SPINDOWN OF ISOLATED NEUTRON STARS: GRAVITATIONAL WAVES OR MAGNETIC BRAKING?

Neutron stars are highly compact stars of typical radius R ∼ 12 km and mass M ∼ 1.5 M_☉ made mostly of degenerate neutron-rich matter at densities up to several times nuclear matter saturation density ρ₀ = 2.5 × 10¹⁴ g cm⁻³. By tracking their long-term thermal and rotational evolution, we can learn about the nature of matter under the crust. For example, Page et al. (2011) have proposed that the thermal history of the neutron star in Cas A may be indicating the recent onset of neutron superfluidity deep in its interior. Recently, Negreiros et al. (2011) have shown how rotational evolution is linked to a reorganization of particle composition in the stellar interior, leading to the switching on of exotic neutrino emission processes. A neutron star is a complex system with intertwined physical properties that can change on relatively short astrophysical timescales.

In this paper, we are concerned with the rotational evolution of a highly magnetized isolated neutron star. The electromagnetic emissions of a neutron star derive from its rotational kinetic energy, and its spindown is usually measured in terms of a braking index, n, which is dependent on the magnetic field configuration (n = 3 for a dipolar field). Neutron stars can also spin down through gravitational wave emissions associated to the r-mode (Andersson 1998). In fact, the observational interest in r-modes comes from the fact that no neutron stars have been found to spin at rates near the maximum allowed frequency (the "break-up frequency"). r-modes offer a possible explanation of this fact: in rotating neutron stars, these modes lose energy through gravitational waves, which carry away angular momentum from the star and act as braking radiation. As the star spins down, its central density increases. This increase could be sufficient to make baryonic matter undergo phase transitions to more exotic phases of strongly interacting matter, such as quark matter, with possible implications for gamma-ray bursts (Berezhiani et al. 2003; Haensel & Zdunik 2003; Drago et al. 2008) and the formation of quark stars, if theoretical conjectures about the absolute stability of strange quark matter are realized in nature (Itoh 1970; Bodmer 1971; Witten 1984). While it is almost certain that this quark phase, if it exists inside neutron stars, cannot be a free gas of quarks (Özel et al. 2010; Weissenborn et al. 2011), an interacting phase of quarks still appears to be consistent with the recent finding of a 2 M_☉ neutron star (Demorest et al. 2010).

The main questions we seek to answer are: given a few initial parameters of the newly born neutron star, such as its spin period, magnetic field, and baryonic mass, can we determine which neutron stars are likely to eventually manifest a quark phase in their interior? If so, how long before the transition to quark matter occurs? In this work, we take a step toward answering these questions by taking a closer look at the rotational evolution of a newly born hot neutron star as it spins down, and we focus on two main driving factors—magnetic braking and gravitational radiation from r-modes. We consider different equations of state (EoSs) for neutron stars, since the EoS at high density is uncertain and lacking strong empirical constraints (although a recent statistical analysis shows that the EoS stiffens at high densities and is consistent with the expected range in certain nuclear parameters; Steiner et al. 2010). Furthermore, second-generation gravitational wave detectors such as Advanced LIGO will soon be operational, and our work provides an update for a similar theoretical study (Ho & Lai 2000) performed several years ago with regard to the LIGO detector. We should note here that in contrast to the assumptions in Ho & Lai (2000), our work includes the effects of magnetic damping of r-modes in the spindown evolution, which leads to quantitatively different results.

In Section 2, we outline the approach and main equations that describe the neutron star's spindown. The theoretical analysis follows in part the works of Ho & Lai (2000), as well as Cuofano & Drago (2010) and Rezzolla et al. (2001b). Section 3 collects our main conclusions from this analysis. In Section 4, we analyze the evolution of the gravitational wave frequency associated with the growing r-mode, along the lines of Owen et al. (1998), and we present results in Section 5. We conclude in Section 6 with a preliminary calculation of the gravitational wave signal from an explosive deconfinement transition (the "Quark-Nova").

