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I. Conformal and disformal transformations
II. RSD in Standard Cosmology and Kaiser Formula
III. Coupled Dark Matter and Modified Kaiser Formula
IV. Concrete Cosmological Examples
V. Forecasts From Future Galaxy Surveys
Disformal transformations Bekenstein (1993)
!3
• Do not introduce second-order terms in the dynamical equations Bettoni+ (2013)
• Generates Beyond-Horndeski terms Bettoni+ (2013)
• Doesn’t change the number of physical d.o.f. Demenech+ (2015)
• Can distort the causal structure between the two space-times
• Special disformal transformations
• Eliminates non-minimal coupling in DBI Galileon theory Zumalacarregui+(2013)
• Horndeski’s theory: invariant under such transformation Bettoni+ (2013)
• Recently, has been used in a coupled DM model Gleyzes+ (2016)
g μν = A(ϕ, X)g μν + B(ϕ, X)∂μϕ∂νϕ
g μν = A(ϕ)g μν + B(ϕ)∂μϕ∂νϕ
X ⌘ �1
2gµ⌫@
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sha1_base64="SiNWi5XUcv0sqymup8fTFHqRdo4=">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</latexit><latexit 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sha1_base64="SiNWi5XUcv0sqymup8fTFHqRdo4=">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</latexit>
Redshift space distortions
!4
Density contrast �g,s = �g �rzvg,z
aH<latexit sha1_base64="1RJ49fBm6fFgIKfZF2aSnIIo75U=">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</latexit><latexit sha1_base64="E5PJgcOeq23N29Z94Aa5rBR6QnY=">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</latexit><latexit sha1_base64="E5PJgcOeq23N29Z94Aa5rBR6QnY=">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</latexit><latexit sha1_base64="esQyHtVSeoNtpGSFlMofD8WsXPs=">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</latexit>
overdensityoverdensity
Real space Redshift space
From real to redshift space:
s = x+vg,z
aHz
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Position
�g = bg�m<latexit sha1_base64="hpq56OVBvtlI0DexdvtkrS4F9L8=">AAACQ3icbVDLSgMxFE3qq9ZXq0s3wSK4KjMi2I1QcOOygn1AOwyZTGYamswMSaZYhvkAv8atfoAf4Te4E7eCaaeLPjwQOJxzb3JyvIQzpS3rE5a2tnd298r7lYPDo+OTau20q+JUEtohMY9l38OKchbRjmaa034iKRYepz1vfD/zexMqFYujJz1NqCNwGLGAEayN5FbrQ59yjd1sKAUKc3SHPDdEy6LIzZTVsOZAm8RekDpYoO3WYHnoxyQVNNKEY6UGtpVoJ8NSM8JpXhmmiiaYjHFIB4ZGWFDlZPPf5OjSKD4KYmlOpNFcXd7IsFBqKjwzKbAeqXVvJv7nDVIdNJ2MRUmqaUSKh4KUIx2jWTXIZ5ISzaeGYCKZyYrICEtMtClw5SZ/whJVpK5kz0XulRjJaKoYUXnFdGevN7VJutcN2/DHm3qruWixDM7BBbgCNrgFLfAA2qADCHgBr+ANvMMP+AW/4U8xWoKLnTOwAvj7B6/lsJc=</latexit><latexit sha1_base64="hpq56OVBvtlI0DexdvtkrS4F9L8=">AAACQ3icbVDLSgMxFE3qq9ZXq0s3wSK4KjMi2I1QcOOygn1AOwyZTGYamswMSaZYhvkAv8atfoAf4Te4E7eCaaeLPjwQOJxzb3JyvIQzpS3rE5a2tnd298r7lYPDo+OTau20q+JUEtohMY9l38OKchbRjmaa034iKRYepz1vfD/zexMqFYujJz1NqCNwGLGAEayN5FbrQ59yjd1sKAUKc3SHPDdEy6LIzZTVsOZAm8RekDpYoO3WYHnoxyQVNNKEY6UGtpVoJ8NSM8JpXhmmiiaYjHFIB4ZGWFDlZPPf5OjSKD4KYmlOpNFcXd7IsFBqKjwzKbAeqXVvJv7nDVIdNJ2MRUmqaUSKh4KUIx2jWTXIZ5ISzaeGYCKZyYrICEtMtClw5SZ/whJVpK5kz0XulRjJaKoYUXnFdGevN7VJutcN2/DHm3qruWixDM7BBbgCNrgFLfAA2qADCHgBr+ANvMMP+AW/4U8xWoKLnTOwAvj7B6/lsJc=</latexit><latexit sha1_base64="hpq56OVBvtlI0DexdvtkrS4F9L8=">AAACQ3icbVDLSgMxFE3qq9ZXq0s3wSK4KjMi2I1QcOOygn1AOwyZTGYamswMSaZYhvkAv8atfoAf4Te4E7eCaaeLPjwQOJxzb3JyvIQzpS3rE5a2tnd298r7lYPDo+OTau20q+JUEtohMY9l38OKchbRjmaa034iKRYepz1vfD/zexMqFYujJz1NqCNwGLGAEayN5FbrQ59yjd1sKAUKc3SHPDdEy6LIzZTVsOZAm8RekDpYoO3WYHnoxyQVNNKEY6UGtpVoJ8NSM8JpXhmmiiaYjHFIB4ZGWFDlZPPf5OjSKD4KYmlOpNFcXd7IsFBqKjwzKbAeqXVvJv7nDVIdNJ2MRUmqaUSKh4KUIx2jWTXIZ5ISzaeGYCKZyYrICEtMtClw5SZ/whJVpK5kz0XulRjJaKoYUXnFdGevN7VJutcN2/DHm3qruWixDM7BBbgCNrgFLfAA2qADCHgBr+ANvMMP+AW/4U8xWoKLnTOwAvj7B6/lsJc=</latexit><latexit sha1_base64="hpq56OVBvtlI0DexdvtkrS4F9L8=">AAACQ3icbVDLSgMxFE3qq9ZXq0s3wSK4KjMi2I1QcOOygn1AOwyZTGYamswMSaZYhvkAv8atfoAf4Te4E7eCaaeLPjwQOJxzb3JyvIQzpS3rE5a2tnd298r7lYPDo+OTau20q+JUEtohMY9l38OKchbRjmaa034iKRYepz1vfD/zexMqFYujJz1NqCNwGLGAEayN5FbrQ59yjd1sKAUKc3SHPDdEy6LIzZTVsOZAm8RekDpYoO3WYHnoxyQVNNKEY6UGtpVoJ8NSM8JpXhmmiiaYjHFIB4ZGWFDlZPPf5OjSKD4KYmlOpNFcXd7IsFBqKjwzKbAeqXVvJv7nDVIdNJ2MRUmqaUSKh4KUIx2jWTXIZ5ISzaeGYCKZyYrICEtMtClw5SZ/whJVpK5kz0XulRjJaKoYUXnFdGevN7VJutcN2/DHm3qruWixDM7BBbgCNrgFLfAA2qADCHgBr+ANvMMP+AW/4U8xWoKLnTOwAvj7B6/lsJc=</latexit>
(linear bias)
ng = ng(1 + �g)<latexit sha1_base64="NLnN2/1en/tws+8s2YjDIjiBK+g=">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</latexit><latexit sha1_base64="NLnN2/1en/tws+8s2YjDIjiBK+g=">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</latexit><latexit sha1_base64="NLnN2/1en/tws+8s2YjDIjiBK+g=">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</latexit><latexit sha1_base64="NLnN2/1en/tws+8s2YjDIjiBK+g=">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</latexit>
number density ave. number densityng :
