Gravitational wave emission from oscillating millisecond pulsars

Transcription

1 doi: /mnras/stu2361 Gravitational wave emission from oscillating millisecond pulsars Mark G. Alford and Kai Schwenzer Department of Physics, Washington University, St. Louis, MO 63130, USA Accepted 2014 November 4. Received 2014 October 30; in original form 2014 April 4 1 INTRODUCTION With the significant sensitivity improvement of forthcoming next generation gravitational wave detectors like the advanced Laser Interferometer Gravitational-Wave Observatory (LIGO; Harry 2010), advanced Virgo (Weinstein 2012) and the Large Scale Cryogenic Gravitational Wave Telescope (LCGT; Kuroda 2010) thereisare- alistic chance that gravitational waves may be directly observed. In addition to transient events such as neutron star and/or black hole mergers or supernovae, which require that an event happens sufficiently near to us during the observation period (Abadie et al. 2012), it is important to also consider continuous sources. Millisecond pulsars are a particularly promising class since they are very old and stable systems and therefore could be reliable sources of gravitational waves. Their fast rotation strongly favours gravitational wave emission (Aasi et al. 2014), and the fact that their timing behaviour is known to high precision (Manchester et al. 2005) 1 greatly simplifies the analysis required to find a signal in the detector data. Emission due to deformation of these objects ( mountains ), which is usually parametrized by an ellipticity, is the standard paradigm for continuous gravitational wave searches (Abbott et al. 2010), but global oscillation modes of a star can also emit copious gravitational waves that could be detectable if the oscillation reaches sufficiently high amplitudes. R-modes are the most interesting class (Andersson 1998; Lindblom, Owen & Morsink 1998; Owen et al. 1998; Andersson & Kokkotas 2001), because they are generically unstable in millisecond pulsars and therefore will be present unless the dissipative damping is strong enough. schwenzer@physics.wustl.edu 1 See: ABSTRACT Neutron stars undergoing r-mode oscillation emit gravitational radiation that might be detected on the Earth. For known millisecond pulsars the observed spin-down rate imposes an upper limit on the possible gravitational wave signal of these sources. Taking into account the physics of r-mode evolution, we show that only sources spinning at frequencies above a few hundred Hertz can be unstable to r-modes, and we derive a more stringent universal r-mode spin-down limit on their gravitational wave signal. We find that this refined bound limits the gravitational wave strain from millisecond pulsars to values below the detection sensitivity of next generation detectors. Young sources are therefore a more promising option for the detection of gravitational waves emitted by r-modes and to probe the interior composition of compact stars in the near future. Key words: asteroseismology gravitational waves stars: neutron pulsars: general. If r-modes arise in a spinning neutron star, they affect the spindown (since they cause the star to lose angular momentum via gravitational radiation) and the cooling (since the damping forces on the r-mode generate heat). To understand the interplay of these effects we have developed (Alford & Schwenzer 2013, 2014a) an effective description of the spin-down evolution where complicated details about the star s interior are absorbed into a few effective parameters. The resulting spin-down can be rather different from that predicted by simpler approaches, and includes strict bounds on the uncertainties in the final results. In this paper we will use this method to analyse the possible r-mode gravitational radiation of old neutron stars. First, however, we provide some background and motivation. R-modes can occur in young or old pulsars. In the case of young sources (Abbott et al. 2008; Wette et al. 2008; Abadie et al. 2010; Aasi et al. 2014) we have analysed their r-mode evolution (Alford & Schwenzer 2014a) and found that r-modes can provide a quantitative explanation for their observed low spin rates. Moreover, the r-mode gravitational emission is expected to be strong, because a large r-mode amplitude would be required to spin-down the known young pulsars to their current low spin frequencies within their lifetimes which are as short as a thousand years. These known pulsars are no longer in their r-mode spin-down epoch, but there may be unobserved young neutron stars, e.g. associated with known supernova remnants such as SN 1987A, that are currently undergoing r-mode spin-down, and several of them would be in the sensitivity range of advanced LIGO (Alford & Schwenzer 2014a), allowing this scenario to be falsified by future measurements. In this paper we focus on old neutron stars which have been spun up by accretion, and we perform an analysis of their expected r-mode gravitational wave radiation. In Alford & Schwenzer (2013) novel r-mode instability regions in spin-down timing parameter space C 2014 The Authors Published by Oxford University Press on behalf of the Royal Astronomical Society

2 3632 M. G. Alford and K. Schwenzer have been derived that allow us to decide if r-modes can be present in old millisecond radio pulsars. As with the thermal X-ray data of observed low-mass X-ray binaries (LMXBs;Haskell et al. 2012; Mahmoodifar & Strohmayer 2013), there are three distinct scenarios to explain the observed timing data. It might be that there is sufficient damping (e.g. currently poorly constrained or overlooked physical processes) that stops r-modes from growing, so that a given source is outside of the r-mode instability region. In this case there will be no r-mode gravitational radiation. The second possibility is the urated scenario where the low-amplitude damping is insufficient to damp r-modes and a given source is inside the instability region. Athird possibilityis theboundary-straddling scenario where during the evolution a source is trapped on the boundary (e.g. due to the presence of exotic forms of matter; Andersson, Jones & Kokkotas 2002; Reisenegger & Bonacic 2003). In the conventional picture of neutron stars with viscous damping most old millisecond pulsars will be inside the instability region and will be undergoing r-mode oscillations, since for expected r- mode uration amplitudes the dissipative heating ensures that fast spinning sources can neither cool nor spin out of the parameter region where r-modes are unstable (Alford & Schwenzer 2013). Yet, some slower spinning sources can escape the instability region and we will determine the limiting frequency. Therefore, there will be gravitational radiation from most old neutron stars in this scenario, and the main purpose of this paper is to find out whether it could be detected on the Earth. If enhanced damping, that