An Argument For A Recursive Cosmology
This thought experiment presents a theoretical framework where the universe's evolution is influenced by a weak, recursive process linking successive cosmic cycles. We formulate this idea as a minor adjustment to the standard theory of cosmological perturbations. This adjustment can be implemented in two mathematically equivalent ways: as a small change to the background evolution of the early universe, or as a slight excitation of its initial quantum vacuum state.
The model produces specific observational consequences. These include repetitive, oscillatory patterns in the primordial power spectrum, a particular form of non-Gaussianity (a measure of how galaxies cluster), and a predicted relationship between the universe's entropy and the cosmological constant. The paper defines a set of pre-registered, binary tests based on upcoming data from telescopes like CMB-S4 and LiteBIRD. The outcome of these tests will determine if the proposed scenario is supported or ruled out by evidence.
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\hypersetup{pdftitle={Recursive Cosmology in the Standard Cosmology Dialect},
pdfauthor={C. L. Vaillant},
pdfsubject={Cosmology, early universe, quantum fields in curved spacetime},
pdfkeywords={inflation, perturbation theory, non-Gaussianity, information theory, recursion}}
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\begin{document}
\title{Recursive Cosmology in the Standard Cosmology Dialect:\\
From Mukhanov--Sasaki Quantization to Information-Theoretic Recursion}
\author{C.~L.~Vaillant}
\affiliation{Institute for Theoretical Cosmology, Somewhere University}
\date{\today}
\begin{abstract}
We recast a recursive picture of cosmic evolution in the standard language of quantum field theory in curved spacetime used in early-universe cosmology. Starting from the canonical second-order action for curvature perturbations and the Mukhanov--Sasaki mode equation, we introduce a small, data-driven modification that encodes emergent information dynamics across cosmological \emph{cycles} (indexed by $n$). This modification can be represented either as a perturbation to the effective potential $z''/z$ (background channel) or an excited initial state (vacuum channel) parameterized by Bogoliubov coefficients. The framework yields log-periodic features in the primordial spectrum, species-tag non-Gaussianity, and an entropy--$\Lambda$ attractor relation, all expressed in standard observables: $\mathcal P_{\RR}(k)$, $B_{\RR}$, and $C_\ell^{XY}$. We provide priors, likelihoods, and Fisher forecasts for CMB-S4/LiteBIRD/21\,cm, and pre-register binary decision criteria that make the scenario falsifiable in upcoming data.
\end{abstract}
\maketitle
% ====================================================
\section{Introduction}
A broadly useful way to evaluate extensions of early-universe physics is to write them strictly in the idiom of inflationary perturbation theory: canonical quantization of curvature modes, in--in formalism for correlators, and direct mapping to CMB/LSS observables.
We adopt that dialect and introduce an \emph{information-theoretic recursion} that acts between cosmological \emph{cycles} (labeled by an index $n$ distinct from conformal time $\tau$). The recursion encodes how coarse-grained information and constraints from a previous cycle can weakly bias the next, producing testable signatures without abandoning the standard toolkit.
% ====================================================
\section{Background Cosmological Dynamics}
We work in spatially flat FLRW with conformal time $\tau$:
\begin{equation}
ds^2 = a(\tau)^2\left(-d\tau^2 + d\bm{x}^2\right),\qquad \cH \equiv \frac{a'}{a},\quad H\equiv \frac{\dot a}{a}.
\end{equation}
For a single adiabatic degree of freedom the curvature perturbation $\RR$ has quadratic action
\begin{equation}
S_2 = \frac{1}{2}\int d\tau\, d^3x\, z^2\left[(\RR')^2 - (\nabla \RR)^2\right],\qquad
z \equiv a\sqrt{2\epsilon}\,\mpl,\;\; \epsilon\equiv -\frac{\dot H}{H^2}.
\end{equation}
Defining the Mukhanov variable $v\equiv z\RR$, Fourier modes obey
\begin{equation}
v_k'' + \left(k^2 - \frac{z''}{z}\right)v_k = 0,
\label{eq:MS}
\end{equation}
with Bunch--Davies (BD) initial condition $v_k\to e^{-ik\tau}/\sqrt{2k}$ as $-k\tau\to\infty$.
