Tools of the Trade (for infinite sums) - Numberphile

Calculating Primordial Black Hole Formation: A Detailed Analysis

Key Concepts:

• Primordial Black Holes (PBHs): Hypothetical black holes formed in the very early universe, potentially contributing to dark matter.
• Normalization (of Probability): Ensuring the total probability of all possible outcomes equals one.
• Infinite Sums & Limits: Mathematical tools used to calculate probabilities and physical quantities in theoretical physics.
• Trigonometric Identities: Relationships between trigonometric functions (sine, cosine, tangent) used to simplify expressions.
• Bounding: Estimating the range of a value by finding upper and lower limits.
• Integrals, Series, and Products (Table of): A reference book containing pre-calculated mathematical formulas and solutions.
• Shine Function (sinh): A hyperbolic trigonometric function.

I. The Problem & Initial Sum

The video details a calculation undertaken to determine the potential number and mass distribution of primordial black holes (PBHs) that may have formed in the early universe. A crucial component of this calculation involves evaluating a complex infinite sum, denoted as I. This sum is defined as:

I = Σ (from N=1 to ∞) [N * sin(Nf) / (α² * sin²(2πN/α) + 2n²)]

where:

• N is the summation variable.
• f is a variable approaching zero, representing a key limit for the calculation.
• α is a constant related to the properties of the black holes.
• sin(x) represents the sine function.
• sin²(x) represents the square of the sine function.

The initial assessment of this sum led the researcher to believe it was intractable, due to the combination of large values of N and a small value of f. The sum arose from the need to normalize a probability distribution, ensuring it sums to one – a fundamental requirement in probability theory.

II. Initial Exploration & Numerical Approximation

Unable to solve the sum analytically, the researcher initially turned to numerical methods. Using a computer, they calculated 1/I for different upper limits of the summation (n = 10, 20, 50). The results revealed a pattern: as f decreased, the value of 1/I approached a constant value before rapidly diverging. This suggested that the sum might converge to a finite value, approximately on the order of six.

III. Establishing Upper and Lower Bounds

To gain further insight, the researcher employed the technique of bounding. This involves finding upper and lower limits for the sum.

• Upper Bound: By focusing on the denominator of the sum, it was observed that α² * sin²(2πN/α) + 2n² is always greater than or equal to 2n². Taking the reciprocal, this implies that I is less than or equal to the sum: Σ (from N=1 to ∞) [N * sin(Nf) / (2n²)]. This simplifies to 1/α² * (π/2 - f/2), resulting in an upper bound of 1/(2π) ≈ 0.159.

• Lower Bound: Utilizing the fact that sin²(2πN/α) is always less than or equal to 1, the denominator is less than or equal to α² + 2n². This leads to I being greater than or equal to the sum: Σ (from N=1 to ∞) [N * sin(Nf) / (α² + 2n²)]. Further manipulation, using a formula from a table of integrals, series, and products, yields a lower bound of 1/(2π).

IV. The Unexpected Result & Proof

The remarkable finding was that the upper and lower bounds were identical: I = 1/(2π). This meant the sum could be determined without actually calculating the infinite sum itself. Consequently, 1/I = 2π ≈ 6.28.

Driven by a desire for rigorous proof, the researcher proceeded to demonstrate this result analytically. By manipulating the original sum using trigonometric identities, they aimed to express it in a form that could be directly evaluated. This involved bringing the sin² term into the numerator and leveraging known identities to simplify the expression.

V. Tools of the Trade & Importance of Mathematical Resources

The researcher emphasized the importance of mathematical resources, specifically referencing the "book of table of integrals, series and products" as a crucial "tool of the trade" for theoretical physicists. This book provides pre-calculated formulas and solutions that can significantly accelerate complex calculations.

VI. Context & Future Work

The calculated value of 1/I (2π) is not the number of primordial black holes. Instead, it's a critical component in calculating the probability distribution function for PBH formation. The researcher acknowledged that the next step – solving for the actual probability distribution – remains an ongoing challenge. However, the successful evaluation of this sum provides confidence that they are on the right track.

VII. Notable Quotes

• “Usually when you put a bound on it, it’ll tell you that okay I has got to be less than some massive number and bigger than a really small number… but to have it bound like that it was just chance.”
• “You always have a feeling you’re on the right track if you’ve got a two pie hanging in there.”

Conclusion:

The video showcases a fascinating example of problem-solving in theoretical physics. Through a combination of numerical exploration, bounding techniques, and the application of trigonometric identities, a seemingly intractable infinite sum was successfully evaluated, yielding a precise result (1/(2π)). This result is a crucial step towards understanding the potential contribution of primordial black holes to the universe’s dark matter content, highlighting the power of mathematical tools and the importance of rigorous analysis in scientific research. The process demonstrates how seemingly small calculations can have significant implications for broader cosmological questions.