---
title: Early Causal Tensor Power Spectrum
url: https://www.emergentmind.com/topics/early-causal-tensor-ect-power-spectrum
type: topic
---

# Early Causal Tensor Power Spectrum

The Early Causal Tensor (ECT) power spectrum characterizes a universal phenomenology for primordial gravitational waves (GWs) sourced by post-inflationary, causality-limited, and sub-horizon mechanisms in the early universe. Such sources include first-order cosmological phase transitions, topological defect networks, and scalar-induced GWs, all exhibiting a white-noise–like $k^3$ scaling at low comoving wavenumber. This infrared behavior emerges generically whenever the source correlation length is finite, yielding a distinct observational imprint on the cosmic microwave background (CMB) $B$-mode polarization and a specific spectral shape in the stochastic GW background. The ECT formalism provides parameterized, model-independent constraints on early-universe physics by mapping observational data directly to fundamental source parameters.

## 1. Theoretical Foundation and Motivation

Inflationary gravitational waves arise from quantum vacuum fluctuations stretched to super-horizon scales, yielding an approximately scale-invariant tensor power spectrum: $P_t^{\rm inf}(k) \propto k^{n_t}$ with $n_t \simeq 0$. In contrast, any gravitational wave background generated post-inflation—such as from bubble collisions, magnetohydrodynamic (MHD) turbulence, cosmic string networks, or large-amplitude small-scale scalar fluctuations—is subject to causal limits: no correlations can exist on scales exceeding the horizon at the time of generation. This causal bound imposes a generic infrared form, first shown in Cai et al. (2019), wherein the tensor power spectrum scales as $P_t(k) \propto k^3$ for $k \ll k_s$, with $k_s$ corresponding to the source’s maximal correlation scale [2601.20967].

If the source is operative and subsequently shuts off at redshift $z_p \gg 10^4$, well before matter–radiation equality, the $k^3$ regime extends across all observable CMB scales due to Silk damping at high $k$. These sources are collectively referred to as Early Causal Tensors (ECTs) [2601.20958, 2601.20967].

## 2. Universal Parameterization of the ECT Power Spectrum

On observational scales relevant to the CMB, the ECT power spectrum is parameterized in terms of the dimensionless tensor power spectrum:
\[
\mathcal{P}_h(k) = \frac{k^3}{2\pi^2} P_h(k) = r_c\,A_s\,\left(\frac{k}{k_0}\right)^3,
\]
where:

- $k_0 = 0.01\,\mathrm{Mpc}^{-1}$ is the pivot scale, selected to align the ECT B-mode amplitude at $\ell \sim 100$ with that of an inflationary spectrum of equal amplitude.
- $A_s = 2.1 \times 10^{-9}$ is the scalar amplitude at $k_0$, as measured from the CMB.
- $r_c$ is the ECT amplitude, defined by $r_c \equiv \mathcal{P}_h(k_0)/A_s = P_h(k_0)/P_s(k_0)$.

This construction maintains direct analogy with the standard inflationary tensor-to-scalar ratio $r$, but for the characteristic $k^3$ ECT spectral shape [2601.20958].

## 3. Physical Origin of the $k^3$ Infrared Scaling

For a generic causal, sub-horizon source with spacetime correlation length $R_*$ and finite duration $\Delta\tau$, the real-space correlation function $\xi(r)$ vanishes for $|r| > R_*$. The Fourier transform yields:
\[
P_h(k) = \int d^3r\,e^{-i \vec{k} \cdot \vec{r}}\,\xi(r) \propto c_0 + c_2\,k^2 + O(k^4),
\]
implying $\mathcal{P}_h(k) \propto k^3$ at $k \ll k_* \equiv R_*^{-1}$. 

A more general derivation via unequal-time correlators of the GW source stress-energy $\Pi(\tau, k)$ confirms this scaling in the small-$k$ limit whenever the source is local and of finite duration [2601.20967]. All such sources, when completed at $z \gg 10^5$, display their $k^3$ regime over the full CMB window ($k_{\mathrm{min}} \sim 10^{-3}\,\mathrm{Mpc}^{-1}$ to $k_{\mathrm{max}} \sim 0.16\,\mathrm{Mpc}^{-1}$).

## 4. CMB $B$-Mode Angular Power Spectrum from ECTs

Tensor perturbations generate $B$-mode polarization through their dynamical impact on the photon-baryon fluid prior to recombination. For a given tensor power spectrum, the angular $B$-mode power is:
\[
C_\ell^{BB} = \int_0^\infty \frac{dk}{k} P_h(k)\, [\Delta_\ell^B(k)]^2,
\]
where $\Delta_\ell^B(k)$ incorporates the transfer function for tensor-to-$B$-mode projection [2601.20967]. Numerical evaluation using Boltzmann solvers such as CLASS is required for precision modeling on all angular scales.

