CMB Temperature vs Redshift Calculator

Compute the CMB photon temperature, peak wavelength, and photon energy at any cosmic epoch from redshift or temperature.

🌡️ CMB Temperature vs Redshift Calculator
Redshift zz = 0.00
0 (today)1100 (recombination)
CMB Temperature2.725 K
K
2.725 K10,000 K
CMB Temperature T(z)
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Scale Factor a
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Peak Wavelength
—
Photon Energy
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Redshift z
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Scale Factor a
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Peak Wavelength
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Photon Energy
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🌡️ What is the CMB Temperature vs Redshift Calculator?

CMB temperature vs redshift refers to the relationship between the temperature of the Cosmic Microwave Background radiation and the redshift z of any cosmic epoch. The formula T(z) = T0 x (1 + z), where T0 = 2.725 K is the present CMB temperature, tells you exactly how hot the universe's radiation background was at any point in cosmic history. This calculator computes that temperature and three derived quantities: the cosmological scale factor a = 1/(1+z), the peak photon wavelength from Wien's displacement law, and the characteristic photon energy E = kBT.

The CMB is the relic radiation from the Big Bang, released at the epoch of recombination about 380,000 years after the Big Bang when the universe cooled enough for protons and electrons to combine into neutral hydrogen. Before recombination the universe was a hot, opaque plasma. Afterward, it became transparent, and the CMB photons have been streaming freely ever since, cooling as the universe expands. The FIRAS instrument on the COBE satellite measured T0 = 2.725 K with extraordinary precision in the 1990s, a value confirmed to six significant figures by subsequent experiments including WMAP and Planck.

Astronomers use the T(z) relation in several key contexts. Galaxy formation modelers use it to set the thermal state of the intergalactic medium at high redshift, where CMB heating competes with cooling from molecular hydrogen. Observers use molecular rotational excitation temperatures of CO and CN in high-redshift absorption systems as direct CMB thermometers, confirming the (1+z) scaling at z up to 3. Particle physicists use the CMB temperature during Big Bang Nucleosynthesis (z approximately 109, T approximately 109 K) to compute the primordial helium and deuterium abundances.

The inverse mode of this calculator is equally useful: given any observed temperature of the CMB or a CMB-heated gas cloud, it returns the redshift and scale factor of that epoch. Enter 3000 K to recover z approximately 1100 (the surface of last scattering), or enter 9270 K to find z approximately 3400 (matter-radiation equality). This makes the tool a quick cross-reference for any standard LCDM timeline calculation.

📐 Formula

T(z)  =  T0 × (1 + z)
T(z) = CMB temperature at redshift z, in Kelvin
T0 = 2.725 K (CMB temperature today, FIRAS/COBE measurement)
z = cosmological redshift (0 = today, 1100 = recombination)
a = 1 / (1 + z) = scale factor (size of universe relative to today)
Inverse:  z  =  T / T0 − 1
Peak wavelength (Wien's law): λmax = b / T, where b = 2897.772 μm·K
Photon energy: E = kB × T, where kB = 8.617 × 10−5 eV/K
Example: At z = 1100, T = 2.725 × 1101 = 3000.2 K; λmax = 2897.772 / 3000.2 = 0.966 μm (near-infrared)

📖 How to Use This Calculator

Steps

1
Select calculation mode - choose From Redshift to enter a redshift z and find the CMB temperature, or From Temperature to enter a temperature and find the corresponding redshift.
2
Enter the redshift or temperature - type a redshift value from 0 (today) to 1100 (recombination) in the input field, or drag the slider. In temperature mode, enter any temperature at or above 2.725 K.
3
Use epoch presets for quick jumps - click Today (z=0), Reionization (z=10), or Recombination (z=1100) to instantly load a cosmologically significant epoch.
4
Read all four outputs - the results panel shows the CMB temperature (or redshift), scale factor a, peak wavelength with spectral band, and photon energy in eV or meV.

💡 Example Calculations

Example 1 — Today's CMB (z = 0)

What is the CMB temperature, peak wavelength, and photon energy right now?

