{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T01:05:08Z","timestamp":1760231108637,"version":"build-2065373602"},"reference-count":36,"publisher":"MDPI AG","issue":"9","license":[{"start":{"date-parts":[[2022,8,30]],"date-time":"2022-08-30T00:00:00Z","timestamp":1661817600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"FCT\u2014Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["DL 57\/2016\/CP1364\/CT0001","2020.07632.BD","PTDC\/FIS-PAR\/31938\/2017","POCI-01-0145-FEDER-031938","UIDB\/04434\/2020","UIDP\/04434\/2020"],"award-info":[{"award-number":["DL 57\/2016\/CP1364\/CT0001","2020.07632.BD","PTDC\/FIS-PAR\/31938\/2017","POCI-01-0145-FEDER-031938","UIDB\/04434\/2020","UIDP\/04434\/2020"]}]},{"name":"FEDER\u2014Fundo Europeu de Desenvolvimento Regional","award":["DL 57\/2016\/CP1364\/CT0001","2020.07632.BD","PTDC\/FIS-PAR\/31938\/2017","POCI-01-0145-FEDER-031938","UIDB\/04434\/2020","UIDP\/04434\/2020"],"award-info":[{"award-number":["DL 57\/2016\/CP1364\/CT0001","2020.07632.BD","PTDC\/FIS-PAR\/31938\/2017","POCI-01-0145-FEDER-031938","UIDB\/04434\/2020","UIDP\/04434\/2020"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>We derive an analytical approximation for the linear scaling evolution of the characteristic length L and the root-mean-squared velocity \u03c3v of standard frictionless domain wall networks in Friedmann\u2013Lema\u00eetre\u2013Robertson\u2013Walker universes with a power law evolution of the scale factor a with the cosmic time t (a\u221dt\u03bb). This approximation, obtained using a recently proposed parameter-free velocity-dependent one-scale model for domain walls, reproduces well the model predictions for \u03bb close to unity, becoming exact in the \u03bb\u21921\u2212 limit. We use this approximation, in combination with the exact results found for \u03bb=0, to obtain a fit to the model predictions valid for \u03bb\u2208[0,1] with a maximum error of the order of 1%. This fit is also in good agreement with the results of field theory numerical simulations, especially for \u03bb\u2208[0.9,1]. Finally, we explicitly show that the phenomenological energy-loss parameter of the original velocity-dependent one-scale model for domain walls vanishes in the \u03bb\u21921\u2212 limit and discuss the implications of this result.<\/jats:p>","DOI":"10.3390\/sym14091799","type":"journal-article","created":{"date-parts":[[2022,8,31]],"date-time":"2022-08-31T02:09:36Z","timestamp":1661911776000},"page":"1799","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Analytical Scaling Solutions for the Evolution of Cosmic Domain Walls in a Parameter-Free Velocity-Dependent One-Scale Model"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-1440-6963","authenticated-orcid":false,"given":"Pedro","family":"Avelino","sequence":"first","affiliation":[{"name":"Departamento de F\u00edsica e Astronomia, Faculdade de Ci\u00eancias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal"},{"name":"Instituto de Astrof\u00edsica e Ci\u00eancias do Espa\u00e7o, Universidade do Porto, Centro de Astrof\u00edsica da Universidade do Porto (CAUP), Rua das Estrelas, PT4150-762 Porto, Portugal"}]},{"given":"David","family":"Gr\u00fcber","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica e Astronomia, Faculdade de Ci\u00eancias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal"},{"name":"Instituto de Astrof\u00edsica e Ci\u00eancias do Espa\u00e7o, Universidade do Porto, Centro de Astrof\u00edsica da Universidade do Porto (CAUP), Rua das Estrelas, PT4150-762 Porto, Portugal"}]},{"given":"Lara","family":"Sousa","sequence":"additional","affiliation":[{"name":"Departamento de F\u00edsica e Astronomia, Faculdade de Ci\u00eancias, Universidade do Porto, Rua do Campo Alegre 687, PT4169-007 Porto, Portugal"},{"name":"Instituto de Astrof\u00edsica e Ci\u00eancias do Espa\u00e7o, Universidade do Porto, Centro de Astrof\u00edsica da Universidade do Porto (CAUP), Rua das Estrelas, PT4150-762 Porto, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2022,8,30]]},"reference":[{"key":"ref_1","first-page":"1387","article-title":"Topology of Cosmic Domains and Strings","volume":"A9","author":"Kibble","year":"1976","journal-title":"J. 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