Gauge dependence of gravitational waves generated from scalar perturbations

TL;DR

This paper investigates the gauge dependence of gravitational waves generated from scalar perturbations in cosmology, revealing that their power spectrum varies with gauge choice and can dominate primordial signals during matter domination.

Contribution

It demonstrates that second-order induced tensor perturbations are gauge dependent and emphasizes the importance of proper modeling for observational strategies.

Findings

Induced tensor modes depend on the gauge choice for scalar perturbations.

During matter domination, induced modes can surpass primordial gravitational waves for certain scales.

Proper gauge-specific modeling is crucial for interpreting gravitational wave measurements from large-scale structure.

Abstract

A tensor-type cosmological perturbation, defined as a transverse and traceless spatial fluctuation, is often interpreted as the gravitational waves. While decoupled from the scalar-type perturbations in linear order, the tensor perturbations can be sourced from the scalar-type in the nonlinear order. The tensor perturbations generated by the quadratic combination of linear scalar-type cosmological perturbation are widely studied in the literature, but all previous studies are based on zero-shear gauge without proper justification. Here, we show that, being second order in perturbation, such an induced tensor perturbation is generically gauge dependent. In particular, the gravitational wave power spectrum depends on the hypersurface (temporal gauge) condition taken for the linear scalar perturbation. We further show that, during the matter-dominated era, the induced tensor modes dominate over the linearly evolved primordial gravitational waves amplitude for $k\gtrsim10^{-2}~[h/{\rm Mpc}]$ even for the gauge that gives lowest induced tensor modes with the optimistic choice of primordial gravitational waves ($r=0.1$). The induced tensor modes, therefore, must be modeled correctly specific to the observational strategy for the measurement of primordial gravitational waves from large-scale structure via, for example, parity-odd mode of weak gravitational lensing, or clustering fossils.

Equations

$$d{s}^2=-a^2\left(1+2A\right)d{\eta }^2-2a^2{B}_{i}d\eta d{x}^i+a^2\left({\gamma }_{ij}+2C_{ij}\right)d{x}^{i}d{x}^j,$$

$$A\equiv \alpha ,B_i\equiv {\beta }_{,i}+B_i^{(v)},C_{ij}\equiv \phi {\gamma }_{ij}+{\gamma }_{,i|j}+C_{(i|j)}^{(v)}+h_{ij},$$

$${\tilde{T}}_{ab}=\tilde{\mu }{\tilde{u}}_a{\tilde{u}}_b+\tilde{p}\left({\tilde{g}}_{ab}+{\tilde{u}}_a{\tilde{u}}_b\right)+{\tilde{\pi }}_{ab},$$

$$\tilde{\mu }\equiv \mu +\delta \mu ,\tilde{p}\equiv p+\delta p,{\tilde{u}}_i\equiv a\Gamma v_i,v_i\equiv -v_{,i}+v_i^{(v)},\Gamma \equiv -{\tilde{n}}_c{\tilde{u}}^c=\frac{1}{\sqrt{1-a^2{\overline{h}}^{ij}{v}_i{v}_j}},$$

$${\tilde{\pi }}_{ij}\equiv a^2{\Pi }_{ij},{\Pi }_{ij}\equiv \frac{1}{a^2}\left({\Pi }_{,i|j}-\frac{1}{3}{\gamma }_{ij}\Delta \Pi \right)+\frac{1}{a}{\Pi }_{(i|j)}^{(v)}+{\Pi }_{ij}^{(t)}+\frac{1}{3}{\gamma }_{ij}{\Pi }_k^k.$$

$${\overline{h}}_{ij}=a^2\left(1+2\phi \right){\gamma }_{ij},{\overline{h}}^{ij}=\frac{1}{a^2(1+2\phi )}{\gamma }^{ij},\Gamma =\frac{1}{\sqrt{1-\frac{v^k{v}_k}{1+2\phi }}},{\Pi }_i^i=\frac{v^j{v}^j}{1+2\phi }{\Pi }_{ij}.$$

$$\frac{1}{a^2}\left({\nabla }_i{\nabla }_j-\frac{1}{3}{\gamma }_{ij}\Delta \right)\left[\frac{1}{a}{\left(a\chi \right)}^{\cdot }-\alpha -\phi -8\pi G\Pi \right]+\frac{1}{a}{\nabla }_{(i}\left\{\frac{1}{a^2}{\left[a^2\left(B_{j)}^{(v)}+a{\dot{C}}_{j)}^{(v)}\right)\right]}^{\cdot }-8\pi G{\Pi }_{j)}^{(v)}\right\}$$

$$+{\ddot{h}}_{ij}+3H{\dot{h}}_{ij}-\frac{\Delta -2K}{a^2}{h}_{ij}-8\pi G{\Pi }_{ij}^{(t)}=n_{ij},$$

$${\ddot{h}}_{ij}+3H{\dot{h}}_{ij}-\frac{\Delta -2K}{a^2}{h}_{ij}-8\pi G{\Pi }_{ij}^{(t)}=s_{ij},$$

$$s_{ij}\equiv {\mathcal{P}}_{ij}^{k\ell }{n}_{k\ell }\equiv n_{ij}-\frac{1}{3}{\gamma }_{ij}{n}_k^k+\frac{1}{2}\left({\nabla }_i{\nabla }_j-\frac{1}{3}{\gamma }_{ij}\Delta \right){\left(\Delta +3K\right)}^{-1}\left[n_k^k-3{\Delta }^{-1}\left(n_{|k\ell }^{k\ell }\right)\right]$$

$$-2{\nabla }_{(i}{\left(\Delta +2K\right)}^{-1}\left[n_{j)|k}^k-{\nabla }_{j)}{\Delta }^{-1}\left(n_{|k\ell }^{k\ell }\right)\right],$$

