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Superfluids and the problem

Jeremy Sakstein University of Pennsylvania

PASCOS 2018 Case Western Reserve University 7th June 2018

arXiv: 1805.05937 w/Justin Khoury & Adam R. Solomon

Things people say about the cosmological constant problem

“The mother of all physics problems” — attrib. Susskind

“The unwanted child of two pillars of twentieth century physics: quantum field theory and ” - Padilla Cosmological constant in QFT

Equivalence principle: all energy gravitates

Including the vacuum:

p AAACJHicZVDLSgMxFM3Ud31VXboJFsGFlhkRdFl041LBWqEpJZNJ29A8xiRTKWEW/oxu9T/ciQs3foRfYPoAtT1w4XDOvXDuiVPOjA3Dz6AwN7+wuLS8UlxdW9/YLG1t3xqVaUJrRHGl72JsKGeS1iyznN6lmmIRc1qPexdDv96n2jAlb+wgpU2BO5K1GcHWS63SLjJMIHOvrTvq5Eh3VcshLWAfk7xVKoeVcAQ4S6IJKYMJrlqlb5QokgkqLeHYmEYUprbpsLaMcJoXUWZoikkPd2jDU4kFNU03eiKH+15JYFtpP9LCkfr3wmFhzEDEflNg2zXT3lD89SR9IEoILBOHkiR36BCioaWFS/KpJLZ91nRMppmlkoyDtDMOrYLDxmDCNCWWDzzBRDP/CyRdrDGxvteiLymarmSW3B5XorASXZ+Uq+eTupbBLtgDByACp6AKLsEVqAECHsEzeAGvwVPwFrwHH+PVQjC52QH/EHz9AK8dpe0= g⇢ ⇠ vac

2 ⇤ren = ⇤ + ⇢vac /M

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 pl

gravitates counter term divergent EFT for cosmology 4 mi -3 ⇤ Cut-off is 10 eV: IR ⌧ M 2

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 pl

2 4 µ 4 4 ⇤ = ⇤ + M C m + C m + + C m + IR UV pl e e µ ··· H H ··· AAACd3icZVFbaxQxFM5Mq9b1tuqjDwYXpbC6zJSCvgjFfVlBoYq7LTS7QyY524bmMiQZZQn5l774N3zTJ7MX1G4PJHyXc+DkS91I4XxR/Mjynd0bN2/t3e7cuXvv/oPuw0cTZ1rLYMyMNPa0pg6k0DD2wks4bSxQVUs4qS+HS//kK1gnjP7iFw1MFT3XYi4Y9Umqupp8SM2cVoFYhd9/jm+vCONJ7H+chVcHcc0bGYlr1ZpIYxoXhxWoCmaH/eGMqBarKt2JEcaNd/1hNVLV6C+vur1iUKwKXwflBvTQpo6r7k/CDWsVaM8kde6sLBo/DdR6wSTEDmkdNJRd0nM4S1BTBW4aVrlE/DwpHM+NTUd7vFL/nwhUObdQdepU1F+4bW8p/vM0fGNGKap5IJzHQF5isrSsCjxubeLnb6ZB6Kb1oNl6kXkrsTd4+QmYCwvMy0UClFmR3oLZBbWU+fRVnRRSuR3JdTA5GJTFoPx02Dt6t4lrDz1Bz9A+KtFrdIRG6BiNEUPf0e9sJ9vNfuVP8xf5/ro1zzYzj9GVyss/7Y7ASA== loopsX

These have to cancel to 60 decimal places to all loops

Cancellation in the UV — extreme UV-sensitivity Cancellation at all loops — radiative instability Self-tuning — no-go’s

Arrange new fields to cancel a large CC

Weinberg: can’t have Poincare-invariance

fine-tuning Weyl-invariance

Padilla + Niedermann: can’t have broken time-translations

fail solar system tests Our idea: break Lorentz spontaneously Four scalars: solutions of EOM (no tuning) 0 a a

AAACGHicZVDLSgMxFM34rPXRqks3wSK4kDIjgm6EohuXFewDOmO5k0nb0CQzJBmlDP0R3ep/uBO37vwNv8C0HVDbAxcO59wL554w4Uwb1/1ylpZXVtfWCxvFza3tnVJ5d6+p41QR2iAxj1U7BE05k7RhmOG0nSgKIuS0FQ6vJ37rgSrNYnlnRgkNBPQl6zECxkrdcsmvD9i9e+kDTwaATbdccavuFHiReDmpoBz1bvnbj2KSCioN4aB1x3MTE2SgDCOcjot+qmkCZAh92rFUgqA6yKbBx/jIKhHuxcqONHiq/r3IQGg9EqHdFGAGet6biL+epI8kFgJklPlRNM78E+xPLCWyaDyXxPQugozJJDVUklmQXsqxifGkJRwxRYnhI0uAKGZ/wWQACoixXRZtSd58JYukeVr13Kp3e1apXeV1FdABOkTHyEPnqIZuUB01EEEpekYv6NV5ct6cd+djtrrk5Df76B+czx85ZZ/L

= ↵t AAACGXicZVDLSgNBEJz1bXytevQyGAQPEnZF0IsQ9OIxglEhG0PvbMcMzswuM7PRsORL9Kr/4U28evI3/AInMaAmBQ1FVTdUV5wJbmwQfHpT0zOzc/MLi6Wl5ZXVNX9949KkuWZYZ6lI9XUMBgVXWLfcCrzONIKMBV7Fd6cD/6qL2vBUXdhehk0Jt4q3OQPrpJbvR7UOv4HjKEYL9OEGWn45qARD0EkSjkiZjFBr+V9RkrJcorJMgDGNMMhsswBtORPYL0W5wQzYHdxiw1EFEk2zGCbv0x2nJLSdajfK0qH696IAaUxPxm5Tgu2YcW8g/noK71kqJaikiJKkX0R7NBpYWhZJfyyJbR81C66y3KJiP0HauaA2pYOaaMI1Mit6jgDT3P1CWQc0MOvKLLmSwvFKJsnlfiUMKuH5Qbl6MqprgWyRbbJLQnJIquSM1EidMNIlT+SZvHiP3qv35r3/rE55o5tN8g/exzdH76Bf = x SO(1, 3) SO(3) 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 !

