Cosmic String Interpretation of NANOGrav Pulsar Timing Data

Pulsar timing data used to provide upper limits on a possible stochastic gravitational wave background (SGWB). However, the NANOGrav Collaboration has recently reported strong evidence for a stochastic common-spectrum process, which we interpret as a SGWB in the framework of cosmic strings. The possible NANOGrav signal would correspond to a string tension $G\mu \in (4 \times 10^{-11}, 10^{-10}) $ at the 68% confidence level, with a different frequency dependence from supermassive black hole mergers. The SGWB produced by cosmic strings with such values of $G\mu$ would be beyond the reach of LIGO, but could be measured by other planned and proposed detectors such as SKA, LISA, TianQin, AION-1km, AEDGE, Einstein Telescope and Cosmic Explorer.


Introduction
Stimulated by the direct discovery of gravitational waves (GWs) by the LIGO and Virgo Collaborations [1][2][3][4][5][6][7][8] of black holes and neutron stars at frequencies f 10 Hz, there is widespread interest in experiments exploring other parts of the GW spectrum. Foremost among these are pulsar timing array (PTA) experiments, which are sensitive to GWs with frequencies f 1/yr. PTA experiments probe the possible existence of a stochastic GW background (SGWB), as might be generated by very different physical phenomena such as astrophysical sources of GWs, e.g., the mergers of supermassive black hole (SMBHs), or cosmological sources, e.g., cosmic strings.
Aggregating pulsar measurements for over a decade, the EPTA [9], PPTA [10] and NANOGrav [11] PTA experiments have pushed their sensitivities down to an energy density Ω GW h 2 10 −9 over frequencies in the range f ∈ (2.5 × 10 −9 , 1.2 × 10 −8 ) Hz. Until recently, there has been no indication of a positive signal above background. However, a recent NANOGrav analysis of 12.5 yrs of pulsar timing dataf [12] reports strong evidence for a stochastic common-spectrum process that may be interpreted as a GW signal with amplitude A ∼ O(10 −15 ) at f ∼ 1/yr. The NANOGrav Collaboration notes that this signal is in apparent tension with previous upper limits on the SGWB in this frequency range, but argues that this is not real, but reflects its improved treatment of the intrinsic pulsar red noise. The NANOGrav signal has no monopole or dipole correlations, as might arise, e.g., from reference clock or solar-system ephemeris systematics, respectively. On the other hand, neither does the signal exhibit quadrupole correlations, which would have been a "smoking gun" for a GW background, and the NANOGrav Collaboration does not claim a detection of GWs.
Nevertheless, we are emboldened to explore the implications of this possible SGWB detection by NANOGrav for cosmic string models, discussing how experiments could confirm or disprove such an interpretation. Upper limits on the SGWB are often quoted assuming a spectrum described by a GW abundance proportional to f 2/3 , as expected for SMBH mergers [13]. However, the cosmic string GW spectrum is not a simple power law, but is convex with an amplitude and a frequency-dependent slope that depend on the parameter, Gµ, where G is the Newton constant of gravitation and µ is the string tension. Any limit (or estimate) of Gµ from any specific experiment must take into account take into account the appropriate slope parameter, which is in general = 2/3 in the characteristic frequency measurement range. Once an allowed (interesting) value of Gµ has been identified, however, the cosmic string prediction for the magnitude and spectral shape of the SGWB is then fixed as a function of frequency, and can then be compared with the sensitivities of other experiments.
In this paper we calculate the effective slope parameter for the timing-residual crosspower spectral density γ (which translates to γ = 5 − β for Ω ∝ f β ) for frequencies in the range (2.5 × 10 −9 , 1.2 × 10 −8 ) Hz used in [12] to make a single-power fit to the NANOGrav 12.5 yr data. The best fit to the NANOGrav data is shown as an orange dashed line in the left panel of Fig. 1 of [12], and the 68% and 95% CL ranges in the (γ, A) plane are shown as orange dashed and dotted ellipses in the right panel of Fig. 1 of [12]. We find that the cosmic string model gives a better fit than does a single power law with γ = 13/3 as suggested by models of SMBH mergers: the one-parameter cosmic string prediction crosses the 68% CL ellipse, whereas the γ = 13/3 line passes outside it though within the 95% ellipse. The GW spectra predicted by the cosmic string model for Gµ ∈ (2 × 10 −11 , 2 × 10 −10 ), the range where it lies within the NANOGrav 12.5 yr 95% CL region in the (γ, A) plane, are all completely compatible with the EPTA upper limit, although some tension with with the PPTA results remains in the upper part of our range. The cosmic string predictions are well within the estimated reaches of the SKA [14], LISA [15,16], TianQin [17,18], AEDGE [19], AION-1km [20], ET [21,22] and CE [23] experiments, but beyond the present and estimated future sensitivities of the LIGO [24][25][26][27] experiment.

