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johne1618

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Hi,

I am interested to hear what people think of the following argument that in an accelerating Universe virtual particles that are separated by the Hubble radius [itex]R(t)[/itex] or more cannot annihilate and thus become real particles. Thus, by the uncertainty principle, particle pairs with energy [itex]E \approx h c / R(t)[/itex] are continually being created in an accelerating Universe giving it an intrinsic temperature that is inversely proportional to its Hubble radius.

Let us assume that, at cosmological time [itex]t_s[/itex], a particle-antiparticle pair pops into existence separated by a distance [itex]d_s[/itex].

The particle pair will annihilate in the time it takes a light signal to travel between them.

If the proper separation distance expands faster or at the same rate as the light signal proper path distance then the particle pair will never annihilate.

Let us consider the situation at the "present" cosmological time [itex]t_p[/itex].

Assume that the scale factor is given by [itex]a(t)[/itex] where [itex]a(t_p) = 1[/itex].

The proper separation distance of the particles at time [itex]t_p[/itex] is given by

[itex] \Large D(t_p) = \frac{a(t_p)\ d_s}{a(t_s)} [/itex].

Now if a light signal is emitted from one particle at time [itex]t_s[/itex], the proper distance it will have traveled by time [itex]t_p[/itex] is given by

[itex] \Large P(t_p) = \int^{t_p}_{t_s} \frac{a(t_p)\ c\ dt}{a(t)} [/itex].

A sufficient condition for the target particle to escape from the light signal is given by

[itex] \Large \frac{d D(t_p)}{d t_p} \ge \frac{d P(t_p)}{d t_p} [/itex] for all [itex]t_p[/itex].

Substituting into the above condition we find

[itex] \Large \frac{\dot{a}(t_p) d_s}{a(t_s)} \ge c [/itex]

Let us assume that [itex]\dot{a}(t)[/itex] is a monotonically increasing function i.e. that [itex]\ddot{a} \ge 0[/itex] so that the expansion of the Universe is accelerating.

Then a sufficent condition for the particles not to annihilate is given by

[itex] \Large \frac{\dot{a}(t_s)d_s}{a(t_s)} \ge c [/itex]

As the Hubble parameter at time [itex]t_s[/itex] is given by

[itex] \Large H(t_s) = \frac{\dot{a}(t_s)}{a(t_s)} [/itex]

and the Hubble radius at time [itex]t_s[/itex] is given by

[itex] \Large R(t_s) = \frac{c}{H(t_s)} [/itex]

we find, finally, that in an accelerating Universe a sufficient condition for the particles to avoid annihilation is that their initial separation at time [itex]t_s[/itex], [itex]d_s[/itex], should obey the relationship

[itex] \Large d_s \ge R_s [/itex]

where [itex]R_s[/itex] is the Hubble radius at time [itex]t_s[/itex].

John

I am interested to hear what people think of the following argument that in an accelerating Universe virtual particles that are separated by the Hubble radius [itex]R(t)[/itex] or more cannot annihilate and thus become real particles. Thus, by the uncertainty principle, particle pairs with energy [itex]E \approx h c / R(t)[/itex] are continually being created in an accelerating Universe giving it an intrinsic temperature that is inversely proportional to its Hubble radius.

Let us assume that, at cosmological time [itex]t_s[/itex], a particle-antiparticle pair pops into existence separated by a distance [itex]d_s[/itex].

The particle pair will annihilate in the time it takes a light signal to travel between them.

If the proper separation distance expands faster or at the same rate as the light signal proper path distance then the particle pair will never annihilate.

Let us consider the situation at the "present" cosmological time [itex]t_p[/itex].

Assume that the scale factor is given by [itex]a(t)[/itex] where [itex]a(t_p) = 1[/itex].

The proper separation distance of the particles at time [itex]t_p[/itex] is given by

[itex] \Large D(t_p) = \frac{a(t_p)\ d_s}{a(t_s)} [/itex].

Now if a light signal is emitted from one particle at time [itex]t_s[/itex], the proper distance it will have traveled by time [itex]t_p[/itex] is given by

[itex] \Large P(t_p) = \int^{t_p}_{t_s} \frac{a(t_p)\ c\ dt}{a(t)} [/itex].

A sufficient condition for the target particle to escape from the light signal is given by

[itex] \Large \frac{d D(t_p)}{d t_p} \ge \frac{d P(t_p)}{d t_p} [/itex] for all [itex]t_p[/itex].

Substituting into the above condition we find

[itex] \Large \frac{\dot{a}(t_p) d_s}{a(t_s)} \ge c [/itex]

Let us assume that [itex]\dot{a}(t)[/itex] is a monotonically increasing function i.e. that [itex]\ddot{a} \ge 0[/itex] so that the expansion of the Universe is accelerating.

Then a sufficent condition for the particles not to annihilate is given by

[itex] \Large \frac{\dot{a}(t_s)d_s}{a(t_s)} \ge c [/itex]

As the Hubble parameter at time [itex]t_s[/itex] is given by

[itex] \Large H(t_s) = \frac{\dot{a}(t_s)}{a(t_s)} [/itex]

and the Hubble radius at time [itex]t_s[/itex] is given by

[itex] \Large R(t_s) = \frac{c}{H(t_s)} [/itex]

we find, finally, that in an accelerating Universe a sufficient condition for the particles to avoid annihilation is that their initial separation at time [itex]t_s[/itex], [itex]d_s[/itex], should obey the relationship

[itex] \Large d_s \ge R_s [/itex]

where [itex]R_s[/itex] is the Hubble radius at time [itex]t_s[/itex].

John

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