<latexit sha1_base64="n9KSpk2MpWae55EEIOphNZIKbns=">AAACMHicbVDLSsNAFJ34rPGV6tLNYBFclUQExVXBjcsK9gFNCJPJpB06mYSZSTGE/Ilb/QC/Rlfi1q9w2mRhWw8MHM65987hBCmjUtn2p7GxubW9s9vYM/cPDo+OreZJXyaZwKSHE5aIYYAkYZSTnqKKkWEqCIoDRgbB9H7uD2ZESJrwJ5WnxIvRmNOIYqS05FuWGyBR8NIvXBHDcXnnWy27bS8A14lTkxao0fWbRsMNE5zFhCvMkJQjx06VVyChKGakNN1MkhThKRqTkaYcxUR6xSJ6CS+0EsIoEfpxBRfq340CxVLmcaAnY6QmctWbi/95o0xFt15BeZopwnH1UZQxqBI47wGGVBCsWK4JwoLqrBBPkEBY6baWLoUzmsoqtVk8V7mXYqSTXFIsS1N356w2tU76V21H88frVseuW2yAM3AOLoEDbkAHPIAu6AEMZuAFvII34934ML6M72p0w6h3TsESjJ9fACKpUg==</latexit><latexit sha1_base64="n9KSpk2MpWae55EEIOphNZIKbns=">AAACMHicbVDLSsNAFJ34rPGV6tLNYBFclUQExVXBjcsK9gFNCJPJpB06mYSZSTGE/Ilb/QC/Rlfi1q9w2mRhWw8MHM65987hBCmjUtn2p7GxubW9s9vYM/cPDo+OreZJXyaZwKSHE5aIYYAkYZSTnqKKkWEqCIoDRgbB9H7uD2ZESJrwJ5WnxIvRmNOIYqS05FuWGyBR8NIvXBHDcXnnWy27bS8A14lTkxao0fWbRsMNE5zFhCvMkJQjx06VVyChKGakNN1MkhThKRqTkaYcxUR6xSJ6CS+0EsIoEfpxBRfq340CxVLmcaAnY6QmctWbi/95o0xFt15BeZopwnH1UZQxqBI47wGGVBCsWK4JwoLqrBBPkEBY6baWLoUzmsoqtVk8V7mXYqSTXFIsS1N356w2tU76V21H88frVseuW2yAM3AOLoEDbkAHPIAu6AEMZuAFvII34934ML6M72p0w6h3TsESjJ9fACKpUg==</latexit><latexit sha1_base64="n9KSpk2MpWae55EEIOphNZIKbns=">AAACMHicbVDLSsNAFJ34rPGV6tLNYBFclUQExVXBjcsK9gFNCJPJpB06mYSZSTGE/Ilb/QC/Rlfi1q9w2mRhWw8MHM65987hBCmjUtn2p7GxubW9s9vYM/cPDo+OreZJXyaZwKSHE5aIYYAkYZSTnqKKkWEqCIoDRgbB9H7uD2ZESJrwJ5WnxIvRmNOIYqS05FuWGyBR8NIvXBHDcXnnWy27bS8A14lTkxao0fWbRsMNE5zFhCvMkJQjx06VVyChKGakNN1MkhThKRqTkaYcxUR6xSJ6CS+0EsIoEfpxBRfq340CxVLmcaAnY6QmctWbi/95o0xFt15BeZopwnH1UZQxqBI47wGGVBCsWK4JwoLqrBBPkEBY6baWLoUzmsoqtVk8V7mXYqSTXFIsS1N356w2tU76V21H88frVseuW2yAM3AOLoEDbkAHPIAu6AEMZuAFvII34934ML6M72p0w6h3TsESjJ9fACKpUg==</latexit><latexit sha1_base64="n9KSpk2MpWae55EEIOphNZIKbns=">AAACMHicbVDLSsNAFJ34rPGV6tLNYBFclUQExVXBjcsK9gFNCJPJpB06mYSZSTGE/Ilb/QC/Rlfi1q9w2mRhWw8MHM65987hBCmjUtn2p7GxubW9s9vYM/cPDo+OreZJXyaZwKSHE5aIYYAkYZSTnqKKkWEqCIoDRgbB9H7uD2ZESJrwJ5WnxIvRmNOIYqS05FuWGyBR8NIvXBHDcXnnWy27bS8A14lTkxao0fWbRsMNE5zFhCvMkJQjx06VVyChKGakNN1MkhThKRqTkaYcxUR6xSJ6CS+0EsIoEfpxBRfq340CxVLmcaAnY6QmctWbi/95o0xFt15BeZopwnH1UZQxqBI47wGGVBCsWK4JwoLqrBBPkEBY6baWLoUzmsoqtVk8V7mXYqSTXFIsS1N356w2tU76V21H88frVseuW2yAM3AOLoEDbkAHPIAu6AEMZuAFvII34934ML6M72p0w6h3TsESjJ9fACKpUg==</latexit>
ng :<latexit sha1_base64="xV5cnxmLF2Lgp9suI8myVzaxnZo=">AAACKHicbVBPS8MwHE3nv1n/bXr0EhyCp9GKoHgaePE4wf3BrYw0TbewNA1JOiyl38KrfgA/jTfZ1U9itvbgNh8EHu/9fsnL8wWjSjvO3Kpsbe/s7lX37YPDo+OTWv20q+JEYtLBMYtl30eKMMpJR1PNSF9IgiKfkZ4/fVj4vRmRisb8WaeCeBEacxpSjLSRXvgoG8oIjvP7Ua3hNJ0l4CZxS9IAJdqjulUdBjFOIsI1ZkipgesI7WVIaooZye1hoohAeIrGZGAoRxFRXraMnMNLowQwjKU5XMOl+ncjQ5FSaeSbyQjpiVr3FuJ/3iDR4Z2XUS4STTguHgoTBnUMF/+HAZUEa5YagrCkJivEEyQR1qallZuCGRWqSG1nr0XulRhikiqKVW6b7tz1pjZJ97rpGv5002g5ZYtVcA4uwBVwwS1ogUfQBh2AAQdv4B18WJ/Wl/VtzYvRilXunIEVWD+/H+umXA==</latexit><latexit sha1_base64="xV5cnxmLF2Lgp9suI8myVzaxnZo=">AAACKHicbVBPS8MwHE3nv1n/bXr0EhyCp9GKoHgaePE4wf3BrYw0TbewNA1JOiyl38KrfgA/jTfZ1U9itvbgNh8EHu/9fsnL8wWjSjvO3Kpsbe/s7lX37YPDo+OTWv20q+JEYtLBMYtl30eKMMpJR1PNSF9IgiKfkZ4/fVj4vRmRisb8WaeCeBEacxpSjLSRXvgoG8oIjvP7Ua3hNJ0l4CZxS9IAJdqjulUdBjFOIsI1ZkipgesI7WVIaooZye1hoohAeIrGZGAoRxFRXraMnMNLowQwjKU5XMOl+ncjQ5FSaeSbyQjpiVr3FuJ/3iDR4Z2XUS4STTguHgoTBnUMF/+HAZUEa5YagrCkJivEEyQR1qallZuCGRWqSG1nr0XulRhikiqKVW6b7tz1pjZJ97rpGv5002g5ZYtVcA4uwBVwwS1ogUfQBh2AAQdv4B18WJ/Wl/VtzYvRilXunIEVWD+/H+umXA==</latexit><latexit sha1_base64="xV5cnxmLF2Lgp9suI8myVzaxnZo=">AAACKHicbVBPS8MwHE3nv1n/bXr0EhyCp9GKoHgaePE4wf3BrYw0TbewNA1JOiyl38KrfgA/jTfZ1U9itvbgNh8EHu/9fsnL8wWjSjvO3Kpsbe/s7lX37YPDo+OTWv20q+JEYtLBMYtl30eKMMpJR1PNSF9IgiKfkZ4/fVj4vRmRisb8WaeCeBEacxpSjLSRXvgoG8oIjvP7Ua3hNJ0l4CZxS9IAJdqjulUdBjFOIsI1ZkipgesI7WVIaooZye1hoohAeIrGZGAoRxFRXraMnMNLowQwjKU5XMOl+ncjQ5FSaeSbyQjpiVr3FuJ/3iDR4Z2XUS4STTguHgoTBnUMF/+HAZUEa5YagrCkJivEEyQR1qallZuCGRWqSG1nr0XulRhikiqKVW6b7tz1pjZJ97rpGv5002g5ZYtVcA4uwBVwwS1ogUfQBh2AAQdv4B18WJ/Wl/VtzYvRilXunIEVWD+/H+umXA==</latexit><latexit sha1_base64="xV5cnxmLF2Lgp9suI8myVzaxnZo=">AAACKHicbVBPS8MwHE3nv1n/bXr0EhyCp9GKoHgaePE4wf3BrYw0TbewNA1JOiyl38KrfgA/jTfZ1U9itvbgNh8EHu/9fsnL8wWjSjvO3Kpsbe/s7lX37YPDo+OTWv20q+JEYtLBMYtl30eKMMpJR1PNSF9IgiKfkZ4/fVj4vRmRisb8WaeCeBEacxpSjLSRXvgoG8oIjvP7Ua3hNJ0l4CZxS9IAJdqjulUdBjFOIsI1ZkipgesI7WVIaooZye1hoohAeIrGZGAoRxFRXraMnMNLowQwjKU5XMOl+ncjQ5FSaeSbyQjpiVr3FuJ/3iDR4Z2XUS4STTguHgoTBnUMF/+HAZUEa5YagrCkJivEEyQR1qallZuCGRWqSG1nr0XulRhikiqKVW6b7tz1pjZJ97rpGv5002g5ZYtVcA4uwBVwwS1ogUfQBh2AAQdv4B18WJ/Wl/VtzYvRilXunIEVWD+/H+umXA==</latexit>
vg ⇡ vm<latexit sha1_base64="TG1epoNpFWK5QlkTb+IDtB85Fu4=">AAACO3icbVBLTwIxGOziC/EFeuTSSEw8kV1jIkcSLx4xkUcCG9ItBRrabtN2CZvNHvw1XvUH+EM8ezNevVvYPQg4SZPJzPe10wkko9q47odT2Nnd2z8oHpaOjk9Oz8qV844OI4VJG4csVL0AacKoIG1DDSM9qQjiASPdYHa/9LtzojQNxZOJJfE5mgg6phgZKw3L1fkwGSgOJykcIClVuIC5wtNhuebW3RXgNvFyUgM5WsOKUxyMQhxxIgxmSOu+50rjJ0gZihlJS4NIE4nwDE1I31KBONF+svpFCq+sMoLjUNkjDFypfzcSxLWOeWAnOTJTvektxf+8fmTGDT+hQkaGCJw9NI4YNCFcVgJHVBFsWGwJworarBBPkULY2OLWbhrNqdRZ6lKyyHKvxZDTWFOs05Ltzttsapt0buqe5Y+3tWYjb7EIquASXAMP3IEmeAAt0AYYPIMX8ArenHfn0/lyvrPRgpPvXIA1OD+/+42t2A==</latexit><latexit sha1_base64="TG1epoNpFWK5QlkTb+IDtB85Fu4=">AAACO3icbVBLTwIxGOziC/EFeuTSSEw8kV1jIkcSLx4xkUcCG9ItBRrabtN2CZvNHvw1XvUH+EM8ezNevVvYPQg4SZPJzPe10wkko9q47odT2Nnd2z8oHpaOjk9Oz8qV844OI4VJG4csVL0AacKoIG1DDSM9qQjiASPdYHa/9LtzojQNxZOJJfE5mgg6phgZKw3L1fkwGSgOJykcIClVuIC5wtNhuebW3RXgNvFyUgM5WsOKUxyMQhxxIgxmSOu+50rjJ0gZihlJS4NIE4nwDE1I31KBONF+svpFCq+sMoLjUNkjDFypfzcSxLWOeWAnOTJTvektxf+8fmTGDT+hQkaGCJw9NI4YNCFcVgJHVBFsWGwJworarBBPkULY2OLWbhrNqdRZ6lKyyHKvxZDTWFOs05Ltzttsapt0buqe5Y+3tWYjb7EIquASXAMP3IEmeAAt0AYYPIMX8ArenHfn0/lyvrPRgpPvXIA1OD+/+42t2A==</latexit><latexit