Vector-Induced Curvature Perturbations

• Vector-induced curvature perturbations are generated when vector fields directly convert fluctuations or induce anisotropic stress to modulate the primordial curvature.
• They yield observable imprints such as quadrupolar statistical anisotropy, anisotropic bispectra, and direction-dependent non-Gaussian correlations in both the CMB and galaxy surveys.
• Theoretical models utilize approaches like the δN formalism and vector curvaton mechanisms, impacting predictions in inflationary scenarios, primordial magnetism, and even primordial black hole formation.

Searching arXiv for papers on vector-induced curvature perturbations and related mechanisms. Taken together, the cited works motivate the broad label vector-induced curvature perturbations for curvature perturbations $\zeta$ or $\mathcal R$ that are generated, modulated, or shifted by vector fields, by the vector part of the stress–energy tensor, or by vector metric perturbations. In this literature, vector sourcing appears in several distinct but related forms: direct conversion of vector fluctuations into $\zeta$ through the $\delta N$ formalism, generation of $\zeta$ from anisotropic stress perturbations induced by primordial Gaussian vector fields, second-order scalar perturbations sourced by first-order vector modes, and frame-dependent shifts of $\zeta$ under vector disformal transformations (Dimopoulos, 2011, Shiraishi et al., 2013, Carrilho et al., 2015, Han et al., 22 Jun 2026). A recurrent theme is that vector sources tend to produce preferred directions, quadrupolar statistical anisotropy, anisotropic bispectra, and distinctive signatures in large-scale structure, the CMB, and, in one recent proposal, primordial-black-hole production.

1. Source terms and basic formalisms

A central route to vector-induced curvature perturbations is the extension of the separate-universe or $\delta N$ formalism to include vector fluctuations. In that case the local expansion depends not only on scalar fields but also on vector components, and one writes

$$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$

At linear order, the vector contribution is $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$, so vector perturbations can directly seed the primordial curvature perturbation (Karciauskas, 2010).

A second route proceeds through anisotropic stress. For a statistically isotropic Gaussian vector field $V_i(\mathbf x)$ with

$\mathcal R$0

the induced anisotropic stress is

$\mathcal R$1

In the comoving gauge, the curvature perturbation is then approximated by

$\mathcal R$2

with logarithmic growth until neutrino compensation. This realizes $\mathcal R$3, so even Gaussian vector fields generate non-Gaussian curvature perturbations (Shiraishi et al., 2013).

A third route is explicitly second-order. In a post-inflationary kination epoch with $\mathcal R$4, the transverse vector metric mode $\mathcal R$5 obeys

$\mathcal R$6

and because $\mathcal R$7 for $\mathcal R$8, one obtains a non-decaying solution

$\mathcal R$9

These first-order vector modes then source second-order scalar perturbations through

$\zeta$0

with $\zeta$1 for $\zeta$2 (Han et al., 22 Jun 2026).

This suggests a useful classification into direct vector conversion, anisotropic-stress mediation, and second-order vector-metric sourcing. The literature develops all three.

2. Inflationary vector sectors and the curvaton mechanism

The best-developed inflationary realization is the vector curvaton paradigm. A standard setup uses a massive Abelian vector field with a Maxwell-type kinetic term, a time-varying kinetic function, and a time-dependent mass,

$\zeta$3

with the physical field

$\zeta$4

For the ansatz $\zeta$5, $\zeta$6, nearly scale-invariant spectra of all three polarizations are obtained for

$\zeta$7

and the gauge-safe choice emphasized in the literature is $\zeta$8, $\zeta$9 (0909.0475).

This choice produces two qualitatively different regimes. If the vector remains light until the end of inflation, $\delta N$0, then $\delta N$1, particle production is strongly anisotropic, and the vector contribution to $\delta N$2 must typically remain subdominant. If instead the field is heavy by the end of inflation, $\delta N$3, then $\delta N$4, statistical anisotropy disappears, and the vector curvaton can be the sole source of $\delta N$5; in that limit one recovers the scalar-curvaton phenomenology with

$\delta N$6

where $\delta N$7 is the vector energy fraction at decay (0909.0475, 0907.1838).

A closely related presentation writes the vector contribution in $\delta N$8 form as

$\delta N$9

and parameterizes the power spectrum as

$\zeta$0

In the light-field limit one typically has $\zeta$1 and hence $\zeta$2 unless the vector contribution is suppressed; in the heavy regime, $\zeta$3, so $\zeta$4 (0909.0475).

The magnitude of the homogeneous vector condensate also matters. In the stochastic treatment of the supergravity-inspired $\zeta$5, $\zeta$6 model, the longitudinal condensate obeys

$\zeta$7

and, for sufficiently many e-folds, one may set at horizon exit

$\zeta$8

This makes the vector-curvaton framework predictive rather than postdictive in the sense emphasized there (Sanchez et al., 2013).