increases with temperature, is present in a source, as is for instance the case for matter quark (Madsen 1998; Andersson et al. 2002; Schwenzer 2012), there can be an r-mode stability window that leads to the third scenario. We also discuss this scenario and show that the gravitational wave emission is similar to the previous case. The detectability of known continuous sources is generally described by the spin-down limit which is, for a specific source with known timing data, the maximum gravitational wave strain that can be emitted by that source. Despite the quite restrictive limits set by the spin-down data, the large spin frequencies of millisecond pulsars could nevertheless lead to a detectable signal. Present gravitational wave detectors like the original LIGO interferometer did not probe the spin-down limit for millisecond pulsars. However, next generation detectors including the advanced LIGO detector will be able to beat the spin-down limit for various sources. Therefore, it is interesting to assess the chance to detect gravitational emission from oscillating millisecond pulsars. We will introduce here the universal r-mode spin-down limit on the gravitational wave strain, which is more restrictive since it takes into account our understanding of the r-mode spin-down and the complete information we have about these systems. Whereas deformations of a given source depend on its evolutionary history and could therefore vary significantly from one source to another, for proposed uration mechanisms the r-mode uration amplitude proves to be rather insensitive to details of a particular source, like its mass or radius (Alford, Mahmoodifar & Schwenzer 2012b; Bondarescu & Wasserman 2013; Haskell, Glampedakis & Andersson2014). The expected gravitational wave strain of a given source can then be strongly constrained by the timing data of the entire set of millisecond pulsars. Using our semi-analytic approach to pulsar evolution, and assuming that the same uration and cooling mechanism (with given power-law dependence on temperature) operates in all the stars, we can then obtain the universal limit in the urated scenario given in equation (12), or the equivalent expression equation (18) in the boundary-straddling scenario. We will see that these are considerably below the standard spin-down limit, indicating that it will be harder than previously expected to see r-mode gravitational waves from these sources. 2 R-MODE SPIN-DOWN OF MILLISECOND PULSARS As described in Alford & Schwenzer (2013, 2014a) the r-mode evolution (Owen et al. 1998) can be discussed within an effective description, which relies on the fact that a compact star appears effectively as a point source and that the relevant material properties integrated over the star have simple power-law dependencies on the macroscopic observables that change during the evolution. The relevant macroscopic quantities are the power P G fedintothermode due to the gravitational wave emission, the dissipated power P D that heats the star and the thermal luminosity L that cools the star: P G = Ĝ 8 α 2,P D = ˆDT δ ψ α φ,l= ˆLT θ, (1) in terms of the rotational angular velocity = 2πf, the core temperature T of the star and the dimensionless r-mode amplitude α defined in Lindblom et al. (1998) and Alford & Schwenzer (2014a). The explicit form of the pre-factors Ĝ, ˆD and ˆL for different damping and cooling mechanisms (Alford & Schwenzer 2013) is given in Table A1 in Appendix A. In the following we study the r-mode evolution when r-modes are unstable because a source is inside the instability region, where the fast r-mode growth has to be stopped by a non-linear dissipative uration mechanism. The boundary-straddling scenario where sources are trapped on the boundary of a stability window can lead to a qualitatively different evolution and we will study this case in detail in Section 4. Even though there are several interesting proposals (Rezzolla, Lamb & Shapiro 2000; Lindblom, Tohline & Vallisneri 2001; Wu, Matzner & Arras 2001; Lin & Suen 2006; Alfordetal. 2012b; Bondarescu & Wasserman 2013; Haskell, Glampedakis & Andersson 2014) it is not yet settled which mechanism will dominate and urate r-modes. For millisecond pulsars we expect moderate uration amplitudes, in which case the pulsar spin-down is determined by the equation (Owen et al. 1998) d dt = 3Ĝα2 (T, ) 7, (2) I in terms of the moment of inertia of the star I and the r-mode uration amplitude α. The observed total spin-down rate will in general be larger since in addition to r-modes there are other spin-down mechanisms, like in particular magnetic dipole radiation, given by the ellipsis. Nevertheless, by assuming that the observed spin-down rate is entirely due to r-modes, observed pulsar timing data allow one to give upper bounds on the r-mode uration amplitude. These bounds are shown for the observed radio pulsars included in the Australia Telescope National Facility (ATNF) data base (Manchester et al. 2005) in Fig. 1 and they require very low uration amplitudes, 10 7 α 10 5 (Alford & Schwenzer 2013; Bondarescu & Wasserman 2013). Similar low values are obtained from pulsars in LMXBs (Alford & Schwenzer 2013; Mahmoodifar & Strohmayer 2013). Moreover, one can see in Fig. 1 that faster spinning sources generally set more stringent bounds on the uration amplitude of r-modes in the considered pulsar. The r-mode uration amplitude can in general depend both on the temperature and the frequency of the star. We use a general parametrization of the uration amplitude with a power-law form α (T, ) = ˆα T β γ, (3)

3 GW emission from oscillating millisecond pulsars f Hz Figure 1. Upper bounds on the r-mode uration amplitude arising from the observed spin-down of the pulsars in the ATNF data base (Manchester et al. 2005). The strongest bound α is obtained for the 533 Hz pulsar J which has a spin-down rate ḟ s 2 (large triangle at lower right, red online). as realized for the proposed uration mechanisms (Wu et al. 2001; Alford et al. 2012b; Bondarescu & Wasserman 2013; Haskell et al. 2014). Here the exponents are fixed (rational) numbers determined by the uration mechanism, whereas the reduced amplitude ˆα is less well known and can also depend on parameters of the particular source, like the mass or the radius. Using this general approach it was found in Alford & Schwenzer (2014a) that the r-mode heating is significant even for small amplitude modes and the thermal evolution is systematically faster than the spin-down. Therefore the star reaches a thermal steady state where the r-mode is urated and the dissipative r-mode heating balances the cooling due to photons and neutrinos. The urated steady state temperature is given by T hc = ( Ĝ ˆα 2 8+2γ ˆL 500 ) 1 θ 2β. (4) This leads to a spin-down equation along this steady state curve (Alford & Schwenzer 2013, 2014a): d dt = 3Ĝθ/(θ 2β) ˆα 