% ====================================================
\section{Quantization and Correlators in Curved Spacetime}
We quantize via $v(\tau,\bm{x})=\int\frac{d^3k}{(2\pi)^3}\left[v_k(\tau)\hat a_{\bm{k}}e^{i\bm{k}\cdot\bm{x}}+\text{h.c.}\right]$,
$[\hat a_{\bm{k}},\hat a^\dagger_{\bm{k}'}]=(2\pi)^3\delta^{(3)}(\bm{k}-\bm{k}')$.
The power spectrum is
\begin{equation}
\mathcal P_{\RR}(k) \equiv \frac{k^3}{2\pi^2}|\RR_k|^2
= \frac{H_*^{2}}{8\pi^2 \epsilon_* \mpl^2}\left(\frac{k}{k_*}\right)^{n_s-1},\qquad n_s-1\simeq -2\epsilon_*-\eta_*,
\end{equation}
and the tensor spectrum $\mathcal P_h(k)=\frac{2H_*^2}{\pi^2 \mpl^2}$ with $r\simeq 16\epsilon_*$.
Interacting $n$-point functions are computed with the Schwinger--Keldysh (in--in) formalism:
\begin{equation}
\vev{Q(\tau)}=\Big\langle 0\Big|\bar T\exp\!\Big(i\!\int_{-\infty}^{\tau}\!\! d\tau_1 H_I(\tau_1)\Big)\,Q_I(\tau)\,
T\exp\!\Big(-i\!\int_{-\infty}^{\tau}\!\! d\tau_2 H_I(\tau_2)\Big)\Big|0\Big\rangle.
\end{equation}
% ====================================================
\section{Information-Theoretic Recursion as a Small Modification}
We encode inter-cycle effects (index $n$) with two equivalent channels:
\paragraph*{(i) Background channel.}
A small recursion potential $\Delta_{\mathrm{rec}}(\tau;n)$ perturbs the effective mass term in Eq.~\eqref{eq:MS}:
\begin{equation}
\frac{z''}{z}\;\mapsto\; \frac{z''}{z} + \Delta_{\mathrm{rec}}(\tau;n),\qquad |\Delta_{\mathrm{rec}}|\ll |z''/z|.
\label{eq:bg_mod}
\end{equation}
Phenomenologically, $\Delta_{\mathrm{rec}}$ acts like a gentle, scale-dependent phase that can generate log-periodic features.
\paragraph*{(ii) Initial-state (vacuum) channel.}
Allow a weakly excited state described by Bogoliubov coefficients $\alpha_k,\beta_k$:
\begin{equation}
v_k = \alpha_k v_k^{\rm BD} + \beta_k v_k^{\rm BD*},\qquad |\alpha_k|^2-|\beta_k|^2=1,\quad n_k=|\beta_k|^2\ll 1.
\label{eq:bog}
\end{equation}
\paragraph*{Information-theoretic driver.}
At the cycle level, the coarse-grained state $U_n$ relaxes by minimizing an informational free energy
\begin{equation}
\frac{dU}{dn} = -\nabla_U H_{\mathrm{info}}(U),\qquad
H_{\mathrm{info}} = S_{\mathrm{thermo}} - \beta I_{\mathrm{ent}} + \lambda D_{\mathrm{KL}}(p_U\Vert p_*),
\label{eq:Hinfo}
\end{equation}
with weakly-informative priors on $(\beta,\lambda)$ and constraints from joint CMB+LSS fits (Sec.~\ref{sec:stats}). The cycle index $n$ is \emph{not} conformal time; it labels successive cosmic epochs.
Thermodynamic consistency implies a monotone approach of the coarse entropy towards an attractor $S_*$:
\begin{equation}
\frac{dS}{dn} = -\frac{\partial H_{\mathrm{info}}}{\partial S} \ge 0,\qquad
S_{n+1} = S_n + \gamma(S_* - S_n),\quad 0<\gamma<2.
\label{eq:entropy}
\end{equation}
A covariant holographic cap bounds inter-cycle correlations:
$I_{\mathrm{ent}}(U_{n+1}\!:\!U_n)\le A_{\mathrm{hor}}/(4\ell_P^2)$.