In a previous paper (Staff et al. 2006), we had determined birth parameters of a neutron star which would support the transition to deconfinement driven solely by magnetic braking. We found that neutron stars that are born with mass M ≳ 1.5 M_☉ and spin period P ≲ 3 ms are the best candidates to reach deconfinement density in their core (assumed to be ∼5ρ₀), given a range of neutron star magnetic fields of 10¹²–10¹⁵ G. This small value of the initial spin period (or large spin frequency) is required since magnetic braking from a rotating magnetic dipole is itself proportional to the third power of the spin frequency. Most of the increase in central density occurs soon after the rapidly rotating neutron star is born.

In that work, we neglected the role of gravitational radiation from r-modes in an ad hoc way—by assuming that neutrons stars are born axisymmetric and remain so. In reality, non-axisymmetric perturbations are expected due to the violent process of a supernova, which leaves the newly born neutron star in a turbulent state (Keil et al. 1996; Akiyama & Wheeler 2006). What then is the role of r-modes and gravitational braking in comparison to magnetic braking as far as the time to deconfinement is concerned? Previous studies on the interplay of r-modes and neutron star magnetic fields have focused on different questions—for example, Lee (2010) has studied the r-mode in magnetized neutron stars with B ⩽ 10¹² G with the aim of understanding X-ray pulsations from local hot spots in accretion-powered pulsars. Ho & Lai (2000) have shown that magnetic fields of B ⩾ 10¹⁴ G can make magnetic braking as important as r-mode spindown, specially for slowly rotating stars, and that Alfvén wave driving of the r-mode can also play a role.

Meanwhile, other works (Rezzolla et al. 2001a, 2001b; Kiuchi et al. 2011; Cuofano & Drago 2010) have focused on the evolution of toroidal magnetic fields generated by the secular r-mode. Essentially, the toroidal field is generated by differential rotation of the stellar fluid associated with the r-mode. The differential rotation is a key feature of the r-mode instability in the nonlinear regime (Levin & Ushomirsky 2001). Rezzolla et al. (2001a) have shown that, for isolated neutron stars, this effect can amplify existing magnetic fields by two orders of magnitude within the time taken by the r-mode to saturate (≈a few hundred seconds). Back-reaction on the r-mode due to this toroidal field implies that there is an associated magnetic damping, which has to be factored into the spin evolution. For neutron stars that are accreting from a binary companion, the build-up of toroidal fields to sizable values is rather slow, and according to Cuofano & Drago (2010), can take up to several hundred years. However, we are interested in this effect on isolated neutron stars. The magnitude of the magnetic damping term F_m is proportional to the integrated time evolution of α²(t) (Rezzolla et al. 2001b), with α being the r-mode amplitude. These authors obtained a strong damping effect by assuming that the background evolution of α(t) is identical to that given by the absence of any magnetic fields. However, we solve the evolution equations for the star's angular frequency Ω and α(t) with magnetic fields and magnetic damping from the start. F_m grows while the mode is unstable, and once it saturates F_m remains constant. This consistent inclusion of F_m leads to a slower evolutionary path for α(t) toward saturation, and hence the magnetic damping is not as effective in suppressing the r-mode. Consequently, for a range of magnetic fields and small r-mode saturation values, we find that this timescale can be much larger than the typical time taken by the neutron star to spin down to typical quark deconfinement density. Therefore, our results clearly indicate that the r-mode, for realistic saturation values, quantitatively affects the spindown evolution of an isolated neutron star. We now discuss the relevant equations for the evolution of the star's rotation rate and r-mode evolution.

