5 Local tides, intrinsic alignment, and gravitational waves
Where we are
Chapter 4 established that the gravitational information available inside a small freely falling laboratory is the electric part of the Riemann tensor, \(\mathcal R_{ij}=R_{\hat 0i\hat 0j}\), measured in the observer’s own tetrad. This chapter collects the consequences.
The local tensor \(R_{\hat 0 i\hat 0 j}\) provides a common language for several effects that are often presented separately. Its scalar-perturbation limit is the tidal operator used in galaxy bias. A galaxy shape can respond to its trace-free part during formation, producing intrinsic alignment. A gravitational wave supplies a tensor contribution to the same local curvature and produces the familiar differential motion of freely falling particles.
The organizing observation is that a local source responds to a tensor representation, not to a mechanism. The response kernel cannot ask whether a trace-free tide was generated by scalar density perturbations or by a passing tensor mode; it sees only \(\mathcal E_{\langle ij\rangle}\) and its history along the worldline. What distinguishes the two cases is the time dependence of the tide and its correlation with other observables, not the response itself.
Three results follow in sequence. First, the scalar limit of \(\mathcal R_{ij}\) reproduces the tidal operator \(K_{ij}\) that Chapter 1 introduced on Newtonian grounds, which closes the argument begun there. Second, extending the functional-response formalism of Chapter 1 to a tensor-valued observable produces intrinsic alignment, with rotational invariance fixing the index structure and leaving a single scalar kernel \(R_E\) that carries the population-dependent internal dynamics. Third, the same construction applied to a gravitational wave shows why the local forcing is \(\ddot h_{ij}^{\mathrm{TT}}\) rather than \(h_{ij}^{\mathrm{TT}}\).
The chapter ends with a distinction that matters for any survey measuring shapes. An observed spin-two field on the sky receives both a local source response, defined here, and the propagation distortion of Chapter 3. The two are indistinguishable by tensor structure alone and must be separated by where the response is defined.
5.1 The scalar tidal field in a local frame
We first recover the Newtonian tide from curvature. The metric in this subsection is written in local physical coordinates rather than in the conformal-coordinate convention of Chapter 2. The calculation isolates the perturbative Riemann tensor before restoring the FLRW scale-factor conversion.
Consider a weak-field metric in local physical coordinates,
\[ \mathrm ds^2=-(1+2\Psi)\mathrm dt^2 +(1-2\Phi)\delta_{ij}\mathrm dx^i\mathrm dx^j, \tag{5.1}\]
where the sign convention for \(\Phi\) is now the common local weak-field convention. The linearized Riemann tensor about Minkowski space is
\[ \begin{aligned} R^{(1)}_{\rho\sigma\mu\nu} =\frac12\big(& \partial_\mu\partial_\sigma h_{\rho\nu} +\partial_\nu\partial_\rho h_{\sigma\mu}\\ &-\partial_\nu\partial_\sigma h_{\rho\mu} -\partial_\mu\partial_\rho h_{\sigma\nu} \big). \end{aligned} \tag{5.2}\]
For Eq. 5.1,
\[ R_{0i0j}=\partial_i\partial_j\Psi +\delta_{ij}\ddot\Phi. \tag{5.3}\]
In the quasistatic Newtonian limit, the time-derivative term is negligible and
\[ R_{0i0j}\simeq\partial_i\partial_j\Psi. \tag{5.4}\]
The geodesic-deviation equation then becomes
\[ \ddot X^i=-\partial^i\partial_j\Psi\,X^j, \tag{5.5}\]
which is the relative acceleration obtained by subtracting the force at the central worldline from the force at a neighboring point.
A constant \(\Psi\) produces no acceleration, and a term linear in position produces the same acceleration for every particle in the laboratory. Both disappear from relative motion. The Hessian is the first part of the potential that survives this subtraction, in agreement with the Fermi-coordinate expansion.
