GRAVITATIONAL WAVE EMISSION AND SPIN-DOWN OF YOUNG PULSARS

Since for a compact star the thermal evolution is faster than the spin-down, the evolution of the star should follow a curve Ω_hc(T) where it is always thermally in a steady state where neutrino cooling equals dissipative heating L_ν + P_G = 0 (Bondarescu et al. 2009). With the semi-analytic expressions (Equations (4) and (8)) this equation can be solved explicitly and yields for the dominant m = 2 r-mode

$$\begin{equation} \Omega _{\rm hc}\!\left(T\right)=\left(\frac{3^{8}5^{2}}{2^{15}}\frac{\tilde{L}\Lambda _{{\rm QCD}}^{9-\theta }T^{\theta -2\beta }}{\tilde{J}^{2}\Lambda _{{\rm EW}}^{4}GM^{2}R^{3}\hat{\alpha }_{{\rm sat}}^{2}}\right)^{1/\left(8+2\gamma \right)}. \end{equation} \tag{ 25 }$$

The evolution should therefore follow this curve until the star leaves the instability region. As can be seen in Figure 2, the boundary of the instability region of an NS has different segments: a low-temperature boundary outside of which shear viscosity damps the mode, and an intermediate-temperature boundary where bulk viscosity is the dominant damping mechanism. Since the bulk viscosity features a resonant behavior and decreases again at very large temperatures T ≳ 10¹¹ K, there a is third high-temperature boundary above which the r-mode is unstable (see Figure 3 of Alford et al. 2012b), but for NSs this segment is quite likely physically irrelevant since the stars cool below it quickly. Semi-analytic expressions for these segments of the boundary of the instability region have been derived in Alford et al. (2012b). The form of the boundary depends strongly on the form of matter (Ho et al. 2011; Haskell et al. 2012). When exotic forms of matter are present, in the relevant temperature regime there are so-called stability windows, caused by the resonant behavior of the bulk viscosity of the corresponding phase, where the star is stable against developing an r-mode up to large frequencies (see Figure 3 of Alford et al. 2012b). In the case in which the steady-state curve formally lies in such a stability window it would clearly be irrelevant for the spin-down.⁴ Then the evolution can take a long time, since the strong heating can push the star out of the instability region before the r-mode reaches a large amplitude, and other spin-down sources will likely dominate. Therefore, we will restrict ourselves here to the case of pure NSs. The general evolution for different forms of matter will be discussed elsewhere.

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In the case of NSs the intermediate-temperature boundary of the instability region has a steeper slope than the steady state curve, which thereby can only intersect the low-temperature boundary. In the case of an NS the low-temperature boundary Ω_lb(T) is determined by the condition τ_S = τ_G and the corresponding semi-analytic solution is (Alford et al. 2012b)

$$\begin{equation} \Omega _{{\rm lb}}\!\left(T\right)=\left(\frac{3^{8}5^{3}}{2^{17}\pi }\frac{\tilde{S}\Lambda _{{\rm QCD}}^{3+\sigma }}{\tilde{J}^{2}{\it GM}^{2}R^{3}T^{\sigma }}\right)^{1/6}. \end{equation} \tag{ 26 }$$

Equating Equations (25) and (26), the final angular velocity Ω_f where the evolution leaves the instability region is given by

$$\begin{eqnarray} \Omega _{f}&=&\left (\left(\frac{3^{8}5^{3}}{2^{17}\pi }\right)^{\theta +\sigma -2\beta }\left(\frac{4\pi }{5}\right)^{\sigma }\right.\nonumber\\ && \times \left.\frac{\tilde{S}^{\theta -2\beta }\tilde{L}^{\sigma }\Lambda _{{\rm QCD}}^{3\theta +9\sigma -6\beta -2\beta \sigma }}{(\tilde{J}^{2}GM^{2}R^{3})^{\theta +\sigma -2\beta }\Lambda _{{\rm EW}}^{4\sigma }\hat{\alpha }_{{\rm sat}}^{2\sigma }}\right)^{\!1/\left(6\theta +8\sigma +2\gamma \sigma -12\beta \right)},\qquad \end{eqnarray} \tag{ 27 }$$

and the corresponding temperature T_f reads

$$\begin{eqnarray} T_{f}&=&\left (\left(\frac{3^{8}5^{3}}{2^{17}\pi }\right)^{2+2\gamma }\left(\frac{5}{4\pi }\right)^{6}\right.\nonumber\\ && \left.\times \frac{\tilde{S}^{8+2\gamma }\Lambda _{{\rm EW}}^{24}\Lambda _{{\rm QCD}}^{-30+6\theta +8\sigma +6\gamma +2\gamma \sigma }\hat{\alpha }_{{\rm sat}}^{12}}{\tilde{L}^{6}(\tilde{J}^{2}GM^{2}R^{3})^{2\left(1+\gamma \right)}}\right)^{\!1/\left(6\theta +8\sigma +2\gamma \sigma -12\beta \right)}\!\!\!. \nonumber \\ \end{eqnarray} \tag{ 28 }$$

Note that, if we had used a different steady-state curve $\Omega _{\rm hc}^{\left(S\right)}$ where the heating comes only from shear viscosity, as assumed in Owen et al. (1998), or if we had included the bulk viscosity as well, the rest of the evolution could be very different, but we would have obtained exactly the same result for the endpoint (Equations (27) and (28)) of the evolution. This surprising feature comes about since the shear viscosity is strongly dominant compared to the bulk viscosity on the left boundary, and this boundary is precisely defined by the condition τ_S = τ_G, which implies that dissipated power equals the power radiated in gravitational waves P_S = P_G. In this sense the result (Equation (27)) is universal and does not depend on assumptions about the dissipative aspects of the saturation mechanism.

The above expressions involve the star constants given in Table 2. In order to give these expressions in a more physical form, we write them relative to a fiducial reference model chosen as a 1.4 M_☉ NS with an APR equation of state (Akmal et al. 1998) given in the first row of Table 2 and denoted by a subscript "fid." In the case of a constant, T- and Ω-independent, saturation amplitude (Owen et al. 1998) β = γ = 0 this gives for a standard NS with shear viscosity due to leptonic processes (Shternin & Yakovlev 2008) and neutrino emission due to modified Urca reactions (Friman & Maxwell 1979)

$$\begin{eqnarray} T_{f}^{\left({\rm NS}\right)}&\approx & (1.26\times 10^{9}\,\mathrm{K})\left(\!\frac{\tilde{S}}{\tilde{S}_{{\rm fid}}}\!\right)^{\!3/23}\!\left(\!\frac{\tilde{L}}{\tilde{L}_{{\rm fid}}}\!\right)^{\!-9/92}\!\left(\!\frac{\tilde{J}}{\tilde{J}_{{\rm fid}}}\!\right)^{\!-3/46}\nonumber \\ && \times \left(\!\frac{M}{1.4\, M_{\odot }}\!\right)^{-3/46}\!\left(\!\frac{R}{11.5\,\mathrm{km}}\!\right)^{-9/92}\!\alpha _{{\rm sat}}^{9/46}, \end{eqnarray} \tag{ 29 }$$

$$\begin{eqnarray} f_{f}^{\left({\rm NS}\right)}& \approx & \left(61.4\,\mathrm{Hz}\right)\left(\!\frac{\tilde{S}}{\tilde{S}_{{\rm fid}}}\!\right)^{\!3/23}\!\left(\!\frac{\tilde{L}}{\tilde{L}_{{\rm fid}}}\!\right)^{\!5/184}\!\left(\!\frac{\tilde{J}}{\tilde{J}_{{\rm fid}}}\!\right)^{\!-29/92}\nonumber \\ && \times \left(\!\frac{M}{1.4\, M_{\odot }}\!\right)^{-29/92}\!\left(\!\frac{R}{11.5\,\mathrm{km}}\!\right)^{-87/184}\!\alpha _{{\rm sat}}^{-5/92}.\quad \end{eqnarray} \tag{ 30 }$$

First, it is interesting to note that all these expressions depend neither on the specific heat nor directly on the bulk viscosity of the considered form of matter. The latter is at first sight surprising since the bulk viscosity is the dominant dissipation mechanism at high temperatures and low amplitudes. But during the spin-down evolution the dissipation is dominated by the appropriate saturation mechanism, while the amount of dissipation is, according to Equations (11) and (19), determined entirely by gravitational physics. This is particularly favorable since it eliminates the complications arising from the fact that the damping due to bulk viscosity requires the density fluctuation of the r-mode (Equation (12)), which is only non-vanishing to second order in Ω (Lindblom et al. 1999) and involves significant uncertainties. Even more striking is the extreme insensitivity of these results to the remaining microphysics included in the constants $\tilde{S}$ and $\tilde{L}$. The neutrino emissivity constant $\tilde{L}$, in particular, depends on poorly known strong interaction corrections to the weak processes (Friman & Maxwell 1979). Yet, the remarkably small power 5/184 arising in the important expression for the final angular velocity $\Omega _{f}^{\left({\rm NS}\right)}$ almost eliminates this uncertainty. The dependence of $\Omega _{f}^{\left({\rm NS}\right)}$ on the shear viscosity constant $\tilde{S}$ is somewhat stronger, but $\tilde{S}$ is dominated by electromagnetic leptonic processes (Shternin & Yakovlev 2008) that are well under theoretical control⁵ and depends on the equation of state only via the lepton densities. The dependence on the parameters $\tilde{J}$, M, and R, which encode macroscopic properties of the star, is stronger, but they vary only within narrow margins. This insensitivity to the underlying parameters is similar to the case of the minimum of the instability region (Lindblom et al. 1998) analyzed previously in Alford et al. (2012b), yet it is fortunately even more pronounced in the physically important case (Equation (30)). The most striking feature of these expressions, however, is that the insensitivity extends even to the dependence on α_sat, which is theoretically uncertain by many orders of magnitude (Alford et al. 2012c; Bondarescu et al. 2009; Lindblom et al. 2001; Kastaun 2011) and therefore represents the main uncertainty in the analysis. Here the similarly small power 5/92 dramatically reduces this dependence and thereby allows a quantitative comparison with observational data below.

