar ^2}}}{2}\frac{{{\partial ^2}}}{{\partial {\phi ^2}}}\\ + \,{e^{6\alpha }}\left( {V\left( \phi \right) + \frac{\Lambda }{{8\pi G}}} \right) - 3{e^{4\alpha }}\left. {\frac{k}{{8\pi G}}} \right)\Psi \left( {\alpha ,\phi } \right) = 0\end{array}\]

    a partial differential equation determining a wave-function not defined in space or time or spacetime, with:

        \[\Psi { \approx _{Heisb}}\exp \left( {i{S_0}\left[ {{h_{ab}}} \right]/\hbar } \right)\psi \left[ {{h_{ab}},\left\{ {{x_n}} \right\}} \right]\]

    and \psi satisfies an approximate Schrödinger equation:

    are clearly quantum in origin. One of the central foundational philosophically pressing problems in physics is to describe a ‘collapse’ dynamics that explains the classical features consistent with astrophysical data. Given the ‘no-time’-property of the Wheeler–DeWitt equation: namely, that it lacks an external time parameter and it lacks a first derivative with an imaginary Schrödinger time-factor, as well as its linearity and symmetrization, we face a deep conflict with the Lindblad equation:

        \[\begin{array}{l}\frac{{d{{\hat \rho }_S}}}{{dt}} = - \frac{i}{\hbar }\left[ {{{\hat H}_S},\hat \rho } \right] + \\\gamma \sum\limits_j {\left[ {{{\hat L}_j}{{\hat \rho }_S}\hat L_j^\dagger - \frac{1}{2}\left\{ {\hat L_j^\dagger {{\hat L}_j},{{\hat \rho }_S}} \right\}} \right]} \end{array}\]

    given that its central properties are time-asymmetry and entanglement-entropic-irreversibility, and whose Lindbladian:

        \[\gamma \left[ {\hat S{{\hat \rho }_S}{{\hat S}^\dagger } - \frac{1}{2}\left\{ {{{\hat S}^\dagger }\hat S,{{\hat \rho }_S}} \right\}} \right]\]

    describes the non-unitary evolution of the density operator, with:

        \[\gamma \equiv 2\pi \int_0^\infty {d\omega J\left( \omega \right)\delta \left( \omega \right)} \]

    Besides the problem of the undefinability of the Lindbladian system-bath interaction:

        \[\left\{ {\begin{array}{*{20}{c}}{{{\hat H}_{SB}} = \hbar \left( {\hat S{{\hat B}^\dagger } + {{\hat S}^\dagger }\hat B} \right)\quad }\\{{{\hat H}_B} = \hbar \sum\limits_k {{\omega _k}\hat a_k^\dagger } {{\hat a}_k}}\end{array}} \right.\]


        \[\left\{ {\begin{array}{*{20}{c}}{\left[ {\hat S,{{\hat H}_S}} \right] = 0}\\{\hat S\left( t \right) = \hat S\quad ;\quad \hat B = \sum\limits_k {g_k^ * } {{\hat a}_k}}\\{\hat B\left( t \right) = {e^{\frac{i}{\hbar }{{\hat H}_B}t}}\hat B{e^{ - \,\frac{i}{\hbar }{{\hat H}_B}t}}}\end{array}} \right.\]

    in the quantum gravitational cosmology context: see Derivation of the Lindblad Equation for technical details, we already face the tripartite conflict of time, which borders on the paradoxical, and is one of the main reasons, outside of M-theory, why quantum gravity theories are unviable. Now, clearly that restricts and limits our theories of wave-functional collapse. My aim here is to show how spontaneous collapse dynamics provides such an explanation, and moreover, with essential use of the k−inflationary action:

        \[\begin{array}{*{20}{l}}{S = - \frac{1}{{16\pi G}}\int {{d^4}} x\sqrt { - g} R + }\\{\int {{d^4}} x\sqrt { - g} p\left( {X,\varphi } \right)}\end{array}\]

    of consequence since the inflaton field is an essential feature of the primordial quantum-to-classical collapse-transition. With

        \[X \equiv \frac{1}{2}{g^{\mu \nu }}{\partial _\mu }\varphi \,{\not \partial _\nu }\varphi \]

    the canonical inflaton kinetic term, and the matter Lagrangian p\left( {X,\varphi } \right) can be varied with respect to the metric to yield the energy-momentum tensor of the inflaton field:

        \[{T_{\mu \nu }} = \left( {\varepsilon + p} \right){w_\mu }{w_\nu } - p{g_{\mu \nu }}\]

    with energy density being \varepsilon = 2Xp,\,\;x - pp = p\left( {X,\varphi } \right) the Lagrangian of the scalar field and {w_\mu } \equiv \varphi /{\left( {1 + X} \right)^{1/2}}.

