The effects of non Bunch-Davies initial conditions on gravitationally produced relics

Here is an explanation of the paper "The effects of non Bunch–Davies initial conditions on gravitationally produced relics," translated into simple language with creative analogies.

The Big Picture: The Universe's "Ghost" Factory

Imagine the early universe as a giant, expanding balloon. Inside this balloon, there are invisible quantum fields—like invisible strings vibrating in the air. According to standard physics, if you start with a perfectly calm, empty string (a "vacuum") and then blow up the balloon really fast (a period called Inflation), the stretching of the balloon itself will shake those strings.

This shaking creates real particles out of nothing. This is called Gravitational Particle Production (GPP). It's like if you pulled a rubber band so fast that it snapped and created little rubber beads.

For decades, scientists have assumed that the universe started with these strings completely still and empty. This "empty start" is called the Bunch–Davies vacuum. The paper asks a simple but revolutionary question: What if the strings weren't empty to begin with? What if they were already vibrating or had some particles in them before the balloon started expanding?

The Core Discovery: It's Not Just "Adding Up"

The authors found that the answer isn't as simple as "Empty Start + New Particles = Total Particles."

Think of it like a crowded dance floor:

The Standard View (Bunch–Davies): The dance floor is empty. The DJ (gravity) starts playing a beat (inflation), and dancers (particles) appear out of nowhere.
The New View (Non-Bunch–Davies): The dance floor is already full of people dancing. When the DJ starts the beat, two things happen:
1. Stimulated Emission: The existing dancers get excited and encourage more people to join the dance. The crowd grows much faster than expected.
2. Pauli Blocking: If the dancers are fermions (like electrons), they can't stand too close to each other. The existing crowd actually stops new people from joining, slowing down the growth.

The paper shows that the final number of "relics" (particles that survived to become Dark Matter) depends heavily on how crowded the dance floor was at the very beginning.

The Two Scenarios Tested

The authors tested two different "what if" scenarios to see how this changes the amount of Dark Matter in the universe today.

Scenario 1: The "Hot Bath" (Thermal Initial Conditions)

Imagine the universe started not as an empty vacuum, but as a hot soup of particles.

The Analogy: Think of trying to fill a swimming pool. In the standard model, you start with an empty pool and turn on a hose. In this model, the pool is already half-full of warm water.
The Result: If you start with a hot soup, you can produce the exact right amount of Dark Matter for a much wider range of particle masses.
- In the standard model, only a very specific "Goldilocks" mass works.
- In this "hot soup" model, you can get the right amount of Dark Matter even if the particles are very light or very heavy. It opens up a huge new playground for where Dark Matter could hide.

Scenario 2: The "Two-Stage Inflation" (The Stop-and-Go Universe)

Imagine inflation didn't happen all at once. Instead, the universe expanded, then paused (a radiation phase), and then expanded again.

The Analogy: Think of a runner.
- Standard Model: The runner sprints for 60 seconds straight.
- Two-Stage Model: The runner sprints, stops to tie their shoe (radiation phase), and then sprints again.
The Result: This "stop-and-go" rhythm changes the music of the universe. It creates two peaks in the distribution of particles instead of one.
- This means the "spectrum" (the mix of particle sizes) looks different.
- It allows for the correct amount of Dark Matter to exist even if the universe's expansion rate was different than we thought. It essentially gives us more "knobs" to tune the universe to get the right result.

Why Does This Matter?

Dark Matter is Elusive: We haven't found Dark Matter yet. We don't know what it is or how heavy it is.
More Possibilities: By realizing that the universe might not have started "empty," the authors show that Dark Matter could be made of particles with many different masses. It removes the strict limits we previously thought existed.
Specific Particles: They found that this effect is huge for Spin-1 particles (like massive photons or vector bosons). For these particles, the "initial crowd" matters a lot. However, for some other particles (like certain fermions), the initial conditions don't change the outcome much because they are "conformal" (they don't feel the expansion as strongly).