To get an idea of the uncertainty in $f_{f}^{\left({\rm NS}\right)}$, we estimate the error ranges of the individual parameters by symmetric logarithmic error bands, i.e., we allow for the corresponding parameter to be smaller or larger than the APR reference star by a given factor. The shear viscosity parameter $\tilde{S}$, due to leptonic processes, (Shternin & Yakovlev 2008) is uncertain within a factor of two of $\tilde{S}_{{\rm fid}}$ and the neutrino emissivity parameter $\tilde{L}$ due to modified Urca processes (Friman & Maxwell 1979) within an order of magnitude of $\tilde{L}_{{\rm fid}}$. For $\tilde{J}$ we have rigorous limits (Equation (14)). An upper limit for the mass of an NS is ≲ 2.5 M_☉, and an upper limit for the radius of a star rotating with millisecond periods is ≲ 15 km. Since the upper bound for $\tilde{J}$ is for a constant density profile, which is far from the situation in an NS, and, due to cancellations, the dependence on the mass and radius turns out to be significantly weaker than these individual errors suggest (see Table 3 below), we consider half the above ranges for the individual logarithmic uncertainties for $\tilde{J}$, M, and R as a more realistic estimate. Under the assumption that these errors are independent we find

$$\begin{equation} f_{f}^{\left({\rm NS}\right)}\approx \left(61.4\pm 9.4\right)\mathrm{Hz}\,\,\alpha _{{\rm sat}}^{-5/92}. \end{equation} \tag{ 31 }$$

We want to stress that this is just an estimate of the likely error range and not a statistical measure of well-defined significance.

Different classes of compact stars, containing other phases of dense matter with different low-energy degrees of freedom, feature different values for the exponents in Equations (2) and (3) and can lead to qualitatively different behavior. An example is a heavy star where direct Urca reactions are kinematically allowed in an inner core and the parameters change according to Table 2, which increases the final frequency beyond the above error range. When exotic forms of matter are present, like hyperon or quark matter, the enhanced dissipation of these phases leads to stability windows (Jaikumar et al. 2008; Alford et al. 2012b; Ho et al. 2011; Haskell et al. 2012). In this case the boundary of the stability window in the relevant temperature regime is determined by the bulk viscosity of the exotic phase, so that Equations (27) and (28) are not valid. The steady-state curve then ends at a higher frequency due to the instability window where the star leaves the unstable regime and cools until it reaches the lower boundary of the stability window. Since at these lower temperatures r-mode heating dominates neutrino cooling- the star is repeatedly pushed out of the instability region once its amplitude becomes large, and the residual spin-down takes much longer (Madsen 2000; Andersson et al. 2002). The important point is, however, that the final frequency, which is in this case given by the minimum of the instability region and for which a similarly precise semi-analytic result exists (Alford et al. 2012b), is systematically larger for exotic forms of matter since the strong bulk viscosity of the exotic phase dominates the shear viscosity down to lower temperatures. Therefore, the above expressions (Equations (30) and (31)) provide lower bounds for the frequency to which an arbitrary compact star can be spun down due to the r-mode instability.

Next let us actually solve the evolution equations to obtain the corresponding evolution timescales. The initial cooling time in the stable regime before the evolution enters the instability region t_sc results from solving the thermal equation (Equation (18)), in the absence of viscous heating, down to the temperature determined by the semi-analytic expression for the intermediate-temperature boundary of the instability region given in Alford et al. (2012b). The result reads

$$\begin{eqnarray} t_{{\rm sc}} & =\frac{1}{\theta -\upsilon -1}\left(\frac{3^{3}5^{2}}{2^{12}7\pi \kappa ^{2}}\right)^{\left(\theta -\upsilon -1\right)/\delta }\nonumber\\ & \quad \times \left(\frac{\tilde{V}_{m}^{\theta -\upsilon -1}\tilde{C}^{\delta }\Lambda _{{\rm QCD}}^{9\left(\theta -\upsilon -1\right)-5\delta }R^{\theta -\upsilon -1}}{\tilde{L}^{\delta }\Lambda _{{\rm EW}}^{4\left(\theta -\delta -\upsilon -1\right)}\left(\tilde{J}^{2}GM^{2}\Omega _{i}^{4}\right)g^{\theta -\upsilon -1}}\right)^{1/\delta } \end{eqnarray} \tag{ 32 }$$

in terms of the initial angular velocity Ω_i and depends on both the bulk viscosity and the specific heat. For a standard NS with shear viscosity due to leptonic processes and neutrino emission due to modified Urca reactions this gives

$$\begin{eqnarray} t_{{\rm sc}}^{\left({\rm NS}\right)} & \approx & (7.83\times 10^{-2}\,\mathrm{s})\left(\!\frac{\tilde{V}}{\tilde{V}_{{\rm fid}}}\!\right)\left(\!\frac{\tilde{J}}{\tilde{J}_{{\rm fid}}}\!\right)^{\!-2}\!\left(\!\frac{\tilde{C}}{\tilde{C}_{{\rm fid}}}\!\right)\left(\!\frac{\tilde{L}}{\tilde{L}_{{\rm fid}}}\!\right)^{\!-1}\nonumber \\ && \times \left(\!\frac{M}{1.4\, M_{\odot }}\!\right)^{\!-2}\!\left(\!\frac{R}{11.5\,\mathrm{km}}\!\right)\!\left(\frac{f_{i}}{\mathrm{kHz}}\right)^{\!-4}. \end{eqnarray} \tag{ 33 }$$

This time is so short compared to the subsequent r-mode evolution that it can safely be neglected.

The spin-down time t_sd is obtained by solving the frequency equation (20) in the saturated regime. Even though for r-mode emission the spin-down evolution is coupled to the thermal evolution when the saturation amplitude is temperature dependent, the knowledge of the path of the evolution within the instability region, determined by the analytic steady-state solution (Equation (25)) of the thermal equation, allows us to derive the effective spin-down equation. Inserting (the inverse of) the steady-state relation (Equation (25)) into the spin-down equation (20), we find

$$\begin{eqnarray} \frac{d\Omega }{dt}= & {-}\frac{2^{17}\pi }{3^{7}5^{2}}\frac{\tilde{J}^{2}GMR^{4}}{\tilde{I}}\left(\frac{2^{15}}{3^{8}5^{2}}\frac{\tilde{J}^{2}G\Lambda _{{\rm EW}}^{4}M^{2}R^{3}}{\tilde{L}\Lambda _{{\rm QCD}}^{9-\theta }}\right)^{\frac{2\beta }{\theta -2\beta }}\nonumber \\ & \times \hat{\alpha }_{{\rm sat}}^{\frac{2\theta }{\theta -2\beta }}\Omega ^{n_{{\rm rm}}}, \end{eqnarray} \tag{ 34 }$$

i.e., despite the coupling with the thermal evolution, the spin-down equation takes a standard power-law form (Shapiro & Teukolsky 1983), yet with an effective braking index

$$\begin{equation} n_{{\rm rm}}=\frac{\left(7 + 2\gamma \right)\theta + 2\beta }{\theta - 2\beta }=7\left(\!\frac{1 + 2\gamma /7 + 2\beta /\left(7\theta \right)}{1 - 2\beta /\theta }\!\right). \end{equation} \tag{ 35 }$$

The correction factor in parentheses changes the braking index from the canonical value, which is 7 for r-mode emission. The correction factor takes the value 1 for a constant saturation amplitude (β = γ = 0), but for general β and γ it deviates from 1 and lowers the braking index, since for proposed saturation mechanisms (Arras et al. 2003; Bondarescu et al. 2009; Bondarescu & Wasserman 2013; Alford et al. 2012c) β, γ < 0. Using these values, we find that the correction factor can decrease the braking index to nearly half of its canonical value and therefore has a significant impact on the spin-down evolution. For example, for mode coupling with damping due to shear viscosity (Bondarescu & Wasserman 2013), where β = −4/3 and γ = −2/3, it reduces to $n_{{\rm rm}}^{({\rm mc})}=4$. This is rather close to the canonical value of 3 for electromagnetic dipole radiation, for which a similar dependence of the braking indices on the details of the spin-down mechanism is well known. In the electromagnetic case many mechanisms have been proposed as well that can explain the deviation of observed braking indices from the canonical value of 3 (see, e.g., Blandford & Romani 1988). Therefore, it should be harder than expected to distinguish these different spin-down scenarios based on their braking indices. Actually, an even lower value for the effective r-mode braking index is obtained for the recently proposed scenario that the cutting of superconducting flux tubes by superfluid vortices can saturate r-modes (Haskell et al. 2013), where β = 0, γ = −3. In this case the braking index takes the extreme value $n_{{\rm rm}}^{({\rm fc})}=1$, which means that the spin-down is not even power-like but logarithmic, and the general solution given below does not apply. However, the spin-down evolution decouples in this case and can be easily solved. The case of the suprathermal bulk viscosity (Alford et al. 2012c) is more complicated and will be studied elsewhere, since the large-amplitude enhancement of the neutrino emissivity would here be relevant to determine the effective braking index. As discussed before, such effects are not relevant unless suprathermal damping saturates the mode, which is not the case for standard bulk viscosity from an NS core.

