I am a physics student interested in pursuing research in early-universe cosmology. Astronomy has always fascinated me and inspired my dream of contributing to our understanding of the Universe through research.
Alongside physics, I enjoy painting, reading, and exploring the natural world. I am especially interested in animal behavior, wildlife, and ocean life, and I plan to document these interests here as part of my learning journey.
The Friedmann–Robertson–Walker (FRW) cosmology successfully describes an expanding universe,
but it cannot explain why the universe appears so homogeneous and isotropic today.
According to the standard Big Bang model, the early universe consisted of many causally
disconnected regions that could not exchange information. Despite this, observations of
the Cosmic Microwave Background (CMB) reveal nearly identical temperatures and densities
across the sky. This contradiction is known as the Horizon Problem.
A causal patch is a region in which all points can communicate with one another.
Since no signal can travel faster than light, the size of such a region is limited
by the distance light can travel during a given interval of time.
In cosmology, it is often useful to replace physical time with conformal time,
defined by
τ = ∫ dt / a(t)
This transformation makes the paths of photons appear as straight lines at 45°
on spacetime diagrams, simplifying the analysis of causal structure.
The particle horizon represents the largest distance from which light could have
reached an observer since the beginning of the universe. It therefore defines the
boundary of the observable universe at a given time.
The event horizon determines the furthest distance from which an observer can ever
receive signals in the future. Unlike the particle horizon, it concerns future
observability rather than past observability.
The comoving Hubble radius, (aH)-1, provides another way of understanding
causal structure. It characterizes the scale over which physical processes can remain
in causal contact at a given moment. During standard radiation- and matter-dominated
expansion, the comoving Hubble radius grows with time.
Around 380,000 years after the Big Bang, electrons and protons combined to form
neutral hydrogen. As a result, photons decoupled from matter and began travelling
freely through space. These photons are observed today as the Cosmic Microwave
Background (CMB).
The CMB is remarkably uniform, exhibiting temperature fluctuations of only about
one part in one hundred thousand. However, calculations based on standard Big Bang
cosmology show that many regions of the CMB sky were never in causal contact before
photon decoupling. Nevertheless, they possess nearly identical temperatures.
This apparent contradiction is the Horizon Problem and serves as one of the primary
motivations for cosmic inflation.
The horizon problem arises because, in standard cosmology, the comoving Hubble radius grows with time.
Consequently, regions that are widely separated today were once separated by more than a Hubble radius
and could never have communicated.
The solution is an early epoch where the comoving Hubble radius shrinks: \[
\frac{d}{dt}(aH)^{-1} < 0
\]
If this phase lasts long enough, regions now far apart were once inside the same Hubble volume and could
thermalize before being stretched apart.
For shrinking Hubble radius:
\[
\frac{d}{da}(aH)^{-1} < 0
\]
Using:
\[
(aH)^{-1} \propto a^{\frac{1}{2}(1+3w)}
\]
This implies:
\[
1 + 3w < 0 \quad \Rightarrow \quad w < -\frac{1}{3}
\]
Ordinary matter does not satisfy this:
Radiation: \( w = \frac{1}{3} \)
Inflation therefore requires negative pressure beyond standard matter.
The particle horizon is:
\[
\chi_{\rm ph} = \int \frac{dt}{a} = \int (aH)^{-1} d\ln a
\]
With:
\[
(aH)^{-1} \propto a^\alpha, \quad \alpha = \frac{1}{2}(1+3w)
\]
If \( w < -\frac{1}{3} \), then \( \alpha < 0 \), and early times dominate:
\[
\tau = \frac{2H_0^{-1}}{1+3w} a^{\frac{1}{2}(1+3w)}
\]
As \( a_i \to 0 \), we get:
\[
\tau_i \to -\infty
\]
This means the universe has effectively infinite conformal past, allowing causal contact across regions
now widely separated.
Inflation replaces the initial singular surface with a reheating surface.
For inflation to solve the horizon problem:
\[
(a_0 H_0)^{-1} < (a_i H_i)^{-1}
\]
During inflation: \( H \approx \text{constant} \)
Number of e-folds:
\[
N = \ln\!\left(\frac{a_f}{a_i}\right)
\]
Required:
\[
N \gtrsim 60
\]
CMB scales exit the horizon about 60 e-folds before inflation ends.
Particle Horizon
\[
\chi_{\rm ph} = \int_{t_i}^{t} \frac{dt'}{a(t')}
\]
Measures total causal past — how far signals could have traveled since the beginning.
Comoving Hubble Radius
\[
(aH)^{-1}
\]
Measures instantaneous communication scale during one Hubble time.
During Inflation:
\[
\chi_{\rm ph} \gg (aH)^{-1}
\]
Inflation stretches previously connected regions beyond the Hubble scale,
without destroying their causal history.