AAACDXicZVDLSsNAFJ34rPVVdelmsAgupCQi6LLoxmUF+4AmlJvJpB07MwkzE6WEfINu9T/ciVu/wd/wC5w+QG0PXDiccy+ce8KUM21c98tZWl5ZXVsvbZQ3t7Z3dit7+y2dZIrQJkl4ojohaMqZpE3DDKedVFEQIaftcHg99tsPVGmWyDszSmkgoC9ZzAgYK7V84OkAepWqW3MnwIvEm5EqmqHRq3z7UUIyQaUhHLTuem5qghyUYYTTouxnmqZAhtCnXUslCKqDfJK2wMdWiXCcKDvS4In69yIHofVIhHZTgBnoeW8s/nqSPpJECJBR7kdRkfun2B9bSuRRMZfExJdBzmSaGSrJNEiccWwSPK4GR0xRYvjIEiCK2V8wGYACYmyBZVuSN1/JImmd1Ty35t2eV+tXs7pK6BAdoRPkoQtURzeogZqIoHv0jF7Qq/PkvDnvzsd0dcmZ3Rygf3A+fwCgDpxk find and AAACDHicZVDLSsNAFJ3UV62vqks3g0VwISURQZdFNy4r2Ac0oUwmt+3QmUmYmSgl9Bd0q//hTtz6D/6GX+CkDajtgQuHc+6Fc0+YcKaN6345pZXVtfWN8mZla3tnd6+6f9DWcaootGjMY9UNiQbOJLQMMxy6iQIiQg6dcHyT+50HUJrF8t5MEggEGUo2YJQYK7X8EAzpV2tu3Z0BLxOvIDVUoNmvfvtRTFMB0lBOtO55bmKCjCjDKIdpxU81JISOyRB6lkoiQAfZLOwUn1glwoNY2ZEGz9S/FxkRWk9EaDcFMSO96OXiryfhkcZCEBllfhRNM/8M+7mlRBZNF5KYwVWQMZmkBiSdBxmkHJsY583giCmghk8sIVQx+wumI6IINba/ii3JW6xkmbTP655b9+4uao3roq4yOkLH6BR56BI10C1qohaiiKFn9IJenSfnzXl3PuarJae4OUT/4Hz+AMsDm/A= so that 2 Tµ⌫ = Mpl⇤⌘µ⌫

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

all-loops renormalised CC This is a description of self-gravitating media 0, a t, xa Coordinates: {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 } ⇠ { } e.g. 3D: 3 3 AAACG3icZVDLSgMxFM3UV62vWpdugkVwIWVGBV0W3bisYB/QqSWTuW1Dk8yQZNQyzKfoVv/Dnbh14W/4BaYPUNsDFw7n3Jvce4KYM21c98vJLS2vrK7l1wsbm1vbO8XdUkNHiaJQpxGPVCsgGjiTUDfMcGjFCogIODSD4dXYb96D0iySt2YUQ0eQvmQ9RomxUrdY8idvpArCDPu1Abs77RbLbsWdAC8Sb0bKaIZat/jthxFNBEhDOdG67bmx6aREGUY5ZAU/0RATOiR9aFsqiQDdSSf/ZvjQKiHuRcqWNHii/p1IidB6JALbKYgZ6HlvLP56Eh5oJASRYeqHYZb6x9gfW0qkYTa3ielddFIm48SApNNFegnHJsLjoHDIFFDDR5YQqpi9BdMBUYQaG2fBhuTNR7JIGicVz614N2fl6uUsrjzaRwfoCHnoHFXRNaqhOqLoET2jF/TqPDlvzrvzMW3NObOZPfQPzucPacehhw== xAAACCnicZVDLSsNAFJ3UV62vqks3g0VwISVRQZdFNy4r2gc0sUwmk3bozCTMTNQS8ge61f9wJ279CX/DL3DSBtT2wIXDOffCucePGVXatr+s0sLi0vJKebWytr6xuVXd3mmrKJGYtHDEItn1kSKMCtLSVDPSjSVB3Gek448uc79zT6SikbjV45h4HA0EDSlG2kg3j3cn/WrNrtsTwHniFKQGCjT71W83iHDCidCYIaV6jh1rL0VSU8xIVnETRWKER2hAeoYKxIny0knUDB4YJYBhJM0IDSfq34sUcaXG3DebHOmhmvVy8dcT5AFHnCMRpG4QZKl7BN3ckjwNspkkOjz3UiriRBOBp0HChEEdwbwXGFBJsGZjQxCW1PwC8RBJhLVpr2JKcmYrmSft47pj153r01rjoqirDPbAPjgEDjgDDXAFmqAFMBiAZ/ACXq0n6816tz6mqyWruNkF/2B9/gD2MZrt