Even though the spin-down evolution is coupled to the thermal evolution when the saturation amplitude is temperature dependent, the knowledge of the path of the evolution within the instability region at saturation (Equation (25)) allows us to solve the spin-down equation (34) fully analytically for the power-law parameterization (Equation (22)), which gives⁶

$$\begin{eqnarray} \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \Omega \!\left(t\right)&=&\left[\vphantom{\left(\!\frac{2^{15}}{3^{8}5^{2}}\frac{\tilde{J}^{2}G\Lambda _{{\rm EW}}^{4}M^{2}R^{3}}{\tilde{L}\Lambda _{{\rm QCD}}^{9-\theta }}\!\right)^{\frac{2\beta }{\theta -2\beta }}}\Omega _{i}^{-\frac{2\left(3+\gamma \right)\theta +4\beta }{\theta -2\beta }}+\frac{2^{18}\pi }{3^{7}5^{2}}\frac{\left(3+\gamma \right)\theta +2\beta }{\theta -2\beta }\frac{\tilde{J}^{2}GMR^{4}}{\tilde{I}}\right.\nonumber \\ &\times &\!\! \left.\!\left(\!\frac{2^{15}}{3^{8}5^{2}}\frac{\tilde{J}^{2}G\Lambda _{{\rm EW}}^{4}M^{2}R^{3}}{\tilde{L}\Lambda _{{\rm QCD}}^{9-\theta }}\!\right)^{\frac{2\beta }{\theta -2\beta }}\!\!\hat{\alpha }_{{\rm sat}}^{\frac{2\theta }{\theta -2\beta }}\left(t-t_{i}\right)\!\right]^{\!-\frac{\theta -2\beta }{2\left(3+\gamma \right)\theta +4\beta }}\!\!. \end{eqnarray} \tag{ 36 }$$

An interesting aspect of this expression is that for a temperature-independent saturation amplitude it does not depend on the cooling behavior, which reflects the previous observation that the thermal and spin-down evolutions decouple in this case. Equation (36) has two limits, at early times where the first term in the brackets dominates and at late times where the second dominates, i.e., initially the amplitude hardly changes, whereas at late times the evolution becomes independent of the initial conditions. The semi-analytic temperature evolution at saturation is then obtained by inserting the frequency solution (Equation (36)) in the inverse of the steady-state curve (Equation (25)). In the late-time limit we find

$$\begin{eqnarray} T\!\left(t\right)&\approx & \left[\left(\frac{3^{7}5^{2}}{2^{18}\pi }\frac{\theta -2\beta }{\left(3+\gamma \right)\theta +2\beta }\frac{\tilde{I}}{\tilde{J}^{2}GMR^{4}t}\right)^{8+2\gamma }\right.\nonumber \\ && \left.\times \left(\frac{2^{15}}{3^{8}5^{2}}\frac{\tilde{J}^{2}G\Lambda _{{\rm EW}}^{4}M^{2}R^{3}}{\tilde{L}\Lambda _{{\rm QCD}}^{9-\theta }}\right)^{2\left(3+\gamma \right)}\hat{\alpha }_{{\rm sat}}^{-4}\right]^{\frac{1}{2\left(3+\gamma \right)\theta +4\beta }}.\qquad \end{eqnarray} \tag{ 37 }$$

For a standard NS in particular we find for the constant saturation model in the late time limit Ω(t) ∼ T(t) ∼ t^−1/6.

For the spin-down time we find then in terms of the gravitational timescale (Equation (4)) evaluated at the final frequency

$$\begin{eqnarray} t_{{\rm sd}} & =-\frac{\theta -2\beta }{4\left(3+\gamma \right)\theta +8\beta }\frac{\tau _{G}(\Omega _{f})}{Q\alpha _{{\rm sat}}^{2}(\Omega _{f})}\left(\!1 - \left(\frac{\Omega _{f}}{\Omega _{i}}\right)^{\!\frac{2\left(3+\gamma \right)\theta +4\beta }{\theta -2\beta }}\!\right)\nonumber \\ & \xrightarrow[{\Omega _{f}\ll \Omega _{i}}]\, -\frac{{\cal C}}{6}\frac{\Omega _{f}}{\dot{\Omega }_{f}}, \end{eqnarray} \tag{ 38 }$$

where the dependence on the saturation model enters only via the constant prefactor

$$\begin{equation} {\cal C}\equiv \frac{1-2\beta /\theta }{1+\gamma /3+2\beta /\left(3\theta \right)}{ =\frac{6}{n_{{\rm rm}}-1}}, \end{equation} \tag{ 39 }$$

which takes the value 1 for a constant saturation amplitude (β = γ = 0). For general β and γ it deviates from 1, but for proposed saturation mechanisms (Arras et al. 2003; Bondarescu et al. 2009; Bondarescu & Wasserman 2013; Alford et al. 2012c) β, γ < 0, so that up to small corrections in Ω_f/Ω_i this simply increases the spin-down time compared to a constant saturation amplitude by a constant factor. Since in hadronic matter θ = 8, these corrections are modest,⁷ and for the exponents predicted by proposed saturation mechanisms (Arras et al. 2003; Bondarescu et al. 2009; Bondarescu & Wasserman 2013; Alford et al. 2012c) it amounts at most to a factor of order one. E.g., in the case of the mode coupling mechanism (Bondarescu & Wasserman 2013) spin-down time is longer compared to the saturation model with a constant amplitude by a factor of 2. Taking into account our ignorance of the complicated r-mode saturation physics, this simple and moderate dependence is most welcome.

The above expression (Equation (38)) looks similar to the relation between the spin-down time for other spin-down mechanisms, like magnetic dipole radiation or gravitational wave emission due to deformations, and the characteristic pulsar timescale $-\Omega _{0}/\dot{\Omega }_{0}$ (Shapiro & Teukolsky 1983). Yet, the important point to note here is that Ω_f and $\dot{\Omega }_{f}$ are the values before the star leaves the instability region, which can be very different from the observed values Ω₀ and $\dot{\Omega }_{0}$ if the observation is performed after the star has left the instability region. The characteristic feature of the r-mode mechanism, namely, that it operates only for a finite time interval, renders the r-mode spin-down time completely independent of the characteristic pulsar timescale and allows the former to be orders of magnitude smaller, since the spin-down rate can strongly drop from a large r-mode rate to a much lower rate of other spin-down mechanisms at the boundary of the instability region.

Inserting the expression for the final frequency (Equation (30)) into Equation (38), this yields for the constant saturation model in the case of a standard NS and for Ω_f ≪ Ω_i

$$\begin{eqnarray} t_{{\rm sd}}^{\left({\rm NS}\right)}&\approx& \left(12.3\mathrm{\, yr}\right)\left(\!\frac{\tilde{S}}{\tilde{S}_{{\rm fid}}}\!\right)^{\!-18/23}\!\!\left(\!\frac{\tilde{L}}{\tilde{L}_{{\rm fid}}}\!\right)^{\!-15/92}\!\!\left(\!\frac{\tilde{J}}{\tilde{J}_{{\rm fid}}}\!\right)^{\!-5/46}\nonumber \\ && \times \left(\!\frac{\tilde{I}}{\tilde{I}_{{\rm fid}}}\!\right)\!\left(\!\frac{M}{1.4\, M_{\odot }}\!\right)^{\!41/46}\!\!\left(\!\frac{R}{11.5\,\mathrm{km}}\!\right)^{\!-107/92}\!\alpha _{{\rm sat}}^{-77/46}.\nonumber \\ \end{eqnarray} \tag{ 40 }$$

Here the dependence on the parameters is stronger than for the endpoint of the evolution (Equations (27) and (28)), so that the duration is considerably more uncertain. In particular, it is very sensitive to the saturation amplitude α_sat. If α_sat becomes too small, this timescale is so long that other spin-down mechanisms will dominate, as is clear from the sizable observed spin-down rate of young pulsars (Manchester et al. 2005). Employing the bounds (Equations (13) and (14)) and the full uncertainty ranges for the other underlying parameters discussed before to estimate the uncertainty in the final frequency (Equation (31)), we find that the spin-down time could be smaller or larger by nearly an order of magnitude.

For the constant saturation model, the results for the evolution of the 1.4 M_☉ NS in temperature–angular velocity space is shown in Figure 2 for different values of the initial rotation angular velocity and the saturation amplitude. As can be seen, the numerical solution given by the solid curves perfectly confirms the above picture. The star initially cools until it reaches the instability region, where the r-mode amplitude starts to grow. Once the amplitude is significant, the star starts spinning down, but since the spin-down time τ_Ω is much longer than the cooling time τ_T, the star cools further until it reaches the steady-state curve,⁸ where the fast temperature evolution is stopped and the star evolves along the steady-state curve on the longer timescale τ_Ω. Consequently, the evolution is completely independent of the initial amplitude α_min required to start the growth in Equation (16). Such initial perturbations will clearly be present immediately after the creation of the NS following a core bounce. As anticipated, the numerical solution also proves that the evolution is to very good accuracy independent of the initial frequency unless it is very close to the final frequency. Therefore, similar to the thermal evolution, the spin-down evolution completely loses its memory of the initial conditions at sufficiently late times. The saturation amplitude in contrast provides the dominant uncertainty for the evolution, and the trajectory is shown for a range of possible saturation amplitudes. Larger values lead to stronger dissipative reheating and therefore spin-down the star at larger temperatures so that the evolution leaves the instability region at lower frequencies closer to the minimum of the instability region. The dots are obtained from the analytic expressions (Equations (29) and (30)), which agree with the numerical solution. Note again that we neglect other spin-down mechanisms like magnetic dipole radiation in this work, which would speed up the evolution and continue the magnetic spin-down after the star leaves the r-mode instability region, but would not alter the curves in Figure 2 within the instability region.

In Owen et al. (1998) the evolution was artificially stopped at a temperature of 10⁹ K since superfluidity and non-perfect fluid effects were expected to invalidate the analysis below. We do not follow this ad hoc prescription here since it is not likely that these effects have such a big influence on the spin-down evolution for the following reasons: One important effect of superfluidity is to affect beta equilibration. Around the critical temperature it is enhanced due to pair breaking, but at temperatures far below and small oscillation amplitudes beta equilibration processes are strongly suppressed. Since the superfluid gap depends strongly on density, the entire star may not be gapped at such temperatures, and in this case in part of the star the weak interactions would hardly be affected and the corresponding parameters $\tilde{L}$ and $\tilde{V}$ would only be moderately reduced. Even more, as has recently been shown in Alford et al. (2012d), superfluidity does not suppress weak processes at all at sufficiently large amplitudes that are within the range we study here. Superfluidity can also lead to an additional source of dissipation via mutual friction. In contrast to the earlier generic analysis referred to in Owen et al. (1998), the later dedicated r-mode analysis (Lindblom & Mendell 2000) showed that it is unlikely that mutual friction has such a dramatic impact on the r-mode instability. In Haskell et al. (2009) this effect was studied, taking into account the large uncertainties in pairing and the microscopic drag parameter. It was found that, depending on these uncertainties, the effect can range from the unlikely extreme that the r-mode is completely suppressed below a certain temperature to a nearly negligible impact on the instability region compared to the ungapped case. Therefore, it is interesting to study the present case of standard dissipation sources in detail, since it sets the limit on the frequency to which r-modes can spin-down a star, as well as on the emitted gravitational wave signal, as discussed below. Any additional dissipation source will reduce the r-mode instability and correspondingly lead to larger final spin frequencies and a reduced gravitational wave emission.