The Takeaway

The paper is a reminder that history matters. In cosmology, we often assume the universe started from a blank slate. This paper argues that if the universe had a "pre-history" with particles already present or a different expansion rhythm, the recipe for creating Dark Matter changes completely.

It's like baking a cake: If you assume you started with a bowl of flour, you get one result. But if you realize you actually started with a bowl of flour plus some pre-mixed batter, you need to adjust your recipe entirely to get the perfect cake. The authors have provided the new recipe, showing us that Dark Matter might be hiding in places we previously thought were impossible.

Here is a detailed technical summary of the paper "The effects of non Bunch–Davies initial conditions on gravitationally produced relics" by Bertuzzo, Salla, and Tesi.

1. Problem Statement

Standard models of Gravitational Particle Production (GPP) assume that quantum fields begin in the Bunch–Davies (BD) vacuum at the onset of inflation. This vacuum state corresponds to a particle-free initial condition ( $|0_{in}\rangle$ ). Under this assumption, the abundance of dark matter (DM) relics produced via the amplification of quantum fluctuations is determined solely by the expansion history and the particle's mass/spin.

However, the early universe may have undergone pre-inflationary dynamics (e.g., thermalization, multi-stage inflation) that populate the field with particles before the "visible" inflationary epoch (the last $\sim 60$ e-folds). The paper addresses the gap in literature regarding how non-Bunch–Davies initial conditions (an initial state $|\psi\rangle$ containing particles) affect the final spectrum and abundance of gravitationally produced relics. Specifically, it investigates whether the final abundance is simply the sum of initial particles and GPP-produced particles, or if quantum interference effects alter the outcome significantly.

2. Methodology

The authors develop a general framework to compute the energy density and number density of relics for arbitrary initial states $|\psi\rangle$ .

Formalism: They utilize the Bogoliubov transformation approach within the Heisenberg picture. The field is expanded in terms of mode functions $u_k(\eta)$ (early time) and $w_k(\eta)$ (late-time adiabatic vacuum).
Key Equations:
- The final number density $n_k$ is derived by matching the Hamiltonian expectation values in the initial and late-time states.
- The result (Eq. 17/18) shows that the final number density depends on:
  1. The initial particle occupation number $\langle n^{in}_k \rangle$ .
  2. The initial coherence/matrix elements $\langle c^{in}_k \rangle = \langle \psi | a_k a_{-k} | \psi \rangle$ .
  3. The Bogoliubov coefficients $\alpha_k, \beta_k$ describing the evolution.
- Crucially, the final abundance is not a simple linear sum. It includes terms proportional to $\langle n^{in}_k \rangle$ (stimulated emission for bosons, Pauli blocking for fermions) and interference terms involving $\langle c^{in}_k \rangle$ .
Scenarios Analyzed:
1. Thermal Initial State: The field is assumed to be in thermal equilibrium with temperature $T$ before visible inflation.
2. Two-Stage Inflation: A scenario with two quasi-de Sitter phases separated by a radiation-dominated era. This creates a situation where modes relevant for DM production exit the horizon during the first stage, evolve through radiation, and re-enter the horizon during the second stage, effectively acting as non-BD initial conditions for the second stage.
Fields Considered:
- Conformally Coupled Fields: Spin-0 (conformal), Spin-1/2, and transverse Spin-1. These are largely unaffected by initial conditions because GPP is negligible until the mass term dominates (Compton crossing).
- Longitudinal Spin-1 (Vector DM): This is the primary focus. The longitudinal mode breaks conformal invariance via the mass term and is produced efficiently before the mass becomes dominant, making it highly sensitive to initial conditions.

3. Key Contributions

General Framework: The paper provides the first comprehensive quantification of GPP for generic initial states, moving beyond the standard BD vacuum assumption.
Quantum Interference: It demonstrates that the final relic abundance is determined by quantum mechanical interference between the initial state and the vacuum fluctuations, leading to "stimulated emission" or "Pauli blocking" effects.
Thermal vs. Coherent States: The authors clarify that while thermal and squeezed states significantly alter the abundance, coherent states (which scale with volume) do not affect the final relic density in the thermodynamic limit, behaving effectively like the BD vacuum.
Two-Stage Inflation Dynamics: They derive analytical and numerical results for a two-stage inflationary model, showing how intermediate radiation phases can shift the peak of the power spectrum and alter the mass-abundance relationship.