1 2

AAACCnicZVDLSsNAFJ34rPVVdelmsAgupCQi6LLoxmVF+4Amlslk2g6dR5iZqCXkD3Sr/+FO3PoT/oZf4KQNqO2BC4dz7oVzTxgzqo3rfjkLi0vLK6ultfL6xubWdmVnt6VlojBpYsmk6oRIE0YFaRpqGOnEiiAeMtIOR5e5374nSlMpbs04JgFHA0H7FCNjpZvHO69Xqbo1dwI4T7yCVEGBRq/y7UcSJ5wIgxnSuuu5sQlSpAzFjGRlP9EkRniEBqRrqUCc6CCdRM3goVUi2JfKjjBwov69SBHXesxDu8mRGepZLxd/PUEesOQciSj1oyhL/WPo55biaZTNJDH98yClIk4MEXgapJ8waCTMe4ERVQQbNrYEYUXtLxAPkULY2PbKtiRvtpJ50jqpeW7Nuz6t1i+KukpgHxyAI+CBM1AHV6ABmgCDAXgGL+DVeXLenHfnY7q64BQ3e+AfnM8f8uma6w== x xAAACCnicZVDLSsNAFJ3UV62vqks3g0VwISUpgi6LblxWtA9oYplMJu3QmUmYmagl5A90q//hTtz6E/6GX+CkDajtgQuHc+6Fc48fM6q0bX9ZpaXlldW18nplY3Nre6e6u9dRUSIxaeOIRbLnI0UYFaStqWakF0uCuM9I1x9f5n73nkhFI3GrJzHxOBoKGlKMtJFuHu8ag2rNrttTwEXiFKQGCrQG1W83iHDCidCYIaX6jh1rL0VSU8xIVnETRWKEx2hI+oYKxIny0mnUDB4ZJYBhJM0IDafq34sUcaUm3DebHOmRmvdy8dcT5AFHnCMRpG4QZKl7At3ckjwNsrkkOjz3UiriRBOBZ0HChEEdwbwXGFBJsGYTQxCW1PwC8QhJhLVpr2JKcuYrWSSdRt2x6871aa15UdRVBgfgEBwDB5yBJrgCLdAGGAzBM3gBr9aT9Wa9Wx+z1ZJV3OyDf7A+fwD0jZrs 1 2 space-time AAACG3icZVDLSgMxFM34rPVV69JNsAgupMyIoMuiG5cV7AM6tWQyt21okhmSjFqG+RTd6n+4E7cu/A2/wExbUNsDFw7n3Jvce4KYM21c98tZWl5ZXVsvbBQ3t7Z3dkt75aaOEkWhQSMeqXZANHAmoWGY4dCOFRARcGgFo6vcb92D0iySt2YcQ1eQgWR9RomxUq9U9idvpArCDPv1IbvzeqWKW3UnwIvEm5EKmqHeK337YUQTAdJQTrTueG5suilRhlEOWdFPNMSEjsgAOpZKIkB308m/GT6ySoj7kbIlDZ6ofydSIrQei8B2CmKGet7LxV9PwgONhCAyTP0wzFL/BPu5pUQaZnObmP5FN2UyTgxIOl2kn3BsIpwHhUOmgBo+toRQxewtmA6JItTYOIs2JG8+kkXSPK16btW7OavULmdxFdABOkTHyEPnqIauUR01EEWP6Bm9oFfnyXlz3p2PaeuSM5vZR//gfP4AZn+hhQ== fluid-space AAACG3icZVDLSgMxFM3UV62vWpdugkVwIWWmCLosunFZwT6gU0smc9uGJpkhyahlmE/Rrf6HO3Hrwt/wC0wfoLYHLhzOuTe59wQxZ9q47peTW1ldW9/Ibxa2tnd294r7paaOEkWhQSMeqXZANHAmoWGY4dCOFRARcGgFo6uJ37oHpVkkb804hq4gA8n6jBJjpV6x5E/fSBWEGfbrQ3ZX7RXLbsWdAi8Tb07KaI56r/jthxFNBEhDOdG647mx6aZEGUY5ZAU/0RATOiID6FgqiQDdTaf/ZvjYKiHuR8qWNHiq/p1IidB6LALbKYgZ6kVvIv56Eh5oJASRYeqHYZb6p9ifWEqkYbawielfdFMm48SApLNF+gnHJsKToHDIFFDDx5YQqpi9BdMhUYQaG2fBhuQtRrJMmtWK51a8m7Ny7XIeVx4doiN0gjx0jmroGtVRA1H0iJ7RC3p1npw35935mLXmnPnMAfoH5/MHaCOhhg== Building blocks X gµ⌫ @ 0@ 0 µ⌫ a b AAACP3icZVDLSgMxFM34tr6qLt0Ei+BCyoyIuhTduKxgtWBqyWRu22CSGZNMpQzzJ/6MbvUH/AJ3Iu7cmakDaj0QODnn3su9J0wEN9b3X7yJyanpmdm5+crC4tLySnV17cLEqWbQZLGIdSukBgRX0LTcCmglGqgMBVyGNyeFfzkAbXiszu0wgbakPcW7nFHrpE51v0XgNuUD3LvOiEyJSnOSUG05FZ3i3+jza/9HUaXSqdb8uj8C/k+CktRQiUan+kGimKUSlGWCGnMV+IltZ8VcJiCvkNRAQtkN7cGVo4pKMO1sdF+Ot5wS4W6s3VMWj9TfHRmVxgxl6ColtX0z7hXij6fgjsVSUhVlJIryjOxgUlhaZlE+tontHrYzrpLUgmLfi3RTgW2MizBxxDUwK4aOUKa5uwWzPtWUWRd5xYUUjEfyn1zs1gO/Hpzt1Y6Oy7jm0AbaRNsoQAfoCJ2iBmoihu7RI3pCz96D9+q9ee/fpRNe2bOO/sD7/AIHPrEu µ ⌫ ⌧ g @ @ E.g. 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 ab µ µ ⌘ ⌘

