Gravitational Waves from the Early Universe Eiichiro Komatsu [Max Planck Institute for Astrophysics] Instituto de Física de Cantabria, Santander June 4, 2019
GW = Area-conserving distortion of distances between two points
Distance between two points in space • Static (i.e., non-expanding) Euclidean space • In Cartesian coordinates
Distance between two points in space • Homogeneously expanding Euclidean space • In Cartesian comoving coordinates “scale factor”
Distance between two points in space • Homogeneously expanding Euclidean space • In Cartesian comoving coordinates “scale factor” =1 for i=j =0 otherwise
Distance between two points in space • Inhomogeneous curved space • In Cartesian comoving coordinates “metric perturbation” -> CURVED SPACE!
<latexit sha1_base64="9t+cOyQWNF6KWd58uc0r9WSx1xk=">ACBHicbVDLSgMxFM3UV62vUZfdBIvgqsy0gm4KRTcuK9gHtNMhk6ZtbJIZkoxQhlm48VfcuFDErR/hzr8xbWehrQcunJxzL7n3BGjSjvOt5VbW9/Y3MpvF3Z29/YP7MOjlgpjiUkThyUnQApwqgTU01I51IEsQDRtrB5Hrmtx+IVDQUd3oaEY+jkaBDipE2km8XeyrmfkJrbtqvwkmfwrF53aewBh3fLjlZw64StyMlECGhm9/9QYhjkRGjOkVNd1Iu0lSGqKGUkLvViRCOEJGpGuoQJxorxkfkQKT40ygMNQmhIaztXfEwniSk15YDo50mO17M3E/7xurIeXkJFGsi8OKjYcygDuEsETigkmDNpoYgLKnZFeIxkghrk1vBhOAun7xKWpWyWy1Xbs9L9asjwoghNwBlxwAergBjRAE2DwCJ7BK3iznqwX6936WLTmrGzmGPyB9fkDw0+W3g=</latexit> <latexit sha1_base64="QeE26pPU6GUXQeIuz5+CkZC7qA=">ACAXicbZDLSgMxFIYz9VbrbdSN4CZYBFdlphV0Uyi6cVnBXqAdh0yaUOTzJBkhDKMG1/FjQtF3PoW7nwb03YW2vpD4OM/53By/iBmVGnH+bYK6tr6xvFzdLW9s7unr1/0FZRIjFp4YhFshsgRgVpKWpZqQbS4J4wEgnGF9P650HIhWNxJ2exMTjaChoSDHSxvLto75KuJ/Supvd1+DIEM1gHTrQt8tOxZkJLoObQxnkavr2V38Q4YQToTFDSvVcJ9ZeiqSmJGs1E8UiREeoyHpGRSIE+WlswsyeGqcAQwjaZ7QcOb+nkgRV2rCA9PJkR6pxdrU/K/WS3R46aVUxIkmAs8XhQmDOoLTOCASoI1mxhAWFLzV4hHSCKsTWglE4K7ePIytKsVt1ap3p6XG1d5HEVwDE7AGXDBWiAG9AELYDBI3gGr+DNerJerHfrY95asPKZQ/BH1ucPaDuVjQ=</latexit> <latexit