2θ/(θ 2β) I ˆL nrm, (5) 2β/(θ 2β) with an effective braking index ( ) 1+2γ/7+2β/(7θ) n rm = 7 (6) 1 2β/θ that depends on the uration mechanism and can be rather different from the generic r-mode spin-down exponent 7. The r-mode urated spin-down equation has the solution ( ) 1 1/z (t) = ( i ) z + 3z Ĝθ ˆα 2θ θ 2β t t i, 2 ˆL 2β I z 2(3 + γ )θ + 4β. (7) θ 2β Figure 2. Schematic evolution of recycled radio pulsars which have been spun up and heated by accretion. Whereas the cooling (horizontal segments) takes less than a million years, the slow spin-down along the steady state curve takes longer than a billion years. The occurrence of a long r-mode spin-down epoch is determined by whether the frequency when accretion ends is below (B) or above (C) the universal r-mode frequency bound f. See the text for details. This solution has two limits. At early times the first term in the parenthesis of equation (7) dominates, so that the star hardly spins down, i.e. i, and at late times the second term dominates, so that the spin-down becomes independent of the initial angular velocity i. The crossover point between these two regimes is determined by the reduced uration amplitude ˆα. For young pulsars the i -independent late time behaviour of the spin-down law is relevant. For some old millisecond pulsars the spin-down rate is so low that the early time regime is realized and they hardly spin-down even over their billion years age. The r-mode evolution equation (7) takes place unless the dissipation, which depends strongly on temperature and frequency, is strong enough to completely damp r-modes. R-modes are only unstable at sufficiently high frequencies: a typical instability region for a neutron star with standard damping mechanisms in a T diagram (Lindblom et al. 1998) is shown in Fig. 2. By standard damping we mean well-established mechanisms, 2 namely shear viscosity due to leptonic and hadronic scattering (Shternin & Yakovlev 2008) and bulk viscosity due to modified Urca reactions (Sawyer 1989). Alford, Mahmoodifar & Schwenzer (2012a) give a general semianalytic expression for the minimum frequency min down to which r-modes can be unstable, and shows that this limit is extremely insensitive to unknown details of the source and the microphysics (see also Lindblom et al. 1998). Fig.2 also shows two qualitatively different evolution trajectories. An accreting millisecond pulsar entering the instability region at point A in Fig. 2 is slowly spun up and kept warm by accretion in a binary system, following the thick vertical line. The r-mode evolution, equation (7), starts when accretion stops. This may occur when the star is spinning slowly (B) or 2 A potential Ekman layer at the crust-core boundary (Lindblom, Owen & Ushomirsky 2000) does not qualitatively change this picture (Alford & Schwenzer 2013).

4 3634 M. G. Alford and K. Schwenzer quickly (C). In either case, even though the star is in the region of T space where according to the standard damping mechanism r-modes are unstable, the star then cools faster than it spins down (following the thin horizontal lines). If accretion brought the star to a high spin frequency (C), then the star cools until it reaches the steady state line (dashed line, given by equation 4) at point (D); it then slowly spins down, following the steady state line, and would only reach the boundary of the instability (E) after time-scales that are longer than the age of known sources. Therefore, we expect such a source not too far below point (D). Fig. 2 shows a steady-state line for low r-mode uration amplitude, in which case the line is high enough that it exits the instability region at a frequency f which is significantly above the minimum frequency min.thismeansthat if accretion leaves the star with a low spin frequency (B), below f, then the star cools in less than a million years (Yakovlev & Pethick 2004) and reaches the boundary of the instability region (F). The value of f is f = ( ˆD θ 2β ˆα 2δ Ĝ θ δ 2β ˆL δ ) 1/(6θ 8δ 12β 2δγ). (8) It was shown in Alford & Schwenzer (2013, 2014a,b) that this expression is extremely insensitive to the microphysical details. Whereas for young sources discussed in Alford & Schwenzer (2014a) neutrino emission is the relevant cooling mechanism to determine the final spin-down frequency, for the low uration amplitudes relevant for millisecond pulsars photon cooling from the surface and damping due to shear viscosity dominates (Alford & Schwenzer 2013) in equation (8). For a given upper bound on ˆα, all sources below this universal r-mode frequency bound f cannot be undergoing r-mode spin-down since they either have been spun out of the instability region or cooled out of it in less than a million years, which is considerably shorter than their billion years age. In contrast, all sources spinning faster than f must be undergoing r-mode spin-down (i.e. they are on the steady-state curve in Fig. 2). The fastest spinning sources (f 600 Hz) could have only left the instability region if the uration amplitude would be as low as α (Alford & Schwenzer 2013, 2014b), which is orders of magnitude below what proposed uration mechanisms can provide (Wu et al. 2001;Alford et al. 2012b; Bondarescu & Wasserman 2013; Haskell et al. 2014). We conclude that fast spinning sources should be emitting gravitational waves via r-mode spin-down and we will determine the required spin frequencies below. Note that the evolution scenario in Fig. 2 is qualitatively different from previous expectations that assumed standard viscous damping and a large uration amplitude, see e.g. Levin (1999). There it was proposed that sources slowly spin-up at low temperatures outside of the instability region (where cooling becomes slow) followed by quick r-mode heating once the source enters the instability region and similarly fast spin-down and cooling segments which complete a cycle. There would then be no gravitational wave emission from known radio pulsars since sources would leave the instability region very quickly (Levin 1999). However, this scenario is not compatible with the well-established hypothesis that LMXBs are the progenitors of millisecond pulsars, because their observed large temperatures and frequencies (Haskell et al. 2012) place them firmly inside the instability region, as shown e.g. in fig. 1 of Alford & Schwenzer (2013). These astrophysical observations are furthermore theoretically explained by deep crustal heating due to pycnonuclear reactions (Brown, Bildsten & Rutledge 1998). For large r-mode uration amplitudes LMXBs either could not spin-up to the high frequencies of observed radio pulsars or would be spun down very quickly, which should rule out this scenario. A modified scenario that might account for the observed spin limit of pulsars would be the presence of enhanced damping that does not increase with temperature, for instance due