The spindown $\dot{\Omega }$ of a neutron star (mass M, radius R) is accompanied by a loss of energy as well as angular momentum. For the radiating star, conservation of total angular momentum J_tot = J_star + (1 − K_j)J_c with K_j a dimensionless constant and J_star = IΩ (where I is the moment of inertia about the rotation axis) yields (Wagoner 2002; Cuofano & Drago 2010)

$$\begin{equation} \frac{dJ_{\rm tot}}{dt}=2J_cF_g+\dot{J_a}-I\Omega F_{\rm mag} . \end{equation} \tag{ 1 }$$

J_c = −K_cα²J_star is the canonical angular momentum of the r-mode to first order in Ω (Friedman & Schutz 1978). F_g is the rate of gravitational radiation associated to the l = m = 2 current multipole taken for an n = 1 polytropic star from Equation (65) of Andersson et al. (2001). This is expected to be a good approximation to a large part of the neutron star interior. $\dot{J_a}$ is the accretion rate onto the star (assumed zero for our case since we consider isolated stars only) and F_mag is the magnetic braking rate (Manchester & Taylor 1977). The dimensionless quantity $K_c=3\bar{J}/2\bar{I}$ is defined from

$$\begin{eqnarray} &&\bar{J}=\frac{1}{MR^4}\int _0^R \rho r^6 dr,\nonumber \\ &&\bar{I}=\frac{8\pi }{3MR^2}\int _0^R\rho r^4 dr, \end{eqnarray} \tag{ 2 }$$

where ρ = ρ(r) is the density profile (assumed radially symmetric) of the star.⁴ J_c evolves in time as a result of competing influences from gravitational damping (which feeds the r-mode), viscosity (which damps the r-mode), and the magnetic damping term F_m. Both bulk and shear viscosities are included in our analysis although for simplicity we keep the star at a uniform temperature of 10⁹ K for the duration of the evolution. Although viscosities (especially bulk viscosity) is strongly temperature-dependent, our approximation is not as drastic as it may seem. A typical cooling profile of a neutron star, driven by neutrino emission from the modified URCA process is given by (Owen et al. 1998)

$$\begin{equation} T_9(t)=\left(\frac{t}{\tau _c}+T_{i,9}^{-6}\right)^{-1/6}, \end{equation} \tag{ 3 }$$

where T₉ is the star's core temperature T in units of 10⁹ K, T_{i, 9} is the initial temperature, and τ_c ≈ 1 yr is the characteristic cooling time from the modified URCA process. If we begin with birth temperatures T ∼ 10¹¹ K, we see that within a few seconds, we have T ∼ 10⁹ K. Since the r-mode does not have a large impact until t ⩾ 10² s, we can approximate T₉ = T/(10⁹ K) ∼ 1. Subsequent cooling is on the timescale of years and is also a small perturbation on our results. Using the evolution equation for J_c, (see Equation (4) of Cuofano & Drago 2010), one can write

$$\begin{eqnarray} \frac{\dot{\Omega }}{\Omega }&=&-2\alpha ^2K_c[K_jF_g+(1-K_j)[F_v+F_m]]-F_{\rm mag} \quad {\rm and}\nonumber\\ \frac{\dot{\alpha }}{\alpha }&=&[F_g-[F_v+F_m]]-\frac{\dot{\Omega }}{2\Omega }, \end{eqnarray} \tag{ 4 }$$

where F_v is the viscous damping rate (Andersson et al. 2001). We solve the two coupled equations above numerically. We use the RNS code (Stergioulas & Friedman 1995) to construct two-dimensional models of rapidly rotating neutron stars. For a given EoS and for a fixed baryonic mass, the RNS code outputs a sequence of neutron star models (with specific gravitational mass, radius, spin frequency, etc.) that have increasing central density and decreasing angular velocity. The fastest model in such a sequence spins at or near Kepler frequency. Note that the magnetic field does not appear in the RNS code, and in any case, its effect on structure at the field values considered here is negligible. The magnetic field only determines the time to deconfinement, unless the spindown is completely r-mode dominated. As in Staff et al. (2006) we assume 5ρ₀ to be a critical density at which quarks in the interior of the neutron star become deconfined—this is not a well-determined number, and may span a range from (4–8)ρ₀ if a mixed phase of quark and nuclear matter is favored (Glendenning 1992). To counter this uncertainty, we have checked our numerical results for a higher putative deconfinement density (ρ ∼ 8ρ₀) and found that it does not change any of our quantitative conclusions by a significant amount.

