Missing one-loop contributions in secondary gravitational waves

Chao Chen, Atsuhisa Ota, Hui-Yu Zhu, and Yuhang Zhu

Phys. Rev. D 107, 083518 – Published 19 April 2023

Abstract

We find several missing one-loop-order contributions in previous considerations about secondary gravitational waves induced at nonlinear order in cosmological perturbations. We consider a consistent perturbative expansion to third order in cosmological perturbations, including higher-order interactions and iterative solutions ignored in the previous literature. Tensor fluctuations induced by the source with two scalar and one tensor perturbations are correlated with the first-order tensor fluctuation and thus give a one-loop order correction to the tensor-power spectrum. The missing loop correction is scale invariant and negative in the superhorizon region, which secondarily reduces the initial primordial tensor-power spectrum prior to the horizon reentry. Such an IR behavior is very different from the autospectrum of second-order induced tensor modes discussed in the previous literature and can be important for the actual gravitational wave measurements. For a sharp peak of scalar fluctuations with $A_{ζ} = 10^{- 2}$ at $k_{*} = 10^{5} h / Mpc$ motivated by the LIGO/Virgo events, we show that the tensor-power spectrum at the cosmic microwave background scale reduces by at most 35%. Hence, the polarization B mode might not be seen because of the reduction of the original tensor spectrum due to the secondary effect of primordial black hole formation.

Received 13 November 2022
Accepted 20 March 2023

DOI:https://doi.org/10.1103/PhysRevD.107.083518

Physics Subject Headings (PhySH)

Gravitational wave sources Inflation

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Chao Chen ^1,*, Atsuhisa Ota ^1,†, Hui-Yu Zhu ^1,2,‡, and Yuhang Zhu ^1,2,§

¹Jockey Club Institute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People’s Republic of China
²Department of Physics, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, People’s Republic of China

^*iascchao@ust.hk
^†iasota@ust.hk
^‡hzhuav@connect.ust.hk
^§yzhucc@connect.ust.hk

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Vol. 107, Iss. 8 — 15 April 2023

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Images

Figure 1
Nine typical one-loop contributions to the tensor-power spectrum. The internal wavy and solid lines represent tensor and scalar propagators, respectively. The standard second-order SIGW $P_{h}^{(22)}$ is included in diagram (a). Diagrams (b) and (c) were studied in Ref. [75], which are subdominant unless the tensor propagator is more enhanced than the scalar one. (d) is the new graph considered in this paper. We ignore (e), (h), and (i) since we only focus on the enhancement of the scalar propagator. The tadpole (g) is zero, and (f) is also negligible in the IR region. Note that we ignored vectors for simplicity.
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Figure 2
Detailed structures of diagrams (a) and (d) shown in Fig. 1. We draw the diagrams following Ref. [78], while the direction of time is implicit for simplicity. The cross circle represents the contraction between two linear fields (i.e., their power spectra), which is also regarded as an “external source” in this system. The red lines are Green functions. The diagrams (a1) and (d) contribute to $P^{(13)}$ , while (a2) is the induced GWs $P^{(22)}$ .
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Figure 3
Left: the absolute value of the one-loop correction $\frac{1}{3} | P_{h}^{(13)} |$ shown in Eq. (31) as a function of $k / k_{*}$ during the RD era. The vertical axis is normalized by the primordial tensor spectrum $P_{h}^{(11)}$ . Brown, red, and blue solid lines denote different times: $y \equiv k_{*} τ = 4 \times 10^{1}, 4 \times 10^{2}$ , and $4 \times 10^{3}$ , respectively. The damping feature for $k / k_{*} ≳ y^{- 1}$ implies the horizon entry. Right: a comparison between the amplitudes of the one-loop corrections $| P_{h}^{(13)} |$ and the standard second-order SIGWs $P_{h}^{(22)}$ during the RD era, which are shown by the red dashed and the cyan solid curves, respectively. The parameters are taken as $y_{D} = 4 \times 10^{3}$ , $A_{ζ} = 10^{- 2}$ , and $P_{h}^{(11)} = 10^{- 10}$ .
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Figure 4
Left: absolute value of the one-loop correction $| P_{h}^{(13)} |$ sourced by the delta-function-like source during MD era. The vertical axis is normalized by the primordial tensor spectrum $P_{h}^{(11)}$ , and $A_{ζ}$ is taken to be $10^{- 4}$ . Brown, red, and blue solid lines denote different times: $y \equiv k_{*} τ = 10^{2}, 10^{3}$ , and $10^{4}$ , respectively. The damping feature for $k / k_{*} ≳ y^{- 1}$ implies the horizon entry. Right: comparison between two types of one-loop SIGWs power spectra. The cyan curve refers to $\frac{1}{4} P_{h}^{(22)}$ while the red dashed line represents $\frac{1}{3} | P_{h}^{(13)} |$ . Here, the dimensionless time $y$ is chosen to be $10^{2}$ , $A_{ζ} = 10^{- 4}$ , and $P_{h} = 10^{- 10}$ .
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Figure 5
Left: current constraints on the curvature perturbations on different scales, including the Planck [94] (red), Lyman- $α$ forest [97] (blue), far infrared absolute spectrophotometer (FIRAS) CMB spectral distortion [98] (orange), and pulsar timing arrays (PTA) constraint on the standard SIGW $P_{h}^{(22)}$ [41] (magenta). We also present constraint from PBH abundance account for dark matter [27, 86] (green) with a conservative value $A_{ζ} = 10^{- 2}$ . Similar plots can also be found in Refs. [27, 41, 86]. The lower horizontal axis corresponds to the peak scale $k_{*}$ for the delta-function-like source. Right: upper limit on tensor-to-scalar ratio variation $| Δ_{r} |$ in terms of the monochromatic PBH mass $M_{PBH}$ . The shadow refers to the parameter space allowed by the left panel.
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Figure 6
Modulation of the superhorizon tensor fluctuation is illustrated. The blue sine curve means the tensor fluctuations whose wavelength is larger than the Hubble scale $H^{- 1}$ . The thin orange curve means the amplitude modulation introduced by the subhorizon scalar fluctuations coupled to the first-order tensor fluctuations. When integrating out the subhorizon scalar fluctuations, we obtain the reduced tensor fluctuations denoted by the thick orange curve.
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