to a viscous boundary layer (Andersson et al. 2000) or mutual friction in a superfluid and superconducting core (Haskell, Andersson & Passamonti 2009; Haskell et al. 2012). In this case r-modes could be completely damped up to the frequencies of observed radio pulsars and it would be very unlikely that any of the known pulsars currently emits gravitational waves (Andersson et al. 2000). However, estimates on mutual friction are still rather uncertain (Haskell et al. 2009, 2012), and we have recently shown that even in the benevolent scenario of a thin viscous boundary layer the damping is very likely not sufficient to explain the fastest spinning sources (Alford & Schwenzer 2013). In Fig. 2 we show the case that crustal heating dominates r-mode heating whereby a source spins up along a vertical trajectory. If r- mode heating dominates, a source would spin-up along roughly the same trajectory along which it subsequently spins down. Another option is that the spin-up stalls since a steady state is reached where the accretion spin-up is balanced by r-mode spin-down (Wagoner 2002). However these different scenarios during the LMXB phase lead to the same qualitative evolution once the accretion ends. Another important aspect is that the evolution scenario in Fig. 2 is not affected by other potential spin-down mechanisms, like in particular magnetic dipole radiation. The spin-down path is completely determined by the thermal steady state which is not affected by other spin-down mechanisms since they generically do not affect the thermal equilibrium of the star. Additional spin-down mechanisms would, however, increase the spin-down rate and therefore we can only give upper limits on the gravitational wave emission below. 3 GRAVITATIONAL WAVE STRAIN R-modes emit gravitational waves due to their time varying current quadrupole moment. The gravitational wave frequency ν emitted by the dominant fundamental (m = 2) r-mode is related to the rotational angular velocity via ν = 2/(3π) (Owen et al. 1998). This is the canonical relation for a non-relativistic slowly rotating star but in a relativistic analysis of a star with a realistic equation of state the gravitational wave frequency can be slightly larger (Idrisy, Owen & Jones 2014), which should be kept in mind below where we give our results using the canonical values. The gravitational wave signal of a given source is described by the intrinsic gravitational wave strain of the detector, which describes the expected signal in a terrestrial detector and can directly be compared to the detector noise. For r-modes it takes the form (Owen 2010) 2 15 π h 0 = 7 JGMR 3 ν 3 α, (9) 5 D where D is the distance to the source. In a recent study of the gravitational wave emission of young sources (Alford & Schwenzer 2014a) it was found that for the large amplitudes required to explain the low spin frequencies of young pulsars, the late time behaviour of the spin-down evolution (equation 7) is relevant and in this case the strain (equation 9) depends only on the age and the distance of the source Wette et al. (2008), and is independent of the uration amplitude. Because of the restrictive bounds on the uration amplitude shown in Fig. 1, for some old radio pulsars the early time limit of the evolution is relevant, where the frequency barely changes and the strain depends linearly on the uration amplitude. In Fig. 2 such a source stays close to its starting point D on the

5 GW emission from oscillating millisecond pulsars 3635 Figure 3. The time evolution of the emitted intrinsic gravitational wave strain amplitude (solid curves) and the endpoints of the gravitational wave emission (dots) (Alford & Schwenzer 2014a) shown for different uration amplitudes and for a fiducial source located at a distance of 1 kpc. The universal late time behaviour is also shown (dotted line). spin-down curve for more than a billion years. However, in general the time evolution is relevant and has to be taken into account. This can be seen in Fig. 3 where the evolution of the gravitational wave strain is shown for uration amplitudes relevant for millisecond pulsars. The dots also show for various amplitudes the end of the gravitational wave emission, where the source spins slowly enough that the r-mode is damped. As seen, for α < 10 5 the time needed for a star to spin out of the r-mode instability region is considerably more than a billion years, longer than the age of these sources. 3.1 Standard r-mode spin-down limit Using the spin-down equation (2) to eliminate the r-mode uration amplitude α in equation (9), i.e. employing the values given in Fig. 1, yields the spin-down limit 3 (Owen 2010;Aasietal.2014): h (sl) 15 GI 0 = 4 D 2, (10) which provides an upper bound that is urated when the entire rotational energy loss is due to the gravitational wave emission and accompanying dissipation caused by r-modes. The spin-down limits for the observed radio pulsar data (Manchester et al. 2005) are shown in Fig. 4 and are represented by inverted triangles. Here the spin-down limits from r-modes (solid triangles) are compared to those for the typically considered case of elliptic star deformations (open triangles), which has recently been studied in detail by the LIGO collaboration (Aasi et al. 2014). R-modes lead to a slightly higher strain but at a lower frequency (Owen 2010). These limits 3 The expression given here is slightly smaller than the estimate given in Owen (2010), since the rotational energy loss goes not entirely into gravitational waves but is partly dissipated to urate the r-mode at a finite amplitude. Figure 4. The standard spin-down limits of known radio pulsars compared to the characteristic strain amplitude for different detector configurations assuming a coherent analysis of a year of data. Open (magenta) triangles show the limits for the standard case of elliptic deformations of the star (Aasi et al. 2014) and filled (red) triangles for the case of r-mode gravitational wave emission. The solid (grey) curve gives the sensitivity of the original LIGO detector and the dashed (blue) and dot dashed (green) curves show the sensitivity of the advanced LIGO detector in the standard and neutron star enhanced mode. The vertical line shows the limiting frequency below which r-modes are absent in a neutron star and the shaded band gives the uncertainty on it using the semi-analytic result (Alford et al. 2012a) and the ranges for the underlying parameters used in Alford & Schwenzer (2014a). are also compared to the detector sensitivity at 95 per cent confidence limit (curves), given by h S h / t (Aasi et al. 95 per cent 2014) in terms of the spectral density S h of the detector strain noise and the observation interval t. To assess Fig. 4 it is important to recall that r-modes are only unstable at sufficiently large frequencies. The lowest frequency at which r-modes are