Figure 3 shows the evolution of the angular velocity as a function of time. As predicted by Equation (40), the spin-down time increases strongly with decreasing saturation amplitude. The points show that the semi-analytic results for this time again agree with the numerical solution.⁹ Here we are mainly interested in the spin-down aspects and only note that, as far as the thermal evolution is concerned, the r-mode effectively causes a delay in the cooling (Alford et al. 2012a). This is significant initially, but, since the cooling slows down strongly, it is irrelevant at much later times t ≫ t_sd.

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Table 3 lists the semi-analytic results for a range of saturation amplitudes for two different NSs—the previously discussed standard star with a canonical mass of 1.4 M_☉ and, motivated by the recent observation (Demorest et al. 2010), also a heavy 2.0 M_☉ star. Strikingly, the final frequency proves nearly independent of the mass and is far less sensitive than Equation (30) naively suggests. This is apparently caused by cancellations due to the implicit mass dependence of the radius and of the parameters $\tilde{J}$, $\tilde{S}$, and $\tilde{L}$. Despite the explicit normalizations in Table 1, these parameters still have a significant implicit dependence on M and R via the density profile of the star. Correspondingly, whereas the dependence on microscopic material properties is explicitly given by the semi-analytic expressions (Equations (27)–(40)), the dependence on star properties like the mass and radius is not properly reflected by the explicit dependencies alone, but the additional implicit dependence via the averaged quantities in Table 1 is crucial. The initial cooling time before the star enters the instability region is amplitude independent and yields for a star spinning at the Kepler frequency the small values t_sc = 96.4 (55.1) ms, which rise to a few hours for frequencies at the lower end of the instability region, which is indeed negligible compared to the long spin-down times. The left part of Table 5 finally compares the spin-down observables for different stars at a fixed amplitude α_sat = 1. We see that, when direct Urca processes are allowed, the final temperature is significantly lower due to the enhanced neutrino cooling. Since the instability region shrinks at lower temperatures, the final frequency is higher and the spin-down time is correspondingly shorter.

Let us now discuss the influence of different saturation models α_sat(T, Ω). Since in the general case (Equation (22)) the amplitude varies over the instability region and the coefficient $\hat{\alpha }_{{\rm sat}}$ is a dimensionful parameter, we have to find analogous parameter values for $\hat{\alpha }_{{\rm sat}}$ to compare saturation models with different values of β and γ among each other and with the constant model discussed so far. Since the evolution spends most of the time in the final stages of the spin-down evolution, it is natural to choose $\hat{\alpha }_{{\rm sat}}$ so that different models have a similar amplitude in this stage. If we compare models defined by $\hat{\alpha }_{{\rm sat}}$, β, and γ that have the same amplitude α_sat at the end of the evolution, then they leave the instability region at the same point given by Equations (27) and (28). This can be easily seen, noting that the conditions τ_S = τ_G and P_S = P_G that determine the endpoint depend only on the saturation amplitude α_sat(T_f, Ω_f) at this point. However, the path within the instability region, determined by the steady-state curve (Equation (25)), and in particular the thermal behavior will depend on β and γ.

Figure 4 shows a few examples of different saturation models for two different saturation amplitudes α_sat = 10⁻⁴ and α_sat = 1 at the end of the evolution. The solid lines are the constant model β = γ = 0 (Owen et al. 1998) discussed above. The dashed curves show the case of mode coupling with damping of the daughter modes due to shear viscosity in a star with an impermeable crust (Bondarescu & Wasserman 2013). In this case β = −4/3 and γ = −2/3, but depending on the particular damping mechanism of the daughter modes, other dependencies are possible (Bondarescu & Wasserman 2013). Since the amplitude is lower at large frequency, the dissipative heating is smaller, the steady-state curve becomes steeper, and the equilibrium is reached at lower temperatures.

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The final, and somewhat extreme, example is the recently proposed dissipation due to the cutting of superconducting flux tubes by superfluid vortices when both neutrons and protons pair in an NS core (Haskell et al. 2013). In this case β = 0, γ = −3. This moves the steady-state curves to even lower temperatures. As discussed before, this gives a braking index of 1, so that the spin-down is not power-like but logarithmic and strongly slowed down. In conclusion, different saturation mechanisms can change both the thermal evolution and the spin-down evolution significantly and generically increase the spin-down time. However, Equation (27) shows that the observed insensitivity to the microscopic parameters is not strongly altered by the saturation model for realistic values |β|, |γ| ≪ θ.

The semi-analytic expressions finally allow us to compare our theoretical results to observational data. In most analyses, e.g., for low-mass X-ray binaries (Ho et al. 2011; Haskell et al. 2012), the observed pulsar frequencies and temperatures are compared to numerical computations of the r-mode instability region. Since the instability boundary is strongly temperature dependent, information on the temperature is required to decide if the star is actually inside the instability region. However, the temperature is unknown for most young pulsars. For stars that spin slow enough that they are clearly below the minimum of the instability region, the knowledge of the frequency is enough to rule out that r-mode oscillations are present. However, there is at least one young star that lies above the minimum of the instability region. Therefore, we will here go beyond such a static analysis and take into account the dynamic evolution and the frequency (Equation (27)) where the star actually leaves the instability region.

In addition to the spin frequencies, also the spin-down rates are known for many young stars. Naturally, this additional information provides a stronger constraint on spin-down models. The time derivative of the frequency is readily obtained from Equation (20). Figure 5 compares the spin-down evolution for different saturation amplitudes to the data for young stars that still have a significant spin-down rate. As can be seen, there is one pulsar (J0537-6910) that spins fast enough that it could be within the instability region of a standard NS whose minimum frequency is denoted by the vertical dotted line to the left.¹⁰ Taking into account the dynamical evolution, the additional knowledge of the spin-down rate alleviates the uncertainty with respect to the saturation amplitude, since larger amplitudes lead to significantly larger spin-down rates. This is shown by the black line, which is formed by the semi-analytic results for the endpoints of the evolution and represents the boundary of the instability region in f–$\dot{f}$ space. The gray band reflects the uncertainty in the underlying parameters (Equation (31)). The fastest pulsar, J0537-6910, is obviously compatible with being outside of the instability region. We conclude that no observed star lies definitely inside the instability region (see also Arras et al. 2003), which confirms the r-mode scenario for a sufficiently large saturation amplitude resulting in a short spin-down time. Moreover, J0537-6910 lies clearly below the line for our APR NS and just at the lower edge of the uncertainty band. Therefore, it is very likely no young star observed so far currently emits gravitational waves because of the r-mode instability. This conclusion is strengthened by the absence of gravitational wave signals from observed young pulsars—including J0537-6910—in the latest LIGO data (Virgo Collaboration et al. 2010). Unfortunately, the measured braking index of n ≈ −1.5 for J0537-6910 (Middleditch et al. 2006) seems to be strongly affected by the glitch activity of the pulsar (Shapiro & Teukolsky 1983) and therefore cannot provide conclusive evidence on the current spin-down mechanism. For other pulsars, the measured braking indices¹¹ n ≲ 3, as shown in Figure 5 for the Crab (Shapiro & Teukolsky 1983) and the Vela (Lyne et al. 1996) pulsars, already proved consistent with a dominant electromagnetic spin-down. For the Crab pulsar, this was further strengthened by the LIGO S5 limit (LIGO Scientific Collaboration et al. 2008), showing that it could emit at most a few percent of the rotational energy loss as gravitational waves.

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The remarkable aspect of Equation (30) for the final frequency of the spin-down is that it is a robust lower bound and is not affected by our ignorance of the star's composition. Although our analysis was performed for a standard NS, any additional exotic forms of matter entail enhanced damping so that their instability regions do not extend to such low frequencies. The insensitivity of the final frequency (Equation (27)) to the underlying parameters therefore provides a quantitative, universal lower bound given by Equation (31). As a consequence, our result shows that the r-mode picture of the spin-down of young pulsars presents a viable explanation for the low spin rates of young pulsars if the saturation amplitude is large enough (α_sat ≳ 0.01) to spin-down the pulsar in a time t_sd < 10⁴ yr. An upper limit for the saturation amplitude α_sat < 1 comes from the observed timing data of J0537-6910, as noted in Owen (2010), since r-modes would have otherwise spun it down to its current frequency in less than a hundred years, which is much shorter than its age estimate (Wang & Gotthelf 1998). Whereas the r-mode picture yields a direct and quantitative explanation for the maximum observed spin frequency, other mechanisms, like the possibility that all stars are already born with lower frequencies or the subsequent spin-down via magnetic dipole radiation, generally only provide probabilistic answers why fast stars with a small initial magnetic dipole moment resulting in a slow spin-down could not form. Despite the general trend that, according to the dynamo mechanism, fast spinning stars feature larger magnetic fields (Bonanno et al. 2006), the required spin-down evolution studies rely on Monte Carlo population synthesis modeling and involve uncertainties that are to some extent exploited to reproduce the statistical aspects of the observed pulsar distribution (Faucher-Giguere & Kaspi 2006). Finally, further indirect evidence for the above r-mode bound comes from pulsars in old and very stable double NS binaries, which are not recycled and correspondingly should not have experienced any appreciable spin-up during their evolution. The fastest of these, J0737-3039A located in the only known double pulsar system (Lyne et al. 2004), has a frequency of f ≈ 44 Hz, which is still intriguingly close to the quantitative r-mode bound (Equation (31)).