1
Set z = 0. Scale factor a = 1/(1+0) = 1.000000 (universe at full present size).
2
CMB temperature: T = 2.725 x (1+0) = 2.7250 K.
3
Peak wavelength (Wien): λmax = 2897.772 / 2.7250 = 1.063 mm, solidly in the microwave band.
4
Photon energy: E = 8.617 x 10−5 x 2.7250 = 0.2348 meV. Each CMB photon today carries less than a quarter of a millielectronvolt.
CMB Temperature = 2.7250 K | Peak = 1.063 mm (microwave) | E = 0.2348 meV
Try this example →

Example 2 — Surface of Last Scattering (z = 1100)

What was the CMB temperature at recombination, when the universe first became transparent?

1
Set z = 1100. Scale factor a = 1/1101 = 0.000908 (universe was about 1100 times smaller in each dimension).
2
CMB temperature: T = 2.725 x 1101 = 3,000.2250 K. At this temperature hydrogen recombines and the universe becomes transparent to CMB photons.
3
Peak wavelength: λmax = 2897.772 / 3000.2250 = 965.85 nm, in the near-infrared. The CMB at recombination was an infrared glow, not microwave.
4
Photon energy: E = 8.617 x 10−5 x 3000.2250 = 0.2585 eV. The thermal energy at recombination is far below hydrogen's 13.6 eV ionization threshold, but the Wien tail still drives significant ionization via the Saha equation.
CMB Temperature = 3,000.2250 K | Peak = 965.85 nm (near-infrared) | E = 0.2585 eV
Try this example →

Example 3 — From Temperature: Finding the Recombination Epoch (T = 3000 K)

If a molecular cloud shows a CMB excitation temperature of 3000 K, what redshift does that correspond to?

1
Switch to From Temperature mode. Enter T = 3000 K.
2
Redshift: z = 3000 / 2.725 − 1 = 1100.917 − 1 = 1,099.9174. This places the observation just below the surface of last scattering.
3
Scale factor: a = 2.725 / 3000 = 0.000908. The universe was about 1100 times smaller than today.
4
Peak wavelength: λmax = 2897.772 / 3000 = 965.92 nm (near-infrared). Photon energy: E = 8.617 x 10−5 x 3000 = 0.2585 eV.
Redshift z = 1,099.9174 | Scale Factor = 0.000908 | Peak = 965.92 nm (near-infrared)
Try this example →