µ⌫ a b b det g @µ @⌫ +AAACDnicZVDLSsNAFJ34rPVVdelmsAiCUhIRdFl047KCfUATymQyaYfOI8xMlBLyD7rV/3Anbv0Ff8MvcNIG1PbAhcM598K5J0wY1cZ1v5yl5ZXVtfXKRnVza3tnt7a339EyVZi0sWRS9UKkCaOCtA01jPQSRRAPGemG45vC7z4QpakU92aSkICjoaAxxchYqXvq40gaPajV3YY7BVwkXknqoERrUPv2I4lTToTBDGnd99zEBBlShmJG8qqfapIgPEZD0rdUIE50kE3j5vDYKhGMpbIjDJyqfy8yxLWe8NBucmRGet4rxF9PkEcsOUciyvwoyjP/DPqFpXgW5XNJTHwVZFQkqSECz4LEKYNGwqIbGFFFsGETSxBW1P4C8QgphI1tsGpL8uYrWSSd84bnNry7i3rzuqyrAg7BETgBHrgETXALWqANMBiDZ/ACXp0n5815dz5mq0tOeXMA/sH5/AEyb5yw 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 ⌘ ··· q

M 2 L = pl R 2⇤ + m2U(X, ⌧,b,building blocks) p g 2 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 ⇥ ⇤

Choose building blocks based on symmetry Coordinates align

3 3

AAACG3icZVDLSgMxFM3UV62vWpdugkVwIWVGBV0W3bisYB/QqSWTuW1Dk8yQZNQyzKfoVv/Dnbh14W/4BaYPUNsDFw7n3Jvce4KYM21c98vJLS2vrK7l1wsbm1vbO8XdUkNHiaJQpxGPVCsgGjiTUDfMcGjFCogIODSD4dXYb96D0iySt2YUQ0eQvmQ9RomxUrdY8idvpArCDPu1Abs77RbLbsWdAC8Sb0bKaIZat/jthxFNBEhDOdG67bmx6aREGUY5ZAU/0RATOiR9aFsqiQDdSSf/ZvjQKiHuRcqWNHii/p1IidB6JALbKYgZ6HlvLP56Eh5oJASRYeqHYZb6x9gfW0qkYTa3ielddFIm48SApNNFegnHJsLjoHDIFFDDR5YQqpi9BdMBUYQaG2fBhuTNR7JIGicVz614N2fl6uUsrjzaRwfoCHnoHFXRNaqhOqLoET2jF/TqPDlvzrvzMW3NObOZPfQPzucPacehhw==

AAACCnicZVDLSsNAFJ3UV62vqks3g0VwISVRQZdFNy4r2gc0sUwmk3bozCTMTNQS8ge61f9wJ279CX/DL3DSBtT2wIXDOffCucePGVXatr+s0sLi0vJKebWytr6xuVXd3mmrKJGYtHDEItn1kSKMCtLSVDPSjSVB3Gek448uc79zT6SikbjV45h4HA0EDSlG2kg3j3cn/WrNrtsTwHniFKQGCjT71W83iHDCidCYIaV6jh1rL0VSU8xIVnETRWKER2hAeoYKxIny0knUDB4YJYBhJM0IDSfq34sUcaXG3DebHOmhmvVy8dcT5AFHnCMRpG4QZKl7BN3ckjwNspkkOjz3UiriRBOBp0HChEEdwbwXGFBJsGZjQxCW1PwC8RBJhLVpr2JKcmYrmSft47pj153r01rjoqirDPbAPjgEDjgDDXAFmqAFMBiAZ/ACXq0n6816tz6mqyWruNkF/2B9/gD2MZrt Unitary gauge: x

0

AAACGHicZVDLSgMxFM34rPXRqks3wSK4kDIjgm6EohuXFewDOmO5k0nb0CQzJBmlDP0R3ep/uBO37vwNv8C0HVDbAxcO59wL554w4Uwb1/1ylpZXVtfWCxvFza3tnVJ5d6+p41QR2iAxj1U7BE05k7RhmOG0nSgKIuS0FQ6vJ37rgSrNYnlnRgkNBPQl6zECxkrdcsmvD9i9e+kDTwaATbdccavuFHiReDmpoBz1bvnbj2KSCioN4aB1x3MTE2SgDCOcjot+qmkCZAh92rFUgqA6yKbBx/jIKhHuxcqONHiq/r3IQGg9EqHdFGAGet6biL+epI8kFgJklPlRNM78E+xPLCWyaDyXxPQugozJJDVUklmQXsqxifGkJRwxRYnhI0uAKGZ/wWQACoixXRZtSd58JYukeVr13Kp3e1apXeV1FdABOkTHyEPnqIZuUB01EEEpekYv6NV5ct6cd+djtrrk5Df76B+czx85ZZ/L = ↵t a a

AAACGXicZVDLSgNBEJz1bXytevQyGAQPEnZF0IsQ9OIxglEhG0PvbMcMzswuM7PRsORL9Kr/4U28evI3/AInMaAmBQ1FVTdUV5wJbmwQfHpT0zOzc/MLi6Wl5ZXVNX9949KkuWZYZ6lI9XUMBgVXWLfcCrzONIKMBV7Fd6cD/6qL2vBUXdhehk0Jt4q3OQPrpJbvR7UOv4HjKEYL9OEGWn45qARD0EkSjkiZjFBr+V9RkrJcorJMgDGNMMhsswBtORPYL0W5wQzYHdxiw1EFEk2zGCbv0x2nJLSdajfK0qH696IAaUxPxm5Tgu2YcW8g/noK71kqJaikiJKkX0R7NBpYWhZJfyyJbR81C66y3KJiP0HauaA2pYOaaMI1Mit6jgDT3P1CWQc0MOvKLLmSwvFKJsnlfiUMKuH5Qbl6MqprgWyRbbJLQnJIquSM1EidMNIlT+SZvHiP3qv35r3/rE55o5tN8g/exzdH76Bf = x