The distinct ECT $k^3$ spectral form yields enhanced $C_\ell^{BB}$ at high multipoles and suppressed power at low multipoles, leading to a blue-tilted $B$-mode spectrum with a characteristic peak at $\ell \sim 10^3$, strong suppression at $\ell \lesssim 10$, and recombination-scale features at $\ell \sim 100$. This morphology provides a clear target for separating ECT signals from inflationary, scale-invariant primordial tensor backgrounds [2601.20958, 2601.20967].

## 5. Observational Constraints on the ECT Amplitude

A contemporary joint likelihood analysis of BICEP/Keck, SPT-3G, SPTpol, Planck, and WMAP $B$-mode data, encompassing $\ell\sim20$–2300, fits simultaneously for the inflationary $r$ and ECT $r_c$ amplitudes, as well as lensing and foreground nuisance parameters. Under uniform priors $r, r_c \in [0,1]$, the data yield no significant detection of the ECT component. The derived $95\%$ confidence upper limit is:
\[
r_c < 0.0077 \quad (95\%~\mathrm{CL})
\]
[2601.20958]. This bound directly constrains the allowed amplitude of any ECT contribution, independent of microphysical source details, as all such sources yield the same $B$-mode spectral shape on CMB scales.

## 6. Mapping ECT Constraints to GW Energy Density and Early-Universe Models

The primordial ECT spectrum, with amplitude $r_c$, implies a present-day stochastic gravitational wave background (SGWB) with spectral density:
\[
\Omega_\mathrm{GW}(f) h^2 = \frac{h^2}{3} \left(\frac{k}{H_0}\right)^2 \mathcal{P}_h(k) \frac{1}{k^2} \propto f^3 \qquad (f \ll f_\mathrm{eq}),
\]
for comoving $k=2\pi f$ today and for modes re-entering the horizon during radiation domination. The $r_c$ limit implies:
\[
\Omega_\mathrm{GW} h^2(f) \lesssim \mathrm{few} \times 10^{-20} \left(\frac{f}{10^{-19}~\mathrm{Hz}}\right)^3
\]
at CMB frequencies, far below the reach of terrestrial and pulsar-timing GW experiments [2601.20958].

A variety of early-universe models map to this constraint, as summarized below:

| Model Class          | Key Parameter(s)         | ECT Constraint Mapping                                  |
|----------------------|-------------------------|---------------------------------------------------------|
| First-order PT       | $\alpha$, $\beta/H$, $v_w$ | $\Omega_{\rm GW}^{\rm PT} \sim 10^{-6}(H/\beta)^2(\alpha/0.1)^2(v_w/0.7)^3$; $r_c$ limit constrains $(H/\beta)^2(\alpha/0.1)^2 \lesssim \mathcal{O}(10^{-2})$ |
| Scalar-induced GW    | $A_\zeta$               | $r_c \simeq A_\zeta^2 / A_s \implies A_\zeta \lesssim 10^{-2}$, impacting PBH formation scenarios       |
| Cosmic strings/defects | $G\mu$               | $r_c \simeq C_s (G\mu/10^{-6})^2$ with $C_s \sim 0.1$–1; $G\mu \lesssim \mathrm{few} \times 10^{-7}$  |
| Other causal mechanisms | coupling constants  | $r_c < 0.0077$ involves bounds $\lesssim\mathcal{O}(10^{-1}-10^{-2})$ on model-specific couplings      |

The amplitude $r_c$ captures all model dependence; the spectral shape is universally set by causality [2601.20958, 2601.20967].

## 7. Distinctions from Inflationary and Loop-Induced Tensor Spectra

Inflationary models yield a nearly scale-invariant ($n_t \simeq 0$) spectrum, in contrast to the blue-tilted ECT form. Theoretical calculations of quantum corrections (loop- or open EFT–based) to the inflationary tensor power spectrum introduce subleading corrections of the form $\Delta_t^2(k) = 2H^2/(\pi^2 M_{pl}^2)\left[1 + \mathcal{O}(H^2/M_{pl}^2)\ln(H/\mu)\right]$, but do not mimic a causal $k^3$ scaling [2206.05797]. The ECT ansatz therefore isolates non-inflationary, early-universe GW sources via their distinct causal spectral shape.

## References

- First constraints on causal sources of primordial gravitational waves from BICEP/Keck, SPTpol, SPT-3G, Planck and WMAP $B$-mode data [2601.20958]
- A Universal CMB $B$-Mode Spectrum from Early Causal Tensor Sources [2601.20967]
- Quantum corrections to the primordial tensor spectrum: Open EFTs & Markovian decoupling of UV modes [2206.05797]

Source: https://www.emergentmind.com/topics/early-causal-tensor-ect-power-spectrum