sha1_base64="iwZxKxQ2qmVFkrUPGUlb4Wp/QF0=">AB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3bpZhN2N4US+iO8eFDEq7/Hm/GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikpeNUMWyWMSqE1CNgktsGm4EdhKFNAoEtoPx/dxvT1BpHsnM03Qj+hQ8pAzaqzU7k2QZeNZv1xq+4CZJ14OalAjka/NUbxCyNUBomqNZdz02Mn1FlOBM4K/VSjQlYzrErqWSRqj9bHujFxYZUDCWNmShizU3xMZjbSeRoHtjKgZ6VvLv7ndVMT3voZl0lqULlojAVxMRk/jsZcIXMiKklClubyVsRBVlxiZUsiF4qy+vk1at6l1Va4/XlfpdHkcRzuAcLsGDG6jDAzSgCQzG8Ayv8OYkzovz7nwsWwtOPnMKf+B8/gCfcI/B</latexit> Four conditions • Gravitational waves shall be: • Transverse : the direction of the oscillation of space is perpendicular to the propagation direction ~ k 3 • This means X k i h ij = 0 3 conditions for h ij i =1 • Area-conserving : the determinant of the distortion in space remains unchanged 3 • This means that the trace vanishes: X h ii = 0 1 condition for h ij i =1 6 components of h ij minus 4 conditions = 2 degrees of freedom
<latexit sha1_base64="7o35sMmYgUabX25v6xOUlfm+4t4=">AB73icbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2m3bpZhN3J0IJ/RNePCji1b/jzX/jts1BWx8MPN6bYWZekEh0HW/ncLa+sbmVnG7tLO7t39QPjxqmTjVjDdZLGPdCajhUijeRIGSdxLNaRI3g7GtzO/cS1EbF6wEnC/YgOlQgFo2ilzqjfQxFx0y9X3Ko7B1klXk4qkKPRL3/1BjFLI6QSWpM13MT9DOqUTDJp6VeanhC2ZgOedSRe0SP5vfOyVnVhmQMNa2FJK5+nsio5ExkyiwnRHFkVn2ZuJ/XjfF8NrPhEpS5IotFoWpJBiT2fNkIDRnKCeWUKaFvZWwEdWUoY2oZEPwl9eJa1a1buo1u4vK/WbPI4inMApnIMHV1CHO2hAExhIeIZXeHMenRfn3flYtBacfOY/sD5/AE0I5AT</latexit> <latexit sha1_base64="hlPrvhryKWK4jM72Qr7KwPo/K0=">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</latexit> <latexit sha1_base64="73F7RS5/Z58dteq4PTugvB0Ykc=">AB6nicbVBNS8NAEJ34WetX1aOXxSIQkmqoMeiF48V7Qe0oWy2k3bpZhN2N0IJ/QlePCji1V/kzX/jts1BWx8MPN6bYWZekAiujet+Oyura+sbm4Wt4vbO7t5+6eCwqeNUMWywWMSqHVCNgktsG4EthOFNAoEtoLR7dRvPaHSPJaPZpygH9GB5CFn1FjpYdg75XKbsWdgSwTLydlyFHvlb6/ZilEUrDBNW647mJ8TOqDGcCJ8VuqjGhbEQH2LFU0gi1n81OnZBTq/RJGCtb0pCZ+nsio5HW4yiwnRE1Q73oTcX/vE5qwms/4zJDUo2XxSmgpiYTP8mfa6QGTG2hDLF7a2EDamizNh0ijYEb/HlZdKsVryLSvX+sly7yeMowDGcwBl4cAU1uIM6NIDBAJ7hFd4c4bw4787HvHXFyWeO4A+czx/rd42O</latexit> + and x modes • If the GW is propagating in the z (i=3) direction, we can write h + h × 0 h × − h + h ij = 0 0 0 0 y x h + h ×