unstable ( min in Fig. 2) (Lindblom et al. 1998; Alfordetal.2012a) shown by the vertical line, sets a strict frequency limit below which no r-mode gravitational wave emission is possible. Despite this restriction, the figure shows that the spin-down limit for several millisecond radio pulsars should be beaten by the advanced LIGO detector. However, even though the pulsars J and J could be significantly above the detector sensitivity they are not promising sources. The pulsar J is actually a young pulsar that has been analysed in detail in Alford & Schwenzer (2014a) where it is shown that although it is slightly above the minimum frequency of the instability region, it is very likely outside of it. The f = 174 Hz pulsar J is the closest and brightest millisecond pulsar and therefore would be a natural target. However, this is also the only non-accreting source for which a temperature estimate is available (Haskell et al. 2012), and this shows that it is outside of the instability region and similarly cannot emit gravitational waves due to r-modes. Moreover, its frequency is below likely values of f (equation 8) so, as we discussed at the end of Section 2 and will analyse in more detail below, it ought to have already cooled out of the instability region for a neutron star with standard damping mechanisms (Alford & Schwenzer 2013). Thinking beyond the advanced LIGO sensitivity

6 3636 M. G. Alford and K. Schwenzer thresholds shown in Fig. 4, we note that planned detectors like the Einstein telescope, which has an order of magnitude higher sensitivity, would be able to detect the gravitational waves that would be emitted from many sources, if r-modes are responsible for the better part of their observed spin-down rate. 3.2 Universal r-mode spin-down limit The spin-down limit for a particular source only takes into account information about that source. Here we will derive a more restrictive limit taking into account the entire data set of radio pulsars. It is based on the observation that proposed r-mode uration mechanisms are very insensitive to the details of a particular source (Alfordet al. 2012b; Bondarescu & Wasserman 2013; Haskell et al. 2014). To make this statement quantitative, we factorize the reduced uration amplitude given in equation (3) by writing ˆα = ˆα (mic) ˆα (mac), (11) where ˆα (mic) depends on the microphysics of the uration mechanism and is source independent, and α (mac) depends on the macroscopic properties of a specific source (mass, radius, etc.), 4 which generically only vary within narrow margins. For a given uration mechanism determined by β and γ in equation (3) and a particular source dependence encoded in ˆα (mac) we can then use equation (5) to determine the reduced microscopic uration amplitudes ˆα (mic) from given pulsar timing data. The smallest value obtained from the entire data set, which is realized for a particular source with frequency f 0 and spin-down rate ḟ 0, can then be used to give a limit for a general source, spinning with frequency f, using equations (4), (5) and (9). We find the universal r-mode spin-down limit h (usl) 15 GI ( ) ḟ 1 0 ˆα (mac) 1 2β/θ ( ) 3+γ +2β/θ f 1 2β/θ 0 = 4 D 2 f 0 ˆα (mac). (12) f,0 0 The first factor is just the standard spin-down limit (equation 10) for the source with the strongest bound on the reduced microscopic uration amplitude, whereas the two others are correction factors involving information on the source to which this limit applies, with exponents β, γ, θ from equations (1) and (3). Note that this result is independent of the details of the cooling mechanism encoded in ˆL although the effective spin-down law (equation 5) that was used to obtain equation (12) depends on it. It depends on the power law exponent θ of the cooling luminosity via the factor 2β/θ, which takes different values depending on whether the cooling is dominated by neutrinos (θ = 8 for modified Urca cooling) or photons (θ = 4). Recently it has been shown (Alford & Schwenzer 2013) that unless the dissipation is so strong that r-modes are completely damped away, radio pulsars should be surprisingly hot due to the strong heating from r-mode dissipation. Therefore, both photon and neutrino cooling can be relevant for observed radio pulsars. Since the detailed properties of particular sources are generally unknown the ratio of the macroscopic parts of the reduced uration amplitudes in equation (12) can only be estimated. However, from our theoretical understanding of compact stars (possible range of masses, radii, 4 To make the factorization unambiguous, we will use the convention that once a set of macroscopic parameters have been chosen, the sourcedependent macroscopic part α (mac) consists of powers of those parameters with no multiplicative pre-factor. For the uration mechanisms considered here there is a known set of macroscopic parameters (mass, radius, etc.) but for generality we do not limit ourselves by writing α (mac) explicitly in terms of them. etc.) we can, for a given uration mechanism, determine bounds on this unknown factor that are tight enough that the universal spindown limit is still considerably more restrictive than the standard spin-down limit. We should note again that the limit (equation 12) applies in the case of standard neutron stars without enhanced damping where r-modes will be present, whereas in the case of enhanced damping, e.g. due to exotic forms of matter (Andersson et al. 2002; Schwenzer 2012) or superfluidity (Gusakov, Chugunov & Kantor 2014), r-modes could be more strongly damped, and this case is discussed in Section 4. The simplest and most often used toy model for r-mode uration (Owen et al. 1998) assumes a constant uration amplitude that is independent of both temperature and frequency, so β = γ = 0. Although realistic models based on an explicit physical uration mechanism have a more complicated dependence this simple case is useful for illustrative purposes. In this case the uration amplitude is also assumed to be independent of the source so α = ˆα (mic), which is given in Fig. 1. The strongest limit α is obtained for the fast pulsar J with f Hz and ḟ s 1. Using this bound in equation (8) shows that in the constant uration model r-mode gravitational wave emission can only be present in sources spinning with frequencies f 225 Hz corresponding to gravitational wave frequencies ν 300 Hz, since slower spinning sources would have left the r-mode instability region (see Fig. 2 and the accompanying discussion). For a consistent evaluation of the universal spin-down limit the frequency f 0 of the comparison source should obviously be larger than this frequency bound, which is the case here. The expression for the universal spindown limit shows that the bounds for other sources scale in this case as (f/f 0 ) 3. Therefore the universal spin-down limits are significantly lower than the standard