sha1_base64="TG1epoNpFWK5QlkTb+IDtB85Fu4=">AAACO3icbVBLTwIxGOziC/EFeuTSSEw8kV1jIkcSLx4xkUcCG9ItBRrabtN2CZvNHvw1XvUH+EM8ezNevVvYPQg4SZPJzPe10wkko9q47odT2Nnd2z8oHpaOjk9Oz8qV844OI4VJG4csVL0AacKoIG1DDSM9qQjiASPdYHa/9LtzojQNxZOJJfE5mgg6phgZKw3L1fkwGSgOJykcIClVuIC5wtNhuebW3RXgNvFyUgM5WsOKUxyMQhxxIgxmSOu+50rjJ0gZihlJS4NIE4nwDE1I31KBONF+svpFCq+sMoLjUNkjDFypfzcSxLWOeWAnOTJTvektxf+8fmTGDT+hQkaGCJw9NI4YNCFcVgJHVBFsWGwJworarBBPkULY2OLWbhrNqdRZ6lKyyHKvxZDTWFOs05Ltzttsapt0buqe5Y+3tWYjb7EIquASXAMP3IEmeAAt0AYYPIMX8ArenHfn0/lyvrPRgpPvXIA1OD+/+42t2A==</latexit><latexit sha1_base64="TG1epoNpFWK5QlkTb+IDtB85Fu4=">AAACO3icbVBLTwIxGOziC/EFeuTSSEw8kV1jIkcSLx4xkUcCG9ItBRrabtN2CZvNHvw1XvUH+EM8ezNevVvYPQg4SZPJzPe10wkko9q47odT2Nnd2z8oHpaOjk9Oz8qV844OI4VJG4csVL0AacKoIG1DDSM9qQjiASPdYHa/9LtzojQNxZOJJfE5mgg6phgZKw3L1fkwGSgOJykcIClVuIC5wtNhuebW3RXgNvFyUgM5WsOKUxyMQhxxIgxmSOu+50rjJ0gZihlJS4NIE4nwDE1I31KBONF+svpFCq+sMoLjUNkjDFypfzcSxLWOeWAnOTJTvektxf+8fmTGDT+hQkaGCJw9NI4YNCFcVgJHVBFsWGwJworarBBPkULY2OLWbhrNqdRZ6lKyyHKvxZDTWFOs05Ltzttsapt0buqe5Y+3tWYjb7EIquASXAMP3IEmeAAt0AYYPIMX8ArenHfn0/lyvrPRgpPvXIA1OD+/+42t2A==</latexit>
Line
of s
ightz
galaxy’s peculiar velocity along the line of sight
Newtonian gauge:
Poisson, continuity and Euler equations (in Fourier space):
EoM for the density contrast
k
aH� 1
<latexit sha1_base64="LmjPHRTKrbNsfiSqVW50LfQaVUo=">AAACMnicbVDNSsNAGNzUvxr/2nr0slgETyURQY8FLz1WsK3QhLLZbNKlm2TZ3ZSGkFfxqg/gy+hNvPoQbpscbOvAwjDzfd8O43FGpbKsD6O2s7u3f1A/NI+OT07PGs3WUCapwGSAE5aIZw9JwmhMBooqRp65ICjyGBl5s4elP5oTIWkSP6mMEzdCYUwDipHS0qTRcgKBcD4rctQroBOG0J402lbHWgFuE7sibVChP2kadcdPcBqRWGGGpBzbFldujoSimJHCdFJJOMIzFJKxpjGKiHTzVfgCXmnFh0Ei9IsVXKl/N3IUSZlFnp6MkJrKTW8p/ueNUxXcuzmNeapIjMuPgpRBlcBlE9CngmDFMk0QFlRnhXiKdBtK97V2yZ9TLsvUZr4oc6/F4NNMUiwLU3dnbza1TYY3HVvzx9t216parIMLcAmugQ3uQBf0QB8MAAYL8AJewZvxbnwaX8Z3OVozqp1zsAbj5xfcMKmz</latexit><latexit sha1_base64="LmjPHRTKrbNsfiSqVW50LfQaVUo=">AAACMnicbVDNSsNAGNzUvxr/2nr0slgETyURQY8FLz1WsK3QhLLZbNKlm2TZ3ZSGkFfxqg/gy+hNvPoQbpscbOvAwjDzfd8O43FGpbKsD6O2s7u3f1A/NI+OT07PGs3WUCapwGSAE5aIZw9JwmhMBooqRp65ICjyGBl5s4elP5oTIWkSP6mMEzdCYUwDipHS0qTRcgKBcD4rctQroBOG0J402lbHWgFuE7sibVChP2kadcdPcBqRWGGGpBzbFldujoSimJHCdFJJOMIzFJKxpjGKiHTzVfgCXmnFh0Ei9IsVXKl/N3IUSZlFnp6MkJrKTW8p/ueNUxXcuzmNeapIjMuPgpRBlcBlE9CngmDFMk0QFlRnhXiKdBtK97V2yZ9TLsvUZr4oc6/F4NNMUiwLU3dnbza1TYY3HVvzx9t216parIMLcAmugQ3uQBf0QB8MAAYL8AJewZvxbnwaX8Z3OVozqp1zsAbj5xfcMKmz</latexit><latexit sha1_base64="LmjPHRTKrbNsfiSqVW50LfQaVUo=">AAACMnicbVDNSsNAGNzUvxr/2nr0slgETyURQY8FLz1WsK3QhLLZbNKlm2TZ3ZSGkFfxqg/gy+hNvPoQbpscbOvAwjDzfd8O43FGpbKsD6O2s7u3f1A/NI+OT07PGs3WUCapwGSAE5aIZw9JwmhMBooqRp65ICjyGBl5s4elP5oTIWkSP6mMEzdCYUwDipHS0qTRcgKBcD4rctQroBOG0J402lbHWgFuE7sibVChP2kadcdPcBqRWGGGpBzbFldujoSimJHCdFJJOMIzFJKxpjGKiHTzVfgCXmnFh0Ei9IsVXKl/N3IUSZlFnp6MkJrKTW8p/ueNUxXcuzmNeapIjMuPgpRBlcBlE9CngmDFMk0QFlRnhXiKdBtK97V2yZ9TLsvUZr4oc6/F4NNMUiwLU3dnbza1TYY3HVvzx9t216parIMLcAmugQ3uQBf0QB8MAAYL8AJewZvxbnwaX8Z3OVozqp1zsAbj5xfcMKmz</latexit><latexit sha1_base64="LmjPHRTKrbNsfiSqVW50LfQaVUo=">AAACMnicbVDNSsNAGNzUvxr/2nr0slgETyURQY8FLz1WsK3QhLLZbNKlm2TZ3ZSGkFfxqg/gy+hNvPoQbpscbOvAwjDzfd8O43FGpbKsD6O2s7u3f1A/NI+OT07PGs3WUCapwGSAE5aIZw9JwmhMBooqRp65ICjyGBl5s4elP5oTIWkSP6mMEzdCYUwDipHS0qTRcgKBcD4rctQroBOG0J402lbHWgFuE7sibVChP2kadcdPcBqRWGGGpBzbFldujoSimJHCdFJJOMIzFJKxpjGKiHTzVfgCXmnFh0Ei9IsVXKl/N3IUSZlFnp6MkJrKTW8p/ueNUxXcuzmNeapIjMuPgpRBlcBlE9CngmDFMk0QFlRnhXiKdBtK97V2yZ9TLsvUZr4oc6/F4NNMUiwLU3dnbza1TYY3HVvzx9t216parIMLcAmugQ3uQBf0QB8MAAYL8AJewZvxbnwaX8Z3OVozqp1zsAbj5xfcMKmz</latexit>
ds2 = �[1 + 2�(t,x)]dt2 + a2(t)[1� 2 (t,x)]dx2<latexit sha1_base64="kxBQsHwBBxnFL9t2d/59czKtSxk=">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</latexit><latexit sha1_base64="kxBQsHwBBxnFL9t2d/59czKtSxk=">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</latexit><latexit sha1_base64="kxBQsHwBBxnFL9t2d/59czKtSxk=">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</latexit><latexit sha1_base64="kxBQsHwBBxnFL9t2d/59czKtSxk=">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</latexit>
vm = �a2H
k2fm�m
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k2
a2� = �4⇡G⇢m�m
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�m +k2
a2vm = 0
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vm � � = 0<latexit sha1_base64="nJpJ53WW+pG03pcHzGXdBQ/Upak=">AAACOnicbVDLSsNAFJ34rPHV6k43g0VwY0mkoBuh4MZlBfuAJoTJZNIMnUnCzKRYQsCvcasf4I+4dSdu/QCnTRe29cDA4dzXmeOnjEplWR/G2vrG5tZ2Zcfc3ds/OKzWjroyyQQmHZywRPR9JAmjMekoqhjpp4Ig7jPS80d303pvTISkSfyoJilxORrGNKQYKS151RMnSFQ+LrzcERzyAl5Cpx1ReAstr1q3GtYMcJXYc1IHc7S9mlHR23DGSawwQ1IObCtVbo6EopiRwnQySVKER2hIBprGiBPp5rNPFPBcKwEME6FfrOBM/TuRIy7lhPu6kyMVyeXaVPyvNshUeOPmNE4zRWJcHgozBlUCp4nAgAqCFZtogrCg2ivEERIIK53bwqZgTFNZujbzp9L3go00mkiKZWHq7OzlpFZJ96pha/7QrLea8xQr4BScgQtgg2vQAvegDToAg2fwAl7Bm/FufBpfxnfZumbMZ47BAoyfXxHOrDc=</latexit><latexit