4. Key Results

A. Impact on Conformal vs. Non-Conformal Fields

For fields where conformal symmetry breaking is solely due to mass (conformal scalars, fermions, transverse vectors), the initial conditions have negligible impact. The initial particle population simply redshifts away before GPP becomes active.
For longitudinal vector modes (massive spin-1), GPP is active well before the Compton crossing. Here, non-BD conditions drastically change the outcome.

B. Thermal Initial Conditions

Spectrum: The momentum spectrum retains the characteristic peak at $k_\star$ (determined by the mass and Hubble scale), but the slope at low momenta changes due to the thermal distribution ( $n_k \propto T/k$ ).
Abundance:
- For high reheating temperatures ( $T_{RH}$ ), the abundance is dominated by the "stimulated production" term. The correct DM abundance can be achieved for a wider range of masses (both lighter and heavier) compared to the BD case.
- For low $T_{RH}$ , the standard BD production is insufficient. The initial thermal population provides the necessary enhancement to reach the observed DM density for masses that would otherwise be excluded.
- Constraints: The parameter space is constrained by the requirement that the initial thermal energy density does not spoil inflation ( $\rho_{DM} < \rho_{inf}$ ). This excludes very low mass regions for high initial temperatures.

C. Two-Stage Inflation

Spectrum Deformation: The presence of an intermediate radiation phase creates a "memory" of the first inflationary stage.
- For light masses ( $k_\star < k_{dR}$ ), the spectrum has a single peak at $k_\star$ , but the amplitude is enhanced.
- For intermediate masses ( $k_{dR} < k_\star < k_{d\Lambda}$ ), the spectrum develops a new peak at $k_{dR}$ (the scale of the radiation phase), and the spectrum behaves as $k^{-5}$ in the intermediate region.
- For heavy masses, two peaks appear (at $k_{dR}$ and $k_\star$ ).
Abundance: This scenario allows the correct DM abundance to be reproduced for vector masses $m \gtrsim 6 \times 10^{-6}$ eV with Hubble parameters ( $H_e$ ) and reheating temperatures ( $T_{RH}$ ) that are significantly smaller than those required in the standard single-stage inflation model.

D. Observational Constraints

Isocurvature: The authors show that for the specific parameters allowing the correct DM abundance in their scenarios, the isocurvature perturbations remain within Planck limits (provided the peak momentum $k_{peak}$ is sufficiently large).
Lyman- $\alpha$ : The free-streaming length of the produced relics is calculated. The results indicate that Lyman- $\alpha$ constraints are generally satisfied for the viable parameter space, as the extra radiation phase tends to decrease the mean velocity of the DM.

5. Significance

This work fundamentally alters the predictive power of GPP as a Dark Matter production mechanism.

Relaxing Mass Constraints: It demonstrates that the "standard" mass predictions for GPP-produced DM (e.g., specific mass windows for vector DM) are not unique. Non-BD initial conditions can shift the viable mass range by orders of magnitude.
Model Independence: It highlights that without precise knowledge of the pre-inflationary history (initial conditions), the abundance of gravitationally produced relics cannot be uniquely predicted from the particle mass alone.
Phenomenology: The findings suggest that future observational constraints on DM (e.g., from CMB isocurvature or Lyman- $\alpha$ ) must be interpreted within a broader context of initial conditions, as the "standard" BD calculation may miss viable regions of parameter space or overestimate constraints.

In conclusion, the paper establishes that initial conditions are a critical, non-trivial factor in gravitational particle production, particularly for massive vector fields, and that the standard Bunch–Davies assumption may lead to an incomplete or incorrect picture of the Dark Matter landscape.