3. Statistical anisotropy, bispectra, and effective local non-Gaussianity

Because a background vector picks out a preferred direction, the power spectrum of the curvature perturbation generally acquires a quadrupolar modulation. A common form is

$\zeta$9

or equivalently

$\zeta$0

In the $\zeta$1 treatment of vector fluctuations, the anisotropy of particle production is encoded by

$\zeta$2

If either $\zeta$3 or $\zeta$4, the generated $\zeta$5 is statistically anisotropic, and the non-linearity parameter also acquires angular modulation. For $\zeta$6, the anisotropic part of $\zeta$7 is dominant over the isotropic one (Karciauskas, 2010).

The vector-curvaton literature emphasizes that the power spectrum anisotropy and bispectrum anisotropy are correlated. In one formulation,

$\zeta$8

with $\zeta$9 for the local shape and $\delta N$0 for the equilateral shape. For $\delta N$1, the bispectrum is dominated by its anisotropic part and peaks in the equilateral configuration; for $\delta N$2, one recovers the usual scalar-curvaton result (0909.0475). Reviews of the paradigm summarize this as a common preferred direction appearing in both the spectrum and bispectrum (Dimopoulos, 2011).

A distinct but related mechanism arises from anisotropic stress induced by primordial Gaussian vector fields. There the curvature bispectrum takes the schematic form

$\delta N$3

with

$\delta N$4

For nearly scale-invariant spectra, its squeezed-limit momentum dependence is exactly that of the local-type ansatz, the shape overlap is high ($\delta N$5), and one may define an effective local parameter $\delta N$6 (Shiraishi et al., 2013).

When the vector field is interpreted as a primordial magnetic field, $\delta N$7, so $\delta N$8 and hence $\delta N$9. The magnetic-induced bispectrum therefore mimics a negative local-type $$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$0 (Shiraishi et al., 2013).

General non-Abelian groups provide another route. With many randomly oriented fields, the anisotropy in the spectrum is suppressed,

$$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$1

while the bispectrum remains anisotropic and local in momentum space, with a typical amplitude $$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$2 (Karciauskas, 2011). A plausible implication is that multi-vector realizations soften the tension between vector sourcing and isotropy bounds without removing vector-specific non-Gaussian structure.

4. Consistency conditions, longitudinal modes, and model-dependent tensions

A persistent issue in vector-induced curvature perturbations is the status of longitudinal modes and potential instabilities. In the model with nonminimal coupling $$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$3, the action is

$$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$4

For a light field during near-de Sitter inflation, the transverse spectrum is

$$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$5

but the spectral energy density of subhorizon vacuum modes contains

$$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$6

The same analysis argues that the theory may nevertheless be healthy in at least some versions, because the negative sign in $$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$7 arises from the nonminimal coupling and only persists while $$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$8; at late times $$\zeta(\mathbf x)=\delta N =N_\phi\,\delta\phi+N_A^i\,\delta A_i +\tfrac12\,N_{\phi\phi}\,\delta\phi^2 +\tfrac12\,N_{AA}^{ij}\,\delta A_i\,\delta A_j+\cdots \, .$$9, and the kinetic term reverts to the usual positive-definite sign (Karčiauskas et al., 2010).

The varying-kinetic-function models were formulated to avoid such pathologies. In that class, the Maxwell kinetic term with $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$0 guarantees the absence of ghosts in the transverse sector, and the choice $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$1, $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$2 was identified as stable and ghost-free throughout inflation (0909.0475).

Even when the vector sector is ghost-free, embedding the time dependence of $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$3 and $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$4 into a complete inflationary model can introduce additional tensions. In the minimal implementation where the inflaton itself controls $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$5 and $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$6, the quadratic fluctuation action contains off-diagonal terms such as $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$7, and the full $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$8 coupled system must be diagonalized. Numerical integration then shows that, except for a narrow window $\zeta_A(\mathbf k)=N_A^i\,\delta A_i(\mathbf k)$9–$V_i(\mathbf x)$0, the interaction with the inflaton generally causes the curvature perturbations to violate statistical isotropy beyond the observational limit (Namba, 2012).

At the level of second-order perturbation theory, vector and tensor contributions also affect the conservation of curvature perturbations. The full evolution equation for the second-order curvature perturbation contains vector quantities such as $V_i(\mathbf x)$1, $V_i(\mathbf x)$2, and explicit couplings to anisotropic stress. On large scales, a conserved quantity exists only if, in addition to the non-adiabatic pressure, the transverse traceless part of the anisotropic stress tensor is negligible. The version exactly conserved under those conditions is the gauge-invariant curvature perturbation defined with the determinant of the spatial part of the inverse metric (Carrilho et al., 2015).