GW spectrum from cosmic strings
Cosmic strings are one-dimensional stable objects described by their characteristic tension µ. They are a common prediction of many extensions of the Standard Model [28] featuring a U (1) symmetry-breaking phase transition in the early universe [29]. They can also arise in superstring theory as cosmologically-stretched fundamental strings [30,31]. We focus mostly on the former case, for which the inter-commutation probability p = 1, and comment on the latter towards the end of the following Section.
We use a simple method of computation of the GW spectrum from a cosmic string network following [32,33] (for an overview, see [34]). We utilise the Velocity-dependent One-Scale (VOS) model [35][36][37], assuming that the length of a loop produced by the network at time t i evolves as where Gµ is the string tension and α the initial loop size. In order to fit recent numerical simulations [38,39], we focus on the largest loops produced by the network, fixing α = 0.1, as these dominate the GW emission. String loops emit at normal oscillation mode frequencies, allowing us to express the frequency measured today from mode k with emission timet as where t 0 is the current time. The GW abundance can then be computed using where we assume emission from cusps: [38][39][40][41][42] 1 . In evaluating the scale factor, we use the number of degrees of freedom predicted by the Standard Model as given by microMEGAS [46]. The lower integration limit t F corresponds to the network formation time, which can be assumed to be an arbitrarily small number for our purposes, as it only controls the high frequency cut-off of the spectrum, whereas we are mostly interested in the low-frequency peak. 2 We calculate the C ef f factor controlling the loop number density using the VOS model as in [33] which gives C ef f = 5.4 and 0.39 during radiation and matter domination, respectively. These values agree quite well with the values predicted by recent numerical simulations [38,39,[49][50][51]. Finally the additional factor 0.1 comes from the same simulations, which find that only this fraction of energy goes into large loops that produce GWs efficiently, whereas the rest goes into the kinetic energy of small loops that is then lost to redshifting.

