Search for the Gravitational-wave Background from Cosmic Strings with the Parkes Pulsar Timing Array Second Data Release

We perform a direct search for an isotropic stochastic gravitational-wave background (SGWB) produced by cosmic strings in the Parkes Pulsar Timing Array (PPTA) Data Release 2 (DR2). We find no evidence for such an SGWB, and therefore place a 95% confidence level upper limit on the cosmic string tension, G μ, as a function of the reconnection probability, p, which can be less than 1 in the string-theory-inspired models or pure Yang–Mills theory. The upper bound on the cosmic string tension is G μ ≲ 5.1 × 10−10 for p = 1, which is about five orders of magnitude tighter than the bound derived from the null search of individual gravitational-wave bursts from cosmic string cusps in the PPTA DR2, and comparable to previous bounds derived from the null search of the SGWB from cosmic strings.

Cosmic strings are linear topological defects that can either form in the early Universe from symmetrybreaking phase transitions at high energies [49][50][51][52] or be the fundamental strings of superstring theory (or one-dimensional D-branes) stretched out to astrophysical lengths [52,53].After their formation, the intersection between cosmic strings can lead to reconnections and form loops, which will then decay due to relativistic oscillation and emit GWs.PTA observations may detect a cosmic string network either through GW bursts emitted at cusps or through the SGWB superposed by radiation from all loops existing through cosmic history.A null detection of the individual GW burst from cosmic strings in PPTA DR2 has been reported in [30], thus placing a 95% upper limit on the cosmic string tension to be Gµ 10 −5 .In this work, we perform the first direct search for the SGWB produced from a network of cosmic strings in the PPTA DR2 data set.The remainder of this paper is organized as follows.In Sec.II, we review the SGWB energy spectrum produced by the cosmic strings.In Sec.III, we describe the data set and methodology used in our analyses.Finally, we summarize the results and give some discussion in Sec.IV.

II. SGWB FROM COSMIC STRINGS
We now review the SGWB produced by cosmic strings following [54].A cosmic string network consists of both long (or "infinite") strings that are longer than the horizon size and loops formed from smaller strings.When two cosmic strings meet one another, they can exchange partners with a reconnection probability p, and form loops. Once loops are formed, they oscillate and decay through the emission of GWs [50], shrinking in size.A cosmic string network grows along with the cosmic expansion and evolves toward the scaling regime in which all the fundamental properties of the system scale with the cosmic time.The scaling regime can be achieved through the formation and subsequent decay of loops.The GW spectrum created by a cosmic string network is exceptionally broadband, depending on the size of the loops created.
We describe the GW energy spectrum of cosmic strings in terms of the dimensionless tension, Gµ, and the reconnection probability, p.Even though p = 1 for classical strings, it can be less than 1 in the string-theory-inspired models.In this work, we adopt the convention that the speed of light c = 1.The dimensionless GW energy density parameter per logarithm frequency as the fraction of the critical energy density is [54] where t 0 is cosmic time today, and ρ gw is the GW energy density per unit frequency that can be computed by with Here, P n is the radiation power spectrum of each loop, and n(l, t) is the density of loops per unit volume per unit range of loop length l existing at time t.The SGWB from a network of cosmic strings has been computed in [54] with a complete end-to-end method by (i) simulating the long string network to extract a representative sample of loop shapes; (ii) using a smoothing model to estimate loop shape deformations due to gravitational backreaction; (iii) computing GW spectrum for each loop; (iv) evaluating the distribution of loops over redshift by integrating over cosmological time; (v) integrating the GW spectrum of each loop over the redshift-dependent loop distribution to get the overall emission spectrum; and at last (vi) integrating the overall emission spectrum over cosmological time to get the current SGWB.The simulations span a large parameter space of Gµ ∈ [10 −25 , 10 −8 ], and f ∈ [10 −15 , 10 10 ]; and the output of the expected energy density spectra has been made publicly available 1 .