In this section, we follow and extend the previous analysis (Owen et al. 1998) and derive general results for the gravitational wave strain based on our analytic solution of the r-mode evolution. An explicit expression for the gravitational wave strain, averaged over polarizations and the orientation and position of a source at distance D on the sky, was given in Owen et al. (1998) and Thorne (1980)

$$\begin{equation} h=\sqrt{\frac{3}{80\pi }}\frac{\omega ^{2}S_{22}}{D}, \end{equation} \tag{ 41 }$$

with the mass current quadrupole moment

$$\begin{equation} S_{22}=\sqrt{2}\frac{32\pi }{15}GM\alpha \Omega R^{3}\tilde{J}. \end{equation} \tag{ 42 }$$

The strain depends on both the time-dependent r-mode amplitude and angular velocity

$$\begin{equation} h\!\left(t\right)=\sqrt{\frac{2^{15}\pi }{3^{5}5^{3}}}\frac{\tilde{J}GMR^{3}\alpha \!\left(t\right)\Omega \!\left(t\right)^{3}}{D}. \end{equation} \tag{ 43 }$$

This expression has three different limits. Initially, the amplitude rises exponentially, whereas the frequency is fixed. Depending on the amplitude, there can be an intermediate period where α is saturated but Ω does not appreciably spin-down yet since the first term in Equation (36) dominates. Finally, at late times, when the second term in Equation (36) dominates, the expression becomes independent of the initial conditions.

At saturation the r-mode amplitude can be replaced using the spin-down Equation (20), which gives

$$\begin{equation} h\!\left(t\right)=\sqrt{\frac{9}{20}\frac{GI\dot{\Omega }\!\left(t\right)}{D^{2}\Omega \!\left(t\right)}}. \end{equation} \tag{ 44 }$$

This expression gives the strain in terms of the present frequency and the spin-down rate. Assuming that r-modes dominate the spin-down, this expression yields for sources with known timing data the standard spin-down limit. This limit, which is analogously obtained, e.g., for gravitational waves due to an ellipticity of the source, is very useful since it gives the maximum signal that can be expected for a given known source. There is only a chance to detect gravitational waves—or alternatively learn something about the source from a non-detection—if the detector sensitivity is lower than the spin-down limit.

However, according to the discussion in Section 5, the observed pulsars do not emit gravitational waves due to r-modes any more, and timing data are not available for other promising young sources, like compact objects in supernova remnants or pulsar spin nebulae. Only the age is usually roughly known. A prediction for the gravitational strain requires therefore an actual solution of the evolution. Previously, a standard spin-down equation with a general breaking index (Shapiro & Teukolsky 1983) has been used to derive the strain as a function of the age of the source (LIGO Collaboration 2008). This expression also applies to the emission due to ellipticity, and at late times it proved to be independent of the saturation amplitude. In the case of r-modes the strong heating generally requires a solution of the coupled set of evolutions (Equations (20) and (21)), though. As discussed in Section 4, the spin-down evolution decouples only in the special case of a constant saturation amplitude. However, due to the r-mode heating the evolution predicted by this toy model can be quite different from that obtained from the coupled system for realistic saturation mechanisms, where the saturation amplitude is generally a function of temperature and frequency. Our analysis leading to the effective spin-down evolution (Equation (34)) confirms that the study performed by LIGO Collaboration (2008) does indeed apply to r-mode gravitational wave emission. The effective braking index (Equation (35)) gives the explicit connection for a realistic r-mode saturation mechanism. Inserting the frequency evolution (Equation (36)) and the general form for the saturation amplitude (Equation (22)), there are significant cancellations, and in the late-time limit, where the frequency is far below the initial frequency, the strain becomes

$$\begin{equation} h\!\left(t\right)\mathop{\longrightarrow}\limits_{\Omega \ll \Omega _{i}}\,\sqrt{\frac{3}{40}\frac{{\cal C}GI}{D^{2}t}}, \end{equation} \tag{ 45 }$$

with the same constant ${\cal C}$ describing the saturation mechanism (Equation (39)) that appeared before in the spin-down time. This expression agrees with the result given in Owen (2010) using the effective braking index (Equation (35)). A striking property of Equation (45) is that it is indeed independent of the unknown saturation amplitude and even of the complete microphysics, including the cooling mechanism. Therefore, the strain of a given source is, up to moderate uncertainties in the moment of inertia (Equation (15)) and the constant factor ${\cal C}$ involving β and γ, completely determined by its distance and its age. We recall that the constant ${\cal C}$ is 1 for a constant saturation model (β = γ = 0), where the expression directly reduces to the result given in LIGO Collaboration (2008) and Owen (2010), and somewhat larger for general saturation models. Because of the square root, the dependence on the saturation model is even weaker, and the result does not change by more than some tens of percent for realistic values |β|, |γ| < 2 predicted by different saturation mechanisms¹² (Arras et al. 2003; Bondarescu et al. 2009; Bondarescu & Wasserman 2013; Alford et al. 2012c). Therefore, the simple expression (Equation (45)) confirms and generalizes the result of LIGO Collaboration (2008) and Owen (2010) to the more complicated case of r-mode emission with realistic saturation mechanisms, where the thermal heating significantly affects the spin-down.

The analytic expression for the cutoff frequency (Equation (27)) also allows us to determine the final gravitational wave strain emitted in the last stage of the evolution h_f ≡ h(Ω_f) in terms of the final frequency Ω_f (Equation (27)). For an NS the final gravitational strain is

$$\begin{equation} h_{f}^{\left({\rm NS}\right)}\approx 7.23\times 10^{-27}\alpha _{{\rm sat}}^{\frac{77}{92}}\left(\frac{\mathrm{Mpc}}{D}\right). \end{equation} \tag{ 46 }$$

The emitted gravitational wave strain during the evolution is shown in Figure 6 for different values of the saturation amplitude. The numerical curves are compared to the analytic results for the minimum gravitational strain, which is emitted at the end of the evolution (Equation (46)) and denoted by the points. As can be seen, small-amplitude r-modes have a lower maximum amplitude but emit low-amplitude gravitational waves over longer times. The plot shows that given the relevant age range of young pulsars, for saturation amplitudes α_sat ≳ 10⁻⁴ the late-time approximation (Equation (45)) should be justified. For much smaller amplitudes the star is hardly spun down by r-modes, but in this case other spin-down mechanisms, like magnetic dipole radiation, will dominate and determine the spin-down Ω(t). This scenario, which is unlikely according to our results in Section 5, is discussed for completeness in Appendix A.

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The comparison of signals to the detector sensitivity is generally done in terms of the intrinsic strain amplitude¹³ h₀, which is in the case of r-modes given by Owen (2010)

$$\begin{equation} h_{0}=\sqrt{\frac{8\pi }{5}}\frac{\tilde{J}{\it GMR}^{3}\left(2\pi \nu \right)^{3}\alpha _{{\rm sat}}}{D}=\sqrt{\frac{25}{3}}h. \end{equation} \tag{ 47 }$$

Using the general constraints on the moment of inertia (Equations (13) and (15)) and the range of different saturation models (Arras et al. 2003; Bondarescu et al. 2009; Bondarescu & Wasserman 2013; Alford et al. 2012c; Owen et al. 1998) yields for a constant saturation amplitude the remarkably definite result of

$$\begin{eqnarray} h_{0} & \approx 2.3_{-0.8}^{+3.5}\times 10^{-27}\sqrt{\frac{1000\,{\rm yr}}{t}}\frac{1\,{\rm Mpc}}{D}, \end{eqnarray} \tag{ 48 }$$

where the uncertainty limits are conservative and should, as discussed before, likely exceed the realistic range.

For an actual detection of gravitational waves, a systematic search strategy is required that takes into account the periodicity of the signal. In particular, this requires a detailed model of the expected signal and its change over the observation interval in order to search for it in the data via a given statistical method, like, e.g., matched filtering. Only for known isolated pulsars with precise timing data, like frequency and spin-down rate, is such detailed knowledge of the phase-coherent signal available. Otherwise, a huge parameter space for whole classes of models with different frequencies, spin-down rates, etc., has to be searched, which greatly complicates the analysis.¹⁴ Reducing the computational cost in such searches is crucial since a coherent search over timescales of order years for such an unconstrained system is still not manageable. Therefore, one important aim of our work is to narrow down the parameter space and to identify the most promising parameter regions for different sources.

Naturally, the likelihood of detecting a continuous signal in gravitational wave searches increases with the observation time. This is usually taken into account by giving the equivalent intrinsic strain amplitude that would be observed within a given search at 95% confidence level. For a coherent search the intrinsic strain amplitude is given by LIGO Scientific Collaboration (2004),

$$\begin{equation} h_{0}^{95\%}=\Theta \sqrt{\frac{S_{h}\!\left(f\right)}{\Delta t}}. \end{equation} \tag{ 49 }$$

It is a property of the particular search analysis and depends on both the power spectral density¹⁵ S_h of the detector noise and the observation time Δt. The equivalent intrinsic strain amplitude (Equation (49)) relies on matched filtering and requires knowledge of the detailed form and time evolution of the coherent signal. Since this might not always be available, the sensitivity obtained using this method is an optimistic estimate for other methods, where, e.g., signals are combined incoherently. For observed isolated pulsars with known timing behavior the prefactor is Θ ≈ 11.4, and for known isolated sources without timing data, like Cas A or hypothetical localized compact sources within known supernova remnants, it is about 35 (LIGO Collaboration 2008). In the following we will discuss the detectability of gravitational waves in terms of the intrinsic strain amplitude, but other sensitivity measures have been used in the literature (Owen et al. 1998), and we will discuss some of them and their reliability in Appendix B.

