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Horizons - Scale-Factor and Light-Cones

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Horizons: See attached for properly formatted questions.

(a) Describe the concept of our past and future light-cone. Explain the meaning of the terms particle horizon distance, event horizon distance and world-line and discuss the difference between time-like and space-like locations.

(b) Show that in an Einstein-de Sitter Universe in which the scale factor a(t) at time t follows a(t) alpha t^(2/3), the particle horizon is at 3ct and the event horizon is at infinity.

(c) Suppose that the scale-factor were given by a(t) alpha exp(mt) where m is a positive constant. Show that the event horizon is finite and that the particle horizon grows exponentially when t >> 1/m.

(d) Explain how such behaviour of the particle horizon might be useful in explaining observations of the cosmic microwave background.

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This solution helps explore scale-factor and light-cones within the context of physics and the expanding universe.

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