$$s_{ij}={\mathcal{P}}_{ij}^{k\ell }{n}_{k\ell }=n_{ij}+\frac{1}{2}\left({\nabla }_i{\nabla }_j-{\gamma }_{ij}\right)n_k^k-2{\Delta }^{-1}{\nabla }_{(i}{n}_{j),k}^k+\frac{1}{2}{\Delta }^{-2}\left({\nabla }_i{\nabla }_j+{\delta }_{ij}\Delta \right)n_{,k\ell }^{k\ell }.$$

$$n_{ij}=\frac{1}{a^3}{\left[a\left(2\phi {\chi }_{,i|j}+{\phi }_{,i}{\chi }_{,j}+{\phi }_{,j}{\chi }_{,i}\right)\right]}^{\cdot }+\frac{1}{a^2}\left(\kappa {\chi }_{,i|j}-4\phi {\phi }_{,i|j}-3{\phi }_{,i}{\phi }_{,j}\right)+\frac{1}{a^4}\left({\chi }_{|i}^{,k}{\chi }_{,j|k}-K{\chi }_{,i}{\chi }_{,j}\right)$$

$$+\frac{1}{a^2}\left[2{\dot{\chi }}_{,i|j}\alpha -H{\chi }_{,i|j}\alpha +{\chi }_{,i|j}\dot{\alpha }-2\left(\alpha +\phi \right){\alpha }_{,i|j}-{\alpha }_{,i}{\alpha }_{,j}-2{\alpha }_{,(i}{\phi }_{,j)}\right]+8\pi G\left(\mu +p\right)v_{,i}{v}_{,j}$$

$$\displaystyle\qquad-{1\over 3}\gamma_{ij}\bigg{\{}{1\over a^{3}}\left[a\left(2\varphi\Delta\chi+2\varphi^{,k}\chi_{,k}\right)\right]^{\displaystyle\cdot}+{1\over a^{2}}\left(\kappa\Delta\chi-4\varphi\Delta\varphi-3\varphi^{,k}\varphi_{,k}\right)+{1\over a^{4}}\left(\chi^{,k|\ell}\chi_{,k|\ell}-K\chi^{,k}\chi_{,k}\right)$$

$$\displaystyle\qquad+{1\over a^{2}}\left[2\alpha\Delta\dot{\chi}-H\alpha\Delta\chi+\dot{\alpha}\Delta\chi-2\left(\alpha+\varphi\right)\Delta\alpha-\alpha^{,k}\alpha_{,k}-2\alpha^{,k}\varphi_{,k}\right]+8\pi G\left(\mu+p\right)v^{|k}v_{,k}\bigg{\}}.$$

$$\kappa \equiv 3H\alpha -3\dot{\phi }-\frac{\Delta }{a^2}\chi ,$$

$$4\pi G\delta \mu +H\kappa +\frac{\Delta +3K}{a^2}\phi =0,$$

$$\kappa +\frac{\Delta +3K}{a^2}\chi -12\pi G\left(\mu +p\right)av=0,$$

$$\dot{\kappa }+2H\kappa -4\pi G\left(\delta \mu +3\delta p\right)+\left(3\dot{H}+\frac{\Delta }{a^2}\right)\alpha =0,$$

$$\dot{\chi }+H\chi -\phi -\alpha -8\pi G\Pi =0,$$

$$\delta \dot{\mu }+3H\left(\delta \mu +\delta p\right)-\left(\mu +p\right)\left(\kappa -3H\alpha +\frac{\Delta }{a}v\right)=0,$$

$$\frac{{\left[a^4\left(\mu +p\right)v\right]}^{\cdot }}{a^4\left(\mu +p\right)}-\frac{1}{a}\alpha -\frac{1}{a\left(\mu +p\right)}\left(\delta p+\frac{2}{3}\frac{\Delta +3K}{a^2}\Pi \right)=0.$$

$$n_{ij\chi }=-\frac{1}{a^2}\left[4\phi {\phi }_{,i|j}+2{\phi }_{,i}{\phi }_{,j}-\frac{1}{3}{\gamma }_{ij}\left(4\phi \Delta \phi +2{\phi }^{|k}{\phi }_{,k}\right)\right]+8\pi G\left(\mu +p\right)\left(v_{,i}{v}_{,j}-\frac{1}{3}{\gamma }_{ij}{v}^{|k}{v}_{,k}\right),$$

$$4\pi G\left(\mu +p\right)av=-\dot{\phi }-H\phi .$$

$${\phi }_{\chi }=\frac{3}{5}C,{\alpha }_{\chi }=-\frac{3}{5}C,{\kappa }_{\chi }=-\frac{9}{5}HC,v_{\chi }=-\frac{2}{5}\frac{1}{aH}C,{\delta }_{\chi }=\frac{6}{5}\left(1-\frac{1}{3}\frac{\Delta }{a^2{H}^2}\right)C,$$

$$\alpha =0,\dot{\phi }=0,\frac{1}{a}{\left(a\chi \right)}^{\cdot }=\phi ,\kappa =-\frac{\Delta }{a^2}\chi .$$

$$n_{ijv}=\frac{1}{a^2}\left(\kappa {\chi }_{,i|j}-2\phi {\phi }_{,i|j}-{\phi }_{,i}{\phi }_{,j}\right)+\frac{1}{a^4}{\chi }_{|i}^{,k}{\chi }_{,j|k}-\frac{1}{3}{\delta }_{ij}\left[\frac{1}{a^2}\left(\kappa \Delta \chi -2\phi \Delta \phi -{\phi }^{,k}{\phi }_{,k}\right)+\frac{1}{a^4}{\chi }^{,k|\ell }{\chi }_{,k|\ell }\right],$$

$${\phi }_v=C,{\alpha }_v=0,{\chi }_v=\frac{2}{5}\frac{1}{H}C,{\kappa }_v=-\frac{2}{5}\frac{\Delta }{a^{2}H}C,{\delta }_v=-\frac{2}{5}\frac{\Delta }{a^2{H}^2}C.$$

