By Radek Wojtak

Cosmologists generally assume that we do not sit at any special place in the Universe when extracting properties about our Universe, such as figuring out its expansion history (for which the Nobel Prize in Physics was awarded in 2011).

The important question we set out to address in our recent paper was whether this is true enough for upcoming major cosmology surveys that we can continue to ignore second order effects on this assumption such as ignoring our local gravitational redshift.

**The wavelength-stretching of photons escaping from the clutches of galaxy clusters**

To step back and get some context: a few years ago a group of astrophysicists from the Dark Cosmology Centre in Copenhagen succeeded in measuring gravitational redshift of galaxies in clusters caused by the gravitational potential well of dark matter haloes -- that is, how the clusters were stretching out the wavelength of light as the photons attempted to escape from their clutches.

This gravitational redshift is a second order effect compared to the shift in emitted wavelength of galaxies caused by the random velocities of galaxies in clusters (translating both to velocity, the gravitational shift is on the order of 15 km/s for a typical mass galaxy cluster, compared to approximately 600 km/s for the random motion of a galaxy bound in the cluster). Thus, the only way to observe a signal was to carry out a joint analysis of redshift data from tens of thousand of galaxy clusters found in the SDSS. After much analysis, the gravitational redshifts were detected as relative shifts between velocity distributions of galaxies at different distances from the cluster centers.

**General Relativity continues its lengthy winning streak -- now on some of the largest scales in the Universe**

Observational confirmation of this effect extended another classical test of general relativity to cosmological scales. When combined with the Pound-Rebka experiment that was carried out on the Earth in 1959, the gravitational redshift has now been measured on scales spanning an unprecedented range: From about 10 meters (on Earth) to about 3 million light years -- a range of distances spanning 21 orders of magnitude. The detection of the gravitational redshift in galaxy clusters also demonstrated the huge physics potential of modern cosmological surveys, which have now reached the level of precision and quality of data that enable us to study the most subtle effects shaping the way light propagates in our Universe.

The gravitational redshift caused by potential wells of dark matter haloes in which the galaxy clusters reside scales up with halo mass and continues to rise even with objects that are not gravitationally bound such as superclusters. The amplitude of the observed matter density fluctuations implies that even larger-scale structures such as megavoids and superclusters can have as much of an impact on the gravitational redshift as the most massive galaxy clusters (which are always defined to be bound).

The figure above shows typical large-scale fluctuations of the gravitational potential computed in a cosmological simulation from the Big Multidark project. The left panel shows a map of the density contrast smoothed at a scale comparable to sizes of superclusters, with small green ‘blobby’ symbols representing dark matter haloes with their sizes scaling up with halo radii. The color strip on the right shows the density contrast scale. The right panel shows the gravitational potential along the reference line from the left panel (which continues from left to right across the periodic boundary condition on the right edge to the next line above, then the last line above that -- the cyan number labels on the line correspond to those on the x-axis in the plot on the right).

A quick comparison of both panels demonstrates that clearly visible peaks and troughs in the potential correspond to large voids and superclusters encountered along the reference line. It is also clear from this picture that the gravitational redshift when light is emitted from an overdensity or a dip in the plot on the right (or a blueshift if light is emitted from a void, or a peak in the plot on the right) can be as large as φ/c^{2} ~ 5x10^{-5} which corresponds to ~15 km/s.

**Teasing out the gravitational redshift from other effects**

We may ask ourselves whether this effect is significant enough to concern us. To answer that question, let us imagine that we measure distances to a number of objects and compare them to the corresponding distances expected for our cosmological model.

There are a number of second order effects which can lead to certain deviations between observations and theory. The main two effects in play are peculiar velocities (i.e. the velocities of galaxies due to small deviations off the straight geometric Hubble flow due to the Universe steadily expanding) and gravitational lensing (i.e. the bending of light caused by by intervening cosmic structures between the source and us). Both effects can slightly shorten or lengthen the observed distances depending on whether an object is moving toward or away from us and whether its light is magnified or demagnified due to over- or under-densities between it and us.