<latexit sha1_base64="iwZxKxQ2qmVFkrUPGUlb4Wp/QF0=">AB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0mqoMeiF48V7Ae0oWy2k3bpZhN2N4US+iO8eFDEq7/Hm/GbZuDtj4YeLw3w8y8IBFcG9f9dgobm1vbO8Xd0t7+weFR+fikpeNUMWyWMSqE1CNgktsGm4EdhKFNAoEtoPx/dxvT1BpHsnM03Qj+hQ8pAzaqzU7k2QZeNZv1xq+4CZJ14OalAjka/NUbxCyNUBomqNZdz02Mn1FlOBM4K/VSjQlYzrErqWSRqj9bHujFxYZUDCWNmShizU3xMZjbSeRoHtjKgZ6VvLv7ndVMT3voZl0lqULlojAVxMRk/jsZcIXMiKklClubyVsRBVlxiZUsiF4qy+vk1at6l1Va4/XlfpdHkcRzuAcLsGDG6jDAzSgCQzG8Ayv8OYkzovz7nwsWwtOPnMKf+B8/gCfcI/B</latexit> ~ z k propagation direction of GW h + =cos(kz) h x =cos(kz)
<latexit sha1_base64="XLil458Y+U4bzO29KFAWiBIo6hk=">ACQXicbZDNS8MwGMZTP+f8mnr0EhyCIx2E/UiD3M4R9wbqNE3XuDQtSqM0n/Ni/+BN+9ePCji1YtZt4NuvCGh9+Tlzd5nIhRqUzxVhaXldW89t5De3tnd2C3v7LRnGApMmDlkoOg6ShFOmoqRjqRIChwGk7o5uJ34gQtKQN9Q4Ir0ADTn1KEZKo0GhY7tuqBI/HST0Pj21PYFwUrEnDKWpbnvRHvXL2tGHn1F4Ba1zO6KwBhsZ6Ce1djoFM2SmRVcFNZMFMGs6oPCs16F4BwhRmSsmuZkeolSCiKGUnzdixJhPAIDUlXS4CIntJlkAKjzVxoRcK3VzBjP6eSFAg5Thw9M0AKV/OexP4n9eNlXfZSyiPYkU4ni7yYgZVCdxQpcKghUba4GwoPqtEPtIp6R06HkdgjX/5UXRKpesSql8d1asXs/iyIFDcAROgAUuQBXcgjpoAgwewSt4Bx/Gk/FmfBpf06tLxmzmAPwp4/sH6cSyIw=</latexit> Equation of motion • Writing Einstein’s gravitational field equation with • We obtain, for a plane wave of GW with the wavenumber k, h ij + k 2 a h ij + 3˙ ¨ ˙ a 2 h ij = 16 π GT GW ij a source of GW
<latexit sha1_base64="XLil458Y+U4bzO29KFAWiBIo6hk=">ACQXicbZDNS8MwGMZTP+f8mnr0EhyCIx2E/UiD3M4R9wbqNE3XuDQtSqM0n/Ni/+BN+9ePCji1YtZt4NuvCGh9+Tlzd5nIhRqUzxVhaXldW89t5De3tnd2C3v7LRnGApMmDlkoOg6ShFOmoqRjqRIChwGk7o5uJ34gQtKQN9Q4Ir0ADTn1KEZKo0GhY7tuqBI/HST0Pj21PYFwUrEnDKWpbnvRHvXL2tGHn1F4Ba1zO6KwBhsZ6Ce1djoFM2SmRVcFNZMFMGs6oPCs16F4BwhRmSsmuZkeolSCiKGUnzdixJhPAIDUlXS4CIntJlkAKjzVxoRcK3VzBjP6eSFAg5Thw9M0AKV/OexP4n9eNlXfZSyiPYkU4ni7yYgZVCdxQpcKghUba4GwoPqtEPtIp6R06HkdgjX/5UXRKpesSql8d1asXs/iyIFDcAROgAUuQBXcgjpoAgwewSt4Bx/Gk/FmfBpf06tLxmzmAPwp4/sH6cSyIw=</latexit> Equation of motion • Writing Einstein’s gravitational field equation with • We obtain, for a plane wave of GW with the wavenumber k, h ij + k 2 a h ij + 3˙ ¨ ˙ a 2 h ij = 16 π GT GW ij a source of GW expansion of the Universe a ff ects h ij