Using the RNS code, we construct sequences of stars (for a given EoS) with constant baryonic mass and decreasing spin such that the non-rotating configuration has a central density equal to 5ρ₀. We then calculate the time the rotating star takes until its central density is within 99% of 5ρ₀. This is a practical way of using Equation (4) to obtain the minimum mass required (for a given EoS) to support deconfinement, since the time taken to reach zero spin is infinite from the magnetic braking being proportional to Ω. Since we are interested in the time it takes for a star to spin down to the critical density for deconfinement, by choosing a sequence where the non-rotating model has a central density equal to the critical density, this time gives us the maximum time the star takes to reach the deconfinement density.

We assume a very small initial amplitude of the r-mode for the fastest spinning star in a sequence, with typical initial α ∼ 10⁻⁶, although our results for those cases where the r-mode saturates are insensitive to this initial value. For the case when the mode does not saturate, we do find mild sensitivity to the initial value of α. From the first of Equation (4), we can calculate the time step between two consecutive neutron star models in the spin-sequence output by the RNS code,

$$\begin{eqnarray} &&\Delta t \equiv t_{i+1}-t_{i}\nonumber\\ &&=\frac{\Omega (t_{i+1})-\Omega (t_i)}{-2K_c\Omega (t_i)\alpha (t_i)^2 [K_j F_g + (1-K_j)(F_v+F_m)] - \Omega (t_i) F_{\rm mag}}.\nonumber\\ \end{eqnarray} \tag{ 5 }$$

The inverse dependence on α implies that for small magnetic fields, where F_mag is small, Δt is large when the mode amplitude is just starting to grow. The RNS code then needs to generate two widely separated rotating configurations within a sequence, leading to low resolution in some parts of the P versus t curves for small magnetic fields (e.g., the curve for B = 10¹² G in Figure 1). However, this is only a problem initially and does not affect the total time to deconfinement. The above equation, along with the second of Equation (4) is then used to determine the time evolution of α. For our analysis we have used four different magnetic field strengths equally spaced on a logarithmic scale from 10¹² G to 10¹⁵ G. We display results for three different equations of state: EoS A, which is composed of only neutrons (Pandharipande 1971) and uses a variational principle to determine the minimum energy state; EoS BBB2, which includes muons and uses a field-theoretic many-body approach (Baldo et al. 1997); and finally EoS APR, which is relatively stiff and admits a mixed phase of quarks and nuclear matter for the heaviest stars (Akmal et al. 1998). All three EoSs generate stable configurations for central densities exceeding 5ρ₀ and can go up to even 8ρ₀. They differ in the details of the density profile, with the softest (EoS A) having higher interior densities for the same mass (more compact). The maximum gravitational mass exceeds 2 M_☉ only for the APR EoS (Akmal et al. 1998), with BBB2 providing a maximum mass of 1.92 M_☉ (Haensel 2003). EoS A is too soft to generate a 2 M_☉ neutron star and may seem inadequate to explain recent observations of such massive compact stars (Demorest et al. 2010). However, if such massive stars are really quark or hybrid stars, we cannot rule out EoS A in this way, and its inclusion is still useful simply to examine the trend of a relatively soft EoS on the gravitational signal from r-mode-driven spindown.

