Diamonds in the sky: the shape of gravitational waves from cosmic string cusps

Author(s)

Auclair, Pierre, Gharib, Chams, Steer, Danièle

Abstract

We carry an in-depth analysis of the gravitational wave (GW) emission of cosmic string cusps, going beyond the seminal work of Damour and Vilenkin. We identify two features that, to the best of our knowledge, have not been described in the literature: first the GWs emission from a cusp, at any given frequency, is not concentrated inside a cone but instead draws a diamond-like shape in the sky. Second, the emission is not maximal in the direction of the cusp velocity, but in two slightly offset directions. We explain these effects in detail, and illustrate them with simplified examples. Then, we study two limiting cases for which the overall radiated power is enhanced: when the diamond is stretched in a particular direction, and when higher order derivatives vanish at the cusp. We statistically estimate the importance of these effects in a population of odd-harmonic loops.

Figures

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Projection onto the plane of $\abs{h}^\mathrm{TT}(\omega,\hat{n}) \equiv \sqrt{{h_{ij}^\mathrm{TT}}^*h_{ij}^\mathrm{TT}}$ for a given GW frequency $\omega$ and different observation directions $\hat{n}$. In this figure the cusp feature sourcing the GWs points towards the center. The colorscale goes linearly from $0$ (light) to the maxima (dark red) which occur when the observer is slightly misaligned with the cusp. The circle shows the standard approximate shape often used in the literature. For more details on the exact configuration see \cref{fig:heatmap_hTT_beta36}.

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Mollenweide projection of the numerical calculation of $\abs{h}^\mathrm{TT}(\omega,\hat{n}) \equiv \sqrt{{h_{ij}^\mathrm{TT}}^*h_{ij}^\mathrm{TT}}$ as a function of the observer's direction $\hat{n}$ . The color scale goes linearly from $0$ (light) to maximum (dark red). We have superimposed the two curves $\mathcal{C}_{\vb{a'}}$ (red) and $\mathcal{C}_{\vb{b'}}$ (green), which correspond to a Kibble-Turok loop~\cite{Kibble:1982cb}.

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Misalignment between the observer's direction $\hat{n}$ and $\mathcal{C}_{\vb{a'}}$. Here we have shown $\mathcal{C}_{\vb{a'}}$ as a great circle, but this is purely for illustrative purposes.

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Representation of relevant geometrical quantities in the plane tangent to the Kibble-Turok sphere at the cusp. The distances $\delta_{a, b}$ can be evaluated by approximating $\mathcal{C}_{\vb{a}', \vb{b}'}$ by their tangent at the cusp

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Representation of relevant geometrical quantities in three dimensions at the cusp.

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: $p = 300$ : $p = 3000$

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: Mollenweide projection of the numerical calculation of the trace-reversed metric $\sqrt{{\bar{h}_{ij}}^*\bar{h}_{ij}}$ of \cref{eq:hII} as a function of $\hat{n}$, for different values of $p$. The beam aperture formula of \cref{eq:beamWidth} is shown in blue. The standard formula \cref{eq:usualBeamWidth} of Ref.~\cite{Damour_2001}, with a proportionality constant of $2$ is shown in gray. : Caption not extracted

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: $\beta = \pi/2$, $p = 100$ : $\beta = -\pi/2$, $p = 100$

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: Numerical calculation of $\abs{h}^\mathrm{TT}(\omega,\hat{n})$ with $\vb{a'}(u)$ (red) and $\vb{b}'(v)$ (green) chosen as orthogonal great circles. The arrows indicate the increasing directions of $u$ and $v$. : Caption not extracted

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Geometrical parametrization used in \cref{s:lumps} for describing the direction of the observer $\hat{n}$ on the Kibble-Turok sphere, with $\mathcal{C}_{\vb{a}'}$ being a great circle at the equator.

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Profile of $\abs{h}^\mathrm{TT}$ (normalized) near the cusp along the axis defined by $\psi = \lambda$, for different values of $p$ ($10$, $50$, $100$, $200$, $1000$, $10~000$). Higher values of $p$ are red, while lower are blue. Here $\beta = \pi/2$.

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: width=.33\textwidth

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: width=.33\textwidth : Examples of odd harmonic loops for various values of the maximal harmonic order $N$. The snapshots are taken at a cusp time.

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: Caption not extracted

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: width=.45\textwidth

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: Probability density of the angle of intersection $\beta$, obtained from $10~000$ realizations of odd harmonic loops of order $N=5$, with $10$ cusps per loop on average. : Caption not extracted

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Complementary cumulative distribution function (blue) of $p_\text{th}$ obtained from $1~000$ odd harmonic loops of order $N=11$ with $22$ cusps per loop on average. The large $p$ region is fit with a power law (orange).