Connection with experimental results
The most recent experimental results from 12.5 yr of NANOGrav data [12] are expressed in terms of a generic power-law signal with characteristic strain given by where f yr = 1yr −1 . The abundance of gravitational waves has the standard form, which can also be recast as a power-law: The experimental analysis was cast in terms of the power law found in the timing-residual cross-power spectral density γ = 3 − 2α = 5 − β, and we adopt this notation. In order to make connection with the experimental results, we approximate the cosmic string spectra with power laws in the range of frequencies where the possible signal was observed. The simple power-law approximation used by NANOGrav [12] was fitted to 5 bins covering roughly f ∈ (2.5 × 10 −9 , 1.2 × 10 −8 ) Hz, with the higher-frequency bins still seemingly dominated by noise in the data. To estimate the prospective cosmic string signal for any given value of Gµ, we fit numerically a power law, see Eq. (3.2), to the calculation of the spectrum described above in the range of interest. We show examples of these fits for two values of Gµ in Fig. 1. However, as also as we also see in the plot, we find that a very good approximation is obtained by simply taking a logarithmic derivative of our cosmic string spectrum to find the slope at the reference frequency f * ≈ 5.6 × 10 −9 Hz. and 95% ranges of (γ, A) fitted to their 12.5 yr data by the NANOGrav collaboration [12]. The grey vertical line at γ = 13/3 represents the slope expected for SMBH mergers, while the points on it mark the upper limits on the amplitude from previously-reported pulsar timing data for that spectrum.
We show in Fig 2 the resulting values of γ and A for a range of Gµ values of interest overlaid on the NANOGrav fit to their 12.5 yr data [12]. We find that values of the string tension Gµ ∈ (4 × 10 −11 , 10 −10 ) give results within the 68% CL range of the NANOgrav fit, while Gµ ∈ (2 × 10 −11 , 3 × 10 −10 ) make predictions within the 95% range. Interestingly, the cosmic string interpretation offers a slightly better fit than SMBH mergers, which predict γ = 13/3 (shown as the vertical gray line in Fig. 2) yielding a fit that is at best within the 95% CL range but outside the 68% range.
spectra that provide 68% and 95% CL fits to the NANOGrav 12.5 yr data. The apparent tension is also visible in Fig. 2, which shows previous PPTA and NANOGrav upper limits on the amplitude of a γ = 13/3 SMBH merger spectrum (vertical grey line) from the earlier pulsar timing data releases cited above. According to the NANOGrav collaboration [12], their new analysis uses improved priors for the intrinsic pulsar red noise. Applying these new priors to older data would ease the previous constraints and tend to reduce the tension. Fig. 4 shows the spectra that fit the new NANOGrav data at the 68% and 95% CLs over an extended frequency range f ∈ (10 −9 , 200) Hz. We also show the current sensitivity of LIGO O2 [27] together with its design sensitivity goal [24][25][26], as well as the projected sensitivities of SKA [14] and the upcoming GW experiments LISA [15,16], TianQin [17,18], AEDGE [19], AION/MAGIS [20,52,53] and ET [21,22]. We see all the next-generation GW experiments should be able to observe cosmic string signals strong enough to fit the current NANOGrav data. However, LIGO would, unfortunately not be able to observe such a signal even after reaching its design sensitivity 3 .
We have focused throughout this Section on cosmic strings with inter-commutation probability p = 1. If these strings originate from superstring theory this probability can be much lower. In a first approximation this just corresponds to the density of strings increasing as p −1 for any given value of the tension, which leads to a similar increase in the amplitude of the GW signal [58]. As a result, the cosmic string curve in Fig. 2 would simply move up in amplitude as A ∝ √ Ω ∝ p −1 . Since the rainbow curve passes close to the top of the NANOGrav 68% CL region, there is little scope for decreasing p while maintaining consistency at the 68% CL, with Ω GW h 2 increasing by < 50%.
Before proceeding to our conclusions, we first mention briefly other possible early uni- verse sources that could potentially fit the new NANOGrav data. One such possibility is primordial inflation [15,59]. Generically, inflation leads to a flat spectrum with γ = 5 that is, however, constrained at by the CMB [60] (at f CM B ≈ 10 −17 Hz) to be orders of magnitude below the amplitude of the observed signal. Modifying the inflationary spectrum to have the observed abundance at PTA frequencies requires roughly β = 0.68 [61], which gives a spectrum at PTA frequencies with γ = 4.32, a value very close to the SMBH merger prediction and again seemingly slightly disfavoured by the current data. The second possibility is a signal from a phase transition. However, these signals typically have much higher frequencies [62,63]. Lowering the frequency requires a transition at lower temperature, leaving only the possibility of a hidden sector model [64], since the frequency cannot be lowered by supercooling [65,66] and models coupling to the Standard Model with such low mass scales would already be observed. Even if a hidden sector model is capable of accommodating a very strong phase transition at a very low temperature, one is still likely to only see a low-frequency slope at PTA frequencies which has β = 3 [67] and hence γ = 2, which is disfavoured by the data. While some exceptions from that scaling exist, they require either an extremely strong transition [68] or modification of cosmological expansion [69], both of which would be extremely difficult to realise at low temperatures without violating other bounds.

Conclusions
We have analysed the GW spectra produced by cosmic string networks, recasting them numerically as power laws in the frequency range f ∈ (2.5×10 −9 , 1.2×10 −8 ) Hz of interest to PTA experiments. This allowed us to express the resulting amplitude and slope as functions of the only free parameter in our model, which is the string tension Gµ. We then use these results to make contact with the recent NANOGrav 12.5 yr [12] data release, which finds evidence of a stochastic common-spectrum process, analysed in terms of power-law modelling, that could be interpreted as a GW background. We find that a cosmic string tension Gµ ∈ (4 × 10 −11 , 10 −10 ) fits the data within the 68% CL region around the best fit while Gµ ∈ (2 × 10 −11 , 3 × 10 −10 ) is compatible with the data at the 95% CL. Cosmic strings provide a better fit to the current data than a GW spectrum from SMBH mergers, which can fit the data at the 95% CL but not the 68% CL. We also show all next-generation GW detectors including SKA, LISA, TianQin, AEDGE, AION and ET will be able to probe the cosmic string spectra that fit the current data, whereas LIGO seems unlikely be able to probe them in the absence of additional cosmological or model features.
A key probe of any GW interpretation of the NANOGrav data would be the appearance of quadrupole correlations, which have not (yet) been detected. Beyond this, measurement of a SGWB background compatible with the shape of spectrum shown in Fig. 4 over a large range of frequencies would provide crucial confirmation of our bold GW interpretation of the NANOGrav 12.5 yr data.