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The following results are obtained upon solving Equation (4) numerically. In Figure 1, the curves show how the period, starting from the Kepler rotation rate at birth, evolves as a function of magnetic field strength, for different EoSs. Note that these curves correspond to structural parameters of that particular stellar configuration which just reaches the putative quark deconfinement density at zero angular velocity—we may call this the critical configuration. For any given EoS, this is how we determine the minimum mass and spin period required for a quark phase to appear as a result of spindown. Unlike Figure 4 of Staff et al. (2006), spindown is no longer always magnetic dominated for arbitrary magnetic fields. For example, for B ≲ 10¹² G, r-mode spindown takes over at a few hundred seconds. As the instability develops, magnetic damping F_m grows and limits the growth of the r-mode. This leads to a plateau in the period that can last up to several years before magnetic damping once again becomes the dominant driver of spindown. It is noteworthy that for the critical configuration, there is no change in the time to deconfinement for any value of B. However, if we choose a heavier mass at birth, the time to deconfinement would be shorter, and can occur while the star is still being spun down as a result of the r-mode. Assuming that even the most massive neutron stars can be transformed to stable quark stars if they reach the deconfinement density (that is, the quark matter EoS should be sufficiently stiff to support such a mass), we may expect that several such neutron stars would have already spun down to the point where they underwent a quark–hadron phase transition, and now contain quark matter in their core.

Our findings here are different from the conclusions of Ho & Lai (2000) who studied the r-mode of magnetized and slowly rotating neutron stars, and found that the r-mode alters the spindown if magnetic fields are less than B ∼ 10¹⁴ G. Due to the inclusion of the magnetic damping term, which was omitted in their work, we find that r-mode-driven effects on spindown are pronounced only for B ≲ 10¹² G, since for higher B fields, magnetic damping effectively kills the r-mode. We emphasize that the effects of magnetic damping for smaller B fields are not as severe as might be expected from the work of Rezzolla et al. (2001a) due to the self-consistent evolution of α in the presence of the magnetic damping term, as explained at the beginning of Section 2.

We also find that the r-mode-driven spindown is more pronounced in a relatively stiff EoS such as EoS APR. This is because F_g, the gravitational damping rate, is larger for a stiffer EoS,⁵ driving the r-mode unstable in a shorter amount of time (effectively spinning the star down to deconfinement density quickly).

The saturation amplitude α_sat of the r-mode assumed in the foregoing analysis is ${\cal O}(1)$. Such a large value is probably unrealistic due to multi-mode coupling and onset of nonlinear effects (Arras et al. 2003; Bondarescu et al. 2009). If we use a smaller saturation value, we find that our conclusions change quantitatively. The r-mode grows more slowly for a smaller α_sat, but continues to be important since the main damping agent F_m, which is roughly proportional to α⁴, is also smaller. As a consequence, the spindown due to the r-mode is weaker, but also lasts longer. This can lead to the star taking a shorter time to reach a given period in the case of smaller B fields. This can be seen clearly in Figure 6, which compares the time taken by the star to spin down to a period P = 3 ms, for α_sat = 0.01 and α_sat = 0.005.

As shown in the previous section, fairly large magnetic fields of 10¹³ G or more are required to make the r-mode irrelevant for spindown. For smaller magnetic fields, the r-mode evolution can be the main agent for spinning down rapidly rotating stars. Following Owen et al. (1998), we can obtain expressions for the growth of the mode amplitude (α) and the corresponding evolution of the frequency of the gravitational wave (f) under certain approximations.

4.1. Saturated Phase

The r-mode amplitude grows rapidly until nonlinear saturation occurs (Arras et al. 2003; Bondarescu et al. 2007) at which point $\dot{\alpha }$ = 0. We can obtain a description of this saturated phase from Equation (4), which implies that the rotation rate evolves according to

$$\begin{equation} \frac{\dot{\Omega }}{\Omega }=\frac{2\alpha ^2K_cF_g-F_{\rm mag}}{1-\alpha ^2K_c(1-K_j)} \,, \end{equation} \tag{ 6 }$$