However, if we consider distances larger than a scale of inhomogeneities in the Local Universe, we expect both effects to average out over many observations distributed in many directions on the sky. The only effect which does not cancel in this averaging is the gravitational redshift caused by the gravitational potential at our own location in the cosmic web. The important question we wanted to address in our recent paper was whether the local gravitational redshift can affect our measurement of cosmological parameters based on the distance-redshift relation. In particular, we focused on a method of measuring cosmological parameter using supernova (SN) Ia data -- that was exactly the one for which the 2011 Physics Nobel was given.

**Using an ensemble of simulations to assess if the local shift is significant**

A simple test assessing the relevance of the local gravitational redshift can be performed by repeating cosmological fits on simulated SN Ia data generated for the same cosmological model and modified by the local gravitational redshift. Ideally, we would like to know the exact large-scale gravitational potential at the Milky Way’s position so that we could simulate the effect of the gravitational redshift as realistically as possible.

Many independent studies of galaxy counts and cluster counts in the Local Universe suggest that the Milky Way is located in an underdense region. We expect then to experience gravitational redshift rather than gravitational blueshift (since in the above plot, we are closer to a peak in the potential field rather than to a trough). Unfortunately, this is probably all that we can deduce from observations, because the current mapping of the Local Universe (out to a few hundred million light-years) is not complete enough for all objects and we cannot ascertain the exact value of the underdensity of our region.

However, instead of referring to observations we can take a more conservative approach and consider a range of all possible gravitational redshifts at locations of Milky-Way-like galaxies found in a large-scale cosmological simulation. This range represents the most conservative prior knowledge of the local gravitational redshift permitted by the currently accepted best cosmological model.

**Yes: the local gravitational redshift does matter**

It turns out the presence of the local gravitational redshift has a noticeable effect on cosmological inference using SN Ia data.

The figure above shows the relative deviations of 2 best-fit cosmological parameters as a function of the gravitational redshift for a flat cosmological model. The relative differences between the actual and measured parameters can be as high as a few percent (vertical lines show the upper limits of the gravitational redshift corresponding to 68% and 95% of the probability distribution for Milky-Way-like galaxies). The bias is higher in fact for the dark energy equation of state parameter *w *(which quantifies departure of dark energy from a cosmological constant, for which *w *= -1 exactly), than for Ω_{M}, a parameter related to total matter density. Unfortunately, *w *is the most crucial cosmological parameter in future dark energy experiments, whose ultimate goal is to verify if dark energy behaves as a cosmological constant or not.

The local gravitational redshift places a roughly 1% limit on the accuracy of measuring cosmological parameters using SN Ia data. Without trying to judge whether or not this is going to be a dominant effect in future observations, it is really remarkable to see that this weak effect manifests itself in a global cosmological fit.

**Will gravitational redshift also affect other cosmological probes?**

A natural follow-up question is whether the gravitational redshift can have a similar effect on other cosmological probes such as Baryon Acoustic Oscillations (or BAO). This depends on how precisely we know the physical scale of objects used to measure distances. In contrast to SN Ia, the standard ruler of the BAO is precisely fixed by Cosmic Microwave Background (CMB) physics. This means that cosmological inference in this case does not need to employ any marginalization over the physical scale which turns out to be the main channel of propagating perturbations due to the gravitational redshift to the errors in the measured cosmological parameters. Therefore, our simple expectation is that BAO observations should not be affected by the local gravitational redshift effect. Interestingly, this may have some consequences for the empirical distance-duality relation based on combining luminosity distances from SN Ia and angular size distances from BAO.

**Can we overcome this hurdle on the path to ever more accurate "Precision Cosmology"?**

It is an open question to be determined in future research whether we might be able to calibrate this effect of our local underdensity out of our cosmological parameter determinations.

These are critical issues to address as the worldwide cosmology community makes an unprecedented effort to conduct upcoming precision cosmology surveys.