In cosmology, subtract the isotropic FLRW background and evaluate the perturbation in the observer’s tetrad. On subhorizon scales,
\[ \delta R_{\hat 0 i\hat 0 j} \simeq\frac{1}{a^2}\partial_i\partial_j\Psi. \tag{5.6}\]
The trace-free scalar tide is
\[ \begin{aligned} \mathcal E_{ij}^{S} &\equiv\delta R_{\hat 0\langle i|\hat 0|j\rangle}\\ &\simeq\frac{1}{a^2} \left(\partial_i\partial_j -\frac13\delta_{ij}\nabla^2\right)\Psi. \end{aligned} \tag{5.7}\]
Using the Poisson equation,
\[ \boxed{ \mathcal E_{ij}^{S} \simeq\frac32H^2\Omega_m(a) K_{ij} } \tag{5.8}\]
with \(K_{ij}\) defined in Eq. 1.18. Thus the Newtonian tidal operator in the bias expansion is the subhorizon limit of a local curvature component.
Here \(\Omega_m=\Omega_m(a)\). The proportionality factor depends on time through \(H^2\Omega_m\), while \(K_{ij}\) is dimensionless by definition. One may formulate a response using either \(K_{ij}\) or the dimensionful curvature \(\mathcal E_{ij}^{S}\), but the corresponding bias coefficient has a different normalization. The functional definition must specify which operator is held fixed.
5.2 Intrinsic shape as a tensor-valued response
An intrinsic shape is a property of the source before its photons propagate to the observer. Unlike the scalar abundance, it carries two spatial indices in the source rest frame and can therefore respond linearly to a trace-free tide. The response formalism used for density bias extends directly once the free tensor indices and the temporal history are retained.
Let \(S_{ij}^{I}\) be a symmetric trace-free intrinsic shape tensor in the local orthonormal rest frame of the galaxy population. As in Chapter 1, average over short-scale formation physics while holding fixed the long-wavelength tidal history. Define the linear response kernel
\[ \mathcal R_{ij}{}^{kl}(\tau;\tau') \equiv \left. \frac{\delta\left\langle S_{ij}^{I}(\tau)\right\rangle_{S\,|\,\mathcal E_L}} {\delta\mathcal E_{kl}(\tau')} \right|_{\mathcal E_L=0}. \tag{5.9}\]
Spatial locality places the perturbation on the past worldline; time nonlocality is retained through \(\tau'\). In an isotropic unperturbed ensemble, the only rank-four tensor mapping one symmetric trace-free tensor into another is the STF projector
\[ \mathcal P_{ij}{}^{kl} \equiv\frac12 (\delta_i{}^k\delta_j{}^l+\delta_i{}^l\delta_j{}^k) -\frac13\delta_{ij}\delta^{kl}. \tag{5.10}\]
Therefore
\[ \mathcal R_{ij}{}^{kl}(\tau;\tau') =R_E(\tau;\tau')\mathcal P_{ij}{}^{kl}, \tag{5.11}\]
and the leading intrinsic response is
\[ \boxed{ S_{ij}^{I}(\tau) =\int^{\tau}\mathrm d\tau'\, R_E(\tau;\tau') \mathcal E_{\langle ij\rangle} \bigl(\boldsymbol{x}_{\mathrm{fl}}(\tau'),\tau'\bigr) +\epsilon_{ij}^{I}+\cdots }. \tag{5.12}\]
Rotational invariance fixes the index structure but not the temporal kernel \(R_E\). The latter contains the internal dynamics of the selected population: a rapidly adjusting system, a fossil response set at formation, and a population with a broad distribution of formation times can all have different kernels while obeying the same tensor symmetry. This is the tensor counterpart of the time nonlocality found for scalar bias in Eq. 1.13, and it carries the same warning. A measurement of the alignment amplitude at one redshift constrains an integral of \(R_E\), not its instantaneous value.
The stochastic shape \(\epsilon_{ij}^{I}\) has zero conditional mean. Reorganizing the time integral perturbatively gives the familiar local-at-observation-time expansion
\[ S_{ij}^{I} =b_K K_{ij} +b_{\delta K}\delta K_{ij} +b_{K\!K} \left(K_{ik}K^k{}_j-\frac13\delta_{ij}K_{kl}K^{kl}\right) +\cdots. \tag{5.13}\]
The normalization of \(b_K\) depends on whether the operator is written as \(K_{ij}\) or as the dimensionful curvature \(\mathcal E_{ij}\). The response definition Eq. 5.9 fixes that convention.