sha1_base64="nJpJ53WW+pG03pcHzGXdBQ/Upak=">AAACOnicbVDLSsNAFJ34rPHV6k43g0VwY0mkoBuh4MZlBfuAJoTJZNIMnUnCzKRYQsCvcasf4I+4dSdu/QCnTRe29cDA4dzXmeOnjEplWR/G2vrG5tZ2Zcfc3ds/OKzWjroyyQQmHZywRPR9JAmjMekoqhjpp4Ig7jPS80d303pvTISkSfyoJilxORrGNKQYKS151RMnSFQ+LrzcERzyAl5Cpx1ReAstr1q3GtYMcJXYc1IHc7S9mlHR23DGSawwQ1IObCtVbo6EopiRwnQySVKER2hIBprGiBPp5rNPFPBcKwEME6FfrOBM/TuRIy7lhPu6kyMVyeXaVPyvNshUeOPmNE4zRWJcHgozBlUCp4nAgAqCFZtogrCg2ivEERIIK53bwqZgTFNZujbzp9L3go00mkiKZWHq7OzlpFZJ96pha/7QrLea8xQr4BScgQtgg2vQAvegDToAg2fwAl7Bm/FufBpfxnfZumbMZ47BAoyfXxHOrDc=</latexit><latexit sha1_base64="nJpJ53WW+pG03pcHzGXdBQ/Upak=">AAACOnicbVDLSsNAFJ34rPHV6k43g0VwY0mkoBuh4MZlBfuAJoTJZNIMnUnCzKRYQsCvcasf4I+4dSdu/QCnTRe29cDA4dzXmeOnjEplWR/G2vrG5tZ2Zcfc3ds/OKzWjroyyQQmHZywRPR9JAmjMekoqhjpp4Ig7jPS80d303pvTISkSfyoJilxORrGNKQYKS151RMnSFQ+LrzcERzyAl5Cpx1ReAstr1q3GtYMcJXYc1IHc7S9mlHR23DGSawwQ1IObCtVbo6EopiRwnQySVKER2hIBprGiBPp5rNPFPBcKwEME6FfrOBM/TuRIy7lhPu6kyMVyeXaVPyvNshUeOPmNE4zRWJcHgozBlUCp4nAgAqCFZtogrCg2ivEERIIK53bwqZgTFNZujbzp9L3go00mkiKZWHq7OzlpFZJ96pha/7QrLea8xQr4BScgQtgg2vQAvegDToAg2fwAl7Bm/FufBpfxnfZumbMZ47BAoyfXxHOrDc=</latexit><latexit sha1_base64="nJpJ53WW+pG03pcHzGXdBQ/Upak=">AAACOnicbVDLSsNAFJ34rPHV6k43g0VwY0mkoBuh4MZlBfuAJoTJZNIMnUnCzKRYQsCvcasf4I+4dSdu/QCnTRe29cDA4dzXmeOnjEplWR/G2vrG5tZ2Zcfc3ds/OKzWjroyyQQmHZywRPR9JAmjMekoqhjpp4Ig7jPS80d303pvTISkSfyoJilxORrGNKQYKS151RMnSFQ+LrzcERzyAl5Cpx1ReAstr1q3GtYMcJXYc1IHc7S9mlHR23DGSawwQ1IObCtVbo6EopiRwnQySVKER2hIBprGiBPp5rNPFPBcKwEME6FfrOBM/TuRIy7lhPu6kyMVyeXaVPyvNshUeOPmNE4zRWJcHgozBlUCp4nAgAqCFZtogrCg2ivEERIIK53bwqZgTFNZujbzp9L3go00mkiKZWHq7OzlpFZJ96pha/7QrLea8xQr4BScgQtgg2vQAvegDToAg2fwAl7Bm/FufBpfxnfZumbMZ47BAoyfXxHOrDc=</latexit>
Sub-horizon approximation:
Growth function
Linear growth rate
Initial density contrast�0 :<latexit sha1_base64="er7koTuNZdb73lX12GVp+PsNIVI=">AAACJ3icbVDNSgMxGEzqX61/rR69BIvgqeyKoHgqePFYwf5Iu5RsNtuGZrMhyRaXpU/hVR/Ap/EmevRNTLt7sK0DgWHm+5LJ+JIzbRznG5Y2Nre2d8q7lb39g8Ojau24o+NEEdomMY9Vz8eaciZo2zDDaU8qiiOf064/uZv73SlVmsXi0aSSehEeCRYygo2VngYB5QYPndthte40nAXQOnELUgcFWsMaLA+CmCQRFYZwrHXfdaTxMqwMI5zOKoNEU4nJBI9o31KBI6q9bJF4hs6tEqAwVvYIgxbq340MR1qnkW8nI2zGetWbi/95/cSEN17GhEwMFSR/KEw4MjGafx8FTFFieGoJJorZrIiMscLE2JKWbgqmTOo8dSV7znMvxZDjVDOiZxXbnbva1DrpXDZcyx+u6s1G0WIZnIIzcAFccA2a4B60QBsQEIEX8Are4Dv8gJ/wKx8twWLnBCwB/vwC+WWlvg==</latexit><latexit sha1_base64="er7koTuNZdb73lX12GVp+PsNIVI=">AAACJ3icbVDNSgMxGEzqX61/rR69BIvgqeyKoHgqePFYwf5Iu5RsNtuGZrMhyRaXpU/hVR/Ap/EmevRNTLt7sK0DgWHm+5LJ+JIzbRznG5Y2Nre2d8q7lb39g8Ojau24o+NEEdomMY9Vz8eaciZo2zDDaU8qiiOf064/uZv73SlVmsXi0aSSehEeCRYygo2VngYB5QYPndthte40nAXQOnELUgcFWsMaLA+CmCQRFYZwrHXfdaTxMqwMI5zOKoNEU4nJBI9o31KBI6q9bJF4hs6tEqAwVvYIgxbq340MR1qnkW8nI2zGetWbi/95/cSEN17GhEwMFSR/KEw4MjGafx8FTFFieGoJJorZrIiMscLE2JKWbgqmTOo8dSV7znMvxZDjVDOiZxXbnbva1DrpXDZcyx+u6s1G0WIZnIIzcAFccA2a4B60QBsQEIEX8Are4Dv8gJ/wKx8twWLnBCwB/vwC+WWlvg==</latexit><latexit sha1_base64="er7koTuNZdb73lX12GVp+PsNIVI=">AAACJ3icbVDNSgMxGEzqX61/rR69BIvgqeyKoHgqePFYwf5Iu5RsNtuGZrMhyRaXpU/hVR/Ap/EmevRNTLt7sK0DgWHm+5LJ+JIzbRznG5Y2Nre2d8q7lb39g8Ojau24o+NEEdomMY9Vz8eaciZo2zDDaU8qiiOf064/uZv73SlVmsXi0aSSehEeCRYygo2VngYB5QYPndthte40nAXQOnELUgcFWsMaLA+CmCQRFYZwrHXfdaTxMqwMI5zOKoNEU4nJBI9o31KBI6q9bJF4hs6tEqAwVvYIgxbq340MR1qnkW8nI2zGetWbi/95/cSEN17GhEwMFSR/KEw4MjGafx8FTFFieGoJJorZrIiMscLE2JKWbgqmTOo8dSV7znMvxZDjVDOiZxXbnbva1DrpXDZcyx+u6s1G0WIZnIIzcAFccA2a4B60QBsQEIEX8Are4Dv8gJ/wKx8twWLnBCwB/vwC+WWlvg==</latexit><latexit sha1_base64="er7koTuNZdb73lX12GVp+PsNIVI=">AAACJ3icbVDNSgMxGEzqX61/rR69BIvgqeyKoHgqePFYwf5Iu5RsNtuGZrMhyRaXpU/hVR/Ap/EmevRNTLt7sK0DgWHm+5LJ+JIzbRznG5Y2Nre2d8q7lb39g8Ojau24o+NEEdomMY9Vz8eaciZo2zDDaU8qiiOf064/uZv73SlVmsXi0aSSehEeCRYygo2VngYB5QYPndthte40nAXQOnELUgcFWsMaLA+CmCQRFYZwrHXfdaTxMqwMI5zOKoNEU4nJBI9o31KBI6q9bJF4hs6tEqAwVvYIgxbq340MR1qnkW8nI2zGetWbi/95/cSEN17GhEwMFSR/KEw4MjGafx8FTFFieGoJJorZrIiMscLE2JKWbgqmTOo8dSV7znMvxZDjVDOiZxXbnbva1DrpXDZcyx+u6s1G0WIZnIIzcAFccA2a4B60QBsQEIEX8Are4Dv8gJ/wKx8twWLnBCwB/vwC+WWlvg==</latexit>
Velocity equation
Standard cosmology: linear perturbation
!5
�m(t, k) = Dm(t)�0(k)<latexit sha1_base64="k19tu4plyujeKj2XTAX3gZMxvtg=">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</latexit><latexit sha1_base64="k19tu4plyujeKj2XTAX3gZMxvtg=">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</latexit><latexit sha1_base64="k19tu4plyujeKj2XTAX3gZMxvtg=">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</latexit><latexit sha1_base64="k19tu4plyujeKj2XTAX3gZMxvtg=">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</latexit>
fm(t) =d logDm
d log a<latexit sha1_base64="TyjrfBRBpb21DHHyi7E4xRMuWbw=">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</latexit><latexit sha1_base64="TyjrfBRBpb21DHHyi7E4xRMuWbw=">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</latexit><latexit sha1_base64="TyjrfBRBpb21DHHyi7E4xRMuWbw=">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</latexit><latexit sha1_base64="TyjrfBRBpb21DHHyi7E4xRMuWbw=">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</latexit>
··δm + 2H ·δm − 4πGδm = 0
Baryons and DM follow similar equations
Kaiser formula
Pg,s(k; t) =⇥bg + µ2fm(t)
⇤2Pm(k; t)
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Galaxy PS in redshift space Matter PS in real space
Linear growth rate measurable by RSD
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
.