$$\hat{\alpha }=\alpha -\frac{1}{a}{\left(a{\xi }^0\right)}^{\prime },\hat{\beta }=\beta -{\xi }^0+{\left(\frac{1}{a}\xi \right)}^{\prime },{\hat{B}}_i^{(v)}=B_i^{(v)}+{\xi }_i^{(v)\prime },\hat{\gamma }=\gamma -\frac{1}{a}\xi ,{\hat{C}}_i^{(v)}=C_i^{(v)}-{\xi }_i^{(v)},$$

$$\hat{\phi }=\phi -\frac{a^{\prime }}{a}{\xi }^0,\hat{\chi }=\chi -a{\xi }^0,\hat{\kappa }=\kappa +\left(3\dot{H}+\frac{\Delta }{a^2}\right)a{\xi }^0,\hat{v}=v-{\xi }^0,\hat{\delta }=\delta -\frac{{\mu }^{\prime }}{\mu }{\xi }^0=\delta +3(1+w)\frac{a^{\prime }}{a}{\xi }^0,$$

$${\chi }_v\equiv \chi -av,{\chi }_{\phi }\equiv \chi -\frac{1}{H}\phi ,{\chi }_{\kappa }\equiv \chi +\frac{\kappa }{3\dot{H}+\frac{\Delta }{a^2}},{\chi }_{\delta }\equiv \chi +\frac{\delta }{3(1+w)H},$$

$${\phi }_v\equiv \phi -aHv,{\phi }_{\chi }\equiv \phi -H\chi ,$$

$$\gamma \equiv 0\equiv C_i^{(v)}.$$

$${\hat{C}}_{ij}$$

$$\hat{\phi }=\phi -\frac{a^{\prime }}{a}{\xi }^0+\frac{1}{2}{\left(\Delta +3K\right)}^{-1}\left[\left(\Delta +2K\right){\mathcal{C}}_k^k-{\mathcal{C}}_{|k\ell }^{k\ell }\right],$$

$$\hat{\gamma }=\gamma -\frac{1}{a}\xi -\frac{1}{2}{\left(\Delta +3K\right)}^{-1}\left[{\mathcal{C}}_k^k-3{\Delta }^{-1}\left({\mathcal{C}}_{|k\ell }^{k\ell }\right)\right],$$

Full text

cosmology: theory — large-scale structure of universe — gravitational waves

††journal: ApJ

1 Introduction

From the beginning of cosmological perturbation theory (Lifshitz, 1946), it is well known that linear order relativistic perturbations can be decoupled into the scalar-, vector- and tensor-type perturbations and the tensor-type perturbations are gauge-invariant in the (spatially homogeneous and isotropic) Friedmann background world model. The gauge dependence, especially the temporal gauge (hypersurface or slicing) dependence, of the scalar-type perturbations is also well known in the literature (Bardeen, 1980). It was Bradeen who suggested a practical strategy of utilizing the gauge dependence as an advantage in analyzing the perturbations. He wrote “The moral is that one should work in the gauge that is mathematically most convenient for the problem at hand.” (Bardeen, 1988)

The natural question arises: among all possible gauge conditions, which one is more relevant for interpreting the physical world? The answer to this question does not depend on the mathematical structure of the gauge: not on the gauge invariance of the variable in the chosen gauge, nor on the explicit gauge-invariance of the combination of perturbation variables. As a matter of fact, all perturbation variables without the gauge-mode ambiguity are gauge-invariant in the senses that their values evaluated in any other gauges remain the same. It is the case for all perturbation variables in the several fundamental gauges introduced in Bardeen (1980, 1988) except for the synchronous gauge. For these gauges, each perturbation variable uniquely corresponds to a gauge-invariant combination of perturbation variables. Again, according to Bardeen “While a useful tool, gauge-invariance in itself does not remove all ambiguity in physical interpretation,” and “Many gauge-invariant combinations of these scalars can be constructed, but for the most part they have no physical meaning independent of a particular time gauge, or hypersurface condition.” (Bardeen, 1988) For a given variable, say density perturbation or velocity perturbation, we can, in fact, construct the variable with infinitely many different gauge conditions, and all of them correspond to the gauge-invariant combinations; this is because the constant-time hypersurface can be deformed in a continuous manner. These statements are also true to the nonlinear order in cosmological perturbation theory (Noh & Hwang, 2004; Hwang & Noh, 2013a). It is, therefore, safe to treat a variable evaluated in different gauges as entirely different variables. Finally, according to Bardeen “Gauge-invariant variables give mathematically unambiguous ways of comparing results obtained in different gauges, but their physical interpretation is not necessarily straightforward, in that it is usually tied to a particular way of slicing the spacetime into hypersurfaces. I know of no way to characterize completely the deviations from homogeneity and isotropy independent of the slicing into spacelike hypersurfaces.” (Bardeen, 1988)

Instead, which gauge-invariant variable corresponds to the one that we measure from observation depends on the nature of the observation. That is, a specification of observation must tell us which gauge-invariant variable or combinations of them is the right one for that particular observation (assuming, of course, that perturbation theory in the Friedmann world model handles the observed phenomena). We discuss this issue further in Section 6.

To the nonlinear order, the three types of perturbations couple to each other in the equation level and the decomposition itself becomes ambiguous. It is because we can introduce many different ways of decomposing the perturbation to scalar-, vector-, and tensor-types (Hwang & Noh, 2013a); hereafter we simply call it the scalar perturbation or scalar mode, etc. Naturally, from the second order, even the tensor perturbation becomes gauge dependent.