The observed intrinsic shear is the screen-projected trace-free tensor
\[ \gamma_{ab}^{I} =\left( P_a{}^iP_b{}^j-\frac12P_{ab}P^{ij} \right)S_{ij}^{I}. \tag{5.14}\]
Projection converts the local three-dimensional tensor into a spin-two field on the sky; it does not make the source response nonlocal.
The screen projection also removes the component along the line of sight and the two-dimensional trace. It is a kinematic step performed after the local response has been defined. Correlations of \(\gamma_{ab}^{I}\) with density or lensing fields then follow from correlations of the long-wavelength tide with those observables.
5.3 A gravitational wave as a local tidal field
The transverse-traceless metric is a convenient coordinate representation of a gravitational wave, but the local detector responds to curvature. Computing \(R_{0i0j}\) makes this statement explicit and permits a direct comparison with the scalar tidal field. The result is independent of whether the same wave is described in TT coordinates or in Fermi coordinates around the detector.
In a local region small compared with the background curvature radius, take
\[ \mathrm ds^2=-\mathrm dt^2+ [\delta_{ij}+h_{ij}^{\mathrm{TT}}(t,\boldsymbol{x})]\mathrm dx^i\mathrm dx^j, \tag{5.15}\]
with
\[ \partial^ih_{ij}^{\mathrm{TT}}=0, \qquad h^{\mathrm{TT}\,i}{}_i=0, \qquad h_{0\mu}=0. \tag{5.16}\]
Substituting these conditions into the general linearized curvature Eq. 5.2,
\[ \begin{aligned} R_{0i0j}^{\mathrm{GW}} &=\frac12 \left( h_{0j,i0}+h_{i0,0j}-h_{00,ij}-h_{ij,00} \right)\\ &=-\frac12\ddot h_{ij}^{\mathrm{TT}}. \end{aligned} \tag{5.17}\]
The local equation of geodesic deviation is therefore
\[ \boxed{ \ddot X^i =\frac12\ddot h^i{}_j{}^{\mathrm{TT}}X^j }. \tag{5.18}\]
The corresponding Fermi metric begins with
\[ g_{\hat 0\hat 0}^{\mathrm F} =-1+\frac12\ddot h_{ij}^{\mathrm{TT}}(T,\boldsymbol{0})X^iX^j +\mathcal O(X^3). \tag{5.19}\]
A spatially constant value of \(h_{ij}^{\mathrm{TT}}\) does not appear in the local metric; the observable source-side effect begins with the curvature, or two derivatives of the tensor perturbation.
For a wave of frequency \(\omega\), the local tidal amplitude therefore scales as \(\omega^2 h_{ij}^{\mathrm{TT}}\). A detector or a galaxy responds after integrating this tidal acceleration against its own dynamical response. The final displacement can be proportional to \(h_{ij}^{\mathrm{TT}}\) for freely falling test masses because the equation of motion is integrated twice in time; the local forcing itself remains the curvature.
5.4 The ring experiment
The ring of freely falling particles separates the coordinate description from the measured effect. The central particle defines the Fermi frame, and the other particles supply deviation vectors. The polarization tensors determine the eigenvectors and eigenvalues of the tidal matrix in the plane transverse to propagation.
Figure placeholder. The ring experiment. A ring of freely falling test particles in the plane transverse to a wave propagating along \(z\), shown at four phases of the cycle for the \(+\) polarization and for the \(\times\) polarization, with the central particle marked as the origin of the Fermi frame and the principal axes of the two cases drawn to show the \(45^\circ\) offset between them.
For a wave propagating in the \(z\) direction,
\[ h_{ij}^{\mathrm{TT}} =\begin{pmatrix} h_+ & h_\times & 0\\ h_\times & -h_+ & 0\\ 0&0&0 \end{pmatrix}. \tag{5.20}\]
Eq. 5.18 gives
\[ \begin{aligned} \ddot X&=\frac12\ddot h_+X +\frac12\ddot h_\times Y,\\ \ddot Y&=\frac12\ddot h_\times X -\frac12\ddot h_+Y. \end{aligned} \tag{5.21}\]
To first order, replace \((X,Y)\) on the right-hand side by the unperturbed position \((X_0,Y_0)\). If \(h_{ij}\) and \(\dot h_{ij}\) vanish at the reference time and the particles are initially at rest relative to the central geodesic, two integrations give
\[ X^i(T) =\left(\delta^i{}_j+\frac12h^i{}_j{}^{\mathrm{TT}}(T)\right)X_0^j. \tag{5.22}\]
A circle becomes an ellipse whose principal axes rotate by \(45^\circ\) between the \(+\) and \(\times\) polarizations. In TT coordinates the particles may remain at fixed coordinates while their proper separation changes; in FNC the central observer describes the same physics as relative acceleration. The common invariant is Eq. 5.17.