f(z)σ8(z)
z
6dFGRS2dFGRS
SDSS MainSDSS LRG
WiggleZBOSS LOWZ
BOSS CMASSVVDS
VIPERSFastSound
ΛCDM
f(R)Cov. GalileonExt. Galileon
DGP|G/G|=
3.5×10-11yr
Okumura et al. (2016)
Standard cosmology: power spectrum
!6
µ = cos ✓ = k · z<latexit sha1_base64="P6NvSgxVM4lFo1rfXq/bBl7bBrQ=">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</latexit><latexit sha1_base64="P6NvSgxVM4lFo1rfXq/bBl7bBrQ=">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</latexit><latexit sha1_base64="P6NvSgxVM4lFo1rfXq/bBl7bBrQ=">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</latexit><latexit sha1_base64="P6NvSgxVM4lFo1rfXq/bBl7bBrQ=">AAACSXicbVDLSgMxFM20Pmp9tbp0EyyCbmRGCroRCm5cVrAP6JSSyaSd0MxkSO4U6zDf4Ne41Q/wC/wMd+LKtNOFbb0QOPece5OT48WCa7DtT6tQ3Njc2i7tlHf39g8OK9WjtpaJoqxFpZCq6xHNBI9YCzgI1o0VI6EnWMcb3830zoQpzWX0CNOY9UMyiviQUwKGGlQu3DDBt9ilUmMXAgZk1gUE0nFmWF9C3j1ng0rNvrTnhdeBswA1tKjmoGqVXF/SJGQRUEG07jl2DP2UKOBUsKzsJprFhI7JiPUMjEjIdD+d/ynDZ4bx8VAqcyLAc/bvRkpCraehZyZDAoFe1Wbkf1ovgeFNP+VRnACLaP7QMBEYJJ4FhH2uGAUxNYBQxY1XTAOiCAUT49JN/oTHOnddTp9y30s24mCqOdVZ2WTnrCa1DtpXl47BD/Vao75IsYRO0Ck6Rw66Rg10j5qohSh6Qa/oDb1bH9aX9W395KMFa7FzjJaqUPwFJdixvw==</latexit>
(Kaiser, 1987)
Dependence on the velocity
Non-minimally coupled dark matter: set-up
!7
Action:
GR DE Matter
Sb =
Zd4x
p�gLb[g, b]
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Baryons: constraints from solar system experiments => minimal coupling
CDM: unknown fundamental nature => non-minimal coupling
Sm = Sb + Sc<latexit sha1_base64="1oJEwsvxVRKMMa2od/xeQiZEqS0=">AAACCnicbZDNSsNAFIVv6l+tf1WXbkZLQRBKIoLdCAU3LivaWmhDmEwn7dCZJMxMhBKyduOruHGhiFufwJ1v47SNoK0HBj7Oncu99/gxZ0rb9pdVWFpeWV0rrpc2Nre2d8q7e20VJZLQFol4JDs+VpSzkLY005x2Ykmx8Dm980eXk/rdPZWKReGtHsfUFXgQsoARrI3llQ9vvLQnBRIZukA5+xk6+WGSeeWKXbOnQovg5FCBXE2v/NnrRyQRNNSEY6W6jh1rN8VSM8JpVuolisaYjPCAdg2GWFDlptNTMlQ1Th8FkTQv1Gjq/u5IsVBqLMyKVYH1UM3XJuZ/tW6ig7qbsjBONA3JbFCQcKQjNMkF9ZmkRPOxAUwkM7siMsQSE23SK5kQnPmTF6F9WnMMX59VGvU8jiIcwBEcgwPn0IAraEILCDzAE7zAq/VoPVtv1vvsa8HKe/bhj6yPb+Y2mRQ=</latexit><latexit sha1_base64="1oJEwsvxVRKMMa2od/xeQiZEqS0=">AAACCnicbZDNSsNAFIVv6l+tf1WXbkZLQRBKIoLdCAU3LivaWmhDmEwn7dCZJMxMhBKyduOruHGhiFufwJ1v47SNoK0HBj7Oncu99/gxZ0rb9pdVWFpeWV0rrpc2Nre2d8q7e20VJZLQFol4JDs+VpSzkLY005x2Ykmx8Dm980eXk/rdPZWKReGtHsfUFXgQsoARrI3llQ9vvLQnBRIZukA5+xk6+WGSeeWKXbOnQovg5FCBXE2v/NnrRyQRNNSEY6W6jh1rN8VSM8JpVuolisaYjPCAdg2GWFDlptNTMlQ1Th8FkTQv1Gjq/u5IsVBqLMyKVYH1UM3XJuZ/tW6ig7qbsjBONA3JbFCQcKQjNMkF9ZmkRPOxAUwkM7siMsQSE23SK5kQnPmTF6F9WnMMX59VGvU8jiIcwBEcgwPn0IAraEILCDzAE7zAq/VoPVtv1vvsa8HKe/bhj6yPb+Y2mRQ=</latexit><latexit sha1_base64="1oJEwsvxVRKMMa2od/xeQiZEqS0=">AAACCnicbZDNSsNAFIVv6l+tf1WXbkZLQRBKIoLdCAU3LivaWmhDmEwn7dCZJMxMhBKyduOruHGhiFufwJ1v47SNoK0HBj7Oncu99/gxZ0rb9pdVWFpeWV0rrpc2Nre2d8q7e20VJZLQFol4JDs+VpSzkLY005x2Ykmx8Dm980eXk/rdPZWKReGtHsfUFXgQsoARrI3llQ9vvLQnBRIZukA5+xk6+WGSeeWKXbOnQovg5FCBXE2v/NnrRyQRNNSEY6W6jh1rN8VSM8JpVuolisaYjPCAdg2GWFDlptNTMlQ1Th8FkTQv1Gjq/u5IsVBqLMyKVYH1UM3XJuZ/tW6ig7qbsjBONA3JbFCQcKQjNMkF9ZmkRPOxAUwkM7siMsQSE23SK5kQnPmTF6F9WnMMX59VGvU8jiIcwBEcgwPn0IAraEILCDzAE7zAq/VoPVtv1vvsa8HKe/bhj6yPb+Y2mRQ=</latexit><latexit sha1_base64="1oJEwsvxVRKMMa2od/xeQiZEqS0=">AAACCnicbZDNSsNAFIVv6l+tf1WXbkZLQRBKIoLdCAU3LivaWmhDmEwn7dCZJMxMhBKyduOruHGhiFufwJ1v47SNoK0HBj7Oncu99/gxZ0rb9pdVWFpeWV0rrpc2Nre2d8q7e20VJZLQFol4JDs+VpSzkLY005x2Ykmx8Dm980eXk/rdPZWKReGtHsfUFXgQsoARrI3llQ9vvLQnBRIZukA5+xk6+WGSeeWKXbOnQovg5FCBXE2v/NnrRyQRNNSEY6W6jh1rN8VSM8JpVuolisaYjPCAdg2GWFDlptNTMlQ1Th8FkTQv1Gjq/u5IsVBqLMyKVYH1UM3XJuZ/tW6ig7qbsjBONA3JbFCQcKQjNMkF9ZmkRPOxAUwkM7siMsQSE23SK5kQnPmTF6F9WnMMX59VGvU8jiIcwBEcgwPn0IAraEILCDzAE7zAq/VoPVtv1vvsa8HKe/bhj6yPb+Y2mRQ=</latexit>
conformal factor
disformal factor
Sc =
Zd4x
p�gLc [g, c]
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gµ⌫ = A(�, X)gµ⌫ +B(�, X)@µ�@⌫�<latexit sha1_base64="mdYcP2QnpW8A0Y8MvU2Xr0CRaDw=">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</latexit><latexit sha1_base64="mdYcP2QnpW8A0Y8MvU2Xr0CRaDw=">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</latexit><latexit sha1_base64="mdYcP2QnpW8A0Y8MvU2Xr0CRaDw=">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</latexit><latexit sha1_base64="mdYcP2QnpW8A0Y8MvU2Xr0CRaDw=">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</latexit>
S = ∫ dx4 − g [ M2Pl
2 R + K(ϕ, X)] + SM
Metric coupling
!8
Energy-momentum conservation
Gμν = 1M2
Pl(T (b)
μν + T (c)μν + T (ϕ)
μν )
∇μT (c)μν = − Qϕν
Q = − 1− g
δ ( − g ℒc)δϕ
= ∇μWμ − Z
Basic equations
Einstein
□ ϕ − Vϕ = QE.o.M.
Z = 12A
Aϕ +AX X (Aϕ − 2BϕX)A − AX X + 2BX X2 T(c) + Bϕ +
BX X (Aϕ − 2BϕX)A − AX X + 2BX X2 Tμν
(c)∂μϕ∂νϕ
Wμ = 12A [2BTμν
(c)∂νϕ − A − 2BXA − AX X + 2BX X2 × (AXT(c) + BXTαβ
(c) ∂αϕ∂βϕ) ∂μϕ]
∇μT (b)μν = 0 ∇μ(T (c)
μν + T (ϕ)μν ) = 0
Background evolution
!9
Evolution equations
··ϕ + 3H ·ϕ + V,ϕ = − Q0·ρc + 3Hρc = Q0·ϕ
Q0 = ρc·ϕ
dΓdt , Γ(ϕ, X ) = ln 2A − AX
·ϕ2 + BX·ϕ4
A − B ·ϕ2
ρc ∝ a− 3eΓ(ϕ,X) m(ϕ, X) = m0eΓ− Γ0
At classical level, interaction can be interpreted as a variation in the coupled particle’s mass
Background coupling
(sub-horizon + quasistatic approx.)