1 2 xAAACCnicZVDLSsNAFJ34rPVVdelmsAgupCQi6LLoxmVF+4Amlslk2g6dR5iZqCXkD3Sr/+FO3PoT/oZf4KQNqO2BC4dz7oVzTxgzqo3rfjkLi0vLK6ultfL6xubWdmVnt6VlojBpYsmk6oRIE0YFaRpqGOnEiiAeMtIOR5e5374nSlMpbs04JgFHA0H7FCNjpZvHO69Xqbo1dwI4T7yCVEGBRq/y7UcSJ5wIgxnSuuu5sQlSpAzFjGRlP9EkRniEBqRrqUCc6CCdRM3goVUi2JfKjjBwov69SBHXesxDu8mRGepZLxd/PUEesOQciSj1oyhL/WPo55biaZTNJDH98yClIk4MEXgapJ8waCTMe4ERVQQbNrYEYUXtLxAPkULY2PbKtiRvtpJ50jqpeW7Nuz6t1i+KukpgHxyAI+CBM1AHV6ABmgCDAXgGL+DVeXLenHfnY7q64BQ3e+AfnM8f8uma6w== AAACG3icZVDLSgMxFM3UV62vWpdugkVwIWWmCLosunFZwT6gU0smc9uGJpkhyahlmE/Rrf6HO3Hrwt/wC0wfoLYHLhzOuTe59wQxZ9q47peTW1ldW9/Ibxa2tnd294r7paaOEkWhQSMeqXZANHAmoWGY4dCOFRARcGgFo6uJ37oHpVkkb804hq4gA8n6jBJjpV6x5E/fSBWEGfbrQ3ZX7RXLbsWdAi8Tb07KaI56r/jthxFNBEhDOdG647mx6aZEGUY5ZAU/0RATOiID6FgqiQDdTaf/ZvjYKiHuR8qWNHiq/p1IidB6LALbKYgZ6kVvIv56Eh5oJASRYeqHYZb6p9ifWEqkYbawielfdFMm48SApLNF+gnHJsKToHDIFFDDx5YQqpi9BdMhUYQaG2fBhuQtRrJMmtWK51a8m7Ny7XIeVx4doiN0gjx0jmroGtVRA1H0iJ7RC3p1npw35935mLXmnPnMAfoH5/MHaCOhhg== 1 AAACG3icZVDLSgMxFM34rPVV69JNsAgupMyIoMuiG5cV7AM6tWQyt21okhmSjFqG+RTd6n+4E7cu/A2/wExbUNsDFw7n3Jvce4KYM21c98tZWl5ZXVsvbBQ3t7Z3dkt75aaOEkWhQSMeqXZANHAmoWGY4dCOFRARcGgFo6vcb92D0iySt2YcQ1eQgWR9RomxUq9U9idvpArCDPv1IbvzeqWKW3UnwIvEm5EKmqHeK337YUQTAdJQTrTueG5suilRhlEOWdFPNMSEjsgAOpZKIkB308m/GT6ySoj7kbIlDZ6ofydSIrQei8B2CmKGet7LxV9PwgONhCAyTP0wzFL/BPu5pUQaZnObmP5FN2UyTgxIOl2kn3BsIpwHhUOmgBo+toRQxewtmA6JItTYOIs2JG8+kkXSPK16btW7OavULmdxFdABOkTHyEPnqIauUR01EEWP6Bm9oFfnyXlz3p2PaeuSM5vZR//gfP4AZn+hhQ== 2

xAAACCnicZVDLSsNAFJ3UV62vqks3g0VwISUpgi6LblxWtA9oYplMJu3QmUmYmagl5A90q//hTtz6E/6GX+CkDajtgQuHc+6Fc48fM6q0bX9ZpaXlldW18nplY3Nre6e6u9dRUSIxaeOIRbLnI0UYFaStqWakF0uCuM9I1x9f5n73nkhFI3GrJzHxOBoKGlKMtJFuHu8ag2rNrttTwEXiFKQGCrQG1W83iHDCidCYIaX6jh1rL0VSU8xIVnETRWKEx2hI+oYKxIny0mnUDB4ZJYBhJM0IDafq34sUcaUm3DebHOmRmvdy8dcT5AFHnCMRpG4QZKl7At3ckjwNsrkkOjz3UiriRBOBZ0HChEEdwbwXGFBJsGYTQxCW1PwC8QhJhLVpr2JKcuYrWSSdRt2x6871aa15UdRVBgfgEBwDB5yBJrgCLdAGGAzBM3gBr9aT9Wa9Wx+z1ZJV3OyDf7A+fwD0jZrs But this is also Lorentz-violating massive

0, a = ↵t + ⇡0, xa + ⇡a {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 } { } g = ⌘ + h

AAACLHicZVDLSgMxFM34rPVVdekmWARBKTMi6EYounFZwT6gU0omc9uGJpkhyShlmA/wZ3Sr/+FGxK0f4BeYPqjaHgicnHMu3HuCmDNtXPfdWVhcWl5Zza3l1zc2t7YLO7s1HSWKQpVGPFKNgGjgTELVMMOhESsgIuBQD/rXQ79+D0qzSN6ZQQwtQbqSdRglxkrtQrHbTn2R+DLJLn0wZPo77k2pTbkldwQ8T7wJKaIJKu3Ctx9GNBEgDeVE66bnxqaVEmUY5ZDl/URDTGifdKFpqSQCdCsdHZPhQ6uEuBMp+6TBI/XvREqE1gMR2KQgpqdnvaH460l4oJEQRIapH4ZZ6p9gf2gpkYbZzCamc9FKmYwTA5KOF+kkHJsID5vDIVNADR9YQqhi9hZMe0QRamy/eVuSN1vJPKmdljy35N2eFctXk7pyaB8doCPkoXNURjeogqqIokf0jF7Qq/PkvDkfzuc4uuBMZvbQPzhfP8eDqZ4= µ⌫ µ⌫ µ⌫ 0 a Unitary gauge: ⇡AAACFHicZVDLSgMxFM3UV62vqks3wSK4kDIjgm4KRTcuK9gHdMaSyWTa0CQzJhmlDPMbutX/cCdu3fsbfoGZdkBtDyQczrkXzj1+zKjStv1llZaWV1bXyuuVjc2t7Z3q7l5HRYnEpI0jFsmejxRhVJC2ppqRXiwJ4j4jXX98lfvdByIVjcStnsTE42goaEgx0kby3Jje2Y38Rw17UK3ZdXsKuEicgtRAgdag+u0GEU44ERozpFTfsWPtpUhqihnJKm6iSIzwGA1J31CBOFFeOg2dwSOjBDCMpHlCw6n6dyNFXKkJ980kR3qk5r1c/PUEecQR50gEqRsEWeqeQDe3JE+DbC6JDi+8lIo40UTgWZAwYVBHMG8IBlQSrNnEEIQlNbdAPEISYW16rJiSnPlKFknntO7YdefmrNa8LOoqgwNwCI6BA85BE1yDFmgDDO7BM3gBr9aT9Wa9Wx+z0ZJV7OyDf7A+fwAALZ6p = ⇡ =0

