Irreducible cosmic production of relic vortons

Author(s)

Auclair, Pierre, Peter, Patrick, Ringeval, Christophe, Steer, Danièle

Abstract

The existence of a scaling network of current-carrying cosmic strings in our Universe is expected to continuously create loops endowed with a conserved current during the cosmological expansion. These loops radiate gravitational waves and may stabilise into centrifugally supported configurations. We show that this process generates an irreducible population of vortons which has not been considered so far. In particular, we expect vortons to be massively present today even if no loops are created at the time of string formation. We determine their cosmological distribution, and estimate their relic abundance today as a function of both the string tension and the current energy scale. This allows us to rule out new domains of this parameter space. At the same time, given some conditions on the string current, vortons are shown to provide a viable and original dark matter candidate, possibly for all values of the string tension. Their mass, spin and charge spectrum being broad, vortons would have an unusual phenomenology in dark matter searches.

Figures

Diagram $(\ell, t)$ for the different types loops/vortons. The left panel is for $G\U = 10^{-16}$ and the right panel for $G\U=10^{-19}$. The dark-dashed vertical line is the time of condensation, when strings become superconducting. The diagonal dark line represents $\ell = \alpha t$ (with $\alpha = 0.1$) the size at which loops are produced. The orange horizontal line shows the value of $\lambda$.

Diagram $(\ell, t)$ for the different types loops/vortons. The left panel is for $G\U = 10^{-16}$ and the right panel for $G\U=10^{-19}$. The dark-dashed vertical line is the time of condensation, when strings become superconducting. The diagonal dark line represents $\ell = \alpha t$ (with $\alpha = 0.1$) the size at which loops are produced. The orange horizontal line shows the value of $\lambda$.


Diagram $(\ell, t)$ for the different types loops/vortons. The left panel is for $G\U = 10^{-16}$ and the right panel for $G\U=10^{-19}$. The dark-dashed vertical line is the time of condensation, when strings become superconducting. The diagonal dark line represents $\ell = \alpha t$ (with $\alpha = 0.1$) the size at which loops are produced. The orange horizontal line shows the value of $\lambda$.

Diagram $(\ell, t)$ for the different types loops/vortons. The left panel is for $G\U = 10^{-16}$ and the right panel for $G\U=10^{-19}$. The dark-dashed vertical line is the time of condensation, when strings become superconducting. The diagonal dark line represents $\ell = \alpha t$ (with $\alpha = 0.1$) the size at which loops are produced. The orange horizontal line shows the value of $\lambda$.


The left panel shows the density parameter $\Omegarelmin$ (today) from the population of \emph{irreducible relaxed} vortons, i.e.~we have assumed that there is no loop at the string forming time ($\Cini=0$). The right panel shows the density parameter $\Omegaprod$ of \emph{produced} vortons derived analytically in equation~\eqref{eq:omegaprodapprox}. The thick green line shows the value $\OmegaDM=0.3$, typical of the current dark matter density parameter. The white patches on these figures correspond to regions of the parameter space where no vortons are present: all loops there are either doomed or proto-vortons. Abundances of these two populations of vortons have not been derived before and constitute an irreducible contribution.

The left panel shows the density parameter $\Omegarelmin$ (today) from the population of \emph{irreducible relaxed} vortons, i.e.~we have assumed that there is no loop at the string forming time ($\Cini=0$). The right panel shows the density parameter $\Omegaprod$ of \emph{produced} vortons derived analytically in equation~\eqref{eq:omegaprodapprox}. The thick green line shows the value $\OmegaDM=0.3$, typical of the current dark matter density parameter. The white patches on these figures correspond to regions of the parameter space where no vortons are present: all loops there are either doomed or proto-vortons. Abundances of these two populations of vortons have not been derived before and constitute an irreducible contribution.