<latexit sha1_base64="tAD74QR0ehNcGl7YTxIV7iqnXJk=">ACJnicdVDLSgMxFM34rPVdelmsAh1U2aqoJtC0Y3LCvYBnVLupJk2NPMguSOUYb7Gjb/ixkVFxJ2fYtoOoq0eCBzOuTfJOW4kuEL+jBWVtfWNzZzW/ntnd29/cLBYVOFsaSsQUMRyrYLigkesAZyFKwdSQa+K1jLHd1M/dYDk4qHwT2OI9b1YRBwj1NALfUKVSjhmeNJoIkTgUQOIv1mJqbV/zyHIaS9QtEqWzOYy8TOSJFkqPcKE6cf0thnAVIBSnVsK8JuMr2SCpbmnVixCOgIBqyjaQA+U91kFjM1T7XSN71Q6hOgOVN/biTgKzX2XT3pAw7VojcV/I6MXpX3YQHUYwsoPOHvFjHD81pZ2afS0ZRjDUBKrn+q0mHoHtB3Wxel2AvRl4mzUrZPi9X7i6Kteusjhw5JiekRGxySWrkltRJg1DySJ7JhLwaT8aL8Wa8z0dXjGzniPyC8fkFVpunkg=</latexit> <latexit sha1_base64="XLil458Y+U4bzO29KFAWiBIo6hk=">ACQXicbZDNS8MwGMZTP+f8mnr0EhyCIx2E/UiD3M4R9wbqNE3XuDQtSqM0n/Ni/+BN+9ePCji1YtZt4NuvCGh9+Tlzd5nIhRqUzxVhaXldW89t5De3tnd2C3v7LRnGApMmDlkoOg6ShFOmoqRjqRIChwGk7o5uJ34gQtKQN9Q4Ir0ADTn1KEZKo0GhY7tuqBI/HST0Pj21PYFwUrEnDKWpbnvRHvXL2tGHn1F4Ba1zO6KwBhsZ6Ce1djoFM2SmRVcFNZMFMGs6oPCs16F4BwhRmSsmuZkeolSCiKGUnzdixJhPAIDUlXS4CIntJlkAKjzVxoRcK3VzBjP6eSFAg5Thw9M0AKV/OexP4n9eNlXfZSyiPYkU4ni7yYgZVCdxQpcKghUba4GwoPqtEPtIp6R06HkdgjX/5UXRKpesSql8d1asXs/iyIFDcAROgAUuQBXcgjpoAgwewSt4Bx/Gk/FmfBpf06tLxmzmAPwp4/sH6cSyIw=</latexit> <latexit sha1_base64="Hh9vfmowV89FN4/WIswMF8l4I0w=">ACA3icbVBNS8NAEN34WetX1JteFotQLyWpgl6EohePFewHNKFstpt26WYTdidCQUv/hUvHhTx6p/w5r9x2+agrQ8GHu/NMDMvSATX4Djf1tLyuraemGjuLm1vbNr7+03dZwqyho0FrFqB0QzwSVrAfB2oliJAoEawXDm4nfemBK81jewyhfkT6koecEjBS1z70GB8hT0uwQsVoVkPxhkpw+m4a5ecijMFXiRuTkoR71rf3m9mKYRk0AF0brjOgn4GVHAqWDjopdqlhA6JH3WMVSiGk/m/4wxidG6eEwVqYk4Kn6eyIjkdajKDCdEYGBnvcm4n9eJ4Xw0s+4TFJgks4WhanAEONJILjHFaMgRoYQqri5FdMBMUmAia1oQnDnX14kzWrFPatU785Ltes8jgI6QseojFx0gWroFtVRA1H0iJ7RK3qznqwX6936mLUuWfnMAfoD6/MHU5mXUw=</latexit> Equation of motion h ij + k 2 a h ij + 3˙ ¨ ˙ a 2 h ij = 16 π GT GW ij a source of GW expansion of the Universe a ff ects h ij • Two tricks: (1) Define “conformal time” dt Z η = a ( t ) and use this instead of time derivatives a ( t ) ∂ ∂ t = ∂ ∂η
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