% ====================================================
\section{Observable Predictions in Standard Form}
\subsection{Power spectrum features}
Both channels lead to oscillatory imprints. A convenient template is the standard log-periodic form
\begin{equation}
\boxed{\;
\mathcal P_{\RR}(k)
= \mathcal P_{0}(k)\left[1-\alpha\,\sin\!\big(\beta \ln(k/k_c)+\phi\big)\right],
\;}
\label{eq:logp}
\end{equation}
with $\mathcal P_0$ the smooth spectrum. In the background channel, $\alpha,\beta,k_c$ map to parameters of $\Delta_{\mathrm{rec}}$; in the initial-state channel they map to $|\beta_k|$ and the phase of $\alpha_k+\beta_k$ in Eq.~\eqref{eq:bog}. Under the recursion picture, the spacing $\Delta\ln k \equiv 2\pi/\beta$ encodes an effective coarse-graining ratio and is predicted to cluster in $\Delta\ln k \in [0.1,0.3]$.
\subsection{Bispectrum and species tagging}
We compute the bispectrum via in--in with cubic interactions (gravity and self-interactions),
\begin{equation}
\vev{\RR_{\bm k_1}\RR_{\bm k_2}\RR_{\bm k_3}}
=(2\pi)^3\delta^{(3)}\!\Big(\sum \bm k_i\Big) B_{\RR}(k_1,k_2,k_3).
\end{equation}
Standard templates:
\begin{align}
B_{\RR}^{\rm loc}&=\frac{6}{5}f_{\rm NL}^{\rm loc}[P_{\RR}(k_1)P_{\RR}(k_2)+\text{2 perm.}],\\
B_{\RR}^{\rm eq}&=6 f_{\rm NL}^{\rm eq}\, \mathcal S_{\rm eq}(k_1,k_2,k_3),\qquad
B_{\RR}^{\rm orth}=6 f_{\rm NL}^{\rm orth}\,\mathcal S_{\rm orth}.
\end{align}
In cyclic/bounce-like constructions one expects $r\ll 10^{-4}$ and small positive local-type $f_{\rm NL}$, while brane-like ekpyrotic scenarios favor small \emph{negative} local-type values (species tag). Our recursion adds the \emph{echo} channel (Sec.~\ref{sec:echo}) that is absent in minimal cyclic models.
\subsection{CMB projection}
\begin{equation}
C_\ell^{XY}=4\pi\int \frac{dk}{k}\,\mathcal P_{\RR}(k)\,\Delta_\ell^{X}(k)\,\Delta_\ell^{Y}(k),\qquad X,Y\in\{T,E\}.
\end{equation}
We also use a TE/EE \emph{coherence score} at the cutoff scale $k_c$: letting $\mathcal{L}(k_c)$ denote multipoles whose kernels peak near $k_c$ and $\mathbb{E}[\Delta_\ell^{(T,E)}(k_c)]$ the normalized responses,
\begin{equation}
\mathcal{C}_{k_c}=\frac{\sum_{\ell\in \mathcal{L}(k_c)}
\hat{C}_\ell^{TE}\,\mathbb{E}[\Delta_\ell^{(T)}]\mathbb{E}[\Delta_\ell^{(E)}]/\sigma_\ell^2}
{\sqrt{\sum_{\ell} (\mathbb{E}[\Delta_\ell^{(T)}])^2/\sigma_\ell^2}
\sqrt{\sum_{\ell} (\mathbb{E}[\Delta_\ell^{(E)}])^2/\sigma_\ell^2}}.
\label{eq:coh}
\end{equation}
% ====================================================
\section{Entropy--\texorpdfstring{$\Lambda$}{Lambda} Relation and Attractor Dynamics}
A recursion-level attractor ties the coarse entropy to the effective cosmological constant,
\begin{equation}
S_*=\frac{3\pi c^5}{G\hbar\,\Lambda_{\rm eff}},
\qquad
\Delta_S \equiv \frac{S_*^{(0)}-S_*^{(\Lambda)}}{S_*^{(\Lambda)}},
\label{eq:SLambda}
\end{equation}
with $S_*^{(0)}$ estimated from photons, neutrinos, and black-hole entropy (GW-informed BH mass functions), and $S_*^{(\Lambda)}$ \emph{computed} from $\Lambda_{\rm eff}$ constrained independently by SNe\,Ia/BAO/chronometers. A pre-registered pass condition is $|\Delta_S|<0.05$ together with $\log B>2.5$ vs.\ $\Lambda$CDM.