The initial conditions in Eq. 5.22 remove homogeneous relative position and velocity modes that are unrelated to the passing wave. Other initial conditions add the corresponding free relative motion. The wave-induced part is still obtained from the same geodesic-deviation equation.
5.5 Intrinsic response to a tensor tide
Because the local response is organized by tensor representation and curvature, the source need not know whether a trace-free tide originated from scalar density perturbations or from a propagating tensor mode. The distinction enters through the spacetime and time dependence of \(\mathcal E_{ij}\) and through the correlations of that tide with other observables.
The response kernel Eq. 5.9 does not distinguish whether the trace-free curvature was generated by scalar or tensor perturbations. For a gravitational wave,
\[ \mathcal E_{ij}^{\mathrm{GW}} =R_{\hat 0 i\hat 0 j}^{\mathrm{GW}} =-\frac12\ddot h_{ij}^{\mathrm{TT}}. \tag{5.23}\]
Hence
\[ S_{ij}^{I,\mathrm{GW}}(\tau) =-\frac12\int^\tau\mathrm d\tau'\, R_E(\tau;\tau') \ddot h_{ij}^{\mathrm{TT}}(\tau')+\cdots. \tag{5.24}\]
For a monochromatic local wave this response carries the expected factor \(\omega^2\) before the temporal response kernel is evaluated. The suppression is an equivalence-principle statement: a local source does not respond to a constant tensor metric perturbation, but it can respond to the associated tidal history.
Whether the final intrinsic shape is additionally suppressed or enhanced depends on the ratio of the wave frequency to the internal dynamical and formation timescales represented by \(R_E\). The curvature factor is universal; the temporal transfer function is population dependent.
5.6 Intrinsic source shape versus propagated ruler shear
The same observed spin-two field can contain a local source response and a propagation distortion. Their tensor appearance alone does not identify the mechanism. The distinction is made by where the response is defined: intrinsic alignment is a functional of local tidal history on the source worldline, whereas ruler shear is a functional of the null geodesic and its endpoints.
Figure placeholder. Intrinsic alignment against ruler shear. A single spacetime diagram carrying both mechanisms: on the source worldline, the local tidal history that sets \(S_{ij}^{I}\), drawn as a tetrad with a distorted galaxy shape; along the null geodesic reaching the observer, the accumulated deflection that produces \(\gamma_{ab}^{\mathrm{ruler}}\). The point of the figure is that both terminate in the same measured ellipticity on the observer’s screen.
The intrinsic tensor Eq. 5.14 is defined at the source. The ruler shear of Chapter 3 is generated by the map from source to observer:
\[ \gamma_{ab}^{\mathrm{ruler}} =-\frac12 \left(P_a{}^iP_b{}^j-\frac12P_{ab}P^{ij}\right)h_{ij}\big|_s -\partial_{\perp\langle a}\Delta x_{\perp b\rangle}. \tag{5.25}\]
The displacement contains line-of-sight integrals and observer terms. Schematically, an observed shape estimator receives
\[ \gamma_{ab}^{\mathrm{obs}} =\gamma_{ab}^{I} +\gamma_{ab}^{\mathrm{ruler}} +\text{selection and calibration responses}. \tag{5.26}\]
Both terms are spin two and can be correlated with the same long mode, but they encode different mechanisms:
\[ \text{local formation response} \neq \text{null-propagation distortion}. \tag{5.27}\]
FNC isolates the first. Cosmic rulers organize the second. A consistent observable prediction requires both.
This separation is also useful for gauge reasoning. The intrinsic tensor is defined in a local orthonormal tetrad and is invariant under a change of cosmological coordinates. The propagated contribution is gauge-invariant only after source, line-of-sight, and observer terms are combined. Their sum can then be projected into whatever shape estimator and selection convention the survey uses.