The most fundamental definition of inflation is that the comoving Hubble radius shrinks with time:
\[
\frac{d}{dt}(aH)^{-1}<0
\]
To relate this condition to the expansion of the universe, note that
\[
(aH)^{-1} = \frac{1}{aH} = \frac{1}{\dot a}
\]
Taking a time derivative:
\[
\frac{d}{dt}(aH)^{-1}
=
\frac{d}{dt}\left(\dot a^{-1}\right)
=
-\frac{\ddot a}{\dot a^{\,2}}
\]
Therefore:
\[
\frac{d}{dt}(aH)^{-1}<0
\qquad \Longleftrightarrow \qquad
\ddot a>0
\]
Thus, inflation is equivalent to a period of accelerated expansion.
The evolution of the Hubble parameter is characterized by the first slow-roll parameter:
\[
\epsilon \equiv -\frac{\dot H}{H^{2}}
\]
It is often convenient to express time evolution in terms of the number of e-folds:
\[
N \equiv \ln a
\]
For which:
\[
dN = d\ln a = H\,dt
\]
Using this relation:
\[
\epsilon = -\frac{d\ln H}{dN}
\]
The derivative of the comoving Hubble radius may be written as:
\[
\frac{d}{dt}(aH)^{-1}
=
-\frac{1}{a}(1-\epsilon)
\]
Inflation therefore requires:
\[
\epsilon < 1
\]
This provides an equivalent and extremely useful criterion for inflation.
For inflation to continue over many e-folds, the Hubble parameter must evolve slowly. Hence:
\[
\epsilon \ll 1
\]
so that \(H\) remains nearly constant during the inflationary era.
The idealized case of de Sitter space corresponds to:
\[
H = \text{constant}
\]
for which:
\[
a(t)\propto e^{Ht}
\]
Since:
\[
\dot H = 0
\]
we have:
\[
\epsilon = 0
\]
A universe with exact de Sitter expansion would inflate forever.
Realistic inflation is instead quasi–de Sitter. In this regime:
\[
0 < \epsilon \ll 1
\]
so the Hubble parameter changes slowly with time.
Inflation eventually ends when:
\[
\epsilon \rightarrow 1
\]
To ensure that inflation persists long enough, one introduces a second slow-roll parameter:
\[
\eta \equiv \frac{d\ln\epsilon}{dN} = \frac{\dot\epsilon}{H\epsilon}
\]
The condition:
\[
|\eta| < 1
\]
implies that the fractional change in \(\epsilon\) per Hubble time is small.
When:
\[
|\eta| \ll 1
\]
\(\epsilon\) remains nearly constant and inflation continues for many e-folds.
Conversely:
\[
|\eta| > 1
\]
means inflation will soon terminate.
The slow-roll conditions can therefore be summarized as:
\[
\epsilon < 1, \quad |\eta| < 1
\]The Horizon Problem
Size of a Causal Patch
Conformal Time
Particle Horizon
Event Horizon
The Hubble Radius
Why Is the CMB So Uniform?
Inflation as the Solution
A Shrinking Hubble Sphere
Matter: \( w = 0 \)
Solving the Horizon Problem
Required Duration: ~60 e-Folds
After reheating: \( H \propto a^{-2} \)
Particle Horizon vs Hubble Radius
Conditions for Inflation
Accelerated Expansion
The Slow-Roll Parameter ε
Quasi–de Sitter Expansion
These notes review the core concepts of linear algebra and build toward the mathematical framework used in quantum computing and quantum mechanics. The focus is on vector spaces, matrices, linear transformations, and their role in describing quantum systems.
A linear equation in multiple variables represents a constraint of the form:
a₁x₁ + a₂x₂ + … + aₙxₙ = b
A system of such equations can be written compactly as:
A x = b
where A is a matrix, x is the vector of unknowns, and b is the output vector.
A matrix is a rectangular arrangement of numbers used to represent linear transformations and systems of equations.
A = [aᵢⱼ]
Matrix multiplication is defined as:
(AB)ᵢⱼ = Σₖ Aᵢₖ Bₖⱼ
A system A x = b may be:
A homogeneous system A x = 0 always has at least the trivial solution x = 0.
In physics, vectors represent quantities with magnitude and direction. In mathematics, vectors are abstract elements of vector spaces.
A system of equations can also be viewed geometrically as a linear combination:
x a₁ + y a₂ = b
meaning we express a vector as a combination of basis vectors.
A vector space consists of:
These structures form the foundation of linear algebra and quantum theory.
Quantum states are represented as vectors in a Hilbert space, and quantum operations are represented as matrices acting on these vectors. This makes linear algebra the mathematical backbone of quantum computing.
A collection of visual work.
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