where α = α_sat is the saturation amplitude of the r-mode. Note that F_m does not appear since it can be eliminated using the second line of Equation (4). Its effect will, however, show up in the growth phase discussed below. Using the fact that Ω = 3πf/2 for the l = m = 2 r-mode, setting K_c = 0.1 (a typical value determined from Equation (2)), and using the frequency-dependent expression for the gravitational wave damping timescale τ_GR ≡ F⁻¹_g, we find

$$\begin{equation} \frac{\dot{f}}{{\rm Hz}^2}\approx -0.35\alpha ^2\left(\frac{f}{1\, {\rm kHz}}\right)^7-0.0055\frac{B_{14}^2R_6^6}{I_{45}}\left(\frac{f}{1\, {\rm kHz}}\right)^{3}, \end{equation} \tag{ 7 }$$

where we have used the expression for τ_GR from Lindblom et al. (1998) and neglected α²K_c(1 − K_j) ≪ 1 in the denominator of Equation (6). B, R, and I are expressed in terms of reduced dimensionless units.

4.2. Growth Phase

In the initial stages of the mode growth phase, the viscosity controls the evolution of the r-mode, but the instability to gravitational waves soon takes over. Viscous damping from bulk viscosity can be large at T ⩾ 10¹¹ K, but as the neutron star cools rapidly on the order of seconds, our approximation of setting T₉ = 1 implies that viscosity does not affect the subsequent spindown behavior or the gravitational waveform during the growth of the r-mode instability. It follows from Equation (4) that during this phase, since α is small,

$$\begin{equation} \frac{\dot{\alpha }}{\alpha }=(F_g-F_m)-\frac{\dot{\Omega }}{2\Omega }. \end{equation} \tag{ 8 }$$

The angular velocity evolves according to the first line of Equation (4). We assume K_j ≈ 0 for the growth phase, which is tantamount to including fully the canonical angular momentum of the r-mode in the star's physical angular momentum. This is only justified if differential rotation is small (Sa & Tome 2005), as is the case when the mode is still small but growing. Then, as before, the proportionality between f and Ω implies

$$\begin{eqnarray} \frac{\dot{f}}{{\rm Hz}^2}&\approx& - 0.0131\alpha ^2(t)\left(\frac{f}{1\, {\rm kHz}}\right)^3-0.0055\frac{B_{14}^2R_6^6}{I_{45}}\left(\frac{f}{1\, {\rm kHz}}\right)^{3}\nonumber\\ &&-\frac{0.202B_{14}^2R_6}{\bar{J}M_{1.4}}\alpha ^2(t)\int _0^t\alpha ^2(t^{\prime })\left(\frac{f(t^{\prime })}{{\rm Hz}}\right)dt^{\prime }, \end{eqnarray} \tag{ 9 }$$

where we have used the expression for F_m from Equation (11) of Cuofano & Drago (2010). We can now use Equations (7) and (9) to obtain the gravitational strain amplitudes and waveforms for the saturated phase and growth phase, respectively.

The strain amplitude h(t) corresponding to the l = m = 2 mode is found by a standard multipole analysis (Thorne 1980). Including the angle-average over the position of sources in the sky,

$$\begin{eqnarray} h(t)&=&\sqrt{\frac{3}{80\pi }}\frac{\omega ^2S_{22}}{D}, \end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray} S_{22}&=&\sqrt{2}\frac{32\pi }{15}\frac{{\it GM}}{c^5}\alpha \Omega R^3\bar{J}, \end{eqnarray} \tag{ 11 }$$

where S₂₂ is the current multipole as given by Equation (3.9) of Owen et al. (1998), the mode frequency ω = 4Ω/3, and D is the source distance, chosen henceforth to have a typical value of 20 Mpc.⁶ Figure 2 shows the time evolution of the strain amplitude h(t) for various magnetic fields and for different EoS for α_sat = 0.5. For magnetic fields B ⩾ 10¹⁴ G, the r-mode withers much before it nears saturation, for any EoS—the strain amplitude is consequently very small. For B ∼ 10¹³–10¹⁴ G, we notice a strong dependence on the EoS. The soft EoS A does not lead to a saturating r-mode while relatively stiffer EoS do. For weaker magnetic fields around B ∼ 10¹² G, the r-mode saturates and displays the behavior shown in Figure 5 of Owen et al. (1998).