spin-down limits since the uration amplitude obtained from the entire data set is lower and the frequencies of most sources are lower than f 0. This is shown in Fig. 5 which compares the universal spin-down limits for the constant uration amplitude model (circles) to the standard spin-down limits (triangles) given before in Fig. 4. The dashed vertical line in Fig. 5 gives the universal r-mode frequency bound (equation 8) below which r-modes cannot be present. Therefore slower spinning sources, which appeared to be rather promising when only taking into account the standard spin-down limit, are entirely excluded, as is denoted by the open symbols. But even for faster spinning sources the universal spin-down limits can be orders of magnitude smaller. Therefore, all limits for this uration mechanism are considerably below the estimated sensitivity of advanced LIGO. R-mode uration amplitudes obtained from realistic mechanisms have a temperature and/or frequency dependence. Moreover, the power law exponents that are found are generally negative and of order one. As an important realistic example we discuss the uration due to mode coupling and the subsequent damping of the daughter modes (Arras et al. 2003). The uration amplitude from mode coupling has recently been revised (Bondarescu & Wasserman 2013), taking into account that the dominant damping source for daughter modes in a neutron star is likely shear viscosity instead of the previously assumed boundary layer damping. The revised uration amplitude could be low enough to be compatible with the restrictive bounds from the observed small spin-down rates given in Fig. 1. In the case of a star with an impermeable crust the uration amplitude is given by (Bondarescu & Wasserman 2013) α = C R = K J κ D 1 JR T f (13)

7 GW emission from oscillating millisecond pulsars 3637 corresponding frequency below which r-modes are excluded is shown by the dashed vertical line. As can be seen the universal spin-down limits are above those for the constant uration model (circles), but in most cases still significantly below both the standard spin-down limits and for all sources they are below the sensitivity of the advanced LIGO detector. For some fast spinning sources the standard spin-down limit is more restrictive but these are far below the advanced LIGO sensitivity. Figure 5. Comparison of different upper bounds on the strain amplitude of known radio pulsars due to r-mode emission. The spin-down limit (red triangles) is obtained from the timing data of an individual source. The universal spin-down limit takes into account that the uration mechanism applies to the entire class of millisecond pulsars, and also provides a lower bound (equation 8) on the frequency. We show universal spin-down limits (green circles) and minimum frequency (green dotted vertical line with uncertainty band) for the toy model of a constant r-mode uration amplitude (Owen et al. 1998). We also show universal spin-down limits (blue rectangles) and minimum frequency (blue dashed vertical line with uncertainty band) for a realistic uration mechanism arising from mode coupling and the damping of the daughter modes by shear viscosity (Bondarescu & Wasserman 2013). For a given uration mechanism, stars below the minimum frequency (open symbols) do not undergo r-mode oscillation. Here the first parenthesis represents ˆα (mic), which can be determined from the spin-down data. The lowest bound on ˆα (mic) is obtained from the f 0 = 336 Hz radio pulsar J Using this bound in equation (8), in the mode coupling model r-modes are only present in sources that spin with frequencies f 160 Hz corresponding to gravitational wave frequencies ν 215 Hz. In the mode-coupling mechanism the uration amplitude has a strong temperature and density dependence which gives a weaker scaling of the universal spin-down limit. As seen in equation (12) the scaling ranges from (f/f 0 ) 3/2 if neutrino cooling dominates to f/f 0 if photon cooling is dominant. To obtain rigorous upper bounds we are conservative and use for each source the weaker of the two constraints obtained from neutrino and photon cooling. The second parenthesis in equation (13) is the source-dependent factor ˆα (mac) which is unknown. To estimate the uncertainty in this factor, we note that the radius of a neutron star R 10 (in units of 10 km) is at present uncertain within 1 R and the factor J, defined in Owen et al. (1998), has been shown in Alford & Schwenzer (2014a) to be strictly bounded within 1/(20π) J 3/(28π). In reality the different macroscopic parameters are not fully independent since they arise from the same solution of the Oppenheimer Volkoff equations. Assuming that they are independent, which can only overestimate the uncertainty on the universal spin-down limit from the source-dependent factor in equation (12), we find ˆα (mac) / ˆα (mac), Including this uncertainty, the results for the universal spin-down limit for uration due to mode coupling are shown as well in Fig. 5 (rectangles) and the 4 SPIN-DOWN ALONG THE BOUNDARY OF A STABILITY WINDOW When enhanced damping is present in a star there can be a stability window where the r-mode is stable up to large frequencies (Madsen 1998; Nayyar & Owen 2006; Haskell & Andersson 2010; Alford et al. 2012a). One example for such a situation is the bulk viscosity damping in hyperonic (Jones 2001; Haensel, Levenfish & Yakovlev 2002) or quark matter (Madsen 1992; Schwenzer 2012), where the resonant temperature which is in standard neutron matter too high to be relevant can be in the astrophysically interesting range. Another extreme example is the recently proposed mode resonance mechanism (Gusakov et al. 2014) where r-modes are stable in a very narrow spike centred at a resonance temperature. Fig. 6 shows an example of how an evolution might occur in the presence of a large stability window, which is for instance realized for quark matter. The source is spun up by accretion from some initial frequency (A) within the stability window and heated by nuclear processes in the crust (Brown et al. 1998). Once the accretion stops (B) it cools and reaches the unstable region (C). In contrast to the situation discussed before, such a scenario in general does not require a strong uration mechanism and the uration amplitude within the instability region can be arbitrarily large. In this case the urated thermal steady state curve is located at large temperatures and the evolution cannot reach the steady state and then oscillates around the low temperature boundary of the stability window, since Figure 6. Schematic evolution of recycled radio pulsars which have been spun up and heated by accretion when a (large) stability window is present. Whereas the cooling (horizontal segments) takes less than a million years, the slow spin-down along the instability boundary takes longer than a billion years.

