The initial states of high frequency gravitons

1. Problem Statement

The paper addresses the ambiguity surrounding the initial quantum states of relic gravitons (primordial gravitational waves) in the context of inflationary cosmology.

• Context: In standard inflationary scenarios, large-scale inhomogeneities are attributed to quantum fluctuations of the vacuum. However, the specific choice of the initial vacuum state is not unique; different canonical transformations of the Hamiltonian can lead to different vacuum definitions.
• The Issue: While the "cosmic no-hair" conjecture suggests inflation erases classical anisotropies, the initial quantum state (vacuum vs. excited states) at the onset of inflation remains a theoretical variable. Previous approaches often rely on speculative "protoinflationary" epochs (pre-inflationary deceleration) to define these states.
• Goal: The author aims to constrain the initial states of gravitons without relying on specific details of the pre-inflationary epoch. Instead, the analysis focuses on the pragmatic moment when different wavelengths of the spectrum cross the comoving Hubble radius during inflation. The objective is to determine if non-vacuum initial states (states with finite energy density and particle multiplicity) are permitted across the frequency spectrum, particularly for high-frequency gravitons.

2. Methodology

The author employs a pragmatic, observation-driven approach to constrain the averaged multiplicity of gravitons ($n_k^{(g)}$) in the initial state.

• Hamiltonian Formalism: The paper begins by establishing the quantum Hamiltonian for relic gravitons (tensor modes) and curvature inhomogeneities (scalar modes). It acknowledges that canonical transformations can alter the explicit form of the Hamiltonian and the definition of the vacuum, but argues that these ambiguities are secondary to the constraints imposed by energy density and observational data.
• Parametrization of Initial States:
  • The initial state is modeled as a state with an averaged multiplicity $n_k^{(g)}$ (number of gravitons per mode), distinct from the vacuum ($n_k=0$).
  • A specific functional form is chosen for the multiplicity to ensure finite energy density:
    $$n_k^{(g)} = n_0 \frac{(k/k_*)^{\beta+1}}{e^{k/k_*} - 1}$$
    where $n_0$, $k_*$, and $\beta$ are parameters. This form ensures exponential suppression at high frequencies ($k \gg k_*$) and power-law behavior at low frequencies.
• Constraint Mechanisms:
  1. Energy Density Bound: The energy density of the initial state, $\rho_{gw}^{(in)}$, must not exceed the background energy density ($3H^2 M_P^2$) at the time a specific scale $k_*$ crosses the Hubble radius.
  2. Tensor-to-Scalar Ratio ($r_T$): The presence of initial gravitons modifies the tensor power spectrum ($P_T$). The ratio $r_T = P_T/P_S$ is constrained by CMB observations (Planck data), which limit $r_T \lesssim 0.03$.
  3. Scale Dependence: The author analyzes constraints as the pivot scale $k_*$ moves from the CMB scale ($k_p \approx 0.002 \text{ Mpc}^{-1}$) to higher frequencies (kHz to THz).
• Alternative Perspective: The paper also briefly considers a "thermal" initial state assigned before inflation, analyzing how the total number of e-folds ($N_t$) affects the survival of non-vacuum states.

3. Key Contributions

• Pragmatic Constraint Framework: The paper shifts the focus from speculative pre-inflationary physics to constraints imposed strictly at the moment of Hubble crossing for observable modes. This avoids assumptions about the "protoinflationary" phase.
• Frequency-Dependent Constraints: The study demonstrates that constraints on initial states are highly frequency-dependent.
  • Low Frequencies (CMB scales): Non-vacuum states are marginally permitted but tightly constrained by $r_T$.
  • High Frequencies (kHz–THz): The constraints become exponentially stringent, effectively forcing the initial state to be the vacuum.
• Resolution of Hamiltonian Ambiguity: The paper clarifies that while Hamiltonian ambiguities exist, the requirement of finite energy density and observational consistency (specifically $r_T$) renders these ambiguities irrelevant for high-frequency gravitons.

4. Key Results

• Low Frequency Regime ($k \sim k_p$):
  • Initial states with finite multiplicity are marginally permitted.
  • The product of the normalization parameter and the integral function, $n_0 I(\beta)$, is constrained to be roughly $< O(10^{10})$.
  • However, the tensor-to-scalar ratio constraint ($r_T < 0.03$) significantly tightens this, requiring the initial state to be very close to the vacuum.
• Intermediate Frequencies (nHz - Pulsar Timing Arrays):
  • As the scale $k_*$ increases (moving to nHz frequencies), the allowed region for non-vacuum parameters shrinks drastically.
  • The condition $n_0 I(\beta) < O(10^{-26})$ is derived, making non-vacuum initial states highly implausible.
• High Frequency Regime (kHz to THz):
  • For the maximal frequency of the spectrum ($k_{max} \sim \text{THz}$), the constraints become extreme.
  • The derived bound is $n_0 I(\beta) < O(10^{-110})$ (or even lower depending on post-inflationary expansion rates).
  • Conclusion: In this regime, the initial state must be the vacuum. Any deviation from the vacuum would violate energy density bounds or observational limits.
• Thermal Initial States:
  • If a thermal state is assumed prior to inflation, it is only consistent with observations if the total number of e-folds ($N_t$) is significantly larger than the e-folds elapsed since the crossing of the largest scales ($N_p$). If $\Delta N = N_t - N_p$ is small ($O(1)$), the resulting tensor-to-scalar ratio exceeds observational bounds.

5. Significance

• Predictive Power for Detectors: The results are crucial for future high-frequency gravitational wave detectors (e.g., electromagnetic detectors, microwave cavities). The paper concludes that high-frequency relic gravitons observed by such experiments must originate from vacuum fluctuations. They cannot carry an imprint of a non-vacuum initial state (like a thermal bath or excited state) because such states are ruled out by the energy density constraints at those scales.
• Robustness of Inflationary Predictions: The findings reinforce the standard inflationary paradigm. Even if the initial conditions were complex, the dynamics of the expanding universe and the requirement of finite energy density "wash out" non-vacuum features at high frequencies, leaving a pure vacuum spectrum.
• Methodological Shift: By decoupling the analysis of initial states from the unknown details of the pre-inflationary epoch, the paper provides a more robust, observationally grounded framework for testing quantum gravity effects in the early universe.

In summary, Giovannini demonstrates that while low-frequency gravitons (CMB scales) might retain a faint memory of non-vacuum initial conditions, high-frequency gravitons are strictly generated from the vacuum, rendering their spectrum a clean probe of the inflationary timeline and post-inflationary expansion history.