Standard Candle Dimming Due to Extra Expansion

[BillSaltLake]

I'm having trouble deriving the amount of dimming expected of standard candles (eg. type 1a supernovae) as a result of dark energy.

Without the presence of dark energy, the standard GR solution (matter-only, at critical density) is that the absolute bolometric brightness of a standard candle varies with redshift z as 1/(1+z - [1+z]^1/2)². This expression is the product of two terms: 1/D_L² multiplied by 1/(1+z)². Here D_L is the "luminosity distance", which is the expected dimming of light due to geometry. As it turns out, D_L is a function of z so that the product simplifies to 1/(1+z - [1+z]^1/2)².

Suppose that recent stretching of space due to dark energy is by a factor of b (b>1). Obviously this would change the redshift, replacing 1+z with b(1+z) for a given distant object. (This factor b is the extra stretch that occurred between the time when a given distant object emitted a photon and the present when the photon is received.) How would the factor b change the geometric distance term?

The observed luminosity at z=1 is only about half of the value 1/(1+z - [1+z]^1/2)². If one simply replaces 1+z with b(1+z), luminosity vs. redshift curve remains the same instead of reducing to about half at z=1.

 

[Chalnoth]

It can't be computed analytically, but must be estimated numerically by evaluating the integral:
$$D_L = c\left(1+z\right) \int_0^z \frac{dz'}{H(z')}$$
(note: this is for flat space)

 

[BillSaltLake]

Thank you. If energy+matter+dark energy adds up to the critical density, then the space is treated as flat. Is that correct? (That is, dark energy doesn't do anything weird to affect the flatness.)

 

[Chalnoth]

Thank you. If energy+matter+dark energy adds up to the critical density, then the space is treated as flat. Is that correct? (That is, dark energy doesn't do anything weird to affect the flatness.)

Yes, this is correct.

 

[BillSaltLake]

Chalnoth, in your expression for D_L, are you sure (1+z) should be there? I get the correct expression for matter-only at critical density: D_L= 3c(t_present)^2/3(t_present^1/3 - t_past^1/3) only if I replace (1+z) with 1.

 

[Chalnoth]

Chalnoth, in your expression for D_L, are you sure (1+z) should be there? I get the correct expression for matter-only at critical density: D_L= 3c(t_present)^2/3(t_present^1/3 - t_past^1/3) only if I replace (1+z) with 1.

Yes. Without that factor, you would be talking about $D_M$, which is the comoving distance (also the proper motion distance). You can read more on the various distance measures used in Cosmology here:

http://arxiv.org/abs/astro-ph/9905116

 

[BillSaltLake]

OK. Makes sense now. D_L^-2 is proportional to the bolometric brightness of a compact source because it already includes the two factors of 1/(1+z) (photon stretch and # of photons per time). I was multiplying the 1/(1+z)² separately into the brightness.