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We now examine the gravitational waveform in the frequency domain $\tilde{h}(f)$ which is the Fourier transform of the time signal. This is useful in estimating the signal-to-noise ratio (S/N) in matched filtering techniques. In the stationary phase approximation,

$$\begin{equation} |\tilde{h}(f)|=\sqrt{\frac{|h(t)|^2}{|\dot{f}|}}. \end{equation} \tag{ 12 }$$

From Equations (7), (9), and (10), we can find $\tilde{h}(f)$ for both the growth phase and the saturated phase. From Figures 3 and 8 (for α_sat = 0.5 and 0.01, respectively), we see that for magnetic fields B ⩽ 10¹³ G, the signal from the r-mode is similar to previous result obtained in the absence of magnetic fields (Owen et al. 1998). A sharp peak at high frequency (≈1.6 kHz) corresponding to the growth phase is seen, followed by a plateau signal at lower frequencies as the r-mode saturates. Note that the curves terminate on the left since at that point, deconfinement density is reached and the subsequent signal from the phase transition needs a detailed analysis beyond the scope of this work. However, we refer the reader to the Appendix for a preliminary treatment of this issue. Previous works (Schenk et al. 2002; Morsink 2002; Brink et al. 2004; Bondarescu et al. 2007) indicate that the saturation amplitude α_sat may be quite small, around 10⁻² instead of order 1. Still, it is useful to contrast both cases: when α_sat ≈ 0.5 and α_sat ≈ 0.01.

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5.1. Signal Detectability

In Figure 4, we present our estimate of the continuous gravitational wave signal from the spindown (assuming a point source at a distance of 20 Mpc), compared to the anticipated sensitivity of Advanced LIGO⁷ and the Einstein Telescope (Hild et al. 2010). The left panel of Figure 4 shows the weighted strain amplitude (1/$\sqrt{\rm Hz}$) versus frequency for the three EoSs studied in this paper at 10¹³ G, compared to the anticipated noise-weighted design sensitivities of Advanced LIGO and the Einstein Telescope. The right panel shows the same for 10¹⁴ G. In both figures, we are assuming α_sat ≈ 0.01. The saturated phase of the curve is at least an order of magnitude above the anticipated sensitivity for Advanced LIGO for all EoSs at 10¹³ G and could therefore be expected to be observed out to 20 Mpc with Advanced LIGO, with a typical S/N of ∼20. At 10¹⁴ G and for the softest EoS, the S/N in Advanced LIGO drops to ∼2. The Einstein Telescope is expected to gain about an order of magnitude in sensitivity compared to Advanced LIGO, and hence all of the above cases should be detectable in principle. Our S/N estimates only provide upper limits since matched filtering is probably impractical for such sources, where spin parameters cannot be promptly measured, though the source location could be known if the supernova is observed. However, we note that most of the signal comes from the saturated phase, which for the case of small α_sat occurs several years after a neutron star's birth in a supernova. Observations of the neutron star's spindown parameters may then be possible, making the matched filtering method more feasible. In the context of r-modes in newly born neutron stars, methods other than matched filtering have been suggested (Brady & Creighton 1998; Owen et al. 2010; Zhu et al. 2011) that would be less computationally intensive.

