Peculiar-velocity distribution functions and 21-cm fluctuations

This paper presents a more accurate calculation of the joint probability distribution function for peculiar velocities at two points, correcting a previous simplification regarding velocity correlations to improve the precision of 21-cm fluctuation predictions with minimal computational cost.

The Gist

The Big Picture: A Cosmic Weather Report

Imagine the early universe as a giant, invisible ocean. In this ocean, there are two main "currents": one made of normal matter (baryons) and one made of dark matter. Usually, these two currents flow together smoothly. However, in the very early universe, they sometimes drifted apart slightly, creating a "relative wind" between them.

Scientists use a special radio signal called 21-cm radiation to map this early universe. Think of this signal like a weather report that tells us how fast stars were forming. The rate at which stars form depends heavily on the speed of that "relative wind." Specifically, it depends on the square of the wind speed (how hard the wind is blowing).

The Problem: A Simplified Map

To predict what this weather report should look like, scientists use computer simulations (specifically a code called Zeus21).

In the past, when calculating how the wind speed at one point in the universe relates to the wind speed at another point, researchers made a simplifying assumption. They treated the wind as if it were perfectly symmetrical in all directions.

The Analogy:
Imagine you are measuring the wind between two trees.

• The Old Way: You assumed the wind blowing directly between the trees was exactly the same as the wind blowing sideways across the trees. You treated the wind as a perfect, uniform sphere.
• The Reality: The wind blowing directly between the trees behaves slightly differently than the wind blowing sideways. It's like how a river flows differently in the center compared to the edges.

The authors of this paper, Ryan Yuran Zhang and Marc Kamionkowski, pointed out that this "perfect sphere" assumption isn't strictly true. The wind has a specific direction, and the math changes slightly depending on whether you are looking at the wind head-on or from the side.

The Solution: A More Precise Calculation

The authors did the hard math to calculate the exact relationship between the wind speeds at two different points. They derived a new, more precise formula that accounts for the difference between the "head-on" wind and the "sideways" wind.

Think of it like upgrading from a flat, 2D map of the ocean to a 3D model that shows the depth and currents accurately.

Does It Matter? (The Results)

You might ask: "If the old way was a simplification, was it wrong enough to ruin our predictions?"

The answer is: Usually, no. But sometimes, yes.

• The General Case: For most places in the universe and most times, the difference between the old "simplified" map and the new "precise" map is tiny—less than a few percent. It's like the difference between measuring a room with a tape measure versus a laser; for most purposes, the tape measure is fine.
• The Special Case: However, the universe is complex. Sometimes, different signals cancel each other out (like noise-canceling headphones). In these specific moments of cancellation, even a tiny error in the math can make a big difference. The authors found that at a specific time in the universe's history (around redshift $z \approx 15$) and for specific distances, the old method could miss the mark more noticeably because of these cancellations.

The Takeaway

The authors didn't discover a new type of star or a new law of physics. Instead, they fixed a small, subtle error in the mathematical recipe used to simulate the universe.

• The Fix: They provided the correct formula to replace the old approximation.
• The Cost: Implementing this fix in the computer code is very easy and only makes the calculation run a few percent slower.
• The Benefit: As our telescopes get better and our measurements become more precise, this small correction ensures that our predictions for the 21-cm signal remain accurate, especially in those tricky moments where signals cancel each other out.

In short: They polished the lens through which we view the early universe, ensuring that when we finally get a crystal-clear picture, our calculations won't be slightly blurry.

Technical Summary

Technical Summary: Peculiar-velocity distribution functions and 21-cm fluctuations

Problem Statement
Predictions for observables involving the cosmological 21-cm background require calculating spatial correlations of star formation rate densities (SFRDs). These SFRDs exhibit a nonlinear dependence on the square of the baryon-dark matter relative peculiar velocity ($v^2$). Previous calculations, such as those implemented in the code Zeus21 (Ref. [9]), relied on a simplifying assumption regarding the joint probability distribution function (PDF) of the peculiar velocity squared at two distinct points. Specifically, these prior works assumed that the correlations of the velocity components parallel and perpendicular to the separation vector between the two points were identical. The authors argue that this assumption neglects a physical subtlety: the correlations of velocity components aligned with the separation vector differ from those of the components transverse to that separation.

Methodology
The authors derive the exact joint PDF for the squares of the peculiar velocity ($v_1^2$ and $v_2^2$) at two points separated by a distance $r$.

1. Velocity Correlations: Starting from a Gaussian random density field, the authors calculate the two-point correlation functions for velocity components. They distinguish between the correlation of components aligned with the separation ($\xi_\parallel$) and those perpendicular to it ($\xi_\perp$). These are derived from the power spectrum $P(k)$ and a transfer function $f(k)$.
2. Joint PDF Derivation: The six components of the two velocity vectors are treated as a multivariate Gaussian distribution with variances $\bar{v}^2$ and specific covariances determined by $\xi_\parallel$ and $\xi_\perp$. The authors integrate this joint velocity PDF over the velocity space, constrained by Dirac delta functions, to obtain the joint PDF for the squared magnitudes $X_1 = v_1^2$ and $X_2 = v_2^2$.
3. Application to SFRDs: The derived PDF is applied to calculate the correlation function for exponential fields $U = e^{-\lambda Y}$, where $Y = v^2/\bar{v}^2$. This represents the correlation required for SFRDs in the Zeus21 framework. The authors derive an exact expression for the correlation function $\xi_U(r)$ that explicitly separates the parallel ($\rho_\parallel$) and perpendicular ($\rho_\perp$) correlation coefficients.

Key Contributions

• Exact Joint PDF: The paper provides the complete analytical expression for the joint PDF of squared peculiar velocities at two points, correcting the approximation used in Ref. [9] which assumed isotropic velocity correlations ($\rho_\parallel = \rho_\perp$).
• Exact Correlation Function: The authors derive the exact two-point correlation function $\xi_U(r)$ for SFRDs (Eq. 19), contrasting it with the approximate form used previously (Eq. 21). The new expression depends on the distinct values of $\rho_\parallel$ and $\rho_\perp$, whereas the approximation uses a single averaged correlation coefficient.
• Implementation Guide: The paper details the specific modifications required to implement this correction in the Zeus21 code, noting that the change involves calculating separate power spectra for parallel and perpendicular components ($P_{\eta,\parallel}$ and $P_{\eta,\perp}$) rather than a single isotropic spectrum.

Results

• Magnitude of Error: The authors find that the error introduced by the previous approximation is generally small, typically less than a few percent for most wavenumbers ($k$) and redshifts ($z$).
• Cancellation Effects: The discrepancy becomes significant only in specific regimes where different contributions to the total 21-cm signal cancel each other out. Specifically, the fractional difference in the 21-cm power spectrum ($\Delta^2_{21}$) increases for a limited range of wavenumbers at redshifts $z \simeq 15$, where such cancellations occur.
• Computational Cost: The implementation of the correct PDF increases the runtime of the Zeus21 code by only a few percent, as it requires two evaluations of the correlation-to-power-spectrum transform ($mcfit$) instead of one.

Significance
The paper concludes that while the error from overlooking the anisotropy in velocity correlations is small compared to other current sources of uncertainty, the correction is easily implemented and necessary for precision. The authors state that as observational capabilities improve and measurements become increasingly precise, this correction may become important, particularly in regimes where signal cancellations amplify the relative impact of the theoretical error. The work ensures that future interpretations of 21-cm fluctuations, specifically those involving velocity-induced acoustic oscillations (VAOs) and kinematic Sunyaev-Zeldovich (kSZ) tomography, are based on the correct statistical treatment of the peculiar velocity field.