Coupled DM: linear perturbation
!10
EoM for the CDM density contrast
• Poisson and baryon equations: unchanged
··δc + 2Heff·δc − 4πGeff ρmδm = 0
• Continuity and Euler equations for CDM: modified
• CDM velocity field
vc(t, k) = − a2Hk2 f eff
c (t) δc(t, k)
Effective gravitational constant
Effective Hubble friction
Effective growth rate
Under the hood: perturbed equations
!11
6
and those for CDM are modified as follows:
�c +k2
a2vc =
�
⇢c
⇣�Q�Q0�c
⌘, (65)
vc � � =Q0
⇢c
⇣��� �vc
⌘. (66)
Using �Q given in equation (62), these become
(1�⌥1)
✓�c +
k2
a2vc
◆= ⌥2
✓�c �
Q0
A��c
◆, (67)
vc � � = �Q0�
⇢c
"⌥1 +⌥3
⌥3vc +
a2
k2
⌥1 +⌥2
⌥3�c �
Q0�
⇢c
⌥2 �⌥3
⌥23
�c
!#, (68)
with
⌥1 =�2
⇢c
AR1
A+R2, ⌥2 =
�2
⇢c
AR2
A+R2, ⌥3 =
A�2
⇢c. (69)
Note that for purely conformal couplings, i.e. AX = B = 0, the functions ⌥1 and ⌥2 vanish. Taking the timederivative of the continuity equations, (63) and (65), and eliminating the metric perturbations, scalar perturbations,and velocity terms, we obtain the following second-order di↵erential equations for baryons and CDM, respectively,
�b + 2H �b � 4⇡G⇢m�m = 0 , (70)
(1�⌥1 �⌥2) �c + 2H(1� E1) �c � 4⇡G
(1� E2) ⇢c�c + (1�⌥1) ⇢b�b
�= 0 (71)
where we have introduced two new functions of background quantities,
E1 = ⌥1 +⌥2 �Q0�
2⇢cH(1�⌥1 �⌥2) +
1
2H
⌥2 +
⌥1⌥2
1�⌥1
!, (72)
E2 = ⌥1 +M2Pl
"2Q0�
⇢2c
⌥2
⌥3+
Q0�
⇢2c⌥3
2⌥2 +⌥2
2⌥1
1�⌥1�
2(�� 2H�)
�
!!�
2Q0�3
⇢3c
A⌥2
⌥23
+2Q2
0�2
⇢3c⌥23
(⌥2 +⌥1⌥3 +⌥2⌥3 �⌥3)
#. (73)
IV. EFFECTIVE GROWTH RATE AND REDSHIFT-SPACE DISTORTIONS
In this section we ...For sub-horizon scales, the system formed by the second-order di↵erential equations (70) and (71) does not depend
on the wavenumber k. Therefore, one can isolate the time dependence from the k dependence of the initial conditionsand express the growing solutions for the baryon and CDM density contrasts as
�I(t,k) = DI(t)�0(k) . (74)
Here, DI(t) is the k-independent growth factor while �0(k) represents the initial density contrasts. Another quantityuseful in describing the evolution of the perturbations is the so-called linear growth rate, defined as
fI(t) ⌘d lnDI
d ln a, (75)
which is quantifies the rate of growth in a Hubble time. The continuity equations, (63) and (65), can then be recastin terms of the growth factors, and the velocity potentials become
vI(t,k) = �a2H
k2fe↵I (t)�I(t,k) . (76)
5
and the perturbation of Q is given by
�Q = (R1 +R2)��c +Q0�c +R1�k2
a2vc +R2
k2
a2��� 3(R1 +R2)� �R3��
+R4��+R3��+R5˙��+R6�� , (49)
where
R1 = !1⇢c , (50)
R2 = �!2�⇢c , (51)
R3 = ⇢c�!0,X , (52)
R4 = �Q0(!0 + !0,X �2)� ⇢c(!0,� + 3!0,X �+ !0,X �� ⇣0,X �) , (53)
R5 = !0,XQ0�2 + ⇢c(!0,� + !0,X �+ !0,X �� ⇣0,X �) , (54)
R6 = !0,�Q0�+ ⇢c(!0,� � ⇣0,�) . (55)
The energy-momentum conservation for baryonic matter yields the standard equation,
�b � 3 +k2
a2vb = 0 , (56)
vb � � = 0 , (57)
while dark matter follows
�c � 3 +k2
a2vc =
�
⇢c
⇣�Q�Q0�c
⌘+
Q0
⇢c�� , (58)
vc � � =Q0
⇢c
⇣��� �vc
⌘. (59)
1. Quasi-static approximation
Currently, data coming from galaxy surveys are restricted to scales much smaller than the cosmological horizon.Moreover, for scales well inside the sound horizon of scalar field perturbations, one can employ the quasi-staticapproximation [32]. E↵ectively, this corresponds to dropping time derivatives, with respect to space derivatives, ofmetric and scalar fluctuations. In the quasi-static approximation, Einstein equations become constraint equations forthe gravitational potentials � and ,
k2
a2 =
k2
a2� = �4⇡G⇢m�m , (60)
while equation (43) gives a constraint for the scalar perturbation,
�Ak2
a2�� ⌘ �
✓A1
k2
a2+m
2�
◆�� = �Q , (61)
where �Q in the quasi-static approximation is given by
�Q = (R1 +R2)��c +Q0�c +R1�k2
a2vc +R2
k2
a2�� . (62)
In equation (61) we have defined A = A(t, k), which in general depends on the wave-number when the Comptonwavelength of the scalar field is smaller than the Hubble horizon scale. When the scalar field consists of a canonicalkinetic term and the scalar mass scale is irrelevant within sub-horizon scales, then we have A = 1 and recover theprevious results. The continuity and Euler equations for baryon are given by
�b +k2
a2vb = 0 , (63)
vb � � = 0 , (64)
5
and the perturbation of Q is given by
�Q = (R1 +R2)��c +Q0�c +R1�k2
a2vc +R2
k2
a2��� 3(R1 +R2)� �R3��
+R4��+R3��+R5˙��+R6�� , (49)
where
R1 = !1⇢c , (50)
R2 = �!2�⇢c , (51)
R3 = ⇢c�!0,X , (52)
R4 = �Q0(!0 + !0,X �2)� ⇢c(!0,� + 3!0,X �+ !0,X �� ⇣0,X �) , (53)
R5 = !0,XQ0�2 + ⇢c(!0,� + !0,X �+ !0,X �� ⇣0,X �) , (54)
R6 = !0,�Q0�+ ⇢c(!0,� � ⇣0,�) . (55)
The energy-momentum conservation for baryonic matter yields the standard equation,
�b � 3 +k2
a2vb = 0 , (56)
vb � � = 0 , (57)
while dark matter follows
�c � 3 +k2
a2vc =
�
⇢c
⇣�Q�Q0�c
⌘+
Q0
⇢c�� , (58)
vc � � =Q0
⇢c
⇣��� �vc
⌘. (59)
1. Quasi-static approximation
Currently, data coming from galaxy surveys are restricted to scales much smaller than the cosmological horizon.Moreover, for scales well inside the sound horizon of scalar field perturbations, one can employ the quasi-staticapproximation [32]. E↵ectively, this corresponds to dropping time derivatives, with respect to space derivatives, ofmetric and scalar fluctuations. In the quasi-static approximation, Einstein equations become constraint equations forthe gravitational potentials � and ,
k2
a2 =
k2
a2� = �4⇡G⇢m�m , (60)
while equation (43) gives a constraint for the scalar perturbation,
�Ak2
a2�� ⌘ �
✓A1
k2
a2+m
2�
◆�� = �Q , (61)
where �Q in the quasi-static approximation is given by
�Q = (R1 +R2)��c +Q0�c +R1�k2
a2vc +R2
k2
a2�� . (62)
In equation (61) we have defined A = A(t, k), which in general depends on the wave-number when the Comptonwavelength of the scalar field is smaller than the Hubble horizon scale. When the scalar field consists of a canonicalkinetic term and the scalar mass scale is irrelevant within sub-horizon scales, then we have A = 1 and recover theprevious results. The continuity and Euler equations for baryon are given by
�b +k2
a2vb = 0 , (63)
vb � � = 0 , (64)
6
and those for CDM are modified as follows:
�c +k2
a2vc =
�
⇢c
⇣�Q�Q0�c
⌘, (65)
vc � � =Q0
⇢c
⇣��� �vc
⌘. (66)
Using �Q given in equation (62), these become
(1�⌥1)
✓�c +
k2
a2vc
◆= ⌥2
✓�c �
Q0
A��c
◆, (67)
vc � � = �Q0�
⇢c
"⌥1 +⌥3
⌥3vc +
a2
k2
⌥1 +⌥2
⌥3�c �
Q0�
⇢c
⌥2 �⌥3
⌥23
�c
!#, (68)
with
⌥1 =�2
⇢c
AR1
A+R2, ⌥2 =
�2
⇢c
AR2
A+R2, ⌥3 =
A�2
⇢c. (69)
Note that for purely conformal couplings, i.e. AX = B = 0, the functions ⌥1 and ⌥2 vanish. Taking the timederivative of the continuity equations, (63) and (65), and eliminating the metric perturbations, scalar perturbations,and velocity terms, we obtain the following second-order di↵erential equations for baryons and CDM, respectively,
�b + 2H �b � 4⇡G⇢m�m = 0 , (70)
(1�⌥1 �⌥2) �c + 2H(1� E1) �c � 4⇡G
(1� E2) ⇢c�c + (1�⌥1) ⇢b�b
�= 0 (71)
where we have introduced two new functions of background quantities,
E1 = ⌥1 +⌥2 �Q0�
2⇢cH(1�⌥1 �⌥2) +
1
2H
⌥2 +
⌥1⌥2
1�⌥1
!, (72)
E2 = ⌥1 +M2Pl
"2Q0�
⇢2c
⌥2
⌥3+
Q0�
⇢2c⌥3
2⌥2 +⌥2
2⌥1
1�⌥1�
2(�� 2H�)
�
!!�
2Q0�3
⇢3c
A⌥2
⌥23
+2Q2
0�2
⇢3c⌥23
(⌥2 +⌥1⌥3 +⌥2⌥3 �⌥3)
#. (73)
IV. EFFECTIVE GROWTH RATE AND REDSHIFT-SPACE DISTORTIONS
In this section we ...For sub-horizon scales, the system formed by the second-order di↵erential equations (70) and (71) does not depend