8 3638 M. G. Alford and K. Schwenzer the strong r-mode dissipation heats the source out of the unstable region, whereas the radiative emission cools it back in again. The magnitude of this oscillation of the r-mode amplitude decreases as the source evolves (Andersson et al. 2002) so that eventually a dynamic boundary-straddling steady state is reached where the heating due to r-modes with a dynamic amplitude α str <α is balanced by cooling (Reisenegger & Bonacic 2003). In the shortinitial oscillatory phase there is strong intermittent emission of gravitational waves, however, in the subsequent spin-down the emission is weaker but continuous. The spin-down can last even longer than in the urated case (Fig. 2), until the r-mode finally decays at the minimum of the instability region (D). Generalizing the result in Alford et al. (2012a), the minimum frequency of the instability region between a damping mechanism that dominates at lower temperatures (l) and one that dominates at higher temperatures (h) is given by ( ) δh ( ) δl 1/((8 ψ l )δ h (8 ψ h )δ l ) min =. (14) 2 ˆD l Ĝ 2 ˆD h Ĝ In general the lower mechanism is shear viscosity which has a negative exponent δ l < 0. Sources that spin with frequencies below min will not emit gravitational waves. Therefore, the minimum frequency min is equivalent to the universal frequency bound f (equation 8) in the case of the urated r-mode evolution. Both for quark matter (Madsen 1998; Alford et al. 2012a; Schwenzer 2012) and hyperonic matter (Haskell & Andersson 2010; Owen 2010), as well as for mode resonance models (Gusakov et al. 2014), min is roughly in the range of frequencies f obtained for different uration mechanisms in Fig. 5. The detailed form and position of the stability window depend strongly on the particular damping mechanism. In Fig. 6 the window is rather broad and extends to large frequencies, but both a very narrow window (Gusakov et al. 2014) or one that extends only to much lower frequencies (Alford et al. 2012a) are possible. In general, a spin-down along the boundary requires that in the considered frequency range the urated thermal steady state curve runs at higher temperatures than the left boundary of the stability window. Whether a urated steady state or a boundary-straddling stage is reached depends further on the initial condition of the evolution and there can be different evolutionary scenarios. If accretion heating is smaller than expected (Brown et al. 1998) sothatlmxbsare dominantly heated by r-modes, the evolution can start to the left of the stability window and sources can be trapped at the lower boundary of the stability window already during spin-up (Gusakov et al. 2014). In contrast, if the evolution first follows along the urated thermal steady state curve and the latter crosses the upper boundary of the stability window, a source can eventually evolve towards the boundary-straddling stage. For the large instability window in Fig. 6 this would for instance be the case for young and hot, rapidly spinning sources, but for different stability windows it might occur at other stages in the evolution. Finally, if the urated thermal steady state curve crosses the low-temperature boundary of the instability window, the evolution can eventually follow the steady state curve again. The parametric power-law dependence of the boundarystraddling amplitude is completely determined by the thermal steady state at the boundary (Reisenegger & Bonacic 2003) andisin general explicitly given by α str = ˆLĜ θ/δ 1 ˆD θ/δ γ, (15) where for a given damping mechanism the frequency exponent γ = (8 ψ)θ/(2δ) 4 is uniquely determined by the radiative cooling and by the (low amplitude) damping mechanism that leads to the stability window. In quark matter the presence of unscreened long-range gauge interactions introduces additional subleading logarithmic temperature dependence of the bulk viscosity and neutrino emissivity (Schwenzer 2012; Alford & Schwenzer 2013) but one can derive an analogous analytic result in this case. For a stability window arising from the strong viscous damping in either hyperonic or quark matter (where ψ = 4, δ = 2andθ = 6) the amplitude depends quadratically on the frequency (Andersson et al. 2002; Reisenegger & Bonacic 2003) and the r-mode amplitude in lowfrequency sources is significantly smaller. For a given model of the compact star, the intrinsic strain amplitude is then directly determined by equation (9) in terms of the boundary-straddling amplitude α str (equation 15), as was previously shown in the special case of quark matter in Reisenegger & Bonacic (2003). However, in order for boundary straddling to present a consistent scenario, the spin-down rates of all observed sources have to be equal or larger than the r-mode spin-down rate (equation 2) that follows for a given source with boundary-straddling amplitude (equation 15), since other spin-down mechanisms, like magnetic dipole emission, increase the observed spin-down rate. This means that the pulsar timing data have to be outside of the recently introduced dynamic instability regions in -space, as given in fig. 1 of Alford & Schwenzer (2013). As in the urated case, where for a given uration mechanism the observed spin-down data set a limit on the large amplitude damping that determines the uration amplitude, in the boundarystraddling case one can therefore obtain for a given mechanism an analogous limit on the required enhanced low-amplitude dissipation that imposes the stability window. Unlike the poorly known uration physics, we have generally a better understanding of the low-amplitude damping that determines stability windows. It can then be challenging to find a stability window that gives an amplitude consistent with the bounds obtained from the timing data. It has been shown that because of the long-ranged nature of strong interactions in strange quark matter (Schwenzer 2012), the enhanced damping provides an example that is consistent with the data. However, there are still significant uncertainties when it comes to the properties of dense quark or hyperonic matter. To derive the universal spin-down limit in the boundarystraddling case, which is analogous to equation (12), the amplitude (equation 15) can be parametrized similarly to the uration amplitude in the form α str = ˆα str γ, with the single exponent γ that determines the (local) form of the stability window and the prefactor ˆα str that encodes quantitative details on the dissipation that determines the position of the stability window. The spin-down rate due to r-modes then becomes R = 3Ĝ I ˆα2 str (8 ψ)θ/δ 1. (16) This mechanism requires R for all data points since additional spin-down mechanisms increase the observed spin-down rate. This sets a bound on the quantity ḟ 0 3Ĝ ˆα 2 str I f (8 ψ)θ/δ 1 0. (17) For a consistent model for the compact star composition this constraint must be fulfilled for all sources. This is the case if the lowest value obtained from the data set, that is reached for a particular source with frequency f 0 and spin-down rate ḟ 0 is realized. Using

