We present a two-dimensional study of star formation at kiloparsec and sub-kiloparsec scales of a sample of local ( z < 0.1 ) Luminous ( 10 ) and Ultraluminous ( 7 ) Infrared Galaxies ( U/LIRGs ) , based on near-infrared VLT-SINFONI integral field spectroscopy ( IFS ) .
We obtained integrated measurements of the star formation rate and star formation rate surface density , together with their 2D distributions , based on Br \gamma and Pa \alpha emission .
In agreement with previous studies , we observe a tight linear correlation between the star formation rate ( SFR ) derived from our extinction-corrected Pa \alpha measurements and that derived from Spitzer 24 \mu m data , and a reasonable agreement with SFR derived from L _ { IR } .
We also compared our SFR _ { Pa \alpha } values with optical measurements from H \alpha emission and find that the SFR _ { Pa \alpha } is on average a factor \sim 3 larger than the SFR _ { H \alpha } , even when the extinction corrections are applied .

Within the angular resolution and sizes sampled by the SINFONI observations , we found that LIRGs have a median-observed star formation rate surface density of \Sigma _ { LIRGs } ^ { obs } = 1.16 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } , and \Sigma _ { LIRGs } ^ { corr } = 1.72 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } for the extinction-corrected distribution .
The median-observed and the extinction-corrected \Sigma _ { SFR } values for ULIRGs are \Sigma _ { ULIRGs } ^ { obs } = 0.16 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } and \Sigma _ { ULIRGs } ^ { corr } = 0.23 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } , respectively .
These median values for ULIRGs increase up to 1.38 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } and 2.90 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } , when only their inner regions , covering the same size as the average FoV of LIRGs , are considered .
For a given fixed angular sampling , our simulations show that the predicted median of the \Sigma _ { SFR } distribution increases artificially with distance , a factor \sim 2-3 when the original measurements for LIRGs are simulated at the average distance of our ULIRGs .
This could have consequences on any estimates of the star formation surface brightness in high-z galaxies , and consequently on the derivation of the universality of star formation laws at all redshifts .

We identified a total of 95 individual star-forming clumps in our sample of U/LIRGs , with sizes that range within \sim 60–400 pc and \sim 300–1500 pc , and extinction-corrected Pa \alpha luminosities of \sim 10 ^ { 5 } –10 ^ { 7 } L _ { \odot } and \sim 10 ^ { 6 } –10 ^ { 8 } L _ { \odot } in LIRGs and ULIRGs , respectively .
The \Sigma _ { SFR } of the clumps presents a wide range of values within 1–90 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } and 0.1–100 M _ { \odot } yr ^ { -1 } kpc ^ { -2 } for LIRGs and ULIRGs .
Star-forming clumps in LIRGs are about ten times larger and thousands of times more luminous than typical clumps in spiral galaxies , which is consistent with expected photon-bounded conditions in ionized nebulae that surround young stellar clusters .
Clumps in ULIRGs have sizes similar ( \times 0.5–1 ) to those of high-z clumps , having Pa \alpha luminosities similar to some high-z clumps , and about 10 times less luminous than the most luminous high-z clumps identified so far .
This could be an indication that the most luminous giant clumps in high-z star-forming galaxies are forming stars with a higher surface density rate than low-z compact ULIRGs .
We also observed a change in the slope of the L-r relation , from \eta = 3.04 of local samples to \eta = 1.88 from high-z observations .
A likely explanation is that most luminous galaxies are interacting and merging , and therefore their size represents a combination of the distribution of the star-forming clumps within each galaxy in the system plus the additional effect of the projected distance between the galaxies .
As a consequence , this produces an overall size that is larger than that of individual clumps , or galaxies ( for integrated measurements )