The response kernel Eq. 5.9 is blind to the origin of the tide it integrates. A trace-free curvature produced by the collapse of a long-wavelength density mode and one produced by a passing gravitational wave enter the same way, with the same universal factor and the same population-dependent kernel \(R_E\). This is a direct consequence of the local-frame argument of Chapter 4: what reaches the source is \(\mathcal E_{\langle ij\rangle}\) and nothing else. Two practical statements follow. Physical mechanisms are distinguished by the time dependence of the tide and by its correlation with other observables, never by the response coefficient alone. And an observed spin-two field is not automatically a source property, since Eq. 5.26 shows it also contains a propagation term that is a functional of the null geodesic rather than of the source worldline.
Conclusion: from spacetime fields to measured structure
These lectures began with a Newtonian statement, \(\delta_g=b_1\delta_m\), and asked what must be added before it becomes a prediction for a relativistic observation. The answer was not a catalogue of small corrections. It was a sequence of maps with different physical roles.
The bias expansion describes how a local population responds to its physical environment. The null-geodesic initial-value problem maps the energy and direction measured by an observer to a source event. The cosmic clock compares the proper age of that event with the age assigned by its observed redshift. Cosmic rulers compare physical separations in the source rest frame with separations inferred from angles and redshifts. Fermi normal coordinates then identify the curvature components that can affect local source physics after the equivalence principle has removed a uniform gravitational field.
The same organizing principle runs through all five lectures:
\[ \boxed{ \text{define the measurement covariantly} \;\longrightarrow\; \text{choose a perturbative representation} \;\longrightarrow\; \text{combine source, propagation, and observer terms} }. \]
Diffeomorphism invariance is therefore not an abstract redundancy that disappears once a gauge has been chosen. It constrains how a fundamental theory becomes a prediction for data. A proposed explanation of the dark universe must pass through this chain: it must determine the local degrees of freedom and their dynamics, their effect on tracers and light propagation, and finally the gauge-independent relations among observed redshifts, directions, shapes, and counts.
Large-scale structure is demanding precisely because it requires all of these ingredients simultaneously. It is also valuable for the same reason. New gravitational fields, modified interactions, nonstandard dark matter, or early-universe initial conditions can be tested only after their spacetime dynamics have been translated into the operational observables constructed here. The framework does not select the new idea. It supplies the standard that a new idea must meet before it can be compared consistently with the observed universe.
Summary
Key ideas. The electric Riemann tensor in the observer’s tetrad is the single object behind scalar tidal bias, intrinsic alignment, and the local effect of a gravitational wave. A scalar abundance cannot respond linearly to a trace-free tide, while a shape tensor can, which is why intrinsic alignment appears at linear order in \(K_{ij}\) whereas scalar clustering feels the tide first at quadratic order. Rotational invariance fixes the index structure of the shape response completely, leaving one scalar kernel that encodes the internal dynamics of the population and remains nonlocal in time. Finally, a local source responds to a tensor representation rather than to a mechanism, so scalar and tensor tides enter identically and are distinguished only by their time dependence and correlations.
Main results.
- The scalar tide as a curvature component, Eq. 5.7, and its identification with the bias operator, Eq. 5.8, closing the argument opened at Eq. 1.18.
- The tensor-valued response kernel Eq. 5.9, its isotropic reduction Eq. 5.11, and the intrinsic shape Eq. 5.12 with its local expansion Eq. 5.13.
- The screen projection Eq. 5.14 converting a local three-tensor into a spin-two field on the sky.
- The gravitational-wave curvature Eq. 5.17, the geodesic deviation Eq. 5.18, and the ring solution Eq. 5.22.
- The intrinsic response to a tensor tide Eq. 5.24, carrying the equivalence-principle factor \(\omega^2\).
- The separation of intrinsic and propagated shear, Eq. 5.26.
Where this leaves these notes. The two notions of observability introduced in the Preface have now both been constructed. Part I produced a nonlocal observable on the past light cone, Eq. 3.38, in which source, propagation, and observer terms combine into a gauge-invariant count. Part II produced local observables in a freely falling frame, where the connection is removed and curvature remains. A complete prediction for a survey requires both, and the last section of this chapter shows why: the same measured spin-two field on the sky contains a local formation response organized by Chapter 4 and a propagation distortion organized by Chapter 3.