In Figure 7, our results for h₀ from gravitational wave emission by r-modes for different NS models are compared to the sensitivity curves¹⁶ for the LIGO detector (S6 run) and the projected advanced LIGO sensitivity (LIGO Scientific Collaboration & Harry 2010), in both the standard ("ZERO_DET_high_P") and the NS optimized mode ("NSNS_Opt"). For each detector, we show the two exemplary cases of a coherent search for known pulsars (where Θ = 11.4) using one year of data (thick) and a search for sources without timing data, i.e., NSs or unseen compact remnants (using Θ = 35) with an analysis of a month of coherent data (thin). The left panel assumes a typical source within the galaxy (D = 10 kpc) and the right panel a source in the local group of galaxies (D = 1 Mpc). The label on the right axis also shows the age of the object according to the direct relation for constant saturation amplitude (Equations (45) and (47)). Let us first discuss this general relation. As can be seen, LIGO could have detected r-mode emission from known galactic pulsars up to an age of about 10,000 yr. According to our analysis in the preceding section, this should have been sufficient to see gravitational wave emission from all such sources that can be seen, but unfortunately there are likely no known young pulsars that are currently in the r-mode phase. For known pulsars, the sensitivity enhancement of advanced LIGO might therefore not help. But in case we detect in the future an even faster pulsar that is less than a hundred years old, then its r-mode emission could be detected with advanced LIGO even if it is extragalactic. There are a few such candidates, including most notably SN 1987A in the Large Magellanic Cloud. For sources without timing data, LIGO would have only seen galactic sources that are at most a few dozens of years old and only within a narrow frequency range. There has not been a galactic supernova for a long time and therefore it is not surprising that LIGO did not make a detection. In contrast, advanced LIGO could, thanks to its flat noise spectrum, detect a galactic source over the entire relevant frequency range and up to an age of more than a thousand years. As discussed before, this is exactly the age range favored by the r-mode scenario for the spin-down of the observed pulsars, and therefore advanced LIGO is a very promising machine for the detection of gravitational waves from r-mode oscillations of young NSs. Moreover, for extragalactic supernovae the sensitivity of advanced LIGO should be sufficient to see r-mode oscillations of a compact remnant in the immediate aftermath up to about a year. For sources as far as the Virgo Cluster of galaxies, as originally discussed in Owen et al. (1998), even with advanced LIGO a detection should only be possible under extremely lucky circumstances, and third-generation detectors like the Einstein telescope (Punturo et al. 2010) will be necessary to take advantage of the much larger number of more distant sources.

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Next consider our theoretical results in Figure 7, which are given for the different NSs listed in Table 2, namely, the 1.4 M_☉ and 2 M_☉ APR stars for which only modified Urca reactions are possible and the 2.2 M_☉ star where direct Urca reactions are present within its core. The steeply descending solid lines show the minimal signal emitted at the end of the r-mode spin-down evolution (where the star spends most of its time) as a function of the frequency, which in turn is a function of the unknown saturation amplitude. The dots show the result for fixed saturation amplitudes of different orders of magnitude ranging from α_sat = 1 (top) to α_sat = 10⁻⁴ (bottom). The dotted curves show the r-mode evolution for the 1.4 M_☉ model for these saturation amplitudes. As expected, the signal is larger and emitted at higher frequencies if the star is younger, but a star also spends far less time in these earlier stages. Although a star of a given age emits gravitational waves with a fixed strain amplitude, depending on the unknown saturation amplitude the frequency of these gravitational waves can be very different. As suggested by the analytic results, the strain depends little on the mass and the details of the composition, and the size of (suppressed) error bands on these curves would be given by the errors in prefactor of Equation (48). This insensitivity holds even for the case of the heavy star where direct Urca emission is possible. As noted before, because of the much stronger cooling in this case, the star spins down at a lower temperature where the boundary of the instability region is at a larger frequency, so that the lower gravitational wave cutoff frequency is considerably higher. However, up to this point the gravitational wave signal is nearly identical to the standard case with modified Urca cooling. This is an important observation since the data on Cas A (Heinke & Ho 2010), which is the only young star for which the cooling has been explicitly observed, clearly point to an enhanced neutrino emission mechanism. The standard explanation (Page et al. 2011; Yakovlev et al. 2011) is emission due to pair breaking of superfluid neutrons, which is in between the cases of modified and direct Urca emission discussed here.

The fact that nearly all known pulsars spin too slowly to emit gravitational waves due to r-modes at the moment requires us to consider other potential sources. Here we give the expected signal for a few examples of known possible sources that could be young enough to emit gravitational waves due to r-modes. These consist of, in addition to the fastest pulsar J0537-6910 discussed above, the young NS Cassiopeia A, from which no pulsation has been observed (Halpern & Gotthelf 2010), and several supernova remnants where no compact object has been detected so far. However, it has recently been suggested that the hard X-ray emission from SN 1957D results from a pulsar wind nebula and the compact remnant that produced it would then be the youngest indirectly observed pulsar (Long et al. 2012). We take into account both the age and the available distance estimate for these sources shown in Table 4. As noted before, the strain amplitude is fixed by these parameters, but the frequency can vary depending on the unknown r-mode saturation amplitude.

Table 4. Expected Gravitational Wave Signal and Expected Range of the Timing Parameters for Different Young Sources in Case There Is an Associated Fast Rotating Compact Object That Spins Down Via the r-mode Mechanism

Source	Age	D	ν	$\dot{\nu }$	h0	hc	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{LIGO}}^{\left({\rm obs}\right)}$	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{aLIGO}}^{\left({\rm obs}\right)}$	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{aLIGO}}^{\left({\rm tot}\right)}$
	(yr)	(kpc)	(Hz)	(10−9 s−2)
J0537-6910	?	52	83	−0.27	3.6 × 10−26	6.5(6.8) × 10−20	2.1(0.2)	28(2.8)	⋅⋅⋅
SN 1987A	25	51.5	[84, 992]	[ − 18, −265]	2.8 × 10−25	0.8(1.2) × 10−19	23(18)	221(236)	2.0(3.2) × 103
SN 1957D	55	4610	[86, 801]	[ − 8.3, −155]	2.1 × 10−27	0.9(1.3) × 10−21	0.2(0.1)	1.6(1.8)	19(33)
G1.9+0.3	≈140	7.7	[89, 545]	[ − 3.3, −59]	8.0 × 10−25	4.9(7.2) × 10−19	57(62)	598(662)	0.9(1.8) × 104
Cassiopeia A	≈300	3.4	[91, 346]	[ − 1.5, −19]	1.2 × 10−24	1.0(1.5) × 10−18	83(106)	906(1015)	1.3(3.7) × 104

Notes. The alternative sensitivity measures in the right part of the table are discussed in detail in Appendix B, and they generally strongly overestimate the detectability since they do not take into account our limited knowledge of the source or the limited observation time. All values are given for a 1.4 M_☉ APR star with α_sat = 0.1 (0.01); for α_sat = 1, r-modes would have already decayed at the current age of these systems. The observable signal-to-noise ratios are computed assuming an observation time Δt = 1 yr. The theoretical total signal-to-noise ratios are here obtained by taking into account the total frequency interval $[\nu _{f},\bar{\nu }]$ that is observable from the star's current age to the end of the r-mode spin-down.

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Because of the significant computational cost of a gravitational search, it is essential to know the spin-down model and the expected timing parameter range for a given source (LIGO Scientific Collaboration et al. 2009; LIGO Collaboration 2008). In gravitational wave searches, the expected signal is usually modeled by a generic spin-down model (Shapiro & Teukolsky 1983)

$$\begin{equation} \dot{f}\sim -f^{n}\quad \Rightarrow \quad \ddot{f}=n\dot{f}^{2}/f \end{equation} \tag{ 50 }$$

with a fixed braking index n. Since the spin-down evolution is coupled to the thermal evolution in the case of r-mode emission, up to now it might have seemed questionable whether the simple model (Equation (50)) is appropriate. However, our general analysis in Section 4 showed that the thermal evolution is always faster than the spin-down (Equation (24)) and can be solved in a first step. Therefore, the effective spin-down (Equation (34)) takes indeed the standard form (Equation (50)), yet with an effective r-mode braking index n_rm given in Equation (35). As discussed, the saturation model can change the braking index from the canonical value 7 to significantly smaller values, e.g., for mode coupling with damping due to shear viscosity (Bondarescu & Wasserman 2013), the braking index is considerably reduced to $n_{{\rm rm}}^{({\rm mc})} = 4$. This is important both for the search for gravitational waves and for the interpretation of the results in case a detection is made. Generally, a hierarchy of braking indices was expected, namely, 3 for electromagnetic dipole emission, 5 for gravitational waves from an ellipticity of the star (or electromagnetic quadrupole emission), and 7 for r-mode emission.¹⁷ Our results show that, analogous to electromagnetic emission, where the observed braking indices are below 3, the possible braking indices due to r-modes are likewise systematically lower and the uncertainty in the unknown saturation mechanism is even larger—to the point where even a clear discrimination between electromagnetic and gravitational wave spin-down becomes questionable. Therefore, it should be hard to determine the spin-down mechanism from current electromagnetic pulsar timing data. However, future gravitational wave observations could change this. Comprehensive information on the source, the gravitational wave amplitude, and the timing data could, due to the distinct r-mode braking indices (Equation (35)) and the quantitative prediction of the r-mode signal (Equation (45)), eventually even allow us to determine the r-mode saturation mechanism observationally.