5. Beyond Abelian curvatons: anisotropic backgrounds, aether sectors, and disformal frame effects

Vector-induced curvature perturbations also arise in theories where the background itself is anisotropic or contains additional vector sectors. In anisotropic inflation from a vector impurity, the action includes a non-minimally coupled massive vector,

$V_i(\mathbf x)$3

and a Bianchi-I geometry with a preferred $V_i(\mathbf x)$4-axis. In slow roll, the anisotropy parameter is

$V_i(\mathbf x)$5

which reduces to $V_i(\mathbf x)$6 for $V_i(\mathbf x)$7. The curvature spectrum becomes

$V_i(\mathbf x)$8

and the “+” tensor polarization is sourced by the curvature perturbation,

$V_i(\mathbf x)$9

This yields a non-zero $\mathcal R$00, TB correlations in the CMB at order $\mathcal R$01, and a net linear polarization of the relic tensor background (0806.2422).

Einstein-aether inflation provides another example. There, quantum fluctuations of the inflaton and aether seed long-wavelength adiabatic and isocurvature scalar perturbations as well as transverse vector perturbations. The adiabatic mode corresponds to curvature perturbations of co-moving slices, but, in contrast with the standard case, it has a non-vanishing anisotropic stress on large scales. Scalar and vector perturbations may therefore leave significant imprints on the CMB (Armendariz-Picon et al., 2010).

Vector disformal transformations add a geometrical variant. For a matter-frame metric

$\mathcal R$02

three cases were analyzed. For a single timelike vector, neither the scalar curvature perturbation nor the gravitational waves are affected: $\mathcal R$03, $\mathcal R$04. For a single spacelike vector, $\mathcal R$05 but gravitational-wave modes are not invariant. For three orthogonal spacelike vectors, both the curvature perturbation and the tensor modes are shifted, with

$\mathcal R$06

This demonstrates that vector-sector contributions can be frame-invariant in some backgrounds and frame-dependent in others (Papadopoulos et al., 2017).

6. Observational signatures: large-scale structure, magnetic correlators, and primordial black holes

The clearest late-time signature identified for anisotropic-stress-sourced curvature perturbations is scale-dependent bias in large-scale structure. In the peak-background split or Press–Schechter approach, the correction to the linear bias is

$\mathcal R$07

so that on large scales

$\mathcal R$08

Integrated perturbation theory confirms that replacing $\mathcal R$09 in the standard expressions reproduces the vector-field result to leading order (Shiraishi et al., 2013).

For primordial magnetic fields, the same framework yields a distinctive sign. Identifying the vector with the comoving magnetic field gives $\mathcal R$10, hence $\mathcal R$11 and $\mathcal R$12. The resulting scale-dependent bias is therefore negative on large scales, which was explicitly highlighted as a smoking-gun sign of primordial magnetism (Shiraishi et al., 2013).

Inflaton–vector couplings also induce mixed non-Gaussian observables. In models with

$\mathcal R$13

one can compute the three-point cross-correlation $\mathcal R$14 and define a magnetic non-linearity parameter $\mathcal R$15 through

$\mathcal R$16

For $\mathcal R$17, the signal is maximized in the flattened configuration, and the analysis gives

$\mathcal R$18

while in the squeezed limit

$\mathcal R$19

This cross-correlation was proposed as a non-Gaussian signature of primordial magnetic fields generated during inflation (Jain et al., 2012).

A recent extension moves the vector-induced mechanism to post-inflationary small scales. During a stiff, or kination, epoch, first-order vector modes sourced by the vector component of the electromagnetic stress tensor remain approximately constant and act as persistent nonlinear sources for second-order scalar perturbations. The resulting curvature power spectrum is

$\mathcal R$20

and near the infrared cutoff of the kination band it scales as

$\mathcal R$21

For the benchmark Ratra-model parameters quoted there, one finds $\mathcal R$22, $\mathcal R$23–$\mathcal R$24, and $\mathcal R$25 in the asteroid-mass window (Han et al., 22 Jun 2026).

Across these applications, the recurring observational structures are a quadrupolar modulation of the scalar spectrum, direction-dependent non-Gaussianity, negative or positive $\mathcal R$26 bias corrections depending on the sign of the effective local amplitude, mixed scalar–vector correlators, and, in the kination scenario, a sharply infrared-enhanced small-scale curvature spectrum. This suggests that vector-induced curvature perturbations are best characterized not by a single template, but by a family of signatures tied to how the vector sector enters the stress tensor, the background, and the conversion to $\mathcal R$27.