SO(1, 3) SO(3)

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 space time internal space time ⇥ G ! ⇥ H Di↵( 1,3) 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 R H ⇢ X = gµ⌫ @ 0@ 0 ↵2g00 E.g. 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 µ ⌫ ! 00

potential for AAACDXicZVDLSgMxFM3UV62vqks3wSK4kJIRQZdFNy4r2Ad0aslkMm1skhmSjFKG+Qbd6n+4E7d+g7/hF5hpB9T2wIXDOffCucePOdMGoS+ntLS8srpWXq9sbG5t71R399o6ShShLRLxSHV9rClnkrYMM5x2Y0Wx8Dnt+OOr3O88UKVZJG/NJKZ9gYeShYxgY6X28C5FKBtUa6iOpoCLxC1IDRRoDqrfXhCRRFBpCMda91wUm36KlWGE06ziJZrGmIzxkPYslVhQ3U+naTN4ZJUAhpGyIw2cqn8vUiy0ngjfbgpsRnrey8VfT9JHEgmBZZB6QZCl3gn0ckuJNMjmkpjwop8yGSeGSjILEiYcmgjm1cCAKUoMn1iCiWL2F0hGWGFibIEVW5I7X8kiaZ/WXVR3b85qjcuirjI4AIfgGLjgHDTANWiCFiDgHjyDF/DqPDlvzrvzMVstOcXNPvgH5/MHLnqcHw== g Lorentz-violating massive gravity

2 Mpl 2 2 2 2 2 2 2 2 = m0h00 +2m1h0i m2hijhij + m3hii 2m4h00hii

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 2 ⇥ ⇤

Familiar theories:

Fierz-Pauli: 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 0 = 0; m1 = m2 = m3 = m4 = m

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 = 0; m = m = m = m =0 condensate: 0 6 1 2 3 4

Unimodular/TDiff: mAAACJnicZVDLSgMxFM34rPVVdSlCsAgupGRqQUEKRTcuK9gHdMqQyaRtaJIZk4xShq78Gd3qf7gTcec3+AWmD1DbAxfOPedeuPcEMWfaIPTpLCwuLa+sZtay6xubW9u5nd26jhJFaI1EPFLNAGvKmaQ1wwynzVhRLAJOG0H/auQ37qnSLJK3ZhDTtsBdyTqMYGMlP3cgfLcs/GIZXXh3CQ6h8JHtT22VysjP5VEBjQHniTsleTBF1c99e2FEEkGlIRxr3XJRbNopVoYRTodZL9E0xqSPu7RlqcSC6nY6fmMIj6wSwk6kbEkDx+rfjRQLrQcisJMCm56e9UbiryfpA4mEwDJMvTAcpt4J9EaWEmk4nLnEdM7bKZNxYqgkk0M6CYcmgqPMYMgUJYYPLMFEMfsLJD2sMDE22awNyZ2NZJ7UiwUXFdybUr5yOY0rA/bBITgGLjgDFXANqqAGCHgEz+AFvDpPzpvz7nxMRhec6c4e+Afn6wekQaQC 1 = m2 = 0; m0 = m3 = m4 =0 Criteria for self-tuning 1. Need four scalars for Minkowski — need symmetries that act on four fields 2. Absence of fine-tuning — self-tuning independent of parameters

3. No massive tensors (m2=0) 2 22

m ⇤ AAACIHicZVDLSgNBEJz1bXxFPXjwMhgEDxp2g6AHD6IXjxFMFNwYZmc7Ojgzu8z0qmHZn9Gr/oc38aif4Rc4eYAaCxqKqm7oqiiVwqLvf3hj4xOTU9Mzs6W5+YXFpfLyStMmmeHQ4IlMzEXELEihoYECJVykBpiKJJxHt8c9//wOjBWJPsNuCi3FrrXoCM7QSe3ymmrXDgL/Kt+p1YoQ4QGNyqFZtMsVv+r3Qf+TYEgqZIh6u/wVxgnPFGjkkll7GfgptnJmUHAJRSnMLKSM37JruHRUMwW2lfcDFHTTKTHtJMaNRtpXf1/kTFnbVZHbVAxv7KjXE388Dfc8UYrpOA/juMjDbRr2LJcsLkY+wc5+Kxc6zRA0HzzSySTFhPbaorEwwFF2HWHcCJeF8htmGEfXacmVFIxW8p80a9XArwanu5XDo2FdM2SdbJAtEpA9ckhOSJ00CCcFeSLP5MV79F69N+99sDrmDW9WyR94n99t8qMS AAACGXicZVBLSwMxGMzWV62vVY9egkXwIGW3CHosevHgoYJ9QLcu2Wy2Dc1jSbKVsvSX6FX/hzfx6sm/4S8wfYDaDgSGmW9gMlHKqDae9+UUVlbX1jeKm6Wt7Z3dPXf/oKllpjBpYMmkakdIE0YFaRhqGGmniiAeMdKKBtcTvzUkSlMp7s0oJV2OeoImFCNjpdB1eVh9qAaa8uDWpmIUumWv4k0Bl4k/J2UwRz10v4NY4owTYTBDWnd8LzXdHClDMSPjUpBpkiI8QD3SsVQgTnQ3nzYfwxOrxDCRyj5h4FT9m8gR13rEI3vJkenrRW8i/nqCPGLJORJxHsTxOA/OYDCxFM/j8UITk1x2cyrSzBCBZ0WSjEEj4WQmGFNFsGEjSxBW1P4F4j5SCBs7ZsmO5C9Oskya1YrvVfy783Ltaj5XERyBY3AKfHABauAG1EEDYDAEz+AFvDpPzpvz7nzMTgvOPHMI/sH5/AFT5qBn m < 10 eV — expect 2 ⇠ 2 (LIGO) 4. No pathologies — ghost, gradient, tachyonic