As discussed before, the lower boundary of the frequency range of the expected gravitational wave signal is rather insensitive to the details of the source and set by where the r-mode evolution exits the instability region (see Figure 2). The corresponding α_sat can be obtained by equating Equation (38) to the known age of the source, which then determines the final frequency. In contrast, the upper boundary is not restricted by gravitational wave emission alone because the r-mode amplitude could in principle be arbitrarily small. Yet, in this case other spin-down mechanisms will dominate and set a limit on the frequency to which a potential pulsar would have been spun down at the age of a considered source. To estimate this effect, we assume again the spin-down model (Equation (50)) with a fixed braking index n. The proportionality constant is related to the characteristic age $t_{c}\equiv -(f/\dot{f})_{0}$ of a given pulsar, and the evolution is then given by

$$\begin{equation} f\left(t\right)\approx \left(\frac{1}{f_{i}^{n-1}}+\frac{\left(n-1\right)t}{t_{c}f_{0}^{n-1}}\right)^{-\frac{1}{n-1}}. \end{equation} \tag{ 51 }$$

For a conservative estimate of the upper limit we assume a braking index of n = 2.5 and choose the initial frequency f_i as the Kepler frequency of our 1.4 M_☉ model. Fiducially choosing the population of young stars as those in the ATNF database with spin-down rate $\dot{f} > 10^{-11}\,{\rm s^{-2}}$, the largest value for $t_{c}f_{0}^{n-1}$ is, as could have been expected, obtained for J0537-6910. Let us assume that the observed population of pulsars is characteristic for the spin-down behavior of so far unobserved sources. This then allows us to estimate an upper limit for the frequency to which other spin-down mechanisms would have spun down a (potentially unobserved) compact star in a considered source even when gravitational wave emission would be negligible for the spin-down. Finally, consider pulsars with a known timing solution, where both the frequency and the spin-down rate are known. If r-mode emission dominates the current spin-down, Equation (20) determines the saturation amplitude (see Figure 5), and in the case of J0537-6910 we find α_sat ≈ 0.084. If other spin-down mechanisms are important as well, α_sat and the resulting gravitational strain would be lower.

Our results are shown in Figure 8. Again the sources without known timing data should be compared to the thin curves, whereas the expected strain for the pulsar J0537-6910 should be compared to the thick curves. As can be seen, there should be a realistic chance to see Cassiopeia A, and potentially a compact remnant in the youngest known galactic supernova G1.9+0.3, from a coherent analysis of a month of LIGO S6 data. Even more strikingly, the frequency range where Cas A is above the detector sensitivity coincides with the expected frequency range for this source. For Cas A, a detailed previous LIGO analysis failed to detect a signal, but imposed strict bounds on a potential r-mode amplitude of this source (LIGO Scientific Collaboration et al. 2010). However, this was done with only 12 days of S5 data. As can be seen by the uppermost thin detector curve in Figure 8, which shows the S5 sensitivity for this shorter observation interval, we find that the signal is just at the detection limit, and even this only over a narrow frequency range, so that it is not too surprising that no detection was made. Cas A should therefore be a prime target for advanced LIGO and, according to our results, it should even be promising to perform another search of the S6 data if a coherent analysis of a whole month of data is computationally feasible. The signal of 1987A is about an order of magnitude smaller, but it would be in the reach of advanced LIGO, which could detect it over the entire expected frequency range (which is larger due to its young age). The youngest candidate pulsar, produced by SN 1957D, would be an ideal target, but it is unfortunately too distant to detect gravitational waves due to r-mode emission with advanced LIGO; more sensitive future detectors will be required. Our results show that as soon as the pulsar is directly detected and timing data become available, this source should be in the range of the forthcoming Einstein telescope (Punturo et al. 2010), which has an order of magnitude higher sensitivity compared to advanced LIGO. If against all odds the fastest directly observed young pulsar J0537-6910 is still in the final stage of its r-mode evolution, there is a good chance that its signal, which was below the threshold of the original LIGO detector, will be detected by advanced LIGO.

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The relevant ranges for the timing parameters, given in Table 4, should help to perform future searches. Since gravitational wave searches rely on the spin-down model (Equation (50)), we give a range estimate for the r-mode braking index n_rm instead of a range for the second frequency derivative. From the currently proposed spin-down mechanisms (Arras et al. 2003; Bondarescu & Wasserman 2013; Alford et al. 2012c; Haskell et al. 2013) we obtain the large range 1 ⩽ n_rm ⩽ 7, where intermediate to higher values are more likely for r-mode emission, so that the previously used range 2 ⩽ n_rm ⩽ 7 (LIGO Scientific Collaboration et al. 2010) presents a viable compromise. If r-mode emission dominates the spin-down, the frequency and spin-down rate are related by Equation (20) and the time variation over the observation interval is given by the semi-analytic solution (Equation (36)). In reality, where other spin-down mechanisms are present as well, there should nevertheless be a strong correlation between frequency and spin-down rate so that it is not necessary to search the complete range in frequency and spin-down rate, but a (narrow) parameter region around the ideal r-mode spin-down behavior should be sufficient. This could strongly reduce the required parameter range in a dedicated r-mode search. In the case of Cas A, the frequency range obtained here by our dynamical analysis roughly agrees with the previously used fiducial range 100 Hz ⩽ ν ⩽ 300 Hz chosen by sensitivity arguments (LIGO Scientific Collaboration et al. 2010). Finally, it is important to note that the saturation amplitudes corresponding to the above relevant gravitational frequency ranges, as denoted by the dots in Figure 8, are precisely in the range 10⁻² ≲ α_sat ≲ 10⁻¹, which is required for an explanation of the low frequencies of observed young pulsars within the r-mode scenario discussed in Section 5. This should provide an incentive to search for gravitational wave emission from r-modes of sufficiently young compact sources.

An alternative way to discuss r-mode emission is via the Fourier spectrum. In principle, this is interesting for periodic sources like NSs since the frequency hardly changes over the short observation interval. The Fourier-transformed gravitational wave strain $\tilde{h}\left(\nu \right)$ in the frequency domain can be obtained in a stationary phase approximation as

$$\begin{equation} |\tilde{h} (\nu )|^{2}=\left|h\!\left(t\right)\right|^{2}\left|\frac{dt}{d\nu }\right|. \end{equation} \tag{ B1 }$$

As seen from Figure 6, there are two qualitatively different stages of the gravitational wave emission. The first arises from the growth phase of the r-mode and results in a narrow spike in $\tilde{h}\!\left(\nu \right)$ at the star's initial frequency. This phase lasts only for a short time of order minutes, and as shown in Owen et al. (1998), there is not enough energy in this pulse to be detectable. Therefore, we neglect this initial part of the spectrum here. In the subsequent saturated phase where the star spins down it emits gravitational waves over a continuous frequency range with strain density

$$\begin{equation} \tilde{h}\!\left(\nu \right)=\sqrt{\frac{9\tilde{I}GMR^{2}}{20D^{2}\nu }}. \end{equation} \tag{ B2 }$$

The quantity that is usually used for a comparison with the detector sensitivity is the characteristic amplitude of the signal defined by $h_{c}\equiv \nu \,\tilde{h}$. In particular, the minimum characteristic amplitude h_{c, f} ≡ h_c(ν_f) reached at the lower frequency boundary of the evolution is interesting. The square root in Equation (B2) makes the dependence of h_{c, f} on the microscopic parameters even weaker than for the final frequency (Equation (30)). However, the expression depends more strongly on macroscopic quantities like the mass and the moment of inertia encoded in the parameter $\tilde{I}$. The rigorous bounds (Equation (13)) show that this dependence is also moderate since all these quantities vary only within roughly a factor of two. For a standard NS we find

$$\begin{eqnarray} h_{c,f}^{\left({\rm NS}\right)}&\approx& 1.6\times 10^{-22}\left(\!\frac{\tilde{S}}{\tilde{S}_{{\rm fid}}}\!\right)^{\!3/46}\!\!\left(\!\frac{\tilde{L}}{\tilde{L}_{{\rm fid}}}\!\right)^{\!5/368}\!\!\left(\!\frac{\tilde{J}}{\tilde{J}_{{\rm fid}}}\!\right)^{\!-29/184}\left(\!\frac{\tilde{I}}{\tilde{I}_{{\rm fid}}}\!\right)^{\!1/2}\nonumber \\ && \times \!\left(\!\frac{M}{1.4\, M_{\odot }}\!\right)^{\!63/184}\!\!\left(\!\frac{R}{11.5\,\mathrm{km}}\!\right)^{\!281/368}\!\!\left(\!\frac{\mathrm{Mpc}}{D}\!\right)\alpha _{{\rm sat}}^{-5/184}.\nonumber \\ \end{eqnarray} \tag{ B3 }$$

Note that this quantity is also nearly independent of the saturation amplitude α_sat (and even increases with decreasing amplitude). The same holds for the microscopic parameters. For example, the dependence on the neutrino emissivity involves the power 5/368, which provides a striking example of how insensitive such macroscopic observables can be to microscopically decisive yet inherently poorly known quantities, since even a drastic change by three orders of magnitude would only have an irrelevant effect of less than 10%.

The detector sensitivity is described by the rms strain noise $h_{{\rm rms}}\!\equiv \!\sqrt{\nu S_{h}\!\left(\nu \right)}$ in terms of the power spectral density S_h of the detector noise. In contrast to Equation (49), this quantity is only a measure of the general sensitivity of the detector but not of a particular search and contains information neither on the source nor on the observation interval. In order for the characteristic amplitude to give a meaningful estimate, the observation time should be comparable to the timescale of the frequency change of the signal (so that all strength of a given frequency is actually detected). Clearly, for small saturation amplitudes where the spin-down time increases drastically with decreasing α_sat (see Equation (40) and Table 3) this is not guaranteed, and therefore the fact that the characteristic amplitude is above the background alone is not sufficient for a detection.

The result for the characteristic amplitude in the saturated regime is given for different stars in Figure 9. Whereas the spectra of the two standard NSs with modified Urca reactions have nearly the same lower frequency cutoff, the heavy star with direct Urca interactions in the core has a higher cutoff >100 Hz, but otherwise features a characteristic amplitude of similar size, as could have been expected from the insensitivity to the neutrino emissivity in Equation (27).