    In a flat Friedmann-Robertson-Walker metric as the background geometry, the above Lagrangian gives us the standard Friedmann equations:

        \[\left\{ {\begin{array}{*{20}{c}}{{H^2} = \frac{1}{{3M_{{\rm{PI}}}^2}}\varepsilon }\\{\dot H = - \frac{1}{{3M_{{\rm{PI}}}^2}}\left( {\varepsilon + p} \right)}\end{array}} \right.\]

    and continuity equation as:

        \[\dot \varepsilon = - 3H\left( {\varepsilon + p} \right)\]

    H the Hubble parameter, \overbrace {}^ \cdot is derivative with respect to cosmic time t and {M_{{\rm{PI}}}} \equiv {\left( {8\pi G} \right)^{ - 1}} the reduced Planck mass, from which we can derive the following:

        \[\begin{array}{l}\dot p = - 3c_s^2H\left( {\varepsilon + p} \right) + \\\dot \varphi \left( {{p_{,\varphi }} - c_s^2{\varepsilon _{,\varphi }}} \right)\end{array}\]

    and the parameter c_s^2 is given by:

        \[c_s^2 \equiv \frac{{p,X}}{{{\varepsilon _{,X}}}} = \frac{{\varepsilon + p}}{{2X{\varepsilon _{,X}}}}\]

    and is the upper limit on the inflationary perturbations cosmic-speed.

    The key reason why k−inflationary cosmology is of import is the implementation of the inflation-phase even when the potential of the inflaton-field either tends to zero or grows very fast, barring slow-rolls. The k−inflation Lagrangian p\left( {\varphi ,X} \right) vanishes as X \to 0 and in the X = 0-region, we can expand it as:

        \[\begin{array}{l}p\left( {\varphi ,X} \right) = K\left( \varphi \right)X + L\left( \varphi \right){X^2}\\ + ...\end{array}\]

    and given that the Lagrangian is a function of the pure kinetic term X, it follows that the evolution equation tends towards an attractor p = - \varepsilon, hence leading to a pure exponential expansion of the universe as required to drive inflation dynamics. Let us deal with the graceful-exit problem. In the X \to 0 limit, we have:

        \[p\left( {\varphi ,X} \right) = K\left( \varphi \right)X + L\left( \varphi \right){X^2}\]

        \[\varepsilon \left( {\varphi ,X} \right) = K\left( \varphi \right)X + 3L\left( \varphi \right){X^2}\]

    If the coefficients K\left( \varphi \right)L\left( \varphi \right) are positive, our Lagrangian would fail to lead to an inflationary solution. In the case of K\left( \varphi \right) < 0, solutions tend toward an inflationary fixed-point {p_{{\rm{fixed}}}} = - {\varepsilon _{{\rm{fixed}}}}, constructible as such: at fixed point \varepsilon + p, we have {X_{p,X}} = 0. One can take L\left( \varphi \right) = 1 without any loss of generality, which yields the 0-th order slow-roll quantities:

        \[\left\{ {\begin{array}{*{20}{c}}{{X_{sr}} = \frac{1}{2}\tilde K\left( {{\varphi _{sr}}} \right)}\\{{\varepsilon _{sr}} = \frac{1}{4}{{\tilde K}^2}\left( {{\varphi _{sr}}} \right)}\\{{H_{sr}} = {{\left( {2\sqrt 3 {M_{{\rm{PI}}}}} \right)}^{ - 1}}\tilde K\left( {{\varphi _{sr}}} \right)}\end{array}} \right.\]

    giving us:

        \[\left\{ {\begin{array}{*{20}{c}}{\delta X/{X_{sr}} \ll 1}\\{\partial {{\left( {\tilde K} \right)}^{ - 1/2}}/\partial \varphi \ll 3/2}\end{array}} \right.\]

    The Hubble k−inflation-slow-roll parameters are:

        \[\left\{ {\begin{array}{*{20}{c}}{{\varepsilon _0} = - \frac{{\dot H}}{{{H^2}}}}\\{{\varepsilon _{n + 1}} = \frac{{\dot \varepsilon }}{{H{\varepsilon _n}}}}\end{array}} \right.\]

    and constrained by:

        \[\left\{ {\begin{array}{*{20}{c}}{{\varepsilon _0} = \left( {3/2} \right)\left( {1 + p/\varepsilon } \right)}\\{{\varepsilon _1} = {H^{ - 1}}\left( {{\rm{In}}\left( {1 + p/\varepsilon } \right)} \right)}\\{{s_0} = {H^{ - 1}}\left( {{\rm{In}}\left( {{c_s}} \right)} \right)}\end{array}} \right.\]

    Such a model needs only two scalar perturbation terms: the inflaton fluctuation \delta \varphi and the metric perturbation \Phi. Note that such a choice of gauge has the property that the scalar perturbations are the same as the gauge invariant quantities: gauge invariant inflaton perturbations and the Bardeen potential respectively. We now define a gauge-invariant scalar \zeta from \delta \varphi and \Phi as such:

        \[\zeta = \Phi + H\frac{{\delta \varphi }}{{\dot \varphi }}\]

    which …