i. conformal and disformal transformations ii. rsd in...

Outline

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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Redshift space distortions

!4

Density contrast �g,s = �g �rzvg,z

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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=">AAACY3icbVBdi9pAFB3Tj7XaD9f2rRSGSkGhSFIWdl+2SO1DH22pH2BEJpOrDs4kYeZGNoT8oP01+1p/QP9HRw20ag8MHM65987hBIkUBl13W3EePX7y9KL6rFZ//uLlq8Zlc2TiVHMY8ljGehIwA1JEMESBEiaJBqYCCeNg3d/54w1oI+LoJ2YJzBRbRmIhOEMrzRt9PwSJbJ77WlFVtPHjukNvae7vT+c/4ozJLzKF4uvfkU5Byy2Xtted2rzRcrvuHvSceCVpkRKD+WWl6ocxTxVEyCUzZuq5Cc5yplFwCUXNTw0kjK/ZEqaWRkyBmeX7SAX9YJWQLmJtX4R0r/67kTNlTKYCO6kYrsyptxP/501TXNzMchElKULEDx8tUkkxprvqaCg0cJSZJYxrYbNSvmKacbQFH10KNyIxh9S1/O6Q+yhGssqM4KbYdeedNnVORp+6nuXfr1q9z2WLVfKWvCdt4pFr0iPfyIAMCSf35IH8ItvKb6fuNJ03h1GnUu68Jkdw3v0Bze64kw==</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=">AAACSXicbVDLSgMxFM20Pmp9tbp0EyyCbmRGCroRCm5cVrAP6JSSyaSd0MxkSO4U6zDf4Ne41Q/wC/wMd+LKtNOFbb0QOPece5OT48WCa7DtT6tQ3Njc2i7tlHf39g8OK9WjtpaJoqxFpZCq6xHNBI9YCzgI1o0VI6EnWMcb3830zoQpzWX0CNOY9UMyiviQUwKGGlQu3DDBt9ilUmMXAgZk1gUE0nFmWF9C3j1ng0rNvrTnhdeBswA1tKjmoGqVXF/SJGQRUEG07jl2DP2UKOBUsKzsJprFhI7JiPUMjEjIdD+d/ynDZ4bx8VAqcyLAc/bvRkpCraehZyZDAoFe1Wbkf1ovgeFNP+VRnACLaP7QMBEYJJ4FhH2uGAUxNYBQxY1XTAOiCAUT49JN/oTHOnddTp9y30s24mCqOdVZ2WTnrCa1DtpXl47BD/Vao75IsYRO0Ck6Rw66Rg10j5qohSh6Qa/oDb1bH9aX9W395KMFa7FzjJaqUPwFJdixvw==</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=">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</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


�c +k2

a2vc =

�

⇢c

⇣�Q�Q0�c

⌘, (65)

vc � � =Q0

⇢c

⇣�� vc

⌘. (66)


(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)


�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)


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)




�I(t,k) = DI(t)�0(k) . (74)


fI(t) ⌘d lnDI

d ln a, (75)


vI(t,k) = �a2H

k2fe↵I (t)�I(t,k) . (76)

6


�c +k2

a2vc =

�

⇢c

⇣�Q�Q0�c

⌘, (65)

vc � � =Q0

⇢c

⇣�� vc

⌘. (66)


(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)


�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)


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)




�I(t,k) = DI(t)�0(k) . (74)


fI(t) ⌘d lnDI

d ln a, (75)


vI(t,k) = �a2H

k2fe↵I (t)�I(t,k) . (76)

6


�c +k2

a2vc =

�

⇢c

⇣�Q�Q0�c

⌘, (65)

vc � � =Q0

⇢c

⇣�� vc

⌘. (66)


(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)


�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)


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)




�I(t,k) = DI(t)�0(k) . (74)


fI(t) ⌘d lnDI

d ln a, (75)


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=">AAACh3icbVFdaxNBFJ2sVdNUbWoffRkaCgWh7orUvgjVivpYpWkL2RhmJ3eTofOxzNwtLsv8xP6A/g5fFZxsVjRtLwxzOPeeO4czWSGFwzi+6UQP1h4+etxd7208efpss7/1/MyZ0nIYciONvciYAyk0DFGghIvCAlOZhPPs8njRP78C64TRp1gVMFZspkUuOMNATfqzOm2W1J+MBYefLYD2NJ/UqVVU+e/NDXnuPX33j6Yv6V/dN1Mx+UGW4Gn6ESSy+8TUT/qDeD9uit4FSQsGpK2TyVanm04NLxVo5JI5N0riAsc1syi4BN9LSwcF45dsBqMANVPgxnVjytPdwExpbmw4GmnD/q+omXKuUlmYVAzn7nZvQd7XG5WYH45roYsSQfPlQ3kpKRq6iJdOhQWOsgqAcSuCV8rnzDKO4RNWNk2vROGWrnv1j6XvFRvFvHKCO98L2SW3k7oLzl7vJwF/fTM4OmxT7JIXZIfskYS8JUfkCzkhQ8LJNflJfpHf0Xr0KjqI2tmo02q2yUpF7/8AKADIRQ==</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=">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</latexit><latexit sha1_base64="bZly34r56QLq4QJbxKGms5t4490=">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</latexit><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>

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

<latexit sha1_base64="0zqroVT8APe7rc0ilNgIC6rPFms=">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</latexit><latexit sha1_base64="0zqroVT8APe7rc0ilNgIC6rPFms=">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</latexit><latexit sha1_base64="0zqroVT8APe7rc0ilNgIC6rPFms=">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</latexit><latexit sha1_base64="0zqroVT8APe7rc0ilNgIC6rPFms=">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</latexit>

• 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!

i. conformal and disformal transformations ii. rsd in...

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