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In Owen et al. (1998), the following expression for the signal-to-noise ratio for matched filtering was given:

$$\begin{equation} \left.\left(\frac{{\rm S}}{{\rm N}}\right)^{\!2}\right|^{\left({\rm tot.}\right)} = 2\int _{\nu _{{\rm min}}}^{\nu _{{\rm max}}}\!\frac{d\nu }{\nu }\left(\!\frac{h_{c}}{h_{{\rm rms}}}\!\right)^{\!2} = \frac{9I}{10D^{2}}\int _{\nu _{{\rm min}}}^{\nu _{{\rm max}}}\!\!\frac{d\nu }{\nu \, S_{h}\!\left(\nu \right)} \end{equation} \tag{ B4 }$$

in terms of the power spectral density S_h of the noise and ranging over a range from the final frequency of the r-mode evolution ν_min = ν_f to the initial frequency ν_max = ν_i at which the star is created. Here the label "(tot.)" has been added to indicate that the total frequency spectrum is included in this quantity. This point will be discussed in more detail below. In Equation (B4) logarithmic frequency intervals are weighted by 1/S_h, since at low frequencies the frequency changes slowly, which makes it easier to detect the periodic signal. After a numerical integration an approximate fit to the result over the relevant parameter ranges for ν_min and ν_max is given for the advanced LIGO detector by¹⁸

$$\begin{eqnarray} \left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{aLIGO}}^{\left({\rm tot.}\right)}&\approx & 4.09\times 10^{23}\sqrt{{\rm Hz}}\frac{\sqrt{GI}}{D}\nonumber\\ & \times& \sqrt{1+0.40\log \left(\frac{\nu _{{\rm max}}}{1000\,{\rm Hz}}\right)+0.36\frac{100\,{\rm Hz}-v_{{\rm min}}}{100\,{\rm Hz}}}. \nonumber\\ \end{eqnarray} \tag{ B5 }$$

Compared to the result given in Owen (2010) for the anticipated second-generation LIGO detector, we find a much weaker dependence on the minimum final frequency and a stronger dependence on the unknown maximum initial frequency. By inserting the result for the minimum frequency (Equation (30)), we see that the uncertainty of this result due to the microphysics is minor. The main uncertainty arises from the moment of inertia, which, using the above bounds, has a maximum theoretical uncertainty given in Equation (15).

The signal-to-noise ratios for different sources at different distances are shown in Table 5. Here we compare the values for sources given by the different considered stars listed in Table 2. These are assumed to be located at different distances, namely, in the Virgo Cluster (20 Mpc), the Local group of galaxies (1 Mpc), and within the Milky Way (30 kpc). Using the actual anticipated noise background of the advanced LIGO detector and without the assumption that the evolution stops at 10⁹ K, our values are slightly larger than those obtained in Owen et al. (1998). Heavier stars tend to give larger values due to the larger moment of inertia, whereas direct Urca processes reduce the signal-to-noise ratio since the thermal steady-state curve is at lower temperatures and its intersection with the boundary of the instability region is correspondingly at a higher frequency that sets the lower gravitational wave frequency cutoff in Equation (B5). However, overall these values are rather similar for the different considered stars. These considerations suggest that theoretically a detection is possible for sources up to large distances in case the gravitational wave signal could be detected over its complete frequency range. However, this is inherently impossible for a continuous r-mode signal.

Table 5. Results for the Spin-down and Gravitational Wave Observables for Different NSs Considered in This Work Obtained from the Semi-analytic Expressions (Equations (28), (27), (38), (46), (B3), and (B4)).

NS	Tf	ff	νf	tsd	hf @ 1 Mpc	hc, f @ 1 Mpc	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{Virgo}}^{\left({\rm tot}\right)}$	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{L.\, g.}}^{\left({\rm tot}\right)}$	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{M.\, w.}}^{\left({\rm tot}\right)}$	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{Virgo}}^{\left({\rm obs}\right)}$	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{L.\, g.}}^{\left({\rm obs}\right)}$	$\left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{M.\, w.}}^{\left({\rm obs}\right)}$
	(K)	(Hz)	(Hz)	(yr)
APR 1.4 M☉	1.26 × 109	61.4	81.8	12.3	7.2 × 10−27	1.6 × 10−22	11.7	234	7794	0.084	1.7	56
APR 2.0 M☉	1.39 × 109	60.3	80.4	4.9	9.7 × 10−27	1.8 × 10−22	14.0	281	9356	0.099	2.0	66
APR 2.21 M☉	3.15 × 108	84.0	112	1.7	2.2 × 10−26	2.1 × 10−22	12.7	254	8460	0.094	1.9	62

Notes. In contrast to the lower mass stars, the 2.21 M_☉ NS is dense enough to allow for direct Urca reactions. All values are given for a large saturation amplitude α_sat = 1. The signal-to-noise ratios are given for the advanced LIGO detector. The total values (tot; Equation (B4)) are obtained by including the entire gravitational wave spectrum with an upper cutoff determined by the kepler frequency. The observational values (obs; Equation (B8)) are given for a fiducial observation interval Δt/t = 10⁻³. They are given for sources at three different distances ranging from the Virgo Cluster (20 Mpc) and the local group of galaxies (1 Mpc) to a source within the Milky Way (30 kpc).

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The above total signal-to-noise ratios do not reflect the detectability of the signal of a given single source over the actual observation period, as noted already in Owen et al. (1998). They would provide such an actual measure for the detectability only for a short multi-wavelength signal that is completely observed, for example, a binary inspiral. However, these ratios cannot give a reasonable measure in the present case where the source is monochromatic and changes slowly with time while it is only observed for a short time interval, since the frequency ranges in Equation (B4) that would only be detectable hundreds of years before or after the present observation can hardly affect the detectability within the limited observation interval. Therefore, a proper signal-to-noise ratio that measures the detectability must take into account the finite duration of the observation and only include the relevant frequencies in Equation (B4).

To find a more realistic signal-to-noise ratio for the detectability in gravitational wave detectors, assume that the frequency of the source changes only slightly during the short observation time interval Δt over which it has an average frequency $\bar{\nu }$. The change $\Delta \nu \ll \bar{\nu }$ during this time interval can be obtained from Equation (20),

$$\begin{equation} \Delta \nu \approx \frac{2}{3\pi }\frac{d\Omega }{dt}\Delta t=\frac{2\bar{\nu }Q\alpha _{{\rm sat}}^{2}}{\tau _{G}\left(\bar{\Omega }\right)}\Delta t. \end{equation} \tag{ B6 }$$

The current angular velocity $\bar{\Omega }$ can be obtained in terms of the current age t of the star from the solution of the spin-down equation (Equation (20)), which yields in the limit $\Omega _{i}\gg \bar{\Omega }$

$$\begin{equation} \Delta \nu \approx -\frac{\bar{\nu }}{6}\frac{\Delta t}{t}. \end{equation} \tag{ B7 }$$

For $\Delta \nu \ll \bar{\nu }$ the integral in Equation (B4) gives an estimate for the actual observable signal-to-noise ratio,

$$\begin{equation} \left.\frac{{\rm S}}{{\rm N}}\right|^{\left({\rm obs.}\right)} \approx \sqrt{\frac{9\tilde{I}GMR^{2}}{10D^{2}}\frac{\Delta \nu }{\bar{\nu }S_{h}\!\left(\bar{\nu }\right)}}=\sqrt{\frac{3GI}{20S_{h}\!\left(\bar{\nu }\right)D^{2}}\frac{\Delta t}{t}}. \end{equation} \tag{ B8 }$$

Analogous to the equivalent intrinsic strain (Equation (49)), this quantity depends on the square root of the observation time interval Δt (see also Watts et al. 2008) and depends on the internal composition of the star only via the moment of inertia with uncertainty range (Equation (15)). Moreover, it depends only on the present (averaged) gravitational wave frequency $\bar{\nu }$ and not the complete spectrum. It is again not directly dependent on the saturation amplitude, which only enters indirectly via the dependence on the current frequency via $S_{h}\!\left(\bar{\nu }\right)$. Yet, we stress again that this expression is only valid during the later evolution where the spin-down becomes sufficiently slow so that $\Delta \nu \ll \bar{\nu }$. The right part of Table 5 shows the values for the advanced LIGO detector for the different stars and a ratio Δt/t = 10⁻³ compared to the previous signal-to-noise estimates. As can be seen in this case, the observable signal-to-noise ratios are more than two orders of magnitude smaller than the total expressions (Equation (B5)). Nevertheless, Equation (B8) still greatly overestimates the detectability since it does not take into account our limited information about the source.

For the advanced LIGO detector the background is nearly constant over the relevant range from 80 to 800 Hz with $S_{h}^{1/2}\approx 4\times 10^{-24}$ and thereby allows us to further approximate to obtain the simple estimate

$$\begin{equation} \left.\frac{{\rm S}}{{\rm N}}\right|_{\mathrm{aLIGO}}^{\left({\rm obs.}\right)}\approx 50\,\sqrt{\frac{\Delta t}{t}}\left(\frac{1\,\mathrm{Mpc}}{D}\right). \end{equation} \tag{ B9 }$$

Although this expression overestimates the detectability under realistic conditions, it might be useful for simple estimates, e.g., for a one-year observation period, comparable to past LIGO runs, a source in the Virgo Cluster (20 Mpc) would, in contrast to the early estimates in Owen (2010), be impossible to detect unless it is extraordinarily young. A source in our local group of galaxies (1 Mpc) in contrast might be detectable if the results obtained assuming matched filtering do not strongly overestimate the detectability for more realistic search methods. A source in the Milky Way (30 kpc) in contrast could be detectable even if a more realistic search strategy has a significantly lower signal-to-noise ratio. As discussed before, since the original LIGO detector had a more than an order of magnitude larger background, in contrast to previous estimates (Owen et al. 1998), it is not surprising that no signal has been detected so far.

Finally, let us estimate likely values for the above signal-to-noise ratios for the examples of possible sources that have been discussed in the main text. As can be seen in Table 4, despite the fact that the observable signal-to-noise ratio (Equation (B9)) is reduced by more than an order of magnitude compared to the total versions (Equation (B5)), the values are still significantly larger than what one would expect from the detailed analysis in terms of h₀ in the main text. Although the relative sizes for the individual sources are similar, the sensitivity (Equation (49)) that properly takes into account our ignorance of the details of the source leads to systematically lower sensitivities. Correspondingly, one should generally be careful with conclusions based on these alternative sensitivity estimates.