In particular, the second-order tensor perturbations generated from the quadratic combinations of linear scalar perturbations (induced tensor perturbations) must depend on the gauge condition, as the linear scalar perturbations depend on the choice of the constant-time hypersurface. There are studies of such induced tensor modes in the literature (Mollerach et al., 2004; Baumann et al., 2007; Ananda et al., 2007; Arroja et al., 2009; Assadullahi & Wands, 2009, 2010; Jedamzik et al., 2010; Alabidi et al., 2013; Saga et al., 2015), but all these studies have been based on one particular gauge condition, the zero-shear gauge in our terminology. Note that, with no entirely clear reason, in the literature, this gauge condition is often termed as the longitudinal, Newtonian, conformal-Newtonian, or Poisson gauge, etc. The zero-shear gauge takes the scalar part of the shear of the normal frame vector to vanish; if we ignore the vector and tensor perturbation, this statement is valid for fully nonlinear orders in perturbation, see Eqs. (B7) and (C7) in Hwang & Noh (2013a).

In this work, we explicitly show the gauge dependence of the power spectrum of the induced tensor perturbations. The main results are summarized in Eq. (61) in a unified form, and shown in Figure 1. This paper is organized as following. In Section 2, we introduce our notations and basic equations. In Section 3, we use the nonlinear gauge transformation to show the gauge dependence of the tensor modes to the second order and relations of the tensor-mode solutions among different gauge conditions, see Eqs. (42) and (58). In Section 4, we present the tensor power spectrum in several gauge conditions in a unified form. We discuss the implication of our result to the future observations in Section 6. We set $c\equiv 1$.

2 Equations

As the metric convention we have

[TABLE]

where the spatial indices of $B_{i}$ and $C_{ij}$ are raised and lowered using the background metric of comoving coordinate $\gamma_{ij}$; for a spatially flat background with $K=0$ we have $\gamma_{ij}=\delta_{ij}$; $i,j,k,\dots$ are the three-dimensional spatial indices and $a,b,c,\dots$ are spacetime indices. We decompose the spatial vector $B_{i}$ and the spatial tensor $C_{ij}$ into the scalar, vector and tensor perturbations as (York, 1973)

[TABLE]

where $B^{(v)|i}_{i}\equiv 0\equiv C^{(v)|i}_{i}$ (transverse vector) and $h^{j}_{i|j}\equiv 0\equiv h^{i}_{i}$ (transverse-traceless tensor); a vertical bar indicates a covariant derivative based on $\gamma_{ij}$ as the metric; like $B_{i}$ and $C_{ij}$, indices of $B^{(v)}_{i}$, $C^{(v)}_{i}$ and $h_{ij}$ are raised and lowered using $\gamma_{ij}$ as the metric; we define $A_{(ij)}\equiv{1\over 2}(A_{ij}+A_{ji})$. We set $\chi\equiv a(\beta+a\dot{\gamma})$; an overdot indicates time derivative based on cosmic time $t$ (defined with $dt\equiv ad\eta$). For the energy-momentum tensor, we have

[TABLE]

where $\widetilde{\mu}$, $\widetilde{p}$, $\widetilde{u}_{c}$ and $\widetilde{\pi}_{ab}$ are the energy density, the pressure, the fluid four-vector and the anisotropic stress, respectively (Ehlers, 1993). We decompose them as

[TABLE]

where $\widetilde{n}_{a}$ is the normal four-vector and $\Gamma$ is the Lorentz factor. Indices of $v_{i}$, $v^{(v)}_{i}$, $\Pi_{ij}$, $\Pi^{(v)}_{i}$ and $\Pi^{(t)}_{ij}$ are raised and lowered using $\gamma_{ij}$ as the metric. We have $v^{(v)|i}_{i}\equiv 0\equiv\Pi^{(v)|i}_{i}$ and $\Pi^{(t)i}_{\;\;\;\;\;i}\equiv 0\equiv\Pi^{(t)j}_{\;\;\;\;\;i|j}$; tildes indicate the covariant quantities. We added $\Pi^{k}_{k}$ term in the decomposition of $\Pi_{ij}$ as we have $\Pi^{k}_{k}\neq 0$ to the nonlinear order. $\overline{h}^{ij}$ is the inverse metric of the ADM (Arnowitt-Deser-Misner) metric $\overline{h}_{ij}$ defined as $\widetilde{g}_{ij}\equiv\overline{h}_{ij}$. For $\gamma\equiv 0\equiv C^{(v)}_{i}$ and $h_{ij}\equiv 0$ we can derive explicit form of the inverse metric and we have (Hwang & Noh, 2013a; Hwang et al., 2016)

[TABLE]

Setting $\gamma\equiv 0\equiv C^{(v)}_{i}$ corresponds to taking the spatial gauge condition without losing any generality and without missing any physics: according to Bardeen “Since the background 3-space is homogeneous and isotropic, the perturbations in all physical quantities must in fact be gauge invariant under purely spatial gauge transformations.” (Bardeen, 1988); this statement is true to fully nonlinear order (Noh & Hwang, 2004; Hwang & Noh, 2013a).

The tracefree ADM propagation equation can be written as

[TABLE]

where $n_{ij}$ indicates pure nonlinear parts, see Eq. (109) of Hwang & Noh (2007); in our notation we absorb the $\Pi^{k}_{k}$-part to the right-hand-side, thus $n_{ij}$ is tracefree. The tensor part of Eq. (6) becomes

[TABLE]

with $s_{ij}$ being the transverse-tracefree projection of $n_{ij}$:

[TABLE]

where ${\cal P}_{ij}^{\;\;\;k\ell}$ is a transverse-tracefree projection operator on a symmetric spatial tensor. For the spatially flat case, $K=0$ we have

[TABLE]

Up to this point the decomposition and equations are valid to fully nonlinear order in perturbation.

