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We present the results of a new field-theoretic simulation of cosmological axion strings, which are eight times longer than previous ones. We have upgraded our simulation of physical strings in Hiramatsu et al. [Phys. Rev. D

The axion is one of the best motivated particles beyond the Standard Model [

Formation of topological defects associated with the broken global PQ U(1) symmetry takes place if the symmetry breaking occurs after the observable Universe exits the horizon during inflation. At first, axion strings form following the PQ phase transition. As the cosmological network of the strings evolves, its energy is released in the form of axion radiation. Later on, the QCD phase transition takes place and a domain wall (DW) stretches out between the axion strings. If the number of DWs attached to each string is unity, the string–DW network is unstable, so that it disappears soon after the QCD phase transition, leaving an additional contribution to the axion CDM [

In this letter, we will focus on the network of axion strings in between the two phase transitions. This subject has already been studied by many different people over the last few decades [

The primary purpose of this letter is to update the result of our previous simulation [

This letter is organized as follows. In the next section, we will first describe the Lagrangian of the PQ scalar that we simulate, as well as the essence of our numerical simulation and analysis. Then in

We adopt the following Lagrangian density for the PQ complex scalar

We assume a flat Universe with its line element given by

We normalize the scale factor at the PQ phase transition (i.e.

Following Ref. [

Given that the physical width of the axion string ^{1}

We start the simulation immediately before the PQ phase transition, and the initial condition of the real and imaginary parts of the PQ field

We integrate the classical equation of motion of

We perform a suite of analyses in order to extract a variety of dynamical quantities associated with the string network. One of the essential techniques in the analysis is identification of axion strings from the discretized data of the PQ field on grids. We adopt the same string identification method as in Ref. [

Furthermore, this time we introduce a novel method to identify string loops from the string points obtained from the string identification method. This is realized by grouping those string points by the friends-of-friends algorithm. We found that setting the physical linking length to

We also implant an estimation method of string speed following Ref. [

This is not exactly the same estimation method as in Ref. [

On the other hand, the energy spectrum of the axion radiation is computed according to the pseudo-power spectrum estimator [

A more detailed description of our simulation and analysis will be presented in a separate paper (Kawasaki et al., manuscript in preparation).

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Evolution of string parameter

On the other hand,

Evolution of the string parameter contributed only from loops with physical circumference lengths less than

The velocity of strings also exhibits some degree of discrepancy from the previous studies. The root mean square of string velocity

Evolution of the rms of the velocity. Colors and symbols are the same as in

In

Differential spectrum between two scale factors (

To quantify the evolution of the spectral peaks, let us define

Evolution of the mean momentum of radiated axions in units of the Hubble expansion rate (see Eq. (

On the other hand, the wavelengths of radiated axions are expected to be related to the correlation length of the strings, which is proportional to

Among the results that we have presented in the previous section, the growth of the string parameter will have the most profound significance. While field-theoretic simulation of physical strings cannot trace ^{4}

The number of axions radiated from strings is proportional to the number of axion strings and inversely proportional to the typical momentum of radiated axions. The former is proportional to

Logarithmic growth in the string parameter is reported in Refs. [

What is more puzzling is the nontrivial dependence of

Such dependence apparently conflicts with the prediction of the one-scale model [

At the moment, the physical origin of the logarithmic growth of the string parameter

We have also computed the spectrum of the axion radiated from the axion strings. The result is largely in agreement with the previous studies [

This research is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Numbers 17H01131 (M.K.), 17K05434 (M.K.), JP25287054 (M.Y.), JP18H04579 (M.Y.), JP15H02082 (J.Y. and T.S.), 18H04339 (T.S.), 18K03640 (T.S.), a Grant on Innovative Areas JP15H05888 (M.Y. and J.Y.), and Ministry of Education, Culture, Sports, Science and Technology (MEXT) KAKENHI Grant Number 15H05889 (M.K.). This work is also supported by the WPI Initiative, MEXT, Japan (M.K.). This research used the computational resources of COMA and Oakforest-PACS provided by the Interdisciplinary Computational Science Program in the Center for Computational Sciences, University of Tsukuba.

Open Access funding: SCOAP

^{1} The choice of

^{2} We note that our previous choice of

^{3} We note that when we fit

^{4} Fitting