5. UV-insensitive — protected by symmetries, good EFT Strategy

• Identify phases of LVMG that satisfy these criteria

• Look for symmetries that protect them

• Determine fluid building blocks

• Solve the CC problem Time-dependent volume-preserving spatial diffs

xa xa + ⇠a(t, xb) a 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 @AAACG3icZVDLSgMxFM34rPVV69JNsAgupMyIoBuh6MZlBfuATi13MmkbmmSGJKMtQz9Ft/of7sStC3/DLzDTFtT2QOBwzr2XkxPEnGnjul/O0vLK6tp6biO/ubW9s1vYK9Z1lChCayTikWoGoClnktYMM5w2Y0VBBJw2gsF15jceqNIskndmFNO2gJ5kXUbAWKlTKPoxKMOAdwD7Q3YPl26nUHLL7gR4kXgzUkIzVDuFbz+MSCKoNISD1i3PjU07ze4STsd5P9E0BjKAHm1ZKkFQ3U4n2cf4yCoh7kbKPmnwRP27kYLQeiQCOynA9PW8l4m/nqSPJBICZJj6YThO/RPsZ5YSaTieS2K6F+2UyTgxVJJpkG7CsYlwVhQOmaLE8JElQBSzf8GkDwqIsXXmbUnefCWLpH5a9tyyd3tWqlzN6sqhA3SIjpGHzlEF3aAqqiGChugZvaBX58l5c96dj+nokjPb2Uf/4Hz+AO5NoT0= ⇠ =0 ! a a a(0, b)det(@ a/@b)=1 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 ! ⇥ ⇤ Building blocks: det(@ A) X gµ⌫ @ 0@ 0 Yb µ Fluids — 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 µ ⌫ p g ⌘ 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

LV M G — hAAACHHicZVDLSsNAFJ34rPUV7dLNYBFclJKIoMuiG5cV7AOaUCaTaTt0ZhJnJkoI+RXd6n+4E7eCv+EXOGkDantg4Nxz7oUzJ4gZVdpxvqyV1bX1jc3KVnV7Z3dv3z447KookZh0cMQi2Q+QIowK0tFUM9KPJUE8YKQXTK8Lv/dApKKRuNNpTHyOxoKOKEbaSEO7NhlmjpM3vPsEhdAMlOZDu+40nRngMnFLUgcl2kP72wsjnHAiNGZIqYHrxNrPkNQUM5JXvUSRGOEpGpOBoQJxovxsFj6HJ0YJ4SiS5gkNZ+rfiwxxpVIemE2O9EQteoX46wnyiCPOkQgzLwzzzGtAr7Akz8J8IYkeXfoZFXGiicDzIKOEQR3BoikYUkmwZqkhCEtq/gLxBEmEtemzakpyFytZJt2zpus03dvzeuuqrKsCjsAxOAUuuAAtcAPaoAMwSMEzeAGv1pP1Zr1bH/PVFau8qYF/sD5/AM2mobg= 00,hii

This describes a finite-temperature superfluid! Time-dependent volume-preserving spatial diffs

Field content:

(!2 = p2) • Two massless luminal tensors AAACFXicZVDLSgNBEJyN7/iKevQyGIQIEnaDoBdB9OIxglHB3cjsbCcZMo9lZlYJS75Dr/of3sSrZ3/DL3CSLKixoKGo6obqilPOjPX9T680Mzs3v7C4VF5eWV1br2xsXhmVaQotqrjSNzExwJmElmWWw02qgYiYw3XcPxv51/egDVPy0g5SiATpStZhlFgntWuhEtAl7cZx2m7s3VWqft0fA/8nQUGqqEDzrvIVJopmAqSlnBhzG/ipjXKiLaMchuUwM5AS2idduHVUEgEmyseph3jXKQnuKO1GWjxWf1/kRBgzELHbFMT2zLQ3En88CQ9UCUFkkodJMszDfRyOLC3yZDiVxHaOopzJNLMg6SRIJ+PYKjyqCCdMA7V84AihmrlfMO0RTah1RZZdScF0Jf/JVaMe+PXg4qB6clrUtYi20Q6qoQAdohN0jpqohSjS6Ak9oxfv0Xv13rz3yWrJK2620B94H99gdJ7V • No vectors !2 =0 • One global scalar 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

Unique theory that satisfies all criteria! UV-insensitivity

Scalar mode propagates due to higher-derivatives in EFT: m4 m2 4 !2 = ap4 0 m2 ✓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 3 ◆

no ghost if >0 UV-physics

4 2 p ! 2 UV-insensitive 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 / M Mixing with gravity 4 2 2 p ! = 2⇣p + 2 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 M