In the following we shall consider only the scalar perturbations as the source of the tensor modes. As the spatial gauge condition we take the $\gamma\equiv 0$ (Bardeen, 1988). Thus, we have $A=\alpha$, $B_{i}=\chi_{,i}/a$ and $C_{ij}=\varphi\gamma_{ij}$. Considering the quadratic combination of linear scalar perturbation, Eq. (71) of Hwang & Noh (2007) gives

[TABLE]

For later convenience, here we summarize the basic set of linear order scalar perturbation equations, see Eqs. (95)-(101) in Hwang & Noh (2007):

[TABLE]

$\kappa$ is a perturbed part of the trace of extrinsic curvature.

2.1 Zero-shear gauge

In the zero-shear gauge we set $\beta\equiv 0\equiv\gamma$, thus $\chi=0$. For $\Pi=0$, we have $\alpha=-\varphi$ to the linear order, and Eq. (10) gives

[TABLE]

where the sub-index $\chi$ indicates the zero-shear gauge. In the zero-shear gauge we have $\alpha=\alpha_{\chi}$, $\varphi=\varphi_{\chi}$, and $v=v_{\chi}$. To the linear order, from Eqs. (11) and (13), we have

[TABLE]

These equations are valid for general $K$.

In the matter dominated era the growing solutions to the linear order for $K=0=\Lambda$ are (Hwang, 1994)

[TABLE]

where $C$ is a constant (in time) coefficient indicating the growing solution.

2.2 Comoving gauge

In the comoving gauge we set $v\equiv 0\equiv\gamma$. In the zero-pressure case with $\Pi=0$ and $K=0$, to the linear order, from Eqs. (11)-(17), we have

[TABLE]

Using this, Eq. (10) gives

[TABLE]

where the sub-index $v$ indicates the comoving gauge. In this gauge we have $\alpha=\alpha_{v}$, $\varphi=\varphi_{v}$, $\chi=\chi_{v}$, and $\kappa=\kappa_{v}$.

In the matter dominated era the growing solutions to the linear order for $K=0=\Lambda$ are (Hwang, 1994)

[TABLE]

The normalization of the growing solution is based on the conserved nature of a variable $\varphi_{v}\equiv C$.

3 Gauge Issue

We consider the gauge transformation $\widehat{x}^{a}=x^{a}+\xi^{a}(x^{e})$. To the linear order, using $\xi_{i}\equiv{1\over a}\xi_{,i}+\xi^{(v)}_{i}$ with $\xi^{(v)|i}_{i}\equiv 0$, we have [see Eq. (250) in Noh & Hwang (2004)]:

[TABLE]

where the prime indicates a time derivative based on $\eta$ and $x^{0}=\eta$; $w\equiv p/\mu$. From these we have constructed gauge-invariant combinations

[TABLE]

which correspond to $\chi$ (the scalar part of the shear of the normal frame vector) in, respectively, comoving gauge ($\chi_{v}$), uniform expansion gauge ($\chi_{\varphi}$), uniform curvature gauge ($\chi_{\kappa}$), and uniform density gauge ($\chi_{\delta}$), and

[TABLE]

are the scalar metric (curvature) perturbation $\varphi$ in comoving gauge ($\varphi_{v}$) and zero-shear gauge ($\varphi_{\chi}$).

To the linear order we fix the spatial (including the scalar and vector) gauge by conditions (Bardeen, 1988)

[TABLE]

Under these gauge conditions, the spatial gauge degrees of freedom are fixed completely with $\xi=0=\xi^{(v)}_{i}$, thus $\xi_{i}=0$, to the linear order. The second order gauge transformation is given from Eq. (231) in Noh & Hwang (2004) as

[TABLE]

where ${\cal C}_{ij}$ indicates pure quadratic parts of the gauge transformation property of $C_{ij}$. From this, using the decomposition in Eq. (2), we can show

[TABLE]

To the second order we can continue taking the spatial and rotational gauge by the same conditions in Eq. (27). These are possible by suitable choices of $\xi$ and $\xi^{(v)}_{i}$ using Eqs. (30) and (31); i.e., the spatial gauge conditions to the second order determine $\xi$ and $\xi^{(v)}$ to the second order as

[TABLE]

Notice that even in the case of vanishing vector perturbation we should not ignore $\xi^{(v)}_{i}$ to the second order. By taking conditions in Eq. (27), we have

[TABLE]

where

[TABLE]

are spatially gauge-invariant combinations to the linear order. We can show that the $\gamma_{ij}$ part in ${\cal C}_{ij}$ does not affect the tensor mode gauge transformation in Eq. (32).

Now, we consider pure scalar perturbation to the linear order. Ignoring the $\gamma_{ij}$ part that do not contribute to the tensor modes, we have

[TABLE]

Using Eqs. (24)-(26) and Eq. (36), we can construct a set of variables ${\cal C}_{ij{\rm x}}$ such that the gauge transformation is given with ${\cal C}_{ij}^{\rm(tensor)}$,

[TABLE]

where ${\rm x}=\chi$, $v$, $\varphi$, $\kappa$, and $\delta$. The explicit expressions for $C_{ij\rm{x}}$ that we consider here are given as following:

[TABLE]

Note that, unlike the gauge-invariant variables $\chi_{\rm x}$ and $\varphi_{\rm x}$ that we have defined earlier, ${\cal C}_{ij{\rm x}}$ is not a gauge-invariant notation. With these new variables, we can show that

[TABLE]

thus

[TABLE]

Using these notations, we can construct a unified form of explicit gauge-invariant combination $h_{ij{\rm x}}$ as

[TABLE]

where $h_{ij{\rm x}}$ is a unified notation of the gauge-invariant combinations; for example, for ${\rm x}=\chi$, $h_{ij\chi}$ is a gauge-invariant combination which is the same as $h_{ij}$ in the zero-shear gauge ($\chi\equiv 0$), and similarly for ${\rm x}=v$ (the comoving gauge), ${\rm x}=\varphi$ (the uniform-curvature gauge), ${\rm x}=\kappa$ (the uniform-expansion gauge), and ${\rm x}=\delta$ (the uniform-density gauge). Therefore, using the relations in Eq. (39) we arrive at a rather simple relation between the tensor perturbations in a general gauge ${\rm x}$ and the ones in the zero-shear gauge (with subscript $\chi$) as,

[TABLE]

Complete sets of solutions for $\chi_{\rm x}$ in all fundamental gauges are presented in Tables of Hwang (1994) for a pressureless medium and Tables of Hwang (1993) for an ideal fluid medium.