The left panel shows the density parameter $\Omegarelmin$ (today) from the population of \emph{irreducible relaxed} vortons, i.e.~we have assumed that there is no loop at the string forming time ($\Cini=0$). The right panel shows the density parameter $\Omegaprod$ of \emph{produced} vortons derived analytically in equation~\eqref{eq:omegaprodapprox}. The thick green line shows the value $\OmegaDM=0.3$, typical of the current dark matter density parameter. The white patches on these figures correspond to regions of the parameter space where no vortons are present: all loops there are either doomed or proto-vortons. Abundances of these two populations of vortons have not been derived before and constitute an irreducible contribution.

The left panel shows the density parameter $\Omegarelmin$ (today) from the population of \emph{irreducible relaxed} vortons, i.e.~we have assumed that there is no loop at the string forming time ($\Cini=0$). The right panel shows the density parameter $\Omegaprod$ of \emph{produced} vortons derived analytically in equation~\eqref{eq:omegaprodapprox}. The thick green line shows the value $\OmegaDM=0.3$, typical of the current dark matter density parameter. The white patches on these figures correspond to regions of the parameter space where no vortons are present: all loops there are either doomed or proto-vortons. Abundances of these two populations of vortons have not been derived before and constitute an irreducible contribution.


The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.

The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.


The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.

The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.


The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.

The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.


The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.

The upper left-hand panel shows the density parameter of relaxed vortons coming only from loops present at the string-forming phase transition, when starting from a Vachaspati-Vilenkin distribution at $t=\tini$. This is the population derived in Ref.~\cite{Brandenberger:1996zp}, that we recover by setting $C=0$ in our equations. The upper right-hand panel shows the numerically evaluated density parameter of the irreducible relaxed vortons $\Omegarel^{\min}$ (to be compared to our analytic estimation in the left panel of figure~\ref{fig:omegaapprox}). The lower left-hand panel shows the density parameter $\Omegarel$ (today) from the population of all \emph{relaxed} vortons (the sum of the upper left and right panels). Thermal history effects are visible on the upper boundary towards the minimum possible values of $1/\NNstar$ and $G\U$. The lower right-hand panel shows the density parameter $\Omegaprod$ today of \emph{produced} vortons derived numerically, and is indistinguishable from our analytic estimation of equation~\eqref{eq:omegaprodapprox} (see right-hand panel of figure~\ref{fig:omegaapprox}). The thick green line corresponds to all density parameter values in the range $[0.2,0.4]$.


The total relic abundance of all vortons starting from a Vachaspati-Vilenkin initial loop distribution, with an initial thermal correlation length $\ellcorr = 1/\sqrt{\U}$, and a one-scale loop production function with $\alpha=0.1$. The green line corresponds to the range of values $[0.2,0.4]$. The different populations contribution is represented in figure~\ref{fig:omegasvv}.

The total relic abundance of all vortons starting from a Vachaspati-Vilenkin initial loop distribution, with an initial thermal correlation length $\ellcorr = 1/\sqrt{\U}$, and a one-scale loop production function with $\alpha=0.1$. The green line corresponds to the range of values $[0.2,0.4]$. The different populations contribution is represented in figure~\ref{fig:omegasvv}.


The total relic abundance of all vortons starting from a Vachaspati-Vilenkin initial loop distribution with various correlation length $\ellcorr$ ranging from the thermal one $1/\sqrt{\U}$ to the Kibble one $\horizon{\tini}$. Each curve represents the value $\Omegatot=0.3$. Domains right of this curve lead to vortons overclosing the Universe, domains on the left are compatible with current cosmological constraints. The upper hatched region corresponds to the irreducible relaxed and produced vortons not affected by the initial conditions.

The total relic abundance of all vortons starting from a Vachaspati-Vilenkin initial loop distribution with various correlation length $\ellcorr$ ranging from the thermal one $1/\sqrt{\U}$ to the Kibble one $\horizon{\tini}$. Each curve represents the value $\Omegatot=0.3$. Domains right of this curve lead to vortons overclosing the Universe, domains on the left are compatible with current cosmological constraints. The upper hatched region corresponds to the irreducible relaxed and produced vortons not affected by the initial conditions.


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