Slowly rotating neutron stars in scalar-tensor theories

Paolo Pani and Emanuele Berti

Phys. Rev. D 90, 024025 – Published 10 July 2014

Abstract

We construct models of slowly rotating, perfect-fluid neutron stars by extending the classical Hartle—Thorne formalism to generic scalar-tensor theories of gravity. Working at second order in the dimensionless angular momentum, we compute the mass $M$ , radius $R$ , scalar charge $q$ , moment of inertia $I$ , and spin-induced quadrupole moment $Q$ , as well as the tidal and rotational Love numbers. Our formalism applies to generic scalar-tensor theories, but we focus in particular on theories that allow for spontaneous scalarization. It was recently discovered that the moment of inertia, quadrupole moment, and Love numbers are connected by approximately universal (i.e., equation-of-state independent) “I-Love-Q” relations. We find that similar relations hold also for spontaneously scalarized stars. More interestingly, the I-Love-Q relations in scalar-tensor theories coincide with the general relativistic ones within less than a few percent, even for spontaneously scalarized stars with the largest couplings allowed by current binary-pulsar constraints. This implies that astrophysical measurements of these parameters cannot be used to discriminate between general relativity and scalar-tensor theories, even if spontaneous scalarization occurs in nature. Because of the well-known equivalence between $f (R)$ theories and scalar-tensor theories, the theoretical framework developed in this paper can be used to construct rotating compact stellar models in $f (R)$ gravity. Our slow-rotation expansion can also be used as a benchmark for numerical calculations of rapidly spinning neutron stars in generic scalar-tensor theories.

Received 23 May 2014

DOI:https://doi.org/10.1103/PhysRevD.90.024025

Authors & Affiliations

Paolo Pani^1,* and Emanuele Berti^2,†

¹CENTRA, Departamento de Física, Instituto Superior Técnico, Universidade Técnica de Lisboa - UTL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal
²Department of Physics and Astronomy, The University of Mississippi, University, Mississippi 38677, USA

^*paolo.pani@tecnico.ulisboa.pt
^†eberti@olemiss.edu

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Vol. 90, Iss. 2 — 15 July 2014

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Images

Figure 1
NS configurations in GR (solid lines) and in two scalar-tensor theories defined by Eq. (3) with $A (Φ) \equiv e^{\frac{β}{2} Φ^{2}}$ and $V (Φ) \equiv 0$ . Dashed lines refer to $β / (4 π) = - 4.5$ , $Φ_{0}^{\infty} = 1 0^{- 3}$ ; dashed-dotted lines refer to $β / (4 π) = - 6$ , $Φ_{0}^{\infty} = 1 0^{- 3}$ . Each panel shows results for three different EOS models (FPS, APR, and MS1). Top-left panel, left inset: relation between the nonrotating mass $M$ and the radius $R$ . In all plots $M$ refers to the Arnowitt—Deser—Misner mass in the Einstein frame; see Appendix pp2 for a discussion. Top-left panel, right inset: relative mass correction $δ M / M$ induced by rotation as a function of the mass $M$ of a nonspinning star with the same central energy density [cf. Eq. (33)]. Top-right panel, left inset: scalar charge $\tilde{q} / M$ as a function of $M$ . Top-right panel, right inset: relative correction to the scalar charge $δ \tilde{q} / \tilde{q}$ induced by rotation as a function of $M$ . Bottom-left panel: Jordan-frame moment of inertia $\tilde{I}$ (left inset) and Jordan-frame quadrupole moment $\tilde{Q}$ (right inset) as functions of $M$ . Bottom-right panel: Jordan-frame tidal ( $\tilde{λ}$ ) and rotational ( ${\tilde{λ}}^{rot}$ ) Love numbers as functions of $M$ .
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Figure 2
EOS-independent relations $\bar{I} (\bar{λ})$ (left) and $\bar{Q} (\bar{λ})$ (right). Solid line styles refer to data in GR and dashed line styles to data for scalarized stars. In each panel, the top inset shows the relation itself; the middle and bottom insets show deviations from universality, as measured by the residual $Δ X = 100 [X / X_{fit} - 1]$ . $Δ_{G R} X$ means that the universal relation is obtained by fitting only pure GR solutions; $Δ_{S T} X$ means that the fit is obtained only from scalarized solutions. The top panels show that both residuals are always smaller than 2%, and typically smaller than 1%, for scalar-tensor theories that are marginally ruled out by binary pulsar observations. The bottom panels show that the residuals in $\bar{Q} (\bar{λ})$ can get as large as $\sim 10 %$ for theories that are already ruled out by experiment at more than $1 σ$ confidence level.
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Figure 3
Dimensionless rotational Love number ${\bar{λ}}^{rot} \equiv {\bar{I}}^{2} \bar{Q}$ as a function of the compactness $M / R$ in a scalar-tensor theory defined by $A (Φ) \equiv e^{\frac{β}{2} Φ^{2}}$ and $V (Φ) \equiv 0$ , for $β / (4 π) = - 4.5$ , $Φ_{0}^{\infty} = 1 0^{- 3}$ and for different tabulated EOS models. The residuals shown in the insets are defined as in Fig. 1, and they are always smaller than a few percent. All quantities refer to the Jordan frame, $M$ is in solar-mass units whereas $R$ is in units of kilometers.
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