4 Fourier analysis

We consider a spatially flat background, where the plane wave solutions are eigenfunctions of the Laplacian operator. It is, then, convenient to work in the Fourier space. We introduce a Fourier decomposition of a tensor perturbation $h_{ij}({\bf x},t)$ as (Mollerach et al., 2004)

[TABLE]

where

[TABLE]

are polarization bases for spin-2 fields; we construct the bases from two transverse unit vectors $e_{i}$ and $\overline{e}_{i}$ (satisfying $|e_{i}|\equiv 1\equiv|\overline{e}_{i}|$ and $e^{i}k_{i}=\overline{e}^{i}k_{i}=e^{i}\overline{e}_{i}=0$). Note that the two polarization bases are orthogonal ($e^{ij}({\bf k})\overline{e}_{ij}({\bf k})=0$) and normalized as $e^{ij}({\bf k})e_{ij}({\bf k})=\overline{e}^{ij}({\bf k})\overline{e}_{ij}({\bf k})=1$. These two bases are sometimes called $e^{+}_{ij}=\sqrt{2}\,e_{ij}$ and $e^{\times}_{ij}=\sqrt{2}\,\overline{e}_{ij}$ in literature; see, for example, Dai et al. (2012). By using the orthogonality of the polarization bases, we have

[TABLE]

For the spatially flat case ($K=0$) and in the absence of the genuine tensor-origin contribution to anisotropic stress $\Pi_{ij}^{(t)}$, which is the case for the standard cosmological models, the gravitational wave equation in Eq. (7) becomes

[TABLE]

with $s_{ij}$ given in terms of $n_{ij}$ in Eq. (9). The other polarization mode $\overline{h}({\bf k},t)$ obeys the equation parallel to Eq. (46) with $\overline{s}({\bf k},t)$ defined with $\overline{e}_{ij}$ instead of $e_{ij}$. In the parity-preserving Universe, the two polarization modes $h({\bf k},t)$ and $\overline{h}({\bf k},t)$ must have the exactly same statistical properties. We, therefore, shall focus only on $h({\bf k},t)$ in what follows. The effect of $\overline{h}({\bf k},t)$ will be taken into account by simply adding the same contribution at the end of the calculation.

We transform Eq. (46) by introducing the new variable $\mathpzc{v}\equiv ah$ and using the conformal-time derivative (denoted by prime) as

[TABLE]

The solution is then given by

[TABLE]

by using the Green’s function

[TABLE]

for $\eta\geq\tilde{\eta}$ and 0 otherwise, as the scalar source at time $\tilde{\eta}$, $s({\bf k},\tilde{\eta})$, only affects the gravitational waves at later times $\eta\geq\tilde{\eta}$. Here, $\mathpzc{v}_{1}$ and $\mathpzc{v}_{2}$ are two linearly independent solutions for the homogeneous part of Eq. (47).

In the flat, matter dominated Universe ($K=0=\Lambda$), we have $a\propto\eta^{2}$, and the solutions to the linear order gravitational waves are (Lifshitz, 1946; Weinberg, 1972)

[TABLE]

where $x\equiv k\eta$. The tensor amplitude $\mathpzc{v}$ is gauge-invariant in the linear order: that is, it is independent of the gauge condition taken for the scalar perturbation. The induced tensor amplitudes appear from the second order as Eq. (48), where the Green’s function is given as

[TABLE]

The induced tensor amplitude depends on the temporal gauge chosen for the scalar perturbations.

In this section, we shall present explicit expressions for the induced tensor amplitude in various different temporal gauge conditions. In particular, we shall calculate the Fourier space expression for the source term $s_{\rm x}({\bf k},\eta)$ with a temporal gauge condition denoted by the subscript ${\rm x}$. We start from the zero-shear gauge and find $s_{\chi}({\bf k},t)$ in the matter-dominated universe, and generalize to the other gauges by using the gauge transformation that we have presented in Sec. 3.

4.1 Zero-shear gauge

In the zero-shear gauge, using Eqs. (18) and (46) we can show

[TABLE]

In the matter dominated era, using Eq. (20) we have

[TABLE]

The general solution to the second order is

[TABLE]

Imposing the initial condition $h_{\chi}=0=h_{\chi}^{\prime}$ at $\eta=0$ we have $c_{2}=0$ and (Mollerach et al., 2004)

[TABLE]

For $x\ll 1$ we have $g={1\over 10}x^{2}$ thus $h_{\chi}={1\over 10}s_{\chi}\eta^{2}$. For $x\gg 1$ we have $g=1$ thus $h_{\chi}=s_{\chi}/k^{2}$.

4.2 Unified expression in other gauges

Solutions in other gauge conditions simply follow from the one in the zero-shear gauge, as Eq. (42) gives

[TABLE]

The linear solutions in matter dominated era for $K=0=\Lambda$ are [see Table 1 of Hwang (1994)]

[TABLE]

which are $\chi$ value evaluated in, respectively, comoving gauge ${\rm x}=v$, uniform curvature gauge ${\rm x}=\varphi$, uniform expansion gauge ${\rm x}=\kappa$, and uniform density gauge ${\rm x}=\delta$. In the case of the comoving gauge (${\rm x}=v$), we can check that the solution of $h_{v}$ derived from the gauge transformation in Eq. (56) coincides with solution directly derived from Eqs. (7), (9) and (22).