9 GW emission from oscillating millisecond pulsars 3639 the imposed bound on ˆα str, we find the universal spin-down limit for boundary-straddling evolution analogous to equation (12): ( ) f (8 ψ)θ/(2δ) 1 h (usl) 0 = h (sl) 0. (18) f 0 Using the bound (equation 17) with the expression for the boundarystraddling amplitude (equation 15) we can constrain the damping that leads to the stability window and therefore get a lower bound on the minimum of the instability window from equation (14). In the case of hyperonic or quark matter, the bulk viscosity from non-leptonic flavour-changing processes gives a frequency exponent of 5 in equation (18) when neutrino cooling dominates, so that the strain of low-frequency sources is substantially weaker. If photon cooling dominates the exponent reduces to 4ι 1 1. The standard spin-down limit shown in Figs 4 and 5 does not feature a pronounced frequency dependence. This means that in the boundary-straddling scenario, bulk viscosity effects suppress the signal of low-frequency sources to levels significantly below the standard spin-down limit. Correspondingly our central conclusion, that the gravitational wave emission from low-frequency sources is considerably weaker, obtained previously from the universal spin-down limit in the urated scenario is even strengthened in the boundary-straddling scenario. In general how far below the standard spin-down limit the signal for high-frequency sources is depends on what fraction of the observed spin-down is due to r-modes and thereby how far the spin-down data is from the boundary of the dynamic instability region (see fig. 1 of Alford & Schwenzer 2013), but for fast spinning sources already the ordinary spin-down limit is quite restrictive. For the mode resonance mechanism (Gusakov et al. 2014), where a very narrow resonant stability spike arises, δ should be large, so that the frequency exponent would be small as in the mode coupling model in Fig. 5. In the limit of a vertical boundary the gravitational wave signal would even inversely depend on the frequency so that high-frequency sources are more efficiently constrained. The analysis of this mechanism is still schematic at this point (e.g. no quantitative determination of the width of the spike has been performed; Gusakov et al. 2014) and consequently we cannot perform a detailed evaluation for this case, yet. Generically, the expected gravitational wave strain will depend on the position of the stability window and if it is located at sufficiently low temperatures (or spin-down rates in the dynamic case) the emission would likewise be substantially reduced compared to the standard spin-down limit. The universal spin-down limits for the gravitational wave strain in the boundary-straddling scenario with a stability window resulting from hyperonic or quark matter are shown in Fig. 7. Since the result is sensitive to the cooling mechanism, the figure shows both the case when cooling due to neutrino emission (yellow diamonds) or photon cooling (cyan squares) dominates. Comparing this figure to the urated case given in Fig. 5 shows these limits are very similar so that the universal spin-down limit strongly restricts the gravitational wave strain irrespective of which of the two evolutionary scenarios is realized and correspondingly both for normal and exotic forms of matter. For neutrino cooling the f 0 = 622 Hz pulsar B gives the most restrictive bound. Using this bound shows that r-mode gravitational wave emission can only be present in sources spinning with frequencies f 214 Hz corresponding to gravitational wave frequencies ν 285 Hz. For photon cooling the f 0 = 336 Hz pulsar J is most restrictive. Here gravitational wave emission can only be present in sources spinning with frequencies f 142 Hz corresponding to gravitational wave frequencies ν 189 Hz. These values for the limiting frequencies are very similar to those obtained previously in the urated spin-down scenario for a standard Figure 7. Universal spin-down limits in the boundary-straddling scenario in the case of enhanced bulk viscosity due to non-leptonic flavour-changing processes in hyperon or quark stars (Reisenegger & Bonacic 2003; Alford & Schwenzer 2013). The results are shown both in case neutrino cooling (yellow diamonds) or photon cooling (cyan squares) dominates, compared to the standard spin-down limits (red triangles). The dashed and dotted vertical lines with corresponding uncertainty bands show for these two cooling scenarios the minimum frequency of the instability region (Alford & Schwenzer 2013), below which there would be no gravitational wave emission. neutron star and are shown with corresponding uncertainty bands in Fig. 7. The pulsar data lies in the transition region between neutrino and photon cooling, see Alford & Schwenzer (2013), so that for fast sources the diamonds and for slower sources the squares should roughly apply, whereas for intermediate sources the limits are in between. The logarithmically enhanced damping due to long-range interactions in quark matter (Schwenzer 2012;Alford& Schwenzer 2013) can even slightly lower these limits. The results for the particular model discussed in Alford & Schwenzer (2013) are nevertheless quite close to those in Fig. 7 since the pulsar data lie very close to the corresponding dynamical instability region. Finally, we note that in the special case where the thermal steady state curve intersects the instability boundary it could be that some of the sources are currently spinning down along the urated thermal steady state and the others spin-down along the boundary of the stability window. This would comprise two different classes of sources. Therefore, it could happen that when the reference source is part of one of these classes, the bounds on sources of the other class underestimate the actual gravitational wave strain. However, if the stability window is very narrow, as in the mode resonance mechanism (Gusakov et al. 2014), it would require fine-tuning of the parameters to obtain such a crossing and it would be even more unlikely that sources are in the short segment where they evolve along the boundary. In the case of a broad stability window, such a scenario cannot be ruled out, but the very similar form and parameter dependences of the two expressions equations (12) and (18) give nearly the same upper bounds as seen in Figs 5 and 7 so that the gravitational wave signal will not be underestimated. Moreover, future thermal X-ray measurements of radio pulsars