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5.2. Saturation at α_sat ≈ 0.5

5.3. Saturation at α_sat ≈ 0.01

For a more realistic value of α_sat, we find that the strain amplitude is about an order of magnitude smaller than for α_sat = 0.5, but the signal persists for a much longer time (several months/years for B ⩽ 10¹⁴ G). This is clear from Figure 7. This is because of the dependence of F_m on B² and α(t). For smaller B fields and α_sat, the weaker magnetic damping allows the r-mode to survive for a longer time, even though mode growth is slower. This eventually leads to a faster spindown for the star, compared to a higher α_sat. To emphasize this effect, in Figure 6, we compare spindown with α_sat = 0.01 and 0.005, for the same initial stellar configuration and the same magnetic field. The crossing of the two curves in that figure illustrates this effect. However, we note from comparing Figures 3 and 8 that $\tilde{h}$ remains relatively independent of α_sat, although both the B = 10¹³ G and B = 10¹⁴ G cases shows a flat segment corresponding to the saturated phase for α_sat = 0.01. Essentially, this is because the decrease in the peak value of h(t) for smaller α_sat is compensated by the corresponding decrease in $\dot{f}$ (e.g., as in Equation (7)). This implies that we require a larger interval of time integration to validate the stationary phase approximation for a smaller α_sat (Equation (12)). Thus, $\tilde{h}$ remains almost unchanged.

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We have studied the role of r-modes in spinning a rapidly rotating and magnetized neutron star down to typical quark deconfinement densities. Apart from the usual spindown associated to magnetic braking, the magnetic damping of r-modes plays an important role in the period evolution of the star. We find that, for realistic (small) values of the r-mode saturation amplitude α_sat, the time to deconfinement is sped up by the r-mode instability when magnetic fields are of the order of 10¹² G or less. This result reflects the strong dependence of the magnetic damping effect on B and the evolution of α. Thus, in contrast to the result in (Ho & Lai 2000), where the r-mode was seen to affect the spindown already for magnetic fields less than B ∼ 10¹⁴ G, the inclusion of the magnetic damping term leads us to conclude that r-mode-driven effects on spindown are pronounced only for B ≲ 10¹² G. We also explored the gravitational wave signal generated during the spindown phase as the r-mode first grows, then saturates—we follow the signal continuously until the quark deconfinement threshold is reached. For realistic values of α_sat, the r-mode saturates and leads to a strong signal in gravitational waves, except in case of very soft EoSs and very large magnetic field (10¹⁵ G or more). There could be a sizable fraction of the total rotational kinetic energy emitted as gravitational waves in this epoch, which can last several years, making it detectable in upcoming second and third generation detectors. Therefore, gravitational waves could be used in this manner to probe the EoS inside neutron stars. As shown here, using r-modes is an alternate way in which the EoS can be probed through gravitational waves.⁸

Gravitational waves are going to open a new window of observation into our universe. Among the many discoveries that will be made, we anticipate the exciting prospect that neutron star spindown will reveal signs of the elusive r-mode instability as well as signatures for the onset of quark matter in the core. In the Appendix, we outline a first preliminary estimate of what the gravitational wave signal from a "Quark-Nova" would look like. It would be interesting to examine this signal with detailed simulations, especially in cases when the Quark-Nova occurs very shortly after the neutron star is born in a supernova—the signature of this "dual-explosion" in gravitational wave detectors would be two very different signals coming from the same source but separated in time by a few days to few weeks, depending on the time delay between the two explosions. Such a signature would be unmistakable in upcoming gravitational wave detectors.

We are grateful to the anonymous referee who pointed out the importance of magnetic damping on the r-modes. We also thank David Shoemaker and Joel Tohline for comments on the manuscript, Sharon Morsink for help with the RNS code, and Sam Koshy for helpful discussions on gravitational wave theory. J.S. and P.J. thank the hospitality of the Department of Physics and Astronomy at the University of Calgary, where part of this work was completed. J.S. is supported, in part, by grant AST-0708551 from the U.S. National Science Foundation and, in part, by grant NNX07AG84G and NNX10AC72G from NASA's ATP program. P.J. and V.C. acknowledge support from California State University Long Beach and the U.S. Army High Performance Computing Research Center. The research of R.O. is supported by an operating grant from the National Science and Engineering Research Council of Canada (NSERC).