From Eqs. (53), (55) and (56) we have the unified expression for the pure second order contribution,

[TABLE]

where

[TABLE]

We have ${1\over aH}={1\over 2}\eta$. For $x\ll 1$ we have $g={1\over 10}x^{2}$ thus $h_{\chi}\propto h_{\rm x}$. Thus, we have $h_{\rm x}=0=h_{\rm x}^{\prime}$ at $\eta=0$. For $x\gg 1$ we have $g=1$ thus $h_{\chi}\ll h_{\rm x}$ except for ${\rm x}=\kappa$.

4.3 Power spectrum: unified expression

Using the definition of power spectra

[TABLE]

we have the expression for the induced tensor power spectrum as

[TABLE]

Note that the factor $1/2$ in Eq. (60) accounts for the two polarization modes whose power spectrum must be equal. Here, we assume that primordial curvature perturbations follow Gaussian statistics. Note that, although it is the same order, the cross-term multiplying linear order and third order scalar perturbations is not present because there is no linear order scalar contribution to the induced tensor mode in the standard Friedmann-Robertson-Walker world models.

5 Spectrum of induced gravitational waves

We calculate the spectrum of induced gravitational waves in the standard $\Lambda$CDM world model adopting the best-fitting cosmological parameters (maximum likelihood values in the table entitled

“base_plikHM_TTTEEE_lowTEB_lensing_post_BAO_H080p6_JLA”) from Planck 2015 (Planck Collaboration et al., 2015): $\Omega_{\rm b}h^{2}=0.022307$, $\Omega_{\rm cdm}h^{2}=0.11865$, $\Omega_{\nu}h^{2}=0.000638$, $\Omega_{\Lambda}=0.69179$, with current Hubble expansion rate of $H_{0}=67.78\,{\rm km/s/Mpc}$. Primordial scalar power spectrum amplitude and spectral index are, respectively, $\mathcal{A}_{s}=2.147\times 10^{-9}$ and $n_{s}=0.9672$ that yield the normalization of matter power spectrum at present time as $\sigma_{8}=0.8166$.

From the Einstein equation in the comoving gauge, we find that

[TABLE]

which relates the power spectrum of $C$ to the usual linear matter power spectrum in the comoving gauge $P_{L}(k)$ as

[TABLE]

We have calculated the power spectrum of induced tensor perturbations in Figure 1, as the gravitational wave energy density parameter per logarithmic interval

[TABLE]

This follows from the 00-component of the energy-momentum tensor of the gravitational waves with $\rho_{\rm GW}\propto\left(h^{\prime}_{\rm x}\right)^{2}$ (Watanabe & Komatsu, 2006). The second approximated is accurate in sub-horizon scales. We show both wavenumber ($k$, along the top $x$-axis) and frequency ($f=kc/2\pi$, along the bottom $x$-axis).

Fig. 1 shows the induced tensor perturbations calculated in zero shear (ZS) gauge and uniform expansion (UE) gauge (dark blue), comoving gauge (Co, dark red), and uniform curvature gauge (UC, dark green) at four different redshifts ($z=6$, $z=5$, $z=2$, and $z=0$ from top, left to bottom, right). We do not present the power spectrum for the pathological uniform density gauge, because the integrand in this gauge blows up on small-scales (for larger $k$). We plot the result down to $k=100~{}[h/{\rm Mpc}]$, just for the presentation purpose. Of course, the second-order perturbation theory, breaks down well before $k\sim 1[h/{\rm Mpc}]$ even for the highest redshift ($z=6$) shown here; see, for example, Jeong & Komatsu (2006).

First of all, we note the gauge dependence of the power spectra of induced tensor perturbations. While zero shear gauge and uniform expansion gauge show the same power spectrum, the power spectra calculated from comoving gauge and uniform curvature gauge are very different. As the induced tensor modes result from the non-linear interactions, the power on large-scales is suppressed and scales as $h_{\rm x}(k)\propto k^{2}$ in $k\to 0$ limit for all cases. Even on these near-horizon scales, however, amplitude are different for all cases.

To facilitate the comparison, we have also shown the power spectrum of linear tensor perturbations with $r=0.1$ with (red solid) and without (black dotted) the damping due to free-streaming neutrinos after neutrino decoupling epoch ($T\simeq 1.5~{}{\rm MeV}$) (Weinberg, 2004). Here, we adopt the damping factor calculation of Watanabe & Komatsu (2006) that the primordial linear gravitational waves $h_{\rm prim}$ are damped by $80.313\%$ for the modes that re-entering horizon during radiation dominated epoch. For larger-scale modes, we estimate the damping factor by linearly re-scaling the small-scale damping factors with the neutrino fraction at the time of the horizon-crossing. That is, damping factor is applied as $h^{(1),({\rm with~{}\nu})}(k,\eta)=[1-0.48582\,\Omega_{\nu}(\eta=k^{-1})]h^{(1),({\rm without~{}\nu})}(k,\eta)$, where $h^{(1)}$ is the amplitude of the linear gravitational waves originated from the primordial Universe. Note that $\Omega_{\nu}=0.40523$ during the radiation-dominated epoch, which gives the desired damping factor of $80.313\%$, but the damping effect dies out for the modes re-enters the horizon during matter and $\Lambda$-dominated epoch when the neutrino fraction is negligibly small.

At higher redshift ($z=6$ and $z=5$), the induced tensor power spectrum is much bigger compared to the linear power spectrum for $k\gtrsim 10^{-2}~{}[h/{\rm Mpc}]$, even for the lowest case for the zero-shear gauge or uniform expansion gauge. While the amplitude of primordial gravitational waves decays as a linear theory tensor mode (without the linear source), in the matter dominated epoch, the amplitude of induced tensor modes stays constant because the induced tensor mode is proportional to the gravitational potential perturbation which stays constant. For lower redshift ($z=2$ and $z=0$), we observe the competition between the cosmological redshift of the linear tensor mode and the damping of gravitational potential in the presence of cosmological constant ($\Lambda$). Note that for the lower redshift, we estimate the induced tensor modes by simply using the result in the matter-dominated era [Eqs. (58)-(59)] and re-scale the gravitational potential power spectrum Eq. (63) with the linear growth factor.