❓ Frequently Asked Questions

What is the CMB temperature today?+
The CMB temperature today is 2.725 K, measured by the FIRAS instrument aboard the COBE satellite (Mather et al. 1994, Fixsen et al. 1996). More precise analyses give T0 = 2.7255 plus or minus 0.0006 K. This temperature corresponds to a peak wavelength of about 1.06 mm, in the microwave band, which is why it is called the Cosmic Microwave Background.
Why does CMB temperature increase with redshift?+
As the universe expands, CMB photons are stretched to longer wavelengths (redshifted). Running time backward means the universe was smaller and the photons were blueshifted to shorter wavelengths and higher energies. Since the photon distribution remains a perfect blackbody, this corresponds directly to a higher temperature. The scaling T proportional to (1+z) is a direct consequence of the photon wavelength stretching with the scale factor a.
What was the CMB temperature at recombination?+
At recombination (z approximately 1100), the CMB temperature was T = 2.725 x 1101 = 3000.2 K. At this temperature the Saha equation predicts that hydrogen transitions from being nearly fully ionized to nearly fully neutral over a narrow range of redshifts. This transition made the universe transparent to CMB photons, creating the surface of last scattering that we observe today.
What is Wien's displacement law and how does it apply to the CMB?+
Wien's displacement law states that the peak wavelength of a blackbody spectrum is lambda_max = b / T, where b = 2897.772 micrometers times Kelvin. For the CMB today (T = 2.725 K), this gives lambda_max = 1063 micrometers = 1.063 mm, in the microwave band. At recombination (T = 3000 K), lambda_max = 966 nm, in the near-infrared. The CMB was originally an infrared glow and has cooled into microwave radiation over cosmic time.
What is the scale factor a and how does it relate to redshift?+
The scale factor a = 1/(1+z) measures the size of the universe relative to today. At z = 0 (today) a = 1. At z = 1 a = 0.5, meaning the universe was half as large in each linear dimension. At recombination (z = 1100) a = 0.000908, meaning the universe was about 1100 times smaller. Physical distances at that epoch were a factor of 1101 smaller than they are today.
What is the photon energy of CMB radiation?+
The characteristic CMB photon energy is E = kB T, where kB = 8.617 x 10-5 eV/K. Today (T = 2.725 K) E = 0.235 meV. At recombination (T = 3000 K) E = 0.259 eV. These energies are far below hydrogen's 13.6 eV ionization threshold, but the large photon-to-baryon ratio (about 109 photons per baryon) means even the Wien tail of the distribution can drive significant ionization.
Has the CMB temperature-redshift relation been directly confirmed?+
Yes. Molecular rotational excitation temperatures of CO and CN in quasar absorption systems at z up to 3 serve as direct CMB thermometers. Measurements by Noterdaeme et al. (2011) at z = 2.4 and Srianand et al. confirm T(z) = T0(1+z) to within a few percent, in excellent agreement with LCDM predictions. These observations rule out alternative cosmologies where photon number is not conserved.
What was the CMB temperature at matter-radiation equality?+
Matter-radiation equality occurred at z approximately 3400, when the energy densities of matter and radiation were equal. At that epoch T = 2.725 x 3401 = 9267.7 K. The peak wavelength was lambda_max = 2897.772 / 9267.7 = 312.7 nm, at the boundary between ultraviolet and visible light. This epoch marks the transition from a radiation-dominated to a matter-dominated universe, which is crucial for understanding the growth of structure.
What does z = 10 correspond to in cosmic history?+
Redshift z = 10 corresponds to approximately 480 million years after the Big Bang, near the end of the Epoch of Reionization when the first quasars and massive galaxies were reionizing the intergalactic medium. At that time the CMB temperature was T = 2.725 x 11 = 29.975 K, with a peak wavelength of 96.7 micrometers in the far-infrared. The CMB heated intergalactic gas at this epoch to a temperature floor of about 30 K.
Can I use this calculator for Big Bang Nucleosynthesis temperatures?+
Yes, by typing any redshift in the text field (not just the slider range). Big Bang Nucleosynthesis occurred at z approximately 4 x 108, when the CMB temperature was about 109 K (1 billion Kelvin) and the universe was about 3 minutes old. Type 400000000 in the z field to explore the BBN epoch. Note that at such extreme temperatures and densities, quantum corrections to the pure blackbody spectrum become important.
Why is the temperature lower limit set to 2.725 K?+
The CMB temperature only decreases as the universe expands. The present value of 2.725 K is the minimum observable CMB temperature; no past epoch had a lower CMB temperature in our observable universe. In the far future the CMB will cool further (T will approach zero as z approaches -1 in an expanding universe), but no astronomical observation can access future epochs. The calculator enforces z greater than or equal to 0 and T greater than or equal to 2.725 K as physical lower bounds.
How does this differ from the Cosmological Redshift Calculator?+
The Cosmological Redshift Calculator focuses on the Doppler interpretation of redshift: it converts spectral line shifts to recession velocities, scale factors, and CMB temperatures. This CMB Temperature calculator focuses on the thermal evolution of the photon gas itself: it returns temperature, peak wavelength, photon energy, and supports the inverse calculation of finding the redshift from a known temperature. Both tools share the T = 2.725(1+z) relation but serve different observational use cases.
Tags: Cmb Temperature Calculator Cosmic Microwave Background Redshift Temperature T(z) = T0(1+z) Calculator Formula Free Online Cosmology Wien's Law

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What is the CMB temperature today?

The CMB temperature today is 2.725 K, measured to high precision by the FIRAS instrument aboard the COBE satellite (Mather et al. 1994, Fixsen et al. 1996). More recent analyses give T0 = 2.7255 +/- 0.0006 K. This temperature corresponds to a peak wavelength of about 1.06 mm, solidly in the microwave band.

How does CMB temperature depend on redshift?

The relationship is T(z) = T0 x (1+z), where T0 = 2.725 K. At redshift z = 1 the CMB was twice as hot (5.45 K), at z = 9 it was ten times hotter (27.25 K), and at z = 1100 it was about 3000 K. This scaling is a direct consequence of photon redshifting as the universe expands.

