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General Doppler Shift Equation and the Possibility of Systematic

Error in Calculation of Z for High Redshift Type Ia Supernovae

Steven M Taylor

smtygc@umsl.edu

Abstract

Systematic error in calculation of z for high redshift type Ia supernovae could help

explain unexpected luminosity values that indicate an accelerating rate of expansion of

the universe.

Introduction

The general form of the relativistic Doppler shift equation is

ν

1

β−

c

With an emission angle

= 0

θ

)cos1 (

0

'

θβγν−=

, (1)

where

2

1

γ=

and

u

=

β

with u being velocity of source.

the general form reduces to the familiar

1

)1 (

β

ν

ν

+

°

2

1

2

0

,

)1 (

β

−

=

. (2)

The condition

vector and is typically assumed for astronomical purposes.

With the assumption redshift parameter is defined as

= 0

θ

corresponds to an emission antiparallel to the source’s velocity

°

= 0

θ

°

=

z

1

)1 (

)1 (

2

1

2

1

−

−

+

β

β

. (3)

Evidence of Accelerating Universe and Possible Systematic Error

The primary evidence of an accelerating rate of expansion of the Universe is that

measurements of apparent magnitude of some high-z, type Ia supernovae are fainter than

would be expected for non-accelerating cosmological models. [1]

Perlmutter and Schmidt of the Cosmology Supernovae Project have noted that along with

other possible sources of systematic errors, gravitational lensing may contribute to a

change in luminosity of high-redshift supernovae. Citing several authors, they note that

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as radiation traverses the large scale structure from where it is emitted and where it is

detected, it could be lensed as it encounters fluctuations in gravitational potential. Some

images could be demagnified as their light passes through under-dense regions. It is also

noted that it would also be possible for a light path to encounter denser regions magnify

the image. It is noted that such an effect may limit the accuracy of luminosity distance

measurements.[2]

A lower luminosity in relation to z is to date the strongest evidence of an accelerated

expansion rate for the Universe. In the same sense that a change in luminosity due to

reasons other than distance, such as gravitational lensing could produce systematic error,

so could a false z measurement.

Lowered luminosity would be consistent with a false z measurement if that measurement

was less redshifted due to reasons extraneous to the expansion rate of the universe as

presented by cosmological models.

Whether by gravitation or other effect, any canting in angle of emission of light from a

receding source will cause an increase in frequency as seen by an observer. Taking the

derivative of the general relativistic Doppler shift equation with respect to θ yields:

'

=

∂

Since is an absolute minimum, any deviation from that angle results in a higher

= 0

θ

Example

According to Perivolaropoulos, the Gold supernova data set of 157 points show that

transition from a decelerating towards and accelerating universe to be at z= 0.46 + 0.13.

Using a graph of apparent magnitude vs. redshift based on the Gold data, a supernova

with an approximate 44 apparent magnitude and measured redshift parameter of

, lies on the curve for an accelerating universe. If it did have a redshift

parameter of , the supernova would lie on curve consistent with a decelerating

universe.[3]

Using equation (3) a measured parameter of

likewise a parameter of corresponds to

30. 1

≅

z

Assuming, for the sake of argument, that the Universe is decelerating and the supernova

does have a parameter of , we can calculate the canting in angle of emission

from that allows an observer to measure a parameter of

= 0

θ

θ γβ

0

ν

θ

sin

∂v

. (4)

°

'

ν .

95. 0

≅

z

30. 1

≅

z

95

. 0

=

. 0

β

≅

z

corresponds to

68

.

58. 0

=β

, and

30. 1

≅

z

°

95. 0

≅

z

.

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Using

ν = 0.4364 Hz.

Likewise with

yields

ν = 0.5156 Hz.

0

ν =1 Hz to simplify, and

68 . 0

=β

(corresponding to z =1.30), equation 2 yields

,

0

ν =1 Hz to simplify, and

58 . 0

=β

(corresponding to z =0.95), equation 2

,

0792 . 0

'=Δv

Hz (5)

Inserting

68. 0

=β

into equation 4, yields

2

1

2

'

) 68. 01 (

sin 68. 0

−

=

∂

∂

θ

θ

v

. (6)

Using 0.0792 Hz for

θ∂

∂

'v (5) into equation 6 yields an emission angle of .

°

≅

90 . 4

θ

Conclusion

Given the large distances that light from high z supernovae travel, and the modest canting

from in emission angle required to help explain decreased luminosity for high

redshift supernovae, the possibility of systematic error in z measurement for high redshift

supernova should be further investigated.

References

1. S Perlmutter, B Schmidt, Measuring Cosmology with Supernova, arxiv:astro-

ph/0303428 v1 (2003)

2. ibid

3. L. Perivolaropoulos, Accelerating Universe: Observational Status and Theoretical

Implications, arXiv:astro-ph/0601014 v2 (2006)

°

= 0

θ