Although the result for zero-shear gauge has been reported in the previous studies (Mollerach et al., 2004; Baumann et al., 2007; Ananda et al., 2007; Arroja et al., 2009; Assadullahi & Wands, 2009, 2010; Jedamzik et al., 2010; Alabidi et al., 2013; Saga et al., 2015), the gauge-dependence as well as total domination of induced tensor modes over primordial gravitational waves signature is the new result in this work. From these figures, it is clear that the induced tensor mode contribution must be understood properly in conjunction with the exact observable that being considered; that is the only way to remove the ambiguity due to gauge choice, and, therefore, to extract truly primordial gravitational waves signature from the large-scale structure observables.

6 Discussion

We have presented the leading order induced tensor power spectrum generated by the quadratic combination of linear scalar perturbation in the matter dominated era. The tensor power spectrum depends on the slicing conditions taken for the linear scalar perturbation. The results are summarized in Eqs. (58) and (61) for the solutions and the power spectra, respectively, in unified forms, and in Figure 1.

First of all, we emphasize again that the tensor power spectrum is gauge dependent as it naturally has to be to the nonlinear order. Comparing the induced tensor power spectrum with the linearly evolved spectrum of the primordial gravitational waves (Figure 1), even with the optimistic value of $r=0.1$, we find that the induced tensor power spectrum dominates over the primordial signature for inter-galactic scales ($k\gtrsim 10^{-2}~{}[h/{\rm Mpc}]$). At high redshifts ($z>2$), this is true for all gauge choices that we have considered in this paper, although the power spectrum for zero-shear gauge and uniform-expansion gauge decays faster on smaller scales than comoving gauge and uniform-curvature gauge.

The gauge-dependence that we observe here is a consequence of the gauge-dependence of the scalar perturbation variables. In the subhorizon scales, the linear order density perturbation equations coincide in all four gauges that we consider here (Bardeen, 1980; Hwang, 1993). The equations for velocity field and gravitational potential, however, depend on the choice of gauge, and these differences propagate to the gauge-dependence of the amplitude of the induced gravitational waves. In the zero-pressure limit the zero-shear gauge and the uniform expansion gauge properly reproduce the exact (non-perturbative) Newtonian equations for the density, velocity and the gravitational potential (Hwang & Noh, 2013b). If we consider the relativistic pressure, the uniform-expansion gauge is better than the zero-shear gauge (Hwang & Noh, 2016). In the sub-horizon scale even to the linear order the uniform-curvature gauge fails to reproduce the Newtonian velocity and gravitational potential (Hwang & Noh, 1999a). The comoving gauge is a curious case. In the zero-pressure limit, the equations for density and velocity exactly coincide with the Newtonian ones even to the second-order perturbations in all scales but do not have the proper gravitational potential (Noh & Hwang, 2004).

Then, an important question arises: which gauge is the right choice for the observation of induced gravitational waves? As mentioned in Section 1, in order to properly address the question one has to specify the observational strategy. First, the frequency range that we have considered here is too low (naturally, of order a Hubble time scale, $f\simeq 10^{-19}-10^{-16}~{}{\rm Hz}$) to be detected from the direct detection methods using interferometers such as LIGO or LISA, or pulsar timing array. Large-scale structure of galaxy distribution offers futuristic, but compelling methods of detecting tensor modes by, for example, parity-odd (B-mode) part of the gravitational weak lensing (Schmidt & Jeong, 2012a), galaxy clustering (Jeong & Schmidt, 2012), cosmic ruler (Schmidt & Jeong, 2012b), as well as clustering fossils (Jeong & Kamionkowski, 2012). Because these observables measure the tensor part of the metric perturbations on scales that we are considering here, and blind about the origin of the tensor perturbation, we need to understand the induced tensor perturbation properly in order to pin down the signatures from primordial gravitational waves. As mentioned earlier, the proper choice of gauge is subject to the exact way that the tensor perturbations are measured from each observable.

This was true even for the linear order density and velocity power spectra. As the behavior of density perturbation depends on the gauge (Lifshitz, 1946; Bardeen, 1980), the power spectrum of it should depend on the gauge as well: in many (but not all) fundamental gauge conditions used in the literature the behavior of density perturbation happens to coincide far inside the horizon (Bardeen, 1980). The issue has been resolved in the density perturbation case by addressing the strategy of measuring the density power spectrum: by observing the photons traveled from galaxies (Yoo et al., 2009; Yoo, 2010; Bonvin & Durrer, 2011; Challinor & Lewis, 2011; Jeong et al., 2012; Yoo, 2014; Jeong & Schmidt, 2015). In the case of cosmic microwave background temperature anisotropy power spectrum the observational strategy of measuring temperature ‘difference’ between different angular directions in the sky makes the observed quantities naturally gauge invariant (Abbott & Wise, 1984; Abbott & Schaeffer, 1986; Hwang & Noh, 1999b).

In this work we only have clarified the gauge dependence of the second order tensor perturbation power spectrum generated by linear scalar perturbation. The issue of which one or combination of variable is the right choice for observed power spectrum is left for future investigation.

J.H. was supported by Basic Science Research Program through the National Research Foundation (NRF) of Korea funded by the Ministry of Science, ICT and Future Planning (No. 2016R1A2B4007964). D.J. was supported by National Science Foundation grant AST-1517363. H.N. was supported by National Research Foundation of Korea funded by the Korean Government (No. 2015R1A2A2A01002791).