% ====================================================
\section{Statistical Inference and Forecasts}
\label{sec:stats}
\subsection{Priors and identifiability}
We adopt weak priors:
\begin{equation}
\xi \sim \mathrm{LogU}(10^{-3},10^{-1}),\quad
\gamma \sim \mathrm{LogU}(10^{-2},10^{-1}),\quad
\eta \sim \mathrm{U}(0.1,1.0),\quad
k_c \sim \mathrm{U}\big([2,5]\times10^{-4}\,\mathrm{Mpc}^{-1}\big),\;
\alpha\sim \mathrm{U}(0,0.05),\;\beta\sim \mathrm{U}(0.5,3),\;\phi\sim \mathrm{U}(-\pi,\pi).
\end{equation}
Leakage-like parameters span decades (log-uniform); $\eta$ is $\mathcal{O}(1)$ (uniform). With current CMB, $\{k_c,\alpha\}$ are identifiable; $\xi/\gamma$ needs polarization and/or 21\,cm to tighten.
\subsection{Likelihood and prior-predictive checks}
We use the Gaussian pseudo-$C_\ell$ likelihood
\begin{equation}
\ln\mathcal{L} = -\tfrac12\sum_{\ell,XY}\frac{\left(C_\ell^{XY}(\bm\theta)-\hat C_\ell^{XY}\right)^2}{\sigma_{\ell,XY}^2},
\end{equation}
and verify by prior-predictive sampling ($10^3$ draws) that the induced $C_\ell$ envelope matches low-$\ell$ variance without high-$\ell$ artifacts.
\subsection{Fisher forecasts}
For CMB-S4/LiteBIRD and a wide-field 21\,cm survey we forecast (68\%):
\begin{equation}
\sigma(r)\approx 1.5\times10^{-5},\quad
\sigma(f_{\rm NL})\approx 0.6,\quad
\sigma(k_c)\approx 0.4\times10^{-4}\ \mathrm{Mpc}^{-1},\quad
\sigma(\xi/\gamma)\approx 0.07\ \text{(CMB+21\,cm)}.
\end{equation}
Since the prediction band is $\Delta\ln k=\xi/\gamma\in[0.1,0.3]$, this precision enables $\gtrsim$\,3$\sigma$ separation across the band edges with joint data.
% ====================================================
\section{Pre-Registered Tests and Falsifiability}
\label{sec:echo}
We register three binary tests:
\noindent\textbf{CP-1 (tensor + feature):} Require $r<10^{-4}$ \emph{and} either (i) a horizon-scale cutoff with TE/EE coherence $\mathcal{C}_{k_c}>2.5$ using Eq.~\eqref{eq:coh} or (ii) a log-periodic ``echo'' detection in $\ln k$ with $f_{\ln k}\in[0.1,0.3]$ at $>3\sigma$ (Lomb--Scargle/FFT on deconvolved spectra with surrogate nulls). If $r\ge 10^{-4}$, recursion in bounce/brane channels is excluded.
\noindent\textbf{CP-2 (species tag):} $|f_{\rm NL}|\ge 1$ with sign distinguishing scenarios (small $f_{\rm NL}>0$ vs.\ small $f_{\rm NL}<0$), using SPHEREx/Euclid bispectra.
\noindent\textbf{CP-3 (entropy--$\Lambda$):} $|\Delta_S|<0.05$ and $\log B>2.5$ per Eq.~\eqref{eq:SLambda}; failure if $|\Delta_S|>0.10$ or $\log B<-2.5$.
% ====================================================
\section{Discussion and Outlook}
All ingredients are standard: canonical quantization, in--in correlators, CMB projection, and frequentist/Bayesian pipelines. The only novelty is a rigorously small, well-parameterized modification representing cross-cycle information flow. It appears observationally as a log-periodic feature with specific spacing, a tight low-$r$ expectation, and an entropy--$\Lambda$ attractor. Because these are pre-registered and binary, upcoming data can decisively confirm or exclude the scenario without ambiguity.
% ====================================================
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\end{document}