on the wavenumber k. Therefore, one can isolate the time dependence from the k dependence of the initial conditionsand express the growing solutions for the baryon and CDM density contrasts as
�I(t,k) = DI(t)�0(k) . (74)
Here, DI(t) is the k-independent growth factor while �0(k) represents the initial density contrasts. Another quantityuseful in describing the evolution of the perturbations is the so-called linear growth rate, defined as
fI(t) ⌘d lnDI
d ln a, (75)
which is quantifies the rate of growth in a Hubble time. The continuity equations, (63) and (65), can then be recastin terms of the growth factors, and the velocity potentials become
vI(t,k) = �a2H
k2fe↵I (t)�I(t,k) . (76)
6
and those for CDM are modified as follows:
�c +k2
a2vc =
�
⇢c
⇣�Q�Q0�c
⌘, (65)
vc � � =Q0
⇢c
⇣��� �vc
⌘. (66)
Using �Q given in equation (62), these become
(1�⌥1)
✓�c +
k2
a2vc
◆= ⌥2
✓�c �
Q0
A��c
◆, (67)
vc � � = �Q0�
⇢c
"⌥1 +⌥3
⌥3vc +
a2
k2
⌥1 +⌥2
⌥3�c �
Q0�
⇢c
⌥2 �⌥3
⌥23
�c
!#, (68)
with
⌥1 =�2
⇢c
AR1
A+R2, ⌥2 =
�2
⇢c
AR2
A+R2, ⌥3 =
A�2
⇢c. (69)
Note that for purely conformal couplings, i.e. AX = B = 0, the functions ⌥1 and ⌥2 vanish. Taking the timederivative of the continuity equations, (63) and (65), and eliminating the metric perturbations, scalar perturbations,and velocity terms, we obtain the following second-order di↵erential equations for baryons and CDM, respectively,
�b + 2H �b � 4⇡G⇢m�m = 0 , (70)
(1�⌥1 �⌥2) �c + 2H(1� E1) �c � 4⇡G
(1� E2) ⇢c�c + (1�⌥1) ⇢b�b
�= 0 (71)
where we have introduced two new functions of background quantities,
E1 = ⌥1 +⌥2 �Q0�
2⇢cH(1�⌥1 �⌥2) +
1
2H
⌥2 +
⌥1⌥2
1�⌥1
!, (72)
E2 = ⌥1 +M2Pl
"2Q0�
⇢2c
⌥2
⌥3+
Q0�
⇢2c⌥3
2⌥2 +⌥2
2⌥1
1�⌥1�
2(�� 2H�)
�
!!�
2Q0�3
⇢3c
A⌥2
⌥23
+2Q2
0�2
⇢3c⌥23
(⌥2 +⌥1⌥3 +⌥2⌥3 �⌥3)
#. (73)
IV. EFFECTIVE GROWTH RATE AND REDSHIFT-SPACE DISTORTIONS
In this section we ...For sub-horizon scales, the system formed by the second-order di↵erential equations (70) and (71) does not depend
on the wavenumber k. Therefore, one can isolate the time dependence from the k dependence of the initial conditionsand express the growing solutions for the baryon and CDM density contrasts as
�I(t,k) = DI(t)�0(k) . (74)
Here, DI(t) is the k-independent growth factor while �0(k) represents the initial density contrasts. Another quantityuseful in describing the evolution of the perturbations is the so-called linear growth rate, defined as
fI(t) ⌘d lnDI
d ln a, (75)
which is quantifies the rate of growth in a Hubble time. The continuity equations, (63) and (65), can then be recastin terms of the growth factors, and the velocity potentials become
vI(t,k) = �a2H
k2fe↵I (t)�I(t,k) . (76)
6
and those for CDM are modified as follows:
�c +k2
a2vc =
�
⇢c
⇣�Q�Q0�c
⌘, (65)
vc � � =Q0
⇢c
⇣��� �vc
⌘. (66)
Using �Q given in equation (62), these become
(1�⌥1)
✓�c +
k2
a2vc
◆= ⌥2
✓�c �
Q0
A��c
◆, (67)
vc � � = �Q0�
⇢c
"⌥1 +⌥3
⌥3vc +
a2
k2
⌥1 +⌥2
⌥3�c �
Q0�
⇢c
⌥2 �⌥3
⌥23
�c
!#, (68)
with
⌥1 =�2
⇢c
AR1
A+R2, ⌥2 =
�2
⇢c
AR2
A+R2, ⌥3 =
A�2
⇢c. (69)
Note that for purely conformal couplings, i.e. AX = B = 0, the functions ⌥1 and ⌥2 vanish. Taking the timederivative of the continuity equations, (63) and (65), and eliminating the metric perturbations, scalar perturbations,and velocity terms, we obtain the following second-order di↵erential equations for baryons and CDM, respectively,
�b + 2H �b � 4⇡G⇢m�m = 0 , (70)
(1�⌥1 �⌥2) �c + 2H(1� E1) �c � 4⇡G
(1� E2) ⇢c�c + (1�⌥1) ⇢b�b
�= 0 (71)
where we have introduced two new functions of background quantities,
E1 = ⌥1 +⌥2 �Q0�
2⇢cH(1�⌥1 �⌥2) +
1
2H
⌥2 +
⌥1⌥2
1�⌥1
!, (72)
E2 = ⌥1 +M2Pl
"2Q0�
⇢2c
⌥2
⌥3+
Q0�
⇢2c⌥3
2⌥2 +⌥2
2⌥1
1�⌥1�
2(�� 2H�)
�
!!�
2Q0�3
⇢3c
A⌥2
⌥23
+2Q2
0�2
⇢3c⌥23
(⌥2 +⌥1⌥3 +⌥2⌥3 �⌥3)
#. (73)
IV. EFFECTIVE GROWTH RATE AND REDSHIFT-SPACE DISTORTIONS
In this section we ...For sub-horizon scales, the system formed by the second-order di↵erential equations (70) and (71) does not depend
on the wavenumber k. Therefore, one can isolate the time dependence from the k dependence of the initial conditionsand express the growing solutions for the baryon and CDM density contrasts as
�I(t,k) = DI(t)�0(k) . (74)
Here, DI(t) is the k-independent growth factor while �0(k) represents the initial density contrasts. Another quantityuseful in describing the evolution of the perturbations is the so-called linear growth rate, defined as
fI(t) ⌘d lnDI
d ln a, (75)
which is quantifies the rate of growth in a Hubble time. The continuity equations, (63) and (65), can then be recastin terms of the growth factors, and the velocity potentials become
vI(t,k) = �a2H
k2fe↵I (t)�I(t,k) . (76)
Continuity & Euler: EoM for the scalar field (QS approx.)
EoM for the CDM density contrast
Coupled DM: linear perturbation
!12
Velocity field vm(t,k) = �a2H
k2fe↵m (t) �m(t,k)
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Actual growth rateDM-DE coupling effect
depending on the conformal and disformal
functionsfm =
d logDm
d log a<latexit sha1_base64="58gFRA+5viMntSRTo0I/nREXsV8=">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</latexit><latexit sha1_base64="58gFRA+5viMntSRTo0I/nREXsV8=">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</latexit><latexit sha1_base64="58gFRA+5viMntSRTo0I/nREXsV8=">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</latexit><latexit sha1_base64="58gFRA+5viMntSRTo0I/nREXsV8=">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</latexit>
f e↵m = fm +�f e↵
m<latexit sha1_base64="k1BkzcANE4x/+CJ3wAGfUxJvGVs=">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</latexit><latexit sha1_base64="k1BkzcANE4x/+CJ3wAGfUxJvGVs=">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</latexit><latexit sha1_base64="k1BkzcANE4x/+CJ3wAGfUxJvGVs=">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</latexit><latexit sha1_base64="k1BkzcANE4x/+CJ3wAGfUxJvGVs=">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</latexit>
CDM’s continuity & Euler
eqs are modifiedρmδm = ρbδb + ρcδc
ρmvm = ρbvb + ρcvc
Evolution of total matter is modified
RSD probes velocity, so….
!13
Cold dark matter
Total matter
f effm = ωcDc f eff
c + ωbDb f effb
ωcDc + ωbDb= fm + ωc
Dc
DmΔfc − ωb
Q0·ϕ
Hρm
Dc − Db
Dm
Δfm
f effc = fc − Υ2
1 − Υ1 (fc − Q0H ·ϕ )
Δfc
Υ2 = Υ2(AX, BX, . . . )
A(ϕ), B(ϕ) ⇒ Δfc = 0
X-dependent contribution
BG coupling
Coupled DM: effective growth rate
Coupled DM: power spectrum
!14
Pg,s(k; t) =⇥bg + µ2f e↵
m (t)⇤2
Pm(k; t)<latexit sha1_base64="bZly34r56QLq4QJbxKGms5t4490=">AAACmHicbVHRbtMwFHXCgBLY1sEbvFxRIXUCTcmEBNI0VMED8FYmuk1qsspxndaqnUT2TbUq8ofywrfgJkGiG1eyfHTOvddHx2kphcEw/OX5D/YePnrcexI8fbZ/cNg/en5pikozPmGFLPR1Sg2XIucTFCj5dak5VankV+nqy1a/WnNtRJH/xE3JE0UXucgEo+ioWf82GM/qWCtYvDN2GK/TemXP8BjOIZY8wymknWzhLcSqujmFOm6erS+KDZWfZcUtZG2TsjfNzbPMDvHYAsRaLJaYuKnx35bharsfZv1BeBI2BfdB1IEB6Wo8O/J68bxgleI5MkmNmUZhiUlNNQomuQ3iyvCSshVd8KmDOVXcJHXj1cIbx8whK7Q7OULD/jtRU2XMRqWuU1FcmrvalvyfNq0w+5jUIi8r5DlrH8oqCVjANm+YC80Zyo0DlGnhvAJbUk0Zul/Z2TRfi9K0roP6tvW9Y6NcboxgxgYuu+huUvfB5elJ5PCP94PRpy7FHnlFXpMhicgHMiLfyJhMCCO/vT1v3zvwX/oj/6v/vW31vW7mBdkp/+IPAl3I0Q==</latexit><latexit sha1_base64="bZly34r56QLq4QJbxKGms5t4490=">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</latexit><latexit sha1_base64="bZly34r56QLq4QJbxKGms5t4490=">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</latexit><latexit sha1_base64="bZly34r56QLq4QJbxKGms5t4490=">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</latexit>