III. THE DATA SET AND METHODOLOGY
The PPTA DR2 [55] data set includes pulse TOAs from high-precision timing observations for 26 pulsars collected with the 64-m Parkes radio telescope in Australia.The data were acquired between 2003 and 2018, spanning about 15 years, with observations taken at a cadence of approximately three weeks [55].Details of the observing systems and data processing procedures are described in [10,55].
To search for the GW signal from the PTA data, one needs to provide a comprehensive description of the stochastic processes that can induce the arrival time variations.The stochastic processes can be categorized as being correlated (red) or uncorrelated (white) in time.A careful analysis of the noise processes for individual pulsars in the PPTA sample has been performed in [56], showing that the PPTA data sets contain a wide variety of noise processes, including instrument dependent or band-dependent processes.Similar to [47], we adopt the noise model developed in [56] to characterize the noise processes.After subtracting the timing model of the pulsar from the TOAs, the timing residuals δt for each single pulsar can be decomposed into (see e.g.[57]) The first term M in the above equation accounts for the inaccuracies in the subtraction of timing model [58], where M is the timing model design matrix obtained from TEMPO2 [59,60] through libstempo2 interface, and is a small offset vector denoting the difference between the true parameters and the estimated parameters of timing model.The second term δt RN is the stochastic contribution from red noise [56], including achromatic spin noise (SN) [61]; frequency-dependent dispersion measure (DM) noise [62]; frequency-dependent chromatic noise (CN) [63]; achromatic band noise (BN) and system ("group") noise (GN) [57].We use 30 frequency components for the red noise of the individual pulsar.The third term δt DET represents deterministic noise [56], including chromatic exponential dips [57], extreme scattering events [62], and annual dispersion measure variations [64].The fourth term δt WN represents white noise (WN), including a scale parameter on the TOA uncertainties (EFAC), an added variance (EQUAD), and a per-epoch variance (ECORR) for each backend/receiver system [65].The last term δt CP is the stochastic contribution due to the common-spectrum process (such as an SGWB), which is described by the cross-power spectral density [66]  where Γ IJ is the Hellings & Downs coefficients [67] measuring the spatial correlations of the pulsars I and J.
Following [68], we use 5 frequency components for the common process among all of the pulsars.We perform the Bayesian parameter inferences based on the methodology in [68,69] to search for the SGWB from cosmic strings in the PPTA DR2 data set.Since it is challenging to obtain a complete noise model for pulsar J0437−4715 and pulsar J1713+0747 [56], we do not include these two pulsars.A summary of the noise model for the 24 pulsars used in our analyses can be found in Table 1 of [47].We quantify the model selection scores by the Bayes factor defined as where Pr(D|M) measures the evidence that the data D are produced under the hypothesis of model M. Model M 2 is preferred over M 1 if the Bayes factor is sufficiently large.As a rule of thumb, BF ≤ 3 implies the evidence supporting the model M 2 over M 1 is "not worth more than a bare mention" [70].In practice, we estimate the Bayes factor using the product-space method [71][72][73][74].
Our analyses are based on the latest JPL solar system ephemeris (SSE) DE438 [75].We first infer the parameters of each single pulsar without including the commonspectrum process δt CP in Eq. ( 4), and then fix the white noise parameters to their max likelihood values from single-pulsar analysis to reduce the computational costs as was commonly done in literature (see e.g.[68,69]).
We use the open-source software packages enterprise [76] and enterprise extension [77] to calculate the likelihood and Bayes factors and use PTMCMCSampler [78] package to do the Markov chain Monte Carlo sampling.Similar to [68,79], we use draws from empirical distributions based on the posteriors obtained from the singlepulsar Bayesian analysis to sample the parameters of red noise and deterministic noise.Using empirical distributions can reduce the number of samples needed for the chains to burn in.The model parameters and their prior distributions are listed in Table I.

IV. RESULTS AND DISCUSSION
We first consider a model in which both the cosmic string tension Gµ and the reconnection probability p are free parameters.Fig. 1 shows the posterior distributions of the Gµ and p parameters obtained from the Bayesian search.The Bayes factor of the model including both the UCP and CS signal versus the model including only the UCP is BF UCP+CS UCP = 0.591 ± 0.008, indicating no evidence for an SGWB signal produced by the cosmic string in the PPTA DR2.We also consider models in which the reconnection probability p is fixed to a specific value while the cosmic string tension Gµ is allowed to vary.The lower panel of Fig. 2 shows the Bayes factor as a function of reconnection probability.For all the values of p ∈ [10 −3 , 1], we have BF UCP+CS UCP 3, confirming that there is no evidence for an SGWB produced by cosmic strings in the PPTA DR2.We, therefore, place 95% confidence level upper limit on cosmic string tension Gµ as a function of reconnection probability p as shown in Fig. 2. The blue shaded region indicates parameter space that is excluded by the PPTA DR2.For p = 1, the upper bound on the cosmic string tension is Gµ 5.1 × 10 −10 , which is about five orders of magnitude tighter than the bound of Gµ 10 −5 [30] derived from the null search of individual gravitational wave burst from cosmic string cusps in the PPTA DR2.Note that Ω gw is enhanced for p < 1, and therefore tighter constraints on Gµ are obtained for p < 1.
To sum up, we have searched for the SGWB produced by a cosmic string network in the PPTA DR2 in the work.We find no evidence for such SGWB signal, and therefore place 95% upper limit on cosmic string tension as a function of reconnection probability.

FIG. 1 . 2 BFFIG. 2 .
FIG.1.One-and two-dimensional marginalized posterior distributions for the cosmic string tension, Gµ, and the reconnection probability, p.We show both the 1σ and 2σ contours in the two-dimensional plot.

TABLE I .
Model parameters and their prior distributions used in the Bayesian inference.