Mixing with gravity cuts off at sort distances Jeans-like instability

Not problematic due to LV

(similar tho ghost condensate) Mixing with gravity 4 2 2 p ! = 2⇣p + 2 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 M 4 1 ⇤strong coupling (⇣M ) 4 AAACUXicZVFNaxRBEO2dGI2bqKsevTRZhAiyzIQFPQa95KAQwU0C6c3S01OzadIfQ3dN4tr0X8qfMZcc9Ed482bvZsG4KWh4vPeKqnpdNkp6zPObTrb2YP3ho43H3c2tJ0+f9Z6/OPS2dQJGwirrjkvuQUkDI5So4LhxwHWp4Kg8/zjXjy7AeWnNV5w1MNZ8amQtBcdETXr77FMyV3wSGMI3dDp4dNZMqbBtmm+mMTIv9Q77Dsjp52RzmjYqng7fnAZWOy5CEcMwxkmvnw/yRdH7oFiCPlnWwaT3m1VWtBoMCsW9PynyBseBO5RCQeyy1kPDxTmfwkmChmvw47C4ONLXialobV16BumCvdsRuPZ+psvk1BzP/Ko2J/9pBi6F1ZqbKrCqioG9pWwupTSquLIJ1u/HQZqmRTDidpG6VRQtncdLK+lAoJolwIWT6RYqzniKCdMndFNIxWok98Hh7qDIB8WXYX/vwzKuDfKKbJMdUpB3ZI/skwMyIoJckWvyk/zq/Oj8yUiW3VqzzrLnJfmvss2/5GO2Cg== ⇠ pl

Need large distances and long times (> Hubble): 1 4 2 TeV rI rI mm tI

AAACInicZVDLSsRAEJz4dn2tehIvg4vgQZZEBD0uetGbgquCWcJk0tHBmUmY6ShrCP6MXvU/vIknwa/wC5xdF9S1oKGo6oauinMpLPr+uzcyOjY+MTk1XZuZnZtfqC8undqsMBzaPJOZOY+ZBSk0tFGghPPcAFOxhLP4er/nn92AsSLTJ9jNoaPYpRap4AydFNVXMDoMrVA0TA3jpYkOqzK8A2RVVG/4Tb8P+p8EA9IgAxxF9c8wyXihQCOXzNqLwM+xUzKDgkuoamFhIWf8ml3ChaOaKbCdsh+houtOSWiaGTcaaV/9fVEyZW1XxW5TMbyyw15P/PE03PJMKaaTMkwSl2iThj3LqDKphj7BdLdTCp0XCJp/P5IWkmJGe33RRBjgKLuOMG6Ey0L5FXNdoWu15koKhiv5T063moHfDI63G629QV1TZJWskQ0SkB3SIgfkiLQJJ/fkkTyRZ+/Be/Fevbfv1RFvcLNM/sD7+AK2WaTk ⇠ ⇣ 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 ⇠ ✓ vac ◆ Consistent IR-modification of gravity! (similar tho ghost condensate) Summary

Evade Weinberg by breaking Lorentz:

0 a a

AAACGHicZVDLSgMxFM34rPXRqks3wSK4kDIjgm6EohuXFewDOmO5k0nb0CQzJBmlDP0R3ep/uBO37vwNv8C0HVDbAxcO59wL554w4Uwb1/1ylpZXVtfWCxvFza3tnVJ5d6+p41QR2iAxj1U7BE05k7RhmOG0nSgKIuS0FQ6vJ37rgSrNYnlnRgkNBPQl6zECxkrdcsmvD9i9e+kDTwaATbdccavuFHiReDmpoBz1bvnbj2KSCioN4aB1x3MTE2SgDCOcjot+qmkCZAh92rFUgqA6yKbBx/jIKhHuxcqONHiq/r3IQGg9EqHdFGAGet6biL+epI8kFgJklPlRNM78E+xPLCWyaDyXxPQugozJJDVUklmQXsqxifGkJRwxRYnhI0uAKGZ/wWQACoixXRZtSd58JYukeVr13Kp3e1apXeV1FdABOkTHyEPnqIZuUB01EEEpekYv6NV5ct6cd+djtrrk5Df76B+czx85ZZ/L

= ↵t AAACGXicZVDLSgNBEJz1bXytevQyGAQPEnZF0IsQ9OIxglEhG0PvbMcMzswuM7PRsORL9Kr/4U28evI3/AInMaAmBQ1FVTdUV5wJbmwQfHpT0zOzc/MLi6Wl5ZXVNX9949KkuWZYZ6lI9XUMBgVXWLfcCrzONIKMBV7Fd6cD/6qL2vBUXdhehk0Jt4q3OQPrpJbvR7UOv4HjKEYL9OEGWn45qARD0EkSjkiZjFBr+V9RkrJcorJMgDGNMMhsswBtORPYL0W5wQzYHdxiw1EFEk2zGCbv0x2nJLSdajfK0qH696IAaUxPxm5Tgu2YcW8g/noK71kqJaikiJKkX0R7NBpYWhZJfyyJbR81C66y3KJiP0HauaA2pYOaaMI1Mit6jgDT3P1CWQc0MOvKLLmSwvFKJsnlfiUMKuH5Qbl6MqprgWyRbbJLQnJIquSM1EidMNIlT+SZvHiP3qv35r3/rE55o5tN8g/exzdH76Bf = x

2 Tµ⌫ = Mpl⇤⌘µ⌫

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

This works for ANY value of the CC! — no fine-thing

Next step: see if this is a cosmological attractor Summary

• New method for self-tuning by breaking Lorentz • Unique theory looks like a finite-temperature superfluid • Pathology-free and UV-insensitive • Consistent IR-modification of gravity • More detail in the paper (arXiv: 1805.05937) • Cosmology paper coming soon