Standard scenario (minimally coupled DM)
Coupled DM
Dm = Dc = Db<latexit sha1_base64="i8IcVrVhWOraJZ0OovJAif6w7Yg=">AAACQXicbZC7TsMwFIadcivh1sLIYqiQmKoEIdEFqRIMjEWiF6mNIsdxW6t2EtlORRRl5mlY4QF4Ch6BDbGy4DYZeuFIlj795xz79+9FjEplWZ9GaWNza3unvGvu7R8cHlWqxx0ZxgKTNg5ZKHoekoTRgLQVVYz0IkEQ9xjpepO7Wb87JULSMHhSSUQcjkYBHVKMlJbcytm9mw4EhzyDt7BgvMBe5lZqVt2aF1wHu4AaKKrlVo3ywA9xzEmgMENS9m0rUk6KhKKYkcwcxJJECE/QiPQ1BogT6aTzv2TwQis+HIZCn0DBubq4kSIuZcI9PcmRGsvV3kz8r9eP1bDhpDSIYkUCnD80jBlUIZwFA30qCFYs0YCwoNorxGMkEFY6vqWb/CmNZO7aTJ9z30s2onEiKZaZqbOzV5Nah85V3db8eF1rNooUy+AUnINLYIMb0AQPoAXaAIMX8ArewLvxYXwZ38ZPPloyip0TsFTG7x89lq7I</latexit><latexit sha1_base64="i8IcVrVhWOraJZ0OovJAif6w7Yg=">AAACQXicbZC7TsMwFIadcivh1sLIYqiQmKoEIdEFqRIMjEWiF6mNIsdxW6t2EtlORRRl5mlY4QF4Ch6BDbGy4DYZeuFIlj795xz79+9FjEplWZ9GaWNza3unvGvu7R8cHlWqxx0ZxgKTNg5ZKHoekoTRgLQVVYz0IkEQ9xjpepO7Wb87JULSMHhSSUQcjkYBHVKMlJbcytm9mw4EhzyDt7BgvMBe5lZqVt2aF1wHu4AaKKrlVo3ywA9xzEmgMENS9m0rUk6KhKKYkcwcxJJECE/QiPQ1BogT6aTzv2TwQis+HIZCn0DBubq4kSIuZcI9PcmRGsvV3kz8r9eP1bDhpDSIYkUCnD80jBlUIZwFA30qCFYs0YCwoNorxGMkEFY6vqWb/CmNZO7aTJ9z30s2onEiKZaZqbOzV5Nah85V3db8eF1rNooUy+AUnINLYIMb0AQPoAXaAIMX8ArewLvxYXwZ38ZPPloyip0TsFTG7x89lq7I</latexit><latexit sha1_base64="i8IcVrVhWOraJZ0OovJAif6w7Yg=">AAACQXicbZC7TsMwFIadcivh1sLIYqiQmKoEIdEFqRIMjEWiF6mNIsdxW6t2EtlORRRl5mlY4QF4Ch6BDbGy4DYZeuFIlj795xz79+9FjEplWZ9GaWNza3unvGvu7R8cHlWqxx0ZxgKTNg5ZKHoekoTRgLQVVYz0IkEQ9xjpepO7Wb87JULSMHhSSUQcjkYBHVKMlJbcytm9mw4EhzyDt7BgvMBe5lZqVt2aF1wHu4AaKKrlVo3ywA9xzEmgMENS9m0rUk6KhKKYkcwcxJJECE/QiPQ1BogT6aTzv2TwQis+HIZCn0DBubq4kSIuZcI9PcmRGsvV3kz8r9eP1bDhpDSIYkUCnD80jBlUIZwFA30qCFYs0YCwoNorxGMkEFY6vqWb/CmNZO7aTJ9z30s2onEiKZaZqbOzV5Nah85V3db8eF1rNooUy+AUnINLYIMb0AQPoAXaAIMX8ArewLvxYXwZ38ZPPloyip0TsFTG7x89lq7I</latexit><latexit sha1_base64="i8IcVrVhWOraJZ0OovJAif6w7Yg=">AAACQXicbZC7TsMwFIadcivh1sLIYqiQmKoEIdEFqRIMjEWiF6mNIsdxW6t2EtlORRRl5mlY4QF4Ch6BDbGy4DYZeuFIlj795xz79+9FjEplWZ9GaWNza3unvGvu7R8cHlWqxx0ZxgKTNg5ZKHoekoTRgLQVVYz0IkEQ9xjpepO7Wb87JULSMHhSSUQcjkYBHVKMlJbcytm9mw4EhzyDt7BgvMBe5lZqVt2aF1wHu4AaKKrlVo3ywA9xzEmgMENS9m0rUk6KhKKYkcwcxJJECE/QiPQ1BogT6aTzv2TwQis+HIZCn0DBubq4kSIuZcI9PcmRGsvV3kz8r9eP1bDhpDSIYkUCnD80jBlUIZwFA30qCFYs0YCwoNorxGMkEFY6vqWb/CmNZO7aTJ9z30s2onEiKZaZqbOzV5Nah85V3db8eF1rNooUy+AUnINLYIMb0AQPoAXaAIMX8ArewLvxYXwZ38ZPPloyip0TsFTG7x89lq7I</latexit>
f e↵m = fm
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RSD measures the effective growth rate (actual growth rate + DM-DE interaction)
Modified Kaiser formula vm ∝ f effm
f effm = fm + Δfm
Growth function & growth rate
Discriminating growth and coupling with RSD: - multiple z measurements / - Cross correlation with other proves (e.g. weak lensing)
Concrete examples
!15
• Canonical scalar field
g μν = Ag μν + B∂μϕ∂νϕ
L� = �1
2@µ�@µ�� V (�)
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V (�) = V0��n, n > 0
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• Coupling
No interaction at background level
(Amendola, 2000 )Model I (conf.)
Model II (conf.)
Model III (disf.)
MPl = 1
A(ϕ) = e− 2αϕ, B = 0
A(ϕ) = e− αϕ2, B = 0
α :
A = 1, B(X) = αX
coupling constant
Tracking behavior
!16
Model I: perturbation
··δc + 2H (1 − ℰ1) ·δc − 4πG [(1 − ℰ2) ρcδc + ρbδb] = 0Evolution of perturbation
ℰ1 = α2
·ϕH
ℰ2 = − 2α2
Effective H
Effective G
=> less frictionα > 0
=> scalar fifth-force (attractive)
constant:ℰ2
interaction affects growth even at early
times
=> Growth enhancement
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5N
0.0002
0.0004
0.0006
0.0008
0.0010
Δfm/fmModel 1
α= 0.05α= 0.075α= 0.1
Δfmfm
Model II: perturbation
!17
··δc + 2H (1 − ℰ1) ·δc − 4πG [(1 − ℰ2) ρcδc + ρbδb] = 0Evolution of perturbation
ℰ1 = α2
ϕ ·ϕH
ℰ2 = − 2α2ϕ2
Effective H
Effective G
=> less frictionα > 0
=> scalar fifth-force (attractive)
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5N
10-8
10-7
10-6
10-5
10-4
10-3
Δfm/fmModel 3
α= -0.1α= 0.05α= 0.1
At early times: ℰ1 ≈ ℰ2 ≈ 0
interaction affects only late-
time behavior
=> Growth enhancement
Δfmfm
Model III: disformal coupling
!18
A(ϕ) = 1, B = αX
Conformal/ disformal functions Q0 = 0 Coupling only at pert. level
··δc + 2H (1 − ℰ1) ·δc − 4πG (1 − ℰ2) ρmδm = 0
ℰ1 = −2α2ρc
·ϕ (3 ·ϕ + 2Vϕ)H ( ·ϕ2 − 2αρc) [(1 − 2α) ·ϕ2 − 2αρc]
Effective H
Effective G
ℰ2 = 2α ·ϕ2
·ϕ2 − 2αρc
c2s > 0 ⇒ α <
·ϕ2
2ρc
-3.0 -2.5 -2.0 -1.5 -1.0 -0.5N
-0.10
-0.08
-0.06
-0.04
-0.02
Δfm/fmModel 2
α= -0.01α= -0.05α= -0.1
Effect of disformal coupling: Suppression of growth
Δfmfm
Forecast: Fisher matrix analysis
!19
F↵� =X
zi
Z kmin
kmin
d3k
(2⇡)3Ve↵(k, zi)
@ lnPobs(k, zi)
@✓↵@ lnPobs(k, zi)
@✓�
Galaxy survey:
• Phase two of the Square Kilometer Array project (SKA2)- Network of radio telescopes in South Africa and Australia- Beginning: 2025- Survey specifications based on Yahya et al. (2015)
• Euclid space satellite- Estimated launch: 2021- Survey specifications based on Amendola et al. (2018)
Fisher matrix analysis: Estimating the constraints from a future experiment
Pobs(~k, z) = NAP(z)�b+ f e↵
m µ2�2
Pm(k, z)e�k2µ2�2
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Forecast: results
!20
1
TABLE I: Fiducial parameters.
Parameter Value Obs.
h 0.67
⌦bh2 0.0222
⌦� 0.6817
↵ 0.04 Model I
↵ 0.04 Model II
↵ -0.1 Model III
�8 0.834
ns 0.962
�NL 7 Nuisance param. [Mpc]
I. DISCUSSION
The redshift window that SKA and Euclid can make measurements are di↵erent. In particular, SKA is sensible
to much lower redshifts. Therefore, if the coupling is e↵ective/active (i.e. changes the evolution with respect to the
uncoupled case) in the redshift region that is not surveyed (outside the window), the survey will be insensitive to the
change in the coupling constant. That happens in disformal model III.
Acknowledgments
We thank Jiro Soda and Toshifumi Noumi for many useful comments and discussions.This work was supported in
part by JSPS Grant-in-Aid for Scientific Research Nos. 17K14304 (D.Y.), 17K14276 (R.K.), 25287054 (R.K. & M.Y.),
26610062 (M.Y.), 15K17632 (T.S.) and 15K17659 (S.Y.), MEXT KAKENHI Grant-in-Aid for Scientific Research
on Innovative Areas ”Cosmic Acceleration” Nos. 15H05888 (M.Y.) and 16H01103 (S.Y.), MEXT KAKENHI Grant
Numbers 17H06357 and 17H06359 (T.S.).
Appendix A: Dispersion relation
The equation of motion for the perturbation of the canonical scalar field is
��+ 3H��+
✓k2
a2+ V��
◆��� �
⇣�+ 3
⌘+ 2V�� = �2Q0�� �Q. (A1)
In particular, for model 2 we have,
Q0 = 0, �Q = 2↵⇢c
�2
k2
a2
⇣�vc � ��
⌘, (A2)
and since there is no anisotropic stress, � = . Hence, eq.(A1) becomes
��+ 3H��+
V�� +
✓1� 2↵
⇢c
�2
◆k2
a2
���� 4��+ 2V��+ 2↵
⇢c
�
k2
a2vc = 0. (A3)
Model ↵ (Fid.) Survey 103 ⇥ �(↵) 104 ⇥ �(h) 104 ⇥ �(⌦bh2) 103 ⇥ �(⌦�) 103 ⇥ �(�8) 103 ⇥ �(ns)
I 0.04Euclid 5.90 9.87 2.90 2.38 6.12 4.22
SKA2 3.72 5.55 1.79 1.13 3.30 2.32
II 0.04Euclid 26.37 6.37 1.52 1.16 4.29 3.38
SKA2 19.99 3.75 1.02 0.80 2.97 2.49
III -0.1Euclid 15.00 5.99 1.43 1.28 4.26 3.55
SKA2 5.36 3.64 0.99 0.88 2.85 2.55
~15% ~ 9%~66% ~50%~15% ~5%
~50%
Ade et al. (2016)
Model III
• Conformal/disformal metric couplings: possibility of defining all evolution equations without specifying the DM component
• Effective growth rate: measured by RSD
- combination of the actual growth rate & DM-DE interaction effects
• Future galaxy surveys: possibility of putting constraints on the interaction (~9%)
Summary
!21
Thank you!