Question

the shape of the universe
if you draw a triangle on a flat piece of paper the internal angles of the triangle add up to 180°. but what happens if you draw a triangle on the surface of a ball? the internal angles add up to more than 180°. and what about the internal surface of a bowl? the internal angles of a triangle add up to less that 180°. now imagine drawing a huge triangle on the \surface\ of the universe. what would the internal angles of the triangle be? the answer depends on whether or not the universe is curved or flat. the answer has important consequences for the shape and fate of the universe.
if the internal angles of your triangle add to be greater than 180°, then the universe is positively curved like a sphere. in this case, the universe is finite but has no edge. it will eventually slow its expansion and collapse back on itself.
if the triangles internal angles add to be equal to 180° then the universe is flat. in this case it will continue to expand forever; the expansion will slow down over an infinite amount of time. current evidence suggests the universe is flat.
if your triangle has internal angles that add to less than 180° then the universe is negatively curved, like the shape of a saddle. it is infinite and will expand forever at an ever - increasing rate.

1. what is the cosmic horizon?
2. how big is the observable universe?
3. explain why the cosmic horizon is much smaller than the universe itself.
4. (a) what are the three components that make up the universe?

(b) what percentage of the universe comprises visible matter?

1. (a) what is the distance to the suns nearest star?

(b) what percentage of the diameter of the milky way galaxy is this?
(c) what is the distance to the nearest major galaxy?

1. (a) describe the path of two parallel light beams in a flat universe.

(b) describe the path of two parallel light beams in a positively curved universe.

1. how do we know dark matter and dark energy exist when they cannot be observed directly?

Answer:

1. Brief Explanation: The cosmic horizon is the boundary of the observable universe. Beyond this distance, light has not had enough time to reach us since the beginning of the universe due to the finite speed of light and the finite age of the universe.

• Answer: The cosmic horizon is the boundary of the observable part of the universe, beyond which light has not had time to reach us.

1. Brief Explanation: The observable universe has a radius of about 46.5 billion light - years. This is based on the expansion of the universe and the time since the Big Bang, taking into account the fact that the universe has been expanding while the light has been traveling towards us.

• Answer: The observable universe has a radius of approximately 46.5 billion light - years.

1. Brief Explanation: The universe is likely infinite or at least much larger than the observable part. The cosmic horizon is limited by the finite speed of light and the finite age of the universe. Light from regions beyond the cosmic horizon has not had enough time to reach us yet, so we can't observe them, making the cosmic horizon much smaller than the entire universe.

• Answer: The cosmic horizon is limited by the finite speed of light and the age of the universe. Light from beyond it hasn't reached us, so it's much smaller than the whole universe.

1. (a) Brief Explanation: The three main components of the universe are ordinary (baryonic) matter, dark matter, and dark energy. Ordinary matter makes up stars, planets, and all visible objects, dark matter provides additional gravitational mass that affects the motion of galaxies, and dark energy is thought to be responsible for the accelerating expansion of the universe.

• Answer: The three components are ordinary matter, dark matter, and dark energy.
• (b) Brief Explanation: Visible matter (ordinary matter) makes up only about 5% of the total energy - density of the universe. Dark matter is about 27% and dark energy about 68%.
• Answer: Approximately 5%.

1. (a) Brief Explanation: The nearest star to the Sun is Proxima Centauri, which is about 4.24 light - years away.

• Answer: About 4.24 light - years.
• (b) Brief Explanation: The diameter of the Milky Way Galaxy is about 100,000 - 180,000 light - years. Taking the lower estimate of 100,000 light - years, the percentage is $\frac{4.24}{100000}\times100\approx0.00424\%$.
• Answer: Approximately 0.00424% (using a Milky Way diameter of 100,000 light - years).
• (c) Brief Explanation: The nearest major galaxy to the Milky Way is the Andromeda Galaxy, which is about 2.5 million light - years away.
• Answer: About 2.5 million light - years.

1. (a) Brief Explanation: In a flat universe, two parallel light beams will continue to travel parallel to each other indefinitely. This is similar to how parallel lines in Euclidean geometry behave.

• Answer: They will continue to travel parallel to each other.
• (b) Brief Explanation: In a positively curved universe (like the surface of a sphere), two initially parallel light beams will eventually converge. This is because the geometry of the space causes them to follow curved paths that meet at some point.
• Answer: They will eventually converge.

1. Brief Explanation: We know dark matter exists because of its gravitational effects on visible matter. For example, the rotation curves of galaxies suggest that there is more mass than we can see (visible matter), and this extra mass is attributed to dark matter. Dark energy is inferred from the accelerating expansion of the universe. Observations of distant supernovae and the cosmic microwave background radiation provide evidence for the existence of dark energy, which is thought to be causing the expansion of the universe to speed up.

• Answer: Dark matter is inferred from its gravitational effects on visible matter (e.g., galaxy rotation curves). Dark energy is inferred from the accelerating expansion of the universe (e.g., supernova observations and cosmic microwave background).