The Cosmic Microwave Background (CMB) is the oldest light in the universe, originating from the time when the universe was just 380,000 years old. By studying the anisotropies (variations) in the CMB, we can gain insights into the motion of objects relative to the CMB rest frame. This blog post will provide a comprehensive guide on how to calculate the velocity of an object in relation to the CMB.

## Understanding the CMB Dipole Anisotropy

The primary method for calculating the velocity of an object in the CMB rest frame is by using the CMB dipole anisotropy. This anisotropy is a result of the Doppler effect caused by the motion of the object relative to the CMB. The CMB dipole anisotropy has a magnitude of approximately 3.4 millikelvin (mK).

The Doppler effect causes a shift in the observed frequency of the CMB photons due to the relative motion between the observer and the CMB. This shift can be expressed as:

$\frac{\Delta f}{f} = \frac{v}{c}$

where $\Delta f$ is the change in frequency, $f$ is the original frequency, $v$ is the relative velocity, and $c$ is the speed of light.

By measuring the magnitude of the CMB dipole anisotropy, we can calculate the velocity of the object in the CMB rest frame using the following equation:

$v = c \times \frac{\Delta T}{T}$

where $\Delta T$ is the observed temperature difference and $T$ is the average CMB temperature (2.725 K).

## Using Supernovae Luminosity Measurements

Another common method for measuring the velocity of the Earth in the CMB rest frame is by using supernovae (SNe) luminosity measurements. SNe provide an accurate probe of peculiar velocities (PVs), which are the deviations from the overall Hubble flow.

The steps involved in this method are:

- Estimate the absolute magnitudes of the SNe using standardized candle techniques.
- Obtain distance estimates to the SNe, either through redshift-distance relations or other methods.
- Calculate the PV of each SNe’s host galaxy by comparing the observed and expected apparent magnitudes.
- The motion of the Solar system will show up as a dipole anisotropy in the SNe-derived PVs.
- The Solar system peculiar velocity can be detected at the 3.5σ level without accounting for correlations in the peculiar velocities.
- When the correlations are correctly accounted for, the SNe data only detect the Solar system peculiar velocity at about the 2.5σ level.

The error bars on the PV estimate can be reduced by using data from upcoming surveys such as GAIA and the Large Synoptic Survey Telescope (LSST). The error bars will be about four times smaller than those of current data, but still significant.

## Practical Measurement of the Earth’s Velocity

To measure the velocity of the Earth in the CMB rest frame practically, one can consider the orbital speed of the Earth, which is approximately 30 km/s. This value can be used as a rough estimate of the Earth’s velocity in the CMB rest frame.

Additionally, statistical analysis of large and complex CMB data sets can be performed to define a fiducial model and calculate its properties. This approach involves the following steps:

- Collect and preprocess the CMB data, including removing foreground contamination and other systematic effects.
- Define a fiducial cosmological model that best fits the observed CMB data.
- Calculate the properties of the fiducial model, such as the CMB power spectrum, to determine the velocity of the Earth in the CMB rest frame.
- Quantify the uncertainties in the velocity estimate by performing a thorough error analysis, taking into account various sources of systematic and statistical errors.

## Numerical Examples and Calculations

To illustrate the calculations involved in determining the velocity of an object in the CMB rest frame, let’s consider a few examples:

**Example 1: Calculating Velocity from CMB Dipole Anisotropy**

Suppose the observed temperature difference in the CMB dipole anisotropy is $\Delta T = 3.4$ mK. Using the formula $v = c \times \frac{\Delta T}{T}$, where $T = 2.725$ K, we can calculate the velocity as:

$v = c \times \frac{\Delta T}{T} = 3 \times 10^8 \times \frac{3.4 \times 10^{-3}}{2.725} = 372$ km/s

**Example 2: Estimating Peculiar Velocity from Supernovae Measurements**

Suppose we have a sample of 100 Type Ia supernovae with the following properties:

– Average absolute magnitude: $M = -19.3$

– Average redshift: $z = 0.05$

– Observed apparent magnitudes: $m_i$

Using the redshift-distance relation and the observed apparent magnitudes, we can estimate the peculiar velocities of the SNe host galaxies. Assuming a Hubble constant of $H_0 = 70$ km/s/Mpc, the peculiar velocity of the $i$-th galaxy can be calculated as:

$v_i^{\text{pec}} = c \times (z_i – \frac{m_i – M + 25}{5})$

By analyzing the dipole anisotropy in the distribution of these peculiar velocities, we can infer the velocity of the Solar system in the CMB rest frame.

**Example 3: Fiducial Model Analysis of CMB Data**

Consider a large CMB dataset from a survey like Planck or WMAP. We can perform the following steps to determine the velocity of the Earth in the CMB rest frame:

- Preprocess the CMB data, removing foreground contamination and other systematic effects.
- Define a fiducial cosmological model that best fits the observed CMB power spectrum.
- Calculate the parameters of the fiducial model, including the amplitude of the CMB dipole anisotropy.
- Using the formula $v = c \times \frac{\Delta T}{T}$, we can determine the velocity of the Earth in the CMB rest frame.
- Quantify the uncertainties in the velocity estimate by performing a thorough error analysis, considering various sources of systematic and statistical errors.

These examples demonstrate the practical application of the methods discussed in this blog post. By following these steps, you can calculate the velocity of an object in relation to the Cosmic Microwave Background.

## Conclusion

Calculating the velocity of an object in the Cosmic Microwave Background rest frame is a crucial task in cosmology and astrophysics. The primary methods involve using the CMB dipole anisotropy and supernovae luminosity measurements, taking into account various factors and uncertainties. By applying these techniques, researchers can gain valuable insights into the motion of the Solar system and other objects relative to the oldest light in the universe.

## References

- Christopher Gordon, Kate Land, Anže Slosar, “Determining the motion of the Solar system relative to the cosmic microwave background using Type Ia supernovae”, Monthly Notices of the Royal Astronomical Society, Volume 387, Issue 1, 11 June 2008, Pages 371–376, https://doi.org/10.1111/j.1365-2966.2008.13239.x
- “How can we practically measure the velocity of earth in Cosmic Microwave Background rest frame?”, (2019-01-29)
- “STATISTICAL ANALYSIS OF THE COSMIC MICROWAVE …”, (2013)
- “Data analysis methods for the cosmic microwave background – HAL”, (2008-06-10)
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