What was the CMB temperature at recombination?

At the surface of last scattering (z approximately 1100), the CMB temperature was T = 2.725 x 1101 = 3000.2 K. This is hot enough for the Saha equation to predict significant hydrogen ionization, which is precisely why recombination happened near that temperature and why the CMB photons were released at z approximately 1100.

What is the scale factor a and how does it relate to redshift?

The scale factor a = 1/(1+z) measures the size of the universe relative to today. At z = 0 (today) a = 1. At z = 1100 (recombination) a = 0.000908, meaning the universe was about 908 times smaller in each linear dimension than it is now.

What does Wien's displacement law predict for the CMB peak wavelength?

Wien's law gives the peak wavelength as lambda_max = b/T, where b = 2897.772 micrometers times Kelvin. Today (T = 2.725 K) the peak is at 1063 micrometers = 1.063 mm, in the microwave band. At recombination (T = 3000 K) the peak was at about 966 nm, in the near-infrared.

What is the photon energy of CMB radiation?

The characteristic CMB photon energy is E = k_B x T, where k_B = 8.617 x 10^-5 eV/K. Today E = 0.235 meV. At recombination E = 0.259 eV. These energies are far below the 13.6 eV hydrogen ionization threshold, but the number of CMB photons is so large that they can still affect hydrogen through the Wien tail.

Has the CMB temperature-redshift relation been directly observed?

Yes. Molecular rotational excitation temperatures of interstellar CO and CN in high-redshift absorption systems provide direct thermometers of the CMB at z > 0. Measurements by Noterdaeme et al. (2011) and others confirm T(z) = T0(1+z) to within a few percent at z up to 3, in excellent agreement with the LCDM prediction.

What is the CMB temperature at matter-radiation equality?

Matter-radiation equality occurred at z approximately 3400, when matter and radiation densities were equal. At that epoch T = 2.725 x 3401 = 9267.7 K. Enter 9267.7 K in the From Temperature mode to confirm: the calculator returns z approximately 3400.

What epoch does z = 10 correspond to cosmologically?

Redshift z = 10 corresponds roughly to the end of cosmic reionization and the beginning of the first galaxies, about 480 million years after the Big Bang. At that time the CMB temperature was T = 2.725 x 11 = 29.975 K, with a peak wavelength of about 96.7 micrometers in the far-infrared.

Can the CMB temperature be used to measure the age of the universe?

Not directly from T alone, because converting z to a lookback time requires integrating the Friedmann equation with cosmological parameters (H0, OmegaM, OmegaLambda). The CMB temperature gives the redshift of an epoch, not its age. Use the Age of Universe or Comoving Distance calculator to convert from z to a cosmic age.

What spectral band is the CMB peak wavelength in at high redshift?

Today the CMB peaks in the microwave band (1.06 mm). By z = 30 (T about 82 K, peak about 35 micrometers) it is in the far-infrared. By z = 1100 (T about 3000 K, peak 966 nm) it is in the near-infrared. By z = 10000 (T about 27,250 K, peak 106 nm) it would be in the ultraviolet. The calculator identifies the spectral band for every result.

Why is the CMB temperature floor set to 2.725 K in the From Temperature mode?

The CMB temperature only decreases as the universe expands (T decreases as a increases). The present value of 2.725 K is the minimum possible CMB temperature; no observable epoch has a lower CMB temperature. The calculator enforces this physical lower bound and returns an error if you enter a temperature below 2.725 K.

📌 Quick Tips

💡The CMB temperature scales exactly as T = 2.725 x (1+z) K, so doubling the redshift approximately doubles the temperature.
💡At recombination (z = 1100) the CMB was at about 3000 K, hot enough to ionize hydrogen, which is why the universe became transparent at that epoch.
💡Use the From Temperature mode to reverse-engineer the redshift of an observed CMB-heated gas cloud or absorption-line system.
💡The peak wavelength shifts from 1.06 mm (microwave) today to 966 nm (near-infrared) at z = 1100, following Wien's displacement law.
💡Matter-radiation equality occurred near z = 3400, when the CMB temperature was about 9270 K. Enter that in the From Temperature mode to explore it.