$site = $contentType.ParentWeb.Site
$publisher = New-Object Microsoft.SharePoint.Taxonomy.ContentTypeSync.ContentTypePublisher $site
$publisher.Publish($contentType)

# Start timer jobs
Start-SPTimerJob -Identity "MetadataHubTimerJob"
Get-SPWebApplication | Get-SPTimerJob -Identity "MetadataSubscriberTimerJob" | Start-SPTimerJob