Revealing Scalar-Induced Gravitational Waves: A Detailed Analysis
Exploring the realm of second-order gravitational waves in the context of perturbation theory, this work delves into the intricate calculations and implications of scalar-induced gravitational waves during radiation domination. The study investigates the direct link between inflation, smallest scales, and the signal of primordial black holes, providing insights into the dynamics and polarization of these gravitational waves.
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SECOND ORDER GRAVITATIONAL WAVES: PAVING THE WAY FOR A FULL CALCULATION RAPHA L PICARD Work based on 2311.14513 with K. A. Malik and 24XX.XXXXX with L.E. PADILLA AND K.A. MALIK
THE POWER SPECTRUM ON SMALL SCALES small scales (end of inflation) CMB scales ?? k (M. Davies)
SET UP ?00= ?2 1 + 2 1 ?0?= 0 = ??0 Metric ? ???= ?2 1 2 1 ??+ ??? 1 ????? ????? 2= 2 ?? 2 ?? ??? Matter ???= + ? ????+ ????
SCALAR INDUCED GRAVITATIONAL WAVES First order scalar perturbations couple and source GWs at second order in perturbation theory (Tomita, Ananda et al, Baumann et al) GW equation of motion: ????? ?+ ????? ? ????? ?= ??? ?? ??? ?? +?? ?? +?? 8 ??= ??? + 4?? ?? 3 1 + ?
SCALAR INDUCED GRAVITATIONAL WAVES Induced during radiation domination They are directly sourced by inflation and therefore a probe of the smallest scales and counterpart signal of PBHs Solution in Fourier space ?3? 2 ? ,???,? ???( ,? ,?) ? ? ? ,? = 4 ?? 3 2 2
SCALAR INDUCED GRAVITATIONAL WAVES Induced during radiation domination They are directly sourced by inflation and therefore a probe of the smallest scales and counterpart signal of PBHs Solution in Fourier space ?3? 2 ? ,???,? ???( ,? ,?) ? ? ? ,? = 4 ?? 3 2 2 Polarization
SCALAR INDUCED GRAVITATIONAL WAVES Induced during radiation domination They are directly sourced by inflation and therefore a probe of the smallest scales and counterpart signal of PBHs Solution in Fourier space ?3? 2 ? ,???,? ???( ,? ,?) ? ? ? ,? = 4 ?? 3 2 2 Kernel
SCALAR INDUCED GRAVITATIONAL WAVES Induced during radiation domination They are directly sourced by inflation and therefore a probe of the smallest scales and counterpart signal of PBHs Solution in Fourier space ?3? 2 ? ,???,? ???( ,? ,?) ? ? ? ,? = 4 ?? 3 2 2 Primordial values
POWER SPECTRUM / SPECTRAL DENSITY = ??? ?3? + ? 2 2 ? ? ?? ?,? h? ,? ? ?3?? ? Plug in equation of motion ?? ,? ~? ,?
SCALAR INDUCED GRAVITATIONAL WAVES The two-point function of second order tensors is proportional to the four-point function of scalars (??)?,? ~ ?? ? ? ? ?,? ?? ?,? ?? ??
SCALAR INDUCED GRAVITATIONAL WAVES What happens for some peaked input power spectra on small scales? (Dirac delta input peak)
INCLUSION OF FIRST ORDER TENSORS ?00= ?2 1 + 2 1 ?0?= 0 = ??0 Metric ? 1 ???= ?2 1 2 1 ??+ 2 ?? + ??? 1 ????? ????? 2= 2 ?? 2 ?? ??? Matter ???= + ? ????+ ????
SECOND ORDER GRAVITATIONAL WAVES First order scalar and tensor perturbations couple and source GWs as they re-enter the horizon during radiation domination GW equation (Zhang et al, Bari et al, Yu et al) ????? ?+ ????? ? ????? ?= ??? ??? ?? +?? ?? +?? 8 ??= ??? + 4?? ?? 3 1 + ? ??= 4 ?????? ??+ 4?? ???? ? ? 4?? ???? ? ?+ 8 ?????? ??+ 4 ? ? ?? + 2?? ???? ?? ??? ??= 8 2 ??+ 8?? ???? + 4 ?? 1 + 3??2 + 1 ??2 2 ???
SECOND ORDER GRAVITATIONAL WAVES First order scalar and tensor perturbations couple and source GWs as they re-enter the horizon during radiation domination They are directly sourced by inflation and therefore give us information about the scalar and tensor power spectrum on smallest scales (??)?,? ~ ?? ? ? ?+???? ?+ ?? ?,? ?? ?,? ?? ??
RESULTS: LOGNORMAL PEAK ? = ? ~????? ? = 10? = 0.01 = 0.1
ARE WE MISSING SOMETHING? 22 ,? ~? ,?
ARE WE MISSING SOMETHING? YES! 22 13 ,? ~? ,? + 2? ,?
THIRD ORDER PERTURBATION THEORY ?00= ?2 1 + 2 1+ 2 1 2?? (2)= ??0 Metric ?0?= ?2 +1 ? 1 2 ???= ?2 1 2 1 2 2 ??+ 2 ?? + ?? 3??? 1 Matter ???= + ? ????+ ???? ????? ????? 3= 3 ?? 2 ?? ???
NOT EVERYTHING AT THIRD ORDER WILL CONTRIBUTE These were studies in the IR regime previously (Chen et al) Ultimately, we correlate the third order solution with a first order tensor. Since first order scalars and tensors do not correlate, we can drop a few terms 2 + ??? 2 + ??? ?2 + ??? 2 + ??? ?2 + ??? 2 3= ??? ???+ ??? ???+ ??? ??? The difficulty: the kernels of source terms containing second order perturbations have to be computed numerically.
CONCLUSION Our work can be used to constrain models of inflation that contain a peak in the power spectrum on small scales for both scalars and tensors. Scalar-tensor induced waves (and tensor-tensor) suffer on small scales. We get an unphysical enhancement of the observable when the primordial input spectrum is not peaked enough. Hopefully, we can fix this by also considering the correlations of first and third order tensors
EXTRA SLIDES (A. Green)
EXTRA SLIDES (Adshead et al)
EXTRA SLIDES (Adshead et al)
